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Zernike Polynomials and Beyond Zernike Polynomials and Beyond "Introduction to Aberrations" ExP x S W R P(xg, 0) z O OA P0 y zg Virendra N. Mahajan The Aerospace Corporation Adjunct Professor El Segundo, California 90245 College of Optical Sciences (310) 336-1783 University of Arizona [email protected] Zernike Lecture 12 April 12 1 References 1. V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd ed. (2011, SPIE Press). 2. M. Born and E. Wolf, Principles of Optics, 7th edition (Cambridge, New York, 1999), Chapter IX, "The Diffraction Theory of Aberrations," pp. 517-553. 3. V. N. Mahajan, “Zernike polynomials and aberration balancing,” SPIE Proc. Vol 5173, 1-17 (2003). 4. V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, D. Malacara ed., 3rd edition (Wiley , 2006). 5. V. N. Mahajan and W. H. Swantner, “Seidel coefficients in optical testing,” Asian J. Phys. 15, 203–209 (2006). 2 Summary • Airy Pattern • Aberrated PSF • Strehl Ratio • Balanced Aberrations • Zernike Circle Polynomials • Seidel Coefficients from Zernike Coefficients • Orthonormal Polynomials for Noncircular Pupils • Annular Pupils and Zernike Annular Polynomials • Use of Circle Polynomials for Noncircular Pupils 3 Airy Pattern • Central bright spot of radius 1.22, called the Airy disc, containing 83.8% of the total power, surrounded by diffraction rings 2 È2Jr1(p ) ˘ 2 2 Ir( ) = Í ˙ , Pr( ccc) =-1 J0 (pp r) - J1 ( r) Î pr ˚ 4 How is the Airy pattern affected when a system is aberrated? • Aberration-free central value for total power P: PSl22 R • Aberration decreases the central value. • Energy flows from the Airy disc into the rings. • Peak does not necessarily lie at the center. • For a small aberration, the relative central value, called the Strehl ratio, can be estimated from its standard deviation.* We illustrate reduction in central irradiance of defocused images, and shift in peak of coma-aberrated images. * Many optics people (incorrectly) use the term rms wavefront error when they actually mean the standard deviation. 5 Defocused PSFs 1.00 Defocus Bd = 0 2 0917-96 W(r) = Bdr 0.75 0.25 ) d 0.50 I (r; B I (r; 1 0.5 10 ¥ 0.25 3 10 ¥ 0.64 2 10 ¥ 0.00 0123 r s. Figure 2-24. Defocused PSF curves for Bd = 1, 2, and 3 (in units of l) have a central value of zero, and have been multiplied by ten for clarity. • Central value of PSF decreases as aberration increases. 6 Defocused PSFs Bd = 0 Bd = 0.5 Bd = 1 Bd = 1.5 Bd = 2 Bd = 3 • Dark spot at the center obtained for integral number of waves of defocus. 7 Coma PSFs 1.00 0.25 Ac = 0 Coma W(r,q) = A r3 cosq c 0915-96 qi = 0 0.75 0.5 ) c ; A i 1 0.50 (a) I (x 0.25 2 3 0.00 – 2.5 0.0 2.5 5.0 xi 3 Figure 2-31. Central profiles of PSFs for coma Acrqcos along the xi axis, where the peak value Ac of the aberration is in units of l. • Central value of PSF decreases and its peak shifts. 8 Coma PSFs Ac = 0.25 Ac = 0.5 Ac = 1 Ac = 2 Ac = 3 3 Figure 2-37. PSFs aberrated by coma Acr cosq are symmetric about the xi (horizontal) axis, where Ac is in units of l. 9 Strehl Ratio* • For a given total power, the central value of the image of a point object is maximum for uniform amplitude and phase of the wave at the exit pupil. • Strehl ratio is the ratio of central irradiances with and without phase/amplitude variations. Aberrated central irraidance I()0 S ==F £ 1 Aberration - free central irraidance I()0 F=0 • Used for aberration tolerancing. • Apodization or amplitude variations also reduce the central value.** *K. Strehl, “Ueber Luftschlieren und Zonenfehler,” Zeitschrift fur Instrumentenkunde 22, 213– 217 (1902). **V. N. Mahajan, "Luneburg apodization problem I," Opt. Lett. 5, 267-269 (1980). 10 How do we determine the Strehl ratio? rr • Pupil function: Pr()pp= A0 exp[] iF() r (Uniform amplitude A0) 2 rrrrr12Û Ê pi ˆ • Aberrated PSF: Ir()ippip= Ù Pr()exp Á - rrdr◊ ˜ l22R ı Ë lR ¯ PS • Aberration-free central value: I()0 = F=0 l22R rr2 I()0 exp[]irdrF()pp S F Ú = = r 2 I()0 F=0 ()Ú drp V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and annular pupils,” J. Opt. Soc. Am. 7, 1258–1266 (1983), 10, 2092 (1993). V. N. Mahajan, “Strehl ratio for primary aberrations in terms of their aberration variance,” J. Opt. Soc. Am. 73, 240–241 (1983). 11 Approximate Expression for Strehl Ratio Aberrated central irraidance I(0) S = = F Aberration- free central irraidance I(0)F=0 rr2 Úexp[]irdrF( pp) = r 2 ( Ú drp) 1 2p 2 ÚÚexp[]iddF()rq , r r q 0 0 = Ê1 2p ˆ 2 Á ÚÚrrqdd˜ Ë0 0 ¯ 1 1 2p 2 idd = 2 ÚÚexp[]F()rq , r r q p 0 0 2 2 = exp(iF) =-exp[]i(FF) 12 • By expanding the exponent and neglecting terms higher than the quadratic, we obtain approximate expressions for a small aberration: 2 Si=-exp[]()FF 2 1 2 =+1 i()FF - --() FF+ .... 2 2 1 2 ~ 1--()FF 2 2 Ê 1 ˆ =-1 s2 (independent of the shape of the pupil) Ë 2 F¯ 2 2 2 2 2 2 sF =-()FF =+-FF2 FF=-FF 2 2 Standard deviation: sF , Variance: sF , rmsF ∫ F 13 Various approximate expressions 2 1 1 S ~ 121- sss2 =-24 + =+S s4 (Maréchal) 1 ( FFF) 4 2 4 F 2 S2 ~ 1 -sF (Nijboer) 1 1 S ~~exp - sss2241 -+ =+S s4 (Mahajan) 3 ( FFF) 2 1 4 F 222 sF =-FF (Phase aberration variance) 1 2p nn1 <>=FF(rq, ) rdd r q p ÚÚ 0 0 l l S ~ 08. fi=s 02. or s ==02. 2 F w 2p 14. 05 • Thus, S = 08. for sw = l 14, regardless of the type of the aberration. 14 Table 2-4. Standard deviation of and aberration tolerance for Seidel or primary aberrations. Aberration FFr(rq,q) sF ASi for= 0.8 2 AA Spherical A r4 ss= l 419. s 35 335. AA Coma A rq3 cos cc= l 496. c 22 283. A Astigmatism A rq22cos a l 351. a 4 BB Defocus B r2 dd= l 406. d 23 346. B Tilt B rqcos t l 703. t 2 • Peak aberration Ai for a given Strehl ratio and, therefore, the same standard deviation, is different for a different aberration. 15 Balanced aberrations: • Since the Strehl ratio is higher for a smaller aberration variance, we mix a given aberration with one or more lower-order aberrations to reduce its variance. • The process of reducing the variance of an aberration by mixing it with an other is called aberration balancing. • Consider, for example, balancing of spherical aberration with defocus: 42 Fr( ) =+ABs rd r • We calculate its variance as follows: 1 2p 1 42 ABsd F= ÚÚ()ABddsdrrrrq+ =+ p 0 0 32 1 2p 2 1 422 F = ÚÚ()ABsdrrrrq+ dd p 0 0 16 1 28 24 6 = 22Ú (ABs rr++d ABds d rrr) 0 ABAB22 =++s d s d [ F2 is the rms value] 53 2 4ABAB22 \ s22=-=++FF2 s d s d F 45 12 6 • Value of Bd for minimum variance: 2 ∂sF 1 0 = =+(BAd s ) fi=- Bd As ∂Bd 6 • Balanced spherical aberration: 2 42 2 A F (rrr) =-A , s = s bs s( ) Fbs 180 • Without balancing with defocus: 2 4 2 4A F (rr) = A , s = s ss Fs 45 17 • Thus, balancing of spherical aberration with defocus reduces its variance by a factor of 16, or the standard deviation by a factor of 4. Hence, the aberration tolerance for a given Strehl ratio increases by a factor of 4. • For example, S = 08. is obtained in the Gaussian image plane for As =l 4. However, the same Strehl ratio is obtained for As = 1 l in a slightly defocused image plane at a distance z such that Bd =-l 4. 2 • Since the longitudinal defocus D=zR - =-8 FBd, the defocused point of 2 maximum irradiance is located at (008,,FAs ) and it is called the diffraction focus. • Similarly, we can balance coma with tilt, and astigmatism with defocus and determine the diffraction focus. A wavefront tilt implies that the diffraction focus lies in the Gaussian image plane. 18 Balancing of Seidel or Primary Aberrations Spherical: r4 , coma: rq3 cos , astigmatism: rq22cos Balanced FFr(rq,q) Diffraction sF ASi for= 0.8 Aberration Focus 42 2 AAss Spherical As rr- 008,,FA = 0.955 l ( ) ( s ) 6 5 13. 2 3 AAcc Coma Ac rr-23cos q(4300FA ,,) = 0.604 l ( ) c 62 849. AA Astigmatism A rq22cos -12 2 aa a ( ) (004,,FAa ) = 0.349 l 2 26 490. = (Aa 22)rqcos • sw is reduced by a factor of 4, 3, and 1.225 for spherical, coma, and astigmatism, respectively. • Aberration tolerance for a given Strehl ratio increases by this factor. 19 1.0 0858-96 0.8 0.6 S 0.4 S3 0.2 S2 S1 0.0 0.00 0.05 0.10 0.15 0.20 0.25 sw Figure 2-8. Strehl ratio as a function of standard deviation sw in units of l. • S3 gives a much better fit than S1 or S2 over a larger range of sw 20 Strehl ratio for small and large aberrations • For a small aberration, an optimally balanced aberration with minimum variance yields the maximum Strehl ratio.
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