Chapter 13 Partially Coherent Imaging, continued
|
|
- Regina Daniels
- 5 years ago
- Views:
Transcription
1 Chapter 3 Partially Coherent Imaging, continued As an example, a common illuminator design is one in which the source is imaged onto the object. This is known as critical illumination source - source has to be quite uniform - could be magnified or de-magnified object Review: In the above approach to analyzing imaging systems, the source is modeled as a collection of independent point sources. The image due to each point source is separately calculated using coherent optics, and then the final image is found by adding the intensities of the individual images from all the source points. This method is known as Abbe s method. An alternative approach is to first find the mutual intensity of the object illumination: ; ξ 2. Under certain conditions, a detailed knowledge of the source is not needed in order to find the mutual intensity of the object illumination. If we know ; ξ 2, then: (3. In this case the image intensity is I i ( uv, = huvξ (, ; h ( uvξ, ; 2 t o t o ( ξ 2 ξ ξ 2 ( ; η 2 dξ dη dξ 2 dη 2 (3.2 Partially_coherent_imaging_pt2_post.fm Chapter 3
2 Illuminator ( x, ỹ source z z 2 condenser lens object Assume that the source coherence is very low, so that the coherence area on the source is small ( S A c compared to the source size: ( «. We further assume that the lens collects a large angle such that the light coherence area on the lens is much smaller than the lens area: The lens aperture can then be viewed as an incoherent source as well. A s ( λz A 2 l» A s (3.3 Let s examine this in detail. The source is assumed to be incoherent, so that the Van Cittert- Zernike theorem applies. The mutual intensity incident on the condenser lens is: J l ( x, ỹ ; x, ỹ 2 = κ ( λz 2 j π exp [( x 2 + ỹ 2 ( x + ỹ ] λz I s ( α, β j 2 exp ( x α+ ỹβ dαdβ λz (3.4 J l '( x, ỹ ; x, ỹ 2 = P c( x, ỹ P c ( x 2, ỹ 2 j π exp - [( x λf + ỹ ( x 2 + ỹ 2 ] Jl ( x, ỹ ; x 2, ỹ 2 (3.5 Here, P c( x, ỹ is the condenser pupil function, such that: (3.6 and represents the pupil aberrations in the condenser. We assume that is very narrow in x, ỹ. Then Wx (, ỹ J l Partially_coherent_imaging_pt2_post.fm Chapter 3
3 j π λ exp [( x z f 2 + ỹ 2 ( x + ỹ ] (3.7 So we can write: where κ J c '( x, ỹ ; x 2, ỹ P ( λz 2 c( x, ỹ 2 x ỹ I S , λz λz (3.8 (3.9 I S I S ( ν x, ν y = F[ I S ( α, β ]. (3.0 is very narrow in x, ỹ, so that J l ' describes an incoherent source with an intensity distribu- κ tion of P. We use the Van Cittert-Zernike theorem to propagate to the object: ( λz 2 c( x, ỹ 2, ξ 2 κ' j π = exp ξ ( λz 2 2 [( 2 + η 2 + η ] ( λz 2 P c( x, ỹ 2 j π exp ( ξx + ηỹ dx dỹ λz 2 From this result we can conclude that: - the mutual intensity at the object is independent of aberrations in the illuminator (3. - the coherence diameter at the object, d λz c λ -, where D c is the diameter of the NA condenser pupil. This will come up again later. Another approach: four dimensional linear systems approach - We will work in the spatial frequency domain. - This will lead to the method used in the program SPLAT for partially coherent imaging. The relation between the object and image mutual intensities D c J i ( u, v, u 2, v 2 = ', ξ 2 hu (, v, ξ h ( u 2, v 2, ξ 2 dξ dη dξ 2 dη 2 (3.2 Partially_coherent_imaging_pt2_post.fm Chapter 3
4 This is a four dimensional superposition integral. If the impulse response is space-invariant, h depends only on the coordinate differences. Then it becomes a convolution integral. The quadratic phase factors in the definition of h interfere with the space invariance of h. These can be dropped for small coherence areas at the object and image planes. Four dimensional Fourier Transform Given a function Jx (, x 2, x 3, x 4, F[ Jx (, x 2, x 3, x 4 ] dx dx 2 dx 3 dx 4 exp[ j2π( x ν + x 2 ν 2 + x 3 ν 3 + x 4 ν 4 ] (3.3 J( x, x 2, x 3, x 4 = J( ν If space invariance holds, then h is a function of 2 variables, hu ( ξ, v η. Define: H( ν F[ hx (, x 2 ] The transfer function involved in the imaging system is (3.4 (3.5 It s easy to show then that H( ν = H( ν H ( ν 3, ν 4 Define the Fourier Transforms of the object and image mutual intensities (3.6 (3.7 (3.8 Then, generalizing the convolution theorem to 4D gives J i ( ν = H( ν H ( ν 3, ν 4 '( ν Recall that H is the coherent transfer function for the imaging lens. H( ν = P( λz i ν, λz i ν 2 which is the imaging system pupil function. Then: (3.9 (3.20 Partially_coherent_imaging_pt2_post.fm Chapter 3
5 J i ( ν = P( λz i ν, λz i ν 2 P ( λz i ν 3, λz i ν 4 '( ν (3.2 Now, let s examine ' - this depends on both the object and illumination. Recall that Then '( ν, ν 4 = ( ξ, ηt o t o ( ξ 2 exp[ j2π( ν ξ + ν 2 η + ν 3 ξ 2 + ν 4 η 2 ]dξ dη dξ 2 dη 2 (3.22 (3.23 Using ξ 2 = ξ + ξ η 2 = η + η '( ν, ν 4 = dξ dη t o exp{ j2π[ ( ν + ν 3 ξ + ( ν 2 + ν 4 η ]} (3.24 The second integral is a Fourier Transform of the product of two functions. It can be expressed as the convolution of transforms. Define T o = F[ t o ]. d ξd η ( ξ, η t ( ξ + ξ η + η exp{ j2π[ ν o 3 ξ + ν 4 η] } The second integral can then be written (using the shift theorem as ( p, qt o ( p ν 3, q ν 4 exp{ j2π[ ξ ( ν 3 p + η ( ν 4 q ]} dpdq The final result is now: '( ν, ν 4 = dξ dη dpdqt o ( p, qt o ( p ν 3, q ν 4 (3.25 (3.26 exp{ j2π[ ( ν + ν 3 ξ ( ν 3 pξ + ( ν 2 + ν 4 η ( ν 4 qη 2 ]} Partially_coherent_imaging_pt2_post.fm Chapter 3
6 '( ν, ν 4 = dpdqt o ( p + γ, q + γ 2 T o ( p γ 3, q γ 4 ( p q (3.27 Recall that if the source is large and incoherent, the illumination of the condenser lens is also incoherent. In this case, ( ξ, η is proportional to the Fourier Transform of P c. So 2 (3.28 where z 2 is the distance from the condenser pupil to the object. Assume that the object spectrum has a certain bandwidth. Then the overlap shown graphically: T o '( p ν 3, p ν 4 ( ν 3 q '( ν, ν 4 T o ( p + ν, q + ν 2 is obtained from ( ν, ν 3 p ( pq, Now combine equations (3.9 and (3.27 J i ( ν ν 4 = H( ν H ( ν 3, ν 4 T o ( p + ν, q + ν 2 (3.29 Recall H is the coherent transfer function of the imaging system, which is given by the pupil function of the imaging lens. This gives us a nice result. Given the condenser pupil, the imaging system pupil function and the object spectrum, we get a four dimensional spectrum of the image mutual intensity. Finally, we want the image intensity or a two dimensional image intensity spectrum. T o ( p ν 3, q ν 4 ( pq, dpdq One way: Transform back to get J i ( u, v ; u 2, v 2 2 Iuv (, = J i ( u, v ; u 2, v 2 Partially_coherent_imaging_pt2_post.fm Chapter 3
7 Alternative: obtain the image intensity spectrum, I i ( ν u, ν v, from J i ( ν ν 4. It is straightforward to show that I i ( v u, v v = J i ( v, v 2, v u v, v v v 2 dv dv 2 Now use equations (3.29 and (3.30: [ let µ p + v ; µ 2 = q + v 2 ] I i ( γ u, γ v = d µ dµ 2 T o ( v, µ 2 T o ( µ v u, µ 2 v v dpdqh ( µ p, µ 2 qh ( µ p v u, µ 2 q v v ( pq, (3.30 (3.3 The transmission cross coefficient only depends on the coherence of the illumination and the imaging pupil function. This involves the integral over the overlap region of the three circles again. This integral is the basis of SPLAT, originally written by K. Toh, M.S The easiest way to use SPLAT is to go to: Click on Simulators, then on SPLAT. There is at least one commercialized version that has undergone some significant improvement in speed and ease of use called PROLITH from Finle Technologies. One nice feature is that given pupil aberrations and illumination partial coherence, the transmission cross coefficients can be calculated and stored. A variety of mask patterns can be calculated rapidly after that. Partial coherence in SPLAT Transmission cross coefficient (TCC In common usage in lithography and other imaging system communities,is the partial coherence parameter: (3.32 How does σ relate to ( pq,? To see this, we go back to the real-space calculation of image intensity: Partially_coherent_imaging_pt2_post.fm Chapter 3
8 I i ( uv, = hu ( ξ, v ηh ( u ξ ξ, v η η (3.33 t o ( ξ, ηt o ( ξ + ξ, η + η ( ξ, ηdξdηd ξd η Here we used a change of variables from an earlier form of this integral. The coherent limit occurs when ( ξ, η is constant, I o. Then (3.34 This holds as long as ( ξ, η is essentially constant over the range of ( ξ, η for which the integrand is appreciable. When ξ, η are greater than the object size, then t o ( ξη, t o ( ξ + ξ, η+ η 0 But h is much narrower. When ξ, η exceed the width of h, then (3.35 hu ( ξ, v ηh ( u ξ ξ, v η η 0 (3.36 Therefore, the illumination may be considered coherent when the illumination coherence area exceeds the area of the point spread function (scaled to the object plane. From our previous result: incoherent] λz d 2 c λ D c NA illuminator [this is valid when the condenser pupil is source D c D p z 2 z o z i Imaging system resolution: d i λz i λ D p NA image exit (3.37 When scaled to the object side: Partially_coherent_imaging_pt2_post.fm Chapter 3
9 The coherent illumination condition is then: d o λz o λ D p NA image entrance (3.38 We define the partial coherence factor: Then the coherent illumination condition is: d c» d o or » (3.39 NA illumination NA illumination NA image input NA image entrance σ (3.40 (3.4 Note that in general, σ. The maximum value of σ occurs when the illuminator completely fills the entrance pupil of the imaging system. SPLAT assumes ( pq, is a constant inside a circle with a radius proportional to σ. σ 0: fully coherent σ = : incoherent This completes our theoretical discussion of partial coherence in imaging. The ramifications will be played out when we discuss applications. Partially_coherent_imaging_pt2_post.fm Chapter 3
Chapter 12 Effects of Partial coherence on imaging systems
Chapter 2 Effects of Partial coherence on imaging systems Coherence effects are described by the mutual intensity function J(x, y ; x 2. (under quasimonochromatic conditions We now have the tools to describe
More informationOptics for Engineers Chapter 11
Optics for Engineers Chapter 11 Charles A. DiMarzio Northeastern University Nov. 212 Fourier Optics Terminology Field Plane Fourier Plane C Field Amplitude, E(x, y) Ẽ(f x, f y ) Amplitude Point Spread
More informationOptics for Engineers Chapter 11
Optics for Engineers Chapter 11 Charles A. DiMarzio Northeastern University Apr. 214 Fourier Optics Terminology Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides11r1 1 Fourier Optics
More informationChapter 5 Wave-Optics Analysis of Coherent Optical Systems
Chapter 5 Wave-Optics Analysis of Coherent Optical Systems January 5, 2016 Chapter 5 Wave-Optics Analysis of Coherent Optical Systems Contents: 5.1 A thin lens as a phase transformation 5.2 Fourier transforming
More information5. LIGHT MICROSCOPY Abbe s theory of imaging
5. LIGHT MICROSCOPY. We use Fourier optics to describe coherent image formation, imaging obtained by illuminating the specimen with spatially coherent light. We define resolution, contrast, and phase-sensitive
More informationMIT 2.71/2.710 Optics 10/31/05 wk9-a-1. The spatial frequency domain
10/31/05 wk9-a-1 The spatial frequency domain Recall: plane wave propagation x path delay increases linearly with x λ z=0 θ E 0 x exp i2π sinθ + λ z i2π cosθ λ z plane of observation 10/31/05 wk9-a-2 Spatial
More informationToday. MIT 2.71/2.710 Optics 11/10/04 wk10-b-1
Today Review of spatial filtering with coherent illumination Derivation of the lens law using wave optics Point-spread function of a system with incoherent illumination The Modulation Transfer Function
More informationChapter 10 Spatial Coherence Part 2
EE90F Chapte0 Spatial Coherence Part Geometry of the fringe pattern: η ( ξ, η ) ξ y Qxy (, ) x ρ ρ r ( ξ, η ) P z pinhole plane viewing plane Define: z + ( x) +( η y) ξ r z + ( x) +( η y) ξ (0.) (0.) ρ
More informationSIMG Optics for Imaging Solutions to Final Exam
SIMG-733-009 Optics for Imaging Solutions to Final Exam. An imaging system consists of two identical thin lenses each with focal length f = f = +300 mm and diameter d = d =50mm. The lenses are separated
More informationDesign and Correction of optical Systems
Design and Correction of optical Systems Part 10: Performance criteria 1 Summer term 01 Herbert Gross Overview 1. Basics 01-04-18. Materials 01-04-5 3. Components 01-05-0 4. Paraxial optics 01-05-09 5.
More information2.71. Final examination. 3 hours (9am 12 noon) Total pages: 7 (seven) PLEASE DO NOT TURN OVER UNTIL EXAM STARTS PLEASE RETURN THIS BOOKLET
2.71 Final examination 3 hours (9am 12 noon) Total pages: 7 (seven) PLEASE DO NOT TURN OVER UNTIL EXAM STARTS Name: PLEASE RETURN THIS BOOKLET WITH YOUR SOLUTION SHEET(S) MASSACHUSETTS INSTITUTE OF TECHNOLOGY
More informationChapter 6 SCALAR DIFFRACTION THEORY
Chapter 6 SCALAR DIFFRACTION THEORY [Reading assignment: Hect 0..4-0..6,0..8,.3.3] Scalar Electromagnetic theory: monochromatic wave P : position t : time : optical frequency u(p, t) represents the E or
More informationChapter 6 Aberrations
EE90F Chapter 6 Aberrations As we have seen, spherical lenses only obey Gaussian lens law in the paraxial approxiation. Deviations fro this ideal are called aberrations. F Rays toward the edge of the pupil
More informationFourier transform = F. Introduction to Fourier Optics, J. Goodman Fundamentals of Photonics, B. Saleh &M. Teich. x y x y x y
Fourier transform Introduction to Fourier Optics, J. Goodman Fundamentals of Photonics, B. Saleh &M. Teich f( x, y) FT g( f, f ) f( x, y) IFT g( f, f ) x x y y + 1 { g( f, ) } x fy { π } f( x, y) = g(
More informationInterference, Diffraction and Fourier Theory. ATI 2014 Lecture 02! Keller and Kenworthy
Interference, Diffraction and Fourier Theory ATI 2014 Lecture 02! Keller and Kenworthy The three major branches of optics Geometrical Optics Light travels as straight rays Physical Optics Light can be
More informationLight Propagation in Free Space
Intro Light Propagation in Free Space Helmholtz Equation 1-D Propagation Plane waves Plane wave propagation Light Propagation in Free Space 3-D Propagation Spherical Waves Huygen s Principle Each point
More informationPhysical Optics. Lecture 7: Coherence Herbert Gross.
Physical Optics Lecture 7: Coherence 07-05-7 Herbert Gross www.iap.uni-jena.de Physical Optics: Content No Date Subject Ref Detailed Content 05.04. Wave optics G Complex fields, wave equation, k-vectors,
More informationChapter 8 Analog Optical Information Processing
Chapter 8 Analog Optical Information Processing June 5, 2013 Outline 8.1 Historical Background 8.2 Incoherent Optical Information Processing Systems 8.3 Coherent Optical Information Processing Systems
More informationImaging Metrics. Frequency response Coherent systems Incoherent systems MTF OTF Strehl ratio Other Zemax Metrics. ECE 5616 Curtis
Imaging Metrics Frequenc response Coherent sstems Incoherent sstems MTF OTF Strehl ratio Other Zema Metrics Where we are going with this Use linear sstems concept of transfer function to characterize sstem
More informationLecture 9: Indirect Imaging 2. Two-Element Interferometer. Van Cittert-Zernike Theorem. Aperture Synthesis Imaging. Outline
Lecture 9: Indirect Imaging 2 Outline 1 Two-Element Interferometer 2 Van Cittert-Zernike Theorem 3 Aperture Synthesis Imaging Cygnus A at 6 cm Image courtesy of NRAO/AUI Very Large Array (VLA), New Mexico,
More informationLens Design II. Lecture 1: Aberrations and optimization Herbert Gross. Winter term
Lens Design II Lecture 1: Aberrations and optimization 18-1-17 Herbert Gross Winter term 18 www.iap.uni-jena.de Preliminary Schedule Lens Design II 18 1 17.1. Aberrations and optimization Repetition 4.1.
More informationSpatial Frequency and Transfer Function. columns of atoms, where the electrostatic potential is higher than in vacuum
Image Formation Spatial Frequency and Transfer Function consider thin TEM specimen columns of atoms, where the electrostatic potential is higher than in vacuum electrons accelerate when entering the specimen
More information4: Fourier Optics. Lecture 4 Outline. ECE 4606 Undergraduate Optics Lab. Robert R. McLeod, University of Colorado
Outline 4: Fourier Optics Introduction to Fourier Optics Two dimensional transforms Basic optical laout The coordinate sstem Detailed eample Isotropic low pass Isotropic high pass Low pass in just one
More informationNature of Light Part 2
Nature of Light Part 2 Fresnel Coefficients From Helmholts equation see imaging conditions for Single lens 4F system Diffraction ranges Rayleigh Range Diffraction limited resolution Interference Newton
More informationHorizontal-Vertical (H-V) Bias, part 2
Tutor52.doc: Version 11/4/05 T h e L i t h o g r a p h y E x p e r t (February 2006) Horizontal-Vertical (H-V) Bias, part 2 Chris A. Mack, Austin, Texas In the last edition of this column we looked at
More informationDesign and Correction of Optical Systems
Design and Correction of Optical Systems Lecture 7: PSF and Optical transfer function 017-05-0 Herbert Gross Summer term 017 www.iap.uni-jena.de Preliminary Schedule - DCS 017 1 07.04. Basics 1.04. Materials
More informationPhysical and mathematical model of the digital coherent optical spectrum analyzer
Optica Applicata, Vol. XLVII, No., 017 DOI: 10.577/oa17010 Physical and mathematical model of the digital coherent optical spectrum analyzer V.G. KOLOBRODOV, G.S. TYMCHYK, V.I. MYKYTENKO, M.S. KOLOBRODOV
More information4: Fourier Optics. Lecture 4 Outline. ECE 4606 Undergraduate Optics Lab. Robert R. McLeod, University of Colorado
Outline 4: Fourier Optics Introduction to Fourier Optics Two dimensional transforms Basic optical laout The coordinate sstem Detailed eample Isotropic low pass Isotropic high pass Low pass in just one
More informationLecture 19 Optical MEMS (1)
EEL6935 Advanced MEMS (Spring 5) Instructor: Dr. Huikai Xie Lecture 19 Optical MEMS (1) Agenda: Optics Review EEL6935 Advanced MEMS 5 H. Xie 3/8/5 1 Optics Review Nature of Light Reflection and Refraction
More informationMicroscopy. Lecture 3: Physical optics of widefield microscopes Herbert Gross. Winter term
Microscopy Lecture 3: Physical optics of widefield microscopes --9 Herbert Gross Winter term www.iap.uni-jena.de Preliminary time schedule No Date Main subject Detailed topics Lecturer 5.. Optical system
More informationHigh-Resolution. Transmission. Electron Microscopy
Part 4 High-Resolution Transmission Electron Microscopy 186 Significance high-resolution transmission electron microscopy (HRTEM): resolve object details smaller than 1nm (10 9 m) image the interior of
More informationMATH 6337: Homework 8 Solutions
6.1. MATH 6337: Homework 8 Solutions (a) Let be a measurable subset of 2 such that for almost every x, {y : (x, y) } has -measure zero. Show that has measure zero and that for almost every y, {x : (x,
More informationPhys 531 Lecture 27 6 December 2005
Phys 531 Lecture 27 6 December 2005 Final Review Last time: introduction to quantum field theory Like QM, but field is quantum variable rather than x, p for particle Understand photons, noise, weird quantum
More informationIntroduction to Interferometer and Coronagraph Imaging
Introduction to Interferometer and Coronagraph Imaging Wesley A. Traub NASA Jet Propulsion Laboratory and Harvard-Smithsonian Center for Astrophysics Michelson Summer School on Astrometry Caltech, Pasadena
More informationSOFT X-RAYS AND EXTREME ULTRAVIOLET RADIATION
SOFT X-RAYS AND EXTREME ULTRAVIOLET RADIATION Principles and Applications DAVID ATTWOOD UNIVERSITY OF CALIFORNIA, BERKELEY AND LAWRENCE BERKELEY NATIONAL LABORATORY CAMBRIDGE UNIVERSITY PRESS Contents
More informationPH880 Topics in Physics
PH880 Topics in Physics Modern Optical Imaging (Fall 2010) Monday Fourier Optics Overview of week 3 Transmission function, Diffraction 4f telescopic system PSF, OTF Wednesday Conjugate Plane Bih Bright
More informationFourier transform = F. Introduction to Fourier Optics, J. Goodman Fundamentals of Photonics, B. Saleh &M. Teich. x y x y x y
Fourier transform Introduction to Fourier Optics, J. Goodman Fundamentals of Photonics, B. Saleh &M. Teich f( x, y) FT g( f, f ) f( x, y) IFT g( f, f ) x x y y + 1 { g( f, ) } x fy { π } f( x, y) = g(
More informationOn the FPA infrared camera transfer function calculation
On the FPA infrared camera transfer function calculation (1) CERTES, Université Paris XII Val de Marne, Créteil, France (2) LTM, Université de Bourgogne, Le Creusot, France by S. Datcu 1, L. Ibos 1,Y.
More informationImproved Ronchi test with extended source
J. Braat and A. J. E. M. Janssen Vol. 6, No. /January 999/J. Opt. Soc. Am. A 3 Improved onchi test with extended source Joseph Braat and Augustus J. E. M. Janssen Philips esearch Laboratories, Professor
More informationAssignment for next week
Assignment for next week Due: 6:30 PM Monday April 11 th (2016) -> PDF by email only 5 points: Pose two questions based on this lecture 5 points: what (and why) is sinc(0)? 5 points: ketch (by hand okay)
More informationTopic 4: Imaging of Extended Objects
V Topic 4: maging of xtended bjects Aim: Covers the imaging of extended objects in coherent and incoherent light. The effect of simple defocus is also considered. Contents:. maging of two points. Coherent
More informationWigner distribution function of volume holograms
Wigner distribution function of volume holograms The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher S.
More informationPRINCIPLES OF PHYSICAL OPTICS
PRINCIPLES OF PHYSICAL OPTICS C. A. Bennett University of North Carolina At Asheville WILEY- INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION CONTENTS Preface 1 The Physics of Waves 1 1.1 Introduction
More informationLinear Systems. : object : image. Describes the output of a linear system. input. output. Impulse response function
Linear Systems Describes the output of a linear system Gx FxHx- xdx FxHx F x G x x output input Impulse response function H x xhx- xdx x If the microscope is a linear system: : object : image G x S F x
More informationAOL Spring Wavefront Sensing. Figure 1: Principle of operation of the Shack-Hartmann wavefront sensor
AOL Spring Wavefront Sensing The Shack Hartmann Wavefront Sensor system provides accurate, high-speed measurements of the wavefront shape and intensity distribution of beams by analyzing the location and
More informationPhysical Optics. Lecture 2: Diffraction Herbert Gross.
Physical Optics Lecture : Diffraction 018-04-18 Herbert Gross www.iap.uni-jena.de Physical Optics: Content No Date Subject Ref Detailed Content 1 11.04. Wave optics G Complex fields, wave equation, k-vectors,
More informationExact Computation of Scalar, 2D Aerial Imagery
Exact Computation of Scalar D Aerial Imagery RL Gordon IBM Corp 580 Rt 5 Hopewell Jct NY 533 ABSTRACT An exact formulation of the problem of imaging a D object through a Köhler illumination system is presented;
More informationThe second-order 1D wave equation
C The second-order D wave equation C. Homogeneous wave equation with constant speed The simplest form of the second-order wave equation is given by: x 2 = Like the first-order wave equation, it responds
More informationMeasurement of spatial coherence of light beams with shadows and digital micromirror device
Measurement of spatial coherence of light beams with shadows and digital micromirror device AMAR NATH GHOSH Master of Science Thesis May 6 Department of Physics and Mathematics University of Eastern Finland
More informationAn intuitive introduction to the concept of spatial coherence
An intuitive introduction to the concept of spatial coherence H. J. Rabal, N. L. Cap, E. Grumel, M. Trivi Centro de nvestigaciones Ópticas (CONCET La Plata - CC) and UD Optimo, Departamento Ciencias Básicas,
More informationAstronomy 203 practice final examination
Astronomy 203 practice final examination Fall 1999 If this were a real, in-class examination, you would be reminded here of the exam rules, which are as follows: You may consult only one page of formulas
More informationEngineering Physics 1 Prof. G.D. Vermaa Department of Physics Indian Institute of Technology-Roorkee
Engineering Physics 1 Prof. G.D. Vermaa Department of Physics Indian Institute of Technology-Roorkee Module-04 Lecture-02 Diffraction Part - 02 In the previous lecture I discussed single slit and double
More informationPhysical Optics. Lecture 3: Fourier optics Herbert Gross.
Phsical Optics Lecture 3: Fourier optics 8-4-5 Herbert Gross www.iap.uni-jena.de Phsical Optics: Content No Date Subject Ref Detailed Content.4. Wave optics G Comple fields, wave equation, k-vectors, interference,
More informationPHY410 Optics Exam #3
PHY410 Optics Exam #3 NAME: 1 2 Multiple Choice Section - 5 pts each 1. A continuous He-Ne laser beam (632.8 nm) is chopped, using a spinning aperture, into 500 nanosecond pulses. Compute the resultant
More informationPhase Retrieval for the Hubble Space Telescope and other Applications Abstract: Introduction: Theory:
Phase Retrieval for the Hubble Space Telescope and other Applications Stephanie Barnes College of Optical Sciences, University of Arizona, Tucson, Arizona 85721 sab3@email.arizona.edu Abstract: James R.
More information4 Classical Coherence Theory
This chapter is based largely on Wolf, Introduction to the theory of coherence and polarization of light [? ]. Until now, we have not been concerned with the nature of the light field itself. Instead,
More informationCONTINUOUS IMAGE MATHEMATICAL CHARACTERIZATION
c01.fm Page 3 Friday, December 8, 2006 10:08 AM 1 CONTINUOUS IMAGE MATHEMATICAL CHARACTERIZATION In the design and analysis of image processing systems, it is convenient and often necessary mathematically
More information* AIT-4: Aberrations. Copyright 2006, Regents of University of California
Advanced Issues and Technology (AIT) Modules Purpose: Explain the top advanced issues and concepts in optical projection printing and electron-beam lithography. AIT-: LER and Chemically Amplified Resists
More informationOptical interferometry a gentle introduction to the theory
Optical interferometry a gentle introduction to the theory Chris Haniff Astrophysics Group, Cavendish Laboratory, Madingley Road, Cambridge, CB3 0HE, UK Motivation A Uninterested: I m here for the holiday.
More informationModulation Transfert Function
Modulation Transfert Function Summary Reminders : coherent illumination Incoherent illumination Measurement of the : Sine-wave and square-wave targets Some examples Reminders : coherent illumination We
More informationThe spatial frequency domain
The spatial frequenc /3/5 wk9-a- Recall: plane wave propagation λ z= θ path dela increases linearl with E ep i2π sinθ + λ z i2π cosθ λ z plane of observation /3/5 wk9-a-2 Spatial frequenc angle of propagation?
More informationFinal examination. 3 hours (9am 12 noon) Total pages: 7 (seven) PLEASE DO NOT TURN OVER UNTIL EXAM STARTS PLEASE RETURN THIS BOOKLET
2.710 Final examination 3 hours (9am 12 noon) Total pages: 7 (seven) PLEASE DO NOT TURN OVER UNTIL EXAM STARTS Name: PLEASE RETURN THIS BOOKLET WITH YOUR SOLUTION SHEET(S) MASSACHUSETTS INSTITUTE OF TECHNOLOGY
More information1 ESO's Compact Laser Guide Star Unit Ottobeuren, Germany Beam optics!
1 ESO's Compact Laser Guide Star Unit Ottobeuren, Germany www.eso.org Introduction Characteristics Beam optics! ABCD matrices 2 Background! A paraxial wave has wavefronts whose normals are paraxial rays.!!
More informationScattering of light from quasi-homogeneous sources by quasi-homogeneous media
Visser et al. Vol. 23, No. 7/July 2006/J. Opt. Soc. Am. A 1631 Scattering of light from quasi-homogeneous sources by quasi-homogeneous media Taco D. Visser* Department of Physics and Astronomy, University
More informationCopyright 2000 by the Society of Photo-Optical Instrumentation Engineers.
Copyright 2 by the Society of Photo-Optical Instrumentation Engineers. This paper was published in the proceedings of Photomask and X-Ray Mask Technology VII SPIE Vol. 466, pp. 172-179. It is made available
More informationEfficient sorting of orbital angular momentum states of light
CHAPTER 6 Efficient sorting of orbital angular momentum states of light We present a method to efficiently sort orbital angular momentum (OAM) states of light using two static optical elements. The optical
More information31. Diffraction: a few important illustrations
31. Diffraction: a few important illustrations Babinet s Principle Diffraction gratings X-ray diffraction: Bragg scattering and crystal structures A lens transforms a Fresnel diffraction problem into a
More informationA Review of Radiation and Optics
A Review of Radiation and Optics Abraham Asfaw 12 aasfaw.student@manhattan.edu May 20, 2011 Abstract This paper attempts to summarize selected topics in Radiation and Optics. It is, by no means, a complete
More informationCoherence and width of spectral lines with Michelson interferometer
Coherence and width of spectral lines TEP Principle Fraunhofer and Fresnel diffraction, interference, spatial and time coherence, coherence conditions, coherence length for non punctual light sources,
More informationPhase Contrast. Zernike Phase Contrast (PC) Nomarski Differential Interference Contrast (DIC) J Mertz Boston University
Phase Contrast Zernike Phase Contrast (PC) Nomarski Differential Interference Contrast (DIC) J Mertz Boston University Absorption Scattering / n wave-number k 2 n = index of refraction / n wave-number
More informationWAVE OPTICS (FOURIER OPTICS)
WAVE OPTICS (FOURIER OPTICS) ARNAUD DUBOIS October 01 INTRODUCTION... Chapter 1: INTRODUCTION TO WAVE OPTICS... 6 1. POSTULATES OF WAVE OPTICS... 6. MONOCHROMATIC WAVES... 7.1 Complex Wavefunction... 7.
More informationAdaptive Optics Lectures
Adaptive Optics Lectures 1. Atmospheric turbulence Andrei Tokovinin 1 Resources CTIO: www.ctio.noao.edu/~atokovin/tutorial/index.html CFHT AO tutorial: http://www.cfht.hawaii.edu/instruments/imaging/aob/other-aosystems.html
More informationMethoden moderner Röntgenphysik I. Coherence based techniques II. Christian Gutt DESY, Hamburg
Methoden moderner Röntgenphysik I Coherence based techniques II Christian Gutt DESY Hamburg christian.gutt@desy.de 8. January 009 Outline 18.1. 008 Introduction to Coherence 8.01. 009 Structure determination
More informationFoundations of Scalar Diffraction Theory(advanced stuff for fun)
Foundations of Scalar Diffraction Theory(advanced stuff for fun The phenomenon known as diffraction plays a role of the utmost importance in the branches of physics and engineering that deal with wave
More informationLecture 27 Frequency Response 2
Lecture 27 Frequency Response 2 Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/6/12 1 Application of Ideal Filters Suppose we can generate a square wave with a fundamental period
More informationLight as a Transverse Wave.
Waves and Superposition (Keating Chapter 21) The ray model for light (i.e. light travels in straight lines) can be used to explain a lot of phenomena (like basic object and image formation and even aberrations)
More informationSolution. For one question the mean grade is ḡ 1 = 10p = 8 and the standard deviation is 1 = g
Exam 23 -. To see how much of the exam grade spread is pure luck, consider the exam with ten identically difficult problems of ten points each. The exam is multiple-choice that is the correct choice of
More informationDISCRETE FOURIER TRANSFORM
DD2423 Image Processing and Computer Vision DISCRETE FOURIER TRANSFORM Mårten Björkman Computer Vision and Active Perception School of Computer Science and Communication November 1, 2012 1 Terminology:
More informationUnited Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency
IC/99/46 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS SPATIALLY PARTIAL COHERENCE
More informationTitle. Statistical behaviour of optical vortex fields. F. Stef Roux. CSIR National Laser Centre, South Africa
. p.1/37 Title Statistical behaviour of optical vortex fields F. Stef Roux CSIR National Laser Centre, South Africa Colloquium presented at School of Physics National University of Ireland Galway, Ireland
More informationDIFFRACTION PHYSICS THIRD REVISED EDITION JOHN M. COWLEY. Regents' Professor enzeritus Arizona State University
DIFFRACTION PHYSICS THIRD REVISED EDITION JOHN M. COWLEY Regents' Professor enzeritus Arizona State University 1995 ELSEVIER Amsterdam Lausanne New York Oxford Shannon Tokyo CONTENTS Preface to the first
More informationRadiometry. Basics Extended Sources Blackbody Radiation Cos4 th power Lasers and lamps Throughput. ECE 5616 Curtis
Radiometry Basics Extended Sources Blackbody Radiation Cos4 th power Lasers and lamps Throughput Radiometry Terms Note: Power is sometimes in units of Lumens. This is the same as power in watts (J/s) except
More informationIntroduction to aberrations OPTI518 Lecture 5
Introduction to aberrations OPTI518 Lecture 5 Second-order terms 1 Second-order terms W H W W H W H W, cos 2 2 000 200 111 020 Piston Change of image location Change of magnification 2 Reference for OPD
More informationDiffraction model for the external occulter of the solar coronagraph ASPIICS
Diffraction model for the external occulter of the solar coronagraph ASPIICS Raphaël Rougeot OCA, Nice 14/05/2018 14/05/2018 R.Rougeot 1 Outline 1) Proba-3 mission and ASPIICS 2) Diffraction from external
More informationEdward S. Rogers Sr. Department of Electrical and Computer Engineering. ECE426F Optical Engineering. Final Exam. Dec. 17, 2003.
Edward S. Rogers Sr. Department of Electrical and Computer Engineering ECE426F Optical Engineering Final Exam Dec. 17, 2003 Exam Type: D (Close-book + one 2-sided aid sheet + a non-programmable calculator)
More informationRecent progress in SR interferometer
Recent progress in SR interferometer -for small beam size measurement- T. Mitsuhashi, KEK Agenda 1. Brief introduction of beam size measurement through SR interferometry. 2. Theoretical resolution of interferometry
More informationLagrangian submanifolds and generating functions
Chapter 4 Lagrangian submanifolds and generating functions Motivated by theorem 3.9 we will now study properties of the manifold Λ φ X (R n \{0}) for a clean phase function φ. As shown in section 3.3 Λ
More informationComputer Vision & Digital Image Processing. Periodicity of the Fourier transform
Computer Vision & Digital Image Processing Fourier Transform Properties, the Laplacian, Convolution and Correlation Dr. D. J. Jackson Lecture 9- Periodicity of the Fourier transform The discrete Fourier
More informationFull field of view super-resolution imaging based on two static gratings and white light illumination
Full field of view super-resolution imaging based on two static gratings and white light illumination Javier García, 1 Vicente Micó, Dan Cojoc, 3 and Zeev Zalevsky 4, * 1 Departamento de Óptica, Universitat
More information= n. Psin. Qualitative Explanation of image degradation by lens + 2. parallel optical beam. θ spatial frequency 1/P. grating with.
Qualitative Explanation of image degradation by lens Mask + 2 lens wafer plane +1 φ 0 parallel optical beam -2-1 grating with θ spatial frequency 1/P Psin φ = n λ n = 0, ± 1, ± 2,... L S P l m P=2L sin
More informationSta$s$cal Op$cs Speckle phenomena in Op$cs. Groupe d op$que atomique Bureau R2.21
Sta$s$cal Op$cs Speckle phenomena in Op$cs Vincent.josse@ins$tutop$que.fr Groupe d op$que atomique Bureau R2.21 Diffrac$on from a rough surface (Fourier speckle) Eample Star imaging through turbulence
More informationROINN NA FISICE Department of Physics
ROINN NA FISICE Department of 1.1 Astrophysics Telescopes Profs Gabuzda & Callanan 1.2 Astrophysics Faraday Rotation Prof. Gabuzda 1.3 Laser Spectroscopy Cavity Enhanced Absorption Spectroscopy Prof. Ruth
More informationOPSE FINAL EXAM Fall 2016 YOU MUST SHOW YOUR WORK. ANSWERS THAT ARE NOT JUSTIFIED WILL BE GIVEN ZERO CREDIT.
CLOSED BOOK. Equation Sheet is provided. YOU MUST SHOW YOUR WORK. ANSWERS THAT ARE NOT JUSTIFIED WILL BE GIVEN ZERO CREDIT. ALL NUMERICAL ANSERS MUST HAVE UNITS INDICATED. (Except dimensionless units like
More informationCoherent x-rays: overview
Coherent x-rays: overview by Malcolm Howells Lecture 1 of the series COHERENT X-RAYS AND THEIR APPLICATIONS A series of tutorial level lectures edited by Malcolm Howells* *ESRF Experiments Division CONTENTS
More informationFourier Optics - Exam #1 Review
Fourier Optics - Exam #1 Review Ch. 2 2-D Linear Systems A. Fourier Transforms, theorems. - handout --> your note sheet B. Linear Systems C. Applications of above - sampled data and the DFT (supplement
More informationEncircled Energy Factor as a Point-Image Quality-Assessment Parameter
Available online at www.pelagiaresearchlibrary.com Advances in Applied Science Research, 11, (6):145-154 ISSN: 976-861 CODEN (USA): AASRFC Encircled Energy Factor as a Point-Image Quality-Assessment Parameter
More informationImage formation in scanning microscopes with partially coherent source and detector
OPTICA ACTA, 1978, VOL. 25, NO. 4, 315-325 Image formation in scanning microscopes with partially coherent source and detector C. J. R. SHEPPARD and T. WILSON Oxford University, Depart of Engineering Science,
More informationChapter 5. Diffraction Part 2
EE 430.43.00 06. nd Semester Chapter 5. Diffraction Part 06. 0. 0. Changhee Lee School of Electrical and Computer Engineering Seoul National niv. chlee7@snu.ac.kr /7 Changhee Lee, SN, Korea 5.5 Fresnel
More informationInterferometry Theory. Gerd Weigelt Max-Planck Institute for Radio Astronomy
Interferometry Theory Gerd Weigelt Max-Planck Institute for Radio Astronomy Plan Introduction: Diffraction-limited point spread function (psf) Atmospheric wave front degradation and atmosperic point spread
More informationHigh-Resolution Transmission Electron Microscopy
Chapter 2 High-Resolution Transmission Electron Microscopy The most fundamental imaging mode of a transmission electron microscope is realized by illuminating an electron-transparent specimen with a broad
More information