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# Class Note for OPTI 596C at UA

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Date Created: 02/06/15

Optical Imaging and Aberrations Virendra N Mahajan Adjunct Protessor The Aerospace Corporation College ot Optical Sciences El Segundo Calitomia 90245 University ot Arizona Sill 336l783 virendranmahajanaeroorg Lecture 6 Imaging With Circular Pupils Aberrated PSFs 61 Lecture 6 Summary Aberrated PSFs Defocused System Axial Irradiance Depth of Focus Focused Beam Optimum Focusing Collimated beam PSFs for Rotationally Symmetric Aberrations o PSF Approximated by a Gaussian Function Symmetry Properties of Aberrated PSFs Rotationally Symmetric Aberrations 62 Aberrated PSF observed at a distance z 2 1 a mg a Ii i Z W JPFP6XpErp39ndrp Ftp Aoewlicbrpl lrpl s a Using normalized quantities p a r y P sexx2132 where irradiance is normalized by the aberrationfree central irradiance PexSex 2R2 we may write 1r 6i z 5 TEZ 2121 f f expilt1gtp 6exp 2 m5prcose epdpde Z 63 Radially symmetric aberration c1gtp6 p Radially symmetric PSF Irz expiCIgtpJ0rt prgt pdp 0 2 281 Defocus wave aberration Reference sphere centered at a distance R Observation plane at a distance z Defocus wave aberration is approximately equal to the difference in the sags of two spheres of radii of curvature z and R 1 1 1 2 2 Wd p 52 EM W Bd a1 i a2 Peak defocus wave aberration Z Ddrp 2TTEWAFP Bglp2 Peak defocus phase aberration 64 2 Bl E l l a2 EN 5 1 N a Fresnelnumber d xz R Z R We will use Bd to represent both the peak wave and phase aberrations and know from the context which peak value it is N is the Fresnel number of the pupil representing the number of half wave Fresnel zones as observed from the focus Irz is asymmetric about the Gaussian image plane zR for three reasons 1 Inversesquare law dependence on z 2 Asymmetry of defocus aberration BdR A BdR A 3 Argument of Jo depends on z 65 Fresnel number of a pupil as observed from a point A is the number of Fresnel39s half wave zones RNM2 R NMZ2 R2 a2 R2NRA R2a2 gt N Defocused PSF and Depth of Focus 1 2 2 0 Z z For small values of N z can be much different from R for Ed to achieve a significant value Hence the system has a large depth of focus and all three factors contribute to asymmetry PSF 1r z For large values of N Bd becomes significant even for small differences between z and R Hence the system has a small depth of focus Let A z R and since zzR he 2 z R 2 zR 2 R 8F2 BdRA 13 BdR A IrR A IrR A Thus for large values of N the defocused PSF is symmetric about the Gaussian image plane 67 Axial irradiance at a distance z when beam is focused at a distance R w 6 1 f expinpzpdp 0 2 S 284 2 SM Bd N51gt Bd2 Z o Axial irradiance at a distance z when a beam is focused at a distance z is equal to Rz2 in units of PexSexAZRZ Hence S represents the Strehl ratio S Axial irradiance at a distance z beam focused at R lt 1 Axial irradiance at a distance z beam focused at z It represents the effect of defocus as an aberration only since the effect of inversesquare law is the same in both cases o Axial irradiance at a certain point depends on two competing factors 1 Increase due to inversesquare law dependence for z lt R 2 Decrease due to defocus aberration ie nonconstructive interference of Huygens39 spherical wavelets 68 Minima of zero irradiance when Bd 2711 Le at z values given by 5 2n z If N s 2 then Rz 21 gt nonzero axial irradiance for z gt R Maxima are obtained by equating the derivative of 10 z with respect to z to zero tanBd2 RzBd2 z R A maximum of axial irradiance lies between two adjacent minima The principal maximum lies between the geometrical focus and the first minimum adjacent to it for which z lt R ie it lies at zp where R lt ltR 12N ZP 69 Axial irradiance for different values of Fresnel number N 20 20 20 D N1O N1OO 15 157 7 157 7 o d 1 10 10 10 7 7 05 05 057 7 00 00 00 i 00 05 10 15 08 09 10 11 12 gtzR gtzR ForN 1zp O6R WmaBd A3 B 2 R2 1 2 SUI d 68 gt 2378 332 sz L6 4 J19HR Large small depth of focus for small large N 610 20 100 O O 8 S 15 7 39 075 N 39 3 g 39 I l g5 10 7 050 T m 39 05 7 025 00 000 00 05 10 15 20 gt zR Figure 2 12 Axial irradiance S a IOz of a beam focused at a distance R with N1 compared with a beam focused at a distance z Minima at zR 13 15 17 etc The ratio of the two curves is the Strehl ratio S Variation of wave aberration in and Strehl ratio S with z are also shown 611 E m ZpIR 4 00 l l l l 0 gtN Figure 210 Variation of zo zp and 1zp with N where zo is the minimum distance of an axial point from the pupil so that IzoIR Principal maximum lies at a distance zp with an irradiance 1zp 612 20 100 m 075 025 000 PrC R is larger for 065 lt rC lt 110 Figure 214 a Focused and defocused irradiance distributions Ir and corresponding encircled powers Prc for a focused beam with N1 The irradiance is normalized by the focalpoint irradiance and encircled power is normalized by the total power Pex The units of rand rC are AF 613 100 085796 075 No D 050 025 000 gtzR Figure 214 b Encircled power Prc in a circle of radius rC for a focused beam with N 1 as a function of the axial distance z from the exit pupil Maximum PrC at z 06R for rC lt 03 and between zp and R for 03 s rC s 1 Location of diffraction focus depends on the criterion of its de nition 614 Optimum focusing of a beam on a target at a given distance z Where should we focus the beam for maximum central irradiance on the target Unnormalized axial irradiance Z PexSexSi1 le2 2 2 84b zz Bd 2 To determine the optimum focusing distance R we take derivative of Ii0z with respect to the R and equate the result to zero 610 z OgtR aR Z Thus a beam focused on the target yields maximum central irradiance on it even though a larger value occurs closer to the pupil for small values of N o Since the target distance is fixed the inversesquare law dependence is fixed Hence the irradiance on the target decreases due to nonconstructive interference of Huygens spherical wavelets if the beam is not focused on it 615 Central irradiance on a target 10 08 g 06 f 04 02 00 05 Figure 2 13 Central irradiance in units of PexSexAzzz on a target at a fixed 10 08 06 04 02 00 05 10 15 20 gt Rz 10 08 06 04 02 00 08 098296 12 distance z from the plane of the pupil when a beam is focused at various distances R The quantity Nz aZAz represents the Fresnel number of the pupil as observed from the target The dashed curves represent a Gaussian beam discussed in Chapter 4 Maximum central irradiance on the target is obtained when the beam is focused on it ie when R z 616 Collimated beam is equivalent to a beam focused at infinity Hence its results can be obtained from those for a focused beam by letting R gt 00 or Fresnel number N gt 0 Bd la2 TEN 1 XKZ R Z 2 E Si for R gtoo z M IOz 410 sin2rra22 z IO Pupil irradiance 6X Maxima IOz 410 at z aZx2n1 n 012 zlargesta2A The Fresnel number Nz aZAz observed from a distance z is 2n 1 which is odd Hence the irradiance is maximum z decreases as n increases Minima IOz 0 at z a22 n1 n o12 zlargesta22x Even number 2 n 1 of Fresnel zones hence zero irradiance 617 100 40 g s quot 8 quots 30 7 i 075 SR Sr 5 n A 20 i 050 T N a g tn42 10 i 025 l l l 00 39 000 000 025 050 075 100 IOz 4losin2rr8z zin units of DZx PexSexAzzz gtIE4z2 Figure 215 Axial irradiance of a collimated beam at a distance z in units of the farfield distance DZA normalized by the exit pupil irradiance 10 compared with that of a corresponding focused beam Ratio of the two 2 represents the Strehl ratio S At z 1 S 4sin2rc8rc4 095 618 Key observations o For z gt aZA the axial irradiance decreases monotonically to zero o For z 2 DZA it decreases approximately as 2 o At z DZA a collimated beam gives an axial irradiance that is 095 times the irradiance at this point if the beam were focused at it ie S 095 A collimated beam yields practically the same irradiance on a target lying in the far eld zzDZA of the exit pupil as a beam focused on it in other words beam focusing does not significantly increase the power concentration on the target z DZA is called the farfield distance of a pupil 619 Axial irradiance of an Aberrated Beam 12 l I I I I I I I yON1 10 I gt z gtzR Beam aberrated by an appropriate amount of spherical aberration yields higher axial irradiance at z values for which it balances the defocus aberration thus reducing its variance Similar results are obtained for astigmatism and a collimated beam gt V N Mahajan quotAxial irradiance of a focused beamquot J Opt Soc Am 22 1814 1823 2005 620 Radially symmetric PSF 2 1m 2R 2 2 R gt J expinp J0n prgt pdp 281 z z 0 Irradiance is in units of the aberrationfree central irradiance PexSexkZR2 and r is in units of AF ARD For a collimated beam BdSexgt z We write irradiance in units of PexSexkzz2 and r in units of AzD Thus 1 2 Irz 4 J expinp2J0Itprpdp 295 0 Irradiance distribution represents the Fresnel or the near led diffraction pattern of a circular pupil For z 2 DZA defocus aberration Bd 5 754 or MS is negligible 621 1 2 Irz 4 J J0nprpdp 2J1rtrrtr2 zz DZA 0 Pm 1J mc Jami o zz DzA is called the farfield condition and the corresponding irradiance distribution is called the Fraunhofer or the far eld diffraction pattern of a circular pupil The difference between the Fresnel and Fraunhofer diffraction patterns is the effect of the defocus aberration Hence Fresnel diffraction may be considered as defocused Fraunhofer diffraction For z gt aZx S gt 04 and 134 for rC lt03 is within 8 of pm siJgmC Jam For z 3 D26x S 2 01 and 134 for rC lt05 is within 5 622 PSFs for Rotationally Symmetric Aberrations AberrationFree PSF 1 2 2J TE 2 1r 4 f 10WPPdP 1 1 0 TE Aberrated PSF 1 2 1 2 Ir4 S4 g expilt1gtpJoWppdp g expilt1gtppdp 132p 52p2 1 01 Defocus aberration c134p 56p4 6p2 1 01 Primary spherical aberration CD6p W20p6 30p4 12p2 1 01 Secondary spherical aberration 138p 370p8 14Op6 90p4 20p2 1 01 Tertiary spherical aberration 623 30 m 00 02 04 06 08 10 Figure 216 Variation of aberration 1 with p For each aberration 01 1 o Since 0 S p S 1 the number of roots of the aberration 1 is n2 Le 13 has a value of zero at 112 different values of p 624 10 m 08 7 7 06 7 7 U 04 7 7 S 02 7 1 5 g 7 c 8m 31m 00 l l l 0 005 010 015 020 025 gt6w Figure 217 Strehl ratio as a functionzof wave aberration ow of rotationally symmetric aberrations Sm 1 o Z is the Marechal approximation The Gaussian expression Sg eXp O2D estimates the Strehl ratio better although it overestimates 32 S4 S6 58 32 has the maximum error whcih is 510 if S 2 038 625 Table 29 Standard deviation of rotationally symmetric aberrations for various Strehl ratios 01 Radians S 132 134 D6 138 10 0 0 0 0 09 03228 03233 03236 03238 08 04671 04687 04695 04702 07 05865 05899 05913 05928 06 06964 07025 07044 07071 05 08034 08139 08159 08203 04 09129 09303 09314 09383 03 10302 10598 10570 10680 02 11646 12203 12037 12216 01 13385 15014 13986 14312 626 1Ollllllll l N 4 4 A 3 D Q Q 4 m 08 7 06 7 04 7 7gt rS PrCS l 02 7 00 Figure 218 Defocused PSFs normalized to unity at the center and encircled power for various values of S Aberrationfree 2 1 curves are included for comparison A Gaussian approximation of the PSFs is also included 32 1 08 06 04 02 01 627 1390llll llll ll llll 0931 96 gt Kr PUC Figure 219 Aberrated PSFs and encircled power for CD4 for various Strehl ratios The PSF approximated by a Gaussian function and the corresponding encircled power are also included Irradiance inside the Airy disc decreases and increases in the bright ring surrounding it However the size of the central bright spot does not change 628 10 l l l l l l l l l l l i l l l l a quot 0 o m 7 W y l I 39 Iquot 00 7 a 7 7 D4 7 90 06 7 39 7 E g 7 7 T 04 7 l 7 lg 7 02 7 7 39039 7 x 7 g X 00 00 05 10 15 20 gt r rC Figure 220 Aberrated PSFs for CD4 normalized to unity at the center and corresponding normalized encircled power Inside the Airy disc the irradiance normalized by its central value is practically the same as for the aberrationfree case 629 Hence irradiance Ir and encircled power Prc within the Airy disc can be estimated easily by multiplying their aberrationfree values with the Strehl ratio PSF within the Airy disc is the Airy pattern multiplied by the Strehl ratio Ir S2J1rcrrcr2 Osrsl Encircled power within the Airy disc is the same as for the Airy pattern multiplied by the Strehl ratio Prc S1 Jgan J12rrrc Osrcsl V N Mahajan Aberrated point spread functions for rotationally symmetric aberrations Appl Opt 22 3035 3041 1983 Balanced aberrations as in Zernike polynomials are considered in this paper 8 Szapiel Aberrationv3riancebased formula for calculating pointspread functions rotationally symmetric aberrations J Opt Soc Am 25 244 251 1986 This paper extends the work to unbalanced aberrations 630 10 l l l l l l l l l l l l l l l A 8 i x39 1 4r 7 Pgrc Rx xx 7 08 i i g 06 i i E g 7 16 T 04 7 i 02 i i gr 00 39 39 39 39 39 3939 39 00 05 10 15 20 gt r rc Figure 221 Aberrated PSFs for CD6 normalized to unity at the center and corresponding normalized encircled power o Again the size of the central bright spot does not change and the irradiance inside the Airy disc normalized by its central value is practically the same as for the aberrationfree case 631 10 l l l l l l l l l l l l l l l l 8 o I quot V o P900 oquot 08 7 gt10 06 7 E Q 7 98 T 04 7 gr 02 7 00 00 05 10 Figure 222 Aberrated PSFs for CD8 normalized to unity at the center and corresponding normalized encircled power Similar result for 138 as for CD4 and 136 632 PSF approximated by a Gaussian function Strehl ratio 5g 6Xpi021 2124 Percent error 1001 SgS is lt 10 percent as long as S i 03 It is negative for the aberrations considered PSF A 2D Gaussian function having a central value of unity and a total power of unity is given by Igr expinr22 O s r51 Gaussian PSF does not exhibit diffraction rings Encircled power Pgrc 1 6Xp rch2 Osrcsl 1390 i i i i i i i i i i ahquot1 i i i i go 039 a 7 Pgrc 08 7 7 1 Jame J are A0 06 7 i E T 04 7 i 7 201 nrrrr 2 7 02 7 i 7 gr 7 00 i i i i i i i i 00 05 10 15 20 Figure 223 Aberrationfree PSF encircled power and their Gaussian approximations 634 Gaussian approximation overestimates PSF especially Prc for rC 2 06 For small values of r the aberrated PSF and the corresponding encircled power are simply scaled by the Strehl ratio Hence their Gaussian approximations may be written IgrS Sexp Icr22 Osrsl PgrCS S1 exp an22 O s rC 51 Substituting for the Strehl ratio we may write I r0q exp O2D exp nr22 Osrsl 2128a Pgrc0q exp O2Dl exp an22 Osrcsl 2128b Equation 2128a describes the Gaussian model for the aberrated PSF 635 Symmetry Properties of Aberrated PSFs 1 Symmetry in the sign of the aberration coefficient 2 Symmetry in the Gaussian image plane 3 Symmetry about the Gaussian image plane 4 Symmetry of axial irradiance If the Fresnel number N is small the PSF about the Gaussian image plane will be asymmetric due to the large depth of focus and the associated inversesquare law dependence on the distance z of the observation plane from the pupil plane ie the PSFs in planes located symmetrically about the Gaussian image plane will not be the same We assume from here on that the Fresnel number is large so that the depth of focus is small and the effect of the variation of the inversesquare law dependence in the vicinity of the Gaussian image plane is negligible 636 Rotationally symmetric aberrations 2 1r 4 expilt1gtpJoWppdp o PSF in the Gaussian image plane is radially symmetric and of course independent of the sign of the aberration Defocused PSF 139 Bd92 1 2 2 IrBd 4 g expinp J0Irrppdp 1r 3d Ir Bd Defocused PSF is symmetric about the Gaussian image plane 637 100 m 075 050 025 000 Figure 224 Defocused PSFs Bd represents the peak value of defocus aberration in units of x The curves for Ed 12 and 3 have been multiplied by ten 638 Defocused PSFs Bd0 Bd05 Bd1 Bd2 Bd3 639 The PSF in the Gaussian image plane for a radially symmetric aberration does not change if the sign of the aberration coefficient is changed since Irlt1gtp Ir ltIgtp Hence the sign of the aberration cannot be determined from its PSF 1 2 11 Bd 4 g XPiCDPBd92JOWPPdP o PSF is not symmetric about the Gaussian image plane since 1r 3d Ir Bd However Ir c139 Bd r c139 Bd Thus aberrated PSFs in symmetrically defocused image planes for two aberrations of the same magnitude but opposite signs are identical 640 Spherical Aberration c1gtp Asp4de2 1 4 2 f 6XplAsp de pdp 0 IOBd 4 10 m 08 06 gt0 Bd 04 02 00 r 39 39 6 5 4 3 2 1 0 1 2 3 Figure 225 Axial irradiance of a beam aberrated by spherical aberration AS in units of x The arrow on the curve for AS 3 is located at the point of symmetry The axial irradiance is shown in a different form in Figure 29 Axial irradiance is symmetric about the point Bd AS ie the point with respect to which the aberration variance is minimum see Lecture 4 p 419 641 100 o I J1 I 03gt II J 01 091696 Balanced Spherical W Asp4 92 075 025 000 Figure 226a PSFs for balanced spherical aberration Various values of AS and the defocused image planes Bd AS corresponding to minimum aberration variance Ed and AS are in units of A and r is in units of AF 642 020 97 1 00 39Aberration free W03 Ad A 1 016 1230 Spherical and Defocus Wp A534 Bap A 1 s I 075 7quot 012 c 008 gtlr Bd J 03925 004 000 000 Figure 226b and c PSFs for spherical aberration Fixed value of AS and Gaussian minimum variance least cofusion and marginal image planes Ed and AS are in units of A and r is in units of AF The curves for Ed O and 2 have been multiplied by 10 in b and 5 in c The factor of 5 for these curves in c can be ignored by using the righthand scale Axial irradiance is symmetric about the point Bd AS Hence for AS 1 10 Bd 0 10 Bd 2 and 10 3 05 10 3 15 643 PSFs for spherical aberration As 1 A Gaussian Bd 0 Min Variance Bd 1 Least Confusion Bd 15 Marginal Bd 2 PSFs in the Gaussian and marginal image planes are different from each other although their central irradiances are equal 644 PSFs for balanced spherical aberration Asp4 p2 Home Work 6 61 Plot and compare the axial irradiance of a beam focused at a distance DZA with that of a corresponding collimated beam Determine the location and the value of its principal maximum 62 Calculate the Strehl ratio for spherical aberration Asp4 for AS 025 05 and 1 A in the Gaussian and minimumaberrationvariance image planes 646

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