SHADOW IMAGING OF GEOSYNCHRONOUS SATELLITES

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1 SHADOW IMAGING OF GEOSYNCHRONOUS SATELLITES by Dennis Michael Douglas Copyright Dennis Michael Douglas 2014 A Dissertation Submitted to the Faculty of the COLLEGE OF OPTICAL SCIENCES In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY In the Graduate College THE UNIVERSITY OF ARIZONA 2014

2 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Dennis Michael Douglas, titled SHADOW IMAGING OF GEOSYN- CHRONOUS SATELLITES and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. José Sasián Date: 30 July 2014 John Greivenkamp Date: 30 July 2014 Jim Schwiegerling Date: 30 July 2014 Final approval and acceptance of this dissertation is contingent upon the candidate s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. Dissertation Director: José Sasián Date: 30 July 2014

3 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the copyright holder. SIGNED: Dennis Michael Douglas

4 4 ACKNOWLEDGEMENTS I would like thank my advisor Professor José Sasián and committee members Professor John Greivenkamp and Professor Jim Schwiegerling. I d also like to thank all the professors and administrative staff at The College of Optical Sciences at The University of Arizona for the opportunity to obtain a superb education in this field. I thank my employer Integrity Applications Incorporated / Pacific Defense Solutions for their support during my graduate studies. Specific individuals I would like to thank include: Dr. Steven Long (one of the authors of the original GEO shadow imaging paper) for his feedback on the development of the physical modeling done in this effort. Dr. Bobby Hunt for the many lunches over the last few years over which I was able to gain insight from his vast knowledge base. Dr. Jerry Johnston for providing comments on the draft versions of my dissertation and asking all the right questions. Riki Maeda for being an inspiring source of enthusiasm regarding all things optical. Wes Freiwald for giving me the employment flexibility and motivation to pursue this endeavor. Benjamin Wheeler for teaching me optics on the fly many years ago. Thank you to Matlab and The University of Arizona for allowing me to obtain a student license containing multiple toolboxes that were extensively utilized in this effort. Thank you to Analytical Graphics Inc. (AGI) for providing a educational license for their satellite toolkit (STK) software package which proved very useful for visualization. I d also like to thank Dr. Carl Maes at the College of Optical Sciences for helping me obtain the AGI STK educational license. Finally, thank you to my family and friends who have supported me during this process. My brief hiatus from social functions is over and I ll be attending birthday parties and beach BBQs again. I hope to see you in the water and catch some good waves with you soon...

5 5 DEDICATION To my wife Takayo and our wonderful sons Kalani and Kainoa.

6 6 TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES ABSTRACT CHAPTER 1 INTRODUCTION Motivation for Shadow Imaging of GEO Satellites Overview and Brief History of Shadow Observations Stellar Populations Stellar Astrometric Population Stellar Photometric Population Angular Extent of Stars Overview of Geosynchronous Satellite Orbits GEO Satellite Positions Note on Shadow Predictions Chapter 1 Summary Outline of Remaining Chapters CHAPTER 2 REVIEW OF SCALAR DIFFRACTION THEORY Preliminaries of Electromagnetism Maxwell s Equations Derivation of the Wave Equation Calculation of Irradiance Pattern from Electric Field Amplitude Time Dependence of Irradiance Pattern Green s Theorem Applied to Scalar Diffraction The Kirchhoff Formulation of Diffraction The Rayleigh-Sommerfeld Formulation of Diffraction Standardized Notation for Aperture and Observations Planes Dirichlet Boundary Conditions Approximations to Diffraction Equations Radiation Approximation Paraxial Approximation Fresnel Approximation Fraunhofer Approximation Huygens-Fresnel Principle Impulse Response and Transfer Function Methodology Fresnel Number Diffraction from a Central Obscuration

7 7 TABLE OF CONTENTS Continued Babinet s Principle and Poisson s Spot Chapter 2 Summary CHAPTER 3 PHYSICAL PARAMETERS Illuminating Source Parameters Stellar Spectral Irradiance Angular Extent of Star Aperture Plane Parameters Propagation Medium Parameters Atmospheric Transmission Atmospheric Refraction and Dispersion Atmospheric Turbulence Notes on Light Propagation Appropriate Diffraction Region per Observational Parameters Sampling in Propagation Planes Test Propagations Demonstrating Environmental Factors Test Sim 1: Monochromatic Simulation Test Sim 2: Polychromatic Simulation Test Sim 3: Extended Source Star Added to Simulation Test Sim 4: Atmospheric Turbulence Test Sim 5: Atmospheric Refraction and Dispersion Shadow Collection Model Shadow Velocity Light Collection Sampling Light Detection and Measurement Process Signal to Noise Ratio (SNR) Chapter 3 Summary CHAPTER 4 SIMULATION PROCESS User Defined Scenario Steps in End-to-End Numerical Simulation Step 1: Generating the Truth Shadow Step 2: Generate the Measured Shadow Step 3: Image Reconstruction for Single Wavelength Block Step 4: Final Image Reconstruction using Multiple Wavelength Blocks Descriptions of Functions Constructed in Matlab Primary Functions Constructed in Matlab Auxiliary Functions Constructed in Matlab Chapter 4 Summary

8 8 TABLE OF CONTENTS Continued CHAPTER 5 CONCLUSIONS AND PATH FORWARD Satellite Models Simulation Input and Output Simulation Group 1: Satellite Shape and Aperture Size Varied Simulation Group 2: Effects of Atmospheric Turbulence Simulation Group 3: Aperture Size and Spectral Width Varied Simulation Group 4: Timing Error Varied Simulation Group 5: Off-Zenith Pointing Varied Simulation Group 6: Source Star Angular Extent Varied Simulation Group 7: Detector Parameters Varied Simulation Group 8: Source Star Brightness Varied Conclusions from Group Simulation Runs Potential for Follow-On Efforts Summary APPENDIX A A.1 Primary Functions A.1.1 Primary Function: imgen.m A.1.2 Primary Function: imcollect.m A.1.3 Primary Function: photgen.m A.1.4 Primary Function: measuregmapd.m A.1.5 Primary Function: proptf.m A.1.6 Primary Function: propinversetf.m A.1.7 Primary Function: forwardprop.m A.1.8 Primary Function: backprop.m A.1.9 Primary Function: forceamp.m A.1.10 Primary Function: forcephase.m APPENDIX B B.1 Auxiliary Functions B.1.1 Auxiliary Function: stari2e.m B.1.2 Auxiliary Function: I2E.m B.1.3 Auxiliary Function: E2I.m B.1.4 Auxiliary Function: phistarangle.m B.1.5 Auxiliary Function: friedparam.m B.1.6 Auxiliary Function: phasescreens.m B.1.7 Auxiliary Function: atmrefract.m B.1.8 Auxiliary Function: P2I.m B.1.9 Auxiliary Function: impowerinterp.m B.1.10 Primary Function: powergen.m B.1.11 Auxiliary Function: shadowvelocity.m REFERENCES

9 9 LIST OF FIGURES 1 LEO and GEO satellite distances from Earth Classical spatial resolution limits Shadow cast by object occulting a star One dimensional shadow observation Two dimensional shadow observation Equatorial coordinate system Positions of the stars in Tycho-2 catalog Position errors of the stars in Tycho-2 catalog Galactic coordinate system Density of stars in the sky per square degree B T and V T magnitudes of the stars in Tycho-2 catalog B T and V T magnitude errors of the stars in Tycho-2 catalog B T V T value for the stars in Tycho-2 catalog Magnitude conversion scale factors B s and V s The quantity V V T representing the Tycho-2 to Vega magnitude conversion V magnitudes and standard errors σ V for stars in the Tycho-2 star catalog Angular extents of 85 stars using the Mark III stellar interferometer Examples of geosynchronous orbits around the Earth Observed motion of GEO satellites from a terrestrial vantage point Fundamental orbital elements for satellites Format for a two line orbital element set Oscillation periods of electric field and Poynting vector amplitudes Kirchhoff s geometry for scalar diffraction foundation Geometry for diffraction at a planar aperture based on Kirchhoff s formulation Rayleigh-Sommerfeld geometry for scalar diffraction foundation Coordinate notation used in propagating scalar diffraction effects Babinet s principle of scalar diffraction as a linear process Example of Babinet s principle using a central obscuration Exoatmospheric spectral irradiance and magnitude of Vega Example of aperture plane transmission function for a representative GEO satellite Atmospheric transmission versus wavelength in baseline model Atmospheric refraction for 500 nm light for observer at sea level versus off-zenith angle

10 10 LIST OF FIGURES Continued 33 Atmospheric dispersion centered around 500 nm for observer at sea level versus off-zenith angle Atmospheric refractive index structure constant versus altitude for Hufnagel-Valley 5/7 HV57 Cn 2 profile Example of randomly drawn phase screens based on atmospheric coherence lengths r o (2 m,1 m,0.5 m,0.25 m) Fresnel number per the physical parameters associated with the diffraction pattern of a star occulted by a GEO satellite Propagation steps performed in simulation for vacuum and atmospheric regions from satellite to observer plane Test Sim 1a: Monochromatic shadow simulation using an open Rect transmission function at the aperture plane Test Sim 1b: Monochromatic shadow simulation using a closed Rect transmission function at the aperture plane Test Sim 1c: Monochromatic shadow simulation using an open Circ transmission function at the aperture plane Test Sim 1d: Monochromatic shadow simulation using an closed Circ transmission function at the aperture plane Test Sim 2: Polychromatic shadow simulation for a center wavelength of λ c 500 nm using an open Rect transmission function Test Sim 3: Source modeled as star with an angular extent of 10 nrad with an open Rect transmission function at the aperture plane Test Sim 4: Atmospheric turbulence applied to shadow simulation using HV57 Cn 2 turbulence profile for 500 nm light Test Sim 5: Atmospheric refraction and dispersion applied to shadow simulation for various off zenith look angles Geometry of light collection system consists of a linear array of circular apertures each coupled to multiple GM-APD detectors Optical throughput for each individual collection system used in model First order ground velocity of west to east moving shadow cast by a GEO satellite based on a terrestrial observer s location Two adjacent collection apertures within the overall linear array and geometry for timing error Example of sampling effects during light collection on an observed normalized ground irradiance pattern Inferred photon fluence per 1000 GM-APD measurement simulations using baseline detector parameters A and B Nominal photon fluence as a function of collection diameter D a size and source star magnitude m v Impact on SNR per varied GM-APD parameters as a function of photon fluence

11 11 LIST OF FIGURES Continued 54 SNR as a function of photon fluence for the GM-APD parameter sets described in Table Step 1 in end-to-end simulation process Step 2 in end-to-end simulation process Step 3 in end-to-end simulation process Step 4 in end to end simulation process D transmission functions of the four satellite models used in the simulation Non-physical satellite GEO-D composed of vertical and horizontal bars used as the baseline resolution metric True and measurement inferred ground irradiance patterns for simulation Group 1 using satellite model GEO-A True and measured number of photons per exposure time for simulation Group 1 using satellite model GEO-A Center profile of true and measured number of photons per exposure time for simulation Group 1 using satellite model GEO-A Signal-to-noise ratios for simulation Group 1 using satellite model GEO-A Reconstructed images for simulation Group 1 using satellite model GEO-A True and measurement inferred ground irradiance patterns for simulation Group 1 using satellite model GEO-B True and measured number of photons per exposure time for simulation Group 1 using satellite model GEO-B Center profile of true and measured number of photons per exposure time for simulation Group 1 using satellite model GEO-B Signal-to-noise ratios for simulation Group 1 using satellite model GEO-B Reconstructed images for simulation Group 1 using satellite model GEO-B True and measurement inferred ground irradiance patterns for simulation Group 1 using satellite model GEO-C True and measured number of photons per exposure time for simulation Group 1 using satellite model GEO-C Center profile of true and measured number of photons per exposure time for simulation Group 1 using satellite model GEO-C Signal-to-noise ratios for simulation Group 1 using satellite model GEO-C Reconstructed images for simulation Group 1 using satellite model GEO-C

12 12 LIST OF FIGURES Continued 76 True and measurement inferred ground irradiance patterns for simulation Group 1 using satellite model GEO-D True and measured number of photons per exposure time for simulation Group 1 using satellite model GEO-D Center profile of true and measured number of photons per exposure time for simulation Group 1 using satellite model GEO-D Signal-to-noise ratios for simulation Group 1 using satellite model GEO-C Reconstructed images for simulation Group 1 using satellite model GEO-D True ground irradiance patterns for simulation Group 2 using satellite model GEO-D Measurement inferred ground irradiance patterns for simulation Group 2 using satellite model GEO-D Reconstructed images for simulation Group 2 using satellite model GEO-D True ground irradiance patterns for simulation Group 3 using satellite model GEO-D Measurement inferred irradiance patterns for simulation Group 3 for D a 0.2 m using satellite model GEO-D Measurement inferred irradiance patterns for simulation Group 3 for D a 0.4 m using satellite model GEO-D Measurement inferred irradiance patterns for simulation Group 3 for D a 0.8 m using satellite model GEO-D Central slice of the photon fluence for simulation Group 3 using satellite model GEO-D Signal to noise ratio for simulation Group 3 for D a 0.2 m using satellite model GEO-D Signal to noise ratio for simulation Group 3 for D a 0.4 m using satellite model GEO-D Signal to noise ratio for simulation Group 3 for D a 0.8 m using satellite model GEO-D Reconstructed images for simulation Group 3 for D a 0.2 m using satellite model GEO-D Reconstructed images for simulation Group 3 for D a 0.4 m using satellite model GEO-D Reconstructed images for simulation Group 3 for D a 0.8 m using satellite model GEO-D Reconstructed images for simulation Group 4 using satellite model GEO-D Reconstructed images for simulation Group 5 for λ 10 nm using satellite model GEO-D

13 13 LIST OF FIGURES Continued 97 Reconstructed images for simulation Group 5 for λ 25 nm using satellite model GEO-D Reconstructed images for simulation Group 5 for λ 50 nm using satellite model GEO-D Reconstructed images for simulation Group 5 for λ 100 nm using satellite model GEO-D Reconstructed images for simulation Group 6 using satellite model GEO-D Reconstructed images for simulation Group 7 using satellite model GEO-D Radiometric quantities and SNR with the source star brightness varied for simulation Group 8 using satellite model GEO-D with λ 10 nm Reconstructed images with the source star brightness varied for Group 8 using an ideal detector and satellite model GEO-D with λ 10 nm Radiometric quantities and SNR with the source star brightness varied for simulation Group 8 using satellite model GEO-D with λ 25 nm Reconstructed images with the source star brightness varied for Group 8 using an ideal detector and satellite model GEO-D with λ 25 nm Radiometric quantities and SNR with the source star brightness varied for simulation Group 8 using satellite model GEO-D with λ 50 nm Reconstructed images with the source star brightness varied for Group 8 using an ideal detector and satellite model GEO-D with λ 50 nm Reconstructed images for a source star of m _ 10 for Group 8 using Baseline A and Baseline B detector parameters and satellite model GEO-D with λ 25 nm Reconstructed images for a source star of m _ 10 for Group 8 using Baseline A and Baseline B detector parameters and satellite model GEO-D with λ 50 nm

14 14 LIST OF TABLES 1 Astrometric and photometric parameters in Tycho-2 star catalog Notation for electromagnetic field and source quantities Notation for locations in aperture and observation planes Notation for electric fields at various locations Sampling parameters used in simulation effort Simulation parameters for Test Sim 1 for monochromatic light Simulation parameters for Test Sim 2 for polychromatic application Simulation parameters for Test Sim 3 for a source star with an angular extent applied Simulation parameters for Test Sim 4 for application of atmospheric turbulence Simulation parameters for Test Sim 5 for application of atmospheric refraction and dispersion Exposure times t based on D a for an observer at the equator Baseline GM-APD parameter values Input parameters for simulation Group Input parameters for simulation Group Input parameters for simulation Group Input parameters for simulation Group Input parameters for simulation Group Input parameters for simulation Group Input parameters for simulation Group Input parameters for simulation Group Limits to spatial resolution in reconstructed images deduced from simulation Group Limits to spatial resolution in reconstructed images deduced from simulation Group 8 using ideal detector parameters

15 15 ABSTRACT Geosynchronous (GEO) satellites are essential for modern communication networks. If communication to a GEO satellite is lost and a malfunction occurs upon orbit insertion such as a solar panel not deploying there is no direct way to observe it from Earth. Due to the GEO orbit distance of 36,000 km from Earth s surface, the Rayleigh criteria dictates that a 14 m telescope is required to conventionally image a satellite with spatial resolution down to 1 m using visible light. Furthermore, a telescope larger than 30 m is required under ideal conditions to obtain spatial resolution down to 0.4 m. This dissertation evaluates a method for obtaining high spatial resolution images of GEO satellites from an Earth based system by measuring the irradiance distribution on the ground resulting from the occultation of the satellite passing in front of a star. The representative size of a GEO satellite combined with the orbital distance results in the ground shadow being consistent with a Fresnel diffraction pattern when observed at visible wavelengths. A measurement of the ground shadow irradiance is used as an amplitude constraint in a Gerchberg-Saxton phase retrieval algorithm that produces a reconstruction of the satellite s 2D transmission function which is analogous to a reverse contrast image of the satellite. The advantage of shadow imaging is that a terrestrial based redundant set of linearly distributed inexpensive small telescopes, each coupled to high speed detectors, is a more effective resolved imaging system for GEO satellites than a very large telescope under ideal conditions. Modeling and simulation efforts indicate sub-meter spatial resolution can be readily

16 16 achieved using collection apertures of less than 1 meter in diameter. A mathematical basis is established for the treatment of the physical phenomena involved in the shadow imaging process. This includes the source star brightness and angular extent, and the diffraction of starlight from the satellite. Atmospheric effects including signal attenuation, refraction/dispersion, and turbulence are also applied to the model. The light collection and physical measurement process using highly sensitive geiger-mode avalanche photo-diode (GM-APD) detectors is described in detail. A simulation of the end-to-end shadow imaging process is constructed and then utilized to quantify the spatial resolution limits based on source star, environmental, observational, collection, measurement, and image reconstruction parameters.

17 17 CHAPTER 1 INTRODUCTION 1.1 Motivation for Shadow Imaging of GEO Satellites Geosynchronous (GEO) satellites are essential for modern communication networks, precision GPS access, weather forecasting, television broadcasts, and many defense applications. Over the past half century more than 1000 GEO satellites have been launched and placed into orbit and the trends indicate that this population will continue to grow at a modest exponential rate. 1 The average price of a GEO satellite is $100 million and the average launch cost is $50 million, with some spacecraft costing substantially more. Over the next decade the GEO satellite market value is forecast at over $100 billion. 2 The world s reliance on GEO satellites cannot be overstated, and this dependence will continue to grow in multiple forms in the years to come. Presently, there does not exist any terrestrial optical telescope large enough to image a GEO satellite with any useful amount of spatial information. This is due to the size of GEO satellites and their large distance from Earth. If a malfunction occurs upon orbit insertion such as a solar panel not deploying or an antenna getting snagged there is no direct way to observe it from Earth. Also radiation from solar flares or impact from orbital debris can potentially render a satellite inoperable. Despite numerous risk reduction and reliability insurance methods these distant satellites remain vulnerable to potential maladies that direct observation through resolved In this effort the term image implies that resolvable spatial features of the object of interest can be extracted from the measurement. A modest telescope/ccd system can fairly easily detect a GEO satellite under favorable conditions but with spatial resolution limited to several 10s of meters.

18 18 imagery would greatly aid in diagnosing and characterizing. Alternatively, low Earth orbiting (LEO) satellites are routinely imaged using various telescopes throughout the world. The distance scale to LEO and GEO satellite orbits is shown in Figure 1. A LEO spacecraft ranges from km from Earth s surface, while a satellite in a GEO is nominally at a distance of 36,000 km. Figure 1 Relative orbital distances of LEO and GEO satellites. While LEO satellites are routinely imaged from terrestrial based telescopes the distance to a GEO satellite orbit severely limits the attainable spatial resolution using conventional imaging techniques. In conventional imaging techniques the attainable spatial resolution is limited by the Rayleigh criterion shown in (1). Here, the smallest resolvable spatial feature is d r, λ is the observation wavelength, D is the light collection aperture diameter, and the distance to the object of interest is denoted by z. d r 1.22 λ D z (1) In some applications the Sparrow criteria represents a slightly better spatial resolution capability in which case the scalar 1.22 in (1) is replaced by The classical spatial resolution limits at the nominal GEO distance as a function of telescope diameter are shown Figure 2 for observing wavelengths of 500 nm, 750 nm, and 1000 nm. To obtain sub-meter spatial resolution a 20 m telescope is required. and a 60 m aperture is needed for resolution capability under 0.3 m. In reality other factors such as atmospheric turbulence, pointing jitter, and static wavefront aberration will further limit the attainable resolution under the best of conditions, as shown in Figure 2.

19 19 Figure 2 Classical spatial resolution limits versus telescope diameter based on Rayleigh and Sparrow criteria for an object at the nominal GEO satellite distance for three observing wavelengths. In reality other factors such as atmospheric turbulence, pointing jitter, and static wavefront aberration will further limit the achievable resolution under the best of conditions. This effort seeks to quantitatively evaluate a method for obtaining high spatial resolution images of GEO satellites from an Earth based observing system by measuring the irradiance distribution on the ground resulting from the occultation of the satellite passing in front of a star. The representative size of GEO satellites combined with the orbital distance results in the ground shadow being consistent with diffraction phenomena associated with the Fresnel region when observed at visible wavelengths. Thus, the size of the shadow of a GEO satellite is on the same order as the satellite itself and intensity ringing is observed in the irradiance distribution. The advantage of shadow imaging is that a redundant set of linearly distributed inexpensive telescopes each coupled to a high speed detector is shown to be a far more effective resolved imaging tool for GEO satellites than even a very large telescope aided by adaptive optics (AO). Given that a would-be shadow observer must be at a precise location at the exact time of the occultation event, it is considered an opportunistic means of obtaining imagery of GEO satellites. In this respect, accurate shadow prediction and a mobile observational system are required to increase the

20 20 possibility for exploiting shadow imaging capability. Still, the potential for obtaining highly resolved GEO satellite imagery from an inexpensive Earth based passive observing platform is exciting. Other non-conventional methods have been proposed to obtain resolved imagery of GEO satellites including baseline interferometric imaging 3, 4 and Fourier telescopy. 5 Optical interferometry in this case uses multiple baselines from telescopes distributed in a 2D fashion that collect solar light reflected from the satellite. Fringe patterns from different telescope source pairs are tracked to observe spatial frequency components of the satellite along different cross sections. Factors such as atmospheric turbulence and satellite size and brightness enhance the difficulty in interferometrically tracking the multiple fringe patterns required to reconstruct an image. In Fourier telescopy multiple laser beams illuminate the satellite and a fringe pattern manifesting from prescribed frequency differences between the sources scans over the satellite. The satellite s spatial frequency content is temporally encoded in the reflected signal intensity and collected on the ground. Fourier telescopy requires a very high degree of coherence in each of the illumination lasers and would likely require an adaptive optics system for the outgoing beams to counteract the turbulence effects. While both of these imaging methods warrant merit they remain extremely challenging and costly. Still, these methods do offer advantages in that they are nonopportunistic and allow imaging capability on demand provided a direct line of sight to the GEO satellite of interest exists. 1.2 Overview and Brief History of Shadow Observations A shadow observation refers to the collection and processing of a diffraction pattern resulting from a resident space object occulting a star. In the context of a conventional lab based diffraction experiments the star that is occulted is the light

21 21 Figure 3 Depiction of a shadow cast by a space object occulting a star. The star light is approximated as a plane wave given the very large distances to stars. The size, orientation, and transmission function of the object determines the spatial distribution of the shadow (irradiance pattern) cast on the Earth s surface. Due to the object s orbit and apparent motion of the stars the shadow traverses across the Earth s surface with a velocity v s. source, the occulting object is an obscuration at the aperture plane, and the observer on the Earth s surface performs a measurement at the observation plane. Given the very large astronomical distances to stars the light incident on the occulting space object is approximated as a plane wave (excluding stars with appreciable angular extents). A depiction of a shadow cast by a space object occulting a star is shown in Figure 3. During the observation the shadow (irradiance pattern) is collected and measured and the transmission function of the object is reconstructed using post processing techniques. The post processing is required because the shadow is not an exact outline of the object due to the diffractive effects originating at the edges of the object from incident light. To obtain a shadow measurement the observer must be in the correct location at the right time. Precise knowledge of the space object s orbit and source star position will determine the nominal location of the shadow at a given time. Atmospheric

22 22 refraction effects must be taken into consideration as they will offset the nominal shadow position based on how far off local zenith the source star is to the observer. For slow moving space objects the velocity of the shadow on the Earth s surface is primarily dictated by the sidereal rate of the celestial sphere (apparent motion of stars) which is 15 arcsec/s from east to west. Thus, the shadow velocity v s on the ground is calculated using (2), with d s representing the distance to the space object and the sidereal rate converted to radians/s inside the tangent function. v s d obj tan `7.27 ˆ 10 5 (2) This results in very large shadow velocities, thus high speed detection devices must be used to successfully obtain measurements of the traversing irradiance pattern. In the Fresnel region the full spatial extent of the shadow is similar to the size of the occulting object. The number of measurement samples n samp over the extent of the shadow is determined by (3) where x obj is the size of the object and f m is the measurement framerate. n samp f m x obj v s (3)

23 23 Figure 4 One dimensional shadow observation. A single aperture is used to collect light over the 2D shadow. A 1D light curve is obtained and reconstructed to determine the occulting object s size. The object s size in the 1D case references the slice through the 2D object that was captured by the single aperture. One Dimensional Shadow Observations A one dimensional (1D) shadow observation uses a single localized light collection aperture pointed at a star and measures the intensity as a function of time as a space object occults the star. The star s measured intensity is nominally constant before the occultation then drops as the object passes between the observer and star, eventually returning to a near constant intensity after the object passes. With high fidelity timing a light curve (plot of intensity versus time) is constructed which can be used to determine the object s diameter. A depiction representing the concept of a 1D shadow observation used to determine an occulting object s projected size is shown in Figure 4. Observing an asteroid occulting a star from a terrestrial location was first proposed in 1952, and the shadow of the asteroid Pallas was first successfully observed in , 7 Shadow observations of asteroids are routinely performed by professional

24 24 and amateur astronomers using single apertures placed at different geographic locations synced with high fidelity timing. Earth based 1D shadow observations remain the most accurate method to calculate an asteroid s size and shape other than measurements made from venturing spacecraft in the solar system. Shadow observations have been made of all the planets in the solar system in studies ranging from atmospheric science to the discovery of new planetary rings around Uranus and Neptune. 8, 9 Techniques have also been developed to study distant objects in the Kuiper Belt and Oort Cloud using serendipitous observations of stellar occultations. 10 Implementing diffraction theory was first proposed in the shape and size calculations of distant objects in the solar system with size/distance values consistent with the near field Fresnel diffraction region by taking into account the intensity ringing at the edge of the observed shadow. 11 For example, on this large scale, an object with a radius of 2 km at a distance of 10 3 AU occulting a star has a Fresnel number of N F «55.5. Shadow observations are also extensively used in the search for exoplanets (planets outside our solar system orbiting other stars). This type of transit observation is typically done by measuring the light curve intensity as the exoplanet passes between the observer and its host star and accounts for the vast majority of exoplanet discoveries to date. Narrow band spectroscopic measurements can also be made during the transit which can reveal characteristics of the atmosphere of the exoplanet. The Kepler spacecraft was launched by NASA in 2009 and utilizes a meter class telescope to capture light curves of stars within the Milky Way galaxy which can be analyzed to potentially indicate the presence of exoplanets. 12 To date (time of publication) Kepler has confirmed the discovery of 974 exoplanets in 77 star systems. Asteroid occultation predictions are updated daily and available online at asteroidoccultation.com and other websites. NASA provides the latest Kepler Mission status including the current number of confirmed

25 25 Figure 5 Two dimensional shadow observation (imaging). An array of individual apertures is used to collect light as the 2D shadow traverses over it. A 2D measurement of the irradiance pattern is obtained and reconstructed to yield an image of the occulting satellite. Two Dimensional Shadow Observations A two dimensional shadow observation consists of a linear array of light collecting apertures coupled to high speed detectors positioned such that the shadow moves orthogonally over the array while the intensity of the star is measured simultaneously through each aperture. The observation yields multiple light curves that can be reassembled to produce an estimate of the spatial intensity distribution of the 2D shadow. For observed diffraction phenomena consistent with the near field Fresnel region the intensity ringing around the shadow s complete extent can be observed and utilized in estimating the 2D transmission function of the object occulting the star. A depiction representing the concept of a 2D shadow observation used to determine an occulting object s projected size is shown in Figure 5. exoplanet discoveries at:

26 26 Shadow Imaging of Geosynchronous Satellites The idea of shadow imaging of geosynchronous satellites was first proposed in the literature in The basic idea is to use a 2D shadow observation to generate an image of the satellite which could not otherwise be obtained using conventional Earth based imaging techniques. The key results demonstrated by Burns, et al are outlined below: (1) The typical size of a GEO satellite (10s of meters) and nominal GEO distance from an observer on Earth of 36,000 km are consistent with near field Fresnel diffraction phenomena using visible wavelengths. (2) A Gerchberg-Saxton phase retrieval algorithm can be implemented to recover the 2D satellite s transmission function which is equivalent to an image of the satellite. The attainable spatial resolution was shown to be less than 1 meter. (3) Based on the sidereal motion of the stars due to the Earth s rotation the shadows of geostationary satellites will move from west to east over the Earth s surface while shadows of geosynchronous satellites will have an additional small northsouth component. (4) The probability of a GEO shadow crossing a linear array of collection apertures can be significantly increased if the system is placed on a track or railroad car such that the array can be moved along the north direction. The probability of collecting a GEO shadow also increases as the eligible source stars are allowed to be dimmer. The idea of shadow imaging of GEO satellites was examined in more detail in a publication released in 2008 by Luu, et al. 14 The key results of this effort are summarized below:

27 27 (1) The concept of spectrally resolved shadow imaging is introduced which seeks to increase the amount of attainable spatial resolution while facilitating higher signal to noise (SNR). This is achieved by splitting the collected light into spectral bins and performing the image reconstruction via a Gerchberg-Saxton based phase retrieval algorithm. The final estimated image results from adding the individual reconstructed images from each spectral bin. This results in near diffraction limited resolution capability. (2) Atmospheric based phenomena such as turbulence and refraction are addressed. The technique of shadow imaging is shown to be very resilient to atmospheric turbulence effects. The spectral binning during collection mitigates the majority of the reconstructed image degradation due to atmospheric dispersion (as long as the spectral bins are sufficiently narrow). (3) Shot noise is implemented into the measurement process and the attainable resolution is calculated as a function of source star brightness. (4) The collection aperture size limiting resolution is 0.61D for a square obscuration, where D is the diameter of an individual collection aperture Both previous papers indicate that shadow imaging of GEO satellites is possible and can provide substantially increased resolution capability as compared to conventional imaging techniques. Together these papers form the stepping stones for the inquiry and research in this effort. While the previous efforts have laid substantial framework a detailed study and performance impact analysis of the various observation variables has not been performed. This dissertation seeks to expand on the published work to date regarding shadow imaging of GEO satellites. The specific questions posed in this effort are summarized below:

28 28 (1) What are the capability impacts to shadow imaging of GEO satellites when considering the following environmental factors? (a) Brightness ratio between source star and sky background (b) Angular extent of source star (c) Atmospheric effects including signal attenuation, refraction/dispersion, and turbulence (d) Velocity of shadow based on the observer s longitude and latitude (2) What are the impacts to shadow imaging performance based on the following light collection parameters? (a) Diameter of light collection apertures and signal attenuation through each beamtrain (b) Spectral binning within each individual collection system (3) What influence does the physical measurement process have on image quality based on the following GM-APD attributes? (a) Photon detection efficiency (b) Noise parameters including dark count rate and afterpulsing (c) Timing errors in detector readout (d) Signal to noise ratio thresholds This dissertation seeks to answer the questions posed above using an end-to-end wave optics based simulation tool specifically created for this effort. The simulation tool produces a shadow image using high fidelity numerical methods based on input values that define the physical scenario. The tool is exceptionally useful in that

29 29 the source star, satellite, environmental, collection, measurement, and image reconstruction parameters can all easily be varied to assess impacts on the resulting image quality. The simulation tool also implements independent stochastic realizations of the atmospheric turbulence and GM-APD measurement process. The remainder of Chapter 1 includes an overview of stellar populations and geosynchronous satellite orbits. The Tycho-2 star catalog is used as the basis for examining the distribution of stellar astrometric and photometric properties with associated uncertainties in 1.3. An upper bound on the distribution of the angular extent of the star population also is quantified in 1.3. Basic attributes and terminology associated with GEO satellite orbits are discussed in 1.4 in addition to a brief discussion on predicting GEO satellite shadow events. 1.5 provides a summary of Chapter 1 as well as content overviews of the subsequent chapters. 1.3 Stellar Populations The Tycho-2 star catalog, 15 containing the brightest 2.5 million stars (as seen from Earth), is used as a basis for statistically quantifying the astrometric and photometric parameters associated with the local galactic stellar population (single stars that can be seen from Earth). The Tycho-2 star catalog was primarily constructed from precision stellar measurements collected by the satellite Hipparcos 16, 17 from 1989 to Tycho-2 contains mean values for stellar parameters describing astrometric positions, proper motions, and two-color photometric magnitudes. Standard errors associated with each of these parameters are also contained in the catalog. The Tycho-2 catalog was downloaded 19 and a Matlab structure was created for this effort to systematically access relevant information from the catalog. While other

30 30 star catalogs do exist, the motivation for using the Tycho-2 catalog stemmed from its public availability, broad sky coverage, and low standard errors. The Tycho-2 star parameters used in this effort are detailed in Table 1. All stars contained in the catalog are in the Milky Way galaxy. Each astrometric and photometric quantity describing a star in the Tycho-2 catalog represents the mean value of multiple measurements (some stars have more measurements taken than others). In the Tycho-2 context; for a given star, the standard error of a measured parameter decreases with a larger number of measurements, as the estimate of the measurement population mean improves. Thus, a star for which many measurements were taken will have low standard errors and more accurate mean values for each stellar parameter. The mean and standard error describe bounds on the measurement process and the standard error can be thought of as the plus/minus uncertainty from the mean value. Table 1 Tycho-2 star catalog astrometric and photometric parameters. The Tycho-2 catalog contains other parameters (which are not explicitly shown here) such as star name identifiers, proper motions, parallax, and astrometric parameters defined in non-equatorial coordinate systems. For this effort the relevant parameters describing each star are classified as astrometric or photometric quantities. Selected Tycho-2 Catalog Parameters Symbol Units Right Ascension (RA) α degree Astrometric Declination (Dec) δ degree Parameters Standard Error (S.E.) in RA σ α milli-arcsec S.E. in Dec σ δ milli-arcsec Photometric Parameters Tycho-2 B Magnitude B T unitless Tycho-2 V Magnitude V T unitless S.E. in Tycho-2 B Magnitude σ BT unitless S.E. in Tycho-2 V Magnitude σ VT unitless Tycho-2 is currently considered one of the recommended star catalogs with respect to astrometric and photometric accuracy. Other recommended star catalogs include, but not limited to, the Naval Observatory Merged Astrometric Dataset (NOMAD), the Washington Comprehensive Dataset (WCD), and the USNO CCD Astrograph Catalog, 2 nd release (UCAC2).

31 Stellar Astrometric Population The astrometric parameters in the Tycho-2 star catalog are given with respect to the International Celestial Reference System (ICRS) 20 and explicitly defined 21 in the epoch J2000. The International Astronomical Union (IAU) currently considers the ICRS as the standard celestial reference frame and defines its origin at the barycenter (center of mass) of the solar system. For convenience, the barycentric reference frame uses angular equatorial coordinates mapped to the Earth s spin orientation. Thus, the stellar coordinates in the Tycho-2 catalog pertain to an observer at the solar system barycenter on January 01, Appropriate position and temporal transformations that place the observer at a unique location on the surface of the Earth at a specific time are readily available in the literature. In this barycentric equatorial coordinate system, the projection of the Earth s equator outward onto the celestial sphere forms the celestial equator. The instantaneous astrometric position coordinates in the ICRS are defined as right ascension praq denoted by α, and declination pdecq denoted by δ and specify the pointing direction towards the star in the epoch J2000. The RA can be thought of as a longitude that defines the angular extent along the celestial equator measured eastward from the vernal equinox. The vernal equinox is one of two points along the ecliptic (apparent path of the Sun on the celestial sphere) that intersects the celestial equator. The Dec can be thought of as a latitude that defines the angle perpendicular to the celestial equator bound by the north celestial pole (NCP) and the south celestial pole (SCP). The equatorial coordinate system is shown in Figure 6. Historically, stellar catalogs define RA in temporal units of hours:min:sec and Dec in angular units of deg:arcmin:arcsec. This effort abandons that historical convention and expresses both RA and Dec in angular units of angular degrees. The distribution of the mean

32 32 measured equatorial coordinates RA and Dec for all stars in the Tycho-2 catalog are shown in Figure 7. Figure 6 Equatorial coordinate system. Left: The vernal equinox is one of two points at which the ecliptic and celestial equator intersect. Right: Right ascension α defines the angle around the celestial equator beginning at the vernal equinox and increasing eastward, ranging from 0 to 360 (right). The declination δ defines the angle perpendicular to the celestial equator, ranging from 90 (at the SCP) to `90 (at the NCP) (right). The location of a star in the celestial sphere is uniquely specified by the astrometric coordinates pα, δ). Figure 7 Histograms of α and δ angular equatorial coordinates for the over 2.5 million stars in the Tycho-2 catalog. The single bin size is 5 for both α and δ. The Tycho-2 star catalog provides a standard error for both RA and Dec denoted by σ α and σ δ, respectfully. The standard errors for each star s equatorial coordi-

33 33 nates are expressed in terms of a great-circle in the Tycho-2 catalog, making it more convenient to apply the errors when performing ground based terrestrial observations of stars. For each star in this context, σ α is defined with respect to the cosine of the measured mean Dec value, while the standard error in σ δ is simply the measured standard error in Dec as shown in (4) and (5). σ α pas measured standard error in αq ˆ cospδq (4) σ δ pas measured standard error in δq (5) For a star in the catalog with δ 85 and σ α 0.70 milli-arcsec, the true uncertainty in RA is σ α 8 milli-arcsec. As a result, the Tycho-2 catalog indicates the true angular stellar position uncertainty on the sky (as seen by a terrestrial based observer) and does not systematically change as a function of Dec. The standard errors σ α and σ δ for all stars in the Tycho-2 catalog appear in Figure 8. Figure 8 will be referenced to gauge the potential for error in the calculation of the shadow position. Figure 8 Histograms of σ α and σ δ angular equatorial coordinate uncertainties for the over 2.5 million stars in the Tycho-2 catalog. The single bin size is 5 milli-arcsec for both σ α and σ δ.

34 34 Star Positions in the Galactic Coordinate System The astrometric positions of the stars can also be expressed in the galactic coordinate system (GCS). 22 The galactic equator is defined as a great circle that most closely represent the plane of the Milky Way and is inclined at an angle of from the barycentric equatorial equator. The north galactic pole (NGP) is located at equatorial coordinates α NGP , δ NGP for the epoch J2000. The galactic latitude coordinate for a star, denoted by b, represents the angle from the galactic equator through the star along a great circle that includes both galactic poles. Thus, the b `90 for a star located at the NGP and b 90 for a star located at the south galactic pole (SGP). The galactic longitude coordinate is denoted by l and represents the angular direction with respect to the galactic center. The galactic coordinate system is shown in Figure 9. Figure 9 Galactic coordinate system shown with respect to equatorial coordinate system. The galactic equator is inclined from the equatorial equator. The galactic coordinates pl, bq are analogous to the equatorial coordinates pα, δq. The galactic coordinates at the galactic center are then defined as l 0 and b 0

35 35 and the corresponding equatorial coordinates for epoch J2000 are α GP and δgp The coordinate transformations for the equatorial coordinate system and GCS are shown in (6) and (7) which are valid for the epoch J The galactic longitude of the NCP is l NGP b sin 1 psinpδ NGP q sinpδq ` cospδ NGP q cospδq cospα α NGP qq l l NCP sin 1 pcospδq cospbq 1 sinpα α NGP qq (6) δ sin 1 psinpδ NGP q sinpbq ` cospδ NGP q cospbq cospl NGP lqsq α α NGP ` sin 1 pcospbq cospδq 1 sinpl NGP lqq (7) The density of stars per square degree was calculated in both the equatorial and galactic coordinate systems and is shown in Figure 10. In the galactic coordinate representation the stars are well distributed over the azimuthal coordinate l and concentrated near the galactic equator at b 0. This is consistent with the spiral disk nature of the Milky Way galaxy. The star density in the equatorial coordinate representation illustrates the inclination between the equatorial and galactic equators. Figure 10 Density of stars per square degree in the equatorial coordinate system (left), and the galactic coordinate system (right). The spiral disk nature of the Milky Way galaxy is evident in the galactic coordinate representation. The equatorial coordinate representation illustrates the inclination between the equatorial and galactic equators.

36 Stellar Photometric Population The Tycho-2 star catalog contains two photometric magnitudes for each star. These mean photometric magnitudes are denoted as B T and V T and are closely related to the more conventional Vega photometric system magnitudes. Histograms of B T and V T for the stars in the Tycho-2 catalog appear in Figure 11. It should be noted that 25 stars in the catalog do not have an associated B T magnitude and 3670 stars in the catalog do not have an associated V T magnitude. For reference, the dimmest star the human eye can typically detect is V T 6.. Figure 11 Histograms of the Tycho-2 magnitudes B T and V T for the over 2.5 million stars in the Tycho-2 catalog. The single bin size is 0.2 magnitude for B T and V T. The standard errors associated with each star s brightness are not symmetric about each photometric mean magnitude. The photometric errors pertain to the standard error on the brighter side (lower magnitude side) of each mean photometric magnitude and are denoted by σ B T and σ V T. The photometric errors on the fainter side (higher magnitude side) of each mean Tycho-2 photometric magnitude are denoted by σ` B T and σ` V T and are determined by (8). 21 Histograms of the standard errors for B T and V T are shown in Figure 12.

37 37 σ` B T σ` V T 2.5 log σ B T (8) σ V T 2.5 log Figure 12 Histograms of the Tycho-2 photometric magnitudes standard errors in B T and V T for the over 2.5 million stars in the Tycho-2 catalog. The single bin size is 0.2 magnitude for σ B T and σ V T. The lighter shaded regions indicate errors towards the brighter side of the mean magnitudes B T and V T while the darker shaded regions indicate errors towards the fainter side of the mean magnitudes. The derived quantity pb T V T q is used to further photometrically classify stars. A star is considered to be on the blue side for pb T V T q ă 0 and on the red side for pb T V T q ą 0. A histogram of the photometric quantity pb T V T q is shown in Figure 13. Due to missing B T and V T magnitudes, there are 3695 stars with no associated pb T V T q value derived from the Tycho-2 catalog. The Tycho-2 data collects used slightly different filter transmission bands from that of the standard astronomical BV filters associated with the commonly used Vega magnitude system. In order to accurately convert the photometric magnitudes in the Tycho-2 catalog to radiometric quantities the B T and V T are first mapped to the more conventional Vega magnitude system and are denoted by B and V. More on magnitude systems and the conversion from photometric magnitudes to radiometric quantities is detailed in Chapter 3.

38 38 Figure 13 Histogram of the Tycho-2 photometric quantity pb T V T q for stars in the Tycho-2 catalog. The single bin size is 0.1 for pb T V T q. The histogram indicates that the majority of the stars are on the red side. The mapping from B T Ñ B and V T Ñ B is a function of the Tycho-2 photometric class indicator pb T V T q and given by (9) and (10). 21 B V ` 0.850pB T V T q ` V s (9) V V T 0.090pB T V T q ` B s (10) The quantities B s and V s above are scale factors for the magnitude conversions and are themselves functions of the Tycho-2 photometric quantity pb T V T q. These scale factors are approximated at discrete points for stars within the range 0.2 ď pb T V T q ď This decreases the total number of stars in the catalog from 2.55 ˆ 10 6 to 2.33 ˆ A spline fit was used to interpolate between the tabulated magnitude conversion factors B s and V s with the range of validity and is shown in Figure 14 as functions of pb T V T q. Analysis of historical stellar spectra data taken at the Vilnius Observatory was examined by Bessell 24 subsequent to the publication of the Tycho-2 star catalog manual and results in a more accurate conversion of V T Ñ V in the range

39 39 Figure 14 Magnitude conversion scale factors B s and V s as functions of Tycho- 2 photometric quantity pb T V T q. The tabulated data points are from the published Tycho-2 manual. A spline fit was used to interpolate between the tabulated values within the range 0.2 ď pb T V T q ď ď pb T V T q ď 2. This decreases the total number of stars in the catalog from 2.55 ˆ 10 6 to 2.42 ˆ The Tycho-2 to Vega magnitude difference quantity pv V T q is shown versus the Tycho-2 photometric class indicator pb T V T q in Figure 15. The difference between the originally published Tycho-2 magnitude conversion and the subsequent Bessell conversion is most pronounced in the range 0 ď pb T V T q ď 1 which is associated with most of the stars in the catalog, as is evident in Figure 13. The magnitude conversions B T Ñ B and V T Ñ V are also a functions of stellar luminosity and spectral class which is not contained in the Tycho-2 catalog (this aspect generally applies to very blue and very red stars for which 0.25 ď pb T V T q ď 2. While well approximated from 0.25 ď pb T V T q ď 2, any extrapolation beyond this range requires a star s luminosity classification. 21 The standard errors towards the bright and faint side of V are denoted by σ V σ` V, respectfully, and are given by (11) and (12) which maintains the Tycho-2 error formalism but for the Vega magnitude system. and

40 40 Figure 15 The quantity V V T representing the Tycho-2 to Vega magnitude conversion is shown versus the Tycho-2 photometric class indicator pb T V T q. The dashed red line indicates the original Tycho-2 magnitude conversion, while the solid black line represents the updated magnitude conversion done by Bessell. σ V b 1.09 `σ V T 2 ` 0.09 `σ B T 2 (11) σ` V 2.5 log σ V (12) Histograms showing V and σ V B and σ B are shown Figure 16. Equivalent histograms for are not explicitly shown because they do not influence the radiometric modeling using the Vega magnitude system. Figure 16 will be referenced to gauge the stellar population suitable for shadow imaging applications based on results pertaining to the source star s limiting visual magnitude given other observational and measurement parameters Angular Extent of Stars The Tycho-2 catalog does contain parallax angles for most of the stars but does not contain any information regarding the distribution in angular extent of the stars. Accurate calculations of the angular extent of a star requires precise interferometric

41 41 Figure 16 Histograms of V (left) and standard errors σ V (right) for stars in the Tycho-2 star catalog. Only stars for which 0.25 ď pb T V T q ď 2 are shown. measurements. This effort will consider the angular diameter distribution of 85 stars measured using the Mark III stellar interferometer which are shown in Figure The sample size is limited and the measurements were made on stars that are very bright and close so the angular extents are biased towards the high side when considering all the stars in the Tycho-2 catalog. Still, this is useful in that an upper bound can be set when examining the impact of a star s angular extent on shadow imaging performance, thus Figure 17 will be referenced in in the following chapters. Figure 17 Angular extents of 85 stars using the Mark III stellar interferometer by Mozurkewich et al. The sample size is limited and the measurements were made on stars that are very bright and generally closer to Earth.

42 Overview of Geosynchronous Satellite Orbits Geosynchronous orbits (GEOs) form a general class of orbits with a period equal to one sidereal day. Geostationary orbits (GSOs) are a subset of GEOs representing circular orbits around the Earth s equator first conceived by Aurthur C. Clark. 26 The altitude of a GSO satellite is km from the Earth s equator in order to maintain an orbital period equal to one sidereal day. 27 Satellites in a pure GSOs remain fixed over a single location on Earth at all times, appearing motionless to terrestrial observers. Many meteorological and communication satellites utilize GSOs so that the pointing of the satellite s terrestrial based receiver(s) can remain fixed. Due to regulatory safety and maneuvering guidelines there exists only a limited number of GSO positions. The inclination for a GEO is defined as the tilt of the orbital plane from the Earth s projected equator and its eccentricity quantifies how much it deviates from a circle (a GSO has a zero inclination and an eccentricity of zero). A GEO satellite with a non zero inclination and zero eccentricity remains above the Earth at a constant longitude while oscillating in latitude with a period of one sidereal day. A terrestrial observer observes an analemma (figure eight) motion (traced over a sidereal day) when the GEO has a non-zero inclination and non zero eccentricity. As the inclination increases the size of the analemma gets larger, and as the eccentricity deviates from zero the pattern is tilted as shown in Figure 18 and Figure GEO Satellite Positions The instantaneous position of the satellite must be known to a high degree of accuracy in order to accurately predict GEO shadow events. While this effort does not seek to develop a rigorous treatment of orbital mechanics the basic terminology is

43 43 Figure 18 Geosynchronous orbits (GEOs) around the Earth. The GEOs are defined by color in this illustration. Red is a pure geostationary orbit (GSO) around the Earth s equator with inclination (i 0) and eccentricity (e 0). The other GEOs are: i 0 e 0.1 (green), i 20 e 0 (orange), and i 20 e 0.1 (purple). The observed motion for a terrestrial observer corresponding to the four orbits is shown in Figure 19. This figure was created using an the Satellite Tool Kit, v10 software suite developed by Analytical Graphics, Inc. using an educational license obtained through the University of Arizona. Figure 19 Observed motion of GEO satellites from a terrestrial vantage point. The groundtracks correspond to the GEOs and color convention in Figure 18. The pure GSO case is illustrated in red. This figure was created using an the Satellite Tool Kit, v10 software suite developed by Analytical Graphics, Inc. using an educational license obtained through the University of Arizona.

44 44 presented. Orbital element sets, or elsets, describe a satellite s instantaneous position and trajectory corresponding to a specific time. The elset can be used to determine the position of the satellite in the near future as long as it does not alter its orbit. The six most fundamental orbital elements are briefly described below. 27 These basic orbital parameters are shown in Figure 20. (1) Semi-Major Axis: Distance from center of orbit ellipse to furthest point in orbit from Earth (apogee). (2) Inclination: Tilt angle of orbital plane from projection of Earth s equator. (3) Eccentricity: Unitless ratio of semi-minor to semi-major axes of orbit ellipse. (4) Right Ascension of Ascending Node: The ascending node is defined as the point along the orbital plane at which the satellite crosses the Earth s projected equator from south to north. Likewise, the descending node is the point along the orbital plane at which the satellite crosses the equator from north to south. All inclined orbits have both an ascending and descending node. The right ascension of the ascending node is defined as the angle in the Earth s equatorial plane measured positively from the vernal equinox to the location of the ascending node. (5) Argument of Perigee: Angle measured from the ascending node to the closest point of the satellite s orbit to Earth (perigee). (6) True Anomaly: Angle measured from the satellite s instantaneous position to perigee. Two Line Element Sets The most common format for specifying an elset for a satellite orbit is a two-line element set (TLE). A TLE is comprised of ten essential orbital parameters (six of

45 45 which are the fundamental orbital elements listed above from which the remaining four are derived). A TLE also contains nine other entries for identification, classification, and flagging purposes. The format for a TLE is shown in Figure 21. Available TLEs are not restricted to GEO satellites and are typically updated daily with many publically available online Note on Shadow Predictions An observer s local sky pointing coordinates in the topocentric coordinate system are defined as azimuth, Az, and elevation, El. The azimuth is measured from due north clockwise and ranges from 0 to 360, while the elevation angle is measured from the local horizon and ranges from 0 to 90. The conversion from star positions Many TLEs can be found online at sites such as CelesTrak: elements/, and Space-Track: Figure 20 Fundamental orbital elements. These parameters pertain to all satellite orbits.

46 46 Figure 21 Format for a two line element set (TLE). A TLE is comprised of ten essential orbital parameters (six of which are considered the fundamental orbital elements from which the remaining four are derived). A TLE also contains nine other entries for identification, classification, and flagging purposes. given in an equatorial based ICRS coordinate system (Tycho-2 catalog for J2000) to a local topocentric coordinate system is well established and routinely exercised by astronomers. Highly accurate free and commercial software exists that allows an observer to point a gimbaled telescope to nearly any star overhead from any location using their local raz, Els pointing coordinates. For an overview of this detailed coordinate transformation see Kaplan, Similarly, a GEO satellite position can be converted from an accurate TLE to local raz, Els pointing coordinates using the formalism detailed by Vallado. 27 Given the distribution of the stars shown in Figure 10 the probability for a shadow event increases substantially if the galactic plane is orientated behind the GEO belt from a given observer s vantage point. As Burns et al pointed out the observed occultation probability also increases significantly if the shadow imaging system is placed on a railroad track km long orientated north/south such that the latitude could be varied. 13 While the development of a GEO satellite shadow prediction calculator is not the focus of this dissertation the foundations are well established in the literature and available software exists to construct such a tool.

47 Chapter 1 Summary The global dependence on GEO satellite operations is well established and the world s reliance on these spacecraft is only anticipated to grow. A vast monetary market value is associated with the construction, launch, deployment, and operation of GEO satellites. However, due to their large orbital range, there does not exist a means for obtaining spatially resolved imagery from an Earth based platform. Imagery of this nature would greatly aid in diagnostic and characterization efforts for both operational and malfunctioning satellites. Earth based shadow observations have been well established in the astronomical community as a means for acquiring information on distant space objects. The notion of utilizing a linear array of light collecting apertures coupled to high speed detectors to collect a shadow from a GEO satellite and reconstruct an image has been proposed by Burns et al,2005 and Luu et al,2008, but still no comprehensive and detailed study of shadow imaging has been published. This dissertation seeks to examine impacts to shadow imaging performance based on a broad set of parameters representing source star, satellite, environmental, light collection, measurement, and image reconstruction attributes Outline of Remaining Chapters The remaining format of this dissertation begins by building on the physical and mathematical framework required to construct the simulation tool. The simulation process is described in detail and shadow imaging performance is evaluated by exercising the simulation to represent multiple observing scenarios. The remaining individual chapters in this dissertation are outlined below: Chapter 2 is titled, Review of Scalar Diffraction Theory and begins by describing

48 48 Maxwell s equations and their role in calculating irradiance (what a detector measures) from a complex electric field amplitude (the form in which light is propagated). The temporal variation of irradiance is examined to ensure it is much less than the anticipated measurement time scales. A review of diffraction theory is developed using Green s theorem per the Kirchoff and Rayleigh-Sommerfeld formulations. Standard notation regarding quantities and dimensional indicators at the aperture and observation planes is introduced which is used throughout the dissertation. Approximations are outlined that lead to the Fresnel diffraction equation which lessens the mathematical and computational overhead when propagating light fields. The Huygens-Fresnel principle is reviewed along with impulse and transfer function methodology and the Fresnel number. The Fresnel transfer function is specifically discussed as it is used as the kernel to propagate light in the simulation tool. Finally, unique attributes regarding diffraction from a central obscuration are reviewed. Chapter 3 is titled, Physical Parameters and forms the mathematical basis for the treatment of physical phenomena involved in the shadow imaging process. The parameters associated with the source star are covered including the conversion from photometric visual magnitude to radiometric irradiance, the mathematical treatment regarding the angular extent of the star. The illumination is discussed regarding the resulting complex electric field emerging from the satellite plane based on the spacecraft s 2D transmission function. The physical and mathematical treatment of the propagation medium is shown regarding atmospheric transmission, refraction/dispersion, and turbulence. The shadow velocity is calculated as a function of observer location. The sampling parameters in numerical simulation are quantified and simple test cases are shown using Rect and Circ functions propagated from the satellite plane to the observation plane to demonstrate the influence of the observing scenario parameters on the ground irradiance. The treatment of the light collection is de-

49 49 scribed including the detailed treatment of the measurement process using GM-APD detectors. The anticipated photon fluence levels are estimated for various observing scenarios in addition to corresponding SNR calculations specific to the GM-APD light detection process. Chapter 4 is titled, Simulation Process and defines the complete physical model in terms of the sequential steps representing the shadow simulation, light collection, measurement process, shadow inference, and image reconstruction. The input/output parameters and physical modeling utility of the primary and auxiliary functions constructed in Matlab that form the simulation engine in the effort are described (Appendices A and B contain the functions themselves). A figure of merit is constructed by which shadow imaging performance can be evaluated by using both resolution and SNR. The simulation is then exercised repeatedly to quantify impacts to shadow imaging performance resulting from: environmental, observational, collection, measurement, and image reconstruction parameters. Chapter 5 is titled, Results and Path Forward and presents results from systematically exercising the end-to-end simulation capability established in the previous chapters using various input parameters. Achievable shadow imaging spatial resolution on a GEO satellite is assessed as a function of collection aperture sizes and spectral binning. Scenario parameters including source star brightness and angular extent, atmospheric turbulence and refraction/dispersion, and GM-APD detector performance are systematically varied and linked to their respect impact on shadow imaging spatial resolution capability.

50 50 CHAPTER 2 REVIEW OF SCALAR DIFFRACTION THEORY The primary foundations of scalar diffraction theory are outlined here in Chapter 2. Diffraction refers to the deviation of light behavior from what is predicted by reflection and refraction as described by geometrical optics theory. The origin of diffraction stems from the confinement of the lateral extent of a wavefront by a physical aperture. The word diffraction was first coined by Grimaldi in 1665 who was able to quantify observations of light behavior that could not be described by standard ray optics theory. 29 Shortly after in 1690, Huygens was able to conceptualize the phenomenon of diffraction with an amazing degree of insight by relating it to an envelope of secondary spherical waves originating at the open aperture areas. 30 It wasn t until almost 200 years later in 1882, after the wave theory of light had gained traction, that Kirchhoff formulated a mathematical foundation for diffraction theory fusing Maxwell s equations with boundary conditions on the electromagnetic field at the aperture. 31 While Kirchhoff s theory agreed remarkably well with experimental observation, inconsistencies in the imposed boundary conditions prevented it from becoming the definitive theory on diffraction. In 1896 Sommerfeld was able to resolve the boundary condition inconsistencies associated with Kirchhoff s theory which resulted in the basis for the modernly accepted Rayleigh-Sommerfeld theory of diffraction. 32 The qualifier scalar refers to the nature of the complex electric field amplitude not being treated as a vector, which neglects the coupling between the electric and magnetic fields near the boundaries immediately behind the aperture plane. This simplification is shown to be valid when the aperture size is much larger than the wavelength of light that is diffracted. For a homogeneous isotropic medium the coupling between the various individual scalar components of the electric and magnetic fields can also be ignored.

51 51 Chapter 2 outlines the foundations of modern diffraction theory beginning with an overview of electromagnetic fields in 2.1 which summarizes Maxwell s equations in and the derivation of the wave equation in 2.1. The calculation of the irradiance from the complex electric field amplitude using the time averaged Poynting vector is outlined in and the time dependence of the irradiance is addressed in Next, Green s theorem is introduced in 2.2 and applied in the developments of the Kirchhoff and Rayleigh-Sommerfeld theories of diffraction in and An internally standardized notation for quantities of interest in the diffraction related geometry is described in 2.3 and the generalized Dirichlet boundary conditions are introduced in this notation in 2.4. A series of approximations that simplify the generalized Rayleigh-Sommerfeld diffraction formula are introduced in 2.5. These approximations include the radiation approximation in which is followed by an overview of the Huygens-Fresnel principle in 2.6. Further approximations are outlined in 2.5 including the paraxial approximation in 2.5.2, the Fresnel approximation in 2.5.3, and the Fraunhofer approximation in Additional concepts in diffraction theory are outlined including the Fresnel number in 2.8, and diffraction from a central obscuration in 2.9 which summarizes Babinet s principle and Poisson s spot in Diffraction theory is a very broad field and this effort seeks to outline only the key foundational elements that directly relate to the topic of Shadow Imaging of Geosynchronous Satellites. The principle sources for the contents of this chapter and its development are (Barrett and Myers), 33 (Goodman), 34 (Born and Wolf), 35 (Milster), 36 and (Mansuripur). 37

52 Preliminaries of Electromagnetism A summary of key parameters and concepts in electromagnetism are introduced which are required in the development of describing scalar diffraction theory. Maxwell s equations are shown in in addition to the constitutive relations. The derivation of the wave equation is detailed in and the calculation of irradiance from the electric field amplitude is outlined in The explicit time dependence of the irradiance is addressed in A summary of the electromagnetic fields, source terms, and selected constants are detailed in Table 2, along with the notation and units used for these quantities Maxwell s Equations Maxwell s equations and the constitutive relations in their most general form are shown below: Gauss s Law (13) E pr, tq ρ pr, tq (13) Ampere s Law (14) B pr, tq J pr, tq ` B D pr, tq (14) Bt Faraday s Law (15) ˆ E pr, tq B B pr, tq (15) Bt Magnetic Monopole Law (16) B pr, tq 0 (16)

53 53 Constitutive Relations (17) D pr, tq ɛ o E pr, tq ` P pr, tq B pr, tq µ o H pr, tq ` M pr, tq (17) For the special case of a source free region Maxwell s equations are simplified by setting the source terms to zero as follows: ρ 0, J 0, P 0, and M 0. The relationship between the permittivity ɛ o and permeability µ o of free space to the speed of light c is given by (18). c c 1 µ o ɛ o (18) Table 2 Notation for electromagnetic field and source quantities. Symbol Description Units E pr, tq Electric field V {m D pr, tq Electric displacement field C{m 2 H pr, tq Magnetic field A{m B pr, tq Magnetic induction field W b{m 2 P pr, tq Polarization of medium C{m 2 M pr, tq Magnetization of medium W b{m 2 ρ pr, tq Free charge density C{m 3 J pr, tq Free current density A{m 2 c Speed of light in vacuum m{s ɛ o Permittivity of free space F {m µ 0 Permeability of free space H{m Derivation of the Wave Equation The scalar wave equation is a fundamental relation that forms the basis for scalar diffraction theory. The derivation is summarized here. The curl of both sides of

54 54 Faraday s law (15) is taken and (17) is used to replace B with H to yield (19). ˆ ˆ E pr, tq µ o B r ˆ H pr, tqs (19) Bt Next, Ampere s law (14) is used is used to replace ˆH, and (17) is again applied as shown in (20). ˆ ˆ E pr, tq µ o B Bt J pr, tq ` ɛ o j B 2 ˆ E pr, tq Bt2 (20) The general vector identity shown in (21) is then invoked to yield (22). ˆ ˆ V p Vq 2 V (21) p E pr, tqq 2 E pr, tq µ o B Bt J pr, tq ` ɛ o j B 2 ˆ E pr, tq Bt2 (22) The generalized wave equation is obtained by applying Gauss s law (13) to (22) in addition to invoking the relation in (18) and rearranging the terms to yield (23). ˆ 2 1 B 2 E pr, tq µ c 2 Bt 2 o B 1 J pr, tq ` ρ pr, tq (23) Bt ɛ o The combined quantities on the right in (23) are known as the source term, denoted by s pr, tq. If the electric field and source term oscillate at a single frequency they become separable functions and can be decomposed as E pr, tq Ñ E o prq e iωt and s pr, tq Ñ s prq e iωt. The term E o prq E ox prq ˆx ` E oy prq ŷ ` E oz prq ẑ is the vector representing the complex electric field amplitude. Using this separability, the time independent version of (23) is obtained known as the Helmholtz scalar wave equation shown in (24). Since the wave equation holds for each component te ox, E oy, E oz u of

55 55 the complex electric field amplitude it can be generalized to apply to a scalar version of the electric field amplitude, denoted by U prq. 2 ` k 2 U prq s prq (24) When the source term is zero, the homogeneous Helmholtz scalar wave equation emerges as shown in (25), which governs an electric field with an oscillating amplitude in a source free region. 2 ` k 2 U prq 0 (25) It should also be noted that the general, Helmholtz scalar, and homogeneous scalar wave equations 23, 24, and 25 also hold for the complex magnetic field in the same manner as the electric field treatment Calculation of Irradiance Pattern from Electric Field Amplitude The diffraction formalism described in the following section 2.2 is concentrated on determining the complex electric field amplitude U prq at an observation plane behind an aperture illuminated by monochromatic light. The observed diffraction pattern is determined by physical measurements of the spatially varying irradiance (optical power per unit area) on the observation plane. In this sense the diffraction pattern is equivalent to the irradiance pattern and the two terms are referred to interchangeably. The calculation of the irradiance from the electric field at the observation plane is described here and applied in the following sections. The electric and magnetic field amplitudes of a monochromatic plane wave are given by (26).

56 56 E pr, tq E o prq cos pk r ωt ` δq B pr, tq B o prq cos pk r ωt ` δq (26) The Poynting vector describes the directional energy flux density of an electromagnetic wave and is given by (27). S pr, tq E pr, tq ˆ H pr, tq (27) The Poynting vector for a monochromatic planewave is described by (28) where the relation between the speed of light and the permittivity and permeability of free space given in (18) is applied after using the constitutive relations in (17). The arbitrary phase term in (28) is denoted by α. S pr, tq ɛ o c 2 E o prq ˆ B o prq cos pk r ωt ` αq 2 (28) The magnetic and electric field amplitudes are proportional and related by (29). B o prq 1 c E o prq (29) Since the time dependence of the Poynting vector in (28) is completely encompassed within the cosine squared term oscillating between 0 and 1, it s time average is 1 2. Thus, the time averaged Poynting vector for a monochromatic planewave is reduced to (30). S prq T 1 2 ɛ oc E o prq 2 (30) To review, E pr, tq is the spatially dependent time varying electric field vector and E o prq is the oscillating complex electric field amplitude vector. In the scalar

57 57 treatment, E o prq is equivalent to U prq in the same manner as the Helmholtz scalar wave equation treats the electric field amplitude as a scalar. Thus (30) can be rewritten as the time averaged irradiance I prq T in the context of the scalar electric field amplitude U prq, as shown in (31). I prq T 1 2 ɛ oc U prq 2 (31) Generally, there s a cos pθq factor if the observation plane is tilted with respect to the incident planewave. This will be examined in the context of the specific cases studied later Time Dependence of Irradiance Pattern The electric field amplitude for a planewave described by (26) oscillates at a temporal frequency of ν ω 2π inside the cosine term. The oscillation frequency of the Poynting vector in (28) is contained within a cosine squared term, and thus oscillates at twice the frequency of the electric field amplitude of the planewave. From a detection point of view this means that the integration time of the detection must be several times greater than the inverse of the Poynting vector s frequency in order to justify the time averaging of the irradiance shown in (31). A plot showing the oscillation period of the electric field and Poynting vector amplitudes is shown in Figure 22. For a common visible wavelength of 500 nm the oscillation period of electric field amplitude is 1.67 ˆ s while the Poynting vector oscillates with a period of 8.33 ˆ s. For longer wavelengths both the electric field and Poynting vector oscillation periods increase but remain very small. This suggests that the time average of the irradiance as described in (31) is valid as long as the detector measurements

58 58 Figure 22 Oscillation periods of electric field and Poynting vector amplitudes. The Poynting vector oscillates at the twice the frequency of the electric field. are made with integration times much larger than 1.0 ˆ s. 2.2 Green s Theorem Applied to Scalar Diffraction This section establishes diffraction theory from the mathematical framework based on Green s Theorem, which incorporates boundary conditions into the Helmholtz scalar wave equation (24). Green s theorem is shown in vector calculus form in (32) for which a surface S encloses a volume V. Scalar functions G and U, as well as their partial derivatives with respect to the outward facing surface normal n, are continuous. 34 V G 2 U U 2 G d 3 r S G B Bn U U B j Bn G ds (32) The partial derivative B Bn can also be written as (33).

59 59 B Bn ˆn (33) The scalar function G in (32) is referred to as the Green s Function. Selecting a suitable Green s function lies at the core of scalar diffraction theory and is outlined the following sections and which describe the Kirchhoff and Rayleigh- Sommerfeld developments of diffraction theory The Kirchhoff Formulation of Diffraction Kirchhoff s formulation of diffraction theory begins with a particular choice of the Green s function G, that satisfies (32) and is developed in the geometry shown in Figure 23. Kirchhoff s choice of the Green s function is a mathematical description of the electric field amplitude U, at point B, inside the volume V bound by surface S, for which a source at A emits a unit amplitude expanding spherical wave. Mathematically, G is represented in the Helmholtz wave equation (24) with the source term s defined as a Dirac delta function with unit amplitude as shown by (34). 2 ` k 2 G δ p r b r a q (34) Simplification of Green s theorem results by substitution of (25) and (34) into (32) and application of the delta function sifting property which eliminates the volume integral portion yielding (35). 33 U paq 1 4π S G B Bn U U B j Bn G ds (35) In Kirchhoff s development the choice for G pbq is an expanding spherical wave given by (36).

60 60 Figure 23 Kirchhoff s geometry for scalar diffraction foundation. Volume V is bound by surface S in which a unit amplitude expanding spherical wave originating at point A is evaluated at point B. The distance from A to B is represented by the vector r ab. G pbq 1 e ikr ab (36) r ab b For which the distance r ab px b x a q 2 ` py b y a q 2 ` pz b z a q 2 with respect to the geometry shown in Figure 23. Substituting (36) into (35) yields (37). U paq 1 4π S e ikr ab r ab B Bn U U B Bn j e ikr vb ds (37) r ab The equation shown in (37) is referred to as the Integral Theorem of Helmholtz and Kirchhoff. This is an important development in that it describes the field at any point A as a function of the boundary values of the wave on any surface completely bounding A. 34 Next the Kirchhoff diffraction formulation is extended to the case of a planar aperture. In this geometry, shown in Figure 24, the bounding surface S is decomposed into S S 1 ` S 2 for which S 1 is the portion of S over the planar aperture, and S 2 is

61 61 the portion of S to the right of the planar aperture represented by a partial sphere of radius R centered at point A. Figure 24 Geometry for diffraction at a planar aperture per Kirchhoff s formulation. The bounding surface S is decomposed into S S 1 `S 2 for which S 1 is the portion of S over the planar aperture, and S 2 is the portion of S to the right of the planar aperture represented by a partial sphere of radius R centered at point A. The point B lies in the open region of the planar aperture. The integral theorem of Helmholtz and Kirchhoff in (35) still holds true except that the surface integral over S is now decomposed into S 1 ` S 2 as shown in (38). U paq 1 4π $ & % S 1 G B Bn U U B j Bn G ds ` S 2 G B Bn U U B j Bn G,. ds - (38) We keep with Kirchhoff s choice of a unit amplitude expanding spherical wave originating from A and proceed with the formalism used by Goodman. 34 The Green s function at any point in the aperture plane S 1 is given by (39), and the Green s function to the right of the aperture plane on S 2 as (40).

62 62 G ps 1 q 1 r ab e ikr ab (39) G ps 2 q 1 R eikr (40) The S 1 portion of (38) was previously determined to be (37) using the Green s function in (39). The S 2 portion of (38) is determined by explicitly solving for B G ps Bn 2q given by (41). ˆ B Bn G ps 2q ik 1 1 R R eikr (41) For large values of R (41) is approximated by (42). B Bn G ps 2q «ikg ps 2 q (42) Converting from the surface integral over S 2 to an integral over solid angle Ω subtended over S 2 from the point A, and using the large R approximation in (42), the S 2 portion of (38) can be written as (43). S 2 G B j ż U U rikgs ds Bn Ω j B R RG Bn U iku dω (43) Continuing with Goodman s formalism, the value of RG is uniform over S The remaining portion of the integrand in (43) vanishes per the Sommerfeld Radiation Condition which holds when U diminishes at least as fast as an expanding spherical wave as represented by (44). j B lim R RÑ8 Bn U iku 0 (44)

63 63 Kirchhoff Boundary Conditions The Kirchhoff boundary conditions stipulate that: 1) In the open aperture region the field U and its normal derivative B U are identical to what they would be without Bn the opaque regions surrounding the aperture. 2) The field U and its normal derivative B U are both zero in the opaque regions surrounding the aperture. Bn These boundary conditions and the result of the Sommerfeld radiation condition allows the S 2 in (38) to vanish resulting a modified form of (37) for which only the portion of the open aperture is represented in the limits of integration as expressed by (45). U paq 1 ij 4π ap G B Bn U U B j Bn G d 2 r (45) Although the Kirchhoff theory of diffraction agrees very well with many experimentally obtained results, there remain inconsistencies in the theory with respect to the Kirchhoff boundary conditions. Stipulating that both U and B U are both zero Bn anywhere on S 1 implies that they both vanish in all space and that the field is zero 33, 34, 35, 36 everywhere behind the aperture The Rayleigh-Sommerfeld Formulation of Diffraction The Rayleigh-Sommerfeld formulation of diffraction resolves the inconsistencies associated with the Kirchhoff boundary conditions by using a variation in the planar aperture geometry and the choice of a different Green s function. The geometry associated with the Rayleigh-Sommerfeld formulation of diffraction from a planar aperture is given in Figure 25.

64 64 Figure 25 Rayleigh-Sommerfeld geometry for scalar diffraction foundation. The bounding surface S is decomposed into S S 1 ` S 2 for which S 1 is the portion of S over the planar aperture, and S 2 is the portion of S to the right of the planar aperture centered at point A. Point sources A and C, symmetrically located on opposite sides of the planar aperture produce expanding spherical waves that are 180 out of phase. In the Rayleigh-Sommerfeld diffraction development the starting point is the simplified version of (38) that is only evaluated over S 1, as shown in (46). The S 2 contribution has been eliminated in an identical manner as shown in the Kirchhoff diffraction development. U paq 1 4π S 1 G B Bn U U B j Bn G ds (46) The Rayleigh-Sommerfeld choice of Green s function is based on two point sources, A and C, symmetrically located on opposite sides of the planar aperture as shown in Figure 25. A and C produce expanding spherical waves and are stipulated to be 180 out of phase. In this context, the Green s function at the aperture location B is given by (47).

65 65 G pbq eikr ba r ba eikrbc r bc (47) Using symmetry, r ba r bc, which implies that G vanishes on the aperture plane, and the field at A can be expressed as (48). U paq 1 ij 4π ap U B Bn G d2 r (48) Evaluation of the partial derivative for the Rayleigh-Sommerfeld choice of Green s function given in (47) yields (49). B Bn G 2 ik 1 j z e ikrba (49) r ba r ba r ba Substitution of (47) and (49) into the sifted Green s theorem (35) results in the Rayleigh-Sommerfeld diffraction formula for a planar aperture (50). ij U paq ap U pbq ik 1 j z e ikrba d 2 r (50) r ba r ba r ba By the choice of an alternative source geometry and Green s function, the Rayleigh-Sommerfeld formulation of diffraction from a planar aperture avoids the physically inconsistent constraint on both U and B U at the aperture plane that the Bn Kirchhoff boundary conditions suggest. The full Rayleigh-Sommerfeld development also introduces an alternative Green s function for which the two point sources oscillate in phase. This choice of Green s function results in B G vanishing at the aperture Bn 33, 34, 35, 36 plane and yields an equivalent result to the field calculation in (48).

66 Standardized Notation for Aperture and Observations Planes At this point the notation used to describe various locations in the aperture and observation planes is standardized and detailed in Table 3. The notation used to describe the various electric fields and irradiance patterns of interest is detailed in Table 4. This notation will be used from this point forward when describing diffraction phenomena from a planar aperture. The notation used in the physical geometry is shown in Figure 26. Table 3 Notation for locations in aperture and observation planes. Symbol Mathematical Definition x o x o x coordinate at aperture plane y o y o y coordinate at aperture plane z o z o z coordinate at aperture plane, z o 0 x x x coordinate at observation plane y y y coordinate at observation plane z a z z coordinate at observation plane r o x 2 a o ` yo 2 2-D vector in aperture plane r b x2 ` y 2 2-D vector in observation plane R r r o 2 ` pz z o q 2 3-D vector from pr o, z o 0q to pr, zq R s b r 2 ` pz z o q 2 3-D vector from pr o 0, z o 0q to pr, zq Table 4 Notation for electric fields at various locations. Symbol Mathematical Definition U inc U inc pr o q Incident electric field to aperture plane U ap U ap pr o q Instantaneous electric field emerging from aperture plane U obs U obs prq Electric field at observation plane 2.4 Dirichlet Boundary Conditions The Dirichlet boundary conditions are introduced using the standardized notation described in Table 3 and Table 4. These boundary conditions which, result from the

67 67 Figure 26 Diffraction coordinate notation. Aperture plane coordinates are given by r o a x 2 o ` y 2 o. Observation plane coordinates are given by r a x 2 ` y 2. The vectors R and R s both span from the aperture plane to the observation plane. The angle subtended by R perpendicular to the aperture plane is denoted by θ such that cos θ z R. Rayleigh-Sommerfeld development in 2.2.2, stipulate that: (1) The field at the aperture plane U ap pr o q is equal to the incident field upon the aperture plane U inc pr o q in the open regions of the aperture. (2) U ap pr o q is equal to zero in the opaque regions of the aperture. These boundary conditions are equivalent to multiplying U inc pr o q by the transmission function of the aperture, t ap pr o q, to obtain U ap pr o q as shown in (51). U ap pr o q U inc pr o q t ap pr o q (51)

68 68 The Rayleigh-Sommerfeld diffraction formula can be further conceptually simplified by explicitly applying the Dirichlet boundary conditions to the notation introduced in Table 3 and Table 4 as shown in (52). U obs prq 1 ij 2π ap U inc pr o q t ap pr o q ik 1 j z e ikr R R R d2 r o (52) The irradiance associated with the Rayleigh-Sommerfeld diffraction formula is obtained by using (31) and is given by (53). I obs prq ɛ oc 8π 2 ij ˇ ap U inc pr o q t ap pr o q ik 1 j z e ikr R R R d2 r o ˇ 2 (53) 2.5 Approximations to Diffraction Equations A series of approximations can be made to the Rayleigh-Sommerfeld diffraction formula given in (52) to make the calculation more straightforward. Each approximation pertains to constraints on the physical parameters and observation regions, thus it must be ensured that the desired diffraction scenario is consistent with the diffraction formula used. The radiation approximation is described in 2.5.1, and the paraxial approximation is outlined in The Fresnel and Fraunhofer approximations are detailed in and Radiation Approximation The Rayleigh-Sommerfeld diffraction formula in (52) is typically simplified using the radiation approximation. This stipulates that z " λ and reduces the complexity of the integral but limits its validity to observational regions that are not extremely

69 69 close behind the aperture plane. Under these conditions and using (54), the term ik 1 R inside the integral is simplified to ik, resulting in the common version of the Rayleigh-Sommerfeld equation in (55).» ik 1 j i2π R λ 1 b r r o 2 ` z 2 fi fl (54) Note that the substitution of k 2π λ is also made. U obs prq 1 ij iλ ap U inc pr o q t ap pr o q z e ikr R R d2 r o (55) The irradiance associated with the radiation approximated Rayleigh-Sommerfeld diffraction formula is given by (56). I obs prq ɛ oc λ 2 ij ˇ ap U inc pr o q t ap pr o q z e ikr R R d2 r o ˇ 2 (56) Paraxial Approximation A further simplification can be made for regions in the observation plane in which θ is very small, or equivalently z " r, for which z Ñ 1 and 1 Ñ 1. The field in R R z this paraxial region of the observation plane is given by (57). U obs prq 1 ij iλ ap U ap pr o q e ikr d 2 r o (57) The irradiance associated with the paraxially approximated Rayleigh-Sommerfeld diffraction formula is given by (58).

70 70 I obs prq ɛ oc λ 2 ij ˇ ap U ap pr o q e ikr d 2 r o ˇ 2 (58) Fresnel Approximation A more usable expression for appropriate observation distances can be obtained by applying the Fresnel approximation which simplifies R in the exponential term in (57) as described in Barrett and Myers. 33 Factoring z out of R yields (59). d ˆ r ro R z 1 ` z 2 (59) Using the binomial expansion of a square root given by `? 1 ` a 1 ` 1 a a2 `... the form of R in (59) can be rewritten as (60) for which the binomial expansion term a r ro z 2. R «z «1 ` 1 2 ˆ r ro z ˆ r ro z 4 `... ff (60) For the quartic term in the binomial expansion of R to be negligible (in addition to subsequent higher order terms) it must hold that the physical parameters maintain that the relation (61) holds. 33 k 8z 3 r r o 4! π 4 ùñ r r o 4! λz 3 (61) The term r r o 2 in (60) is expanded as (62). r r o 2 r 2 ` r 2 o 2 pr r o q (62)

71 71 Substituting the truncated binomial expansion of R given in (60) into (57) the Fresnel diffraction formula is obtained and given by (63). U obs prq 1 iλz eikz e iπ ż r2 λz 8 ˆ U ap pr o q e iπ r 2 o λz ro r e i2πp λz q d 2 r o (63) Recognizing that the Fresnel diffraction formula (63) takes the form of a 2D Fourier transform it can be rewritten as (64) assuming the variable transformation of ρ Ñ r λz. U obs prq 1 iλz eikz e iπ r2 λz F 2 # ˆ U ap pr o q e iπ r 2 o λz + ρñ r λz (64) As previously stated, the irradiance on the observation plane is analogous to the time averaged Poynting vector given in 31. The irradiance pattern at the observation plane for Fresnel diffraction is given by 65, which is greatly simplified due to all of the leading complex exponentials vanishing. I obs prq ɛ # oc 2 pλzq 2 ˇ U ap pr o q e iπ ˇF2 ˆ r 2 o λz + ρñ r λz ˇ 2 (65) Fresnel Diffraction Formula in Cartesian Coordinates The same Fresnel diffraction formulas (63) and (64) can be expressed in Cartesian coordinates by (66) and (67), respectively. ˆ U obs px, yq 1 iλz eikz e iπ ij x 2`y 2 `8 λz 8 ˆ U ap px o, y o q e iπ x 2 o`y2 o λz e i2πp xxo yyo λz q dxo dy o (66)

72 72 This takes the form of a 2D Fourier transform as follows. ˆ U obs px, yq 1 iλz eikz e iπ # ˆ x 2`y 2 λz F 2 U ap px o, y o q e iπ + x 2 o`y2 o λz ξñ x λz,ηñ x λz (67) The irradiance at the observation plane for Fresnel diffraction in Cartesian coordinates is given by 68. I obs px, yq ɛ # oc 2 pλzq 2 ˇ U ap px o, y o q e iπ ˇF2 ˆ + x 2 o`y2 o λz ξñ x λz,ηñ x λz ˇ 2 (68) Fraunhofer Approximation For large observational distances z, as compared to the radius of the open aperture a, the Fraunhofer approximation can be used to reduce the diffraction formula to its simplest form. The Fraunhofer approximation is mathematically described by (69). 34 z " ka2 2 Ñ z " πa2 λ (69) Which reduces the exponential in the integral for the Fresnel diffraction equation (64) to unity as shown in (70). ˆ e iπ ro 2 λz Ñ 1 (70) This results in the Fraunhofer diffraction equation which indicates that, at appropriate observational distances with respect to the open aperture size, the field at the observation plane is described by a scaled version of the Fourier transform of U ap pr o q as shown in (71).

73 73 U obs prq 1 iλz eikz e iπ r2 λz F 2 tu ap pr o qu ρñ r λz (71) The irradiance pattern at the observation plane for Fraunhofer diffraction is given by (72) I obs prq ɛ oc 2 pλzq 2 ˇˇˇF2 tu ap pr o qu ρñ r ˇ ˇ2 λz (72) Fraunhofer Diffraction Formula in Cartesian Coordinates Equivalently in Cartesian coordinates, the Fraunhofer diffraction formula is given by (73). ˆ U obs px, yq 1 iλz eikz e iπ x 2`y 2 λz F 2 tu ap px o, y o qu ξñ x λz,ηñ x λz (73) The orthogonal aperture size dimensions denoted by s x and s y stipulated to adhere to the condition shown in (74) in order for (73) to be valid. z " k `s 2 x ` s 2 y 2 Ñ z " π `s 2 x ` s 2 y λ (74) The irradiance at the observation plane for Fraunhofer diffraction in Cartesian coordinates is given by 75. I obs px, yq ɛ oc 2 pλzq 2 ˇˇˇF2 tu ap px o, y o qu ξñ x ˇ ˇ2 λz,ηñ x λz (75)

74 Huygens-Fresnel Principle The term z R in the radiation approximated Rayleigh-Sommerfeld diffraction formula given in (55) can also be expressed as cos θ, with θ being the angle subtended by R perpendicular to the open area of the aperture plane as shown in Figure 26. This substitution leads to the diffraction integral form associated with the Huygens-Fresnel principle which is mathematically represented by (76). U obs prq 1 ij iλ ap U ap pr o q cos θ eikr R d2 r o (76) The Huygens-Fresnel principle describes the field emerging from the aperture plane, U pr o q, as a collection of expanding spherical waves originating from an infinite number of point sources in the open area of the aperture plane. In this context, the observed field at a location behind the aperture plane can be thought of as a superposition of these expanding spherical waves and will vary as a function of r in the observation plane. As alluded to in the beginning of 3, Huygens conceptualized this physical description of diffraction phenomena well before the mathematical foundation theories of Kirchhoff and Rayleigh-Sommerfeld were developed. 2.7 Impulse Response and Transfer Function Methodology Diffraction can also be mathematically described by using impulse response and transfer function methodology. This is a particularly convenient means to calculate diffraction patterns using numerically based computational methods. The convolution integral for a linear space invariant system is the starting point for this methodology and is expressed as (77).

75 75 g 2 px 2, y 2 q 8ij 8 g 1 px 1, y 1 q h px 2 x 1, y 2 y 1 q dx 1 dy 1 (77) The convolution integral in (77) can also be expressed as (78) using the convolution theorem, which is equivalent to (79) in spatial frequency coordinates. In this notation, G pξ, ηq is the 2D Fourier transform of g px, yq expressed as G pξ, ηq F 2 tg px, yqu such that rx, ys are in spatial units and rξ, ηs are in inverse spatial units also referred to as spatial frequency units. g 2 px, yq g 1 px, yq h px, yq (78) G 2 pξ, ηq G 1 pξ, ηq H pξ, ηq (79) A computationally convenient way of expressing this is shown in (80), for which the inverse Fourier transform is used such that g px, yq F 1 2 tf 2 tg px, yquu. g 2 px, yq F 1 2 tf 2 tg 1 px, yqu F 2 th px, yquu (80) This convolution methodology can be extended to expressing the electric field amplitude at the observation plane given the Fresnel diffraction formula. In 2D coordinates and using our standard notation, (66) can be rearranged as (81). U obs px, yq 1 iλz eikz ij`8 8 U ap px o, y o q e ik 2zrpx x oq 2`py y oq 2 s dxo dy o (81) This now takes the form of the convolution integral in (77) and can be extended to (78) such that the convolution kernel h px, yq is given by (82).

76 76 h px, yq 1 iλz eikz e ik 2zpx 2`y 2 q (82) The convolution kernel for Fresnel diffraction given in (82) is commonly referred to as the Fresnel impulse response function. The associated transfer function H pξ, ηq is the Fourier transform of h px, yq and shown in (83). H pξ, ηq e ikz e iπλzpξ2`η 2 q (83) Using our standard notation, the electric field amplitude can be expressed in terms of either h px, yq or H pξ, ηq as shown in (84) and (85). Note that (84) is equivalent to the convolution U obs px, yq U ap px o, y o q h px, yq, and (85) is equivalent to the product U obs pξ, ηq U ap pξ, ηq H pξ, ηq. U obs px, yq F 1 2 tf 2 tu ap px o, y o qu F 2 th px, yquu (84) U obs px, yq F 1 2 tf 2 tu ap px o, y o qu H pξ, ηqu (85) Equation (83) is utilized extensively in this effort when numerically computing diffraction patterns, particularly in Chapter 4 and Chapter 5 during the end-to-end simulation process. 2.8 Fresnel Number A common means of determining which diffraction equation is suitable given observational parameters is by quantifying the Fresnel Number. 38 The Fresnel number, denoted by N f, is a dimensionless parameter defined by (86), for which a is the open aperture radius.

77 77 N f a2 λz (86) Evaluation of N f serves as a general indicator as to which diffraction region best defines the scenario. If N f ď 1 the observation plane is said to be in the far field for which the Fraunhofer diffraction equation given in (71) is a suitable means for determining the diffraction pattern. For cases in which N f ą 1 the observing plane is said to be in the near field for which use of the Fresnel diffraction equation given in (64) is a appropriate. 2.9 Diffraction from a Central Obscuration Thus far diffraction theory has been developed using a planar aperture consisting of an open region in the central portion that is completely surrounded by a nontransparent region. The described diffraction formalism also holds for an alternative configuration in which the central aperture region completely blocks incident light and the surrounding portion is totally transmissive. To resolve this apparent inconsistency Babinet s Principle and Poisson s Spot are used to show the developed scalar diffraction formalism applies to both open and centrally obscuring aperture geometries Babinet s Principle and Poisson s Spot Babinet s principle 39 introduces the concept of aperture algebra based on scalar diffraction being a linear process. The linear nature of diffraction stems from the Huygens-Fresnel principle described in 2.6 which physically associates the diffraction phenomena with a linear superposition of expanding spherical waves originating from the open regions in the aperture plane evaluated at the observation plane. Due to

78 78 this linearity, the aperture itself can be decomposed into components for which the sum of the diffraction pattern from each of the individual components is equivalent to the diffraction pattern of the full aperture. Babinet s Principle and aperture algebra are illustrated in Figure 27. This is a simple yet powerful tool in that the diffraction pattern resulting from a complex aperture can be determined by the superposition of diffraction patterns from simpler component apertures. Figure 27 Babinet s principle of scalar diffraction as a linear process. Top: The diffraction pattern resulting from aperture A is equivalent to the sum of diffraction patterns resulting from apertures B and C. Bottom: Aperture algebra allows the diffraction pattern resulting from C to be expressed by the difference in diffraction patterns resulting from A and B, ie) C A B. For each aperture the outermost region is fully opaque. Up to this point we have only considered diffraction from apertures with a fully opaque outer region such as in Figure 27. The application of Babinet s principle helps to extend the developed diffraction formalism to obscurations surrounded by a transmissive region. The term aperture is more loosely applied for these geometries, but the means of determining the resulting diffraction problem remains unchanged. Consider a plane wave normally incident onto a circular obscuration with no additional

79 79 obscurations in the aperture plane. A conceptually convenient method of determining the resulting diffraction pattern behind the circular obscuration is by applying aperture algebra as shown in Figure 28, for which the diffraction pattern resulting from the central obscuration F is equal to the diffraction pattern resulting from the circular aperture E subtracted from the diffraction pattern resulting from the unperturbed incident plane wave D. Poisson showed that in the Fresnel region, for which N f ą 1 is maintained, the on-axis irradiance behind the F is the same as the on-axis irradiance for an otherwise unperturbed plane wave. This nonintuitive on-axis irradiance near field phenomenon behind a centrally located circular obscuration came to be known as the Spot of Arago, or Poisson s Spot. 36 In this scenario, the off-axis irradiance, however, is not constant as a function of observing plane distance behind the obscuration. Figure 28 A circular central obscuration surrounded by a transmissive region described by aperture algebra per resulting diffraction patterns. Top: The irradiance pattern from the fully open aperture D is equivalent to the addition of irradiance patterns resulting from apertures E and F. Bottom: The diffraction pattern resulting from the central obscuration F can be determined by the difference between diffraction patterns resulting from D and E, ie) F D E.

80 Chapter 2 Summary An overview of the foundational development of scalar diffraction theory was presented in a summarized fashion. The Helmholtz scalar wave equation was derived from Maxwell s equations. Green s theorem was utilized and boundary conditions were imposed on the electromagnetic field which lead to a mathematical framework for which scalar diffraction from a planar aperture could be described as outlined by the Kirchhoff and Rayleigh-Sommerfeld theories. The scalar complex electric field amplitude was shown to facilitate the calculation of the irradiance pattern at the observation plane. A series of approximations were described which allowed the most generalized Rayleigh-Sommerfeld diffraction formula to be considerably simplified for specific scenarios that constrain the observing parameters. The application of diffraction theory was shown to extend from open apertures to central obscurations using Babinet s principle. This outline of scalar diffraction theory serves as the basis for the subsequent Chapters which concentrate on the numerical simulation of scalar diffraction patterns from GEO satellites.

81 81 CHAPTER 3 PHYSICAL PARAMETERS The detailed physical parameters associated with the diffraction problem, light collection methodology, and radiation measurement process are described in this Chapter. The chapter begins with the treatment of light propagation from the satellite plane to the observation plane using an energy conserving radiometric foundation. The photometric to radiometric conversion of starlight illuminating the satellite is described in 3.1, in addition to the treatment regarding the star s angular extent. The aperture plane is represented by a satellite of finite extent with properties detailed in 3.2. The parameters associated with the propagation medium including vacuum space and the Earth s atmosphere are outlined in 3.3. The numerical sampling used in the wave propagation is quantified in 3.4, and a series of simple propagations from the satellite plane to the ground are shown in 3.5 using simple Rect and Circ functions. These simple propagations demonstrate the accuracy of the overall diffraction model with respect to environmental factors and spectral bandwidth. The model used to describe the collection of light with a linear array of ground based telescopes is detailed in 3.6. The light measurement process unique to GM-APD detector technology is described in 3.7 along with the treatment of the signal to noise calculation. The chapter concludes by showing representative photon fluence (photons/pix/s) levels and associated SNR metrics for anticipated shadow imaging scenarios using established baseline GM-APD parameters. The satellite plane in this case is synonymous with the term aperture plane which describes the plane at which the incident light is forced to experience the boundary conditions described in 2.4.

82 Illuminating Source Parameters The properties of the illumination source include the stellar spectral irradiance at the aperture plane and angular extent of the star. The spectral irradiance at the satellite location is described in The treatment of the angular extent of the illuminating star is detailed in Stellar Spectral Irradiance The spectral irradiance from an illuminating star at the satellite location is determined by the star s V magnitude. High resolution spectrophotometric exoatmospheric (outside the Earth s atmosphere) measurements of the star Vega taken by the Hubble Space Telescope (HST) and processed by the Space Telescope Science Institute (STSCI) form the basis for the radiometric model of the illumination source. These measurements offer the most accurate calculation to date for the apparent V magnitude of Vega as V vega The exoatmospheric spectral irradiance from Vega calculated by STSCI is shown in the top of Figure 29. The monochromatic Space Telescope Magnitude (STMAG) system defines a star with constant spectral irradiance per unit wavelength to have a uniform STMAG. The STSCI data reduction correlates an absolute exoatmospheric irradiance from Vega of I λ pλ o q 3.46 ˆ W m 2 at λ o nm. The monochromatic AB magnitude system defines a star with constant spectral irradiance per unit frequency to have a uniform AB magnitude. 41 The AB magnitude system has been adopted by the Sloan Digital Sky Survey (SDSS) and other ground based photometric data collection systems and is utilized in this effort. The AB system defines the absolute exoatmospheric spectral irradiance from Vega as I ν pλ o q 3.56 ˆ W m 2 Hz 1. Exoatmospheric Vega spectrum obtained from the Space Telescope Science Institute FTP server: ftp://ftp.stsci.edu/cdbs/current_calspec

83 83 Figure 29 Top: Exoatmospheric (outside of Earth s atmosphere) spectral irradiance from Vega as measured by the Hubble Space Telescope (HST) and processed by the Space Telescope Science Institute (STSCI). Bottom: Spectral magnitude of Vega based on STSCI exoatmospheric spectral irradiance, m vega p555.6 nmq The flux conversion f ν pλ 2 {cq f λ is used to transition from I λ pλq shown in the top of Figure 29 to I ν pλq such that the AB magnitude m vega pλq can be calculated by (87). The spectral AB magnitude of Vega is shown in the bottom of Figure 29. m vega pλq 2.5 log I ν pλq I ν pλ o q 2.5 log λ2 I λ pλq λ 2 o I λ pλ o q (87) The exoatmospheric spectral irradiance of a star with a V magnitude of m _ is then given by (88). Note that the notation changes slightly at this point in that V magnitude is now represented by m _ which is used to describe the source star s brightness. I S px o, y o ; λq I ν pλq 10 mvegapλq m_ 2.5 (88)

84 84 Here, I S px o, y o ; λq is the irradiance at the satellite from the star which considered to be uniform over all aperture plane coordinates px o, y o q. The amplitude of the monochromatic scalar electric field U S px o, y o ; λq incident to the aperture plane is calculated from the exoatmospheric stellar spectral irradiance I S px o, y o ; λq based on (31) and shown in (89). U S px o, y o ; λq d 2 I S px o, y o ; λq c ɛ o (89) Note that U S px o, y o, λq represents the amplitude of the scalar electric field which is uniform over the aperture plane. If the star light is modeled as a normally incident planewave the electric field at the aperture plane does not contain a phase term. The extension from a monochromatic to polychromatic treatment of the electric field amplitude is done by integrating over wavelength. Thus, the Fresnel transfer function propagator described in (85) can be extended to a polychromatic calculation by integrating over a spectral band as shown in (90). The detailed treatment of the propagation planes and spatial sampling is described in 3.4. U 2 px, y; λ 1 Ñ λ 2 q żλ2 λ1 F 1 2 tf 2 tu 1 px o, y o ; λqu H pρ, z; λqu dλ (90) Angular Extent of Star In most cases stars are modeled as point sources and are assumed to be an infinite distance away. In reality stars do exhibit a small angular extent based on diameter and distance as evidenced by the measurements shown in Figure 17. For a single planewave tilted from the z-axis by α with tilt orientation θ in the xy plane the phase term is given by (91). 42

85 85 φ tilt px o, y o, λq k rx o cos pθq ` y o sin pθqs tan pαq (91) This effort models the angular extent(diameter) of a star α S as a continuum of radially distributed tilted incident planewaves incident to the aperture plane. The radially tilted phase term for the star s angular extent is given by (92). φ S px o, y o q żα S ż 2π k rx o cos pθq ` y o sin pθqs tan pαq dθ dα (92) 0 0 Thus, the complex scalar electric field amplitude incident onto the aperture plane from a star with an angular extent of α S is given by (93). U S px o, y o, λq U S px o, y o, λq e iφ Spx o,y oq (93) In this formalism the source parameters are fully described by the V magnitude of the star m _ and the angular extent of the star α S. 3.2 Aperture Plane Parameters The transmission function t ap px o, y o q at the aperture plane is modeled as a 2D projection of a 3D satellite bounded by a transparent window of finite extent. The shape of t ap px o, y o q is determined by the satellite s physical orientation as viewed from the observer s line of sight. The axial location of the aperture plane is defined by the satellite s center of mass and is by definition set to z 0 such that the depth of the transmission function per the true 3D nature of the satellite is ignored. The satellite s projection is constrained to be fully opaque and is centered within the aperture plane such that t ap px o, y o q is a binary function. The satellite s projection within t ap px o, y o q is constrained to be monolithic (all opaque regions are connected) yet need not be

86 86 mathematically continuous (sharp edges between opaque and transmissive regions are supported). Also, the outer transmissive extent of t ap px o, y o q is well beyond the centralized opaque region of the satellite. The satellite s transmission function is also modeled as being uniform over wavelength, thus all light is blocked by the opaque regions. An example of t ap px o, y o q is shown in Figure 30. Figure 30 Example of aperture plane transmission function t ap px o, y o q for a representative GEO satellite. The transmission function is binary with a fully opaque region representing the projection of the satellite determined by the observer s line of sight centered within the aperture plane. The extent of t ap px o, y o q is finite with a fully transmissive outer region. Thus, the complex scalar electric field amplitude at the aperture plane U ap px o, y o, λq with transmission function t ap px o, y o q illuminated by a star is given by (94). U ap px o, y o ; λq U S px o, y o ; λq t ap px o, y o q (94) Here, the electric field amplitude emerging from the aperture plane is given in a monochromatic form. The polychromatic form can be obtained by integrating (94) over wavelength.

87 Propagation Medium Parameters The propagation medium from the aperture plane at the satellite to the observation plane at the Earth s surface is split into two regions consisting of: 1) A vacuum region from the satellite to the top of the Earth s atmosphere, and 2) A layered atmospheric region from the top of the Earth s atmosphere to the Earth s surface (observation plane). The vacuum region of propagation does not introduce any additional phase terms or external influence on the nominally propagated field from the aperture plane to the top of the Earth s atmosphere. Thus, we assume that in this vacuum region the propagated field does not experience any additional diffractive effects such as scatter and is not attenuated. The propagation region from top of the Earth s atmosphere to the Earth s surface does introduce additional effects to the nominally propagated field. The physical nature and behavior of the atmosphere is a broad and in-depth topic. In this section the basic formalism is summarized as it relates to the atmosphere s influence on the resulting diffraction pattern pertaining to our observing scenario. The attenuation of light propagating through the atmosphere results in a net transmission loss and is covered in The net directional deviation of the polychromatic waveband alters the location on the Earth s surface at which the shadow of the satellite lands due to the atmosphere s index of refraction gradient and is addressed in in addition to the dispersion of polychromatic light about a given reference wavelength. Atmospheric turbulence and its influence on the nominal diffraction pattern are outlined in These combined atmospheric effects introduce an attenuation term, a spatial offset, and multiple phase terms to the scalar electric field as it propagates through the Earth s atmosphere.

88 Atmospheric Transmission Light propagating through the atmosphere is attenuated in intensity due to the chemical composition and density variations within the atmosphere. The magnitude of attenuation through the atmosphere is a function of wavelength and propagation distance. For this effort, a single atmospheric transmission case was calculated using the MODTRAN (MODerate resolution TRANsmission model) radiative transfer code. 43, 44, 45 MODTRAN uses radiative transfer equations while incorporating molecular and particulate absorption/emission/scattering phenomena and outputs the atmospheric transmission and sky radiance for a given observing scenario. A single nominal case scenario was run for which the observation took place at the equator at sea level (altitude h 0 m) at midnight local time looking straight up at zenith. The output for this observing case will serve as the baseline atmospheric transmission T atm pλ, θ _ 0q for modeling efforts and is shown by the black curve in Figure 31. A simple airmass model shown in (95) is used to uniformly scale T atm pλ, θ _ 0q based on an off zenith look angle θ _. 46 A _ cos pθ _ q ` 0.025e 11 cospθ_q 1 (95) The colored curves in Figure 31 represent uniformly scaled atmospheric transmissions per off zenith look angle using this airmass model. In reality, T atm pλ, θ _ q is not a truly uniform function of θ _ and the nominal zenith transmission attenuates based on wavelength as well. This effect is ignored for simplicity and the atmospheric transmission T atm pλ, θ _ q is given by (96). T atm pλ, θ _ q 1 A _ T atm pλ, θ _ 0q (96)

89 89 Figure 31 Atmospheric transmission versus wavelength calculated using MOD- TRAN. The observing scenario corresponds to a ground observer at the equator at sea level at midnight local time. The black curve corresponds to a zenith look angle (baseline calculation), the colored curves are scaled by the inverse of the airmass A 1 _. The narrow attenuation regions correspond to molecular absorption bands in the atmosphere. The nominal black curve in Figure 31 forms the tabulated function quantifying the atmospheric transmission T atm pλ, θ _ 0q. The effective atmospheric attenuation per off zenith look angle θ _ is accounted for by inserting T atm pλ, θ _ q inside the integral in (90) when calculating the polychromatic electric field amplitude per the Fresnel transfer function Atmospheric Refraction and Dispersion Light incident onto the Earth from space undergoes an angular displacement due to the refractive index of the atmosphere. This angular displacement is a function of wavelength, the observer s altitude and line of sight from zenith, and the local environmental conditions at the observer s location. In typical astronomical applications the macroscopic refractive effects of the atmosphere require an angular pointing offset be applied to the telescope such that the object of interest is centered in the sensor s

90 90 field of view. A complexity associated with shadow imaging is that the line of sight to the source star is altered in addition to the spatial location on the Earth s surface that the shadow lands. Thus, to collect a shadow image both a pointing offset and a location shift must be applied to the observation platform with respect to the nominal vacuum space geometry. For the purposes of calculating refraction effects the atmosphere is modeled to contain an outer stratosphere region and inner troposphere region. 47 The altitude of the top of the stratosphere is modeled as h s 80 km and the altitude at the top of the troposphere is set to h t 11 km, with the tropopause layer representing the distinction between the two regions. The total angular deviation of light through the 48, 49 atmosphere is given by a two part integral shown in (97). For a given set of observation point environmental specifications (latitude φ lat, temperature in kelvin T obs, pressure in millibars P obs, and relative humidity Rh obs ), the atmospheric refraction can be characterized by wavelength in microns λ µm, and off zenith look angle Z o. Θ atm pλ um, Z o q żz t Z o f pzq dz ` żz s Z t f pzq dz (97) Here, Z t is the off-zenith angle in the troposphere and Z s in the stratosphere. The integrand in (97) is given by (98) for which r R C ` h. Here, R C is the nominal radius of the Earth equal to 6378 km. f pzq rpzq dnpzq drpzq npzq ` rpzq dnpzq drpzq (98) The calculation of the refraction integral is done iteratively via numerical inte- 48, 49 gration in 1 m steps through the full atmosphere. using a nominal set of environmental conditions. The angular offset and spatial displacement of light due to refraction through the atmosphere at 500 nm for an observer at sea level is shown in

91 91 Figure 32. Figure 32 Atmospheric refraction for 500 nm light for observer at sea level. Top: Angular deviation of a star from the perspective of a ground observer. Bottom: Spatial offset of shadow on the Earth s surface. Atmospheric Dispersion The previous discussion on atmospheric refraction pertained to the bulk displacement of the shadow on the Earth s surface for light at 500 nm. Because a polychromatic diffraction pattern is measured, the dispersion of the wavelengths relative to this 500 nm reference point must also be taken into account. This is done using (97) to calculate the angular offsets per wavelength relative to the reference wavelength using the dispersion angle calculation shown in (99). The dispersion centered around 500 nm light through the atmosphere is shown in Figure 33. Θ disp pλq Θ atm pλ 0.5 µmq Θ atm pλq (99) Simpler analytic based calculations of refraction through the atmosphere exist, however the numerical integration described in the Explanatory Supplement to the Astronomical Almanac is considered to be the most accurate method.

92 92 Figure 33 Atmospheric dispersion centered around 500 nm. Top: Angular offset of light through the Earth s atmosphere. Bottom: Spatial deviation of shadow on Earth s surface. The observation platform is at sea level and is prepositioned to be centered on light at 500 nm. While the effect of the bulk refraction is taken into account during the placement of the observing platform ( prepositioned to be centered on light at 500 nm) the dispersive effect around the center wavelength must be accounted for by shifting each monochromatic diffraction pattern by the appropriate amount per (97) and (99) Atmospheric Turbulence When describing turbulence the atmosphere is modeled as a collection of volume pockets referred to as turbulent eddies of constant refractive index and with associated characteristic sizes. The sizes of the eddies along a given line of sight are determined by the dynamics of the local atmosphere based on spatially varying pressures, densities, and temperatures. The structure of the atmosphere describes the sizes of the eddies with outer scale L o quantifying the largest eddy size and the inner scale l o the smallest eddy size. Kolmogorov theory forms the basis for much of the modern study of the atmosphere and maintains there exists a continuum of eddy

93 93 sizes that spans from the l o to L o. 50 Furthermore, Kolmogorov theory states that the structures within the atmosphere are statistically isotropic, homogeneous, and independent of the macroscopic structures. 51 The refractive index structure function δ n pr, hq is shown in (100) and represents the difference in refractive index at two points in the local atmosphere separated by r a at altitude h, such that l o! r a! L o. δ n pr a, hq C 2 n phq r 2{3 a (100) Atmospheric C 2 n Profile The refractive index structure constant Cn 2 phq in units of m 2{3 describes the turbulence strength based on the statistical properties of the refractive index variation. Due to the large variation in climate and weather patterns on Earth, Cn 2 phq is strongly dependent on geographical location and altitude and is generally derived from an average of atmospheric measurements over the course of many years. 47 A variety of methods exist to measure C 2 n phq profiles that involve wavefront and differential image motion sensors, and scintillometers. 50 C 2 n phq profiles are sometimes described by piecewise functions to represent the layered nature of the atmosphere and typically do not extend beyond an altitude of 25 to 30 km above sea level at which atmospheric turbulence effects become negligible. While a large set of widely published C 2 n phq profiles exists in the literature this effort will exclusively use the Hufnagel-Valley 5/7 (HV57) Cn 2 47, 50 phq profile. 2 ˆ21 HV 57Cn 2 phq 5.94h 10 e h 1000 ˆ10 53 `2.7e h 1500 ˆ10 16 `1.7e h ˆ (101) 27

94 94 The HV57 C 2 n phq profile is derived from the more generalized Hufnagel-Valley profile for which a specific ground layer turbulence strength and wind velocity at the 52, 53, 54 tropopause are chosen. The HV57 C 2 n phq refractive index structure constant as a function of altitude is shown explicitly in (101) and graphically in Figure 34. Figure 34 Atmospheric refractive index structure constant versus altitude for Hufnagel-Valley 5/7 HV57 Cn 2 phq model used as the baseline turbulence profile in this effort. The profile exhibits stronger turbulence at lower altitudes and is consistent with a multi layered atmosphere with surface and planetary boundary layers and a tropopause wind sheering layer near h 10 km. Atmospheric Coherence Length The Fried parameter denoted by r 0 describes the atmospheric coherence length at which turbulence is considered to become significant in beam propagation applications. 55 Conceptually, the coherence length can be thought of as a column, of cross sectional length r 0, extending through a defined vertical section in the atmosphere for which the perturbations of index of refraction cause negligible phase changes to a propagated beam. The derivation of r 0 follows from the Kolmogorov atmospheric The Fried parameter r 0 is typically referenced in the context of imaging applications and de-

95 95 model and is a function of wavelength, the off zenith look angle, and the integrated Cnphq 2 profile. Beam propagation methods typically use a layered atmospheric model due to the high dynamic range and nonlinear nature of Cnphq 2 profiles as seen in Figure 34. When using a layered atmospheric model r 0 is defined over a specific altitude range h 1 Ñ h 2 as shown by (102). When describing the effective r 0 through the full atmosphere the limits of integration in (102) are changed to 0 Ñ 25 km, using intermediate propagation plane coordinates px a, y a q, and wavenumber k 2π{λ.» fi żh 2 r k2 Cn 2 phq dhfl cos pθ _ q h 1 3{5 (102) The next step is to construct atmospheric phase screens based on (102) to represent layers of the atmosphere that can be applied to the Fresnel transfer function propagation method. The atmospheric phase term φ atm px a, y a q is the phase screen representing the layer of the atmosphere between altitudes h 1 and h 2 and given by (103). φ atm px a, y a ; λq 8ij 8 Φ φ pξ, ηq e i2πpξxa`ηyaq dξ dη (103) The term Φ φ pξ, ηq represents the power spectral density (PSD) in spatial frequency space that constrains all random realizations of φ atm px a, y a q. The spatial scribes when the RMS wavefront error of a given beam diameter at a telescope aperture is within 1 radian. 55 A ground based observer using a telescope diameter D ď r 0 will maintain the ability to image at or near the diffraction limit (assuming all other factors support this as well). For a telescope diameter D ą r 0 diffraction limited imaging performance can no longer be achieved because the telescope pupil is now sampling multiple coherent paths through the atmosphere which increases the RMS wavefront error of the incident light received at the entrance pupil. Without mitigating factors the story becomes bleak very quickly in that the fundamental resolution limit λ{d no longer applies when imaging through the atmosphere. Ground based observers looking at objects in space are thus confronted with balancing the received signal and attainable image quality. The received signal increases as the light collection area increases but the image quality degrades for larger ratios of D{r 0. As a result modern astronomical observatories are required to implement adaptive optics systems and/or specialized image post processing techniques to help overcome the limitations imposed by the atmosphere when imaging space objects with large ground based apertures.

96 96 frequency PSD term is a function of r 0 and given by (104). 50 Φ φ pξ, ηq r 5{3 0 `ξ2 ` η 2 11{5 (104) There exists ample literature regarding the topic of generating atmospheric phase screens and under which conditions various methods best apply. While this effort baselines on a Fourier series phase screen generation technique outlined by Schmidt other methods exist including using Zernike coefficients, wavelet bases, etc. 56 Examples of turbulence phase screens using the randomly drawn Fourier series approach for various r 0 values are shown in Figure 35. At each layer in the atmosphere the appropriate phase term is applied to the complex electric field U 1a px a, y a, λq using (105) and the resulting field U 1b px a, y a, λq is then propagated to the next layer and another phase screen is applied and so forth until the observation plane is reached. The number of atmospheric layers in the model is fixed at ten. U 1b px a, y a ; λq U 1a px a, y a, λq e iφatmpxa,ya;λq (105) 3.4 Notes on Light Propagation Numerical simulation serves two main purposes in this effort in the context of light propagation: (a) To produce realistic GEO satellite shadows in the form of diffraction patterns on the Earth s surface to serve as a true shadow event. (b) To solve the inverse problem required to estimate the true transmission function (image) of the satellite at the aperture plane.

97 97 Figure 35 Example of randomly drawn phase screens based on atmospheric coherence lengths r o [2 m,1 m,0.5 m,0.25 m]. The phase has been wrapped to r0, 2πs and the spatial extent has been zoomed to the center 20 by 20 meters for visualization. The atmospheric phase screens are generated using the numeric formalism outline by Schmidt, Purpose (a) uses forward propagation from the aperture plane to the observation plane (Earth s surface). Purpose (b) utilizes forward and back propagation iteratively to retrieve the phase information lost during the collection of the shadow. Both of these propagation directions are detailed in Chapter 4 along with the explicit functions created in Matlab that apply each of the physical phenomena described above to the model.

98 Appropriate Diffraction Region per Observational Parameters The physical parameters associated with the geometry and means of observation of a GEO satellite occulting a star govern which equation(s) developed in Chapter 2 are valid for calculating the diffraction pattern. These physical parameters can be reduced to: (1) The physical size of the satellite as projected onto the observer s line of sight. (2) The distance from the satellite to the observer on the Earth s surface. (3) The wavelength at which the measurement is taken. The Fresnel number as described in 2.8, defined by (86), is used to determine the appropriate treatment of the GEO shadow diffraction problem in the context of the constrained physical parameters described above. A set of loose, yet realistic, observing parameters was initially examined per N f to determine the appropriate constraints to apply moving forward. The Fresnel number versus wavelength is plotted in Figure 36 with satellite size and observing distance discretely varied. Figure 36 indicates that a large number of observing scenarios are consistent with N f ą 1 ensuring Fresnel diffraction in the near field region offers the appropriate treatment for the problem. Thus, this effort is primarily based on Fresnel diffraction phenomena and will use the Fresnel Transfer function shown in (83) and (85) as the propagation kernel. This is particularly beneficial for image reconstruction based on phase retrieval algorithms due to the presence of high order ringing features in the diffraction pattern. As Chapters 4 and 5 will reveal, the more smoothed out nature of a Fraunhofer diffraction pattern is less facilitating in terms of resolved image reconstruction in this context.

99 99 Figure 36 Fresnel number per the physical parameters associated with the diffraction pattern of a star occulted by a GEO satellite as observed on the Earth s surface. Scenarios for which N f ą 1 represent the near field Fresnel region, while parameter combinations resulting in N f ă 1 are considered to be in the far field Fraunhofer region. The nominal GEO distance of 36,000 km is used for the majority of this effort Sampling in Propagation Planes The transformation from an analytic function f px, yq to a numerically sampled function is shown in (106). The integer sample indices for the tx, yu coordinates are given by tm, nu, and the sampling interval along the x and y directions are given by x and y. f px, yq ñ f pm x, n yq (106) The finite number of sample intervals x and y over their respective dimensions is given by M and N such that the sample indices span the range m M 2,..., 0,..., M 2 1 and n N 2,..., 0,..., N 2 1. The finite physical size of the sampled dimensions in x and y is given by L x and L y such that L x M x and L y N y. The support of the function f px, yq pertains to the physical region in

100 100 which the values of f px, yq are considered significant to the solution of the particular problem. Maintaining that supports S x and S y are within the sampled region of the function requires that S x ă L x and S y ă L y. A function for which the spatial frequency content is limited to a finite range of frequencies is said to be bandlimited. The bandwidth of the spatial frequency spectrum along the x and y dimension is given by B x and B y. The Shannon-Nyquist sampling theorem states that if a function is bandlimited its continuous content can be recovered exactly if the sample intervals are smaller than a particular value, as expressed in (107). Table 5 Sampling parameters used in simulation. A square propagation plane with a 100 m width and single sample size 0.1 m is held constant throughout the simulation. The propagation step sizes through the vacuum and atmospheric regions were chosen such that the minimum sampling criteria ă λz L is maintained in the simulation. Symbol Description Value Units L ap L 1D length in aperture plane 100 m L obs L 1D length in observation plane 100 m M number of samples along L 1000 unitless L x y physical single sample length M px o, y o q aperture plane coordinates L 2 2 px, yq observation plane coordinates L ξ η spatial frequency sample size /m L pξ o, η o q aperture plane frequencies 1 1 1/m 2 L 2 L pξ, ηq observation plane frequencies 1 1 1/m 2 L 2 L z total propagation distance 3.6 ˆ 10 7 m z vac total vacuum distance ˆ 10 7 m z vac single vacuum step prop. distance ˆ 10 6 m z atm total atmospheric distance 2.5 ˆ 10 4 m z atm single atm. step prop. distance 2.5 ˆ 10 3 m min pλ, zq minimum spatial sampling λz{l m min p400 nm, z vac q min. vac. sampling λ 400 nm 7.2 ˆ 10 3 m min p1000 nm, z vac q min. vac. sampling λ 1000 nm 1.8 ˆ 10 2 m min p400 nm, z atm q min. atm. sampling λ 400 nm 1.0 ˆ 10 5 m min p1000 nm, z atm q min. atm. sampling λ 1000 nm 2.5 ˆ 10 5 m

101 101 x ă 1 2B x y ă 1 2B y (107) Aliasing results when the Shannon-Nyquist criteria is not met leading to the continuous function s undersampled high frequency components appearing as low frequency content in the sampled function. The Nyquist criteria rearranges (107) to determine the highest attainable frequencies f xq and f yq given the sampling intervals x and y and is shown in (108). 42 f xq 1 2 x f yq 1 2 y (108) This effort makes use of a consistent set of sampling parameters, as detailed in Table 5. For convenience, all propagation planes use the same 1D length L 100 m, number of samples M 1000, and sample size 0.1 m/sample. The Fresnel transfer function method, as shown in (83), is exclusively used as the propagator for numerically calculating the electric field amplitudes and resulting diffraction patterns at the observation plane (and at intermediate locations). The primary sampling 42, 56 constraint for the Fresnel transfer function propagation method is given by (109). The use of a single set of sampling parameters in all planes along the propagation is particularly useful when integrating over a spectral band so that polychromatic irradiance patterns can be calculated. min pλq ă λz L (109) Given the wavelength range λ P r400 nm, 1000 nms and nominal propagation distance z 36, 000 km the sampling criteria given in (109) will be violated if the electric field amplitude is propagated from the aperture plane to the observation plane in a

102 102 Figure 37 Multiple propagation steps performed in simulation in vacuum and atmospheric regions over a total distance of z 3.6 ˆ 10 7 m. In the vacuum region 20 propagation steps of z vac ˆ 10 6 m are performed. In the atmospheric region 10 propagation steps of z atm 2.5 ˆ 10 3 m are performed and at each plane a unique randomly drawn turbulence phase screen is applied. The phase term for the angular extent of the star is applied at the aperture plane, and the dispersion effects are applied by shifting each monochromatic ground irradiance pattern by the appropriate amount before assembling the polychromatic irradiance through summation. single step. Thus, to conserve the simplicity of a fixed array and sample size for both the aperture and observation planes multiple numerical propagations are performed such that the sampling constraint in (109) is maintained and the full desired propagation distance is achieved. This multistep propagation method introduces a series of intermediate evaluation planes at which the complex electric field amplitude is computed and further propagated until the observation plane is reached. The region between the satellite and top of the Earth s atmosphere is modeled as vacuum and the region from the top of the Earth s atmosphere to the surface of the Earth is characterized by the HV57 Cn 2 turbulence profile. In the vacuum region 20 propagation steps of z vac ˆ 10 6 m are performed. In the atmospheric region 10 propagation steps of z atm 2.5 ˆ 10 3 m are performed and at each plane a unique randomly

103 103 drawn turbulence phase screen is applied. The phase term for the angular extent of the star is applied at the aperture plane, and the dispersion effects are applied by shifting each monochromatic ground irradiance pattern by the appropriate amount before assembling the polychromatic irradiance through summation. The flow of the multistep propagation method is depicted in Figure Test Propagations Demonstrating Environmental Factors A series of simulations are performed that forward propagate light from the satellite plane to the observation plane to illustrate the effects of the physical parameters described in 3.1, 3.2, and 3.3. Simple Rect and Circ functions are used for the transmission at the satellite plane to demonstrate the accuracy and radiometric validity of the forward simulation based on known diffraction patterns. Note that these simulations do not contain any phenomena resulting from light detection or measurement.

104 Test Sim 1: Monochromatic Simulation The input parameters associated with a monochromatic simulation at 500 nm using basic Rect and Circ transmission functions at the aperture plane are shown in Table 6. The open Rect transmission function yields an observation plane irradiance profile with 12 peaks which is consistent with the expected result for a Fresnel number of N f 12.5, as demonstrated by Figure 38. The closed Rect transmission function shown in Figure 39 also produces 12 peaks in the central observation plane irradiance profile although highly attenuated. The open Circ transmission function produces the expected irradiance pattern at the observation plane with a peaked on-axis intensity as shown in Figure 40. Figure 41 pertains to a closed Circ transmission function and shows the observation plane irradiance pattern containing the on-axis peak known as Poisson s Spot described in Table 6 Simulation parameters for Test Sim 1 for monochromatic light. Parameter Value(s) Note wavelength 500 nm monochromatic magnitude of source star m _ 6 bright star aperture transmission open Rect function (Figure 38) square width = 30 m closed Rect function (Figure 39) square width = 30 m open Circ function (Figure 40) circle diameter = 30 m closed Circ function (Figure 41) circle diameter = 30 m angular extent of star 0 point source zenith angle 0 no atmospheric refraction atmospheric turbulence 0 no atmospheric turbulence

105 105 Figure 38 Test Sim 1a: Monochromatic shadow simulation using an open Rect transmission function at the aperture plane. Top: Aperture plane transmission function and irradiance pattern. Bottom: Observation plane irradiance pattern and profile. The Fresnel number is N f 12.5 for 500 nm light which is consistent with the number of expected peaks in the irradiance profile for a open Rect aperture transmission function with a half width of 15 m and a propagation distance of 36 Mm.

106 106 Figure 39 Test Sim 1b: Monochromatic shadow simulation using a closed Rect transmission function at the aperture plane. Top: Aperture plane transmission function and irradiance pattern. Bottom: Observation plane irradiance pattern and profile.

107 107 Figure 40 Test Sim 1c: Monochromatic shadow simulation using an open Circ transmission function at the aperture plane. Top: Aperture plane transmission function and irradiance pattern. Bottom: Observation plane irradiance pattern and profile.

108 Figure 41 Test Sim 1d: Monochromatic shadow simulation using an closed Circ transmission function at the aperture plane. Top: Aperture plane transmission function and irradiance pattern. Bottom: Observation plane irradiance pattern and profile. The on-axis intensity at the observation plane is consistent with Poisson s spot for a circular obscuration at the aperture plane. 108

109 Test Sim 2: Polychromatic Simulation The input parameters associated with a polychromatic simulation centered at 500 nm are shown in Table 7. An open Rect transmission function is used and polychromatic irradiance patterns at the observation plane are shown for different spectral widths in Figure 42. The overall intensity of the irradiance patterns increase as the wavelength band gets wider while the pattern itself gets washed out. Wider spectral bands yield fewer fine features in the irradiance pattern. Table 7 Simulation parameters for Test Sim 2 for polychromatic application. Parameter Value(s) Note wavelength λ 11 nm, λ 51 nm polychromatic λ 101 nm, λ 201 nm centered at 500 nm magnitude of source star m _ 6 bright star aperture transmission open Rect function square width = 30 m angular extent of star 0 point source zenith angle 0 no atmospheric refraction atmospheric turbulence 0 no atmospheric turbulence

110 110 Figure 42 Test Sim 2: Polychromatic shadow simulation for a center wavelength of λ c 500 nm using an open Rect transmission function. Top: Irradiance patterns at observation plane for λ 11 nm, and λ 51 nm. Bottom: Irradiance patterns at observation plane for λ 101 nm, and λ 201nm. While the overall intensity increases as the spectral band widens, the fine features within the narrow band patterns become washed out and less apparent.

111 Test Sim 3: Extended Source Star Added to Simulation A small angular extent is applied to the source star resulting in a phase term in the complex electric field at the aperture plane. The simulation parameters are shown in Table 8. The resulting irradiance pattern and phase at the aperture and observation planes are shown in Figure 43 using an open Rect transmission function. The angular extent of the star results in a loss of fine spatial features in the central potion of the irradiance pattern. Table 8 Simulation parameters for Test Sim 3 for a source star with an angular extent applied. Parameter Value(s) Note wavelength λ 11 nm centered at 500 nm magnitude of source star m _ 6 bright star aperture transmission open rect function (Figure 43) square width = 30 m angular extent of star 10 nrad radially distributed plane waves zenith angle 0 no atmospheric refraction atmospheric turbulence 0 no atmospheric turbulence

112 112 Figure 43 Test Sim 3: Source modeled as star with an angular extent of 10 nrad with an open Rect transmission function at the aperture plane. Top: Aperture plane irradiance pattern and phase. Bottom: Observation plane irradiance pattern and zoomed view of irradiance pattern. When no stellar angular extent is applied in the model the phase at the aperture plane is uniformly zero. The angular extent of the source star results in a loss of fine spatial features near the center of the irradiance pattern at the observation plane.

113 Test Sim 4: Atmospheric Turbulence Atmospheric turbulence is applied to the simulation using the HV57 C 2 n profile for an observer at sea level. The turbulence is applied using 10 phase screens equally spaced through the atmosphere. The simulation parameters are shown in Table 9. The irradiance patterns at the observation plane are shown in Figure 44 using open and closed Rect and Circ transmission functions at the aperture plane. The addition of turbulence to the simulation yields a grainy structure to the irradiance patterns. The peak irradiance is higher than for the case of no atmospheric turbulence, but the sum of the energy remains constant. Table 9 Simulation parameters for Test Sim 4 for application of atmospheric turbulence. Parameter Value(s) Note wavelength 500 nm centered at 500 nm magnitude of source star m _ 6 bright star aperture transmission open Rect function square width = 30 m closed Rect function square width = 30 m open Circ function circle diameter = 30 m closed Circ function circle diameter = 30 m angular extent of star 0 point source zenith angle 0 deg no atmospheric refraction atmospheric turbulence HV57 CN 2 profile atmospheric turbulence applied

114 114 Figure 44 Test Sim 4: Atmospheric turbulence applied to shadow simulation using HV57 Cn 2 turbulence profile for 500 nm light. Top: Irradiance pattern at observation plane using open and closed Rect transmission functions. Bottom: Irradiance pattern at observation plane using open and closed Circ transmission functions. The addition of turbulence to the simulations yields a grainy structure to the irradiance patterns. The peak irradiance is higher than for the case of no atmospheric turbulence, but the sum of the energy remains constant.

115 Test Sim 5: Atmospheric Refraction and Dispersion Atmospheric refraction is applied in a monochromatic fashion at 500 nm to the simulation for off zenith angles 25, 45, and 75 deg and shown assuming a fixed observer location that does not shift to center the shadow. The simulation parameters are shown in Table 10. The monochromatic irradiance patterns at the observation plane for these off zenith angles are shown in Figure 45. Also shown is a centered polychromatic irradiance pattern using 400 nm to 1000 nm light in 100 nm steps for an off zenith angle of 75 deg to illustrate the dispersive effect about 500 nm. An open Rect transmission function is used at the aperture plane for each case. The monochromatic refractive shifts and dispersion in the irradiance patterns are consistent with the displacements shown in Figure 32 and Figure 33. Table 10 Simulation parameters for Test Sim 5 for application of atmospheric refraction and dispersion. Parameter Value(s) Note wavelength 500 nm shifted from nominal nm in 100 nm steps centered at 500 nm magnitude of source star m _ 6 bright star aperture transmission open Rect function square width = 30 m angular extent of star 0 point source zenith angle 25, 45, 75 deg atmospheric refraction atmospheric turbulence none atmospheric turbulence not applied

116 Figure 45 Test Sim 5: Atmospheric refraction and dispersion applied to shadow simulation for various off zenith look angles. Top: Irradiance pattern at observation plane using open rect transmission function for 500 nm light with off zenith angles of 25 deg and 45 deg. Bottom: Irradiance pattern at observation plane for 500 nm light and off zenith angle of 75 deg, and the centered irradiance pattern for nm light in 100 nm steps for an off zenith angle of 75 deg. 116

117 Shadow Collection Model The basic geometry of the light collection system is shown in Figure 46 which consists of a linear array of apertures extending from north to south. The collection system is prepositioned based on the observing scenario and shadow prediction and remains fixed during the light collection. For simplicity, each aperture over the array is constrained to have the same diameter. The number of individual apertures is set such that the array is 100 m long regardless of the aperture diameter. This is done to make the collection area consistent with the spatial extent L of the propagation planes. Each individual collection aperture is coupled to an array of geiger-mode avalanche photo-diode (GM-APD) detectors, each with an independent readout. Figure 46 Geometry of light collection system consists of a linear array of circular apertures each coupled to multiple GM-APD detectors. The exposure time is set to the time it takes a point on the shadow to traverse a single collection aperture. Each collection system in the array is identical and is assumed to support the simultaneous detection of multiple spectral bands through a single aperture. The details of the beamtrain will depend on how narrow each spectral band is that is sent to an individual detector element. For narrow spectral band splitting ( λ ď 10

118 118 nm) a spectrometer system would likely best facilitate the wavelength mapping to the individual detectors. For wider spectral band splitting ( λ ě 25 nm) a series of cascading beamsplitters is a likely method for the wavelength division. The effective spectral transmission T a pλq for the optical beamtrain will depend on the method of wavelength division. The nominal curve for T a pλq used in this effort is shown in Figure 47. An anticipated follow on to this dissertation would focus on the details of an optimized optical design based on the results pertaining to the performance thresholds of shadow imaging and optimal spectral width each GM-APD detector receives. Figure 47 model. Optical throughput for each individual collection system used in Shadow Velocity The distance to the satellite is larger for observers at non zero latitudes and longitudes that differ from the nominal GEO satellite s longitude. Using a spherical earth model (in reality the Earth is an oblate sphere) the distance to the satellite d sat is calculated by (110) with the mean radius of the Earth given by R C 6, 367 km and the distance from the center of the Earth to the GEO range a G R C ` 36, 000 km. 27 The observer s latitude is given by φ lat and relative difference in longitude from

119 119 Figure 48 First order ground velocity of west to east moving shadow cast by a GEO satellite based on a terrestrial observer s location. The minimum velocity occurs when the observer is on the equator directly under the satellite. The shadow velocity increases as the observer moves in latitude and relative longitude. This assumes a star is present such that the shadow crosses over the observer s location. The calculation uses a spherical Earth and neglects the satellite s motion deviating from a pure GSO. the satellite by φ long. d sat b a 2 G ` 2a GR C cos pφ long q ` cos pφ lat q ` R 2 C (110) Assuming a star is suitably located, the GEO satellite s shadow velocity at the observer s location is calculated to first order using (2). The shadow velocity as a function of observer location is shown in Figure Light Collection Sampling The spacing between the telescopes is assumed to be zero in the model. The exposure time t of all the detectors is identical and determined by the transit time of the shadow over an individual aperture diameter D a as shown in the left side of Figure 49. Thus, the exposure time t is given by (111), for which v s is determined by (2) and (110) based on the observer s geographic location and D a.

120 120 t D a v s (111) Figure 49 Left: Two adjacent collection apertures within the overall linear array. The spacing between the adjacent circular apertures is assumed to be zero in the model. The exposure time of all the detectors is identical and set to the transit time of the shadow over an individual aperture based on v s and D a. Right: Timing error is introduced by applying a horizontal shift to each individual collection aperture based on a random Poisson draw with a mean at the nominal error free aperture location. Due to the sampling grid spacing of 0.1 m the smallest increment in timing error in the model is limited to approximately 38 µs. Exposure times for various aperture diameter sizes for an observer on the equator are shown in Table 11. A timing error σ t representing the potential for detector readout delay and synchronization issues is added to the model by horizontally displacing each collection telescope via a random draw from a Poisson distribution with a mean representing the error free nominal telescope position. Due to the sampling grid spacing of 0.1 m the smallest increment in timing error in the model is limited to approximately 35 µs. The right side of Figure 49 illustrates how the timing error is applied via a

121 121 Table 11 Exposure times t based on D a for an observer at the equator. The sidereal rate of the stars is 15 arcsec/s which translates to a shadow velocity v s on the Earth of 2618 m/s assuming a stationary satellite 36,000 km away. The number of apertures is chosen such that the array is 100 m long along the north/south direction to be consistent with the spatial extent L of the propagation planes. Aperture Diameter D a (m) Exposure Time t (s) Number of Apertures M ap ˆ ˆ ˆ ˆ ˆ ˆ horizontal shift in the individual telescope locations. The impact of collection aperture diameter on the as-observed normalized irradiance is shown in Figure 50. This pertains to the satellite plane transmission function shown in Figure 30, with zero gap between adjacent telescopes and no timing error applied. Figure 50 clearly illustrates the loss in spatial content and contrast as the telescope size increases.

122 122 Figure 50 Top: Example of sampling effects during light collection on the observed normalized ground irradiance pattern using the satellite plane transmission function shown in Figure 30. The top left is the true irradiance pattern which is then spatially sampled using a linear array of apertures of varying diameter sizes for λ 500 nm. The observation plane grid is zoomed in to the central +/-30 m region to better illustrate the collection sampling effects. Bottom: Corresponding normalized irradiance profiles illustrating the loss of resolution and contrast due to spatial sampling during light collection. As the aperture diameter increases there is a loss of contrast in the irradiance profile. Note: Detector measurement effects are not applied in these illustrations.

123 Light Detection and Measurement Process The devices responsible for the physical detection and measurement of light are modeled as a set of GM-APDs optically coupled to each individual collection system. Incoming light is spectrally divided and each wavelength block Λ j is sent to a single GM-APD detector element. GM-APD detectors were chosen due to their ability to measure very low light levels at high frame rates. Also referred to as single photon avalanche diodes (SPADs) GM-APD technology is at the forefront for low light and low contrast measurements. Existing applications using GM-APD devices include light detection and ranging (LIDAR) and adaptive optics (AO). 57, 58 GM-APD arrays have also been developed for conventional imaging applications. 59 The format of the GM-APD pixels could either be arranged in an 1D or 2D array of closely spaced pixels or as individually spaced single pixel elements depending on the optimal wavelength binning to each pixel. Overview of GM-APD Physics The photon measurement process in a GM-APD is inherently different from a CCD, CMOS, or regular photo-diode detector. A brief overview of how a GM-APD works and the physical parameters that influence the measurement result are outlined here. An APD refers to a reverse biased P/N junction that operates under a strong electric field (if the electric field is not suitably strong the device is simply a photodiode). When an incident photon generates a electron/hole pair the strength of the electric field is sufficient to accelerate the electron to a state of high kinetic energy. The electron is then capable of creating other electron/hole pairs in the lattice structure through impact ionization. These newly freed electrons are also accelerated by the

124 124 strong electric field, each creating more electron/hole pairs. This is referred to as an avalanche effect in the photo-diode. The breakdown voltage of the APD describes the threshold field strength for balancing the creating of electron/hole pairs with the loss of electrons accelerated out of the region which manifest in a detectable current. An APD operating with a reverse bias below the breakdown voltage is said to be in linear mode (LM) in which case the population of electron/hole pairs declines. Thus for a LM-APD the output current from the device is proportional to the incident photon flux. In contrast, a GM-APD is reverse biased above the breakdown voltage. In this case the electron/hole pairs in the P/N junction grow exponentially in population because they are being created faster than they are accelerated away. Resistance in the diode causes the voltage to drop to the breakdown value and a steady state in electron/hole pair creation and exit is achieved manifesting in a very stable output current when no incident light is present. For a GM-APD in this steady state the absorption of even a single photon results in a detectable increase in output current due to an avalanche event. Returning unaided to the steady state after the avalanche would take an unacceptable amount of time so the diode is quenched by stopping the avalanche process so that another single photon detection can occur. Passive quenching is done by adding resistance to the diode to limit the current flow so that it can be charged back up to the original reverse bias voltage. Active quenching is done by sensing when the avalanche begins and quickly cycling the reverse bias voltage below and above the breakdown voltage such that the steady state is again achieved and another photon can be detected. Because quenching is required to return the diode to a state capable of detecting more light GM-APDs generally operate in a gated mode. In this case a large number of gates (windows of light sensitivity) are each separated by the dead time required for quenching. A list of parameters unique

125 125 to GM-ADPs and frequently used in this effort are described below. (1) Dead Time (t d ): The dead time quantifies how long it takes the GM-APD to be quenched after an avalanche and then returned to nominal geiger mode. This accounts for the quenching time, the time the device remains below the breakdown voltage (hold off time), and the time it takes to bring the voltage back above the breakdown threshold (reset time). (2) Gate Time (t g ): The gate time refers to the time in which the device is in the nominal steady state and capable of detecting an avalanche event. Thus the gate time is when the device is sensitive to light. The dead time separates two successive gates. (3) Number of Gates (n g ): The number of gates is the number of measurement windows within the total exposure time. This is calculated by n g t{pt g ` t d q, or n g c d pt{t g q for which c d is the duty cycle efficiency which takes into account the dead time. (4) Exposure Time (t): The exposure time is the duration of the device measurement and consists of multiple discrete units of t g ` t d. (5) Photon Detection Efficiency (P d ): The P d combines the quantum efficiency (QE) and spatially dependent detection efficiency on the active area of the GM-APD pixel. Due to the on-pixel resister the probability of a photon being absorbed varies over the spatial extent of the pixel much like a CMOS pixel. The P d is a unitless fraction below unity. (6) Dark Count Rate (N D ): The dark count rate quantifies the number of excited electrons that trigger an avalanches without a photon being absorbed. This stems

126 126 from electron/hole pairs created from thermally induced effects or material defects. The DCR is generally given in units of electrons/s/pix or Hz. (7) Afterpulsing: Afterpulsing describes the release of a trapped electron that causes an avalanche within a single gate time. Generally the electron is trapped during a gate and then released during the next gate. As the dead time increases between gates the probability of afterpulsing P ap decreases. In effect a GM-APD simply makes a binary measurement (yes or no) as to an avalanche occurring within a single gate and repeats the process over many gates. The measurement result is a linear array of 1s and 0s that quantify the probability of an avalanche event within a single gate from which the incident photon flux can be inferred. To accurately simulate the GM-APD measurement process this effort utilizes the recent work by Kolb which describes the means for accounting for all of the detector parameters listed above in a stochastically bound measurement realization. 60 It s important to note that the formalism summarized here pertains only to actively quenched GM-APDs in which only one avalanche can occur per single gate (requiring the reset process to be controlled by external clocking). This is important because the process described here will differ slightly for GM-APD devices using passive quenching and/or for detectors in which multiple avalanches can occur within a single gate. The average number of incident signal photons per single gate n s is given by (112) where N S is the total signal photon fluence (photons/s/pix) after the light passes through the optical beamtrain. Similarly the average number of incident sky background photons per gate n b is given by (113) and the average dark count events per gate n d by (114). Here, N B is the total photon fluence from the sky background onto the pixel.

127 127 n s t g N S (112) n b t g N B (113) n d t g N D (114) The total signal photon fluence N S onto a GM-APD pixel is explicitly given by (115), in which Ipr; Λ j q is the portion of the polychromatic ground irradiance pattern for block Λ j of the cast shadow captured by a particular collection telescope. Here, the λ hc term accomplishes the photoelectric conversion. N S π 2 żλ 2 ˆDa Ipr; λq 2 λ 1 ˆ λ hc T a pλq dλ (115) The total sky background photon fluence N B is given by (116) which is based on the brightness of the sky m B given in units of visual magnitude per square arcsec and the angular FOV θ a of an individual collection system. Note that N B is independent of the shadow irradiance. N B π 2 ˆDa żλ 2 ˆθa I ν pλq 2 λ 1 ˆ 10 pm vegapλq m B q{2.5 λ hc T atm pλ, θ _ q T a pλq dλ The average number of effective photons per gate n a is given by (117). (116) n a P d pn s ` n b q ` n d (117)

128 128 Measurement Neglecting Afterpulsing Neglecting afterpulsing, the probability that a given gate will have n p photons is given by the Poisson distribution with a mean of n a as shown by (118). The Poisson distribution also constrains the number of photons to be n ě 0 and discrete. e na n np a P pn p q n p! (118) It follows that the probability P p1q that a given gate has n p ě 1 photons is given by (119), which is equivalent to the avalanche probability during the gate. P p1q P pn p ě 1q 1 e na (119) The result in (119) is also equivalent to the ratio of the total number of recorded 1s (avalanche events) n 1s in all the gates to the total number of gates n g, thus for a given gate P pn p ě 1q n 1s n g. To calculate n 1s given n a, a Bernoulli process is simulated for each i th gate with a binary outcome (avalanche=1 or no avalanche=0) as shown by (120), where R i is a uniformly drawn random number from 0 to 1 with a new seed generator for each gate since each trial is independent and fresh. $ & 1 if R i ď n a c i (120) % 0 if R i ą n a The total resulting n 1s are calculated from the sum of the individual gate trials shown by (121). n ÿ g n 1s c i (121) i 1

129 129 For the case of no afterpulsing, the measured (inferred) photon fluence resulting from a single exposure time t containing n g gates is given by (122). 60 Thus, the stochastic nature of the Bernoulli trials introduces a randomness to each measurement. ˆ j m p pno afterpulsingq ln 1 n1s ng n d (122) n g t g Measurement with Afterpulsing When taking into account afterpulsing the measurement process gets more complicated. During the first gate, the probability for an afterpulsing event is zero because there was no preceding gate in which an electron could be trapped. Thus for gate 1 the probability for an avalanche is remains given by (119). The second gate does have afterpulsing potential and the avalanche probability is given by (123). P pn p ě 1q `1 e na `1 P ap e na (123) During each successive gate the avalanche probability increases due to the potential for more afterpulsing events given that there are more preceding gates in which trapping can occur. The probability for an avalanche during the i th gate is given by (124). P i pn p ě 1q `1 ÿ e na ng i 1 `Pap e na i 1 (124) To simulate this effect, the first gate value is calculated using (120), and the subsequent gate values are determined by (125)

130 130 $ & 1 if R a i ď n a OR R b i ď P ap c i 1 c i (125) % 0 if R a i ą n a AND R b i ą P ap c i 1 The total resulting n 1s are calculated from the sum of the individual gate trails just as in the no afterpulsing case shown by (121). The measured (inferred) photon fluence resulting from a single exposure time t containing n g gates when accounting for afterpulsing is given by (126). 60 «1 n 1s ff n m p pwith afterpulsingq ln g n 1 P 1s n d ap n g n g t g (126) It is clear that when P ap 0 the expression in (126) simplifies to the case with no afterpulsing (122). The simulated measurement process is inherently random and is governed by the GM-APD parameters and incident photon fluence and realized through a series of Bernoulli trials. To illustrate this, 1000 measurement trials were simulated for different photon fluence levels using the two Baseline GM-APD parameter sets described in Table 12 and the results are shown in Figure 51. For each trial, the exposure time is fixed at t 1.53 ˆ 10 4 s per the time it takes for a shadow on the equator to traverse a 0.4 m telescope (see Table 11). The gate time is also fixed with a value of t g 10 6 s, thus the number of gates in this example is n g 145 assuming a 95% duty cycle Signal to Noise Ratio (SNR) The signal to noise ratio (SNR) is an important metric in light detection systems. The form of the SNR calculation is based on the physical mechanisms used by the detector. For all light detectors the SNR is fundamentally constrained by the Shot noise limit which is the square root of the number of signal photons incident onto the

131 131 Figure 51 Inferred photon fluence per 1000 GM-APD measurement simulations using baseline detector parameters A and B for various photon fluence levels. When the fluence level is much higher than t 1 the measurement trials yield a result very close to the true fluence level. As the fluence gets closer to the reciprocal of the exposure time the number of measurement results indicating zero inferred fluence increases. In practice, the zero value measurements can be reduced by decreasing the gate time. Also note that the inferred fluence is offset by N D when not equal to zero. detector (assuming a detection efficiency of unity). The SNR of the measurement with no afterpulsing is calculated by (127) where P p1q is the probability of an avalanche occurring during a single gate given by (119). 60 SNR(no afterpulsing) P? d n s ng b P p1q p1 P p1qq (127) The SNR of the measurement when taking into account afterpulsing is also more complicated and is calculated by (128). 60

132 132 Figure 52 Photon fluence as a function of collection diameter D a size and source star magnitude m v. The spectral widths λ r1, 5, 10, 25s are all centered at 500 nm. Per cross referencing Figure 54: A photon fluence on the detector of 1.5 ˆ 10 4 (light blue color mapping) corresponds to an SNR=1 threshold using GM-APD Baseline B parameters. The fluence is twice this for an SNR=1 using Baseline A parameters. SNR(with afterpulsing)? P d n s ng ı cp A P B 1 ` 2 Pape na 1 P ape na 1 P ap P B p1 P app A q 2 ı (128) The terms P A and P B represent the steady state probabilities (A Ñ 0 and B Ñ 1) recording an avalanche using a very large number of gates given by (129) and (130).

133 133 Table 12 Baseline GM-APD parameter values. Two sets of parameter values for a GM-APD detector are used. Baseline A represents current capability, and Baseline B is a reasonable forecast for increased performance through manufacturing and commercialization for an actively quenched externally clocked GM-APD. The gate time and number of gates are allowed to vary and will be used to increase performance based on the observing scenario. GM-APD Parameter Symbol Baseline A Baseline B duty cycle c d 95% 95% gate time t g 10 6 s 10 6 s number of gates n g variable variable exposure time t shadow limited shadow limited photon detection efficiency P d dark count rate N D 250 Hz 50 Hz probability of afterpulsing P ap P A e na p1 P ap q 1 P ap e na (129) P B 1 e na 1 P ap e na (130) A series of plots was constructed that independently gauge the impact on SNR per GM-APD parameter and are shown in Figure 53. Baseline GM-APD Parameters This effort will use the two sets of GM-APD parameters which are listed in Table 12. Baseline A represents current capability, and Baseline B is a reasonable forecast for increased performance through manufacturing and commercialization for an actively quenched externally clocked GM-APD. The gate time and number of gates are allowed to vary and will be used to increase performance based on the observing scenario. The SNR versus photon fluence for each baseline GM-APD parameter set appears in Figure 54.

134 134 Figure 53 Impact on SNR per varied GM-APD parameters as a function of photon fluence. Upper Left: Higher photon detection efficiency P D results in increased SNR at low fluence levels while also extending the peak SNR to higher fluence levels. Upper Right: Lower dark count rates N D increase the SNR at low fluence levels but have very little SNR impact at higher fluences (the SNR plot in this case is zoomed into the low fluence region). Bottom Left: As the probability of afterpulsing decreases the peak SNR rises but remains associated with the same fluence level. A small increase in SNR at low fluence levels is also observed. Bottom Right: As the gate time t g decreases the number of gates n g increases over a fixed exposure time t. This results in a higher peak SNR that extends to higher fluence levels. Anticipated Signal Photon Rate for Shadow Imaging Scenarios The estimated photon fluence for a set of representative shadow imaging scenarios are shown in Figure 52. A photon fluence on the detector of 1.5 ˆ 10 4 (light blue color mapping) corresponds to an SNR=1 threshold using GM-APD Baseline B

135 135 Figure 54 SNR as a function of photon fluence for the GM-APD parameter sets described in Table 12. Baseline A represents current GM-APD capability and Baseline B corresponds to foreseeable detector performance levels in the future. parameters. The fluence is twice this for an SNR=1 using Baseline A parameters. This is encouraging because it indicates that there is no fundamental SNR constraint for stars as dim as m _ 10 using 0.4 m diameter collection telescopes for spectral binning of 10 nm. The end-to-end simulations will provide the true limits, but again, this is encouraging. 3.8 Chapter 3 Summary The mathematical treatment of the physical phenomena associated with the starlight, diffraction at the satellite plane, and light propagation through space and the Earth s atmosphere was fully developed in 3.1, 3.2, and 3.3. A series of test propagations was demonstrated in 3.5 using simple Rect and Circ transmission functions place at the satellite plane which demonstrated the impacts on the ground irradiance pattern from the source star and atmosphere, while maintaining radiometric accuracy. The collection of light was described in 3.6 including the parameter vari-

136 136 ables that the model accounts for such as telescope aperture diameter and relative separation between telescopes, timing error, and spectral binning. The light detection/measurement process and SNR calculation were detailed in 3.7. At this point the primary physical modeling is complete. The manifestation of this mathematical development into an end-to-end simulation capability is accomplished next in Chapter 4 through the construction of multiple interacting functions in Matlab. This network of numeric functions, representing all of the physical phenomena described in Chapter 3, will serve as the simulation engine used in the performance assessment of GEO satellite shadow imaging capability.

137 137 CHAPTER 4 SIMULATION PROCESS The end-to-end simulation process is presented in a detailed fashion here in Chapter 4 using the underlying physics and mathematical treatment described in Chapter 2 and Chapter 3. The numerical simulation process is initiated using a set of user defined parameters that describe a particular scenario. These user defined input parameters are described in 4.1 in the context of the explicit Matlab input format. The end-to-end simulation consists of four main steps which are outlined in 4.2. The numerical functions constructed in Matlab that form the engine for the numerical simulation process are outlined in a sequential fashion in 4.3 and shown explicitly in Appendix A and Appendix B. 4.1 User Defined Scenario The simulation process uses input parameters defined in Matlab based on a user specified scenario. These input parameters represent the physical variables developed in Chapter 2 and Chapter 3. The specific scenario is defined prior to each simulation. (1) Numerical sampling parameters: These parameters define the discrete sampling in the propagation planes including the support extent L, number of samples M, and grid spacing dx as outlined in The user specified numerical sampling parameters are explicitly defined in Matlab in Code Example 1.

138 138 Code Example 1 User defined numerical sampling parameters. 1 % % Numerical sampling parameters 3 L = 100; % side length of support extent [m] 4 M = 1000; % number of samples in x and y 5 dx = L/M; % single sample interval [m/pix] 6 x = -L/2:dx:L/2-dx; % x plane coordinates [m] 7 y = x; % y plane coordinates [m] 8 [X,Y] = meshgrid(x,y); % build meshgrid of plane coordinates (2) 2D satellite transmission function: This defines the binary transmission function described in 3.2 representing the satellite, which can be selected from a number of options. These options include mock satellites GEO1, GEO2, and GEO3 at different orientations to the observer s line of sight. The user may also select a Rect/Circ function for diagnostic purposes, or a barsat composed of horizontally/vertically orientated resolution bars. The binary contrast of the 2D transmission function can also be reversed. The format for selecting the 2D transmission function in Matlab is shown in Code Example 2. Code Example 2 User defined satellite transmission function. 1 % % 2D Satellite transmission function selection 3 % ap = GEO1A; ap = GEO2A; ap = GEO3A; 4 % ap = GEO1B; ap = GEO2B; ap = GEO3B; 5 % ap = GEO1C; ap = GEO2C; ap = GEO3C; 6 % Test case options for Rect or Circ aperture 7 % w = 10; % aperture half width [m] 8 % ap = rect(x/(2*w)).*rect(y/(2*w)); % rect aperture function 9 % ap = circ(w/dx,[m,m]); % circ aperture function 10 % Test case options for bar targets 11 % ap = smallbarsh; medbarsh; largebarsh; 12 % ap = smallbarsv; medbarsv; largebarsv; 13 ap = GEO1A; 14 % ap = ap; % option to reverse binary contrast (3) Source star, environmental, and observation parameters: These parameters define the source star magnitude and angular extent per 3.1, in addition to environmental and observing parameters described in 3.3. The input format for defining

139 139 these parameters in Matlab is shown in Code Example 3. Code Example 3 parameters. User defined source star, environmental, and observation 1 % % Source star parameters 3 mv = 9; % visual magnitude of star [] 4 alpha = 1e-50; % angluar extent of star [rad] 5 % % Environmental parameters 7 atmt scl = 1; % scaling for nominal atm trans. lookup table 8 sclturb = 1; % scaling for nominal HV57 CN2 profile 9 mvb = 20; % sky brightness [mv/arcsecˆ2] 10 % % Observation parameters 12 zen = 0; % off zenith look angle angle [rad] 13 alt = 0; % observation altitude [m] 14 lat = 0; % observer's latitude [deg] 15 long = 0; % observer's relative longitude to satellite [deg] (4) Light collection parameters : These parameters define the light collection system per including the telescope aperture diameter and optical transmission scaling. The spectral range for each wavelength block Λ j is also specified along with the reference wavelength for atmospheric refraction/dispersion. The input format for defining these parameters in Matlab is shown in Code Example 4.

140 140 Code Example 4 User defined light collection parameters. 1 % % Collection system parameters 3 IFOV = 1; % single pixel FOV [arcsec] 4 syst scl = 1; % scaling for nominal optical trans.... lookup table 5 pixsamp = 4; % pixel collection sampling [pix] (1 pix... = 0.1 m) 6 Da = pixsamp*0.1; % collection aperture diameter [m] 7 numap = floor(m/pixsamp); % number of collection apertures... in array [#] 8 pixerr = 0; % shift in telescope position per timing... error [pix] 9 % % Wavelength range over all blocks 11 wvlref m = 500e-9; % ref. wavelength for refraction and... dispersion [m] 12 wvlblk start nm = 400; % starting wavelength [nm] 13 wvlblk end nm = 500; % ending wavelength [nm] 14 wvlblk nm = 10; % wavelength range in each block [nm] 15 nb = (wvlblk end nm-wvlblk start nm)/wvlblk nm; % # of wvl blks (5) GM-APD detector parameters: These parameters define the GM-APD detector described in 3.7. The exposure time is also defined here based on the shadow velocity per The input format for defining these parameters in Matlab is shown in Code Example 5. Code Example 5 User defined GM-APD detector parameters. 1 % % Timing parameters 3 vs = shadowvelocity dmd(lat,long); % shadow velcoity [m/s] 4 % % GM-APD detector parameters: 6 DCR = 250; % dark count rate [#/s] 7 PDE = 0.25; % probability of photon absorbing [#/s] 8 p aft = 0.20; % probability of afterpulsing [] 9 t exp = Da/vS; % exposure time [s] 10 t gate = 10e-7; % gate length [s] 11 dutyc = 0.95; % duty cycle []

141 Steps in End-to-End Numerical Simulation The end-to-end simulation process is comprised of four main steps. Step 1 generates a polychromatic irradiance pattern that is used to represent the true shadow on the Earth s surface incident onto the light collection system and is detailed in Step 2 produces an inferred irradiance pattern based on the light collection and GM-APD measurement processes and is outlined in Step 3, as described in 4.2.3, performs the image reconstruction process for a single wavelength block based on the inferred irradiance determined by Step 2. Step 4 creates the final image by stacking the reconstructed images of all the wavelength blocks and is summarized in Step 1: Generating the Truth Shadow The true polychromatic shadow is generated in Step 1 as the sum of the monochromatic irradiance patterns within a given wavelength block Λ j via numerical simulation. The primary function imgen.m is called multiple times to generate the monochromatic shadows Ipr; λ i q which are summed to produce the polychromatic shadow Ipr; Λ j q as shown by (131). Iÿ Ipr; Λ j q Ipr; λ i q (131) i 1 A user defined wavelength block Λ j specifies the end points and width of the spectral band of the polychromatic shadow per (132). Λ j rλ i, λ i`1, λ i`2,..., λ I s (132) The process within Step 1 for generating the truth shadow is shown in Figure 55.

142 142 Figure 55 Step 1 in end-to-end simulation process. The true polychromatic shadow Ipr; Λ j q at the observer location is generated by summing the monochromatic irradiance patterns Ipr; λ i q. The function imgen.m is called multiple times to generate the monochromatic shadows Ipr; λ i q. A script representing Step 1 is shown in Matlab format in Code Example 6. The output for Step 1 is denoted by pi truth which represents the true polychromatic ground irradiance Ipr; Λ j q. Note that the functions imcollect.m and photgen.m appear in the loop over λ i within Step 1 to increase processing speed.

143 143 Code Example 6 Simulation Step 1 in Matlab format: The truth shadow Ipr; Λ j q is generated for each wavelength block. 1 % % STEP 1: Generate truth irradiance 3 % % Preallocate zeros to initial polychromatic arrays for indexing... speed 5 pi truth = zeros(m,m); 6 pp samp = zeros(numap,numap); 7 ppesig = zeros(numap,numap); 8 ppesky = zeros(numap,numap); 9 matlabpool(8); % initiate parallel processing for... computational speed 10 parfor j = 1:nB; % loop over each spectral block 11 b = wvlblk start nm+(j-1)*wvlblk nm 12 pi truth{j}=zeros(m,m); 13 pp samp{j}=zeros(numap,numap); 14 ppesig{j}=zeros(numap,numap); 15 ppesky{j}=zeros(numap,numap); 16 % Wavelength range within single block 17 wvl start nm = b; % starting wvl [nm] 18 wvl end nm = b + wvlblk nm - 1; % ending wvl [nm] 19 wvldelta = 1+(wvl end nm - wvl start nm); % wvl [nm] 20 wvl mid nm{j}=round(wvl start nm+wvldelta/2);% mid blk wvl [nm] 21 for w = wvl start nm:1:wvl end nm; % loop over wvls within block 22 wvl m = w*1e-9; % wvl [nm] 23 % Indexed parameters 24 wvl idx = round(1e9*wvl m); % wvl index 25 syst= systrans(wvl idx)*syst scl ; % system transmission at wvl 26 vexo = VegaExoSpectIrrad(wvl idx,2);% spectral irradiance at wvl 27 vmag = vegamag(wvl idx); % AB magnitude of Vega at wvl 28 atmt = atmtrans(wvl idx)*atmt scl; % atm transmission at wvl 29 % Generate monochromatic "TRUTH" ground irradiannce pattern 30 [mi truth] =... imgen dmd(ap,wvl m,wvlref m,mv,alpha,zen,alt,sclturb,atmt,vexo) 31 % Monochromatic power collection per aperture sampling 32 [mp samp] = imcollect dmd(mi truth,pixsamp,pixerr); 33 % Monochromatic signal and sky background photon rates 34 [mpesig,mpesky] =... photgen dmd(mp samp,wvl m,vexo,vmag,syst,da,ifov,mvb); 35 % Polychromatic sums 36 pi truth{j} = pi truth{j} + mi truth; % irradiance [1000 x 1000] 37 pp samp{j} = pp samp{j} + mp samp; % sampled power [numap... x numap] 38 ppesig{j} = ppesig{j} + mpesig; % sig photon rate [numap... x numap] 39 ppesky{j} = ppesky{j} + mpesky; % sky photon rate [numap... x numap] 40 end 41 end 42 matlabpool close; % close parallel processing

144 Step 2: Generate the Measured Shadow The measured or inferred shadow I M pr; Λ j q for each Λ j is generated in Step 2 using the true shadow Ipr; Λ j q, collection system parameters, and detector characteristics. The true monochromatic irradiance patterns Ipr; λ i q along with the collection parameters are fed into imcollect.m which produces the true collected monochromatic power patterns P pr; λ i q. The size of the square 2D arrays P pr; λ i q is denoted by M ap and is determined by D a per (133) and the grid spacing of the collection sampled shadow is set equal to D a. M ap L{D a (133) The monochromatic power patterns are then input to photgen.m which generates the monochromatic signal and sky background photon rate patterns N S pr; λ i q and N B pr; λ i q which are independently summed to yield their polychromatic equivalents N S pr; Λ j q and N B pr; Λ j q. The polychromatic signal and background photon rate patterns are then fed into measuregmapd.m which yields a stochastic realization of the measured polychromatic photon rate pattern N M pr; Λ j q based on the development in 3.7. The monochromatic measured photon rate patterns N M pr; λ s i q are obtained by (134) which approximates that each λ i within Λ j hold equal weighting in forming N M pr; Λ j q. N M pr; s λ i q «1 λ N Mpr; Λ j q (134) Each N M pr; λ i q is discretized and input to powergen.m converting them to measured power patterns P M pr; λ i q which are then interpolated to 2D arrays of size M by M. Each inferred P M pr; λ i q is then converted to measured irradiance I M pr; λ i q.

145 145 The inferred polychromatic irradiance pattern I M pr; Λ j q is then obtained by (135). Iÿ I M pr; Λ j q I M pr; λ i q (135) i 1 The sequential process within Step 2 for generating the measured (inferred) shadow is shown in Figure 56. Note that the interpolation from the sampled array size of M ap by M ap to M by M is required to propagate the electric field resulting from the converted inferred irradiance pattern in Step 3. A script representing Step 2 is shown in Matlab format in Code Example 7. The output for Step 2 is the measured or inferred irradiance pi meas I M pr; Λ j q. Note that for computational speed some of the quantities produced in Step 2 as indicated by Figure 56 are actually embedded in the wavelength loop in Step 1 as constructed in Matlab. These include the true polychromatic sampled power pp samp P pr; Λ j q and the signal and sky background photons rates ppesig N S pr; Λ j q and ppesky N B pr; Λ j q.

146 146 Figure 56 Step 2 in end-to-end simulation process. The true shadow Ipr; Λ j q is sampled and attenuated per the collection parameters to yield the total signal and sky background photon fluence onto each detector. The function measuregmapd.m generates a stochastically based measurement realization which represents the inferred (measured) polychromatic irradiance pattern I M pr; Λ j q.

147 147 Code Example 7 Simulation Step 2 in Matlab format: The measurement inferred irradiance I M pr; Λ j q is generated for each wavelength block. 1 % % STEP 2: calculate measured irradiance 3 % for j = 1:nB 5 % Preallocate zeros to initial polychromatic arrays for indexing... speed 6 pphot meas{j}=zeros(numap,numap); 7 SNR gapd{j}=zeros(numap,numap); 8 % Polychromatic measurement [phot/sec]and SNR 9 [pphot meas{j},snr gapd{j}] =... measuregmapd dmd(ppesig{j},ppesky{j},gapd detspecs); 10 % APPROXIMATION: monochromatic inference 11 mphot meas = pphot meas{j}/wvlblk nm; % [phot/sec] 12 % Preallocate polychromatic irradiance map 13 pi meas{j} = zeros(m,m); 14 end 15 matlabpool(8); % initiate parallel processing for computational... speed 16 parfor j = 1:nB 17 pi meas{j}=zeros(m,m); 18 b = wvlblk start nm+(j-1)*wvlblk nm 19 % Wavelength range within single block 20 wvl start nm = b; % starting wvl [nm] 21 wvl end nm = b + wvlblk nm - 1; % ending wvl [nm] 22 wvldelta = 1+(wvl end nm - wvl start nm); % wvl [nm] 23 wvl mid nm{j} = round(wvl start nm+wvldelta/2); % mid wvl for blk... [nm] 24 for w = wvl start nm:1:wvl end nm 25 wvl m = w*1e-9; % wvl [m] 26 % Generate measured sampled power map 27 [mp samp] = powergen dmd(mphot meas,wvl m); 28 % Monochromatic measured power interpolated to 1000 x 1000 grid 29 [mp meas] = impowerinterp dmd(mp samp,pixsamp); % [W] 30 % Measured monochromatic irradiance 31 [mi meas] = P2I dmd(mp meas,dx); % [W/mˆ2/nm] 32 % Polychromatic sum 33 pi meas{j} = pi meas{j} + mi meas; % measured polychromatic... irradiance 34 end 35 end 36 matlabpool close; % close parallel processing mode Step 3: Image Reconstruction for Single Wavelength Block The inferred shadow I M pr; Λ j q generated by Step 2 is used as the input model to a Gerchberg-Saxton phase retrieval (GSPR) algorithm to estimate the 2D transmission

148 148 function of the satellite. 61 First I M pr; Λ j q is converted to I M pr; s λ i q where s λ i is given by (136). sλ i 1 Iÿ λ i (136) I In effect I M pr; Λ j q ñ I M pr; λ s i q is converting a polychromatic irradiance pattern to a monochromatic irradiance pattern with the same intensity. This is done to increase contrast in the GSPR algorithm but at the cost of losing some of the spatial details (generally a single nanometer spectral bandwidth does not yield sufficient SNR). The GSPR based algorithm in Step 3 uses the following sub steps: i 1 (a) Propagate an initial guess of the satellite transmission function g q 0 pr o q to the observation plane using forwardprop.m. (b) Replace the electric field amplitude at the observation plane with the amplitude derived from the measured irradiance pattern I M pr; s λ i q using forceamp.m to yield u q pr; s λ i q. (c) Back propagate u q pr; s λ i q using backprop.m from the observation to satellite plane. (d) Replace the electric field phase at the satellite plane with the phase from the electric field of the source star φ S pr o ; s λ i q using forcephase.m. (e) Repeat sub steps (a) to (d) for the next iteration. Sub steps (a) to (d) constitute a single iteration. The next iteration propagates u q`1 pr o ; λ s i q to the observation plane. Note the use of forwardprop.m and back- Prop.m do not include atmospheric effects but do use the same intermediate propagation steps as imgen.m. After a user defined number of iterations Q the estimate

149 149 of the satellite transmission function gq 1 pr o; Λ j q is obtained. Note that the satellite transmission function is considered the image estimate for Λ j. The iterative GSPR algorithm within Step 3 for generating the estimated image for Λ j is shown in Figure 57. Figure 57 Step 3 in end-to-end simulation process. A Gerchberg-Saxton phase retrieval algorithm is used in an iterative fashion to form the reconstructed image. An initial guess of the 2D satellite transmission function is used to seed the algorithm. The field is propagated from the satellite plane to the ground and the amplitude is replaced by the measurement inferred field amplitude. The updated field is then back propagated from the ground to the satellite plane and the phase is replaced by the satellite plane phase. This process is repeated iteratively Q times to yield the reconstructed image for wavelength block Λ j. A script representing Step 3 is shown in Matlab format in Code Example 8. The output for Step 3 is the reconstructed image img g 1 Q pr; Λ jq for each wavelength block.

150 150 Code Example 8 Simulation Step 3 in Matlab format: The reconstructed image gq 1 pr; λ s i q is generated for each wavelength block. 1 % % STEP 3: Iterative image reconstruction using Gershberg-Saxton... algorithm 3 % matlabpool(8); % initiate parallel processing for computational... speed 5 parfor j = 1:nB 6 ap est initial{j} = rand(1000)*mean(mean(pi meas{j})); 7 ap est = ap est initial{j}; 8 % mid wavelength in block for reconstruction algorithm 9 wvl m{j} = wvl mid nm{j}*1e-9; % [m] 10 % convert measured wvl block irradiance to scalar electric field 11 u meas{j} = I2E dmd(pi meas{j}); 12 Q = 100; % total number of iterations 13 for q = 1:Q; % iterate Q times in loop 14 % forward propagate estimated aperture transmission function 15 [u est 1] = forwardprop dmd(i2e dmd(ap est),wvl m{j},alt); 16 % replace electric field amp. based on meas and maintain... phase 17 u est 2 = forceamp dmd(u est 1,u meas{j}); 18 % back propagate to satellite plan 19 u est 3 = backprop dmd(u est 2,wvl m{j},alt); 20 % replace phase with satelite plane phase and maintain... amplitude 21 [u est 4,phi ap] =... forcephase dmd(u est 3,alpha,wvl m{j},l,m); 22 % estimated image after current iteration 23 ap est = E2I dmd(u est 4); 24 end 25 img{j} = ap est; % reconstructed image for wvl blk j 26 end 27 matlabpool close; % close parallel processing mode

151 151 Note on Phase Retrieval Algorithm The GSPR algorithm shown utilized in Step 3 is widely used in imaging applications of various forms. This effort uses a fairly simple incarnation of the GSPR algorithm which is shown to work very well. Variations to the algorithm can be made including imposing additional constraints at the satellite plane. Such constraints could include: support constraints (transmission is unity beyond a specified radial distance from the center), and/or binary constraints (forcing the estimated satellite transmission function to be either 1 or 0 based on a specified threshold). 62 Such constraints were imposed on test runs but the image reconstruction proved to converge faster and more accurately for GEO satellite shadow imaging cases without the constraints applied. In addition, non physical artifacts were frequently present in the reconstructed images when these types of constraints were applied at the satellite plane. Overall, the GSPR algorithm demonstrated fairly rapid convergence within 10 iterations for cases that possessed SNR values greater than 1.5. The number of iterations is nominally set to either 50 or 100. A few different cases for the initial guess of the satellite transmission function g q 0 pr o q were attempted, including small centered Circ and Rect functions as well as random shapes. Ultimately, the fastest and most accurate image reconstruction occurred when a uniform distribution (each matrix element randomly drawn from 0-1) multiplied by the mean of the inferred irradiance for a given wavelength block was selected as the initial guess Step 4: Final Image Reconstruction using Multiple Wavelength Blocks The final step in the end to end simulation process is the construction of the estimated image using the image estimates from each wavelength block g 1 Q pr o; s Λ j q.

152 152 The final estimated image g 1 pr o q is generated by stacking the estimated images from each wavelength block using objsum.m as shown in (137). Jÿ g 1 pr o q gpr o ; Λ j q (137) j 1 The summing of each estimated wavelength block image is shown in Figure 58. Figure 58 Step 4 in end to end simulation process. The final estimated image g 1 pr o q is obtained by summing each estimated wavelength block image g 1 pr o ; Λ j q using objsum.m. A script representing Step 4 is shown in Matlab format below in Code Example 9. The output for Step 4 is the final reconstructed image IMG g 1 pr o q.

153 153 Code Example 9 Simulation Step 4 in Matlab format: The final image g 1 pr o q is formed by adding the reconstructed images from each wavelength block. 1 % % STEP 4: Final image formation 3 % % Preallocate zeros to initial polychromatic arrays for indexing... speed 5 IMG = zeros(m,m); 6 for j = 1:nB; % loop over wvl blks 7 IMG = IMG + img{j}; % summation process 8 end 9 save(strcat(f2,runnum IMG),'img','IMG'); % save final image 4.3 Descriptions of Functions Constructed in Matlab Multiple function were created in Matlab specific to this effort. These functions serve as the engine for the end-to-end simulation process and are called repeatedly throughout Steps 1-4 in the scripts shown above in Code Examples 6, 7, 8, and 9. Each function represents a unique physical process or calculation and is assembled external to the scripts from which it is are called. This is done for processing speed and modularity. These independent functions are loosely categorized as primary functions which are described in 4.3.1, and auxiliary functions which are described in Note that a number of Matlab toolboxes (add-on packages to the baseline Matlab functionality) are required to run the simulation in addition to the functions specifically created for this effort by the author. These add-ons include the Image Processing Toolbox, Statistics Toolbox, and Mapping Toolbox Primary Functions Constructed in Matlab Below is a list of the primary functions constructed in Matlab that are used in the end-to-end simulation process. The computations within each function are described in a sequential fashion per the formalism established in the preceding chapters. The primary functions are explicitly shown in Appendix A.

154 154 (1) imgen.m: Generates each true monochromatic irradiance pattern Ipr; λ i q on the ground of size M by M with a grid spacing of based on user defined inputs. In this effort M 1000 and 0.1 m. imgen.m is used within the wavelength loop over λ i in Step 1 for each wavelength block Λ j. The sequential sub steps within imgen.m are outlined below and the function is shown explicitly in A.1.1. (a) The exoatmospheric source star complex electric field at the satellite plane U S pr o ; λ i q is calculated per (89) using stari2e.m. The field is then multiplied by the satellite transmission function gpr o q per (94) to yield the field emanating from the satellite plane. (b) The phase of the star s electric field at the satellite plane φ S pr o ; λ i q is calculated based on its angular extent using phistarangle.m and implemented into U S pr o ; λ i q via (93) prior to being propagated from the satellite plane. (c) The monochromatic field U S pr o ; λ i q is propagated from the satellite plane to the top of the atmosphere in 20 evenly spaced steps per (83) using proptf.m. (d) The atmosphere is divided into 10 evenly spaced regions and the atmospheric coherence length r o pλ i q is calculated for each region by summing over the HV57 Cn 2 profile per (102) using friedparam.m. A unique and stochastically based phase screen is generated for each of the 10 atmospheric layers using phasescreens.m. (e) The field is then propagated from the top of the atmosphere through each successive turbulence layer to the ground location using proptf.m. The random phase at each turbulence layer is applied to the electric field per (105).

155 155 (f) The field at the observer location is shifted in one dimension based on the magnitude and user defined spectral origin of the atmospheric dispersion using atmrefract.m per (99). It is assumed that the bulk displacement from the atmospheric refraction has been accounted for in the positioning of the observer and collection system. (g) The electric field is converted to irradiance using E2I.m and then multiplied by the atmospheric transmission per (96) yielding the output Ipr; λ i q which is used as the true monochromatic ground shadow for λ i in Step 1 outlined in (2) imcollect.m: Generates each collected monochromatic sampled power pattern P pr; λ i q on the ground of size M ap by M ap with a grid spacing determined by D a. For processing speed imcollect.m is used within the wavelength loop over λ i in Step 1, although it is conceptually represented in Step 2 as shown by Figure 56. The sequential sub steps within imcollect.m are outlined below and the function is shown explicitly in A.1.2. (a) Converts true monochromatic ground irradiance Ipr; λ i q to power by multiplying each array value by 2 yielding the M by M true ground monochromatic power pattern. (b) Creates a Circ function with diameter in pixel space equivalent to D a. The Circ function is shifted over the M by M power array in equivalent pixel steps to D a and multiplied with each subregion to simulate the collection of light from a linear array of circular apertures over which the shadow scans. (c) If pixerr is non zero the sample shift of the Circ function in column space is perturbed according to a Poisson distribution with a mean equaling the user

156 156 defined pixerr value. This represents any potential timing error σ t described in (d) The power sub regions are reassembled into a single array of size M ap by M ap and grid spacing determined by D a yielding the true collected monochromatic power pattern P pr; λ i q. (3) photgen.m: Generates as-collected monochromatic signal and sky background photon arrival rates N S pr; λ i q and N B pr; λ i q for each collection aperture per the spatial sampling determined by D a. photgen.m is used within the wavelength loop in Step 1 for processing speed. The sequential sub steps within photgen.m are outlined below and the function is shown explicitly in A.1.3. (a) The monochromatic signal photon rate N S pr; λ i q is calculated per collection aperture by multiplying P pr; λ i q by λ i { hc. The attenuation due to the beamtrain optical transmission T opt pλ i q is also applied. This step is analogous to (115) prior to applying the detector parameters. (b) The monochromatic sky background photon rate N B pr; λ i q is calculated based on the sky brightness per collection aperture using (116). (4) measuregmapd.m: Generates the a stochastically based realization of the measured polychromatic photon rate N M pr; Λ i q for each collection aperture within a single wavelength block. N S pr; λ i q and N B pr; λ i q are inputs to measuregmapd.m along with the GM-APD detector parameters. The sequential sub steps within measuregmapd.m are outlined below and the function is shown explicitly in A.1.4. (a) Calculates photon fluence per single gate with exposure time t per (112), (113), and (114).

157 157 (b) Calculates a single realization of the measured polychromatic photon rate using (126) based on user defined detector parameters and average number of incident photons per gate. (c) Calculates the SNR for each measurement realization. (5) proptf.m: Serves as the engine for forward propagating the complex scalar electric field from plane to plane in Step 1 and Step 4. The electric field U 1 pr; λ i q, support L, wavelength λ i, and propagation distance z are the inputs to proptf.m. The sequential sub steps within proptf.m are outlined below and the function is shown explicitly in A.1.5. (a) The Fresnel transfer function Hpξ, η; λ i q is calculated per (83). (b) A Fourier transform of the input field U 1 pr; λ i q is performed to yield U 1 pξ, η; λ i q which is then multiplied by Hpξ, η; λ i q. (c) The inverse Fourier transform of the product U 1 pξ, η; λ i qhpξ, η; λ i q is calculated to yield the output field U 2 pr; λ i q at a plane located a distance z away. This is equivalent to (85) and generalized for any propagation distance. (6) propinversetf.m: Serves as the engine for back propagating the complex scalar electric field from plane to plane in Step 4. The electric field U 2 pr; λ i q, support L, wavelength λ i, and propagation distance z are the inputs to proptf.m. Note that this is the inverse of the function propfresneltrans.m. The sequential sub steps within propinversetf.m are outlined below and the function is shown explicitly in A.1.6. (a) The Fresnel transfer function Hpξ, η; λ i q is calculated per (83).

158 158 (b) A Fourier transform of the input field U 2 pr; λ i q is performed to yield U 2 pξ, η; λ i q which is then divided by Hpξ, η; λ i q. (c) The inverse Fourier transform of the quotient U 2 pξ, η; λ i q{hpξ, η; λ i q is calculated to yield the output field U 1 pr; λ i q at a plane located a distance z away. This is equivalent to the inverse of (85) and generalized for any propagation distance. (7) forwardprop.m: Performs the forward propagation in Step 4 from the satellite plane to the observation plane for each iteration within the GSPR algorithm. The inputs to forwardprop.m are the electric field (for the q th iteration) at the satellite plane U q pr o ; λ s i q, the propagation wavelength λ s i, and the observer s altitude h. Note that forwardprop.m does not apply any atmospheric effects but does utilize the same propagation planes as the function imgen.m. The sequential sub steps within forwardprop.m are outlined below and the function is shown explicitly in A.1.7. (a) Determines spacing of the propagation planes based on observer s altitude. The is done in the same fashion as imgen.m per Figure 37. (b) Propagates U q pr o ; λ s i q from the satellite plane to the observation plane using proptf.m through intermediate propagation planes to yield U q pr; λ s i q. (8) backprop.m: Performs the back propagation in Step 4 from the observation plane to the satellite plane for each iteration within the GSPR algorithm. The inputs to backprop.m are the electric field (for the q th iteration) at the observation plane U q pr; λ s i q, the propagation wavelength λ s i, and the observer s altitude h. Note that backprop.m does not apply any atmospheric effects but does utilize the same propagation planes as the function imgen.m, but in reverse order. The

159 159 sequential sub steps within backprop.m are outlined below and the function is shown explicitly in A.1.8. (a) Determines spacing of the propagation planes based on observer s altitude. The is done in the same fashion as imgen.m per Figure 37, but in reverse order. (b) Propagates U q pr; λ s i q from the observation plane to the satellite plane using propinversetf.m through the intermediate propagation planes to yield U q pr o ; λ s i q. (9) forceamp.m: Applies the amplitude constraint at the observation plane for each iteration in the GSPR algorithm in Step 4. The inputs to forceamp.m are the raw forward propagated electric field at the observation plane U qr pr; λ s i q and the amplitude of the electric field ˇˇUM pr; λ s i qˇˇ inferred from the measured irradiance pattern I M pr; Λ j q. Note the subscript U qr denotes the raw forward propagated electric field at the observation plane prior to the imposed amplitude constraint. The sequential sub steps within forceamp.m are outlined below and the function is shown explicitly in A.1.9. (a) Calculates the amplitude of the measurement inferred scalar electric field at the observation plane ˇˇUM pr; λ s i qˇˇ. (b) Replaces the amplitude of the raw forward propagated electric field with the measurement inferred scalar electric field amplitude at the observation plane using U q pr; λ s i q ˇˇUM pr; λ s i qˇˇ qpr; λ s eiφ i q. Here, φ q pr; λ s i q is the phase from the forward propagated electric field at the observation plane during the q th iteration.

160 160 (10) forcephase.m: Applies the phase constraint at the satellite plane for each iteration in the GSPR algorithm in Step 4. The inputs to forcephase.m are the raw back propagated electric field at the satellite plane U qro pr o ; λ s i q, the angular extent of the source star α S, the mean wavelength λ s i for the given spectral block Λ j, the support size L, and array size M. Note the subscript U qro denotes the raw back propagated electric field at the satellite plane prior to the imposed phase constraint. The sequential sub steps within forcephase.m are outlined below and the function is shown explicitly in A (a) Calculates the phase φ S pr o ; λ s i q from the star using the function phistarangle.m. (b) Replaces the phase of the raw back propagated electric field with the source star phase at the satellite plane using U q pr o ; λ s i q ˇˇUqr pr o ; λ s i qˇˇ eiφ S pr o; λ s i q Auxiliary Functions Constructed in Matlab A number of auxiliary functions were also constructed in Matlab that support Steps 1-4 in the end-to-end simulation process detailed in 4.2. In many cases the auxiliary functions are called by the primary functions. Below is a list of these auxiliary functions and descriptions of their sequential calculations. The auxiliary functions are explicitly shown in Appendix B. (1) stari2e.m: Converts the exoatmospheric spectral irradiance from the source star to electric field amplitude at the satellite plane. The inputs to stari2e.m are the source star s visual magnitude m _, λ i, and the exoatmospheric spectral irradiance of Vega I ν pλq. The sequential sub steps within stari2e.m are outlined below and the function is shown explicitly in B.1.1.

161 161 (a) The AB magnitude of Vega m vega pλ i q is calculated using (87). (b) The amplitude of the electric field U S pr o ; λq incident onto the satellite plane is calculated per (89). Note that the incident field from the star U S pr o ; λq is assumed to be spatially uniform onto the satellite plane. (2) I2E.m: Converts irradiance to electric field amplitude. I2E.m is similar to stari2e.m but does not pertain to a source star and can be used polychromatically. Note, this calculation yields no phase information. The input to I2E.m is irradiance Iprq. The sequential sub steps within I2E.m are outlined below and the function is shown explicitly in B.1.2. (a) The electric field amplitude is computed using the relation in (31) when solving for U prq. (3) E2I.m: Converts the complex scalar electric field to irradiance. Note that this is done without retaining any phase information. The input to E2I.m is electric field amplitude Uprq. The sequential sub steps within E2I.m are outlined below and the function is shown explicitly in B.1.3. (a) The irradiance Iprq is computed explicitly per (31). (4) phistarangle.m: Generates the phase term φ S pr o ; λ i q of the electric field onto the satellite plane, representing the source star s angular extent. The inputs to phistarangle.m are L, M, λ i and the terms R S and Θ S. Note that the input parameters R S and Θ S are precomputed in the function imgen.m using a polar to Cartesian coordinate transformation based on the linear angular extent of the source star α S. The sequential sub steps within phistarangle.m are outlined below and the function is shown explicitly in B.1.4.

162 162 (a) Builds the appropriate grid size and sampling structure and projects the propagation plane to a circle. (b) Calculates the phase term φ S pr o ; λ i q representing the angular extent of the source star per (92). (5) friedparam.m: Calculates the Fried parameter r 0 pλ i q based on the HV57 C 2 N turbulence profile. The inputs to friedparam.m are λ i, off zenith pointing angle θ _, and the integrated C 2 N range and integrated C 2 N between two successive turbulence layers. The altitude value are precomputed in imgen.m and explicitly fed into friedparam.m. The sequential sub steps within friedparam.m are outlined below and the function is shown explicitly in B.1.5. (a) The Fried parameter r 0 pλ i q is calculated per (102) using the pre-integrated C 2 N profile between two successive turbulence layers. (6) phasescreens.m: Generates low and high frequency turbulence phase screens that are added in imgen.m to yield the effective phase screen for a given turbulence layer. The inputs to phasescreens.m are r 0 pλ i q, M,, and the inner/outer turbulence scales l o and L o. Note that phasescreens.m is assembled based on the explicit syntax published by Schmidt, 2010 pertaining to the function ft sh phase screen.m in the book Numerical Simulation of Optical Wave propagation, with examples in Matlab. 56 For a detailed description of this function refer to the citation. The sequential sub steps within phasescreens.m are outlined below and the function is shown explicitly in B.1.6. (a) Generates high frequency phase screen per the function ft phase screen.m given in Schmidt, 2010 based on (103) and (104) using random draws of Fourier coefficients.

163 163 (b) Generates low frequency phase screen by summing frequency grids that are a subharmonic of the nominal grid sampling. This is also done per (103) and (104) using randomly drawn Fourier coefficients. (7) atmrefract.m: Calculates the net refraction angle Θ atm pθ _, λ i q and monochromatic ground shadow displacement from the atmosphere. The inputs to atm- Refract.m are λ i, θ _, and the observer s environmental state and altitude h. Note that atmrefract.m was assembled in Matlab for this effort based on the Fortran code published by Hohenkerk et al, A concise mathematical description of the function is also offered by Seidelmann, The sequential sub steps within atmrefract.m are outlined below and the function is shown explicitly in B.1.7. (a) Formats all input parameters into derived quantities for efficient implementation into subsequent calculations. (b) Sequentially solves for rpzq in (98) through both the troposphere and stratosphere in 1 m step sizes using numerical integration. (c) Calculates the net monochromatic refraction angle Θ atm pθ _, λ i q per (98) by summing the individual troposphere and stratosphere refractive contributions. (d) Computes the net monochromatic ground shadow displacement based on Θ atm pθ _ ; λ i q and the observer s altitude. (8) P2I.m: Converts a power pattern to an irradiance pattern. The inputs to P2I.m are the power pattern P prq and grid spacing. Note that this calculation is independent of wavelength. The sequential sub steps within P2I.m are outlined below and the function is shown explicitly in B.1.8.

164 164 (a) Performs the power to irradiance conversion by dividing each element in the power pattern P prq by 2. This yields the irradiance pattern in units of W/m 2. (9) impowerinterp.m: Interpolates the measured power pattern P M pr; λ i q of size M ap by M ap to size M by M. The inputs to impowerinterp.m are P M pr; λ i q and D a in multiples of grid spacing. Note that for the simulations shown in this effort M 1000, thus each sampled measured power pattern is always interpolated to a 1000 by 1000 grid regardless of D a. This is done to ensure the inferred measurement pattern is suitable for propagation in the GSPR algorithm after it is ultimately converted to a scalar electric field amplitude. The sequential sub steps within impowerinterp.m are outlined below and the function is shown explicitly in B.1.9. (a) Determines the number of interpolation points n to insert between each sampled value using n log 2 pm{m ap q, and then performs the 2D interpolation using the interp2 function embedded within Matlab. (b) Scales each element of the 2D interpolated power by Nř ř P M pr; λ i q{ N PM 1 pr; λ iq, where PM 1 pr; λ iq is the raw interpolated power and n n the summation is over all array elements. This ensures that the total power is conserved after the interpolation process. (10) powergen.m: Calculates the effective power pattern P M pr; Λ j q for wavelength block Λ j based on the number of total measured photons N M pr; s λ i q. The inputs to powergen.m are N M pr; s λ i q, and s λ i. Note that powergen.m is performed after the approximation 1 λ N Mpr; Λ j q Ñ N M pr; s λ i q. The sequential sub steps within powergen.m are outlined below and the function is shown explicitly in

165 165 B (a) Determines the measured power pattern using P M pr; Λ j q hc sλ i N M pr; s λ i q. (11) shadowvelocity.m: Determines the ground shadow velocity v s based on the observer s geographic location. The inputs to shadowvelocity.m are the observer s latitude φ lat and relative longitude φ long. Note that φ long is the absolute longitude difference between the GEO satellite and the observer. The sequential sub steps within shadowvelocity.m are outlined below and the function is shown explicitly in B (a) The distance d s from the observer to the GEO satellite is calculated per (110). (b) The velocity v s from the ground shadow is calculated per (2). 4.4 Chapter 4 Summary A detailed description of the end-to-end simulation process was presented here in Chapter 4 using the underlying physics and mathematical treatment described in Chapter 2 and Chapter 3. The user defined input parameters that define a given GEO satellite shadow imaging scenario were shown in 4.1. The four steps that comprise end-to-end simulation were individually detailed in 4.2 with accompanying process flow diagrams. The custom functions created in Matlab specific to this effort were sequentially described in 4.3. At this stage the simulation process has been fully developed. The next stage is to systematically exercise the simulation capability in Chapter 5 to produce results that can be analyzed to quantify shadow imaging performance thresholds based on a number of parameters. These results are to include reconstructed images and resolution based figures of merit, in addition to establishing the associated limitations.

166 166 CHAPTER 5 CONCLUSIONS AND PATH FORWARD Chapter 5 is the culmination of this effort. Several simulations are performed to examine the impacts to shadow imaging performance based on input parameters representing various foreseeable scenarios. The satellite models used in the simulation are described in 5.1. The input to and output from the simulation are outlined in 5.2, in addition to descriptions of each scenario investigated. Conclusions drawn from the results of the simulations are discussed in 5.3. Potential follow on efforts to this dissertation are suggested with notes from the author in 5.4. Finally, a summary of the complete work is presented in Satellite Models The 2D transmission function of the satellite is represented by one of four selectable models: GEO-A, GEO-B, GEO-C, or GEO-D. GEO-A, GEO-B, and GEO-C are loosely based on representative GEO satellites contained in the Analytical Graphics Inc. (AGI) satellite toolkit (STK) model library. These were chosen such that the geometry and symmetry of the model satellites have diversity. The GEO-D model is a non physical satellite that was constructed by the author to represent a resolution target. The 2D transmission functions of these four satellite models are shown in an enlarged fashion in Figure 59. Note that the true size of the 2D transmission functions in the model have side lengths of 100 m as in Figure 30. The non-physical satellite GEO-D is composed of vertical and horizontal bars with widths and spacings varying from 1 m to 0.1 m. Patterns containing 4, 5, 6, and 7

167 167 Figure 59 Enlarged view of 2D transmission functions of the four satellite models used in the simulation. GEO-A, GEO-B, and GEO-C are loosely based on representative GEO satellites contained in the Analytical Graphics Inc. (AGI) satellite toolkit (STK) model library. These were chosen such that the geometry and symmetry of the model satellites have diversity. The GEO-D model is a non-physical satellite that was constructed by the author to represent a resolution target. Note that the true size of the 2D transmission functions in the model have side lengths of 100 m as in Figure 30. spokes with a 3 m radius were also added to diversify the transmission function. A detailed view of GEO-D is shown in Figure 60 with accompanying notation detailing the bar and spoke features. Note the horizontal and vertical bar patterns in GEO-D are identical and rotated 90 deg with respect to each other. GEO-D will be used to quantify the attainable resolution for each given scenario that is simulated.

168 168 Figure 60 The non-physical satellite GEO-D is composed of vertical and horizontal bars with widths and spacings varying from 1 m to 0.1 m. Patterns containing 4, 5, 6, and 7 spokes with a 3 m radius were also added to diversify the transmission function. Note the horizontal and vertical bar patterns in GEO-D are identical and rotated 90 deg with respect to each other. GEO-D will be used to quantify the attainable resolution for each given scenario that is simulated. 5.2 Simulation Input and Output Multiple simulations were performed to represent different scenarios. Each simulation has a unique associated set of parameters as described in 4.1. Each simulation is run using spectral end points of 400 nm to 900 nm regardless of the binning within each wavelength block Λ j. Thus, if the spectral binning in each Λ j is 10 nm the simulation yields 50 image reconstructions that are summed to produce the final estimated image. Likewise, if the spectral binning is 25 nm in each Λ j then 20 image reconstructions are summed to produce the final estimated image. Note that the spectral binning in Λ j is always chosen such that it results in an integer number of reconstructed images to be summed in Step 4 shown in Also note that the optical transmission shown in Figure 47 and zenith angle dependant atmospheric transmission shown in Figure 31 are always applied in the simulation (even if the

169 169 atmospheric turbulence is turned off). The output of a single simulation yields the following quantities and metrics: (1) The irradiance pattern of the true ground shadow for each wavelength block Ipr; Λ j q. This is the output from simulation Step 1 detailed in (2) The measurement inferred irradiance pattern for each wavelength block I M pr; Λ j q. This is the output from simulation Step 2 detailed in (3) The individual reconstructed image estimates for each wavelength block g 1 Q pr o; Λ j q. This is the output from simulation Step 3 detailed in (4) The final reconstructed image estimate g 1 pr o q. This is the output from simulation Step 4 detailed in (5) Several diagnostic and radiometric based outputs are also generated by each simulation, including: (a) The signal photon rate N S pr; Λ j q and sky background photon rate N B pr; Λ j q incident onto the GM-APD detector pixels for each wavelength block for each exposure time t. (b) The inferred total measured monochromatic photon rate N M pr; λ s i q associated with the spectral center λ s i of each wavelength block. (c) The spatial distribution of the SNR over the measured ground shadow for each wavelength block. The SNR distributions for each simulation are denoted by S M pr; Λ j q. The simulations that were run are divided into multiple groups such that impacts from independent input parameters can be assessed. Simulation Groups 1-7 use a bright source star(m _ 6) to examine the input parameter effects in the pressence of

170 170 high SNR. Keep in mind that m _ 6 is the dimmest star the average human eye can see on a dark night, but is considered to be very bright by astronomical standards as evidenced by Figure 16. Simulation Group 1, appears in 5.2.1, and varies the shape of the satellite and the aperture size D a of the telescopes in the linear collection array. The impact on shadow imaging resolution from atmospheric turbulence is examined in simulation Group 2 with a detailed description appearing in The effects of varying the spectral width of the wavelength blocks are studied in simulation Group 3 with results displayed in Simulation Group 4 briefly examines the effects of timing error with results displayed in Impacts of off-zenith pointing are studied in simulation Group 5 with results shown in Simulation Group 6 varies the source star angular extent with results presented in The impacts from detector parameters are gauged in Simulation Group 7 with results shown in Next, the impacts from the source star brightness are detailed in Group 8 with results presented in Group 8 also examines which user controlled input parameters are best adjusted to maintain resolution capability as the source star becomes dimmer.

171 171 Note on Image Processing The reconstructed shadow image of satellite model GEO-D is used as the primary source for determining the resolution capability for a given simulated scenario. The shadow image is formed using 50 iterations of the GSPR algorithm detailed in and wavelength block image summing outlined in No other image processing methods are performed in this effort. A multitude of image processing techniques exist that have potential for enhancing the reconstructed shadow image quality. While outside the scope of this effort, application of advanced image processing for shadow imaging of GEO satellites is anticipated to follow this effort. Note on Computer Processing Time A Dell Precision M4700 laptop with a 2.8 GHz processor and 16 MB of RAM running Windows 7 Professional was used to run the simulations in Matlab. The simulations distribute independent loop indices operations using Matlab s parallel processing parfor command over 8 cores in the computer. The end-to-end simulation time for a single scenario with inputs of D a 0.4 m, λ in Λ j 10 nm, and P ap 0 is 7200 seconds. When D a scaled by 0.5X and 2X the simulation time scales by 1.3X and 0.8X, respectfully. When λ in Λ j is scaled by 0.5X and 2X the simulation time scales by 4X and 0.5X, respectfully. When P ap 0 the simulation time in increased by 4X Simulation Group 1: Satellite Shape and Aperture Size Varied Simulation Group 1 uses a bright source star, ideal detector parameters, and ideal observing parameters while varying the shape of the satellite and aperture size D a of

172 172 the telescopes in the linear collection array. Atmospheric turbulence is not applied in Group 1. The input parameters in simulation Group 1 are defined in Table 13. Note that Group 1 simulations were run for each satellite model described in 5.1. Subsequent Group simulations are only run for satellite model GEO-D. A series of results are shown using the Group 1 input parameters for satellite model GEO-A which corresponds to collection aperture sizes D a 0.2 m, D a 0.4 m, and D a 0.8 m. Satellite model GEO-A represents a horizontally symmetric object at the plane of diffraction. The true irradiance Ipr; Λ nm q and measured irradiance I M pr; Λ nm q are shown in Figure 61. The true number of photons per exposure t rn S pr; Λ nm q ` N B pr; Λ nm qs and the measured number of photons per exposure tn M pr; Λ nm q onto the corresponding wavelength block GM-APD pixel are shown in Figure 62. A center slice of the true and measured pixel incident photon patterns are shown in Figure 63 which demonstrates the impacts of measurement noise. The SNR for each collection aperture and exposure time is shown in Figure 64. The reconstructed shadow images for the various aperture sizes using the Group 1 input parameters are shown in Figure 65.

173 173 Table 13 Input parameters for simulation Group 1. The input parameters varied in Group 1 are indicated in red. Parameters Sim Group 1 Sampling Value support length L 100 m grid spacing 0.1 m grid samples M 1000 Object Value Satellite GEO-A Satellite GEO-B Satellite GEO-C Satellite GEO-D Source Star Value magnitude m _ 6 angular extent α 0 nrad Environmental Value turbulence Cn 2 none sky brightness m B 20 m _ {arcsec 2 Observational Value off-zenith θ _ 0 deg latitude φ lat 0 deg longitude φ long 0 deg altitude h 0 m Parameters Cont. Sim Group 1 Collection Value aperture size D a 0.2 m aperture size D a 0.4 m aperture size D a 0.8 m field of view θ a 1 arcsec timing error σ t 0 ms Spectral Value beginning λ 400 nm ending λ 900 nm width of Λ j 10 nm number of blocks J 50 Detector Value exposure time t 77 µs exposure time t 154 µs exposure time t 308 µs dark count rate N D 0 Hz detection efficiency P d 1 afterpulsingp ap 0 gate time t g 1 µs duty cycle c d 1 Reconstruction Value GSPR iterations Q 50

174 174 Figure 61 True and measurement inferred irradiance for simulation Group 1 using satellite model GEO-A. The irradiance is in units of W {m 2 and shown using the same color scale on the right of each plot. The top left plot contains the true polychromatic irradiance Ipr; Λ nm q. The top right shows the measurement inferred irradiance I M pr; Λ nm q using collection aperture sizes D a 0.2 m, and the bottom row corresponds to D a 0.4, and D a 0.8 m. The irradiance patterns are zoomed into the center 60 m by 60 m regions of the full 100 m by 100 m plane. The fine structure of the irradiance pattern becomes more washed out as D a increases due to the light collection sampling. Small scale diffraction features are lost when collected by an aperture size larger than the features themselves.

175 175 Figure 62 True and measured number of photons per exposure time for simulation Group 1 using satellite model GEO-A. The spatial sampling corresponds to the number of collection apertures accross the full support L 100 m, and the number of exposure times is set equal to the number of apertures. The true number of photons per exposure time t rn S pr; Λ nm q ` N B pr; Λ nm qs onto the corresponding GM-APD pixel is shown on in the top plot. The measurement inferred number of photons per exposure time tn M pr; Λ nm q is shown on the bottom row using collection aperture sizes D a 0.2 m, D a 0.4, and D a 0.8 m. Figure 63 Center profile of true and measured number of photons per exposure time for simulation Group 1 using satellite model GEO-A. The pressence of measurement noise is more evident at low signal levels.

176 176 Figure 64 Signal-to-noise ratios for simulation Group 1 using satellite model GEO-A. The top plot shows the spatial distribution of the SNR over the collected light pattern for each collection aperture sizes D a 0.2 m, D a 0.4, and D a 0.8 m. The SNR is a unitless quantity and indicated by the color bars to the right of each plot. Note that the color scaling for the SNR is different for each plot in order to stretch the dynamic range of the display. The bottom row shows the center slice of the SNR profile for each collection aperture size. The SNR increases as the aperture size gets bigger due to each light collection area expanding in addition to the longer associated exposure time.

177 177 Figure 65 Reconstructed images for simulation Group 1 using satellite model GEO-A. The upper left plot is the true GEO-A satellite model. The upper right reconstructed image represents a collection system using aperture size D a 0.2 m. The bottom left reconstructed image represents a collection system using aperture size D a 0.4 m. The bottom right reconstructed image represents a collection system using aperture size D a 0.8 m. Each displayed image is normalized on a scale from zero to one and is the result of summing J 50 individual wavelength block reconstructed images. The number of iterations in the GSPR algorithm for each wavelength block was fixed at Q 50. The decrease in spatial resolution can be seen qualitatively as D a increases. While the resolution decreases the bulk contrast increases for larger aperture sizes due to the increased SNR.

178 178 Group 1 Results using GEO-B A series of results are shown using the Group 1 input parameters for satellite model GEO-B which corresponds to collection aperture sizes D a 0.2 m, D a 0.4 m, and D a 0.8 m. Satellite model GEO-B represents a horizontally asymmetric object at the plane of diffraction. The true irradiance Ipr; Λ nm q and measured irradiance I M pr; Λ nm q are shown in Figure 66. The true number of photons per exposure t rn S pr; Λ nm q ` N B pr; Λ nm qs and the measured number of photons per exposure tn M pr; Λ nm q onto the corresponding wavelength block GM-APD pixel are shown in Figure 67. A center slice of the true and measured pixel incident photon patterns are shown in Figure 68 which demonstrates the impacts of measurement noise. The SNR for each collection aperture and exposure time is shown in Figure 69. The reconstructed shadow images for the various aperture sizes using the Group 1 input parameters are shown in Figure 70.

179 179 Figure 66 True and measurement inferred irradiance for simulation Group 1 using satellite model GEO-B. The irradiance is in units of W {m 2 and shown using the same color scale on the right of each plot. The top left plot contains the true polychromatic irradiance Ipr; Λ nm q. The top right shows the measurement inferred irradiance I M pr; Λ nm q using collection aperture sizes D a 0.2 m, and the bottom row corresponds to D a 0.4, and D a 0.8 m. The irradiance patterns are zoomed into the center 60 m by 60 m regions of the full 100 m by 100 m plane. The fine structure of the irradiance pattern becomes more washed out as D a increases due to the light collection sampling. Small scale diffraction features are lost when collected by an aperture size larger than the features themselves.

180 180 Figure 67 True and measured number of photons per exposure time for simulation Group 1 using satellite model GEO-B. The spatial sampling corresponds to the number of collection apertures accross the full support L 100 m, and the number of exposure times is set equal to the number of apertures. The true number of photons per exposure time t rn S pr; Λ nm q ` N B pr; Λ nm qs onto the corresponding GM-APD pixel is shown on in the top plot. The measurement inferred number of photons per exposure time tn M pr; Λ nm q is shown on the bottom row using collection aperture sizes D a 0.2 m, D a 0.4, and D a 0.8 m. Figure 68 Center profile of true and measured number of photons per exposure time for simulation Group 1 using satellite model GEO-B. The pressence of measurement noise is more evident at low signal levels.

181 181 Figure 69 Signal-to-noise ratios for simulation Group 1 using satellite model GEO-B. The top plot shows the spatial distribution of the SNR over the collected light pattern for each collection aperture sizes D a 0.2 m, D a 0.4, and D a 0.8 m. The SNR is a unitless quantity and indicated by the color bars to the right of each plot. Note that the color scaling for the SNR is different for each plot in order to stretch the dynamic range of the display. The bottom row shows the center slice of the SNR profile for each collection aperture size. The SNR increases as the aperture size gets bigger due to each light collection area expanding in addition to the longer associated exposure time.

182 182 Figure 70 Reconstructed images for simulation Group 1 using satellite model GEO-B. The upper left plot is the true GEO-B satellite model. The upper right reconstructed image represents a collection system using aperture size D a 0.2 m. The bottom left reconstructed image represents a collection system using aperture size D a 0.4 m. The bottom right reconstructed image represents a collection system using aperture size D a 0.8 m. Each displayed image is normalized on a scale from zero to one and is the result of summing J 50 individual wavelength block reconstructed images. The number of iterations in the GSPR algorithm for each wavelength block was fixed at Q 50. The decrease in spatial resolution can be seen qualitatively as D a increases. While the resolution decreases the bulk contrast increases for larger aperture sizes due to the increased SNR.

183 183 Group 1 Results using GEO-C A series of results are shown using the Group 1 input parameters for satellite model GEO-C which corresponds to collection aperture sizes D a 0.2 m, D a 0.4 m, and D a 0.8 m. Satellite model GEO-C exhibits horizontal and vertical asymmetry at the plane of diffraction. The true irradiance Ipr; Λ nm q and measured irradiance I M pr; Λ nm q are shown in Figure 71. The true number of photons per exposure t rn S pr; Λ nm q ` N B pr; Λ nm qs and the measured number of photons per exposure tn M pr; Λ nm q onto the corresponding wavelength block GM-APD pixel are shown in Figure 72. A center slice of the true and measured pixel incident photon patterns are shown in Figure 73 which demonstrates the impacts of measurement noise. The SNR for each collection aperture and exposure time is shown in Figure 74. The reconstructed shadow images for the various aperture sizes using the Group 1 input parameters are shown in Figure 75.

184 184 Figure 71 True and measurement inferred irradiance for simulation Group 1 using satellite model GEO-C. The irradiance is in units of W {m 2 and shown using the same color scale on the right of each plot. The top left plot contains the true polychromatic irradiance Ipr; Λ nm q. The top right shows the measurement inferred irradiance I M pr; Λ nm q using collection aperture sizes D a 0.2 m, and the bottom row corresponds to D a 0.4, and D a 0.8 m. The irradiance patterns are zoomed into the center 60 m by 60 m regions of the full 100 m by 100 m plane. The fine structure of the irradiance pattern becomes more washed out as D a increases due to the light collection sampling. Small scale diffraction features are lost when collected by an aperture size larger than the features themselves.

185 185 Figure 72 True and measured number of photons per exposure time for simulation Group 1 using satellite model GEO-C. The spatial sampling corresponds to the number of collection apertures accross the full support L 100 m, and the number of exposure times is set equal to the number of apertures. The true number of photons per exposure time t rn S pr; Λ nm q ` N B pr; Λ nm qs onto the corresponding GM-APD pixel is shown on in the top plot. The measurement inferred number of photons per exposure time tn M pr; Λ nm q is shown on the bottom row using collection aperture sizes D a 0.2 m, D a 0.4, and D a 0.8 m. Figure 73 Center profile of true and measured number of photons per exposure time for simulation Group 1 using satellite model GEO-C. The pressence of measurement noise is more evident at low signal levels.

186 186 Figure 74 Signal-to-noise ratios for simulation Group 1 using satellite model GEO-C. The top plot shows the spatial distribution of the SNR over the collected light pattern for each collection aperture sizes D a 0.2 m, D a 0.4, and D a 0.8 m. The SNR is a unitless quantity and indicated by the color bars to the right of each plot. Note that the color scaling for the SNR is different for each plot in order to stretch the dynamic range of the display. The bottom row shows the center slice of the SNR profile for each collection aperture size. The SNR increases as the aperture size gets bigger due to each light collection area expanding in addition to the longer associated exposure time.

187 187 Figure 75 Reconstructed images for simulation Group 1 using satellite model GEO-C. The upper left plot is the true GEO-C satellite model. The upper right reconstructed image represents a collection system using aperture size D a 0.2 m. The bottom left reconstructed image represents a collection system using aperture size D a 0.4 m. The bottom right reconstructed image represents a collection system using aperture size D a 0.8 m. Each displayed image is normalized on a scale from zero to one and is the result of summing J 50 individual wavelength block reconstructed images. The number of iterations in the GSPR algorithm for each wavelength block was fixed at Q 50. The decrease in spatial resolution can be seen qualitatively as D a increases. While the resolution decreases the bulk contrast increases for larger aperture sizes due to the increased SNR.

188 188 Group 1 Results using GEO-D A series of results are shown using the Group 1 input parameters for satellite model GEO-D which corresponds to collection aperture sizes D a 0.2m, D a 0.4 m, and D a 0.8 m. Satellite model GEO-D represents an object for which resolution metrics can be assessed. The true irradiance Ipr; Λ nm q and measured irradiance I M pr; Λ nm q are shown in Figure 76. The true number of photons per exposure t rn S pr; Λ nm q ` N B pr; Λ nm qs and the measured number of photons per exposure tn M pr; Λ nm q onto the corresponding wavelength block GM-APD pixel are shown in Figure 77. A center slice of the true and measured pixel incident photon patterns are shown in Figure 78 which demonstrates the impacts of measurement noise. The SNR for each collection aperture and exposure time is shown in Figure 79. The reconstructed shadow images for the various aperture sizes using the Group 1 input parameters are shown in Figure 80.

189 189 Figure 76 True and measurement inferred irradiance for simulation Group 1 using satellite model GEO-D. The irradiance is in units of W {m 2 and shown using the same color scale on the right of each plot. The top left plot contains the true polychromatic irradiance Ipr; Λ nm q. The top right shows the measurement inferred irradiance I M pr; Λ nm q using collection aperture sizes D a 0.2 m, and the bottom row corresponds to D a 0.4, and D a 0.8 m. The irradiance patterns are zoomed into the center 60 m by 60 m regions of the full 100 m by 100 m plane. The fine structure of the irradiance pattern becomes more washed out as D a increases due to the light collection sampling. Small scale diffraction features are lost when collected by an aperture size larger than the features themselves.

190 190 Figure 77 True and measured number of photons per exposure time for simulation Group 1 using satellite model GEO-D. The spatial sampling corresponds to the number of collection apertures accross the full support L 100 m, and the number of exposure times is set equal to the number of apertures. The true number of photons per exposure time t rn S pr; Λ nm q ` N B pr; Λ nm qs onto the corresponding GM-APD pixel is shown on in the top plot. The measurement inferred number of photons per exposure time tn M pr; Λ nm q is shown on the bottom row using collection aperture sizes D a 0.2 m, D a 0.4, and D a 0.8 m. Figure 78 Center profile of true and measured number of photons per exposure time for simulation Group 1 using satellite model GEO-D. The pressence of measurement noise is more evident at low signal levels.

191 191 Figure 79 Signal-to-noise ratios for simulation Group 1 using satellite model GEO-D. The top plot shows the spatial distribution of the SNR over the collected light pattern for each collection aperture sizes D a 0.2 m, D a 0.4, and D a 0.8 m. The SNR is a unitless quantity and indicated by the color bars to the right of each plot. Note that the color scaling for the SNR is different for each plot in order to stretch the dynamic range of the display. The bottom row shows the center slice of the SNR profile for each collection aperture size. The SNR increases as the aperture size gets bigger due to each light collection area expanding in addition to the longer associated exposure time.

192 192 Figure 80 Reconstructed images for simulation Group 1 using satellite model GEO-D. The upper left plot is the true GEO-D satellite model. The upper right reconstructed image represents a collection system using aperture size D a 0.2 m. The bottom left reconstructed image represents a collection system using aperture size D a 0.4 m. The bottom right reconstructed image represents a collection system using aperture size D a 0.8 m. Each displayed image is normalized on a scale from zero to one and is the result of summing J 50 individual wavelength block reconstructed images. The number of iterations in the GSPR algorithm for each wavelength block was fixed at Q 50. The decrease in spatial resolution can be seen qualitatively as D a increases. While the resolution decreases the bulk contrast increases for larger aperture sizes due to the increased SNR.

193 Simulation Group 2: Effects of Atmospheric Turbulence Simulation Group 2 explicitly examines the effects of atmospheric turbulence on the reconstructed image resolution. A bright source star, ideal detector parameters, and near ideal observing parameters are used as simulation inputs. Group 2 cases are run for aperture diameters D a 0.2 m and D a 0.4 m, each with and without turbulence applied. The input parameters in simulation Group 2 are defined in Table 14. Note that Group 2 simulations were only run for satellite model GEO-D. The atmospheric turbulence effects are first assessed by examining the impact on the true ground irradiance Ipr; Λ nm q shown in Figure 81 and measured irradiance I M pr; Λ nm q shown in Figure 82. Reconstructed shadow images of GEO-D are shown in Figure 83 to quantify impacts on resolution.

194 194 Table 14 Input parameters for simulation Group 2. The input parameters varied in Group 2 are indicated in red. Parameters Sim Group 2 Sampling Value support length L 100 m grid spacing 0.1 m grid samples M 1000 Object Value Satellite GEO-D Source Star Value magnitude m _ 6 angular extent α 0 nrad Environmental Value turbulence Cn 2 HV57 turbulence Cn 2 none sky brightness m B 20 m _ {arcsec 2 Observational Value off-zenith θ _ 0 deg latitude φ lat 0 deg longitude φ long 0 deg altitude h 0 m Parameters Cont. Sim Group 2 Collection Value aperture size D a 0.2 m aperture size D a 0.4 m field of view θ a 1 arcsec timing error σ t 0 ms Spectral Value beginning λ 400 nm ending λ 900 nm width of Λ j 10 nm number of blocks J 50 Detector Value exposure time t 77 µs exposure time t 154 µs dark count rate N D 0 Hz detection efficiency P d 1 afterpulsingp ap 0 gate time t g 1 µs duty cycle c d 1 Reconstruction Value GSPR iterations Q 50

195 195 Figure 81 True ground irradiance patterns for simulation Group 2 using satellite model GEO-D. The top row shows the true irradiance Ipr; Λ nm q for D a 0.2 m, D a 0.4 m with no atmospheric turbulence applied in the simulation. The bottom row shows the true irradiance Ipr; Λ nm q for the same aperture sizes with atmospheric turbulence applied. The irradiance is in units of W {m 2 and shown using the same color scale on the right of each plot. A fine grainy structure is observed in the true ground irradiance when the turbulence is applied in the simulation.

196 196 Figure 82 Measurement inferred ground irradiance patterns for simulation Group 2 using satellite model GEO-D. The top row shows the measured irradiance I M pr; Λ nm q for D a 0.2 m, D a 0.4 m with no atmospheric turbulence applied in the simulation. The bottom row shows the measured irradiance I M pr; Λ nm q for the same aperture sizes with atmospheric turbulence applied. The irradiance is in units of W {m 2 and shown using the same color scale on the right of each plot. The salt and pepper features from measurement noise dominates over the fine grainy structure observed from the atmospheric turbulence effects seen in the true ground irradiance in Figure 81.

197 197 Figure 83 Reconstructed images for simulation Group 2 using satellite model GEO-D. The top row shows reconstructed images with no atmospheric turbulence applied in the simulation for D a 0.2 m and D a 0.4 m. The bottom row shows reconstructed images corresponding to these same aperture sizes with atmospheric turbulence applied. While the impact to resolution is evident between the D a =0.2 m and D a 0.4 m cases, there is only a very small impact from the turbulence. This is partially attributed to the measurement noise effects dominating over the fine grainy structure from the turbulence in the irradiance pattern as seen in Figure 81 and Figure 82.

198 Simulation Group 3: Aperture Size and Spectral Width Varied Simulation Group 3 examines the combined influence of D a and spectral binning width λ in each wavelength block Λ j. Group 3 studies aperture sizes of D a 0.2 m, D a 0.4 m, and D a 0.8 m. For each D a value, spectral binning width cases for λ 5 nm, λ 10 nm, λ 25 nm λ 50 nm, and λ 100 nm are simulated. This results in 15 scenarios which offers insight into the dynamic between D a and λ with respect to shadow image resolution. The input parameters in simulation Group 3 are defined in Table 15. The spectral binning is first assessed by examining the impact on the true ground irradiance Ipr; Λ j q shown in Figure 84. The measured irradiance I M pr; Λ j q for D a 0.2 m and each spectral binning case is shown in Figure 85. The measured irradiance I M pr; Λ j q for D a 0.4 m appears in Figure 86, and for D a 0.8 m in Figure 87. The photon fluence for the wavelength block beginning at 500 nm is shown in Figure 88 for each D a and spectral binning case. The SNR is shown in Figure 89, Figure 90, and Figure 91 for each of the aperture sizes. Finally, the reconstructed imagery of GEO-D is presented in Figure 92, Figure 93, and Figure 94 for each of the aperture sizes.

199 199 Table 15 Input parameters for simulation Group 3. The input parameters varied in Group 3 are indicated in red. Parameters Sim Group 3 Sampling Value support length L 100 m grid spacing 0.1 m grid samples M 1000 Object Value Satellite GEO-D Source Star Value magnitude m _ 6 angular extent α 0 nrad Environmental Value turbulence Cn 2 HV57 sky brightness m B 20 m _ {arcsec 2 Observational Value off-zenith θ _ 0 deg latitude φ lat 0 deg longitude φ long 0 deg altitude h 0 m Parameters Cont. Sim Group 3 Collection Value aperture size D a 0.2 m aperture size D a 0.4 m aperture size D a 0.8 m field of view θ a 1 arcsec timing error σ t 0 ms Spectral Value beginning λ 400 nm ending λ 900 nm width of Λ j 5 nm width of Λ j 10 nm width of Λ j 25 nm width of Λ j 50 nm width of Λ j 100 nm number of blocks J 100 number of blocks J 50 number of blocks J 20 number of blocks J 10 number of blocks J 5 Detector Value exposure time t 77 µs exposure time t 154 µs exposure time t 308 µs dark count rate N D 0 Hz detection efficiency P d 1 afterpulsingp ap 0 gate time t g 1 µs duty cycle c d 1 Reconstruction Value GSPR iterations Q 50

200 200 Figure 84 True ground irradiance patterns for simulation Group 3 using satellite model GEO-D. The top plot shows the true irradiance Ipr; Λ nm q for λ 5 nm. The middle row shows the true irradiance Ipr; Λ nm q for λ 10 nm, and Ipr; Λ nm q for λ 25 nm. The bottom row contains the true irradiance Ipr; Λ nm q for λ 50 nm, and Ipr; Λ nm q for λ 100 nm. The irradiance is in units of W {m 2 and shown using using a different color scale on the right of each plot. As λ for the wavelength block gets larger the irradiance increases yet the structure of the true diffraction pattern loses spatial fidelity.

201 Figure 85 Measurement inferred irradiance patterns for simulation Group 3 for D a 0.2 m using satellite model GEO-D. The top plot shows the measured irradiance I p r; Λ nm q for λ 5 nm. The middle row shows the measured irradiance I M pr; Λ nm q for λ 10 nm, and I M pr; Λ nm q for λ 25 nm. The bottom row contains the measured irradiance I M pr; Λ nm q for λ 50 nm, and I M pr; Λ nm q for λ 100 nm. The irradiance is in units of W {m 2 and shown using using a different color scale on the right of each plot. As λ for the wavelength block gets larger the irradiance increases yet the structure of the true diffraction pattern loses spatial fidelity. In addition, the measurement noise is apparent and seen as a high frequency grainy structure. 201

202 Figure 86 Measurement inferred irradiance patterns for simulation Group 3 for D a 0.4 m using satellite model GEO-D. The top plot shows the measured irradiance I p r; Λ nm q for λ 5 nm. The middle row shows the measured irradiance I M pr; Λ nm q for λ 10 nm, and I M pr; Λ nm q for λ 25 nm. The bottom row contains the measured irradiance I M pr; Λ nm q for λ 50 nm, and I M pr; Λ nm q for λ 100 nm. The irradiance is in units of W {m 2 and shown using using a different color scale on the right of each plot. As λ for the wavelength block gets larger the irradiance increases yet the structure of the true diffraction pattern loses spatial fidelity. In addition, the measurement noise is apparent and seen as a high frequency grainy structure. 202

203 Figure 87 Measurement inferred irradiance patterns for simulation Group 3 for D a 0.8 m using satellite model GEO-D. The top plot shows the measured irradiance I p r; Λ nm q for λ 5 nm. The middle row shows the measured irradiance I M pr; Λ nm q for λ 10 nm, and I M pr; Λ nm q for λ 25 nm. The bottom row contains the measured irradiance I M pr; Λ nm q for λ 50 nm, and I M pr; Λ nm q for λ 100 nm. The irradiance is in units of W {m 2 and shown using a different color scale on the right of each plot. As λ for the wavelength block gets larger until λ 50 nm at which point the dynamic range of the measured irradiance begins to decrease. This is due to the low SNR due to the very high photon fluence as seen in Figure 88 and Figure

204 204 Figure 88 Central slice of the photon fluence for simulation Group 3 using satellite model GEO-D. The top plot shows the central slice of the photon fluence for D a 0.2 m and each λ. The middle plot shows the central slice of the photon fluence for D a 0.4 m and each λ. The bottom plot shows the central slice of the photon fluence for D a 0.8 m and each λ. The increase in fluence is expected as D a and λ increase. Note that the fluence surpasses 2 ˆ 10 6 phot/sec/pixel in many of the wide band D a 0.8 m cases which is the threshold at which the SNR begins to drop using a gate time of t g 1 µs as shown in Figure 53. This fluence threshold is seen to manifest in a low SNR in Figure 91.

205 205 Figure 89 Signal to noise ratio for simulation Group 3 for D a 0.2 m using satellite model GEO-D. The top plot shows the SNR for λ 5 nm. The middle row shows the SNR for λ 10 nm, and SNR for λ 25 nm. The bottom row contains the SNR for λ 50 nm, and SNR for λ 100 nm. The increase in fluence is expected as D a and λ increase. The SNR is a unitless quantity and shown using a different color scale on the right of each plot.

206 206 Figure 90 Signal to noise ratio for simulation Group 3 for D a 0.4 m using satellite model GEO-D. The top plot shows the SNR for λ 5 nm. The middle row shows the SNR for λ 10 nm, and SNR for λ 25 nm. The bottom row contains the SNR for λ 50 nm, and SNR for λ 100 nm. The increase in fluence is expected as D a and λ increase. The SNR is a unitless quantity and shown using a different color scale on the right of each plot. Note that because the fluence surpasses 2ˆ10 6 phot/sec/pixel for D a 0.4 m and λ 100 nm SNR begins to drop when using a gate time of t g 1 µs as indicated by Figure 53.

207 207 Figure 91 Signal to noise ratio for simulation Group 3 for D a 0.8 m using satellite model GEO-D. The top plot shows the SNR for λ 5 nm. The middle row shows the SNR for λ 10 nm, and SNR for λ 25 nm. The bottom row contains the SNR for λ 50 nm, and SNR for λ 100 nm. The increase in fluence is expected as D a and λ increase. The SNR is a unitless quantity and shown using a different color scale on the right of each plot. Note that because the fluence surpasses 2 ˆ 10 6 phot/sec/pixel for D a 0.8 m and λ 25 nm SNR begins to drop when using a gate time of t g 1 µs as indicated by Figure 53. At D a 0.8 m using λ 100 nm the SNR is near zero for all but the central most region of the measured irradiance pattern.

208 208 Figure 92 Reconstructed images for simulation Group 3 for D a 0.2 m using satellite model GEO-D. The upper left plot is the true 2D transmission function of GEO-D. The upper right plot is the final reconstructed image for D a 0.2 m using λ 5 nm. The middle row contains the reconstructed images for D a 0.2 m using λ 10 nm, and D a 0.2 m using λ 25 nm. The bottom row contains the reconstructed images for D a 0.2 m using λ 50 nm, and D a 0.2 m using λ 100 nm. There is a clear trend in the loss of spatial resolution as λ increases.

209 209 Figure 93 Reconstructed images for simulation Group 3 for D a 0.4 m using satellite model GEO-D. The upper left plot is the true 2D transmission function of GEO-D. The upper right plot is the final reconstructed image for D a 0.4 m using λ 5 nm. The middle row contains the reconstructed images for D a 0.4 m using λ 10 nm, and D a 0.4 m using λ 25 nm. The bottom row contains the reconstructed images for D a 0.4 m using λ 50 nm, and D a 0.4 m using λ 100 nm. There is a clear trend in the loss of spatial resolution as λ increases.

210 210 Figure 94 Reconstructed images for simulation Group 3 for D a 0.8 m using satellite model GEO-D. The upper left plot is the true 2D transmission function of GEO-D. The upper right plot is the final reconstructed image for D a 0.8 m using λ 5 nm. The middle row contains the reconstructed images for D a 0.8 m using λ 10 nm, and D a 0.8 m using λ 25 nm. The bottom row contains the reconstructed images for D a 0.8 m using λ 50 nm, and D a 0.8 m using λ 100 nm. There is a clear trend in the loss of spatial resolution as λ increases. The image corresponding to D a 0.8 m using λ 100 nm is very faint with low resolution due to the low SNR shown in Figure 91.

211 Simulation Group 4: Timing Error Varied Simulation Group 4 investigates the impact of timing error σ t of the synchronization of the GM-APD readouts in each of the linearly distributed collection apertures. The Group 4 simulations represent scenarios with a fixed aperture size of D a 0.4 m and timing errors of σ t 0 ms, σ t 38 µs, and σ t 77 µs. These temporal errors correspond to the time it takes the shadow to move over a fraction of D a for an observer on the equator directly under the satellite. Explicitly these timing errors correspond to: σt v s Ñ Da σt, and 4 v s Ñ Da. The input parameters in Group 4 are defined in Table Reconstructed shadow images of GEO-D are shown in Figure 95 to quantify impacts on resolution from synchronization timing error σ t.

212 212 Table 16 Input parameters for simulation Group 4. The input parameters varied in Group 4 are indicated in red. Parameters Sim Group 4 Sampling Value support length L 100 m grid spacing 0.1 m grid samples M 1000 Object Value Satellite GEO-D Source Star Value magnitude m _ 6 angular extent α 0 nrad Environmental Value turbulence Cn 2 HV57 sky brightness m B 20 m _ {arcsec 2 Observational Value off-zenith θ _ 0 deg latitude φ lat 0 deg longitude φ long 0 deg altitude h 0 m Parameters Cont. Sim Group 4 Collection Value aperture size D a 0.4 m field of view θ a 1 arcsec timing error σ t 0 µs timing error σ t 38 µs timing error σ t 77 µs Spectral Value beginning λ 400 nm ending λ 900 nm width of Λ j 10 nm number of blocks J 50 Detector Value exposure time t 154 µs dark count rate N D 0 Hz detection efficiency P d 1 afterpulsingp ap 0 gate time t g 1 µs duty cycle c d 1 Reconstruction Value GSPR iterations Q 50

213 213 Figure 95 Reconstructed images for simulation Group 4 using satellite model GEO-D. The top left plot shows the true 2D transmission function of satellite model GEO-D. The top right plot is the reconstructed shadow image using D a 0.4 m and σ t 0. The bottom left plot is the image using D a 0.4 m and σ t 38 µs and the bottom right plot is the image using D a 0.4 m and σ t 77 µs. A minor resolution decrease in the horizontal direction is apparent for σ t 38 µs, while the horizontal resolution for σ t 77 µs is severely impacted. The resolution in the vertical direction is not impacted from the timing error.

214 Simulation Group 5: Off-Zenith Pointing Varied Simulation Group 5 studies the relationship between off zenith pointing angle θ _ and spectral binning width λ in each wavelength block Λ j using a fixed aperture size D a 0.4 m. Scenarios in which θ _ 0 deg, θ _ 30 deg, θ _ 45 deg, and θ _ 60 deg are each simulated using Λ j spectral widths of λ 5 nm, λ 10 nm, λ 25 nm λ 50 nm, and λ 100 nm. These 15 scenarios are used to explore the dynamic between θ _ and λ with respect to reconstructed shadow image resolution. The input parameters in Group 5 are defined in Table 17. The reconstructed imagery of GEO-D for λ 10 nm is presented in Figure 96, for λ 25 nm in Figure 97, for λ 50 nm in Figure 98, and for λ 100 nm in Figure 99. Note that the effects the off zenith pointing angle have on the reconstructed images is only seen along the vertical axis.

215 215 Table 17 Input parameters for simulation Group 5. The input parameters varied in Group 5 are indicated in red. Parameters Sim Group 5 Sampling Value support length L 100 m grid spacing 0.1 m grid samples M 1000 Object Value Satellite GEO-D Source Star Value magnitude m _ 6 angular extent α 0 nrad Environmental Value turbulence Cn 2 HV57 sky brightness m B 20 m _ {arcsec 2 Observational Value off-zenith θ _ 0 deg off-zenith θ _ 30 deg off-zenith θ _ 45 deg off-zenith θ _ 60 deg latitude φ lat 0 deg longitude φ long 0 deg altitude h 0 m Parameters Cont. Sim Group 5 Collection Value aperture size D a 0.4 m field of view θ a 1 arcsec timing error σ t 0 ms Spectral Value beginning λ 400 nm ending λ 900 nm width of Λ j 10 nm width of Λ j 25 nm width of Λ j 50 nm width of Λ j 100 nm number of blocks J 50 number of blocks J 20 number of blocks J 10 number of blocks J 5 Detector Value exposure time t 154 µs dark count rate N D 0 Hz detection efficiency P d 1 afterpulsingp ap 0 gate time t g 1 µs duty cycle c d 1 Reconstruction Value GSPR iterations Q 50

216 216 Figure 96 Reconstructed images for simulation Group 5 for λ 10 nm using satellite model GEO-D. The top plot is the true 2D transmission function of GEO-D. All reconstructed images pertain to D a 0.4 m. The middle row contains the reconstructed images for λ 10 nm at θ _ 0 deg, and λ 10 nm at θ _ 30 deg. The bottom row contains the reconstructed images for λ 10 nm at θ _ 45 deg, and λ 10 nm at θ _ 60 deg. At λ 10 nm the impact on spatial resolution as θ _ increases is nearly indistinguishable.

217 217 Figure 97 Reconstructed images for simulation Group 5 for λ 25 nm using satellite model GEO-D. The top plot is the true 2D transmission function of GEO-D. All reconstructed images pertain to D a 0.4 m. The middle row contains the reconstructed images for λ 25 nm at θ _ 0 deg, and λ 25 nm at θ _ 30 deg. The bottom row contains the reconstructed images for λ 25 nm at θ _ 45 deg, and λ 25 nm at θ _ 60 deg. At λ 25 nm the impact on spatial resolution as θ _ increases is very subtle.

218 218 Figure 98 Reconstructed images for simulation Group 5 for λ 50 nm using satellite model GEO-D. The top plot is the true 2D transmission function of GEO-D. All reconstructed images pertain to D a 0.4 m. The middle row contains the reconstructed images for λ 50 nm at θ _ 0 deg, and λ 50 nm at θ _ 30 deg. The bottom row contains the reconstructed images for λ 50 nm at θ _ 45 deg, and λ 50 nm at θ _ 60 deg. At λ 50 nm the impact on spatial resolution as θ _ increases remains very small.

219 219 Figure 99 Reconstructed images for simulation Group 5 for λ 100 nm using satellite model GEO-D. The top plot is the true 2D transmission function of GEO-D. All reconstructed images pertain to D a 0.4 m. The middle row contains the reconstructed images for λ 100 nm at θ _ 0 deg, and λ 100 nm at θ _ 30 deg. The bottom row contains the reconstructed images for λ 100 nm at θ _ 45 deg, and λ 50 nm at θ _ 60 deg. At λ 100 nm the impact on spatial resolution as θ _ increases remains very small.

220 Simulation Group 6: Source Star Angular Extent Varied Simulation Group 6 demonstrates the influence of the source star s angular extent α on the reconstructed shadow image resolution. The Group 6 simulations represent scenarios with a fixed aperture size of D a 0.4 m and source star angular extents of α 0 nrad, α 10 nrad, and α 20 nrad. The input parameters in Group 6 are defined in Table 18. The reconstructed shadow images of GEO-D shown in Figure 100 are assessed to quantify impacts on shadow imaging resolution from a source star with a large angular extent. Table 18 Input parameters for simulation Group 6. The input parameters varied in Group 6 are indicated in red. Parameters Sim Group 6 Sampling Value support length L 100 m grid spacing 0.1 m grid samples M 1000 Object Value Satellite GEO-D Source Star Value magnitude m _ 6 angular extent α 0 nrad angular extent α 10 nrad angular extent α 20 nrad Environmental Value turbulence Cn 2 HV57 sky brightness m B 20 m _ {arcsec 2 Observational Value off-zenith θ _ 0 deg latitude φ lat 0 deg longitude φ long 0 deg altitude h 0 m Parameters Cont. Sim Group 6 Collection Value aperture size D a 0.4 m field of view θ a 1 arcsec timing error σ t 0 ms Spectral Value beginning λ 400 nm ending λ 900 nm width of Λ j 10 nm number of blocks J 50 Detector Value exposure time t 154 µs dark count rate N D 0 Hz detection efficiency P d 1 afterpulsingp ap 0 gate time t g 1 µs duty cycle c d 1 Reconstruction Value GSPR iterations Q 50

221 221 Figure 100 Reconstructed images for simulation Group 6 using satellite model GEO-D. The top left plot shows the true 2D transmission function of satellite model GEO-D. The top right plot is the reconstructed shadow image using D a 0.4 m and α 0 nrad. The bottom left plot is the image using D a 0.4 m and α 10 nrad and the bottom right plot is the image for a source star with α 20 nrad. A small global morphology is observed for the α 10 nrad case along with a small decrease in resolution. In the α 10 nrad case the morphology is much more pronounced and the resolution is significantly decreased.

222 Simulation Group 7: Detector Parameters Varied Simulation Group 7 examines the sensitivity of the Baseline A and Baseline B GM-APD detector parameters on the reconstructed shadow image resolution when using a bright source star. The Group 7 simulations represent scenarios with a fixed aperture size of D a 0.4 m using ideal detector parameters in addition to detector parameter sets Baseline A and Baseline B described in The input parameters in Group 7 are defined in Table 19. The reconstructed shadow images of GEO-D shown in Figure 101 and used to quantify impacts on shadow imaging resolution based on GM-APD detector parameters when using a bright source star.

223 223 Table 19 Input parameters for simulation Group 7. The input parameters varied in Group 7 are indicated in red. Parameters Sim Group 7 Sampling Value support length L 100 m grid spacing 0.1 m grid samples M 1000 Object Value Satellite GEO-D Source Star Value magnitude m _ 6 angular extent α 0 nrad Environmental Value turbulence Cn 2 HV57 sky brightness m B 20 m _ {arcsec 2 Observational Value off-zenith θ _ 0 deg latitude φ lat 0 deg longitude φ long 0 deg altitude h 0 m Parameters Cont. Sim Group 7 Collection Value aperture size D a 0.4 m field of view θ a 1 arcsec timing error σ t 0 ms Spectral Value beginning λ 400 nm ending λ 900 nm width of Λ j 10 nm number of blocks J 50 Detector Value exposure time t 154 µs gate time t g 1 µs Ideal: dark count rate N D 0 Hz detection efficiency P d 1 afterpulsingp ap 0 duty cycle c d 1 Baseline A: dark count rate N D 250 Hz detection efficiency P d 0.25 afterpulsingp ap 0.20 duty cycle c d 0.95 Baseline B dark count rate N D 50 Hz detection efficiency P d 0.50 afterpulsingp ap 0.05 duty cycle c d 0.95 Reconstruction Value GSPR iterations Q 50

224 224 Figure 101 Reconstructed images for simulation Group 7 using satellite model GEO-D. The top left plot shows the true 2D transmission function of satellite model GEO-D. The top right plot is the reconstructed shadow image using D a 0.4 m and ideal detector parameters. The bottom left plot is the image using D a 0.4 m and the Baseline A detector parameters. The bottom right shows the reconstructed shadow image using D a 0.4 m and the Baseline B detector parameters. When using a bright source star the shadow image resolution is nearly identical when using ideal, Baseline A, or Baseline B detector parameters. However, the contrast is highest when using the ideal detector parameters and lowest when using the Baseline A parameters.

225 Simulation Group 8: Source Star Brightness Varied Simulation Group 8 investigates how dim the source star can be before the image reconstruction fails to yield resolvable features in satellite model GEO-D. The Group 8 simulations represent scenarios with a fixed aperture size of D a 0.4 m using ideal, Baseline A, and Baseline B detector parameters. The spectral width of the wavelength blocks are also varied and the source star magnitude is varied from m _ 8 to m _ 11 in unit steps. The input parameters in Group 8 are defined in Table 20. For each λ in Λ j ideal detector case, multiple quantities of interest were assembled in a quad format including the true ground irradiance Ipr; Λ j q, the measured ground irradiance I M pr; Λ j q, the photon fluence profile, and the SNR profile. For the λ 10 nm case this is displayed in Figure 102, for λ 25 nm in Figure 104, and for λ 50 nm in Figure 106. The image reconstructions for corresponding to ideal detector parameters as a function of source star brightness m _ are displayed in Figure 103 for the λ 10 nm case, in Figure 105 for the λ 25 nm case, and Figure 107 for the λ 50 nm case. The detector parameters are varied for the case of a dim source star of m _ 10 using λ 25 nm in Figure 108 and for λ 50 nm in Figure 109.

226 226 Table 20 Input parameters for simulation Group 8. The input parameters varied in Group 8 are indicated in red. Parameters Sim Group 8 Sampling Value support length L 100 m grid spacing 0.1 m grid samples M 1000 Object Value Satellite GEO-D Source Star Value magnitude m _ 8 magnitude m _ 9 magnitude m _ 10 magnitude m _ 11 angular extent α 0 nrad Environmental Value turbulence Cn 2 HV57 sky brightness m B 20 m _ {arcsec 2 Observational Value off-zenith θ _ 0 deg latitude φ lat 0 deg longitude φ long 0 deg altitude h 0 m Spectral Value beginning λ 400 nm ending λ 900 nm width of Λ j 10 nm width of Λ j 25 nm width of Λ j 50 nm number of blocks J 50 number of blocks J 20 number of blocks J 10 Parameters Cont. Sim Group 8 Collection Value aperture size D a 0.4 m field of view θ a 1 arcsec timing error σ t 0 ms Detector Value exposure time t 154 µs gate time t g 1 µs Ideal: dark count rate N D 0 Hz detection efficiency P d 1 afterpulsingp ap 0 duty cycle c d 1 Baseline A: dark count rate N D 250 Hz detection efficiency P d 0.25 afterpulsingp ap 0.20 duty cycle c d 0.95 Baseline B dark count rate N D 50 Hz detection efficiency P d 0.50 afterpulsingp ap 0.05 duty cycle c d 0.95 Reconstruction Value GSPR iterations Q 50

227 227 Figure 102 Radiometric quantities and SNR with the source star brightness varied for simulation Group 8 using satellite model GEO-D. The displayed quantities pertain to D a 0.4 m and λ 10 nm using ideal detector parameters. The source star brightness is varied from m _ 8 to m _ 11 in unit steps for each quantity displayed. The top right plot represents the true ground irradiance Ipr; Λ nm q, and the top right plot is the measurement inferred irradiance I M pr; Λ nm q. The bottom left plot is the photon fluence profile through the middle of the shadow pattern, and the bottom right plot is the central SNR profile. The corresponding image reconstructions for these scenarios appears in Figure 103.

228 228 Figure 103 Reconstructed images with the source star brightness varied for Group 8 using an ideal detector and satellite model GEO-D. The displayed images pertain to D a 0.4 m and λ 10 nm. The middle row contains reconstructed images for scenarios with source star brightnesses of m _ 8 and m _ 9. The bottom row contains reconstructed images for scenarios with source star brightnesses of m _ 10 and m _ 11. The corresponding radiometric quantities and SNR for these cases appear in Figure 102.

229 229 Figure 104 Radiometric quantities and SNR with the source star brightness varied for simulation Group 8 using satellite model GEO-D. The displayed quantities pertain to D a 0.4 m and λ 25 nm using ideal detector parameters. The source star brightness is varied from m _ 8 to m _ 11 in unit steps for each quantity displayed. The top right plot represents the true ground irradiance Ipr; Λ nm q, and the top right plot is the measurement inferred irradiance I M pr; Λ nm q. The bottom left plot is the photon fluence profile through the middle of the shadow pattern, and the bottom right plot is the central SNR profile. The corresponding image reconstructions for these scenarios appear in Figure 105.

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