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# Review Sheet for OPTI 596C at UA

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Date Created: 02/06/15

Optical Imaging and Aberrations Virendra N Mahajan Adjunct Professor The Aerospace Corporation College of Optical Sciences El Segundo California 90245 University of Arizona 310 336 1783 virendranmahajanaeroorg Imaging Through Atmospheric Turbulence Imaging Through Atmospheric Turbulence In groundbased astronomy a plane wave of uniform amplitude and phase representing the light from a star is incident on the atmosphere As the wave propagates through the atmosphere it undergoes both amplitude and phase variations due to the random inhomogeneities of the refractive index of air In nearfield imaging L ltlt Dzit turbulence primarily introduces aberrations and a short exposure image provides a better resolution In far eld imaging Lgtgt Dzit turbulence introduces not only aberrations but also amplitude variations called scintillations Wavefront gets distorted ie it becomes nonplanar with nonuniform amplitude across it and the distortions vary randomly with time Distorted wavefront of nonuniform amplitude is incident on a groundbased imaging system Summary Imaging Through Atmospheric Turbulence Long Exposure Image Mutual Coherence Function and Wave Structure Function Atmospheric Coherence Length MTF Reduction Factor Phase Aberration in Terms of Zernike Polynomials Short Exposure Image Adaptive Optics LongExposure Image Pupil function of the overall system ie including the effects of atmospheric turbulence representing the wavefront at the exit pupil Pia PLlrplPRlal Instantaneous irradiance distribution ofthe star image formed by the overall system a 211139 o e a JPltrP exp E r x drp Timeaveraged PSF 65 vim PLlalpzlaxPRmPltagtexpl lta w P 2 2X sex 73122 Mutual coherence function WW swim ll Kolmogorov turbulence Ebwr 688rr053 583 Wave structure function L 06 r0 01847k1392JC2zL53dz 584 0 r0 is a characteristic length of turbulence representing its coherence length or diameter called Fried s coherence length If the line of sight makes an angle 0 with the zenith then the path length L through turbulence increases by sece or r0 decreases by sece 5 gum WWW cw r la ai Since the refractive index structure parameter C3 at a given site decreases with altitude the integral in Eq 584 has a higher numerical value when observing a space object from ground looking upwards than when observing a ground object from space looking downwards Because of a Fourier transform relationship between the PSF and the OTF we identify f1 imAR with a spatial frequency Vi a mi Substituting and carrying out the integration over i ILm minim exp 9bwxRVLexp 21in adv er Sal PLPZ 7 RVd1 Turbulencefree OTF Using normalized quantities r gag10 f gym and v mF 10 PeXSeX7 2R2 I 41tlr7 exp ZTci7 dV Timeaveraged PSF up no exp ngvD Timeaveraged OTF Correspondingly r0 is smaller when a satellite is observed from ground compared to when a ground object is observed from a satellite Consequently image degradation is much smaller when a ground object is observed from space than when a space object is observed from ground In the first case the object is near the region of turbulence and it is observed from far away In the second case the object is away from the region ofturbulence but it is observed from nearby This is similar to when an object behind a diffuse shower glass is observed One can see some detail in the object when it is in contact with the shower glass However as soon as the object is moved slightly away from the shower glass it appears only as a halo illustrating complete loss of image resolution Reciprocity of wave propagation does hold That is why wavefront errors of a wave from a point source in space can be corrected with a deformable mirror on ground yielding a diffraction limited neglecting any measurement or correction error beam in space MCF for Kolmogorov turbulence PRP 13133 exp Ema exp 344rr053 Timeaveraged OTF of the overall system TlVDr0 no expl wltv0 TLVTaVDr0 TavDr0 exp 344vDr053 exp 344vi7LRr053 594 TavDr0gt is the longexposure MTF reduction factor associated with atmosphericturbulence Ta is independent of the pupil diameter D as it should be Since exp 344003 atmospheric turbulence reduces the overall system MTF for a spatial frequency v rOD by a factor of 003 Since Ta represents the mutual irradiance function of the wave its magnitude describes the degree of spatial coherence of the wave and thus the visibility of fringes formed in a Young s twopinhole experiment Hence r0 represents a spatial coherence length of the wave so that its degree of coherence corresponding to two points on it separated by r0 is 003 or that the visibility of the fringes formed by the secondary waves from these points is only 003 o l l l l l l 0841 96 4 ltSgt O l l 001 l l l 0 a Der Figure 56b Timeaveraged Strehl ratio Strehl ratio decreases to zero monotonically as Dro increases For a given value of D Strehl ratio decreases rapidly as r0 decreases Even when r0 is as large as D the Strehl ratio is only 0445 11 Application to Circular Pupils Assume that the turbulencefree system is aberration free with uniform transmittance Timeaveraged PSF of the overall system 1 1r Dr0gt 8 I ltTv Dr0 102139crvvdv 0 TvDr0 TLVexp 344vDr053 TLV cos 1v v1 v212 0 S v 51 Timeaveraged Strehl ratio 1 SDr0gt IODr0gt 8ITvDr0vdv 0 10 Unnormalized timeaveraged central irradiance momr0 2533 Smre 33 Dr0 SE nr024 is the coherent area of the atmosphere 21 nlDro Dr02SDr0gt JTLV6XP 3 44VDr053Vdv 599 0 Whereas the Strehl ratio represents the central irradiance normalized by its aberrationfree value 71 represents the central irradiance normalized by the aberrationfree value when the pupil diameter is r0 12 Dru2 lt5 4 T l 01 um i i l 01 1 10 aDer Figure 56c Timeaveraged central irradiance for a fixed total power represented by n Aberrationfree value of n increases as Dr02 as illustrated by the straight line For small values of Dro aberrated value of n also varies as Dr02 As D increases n increases much more slowly with a negligible increase for Dro gt 5 13 Purpose of a large groundbased telescope has therefore generally not been better resolution before the advent of adaptive optics but to collect more light so that dim objects may be observed In the case of a laser transmitter with a fixed value of laser power P the central irradiance on a target will again be limited to its aberrationfree value for an exit pupil of diameter r0 no matter how large the transmitter diameter D is 15 Limiting performance n astronomical obsenations PM increases as D increases However if the observation is made against a uniform background then the background irradiance in the image also increases with D as D2 Hence the detectability of a point object is limited by turbulence to a value corresponding to an exit pupil of diameter r0 no matter how large the diameter D ofthe exit pupil is At visible wavelengths r0 10 cm Consequently the performance of a groundbased telescope making astronomical observations in the visible region is limited primarily by atmospheric turbulence to a telescope of diameter D 10 cm 14 Phase Aberration in Terms of Zernike Circle Polynomials So far we have calculated performance degradation by propagation through atmospheric turbulence without any knowledge of the phase aberrations introduced by it Now we expand the phase aberration introduced by atmospheric turbulence in terms of orthonormal Zernike circle polynomials P7 9 NM ZaJZJpve J Circle polynomials may be written ZevenJp 1l2nl 1RV pcosm i m 0 Zodep 1l2nl 1RV psinm i m 0 Zjp0 1lni1R2p m 0 jis even or odd depending on the value ofn 16 Polynomials varying as sinme are also included here since the turbulent atmosphere does not have an axis of rotational symmetry lndexj is a polynomialordering number and is a function of both n and m Polynomial number ordering Polynomial with a lower value of n is ordered first For a given value of n a polynomial with a lower value of m is ordered first Evenjcorresponds to a symmetric polynomial varying as cosme Oddj corresponds to an antisymmetric polynomial varying as sinme When m 0 j may be even or odd depending on the value of n 17 j n m ZjP9 13 4 2 m4p4 3p2sin29 14 4 4 mp4 00549 15 4 4 Mp4 sin49 16 5 1 m10p5 12p3 3pcose 17 5 1 m10p5 12p3 3psin9 18 5 3 mlt5p5 4P3cos30 19 5 3 m5p5 4p3sm3e 2o 5 5 mp5cosse 21 5 5 mp5smse 22 6 o 20p5 30p4 12p2 1 23 6 2 m15p6 20p4 6p2siI12 24 6 2 m15p5 20p4 6p200520 19 Table 54 Orthonormal Zernike circle polynomials ZJp 9 j n m ZjP9 1 o o 1 2 1 1 2130059 3 1 1 Zpsine 4 2 0 5sz 1 5 2 2 pz sin29 6 2 2 pz 00520 7 3 1 433p3 2psiI1 8 3 l JE3p3 2pcos 9 3 3 M3 61136 10 3 3 13133 00530 11 4 o 6p4 6p2 1 12 4 2 mm 3p200520 18 25 6 4 m6p5 5P4sin40 26 6 4 M6p6 5p4cos4e 27 6 6 mp5 511160 28 6 6 Mp6 00569 29 7 1 435p7 60p5 30p3 4psiI19 30 7 l 435p7 60p5 30p3 4pcos i 31 7 3 421p7 30p5 10p3sm3e 32 7 3 421p7 30p5 10p3cos3e 33 7 5 47p7 6p5sm50 34 7 5 47p7 6p500559 35 7 7 437 51176 36 7 7 4p7 00570 37 8 o 370p8 140p6 90p4 20p2 1 20 Correlation of Zernike Coefficients Orthonormality of Zernike polynomials 1 2 1L 1 1 I 0 Zjp9ZJIp0pdpd9I pdpde 5er 0 0 N Expansion coefficients 1 4 11 i 1 p9ZJp9pdpd9 Cross correlation of the expansion coefficients ajaJr 112 1 d6 1 dB Z Zj3913 1112Id i da39z jrmwl B l 21 Table 5 5 Correlation of Zernike polynomial expansion coefficients for nearfield propagation in units of D7053 Correlation Zernike Value Order Correlation Coefficient Pairs n n thaw a a2 1 1 449x10 1 a3 1 ag 2 2 232x10 2 02087 113117 1 3 142x10 2 11 11 a3 1120 3 3 619x10 3 1411117 1511137 151112 2 4 3 88gtlt10 3 11217 11227 11237 11247 1125 4 4 245x10 3 11711177 84167 0911197 11101118 3 5 156x10 3 2 2 2 2 2 2 3 1157 1177 187 1197 0207 021 5 5 139x 1110227 1120247 1130237 1140257 115025 4 5 760x10 4 23 Zernike coefficients are not statistically independent For a given value of n the crosscorrelations and therefore autocorrelations do not depend on the value of m For a given value of n the correlation values decrease rapidly as the order difference n n increases First nonzero crosscorrelations are 12118 and 13117 Le the tiltcoma crosscorrelations 22 Table 5 6 Correlation of Zernike polynomial expansion coefficients for nearfield propagation in units of D7053 for n1 and n Zl Order Correlation Zernike Difference Value Order Correlation n n n n aagt Coef cient J J Pairs 0 1 1 1 0 449x10 1 0208 13217 1 3 2 142x10 2 121115 131117 1 5 4 754x10 4 121130 131129 1 7 6 952x10 5 W45 131147 1 9 8 861x10 7 121158 131157 1 11 10 1A41gtlt10 7 121192 13093 1 13 12 324x10 8 1211122 1311121 1 15 14 929x10 9 1211154 1311155 1 17 16 314x10 9 24 Modal correction and residual error Residual aberration variance when the first J modes are corrected 1 Zn J 2 A n 1f I p9 Zaijp9 pdpde j1 O O J lt 2gtlta12gt 2lta3gt Ar 2 lt02 J2 No correction A1 E 531 Since r0 732 variance of the wave aberration is independent of 7b as expected in the absence of atmospheric dispersion x and y tilts corrected short exposure image A3 0134Dr053 Tiltcorrected variance is reduced by a factor of 1030134 2 77 25 Strehl Ratio Longexposure image eXp 531 is not a good approximation for estimating the Strehl ratio lt51Dr0gt eXp 531 eXp1O3Dr053 Even for a small value of DrO 1 51gt 0357 compared to a true value of 0445 51gt underestimates the Strehl ratio more and more as DrO increases Much better approximation yielding a slight overestimation lt52 IDFD 1Dr053 12 27 Table 57 Variance of residual phase errors for nearfield propagation in units of DrO 53 A1 10299 A2 0582 A3 0134 A4 0111 A5 00880 A6 00648 A7 00587 A8 00525 A9 00463 A10 00401 All A z 02944J E2 For large J A12 00352 A13 00328 A14 00304 A15 00279 A16 00267 A17 00255 A18 00243 A19 00232 A20 00220 A21 26 Figure 512 Variation of timeaveraged Strehl ratio S with DrO gt DrO 28 ShortExposure Image A timeaveraged shortexposure image is obtained if the tilt is corrected in real time for example with a steering mirror Shortexposure or tiltcorrected aberration variance 2 53 0 0134Dr0 exp 031 with 031 0134Dr053 approximates the true value of Strehl ratio reasonably well for DrO s 6 Variance of angle of arrival 358 2 5 3 03mg RD j Dr0 29 Random Aberration Example o tilt 57v tilt I b c Figure 515 Aberration introduced by atmospheric turbulence for Dr0210 a Aberration shape b Aberration interferogram c lnterferogram with 25 k of wavefront tilt 31 100 10 2 gtDr0 Figure 513c Nearfield tiltcorrected timeaveraged Strehl ratio St The uncorrected Strehl ratio ltSgt is shown to illustrate the improvement made by tilt correction 3O ShortExposure PSFs for DrO 10 Each image is broken up into small spots called speckles which is a characteristic of random aberrations Angular size of a speckle is approximately equal to MD Angular size of the overall image is approximately equal to krO For a given value of D the overall image size increases as r0 decreases showing the effects of what astronomers ca seeing For a given value of r0 the image size is approximately constant but the size of a speckle decreases as D increases Thus an increase in D does not significantly improve the resolution of the system as determined by the overall size of the image 32 Adaptive Optics Correction of wavefront errors in near real time by using a steering mirror and a deformable mirror is called adaptive optics Steering mirror with only three actuators corrects the large 6 and y wavefront tilts also called tip and tilt Deformable mirror deformed by actuating an array of actuators attached to it corrects the wavefront deformation Actuator signals are independent of the optical wavelength provided atmospheric dispersion is negligible Zonal approach In the zonal approach signals for the actuators are determined by sensing the wavefront errors with a wavefront sensor in a closed loop to minimize the variance of the residual errors Zonal approach has the advantage that the rate of correction is limited only by the rate at which the wavefront errors can be sensed and the actuators can be actuated 33 References 1 D Fried Optical resolution through a randomly inhomogeneous medium for very long and very short exposures J Opt Soc Am 56 1372 1379 1966 2 R J Noll Zernike polynomials and atmospheric turbulence J Opt Soc Am 66 207 211 1976 3 Gm Dai and V N Mahajan Zernike annular polynomials and atmospheric turbulence J Opt Soc Am A 24 139 155 2007 4 J W Hardy Adaptive Optics forAstronomical Telescopes Oxford New York 1998 5 R K Tyson Introduction to Adaptive Optics SPIE Press Bellingham Washington 1999 6 V N Mahajan J Govignon and R J Morgan Adaptive optics without wavefront sensors SPIE Proc Vol 228 Active Optical Devices and Applications 6369 1980 35 However the amount of light that is used by the wavefront sensor in sensing the wavefront errors is lost from the image Moreover for imaging an isoplanatic extended object wavefront sensing requires a point source in its vicinity Modal approach In the modal approach the actuators are actuated to produce Zernike modes eg focus two modes of astigmatism two modes of coma etc iteratively until image sharpness 11277 where 7 is the image irradiance distribution is maximized In the modal approach there is no loss of light but the rate or the bandwidth of correction can be slow due to its iterative nature especially when turbulence is severe and a large number of modes must be corrected However no point source is needed since the modal approach is applicable to the extended object itself 34

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