Radiometry TD Cursus/option : 2A Date de mise à jour : 5 sept 2018 Année scolaire : 2018-2019 Intervenants : Antoine Glicenstein 1
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Institut d Optique Graduate School Radiometry TD 1 Learning objectives: To know how to use different radiometric relationships to model the path of light from the source to the illuminated object and finally to the detector. Being able to evaluate the geometrical extent of a simple optical system. Illustrate the difference between a Lambertian surface and a mirror. Exercise 1: Laser ranging of the moon with pulse laser The very precise measurement of the Earth-Moon distance by laser is an experiment that began in the 1970s and is still being carried out at the present time, in particular as it makes it possible to verify the fundamental physical laws of gravity. The system is the following: a pulsed laser emitter, on Earth, is directed at the Moon. The main characteristics of the laser emitter are the following: peak power P peak = 100 MW, total beam divergence 2 = 10 rad pulse duration = 1 ns and wavelength = 1 m. Côte d Azur Observatory. Main site for the Moon laser-ranging experiment. Laser intensity is supposed to be uniform inside the beam. A telescope, situated close by the laser emitter, is observing the laser spot on the moon. It is made up of a telescope of diameter D op = 1.4 m open at f/5, and of perfect transmittance. At its focus, there is a detector of diameter D d = 150 m. 1) What is the peak radiant intensity I laser of the laser beam? 2) Give the expression of the peak irradiance E laser of the laser on the moon (the illuminated area of the moon is supposed to be normal to the incident laser beam). Deduce the peak radiance L Moon reflected by the Moon. The ground on the moon is covered with rocks and dust, its reflection is Lambertian with a very low albedo R d = 10% at laser wavelength. 3) From this radiance, calculate the flux received by the telescope for each pulse reflected by the Moon (distance earth - moon = 380 000 km). Justify the geometrical extent used. 4) At the return of each pulse, how many photons are received by the detector? Conclude that the experiment is almost impossible in these conditions. 5) What would be your answer if the detector diameter was twice smaller? In reality, the laser illuminates a zone of the moon where a set of cube corners has been installed by the Apollo 11 mission. The area of the set of cube corners is S RR = 1 m 2 ; its reflectance 100%, and the laser light is being retro reflected inside a cone of total divergence RR = 0.1 mrad. 3
6) What is the peak radiance L RR of the laser light that is retro-reflected by the cube corners? Set of cube corners for the Lunar Laser Ranging Experiment (Apollo 11 mission). 7) What is the number of laser photons from the cube corners that are received by the detector at the return of each pulse? Compare to the previous case. 4
Institut d Optique Graduate School Radiometry TD 2 Learning Objectives: Illustrate the limitations of a broad light source. Being able to evaluate the luminance of a black body. Knowing how to evaluate the geometrical extent of a system in several optical configurations. Exercise 1. Solar concentrator We want to calculate the maximum theoretical efficiency of a solar concentrator shown below. The optics of this solar concentrator will be modelled by a thin lens. s Solar reflector at Odeillo, Languedoc-Roussillon. P = 1 MW Sun Spupil Simage 1) Give the expression of the irradiance E pupil due to the Sun on the pupil and the irradiance E image in the image plane. 2) Deduce the maximum theoretical efficiency, defined by C = E image / E pupil. Do the numerical application. Remember that the angular diameter of the Sun is equal to s = 0.5. Exercise 2. Pictures taken from a plane (from the 2017 exam) A pinhole camera is a simple box with one side pierced by a tiny hole that lets in light. On the opposite surface is formed the inverted image of the object. This system was probably used to make the first known photograph of history, by Joseph Nicéphore Niépce in 1827. Pinhole Image Object From a geometrical optic point of view, the small size of the hole ensures the stigmatism and therefore the quality of the image. This very simple system does not present any aberration and gives a clear image whatever the distance of the object! The counterpart is that, because of the very small size of the hole collecting the light, the time necessary to impress the photosensitive surface is extremely long: several hours for the historical photography of Joseph Niépce. The objective of this exercise is to radiometrically compare this pinhole with modern cameras. 5
The pinhole camera is a cubic box of side s length L with a circular pinhole of diameter D t. We place a visible sensor composed of 2048x2048 pixels centered at the back of the box. The pixels are square with a side p = 10 μm. We photograph an object located at a distance d. 1) Give the expression of the geometrical extent of a beam limited by the pinhole input hole and completely illuminating a pixel at the center of the sensor (on the optical axis). 2) Consider a point on the object located on the optical axis. If the pinhole was infinitely small, the light of this point object would fall on only one pixel (neglecting diffraction), the one located on the optical axis (stigmatism). Give the maximum diameter of the hole as a function of d, L and p to ensure this stigmatism: that is, no light from this object point falls on another pixel. You will be assumed in the following questions that this condition is respected. The photographed object, the roof of a house as in the historical photograph of Joseph Niépce, is supposed to be Lambertian with an albedo = 30% and is illuminated at 45 by the sun in the configuration shown in the figure. 3) The sun is a black body of temperature T sol = 5900 K. Give an approximate expression of the luminance of the sun L sol in the spectral band Δ = 0,2 μm (centered around 500 nm). 4) Knowing that the angular diameter in the sky is sol = 0.5, give the expression of the illuminance of sunlight on the roof. Deduce the luminance L obj reflected by the roof. 5) The roof has a surface of 10 m 2. By carefully justifying the geometrical extent used, give the expression and the numerical application of the flux received by the pixel in the center of the sensor. You will take L = 50 cm, d = 5 m and D t =5 µm. 6) The pinhole is now replaced by a photographic objective with an aperture N = 8, a transmission T = 80% and a focal length equal to the size L of the box. Give the expression and the numerical application of the flux collected in this configuration. Make the ratio in terms of flux of these two configurations and conclude. 6
Institut d Optique Graduate School Radiometry TD 3 Learning objectives: Knowing how to write an optical design problem in term of geometrical extent. Being able to write the thermal equilibrium of a system and to use Stefan's law to deduce its temperature. Understand the concept of emissivity. Exercise 1. Choice of a source for a collimator During a lab work, you are asked to choose a light source for illuminating a collimator. This collimator is open at f/2, with a circular aperture of diameter D = 3 mm in its object plane. The radiance inside the aperture must be uniform, with a minimum radiance level equal to L min = 10 5 Wm -2 sr -1, on a spectral bandwidth, centred around = 0,9 m, which must be at most equal to = 100 nm You have the choice between two possible sources: a Light Emitting Diode (LED) and an incandescent source coupled with a band-pass filter. 1) The LED emitting surface, is circular with a diameter D led= 500 m. The total flux emitted by the LED over a half space is equal to F led= 100 mw, inside the specified spectral bandwidth. What is the value of this radiance and is it enough for the specification of the optical system? 2) If yes, describe a classical optical set-up (such as a condenser conjugating the diode onto the hole) that could satisfy the specification on the geometrical extent of the collimator? 3) The second possible choice is an incandescent source. It can be considered as a rectangular plane Lambertian surface of area a x b (2 mm x 5 mm) at a temperature T=3200 K with an emissivity = 0,4 inside the specify bandwidth. As previously, describe some simple set-up transforming the geometry of the beam from the incandescent source into one that satisfies the specification on the geometrical extent of the collimator 4) What will be the radiance at the entrance aperture of the collimator, inside the spectral bandwidth, if the maximum transmittance of the band-pass filter in front of the incandescent source is T max (= 80%)? Exercise 2. Equilibrium temperature of a satellite We need to calculate the temperature of a satellite, considered as a blackbody, lighted by the Sun. We will neglect the contribution of the Earth or the Moon. Planck Satellite dedicated to the study of the cosmic microwave background using bolometers cooled down at 0.1 K. 1) Write the condition of thermal equilibrium of the satellite. Deduce the expression of the temperature T of the satellite as a function of the temperature of the sun T sun = 5900 K and its angular diameter = 30' as well as the spectral emissivity of the satellite (the spectral emissivity being equal to the spectral absorption here). 7
Find the temperature of the satellite for the following models: A disc with the same spectral emissivity on each side, facing the Sun. A sphere with a spectral emissivity To reach a much lower temperature (when a cryogenic instrument is found on the satellite for example), A solution is to use a coating with a high reflectivity in the visible wavelengths, and a high emissivity in the I.R. Consider a sphere with = 0 for < 4 µm and = 1 for > 4 µm. Take L sol ( 4 µm) = 2.10 5 W.m -2.sr -1 8
Institut d Optique Graduate School Radiometry TD 4 Learning objectives: To know how to quantify the losses during the propagation of light in the hypothesis of simple scattering. To be able to evaluate the luminance of a scattering medium. Exercise 1. Underwater observation of fish In a calm sea, with a sun at zenith, a diver at a depth of z h = 5 m is observing marine life beneath him. In the visible, the solar illuminance at sea level is E 0 = 100 000 lux. For a mean wavelength of 550 nm, the absorption and scattering coefficients of sea water are respectively equal to µ a = 0. 08 m -1 and µ s = 0. 004 m -1. We will neglect light losses due to the reflectance of air-water interface. 1) Within the single scattering hypothesis, give the expression of solar illuminance at depth z below the sea level. 2) Supposing that the sea is homogeneous with isotropic scattering, what is the expression of the phase function of the medium? Deduce the elementary luminance dl z of the light being scattered by a horizontal sea layer, of thickness dz, situated at depth z. 3) Still within the single scattering hypothesis, give the expression of the apparent luminance dl a of this sea layer, seen by the diver. Deduce the total luminance L hp of the ocean between z h and some depth z p. 4) Give the expression of L h, for z p =, which is the apparent luminance seen by the diver of the bottom of the sea (supposed to be very deep), when there is no object immersed beneath him. 5) Beneath the diver, at some depth z p below sea level, there are some fish that will be modeled as plane, Lambertian objects, of albedo R in the visible. What is the apparent luminance L hp of such fish for the diver, i.e., the luminance due to the reflection of solar light by the fish and to the scattered light from the sea water that is in between the fish and the observer? What contribution to the luminance are we neglecting here? 6) What is the apparent contrast of a white fish (R = 1) and that of a black fish (R = 0), when they are swimming below the diver at depth z p (with respect to the sea surface). We will define apparent contrast as C app = (L hp - L h ) / L h 7) It is considered that for visual detection, the minimum absolute value for the apparent contrast must be higher than 2%. In these conditions, how far away may the diver expect to observe white fish, black fish? Which one will be detected the farthest? 9
Institut d Optique Graduate School Radiometry TD 5 The following exercise is based on an exam of previous years. To train and verify that you have the necessary knowledge, try to solve the exercise in groups of 3 or 4 and only asking the help of the teacher when you are really stuck. Exercice: BepiColombo mission (excerpt from 2015 exam) The European Space Agency BepiColombo mission to the planet Mercury is scheduled for launch in 2017 and to arrive around Mercury in 2024. Among the many instruments that will equip the spacecraft, there is a thermal infrared spectral imaging system called MERTIS. This spectral imaging system works in push broom mode (see Figure 1). The detector is a single line of 120 square pixels of size p = 35 μm, in the focal plane of a telescope with a focal length f = 50 mm, a numerical aperture N = 2 and a transmission T op = 50 %. To acquire an image during the overflight of Mercury, the satellite's movement in one direction is combined with the signal acquired by the pixel line. Thus, at regular time intervals, 1-dimensional images are acquired. All these 1-dimensional images are combined to make a two dimensional image. Finally, a diffraction grating (not shown here) allows to select a spectral detection band with a total spectral width of Δ = 100 nm, between 7 and 14 microns, in which the image will be acquired. Pixel line Telescope Movement of the spacecraft on its orbit Line area imaged on the ground Figure 1: Push-broom imaging principle 10
The scientific objective of this instrument is to study the chemical and mineralogical composition of Mercury's surface. The rocks are differentiated by their emissivity which is the parameter that has to be measured as precisely as possible with this instrument. The probe has an average altitude of H = 750 km. The atmosphere of Mercury is almost nonexistent and will be neglected. 1) Calculate the size a x L of the area imaged on Mercury's surface by the pixel line. What is the spatial resolution in the image? Do the numerical application. 2) The surface of Mercury is considered as a gray body at a temperature T = 440 K and an average emissivity = 0.9. Calculate numerically the spectral radiance of the surface of Mercury for a central wavelength of detection 0 = 10 µm. Derive an approximate numerical value of the surface radiance L surf integrated in the spectral band of the radiometer Δ around 0. 3) Give the expression, with justification, of the flux F surf received by a pixel in the center of the detector observing the surface of Mercury, in the spectral band of the detector. Do the numerical application. 4) Given that the movement of the probe BepiColombo on its orbit causes a displacement of the line of sight on the ground at a speed v = 2.6 km/s, calculate the time during which a given point on the surface is seen by a pixel. Deduce the number of photons received by a pixel from a given area of the surface. Do the numerical application. 5) In reality, the radiometer also receives stray light from the Sun reflected by the surface of Mercury. We assumed the surface to be Lambertian of albedo = 0.1. Give the expression of this flux F par received by one pixel of the radiometer in the spectral band Δ for the same central wavelength 0 = 10 µm. We will consider the sun at the zenith, with a temperature T sol = 5900 K, and an angular diameter of = 3.3 in the sky of Mercury. Do the numerical application. 6) Give the expression of the signal to noise ratio SNR = ΔF surf/ F par where ΔF surf is the variation of flux emitted by the surface for a variation Δ of its emissivity (assuming F par remains constant). Derive an estimate of the smallest variation Δ which can be detected for a minimum SNR of 5. 11
Institut d Optique Graduate School Radiometry TD 6 The following exercise is based on an exam of previous years. To train and verify that you have the necessary knowledge, try to solve the exercise in groups of 3 or 4 and only asking the help of the teacher when you are really stuck. Exercice. Study of an atmospheric LIDAR (excerpt from 2013 exam) The principle of a laser based remote sensing system (LIDAR), for atmospheric sensing, is to send a laser pulse in the atmosphere and analyze the flux that is back-scattered on the ground as a function of time. Looking at the amplitude and the evolution of the scattered flux, it is then possible to extract a lot of information on the chemical composition, concentration, speed...etc. of the molecules or particles in the atmosphere. Also, using a very short laser pulse allows to probe only a given thin atmospheric layer. LIDAR is used extensively for atmospheric research and meteorology. Atmospheric scattering Laser Laser pulse Field of view of the telescope Telescope Detector The LIDAR developed by the LMD laboratory at Polytechnique, is based on a pulse laser with a peak power Pc = 4 MW at a wavelength of = 532 nm and a pulse duration of = 7 ns. 1) The atmosphere, at the laser wavelength, has an extinction of = 0.02 km -1. The LIDAR sends a laser pulse, straight up in the atmosphere. Using the single scattering hypothesis, give the expression of the laser flux F(H) at an altitude H. 2) We will consider an atmospheric layer at an altitude H 0, which has a high concentration of CO 2, with a thickness h << H and a scattering coefficient s different from the rest of the atmosphere. We will neglect the absorption of this layer at the laser wavelength. Give the expression of the total (in all direction) flux F d,total scattered by this layer as a function of F(H 0), s, and h. 3) The CO2 molecules are characterized by a phase function p( ), with the scattering angle compared to the incident angle. Give the expression of the scattered intensity I d by the atmospheric layer, in a given direction, as a function of F d,total and p( ). 12
The optical system used by this LIDAR to collect the back-scattered flux from the atmospheric layer, is a telescope with a mirror of diameter D opt = 20 cm and a f-number N = 4. The detector is a circular photodiode placed in the focal plane of the system, with a diameter D det = 1.5 mm. The overall transmission of the optical system is T 0 = 50 %. This telescope is closed to the laser and looking straight up ( = π). 4) The total angle of divergence of the laser is equal to e= 0.5 mrad. Justify that the optical system works in a flux collector configuration whatever is the altitude H 0 of the atmospheric layer. 5) Finally, staying in the single scattering hypothesis, give the complete expression of the flux F r received by the detector coming from the atmosphere. This equation is called the LIDAR equation. 6) Show that for H 0 = 10 km, this equation can be numerically written: F r (H = 10km) = 8.8 10 4. μ s. p(θ) 7) The CO 2 molecules have a phase function: p(θ) = 3 (1 + 4 cos2 (θ)) How do we call this scattering regime and why are we in this regime? 8) The value of the scattering coefficient s of CO 2 molecules can be computed by : μ s = 1.6 10 2. P(atm) T(K). 1 λ( m) 4 With laser wavelength (unit: micron), P the atmospheric pressure (unit : atm), T the atmospheric temperature (unit : Kelvin) at the altitude H. Using the graphs given in annex, give the numerical value of the scattering coefficient for an altitude of 10 km. 9) Deduce the number of photons received by the LIDAR detector coming from the CO 2 layer, for every pulse send in the atmosphere. Do the numerical application 13
Altitude (km) Altitude (km) Julien Moreau IOGS - 2018 40 35 30 25 20 15 10 5 0 10-3 10-2 10-1 10 0 Pressure (atm) Atmospheric pressure, as a function of altitude. 40 35 30 25 20 15 10 5 0-80 -60-40 -20 0 20 Temperature ( C) Mean temperature ( C) of the atmosphere as a function of altitude. 14