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Lens Design OPTI 517 Seidel Aberration Coefficients Lens Design OPTI 517 Seidel aberration coefficients Prof. Jose Sasian OPTI 517 Fourth-order terms 2 2 W H, W040 W131H W222 H 2 W220 H H W311H H H W400 H H Spherical aberration Coma Astigmatism (cylindrical aberration!) Field curvature Distortion Piston Prof. Jose Sasian OPTI 517 Coordinate system Prof. Jose Sasian OPTI 517 Spherical aberration h 2 h 4 Z 2r 8r 3 u h y1 y 2r W n'[PB'] n'[PA' ] n'[PB'] n[PB] We have a spherical surface of radius of curvature r, a ray intersecting the surface at point P, intersecting the reference sphere at B’, intersecting the wavefront in object space at B and in image space at A’, and passing in image space by the point Q’’ in the optical axis. The reference sphere in object space is centered at Q and in image space is centered at Q’ Question: how do we draw the first order Prof. Jose Sasian OPTI 517 marginal ray in image space? [PQ]2 s Z 2 h 2 s 2 2sZ Z 2 h 2 2 4 4 2 h h h h 2s 3 2 2r 8r 4r [PB] [OQ] [PQ] s 2 1 2 2 s h 2 1 1 h 4 1 1 h 4 1 1 2 2 s r 8r s r 8s s r 2 4 2 h s h s s 1 1 1 u s 2 r 4r 2 s 2 r h y1 y 2r Prof. Jose Sasian OPTI 517 Spherical aberration ' ' [PB] [OQ] [PQ] [PB'] [OQ ] [PQ ] 2 2 2 4 y 2 u 1 1 y 4 1 1 y u 1 1 y 1 1 1 y 1 y ' 2 ' 2 2 2r s r 8r s r 2 2r s r 8r s r 4 2 4 2 y 1 1 y 1 1 8s ' s ' r 8s s r W n '[PB'] n[PB] y 2 u 1 1 1 1 1 y n ' n ' 2 r s r s r y 4 1 1 1 1 n ' n 2 ' 8r s r s r 2 2 y 4 n ' 1 1 n 1 1 ' ' 8 s s r s s r Prof. Jose Sasian OPTI 517 Spherical aberration u y / s u ' y / s ' A 0 A ni n y// s y r 1 2 u W040 A y 8 n Prof. Jose Sasian OPTI 517 Petzval field curvature W220 P Surface of radius r Image Object plane plane S’ ss’ Stop S y0 We locate the aperture stop at the center of curvature of the spherical surface. With being the object height, then the inverse of the distance along the chief ray from the off-axis object point to the surface is: Prof. Jose Sasian OPTI 517 Petzval field curvature 11 1 S 222 y rrsy 0 0 rrs 1 2 rs 11 1 y 2 1 y 2 rrs 1 0 s 1 0 2 2 rss 2 rs 11yy22 11 00 11 srsss22 11 2 rs sr 2 2 111uuy0 Ж 1 22 sirssynri22 111u'2Ж ''2'2' Prof. Jose Sasian Ssy 2 nri OPTI 517 Petzval field curvature By inserting the 1/s in the quadratic term of W 2 ' y ' 11 11 W n [PB'] n[PB] Wn n 2 Sr' Sr 2 y u ' 1 1 1 1 22 1 yn n ''11ЖЖ ' nu nu 2 r s r s r '2 ' 2 44nri nri 4 y ' 1 1 1 1 2 n n 1 Ж ' 8r 2 s ' r s r uu 4 Ar 4 ' 2 2 y n 1 1 n 1 1 11Ж 2 ' ' 8 s s r s s r 4 rn 6 WO220P A 0 11Ж 2 W220P 4 rn Prof. Jose Sasian OPTI 517 Aberration function at CC Exit pupil Surface Chief ray CC 2 W H, W040 W220P H H Prof. Jose Sasian OPTI 517 New aberration function upon stop shifting WHold , yyyH WHnew , old shift old old yold shift H yold New stop Old stop location location yold WHnew,, WH old shift WH old , H yold Expansion about the new chief ray height 2 W H, W040 W220P H H yE A shift H H yE A Prof. Jose Sasian OPTI 517 Quadratic term A A shift shift H H A A 2 A A 2 H H H A A W220P H H W220P H H 2 A A 2 2 W220P H H H W220P H H A A Prof. Jose Sasian OPTI 517 Quartic term 2 2 A A 2 H H H shift shift A A 2 A A 2 H H H A A 2 A A 2 4 H 4 H A A 3 4 A A A 2 2 H H 4 H H H H H A A A Prof. Jose Sasian OPTI 517 Graphical view yE Old Pupil New pupil Prof. Jose Sasian OPTI 517 All terms 2 2 W H, W040 W131H W222 H 2 W220 H H W311H H H W400 H H W040 W040 A W 4 W 131 A 040 2 A W222 4 W040 A 2 A W220 2 W040 W220P A 3 A A W311 4 W040 2 W220P A A 4 2 A A Piston W400 W040 W220P Prof. Jose Sasian A A OPTI 517 For as system of two surfaces Exit pupil becomes entrance pupil for next surface. Exit pupil for surface J Prof. Jose Sasian Entrance pupil for surface J+1 OPTI 517 Fourth-order contributions For a given system we add the OPD contributed by each surface. An issue is that because of pupil aberrations we do not know the pupil coordinates of the ray at previous exit pupils. However, the error in knowing the ray pupil coordinates leads to six-order aberrations. To fourth-order we do not have other fourth-order terms to account for. We are assuming we do not have second-order aberrations. Otherwise these will generate Prof. Jose Sasianother fourth-order terms. OPTI 517 Order of Error • We know the ray heights to first-order • There is an error on the ray heights y and y-bar of third order • If the third order error is accounted for, it leads to sixth-order terms yy y3 yy44 y 6 446 W040y W 040 y W 040 y Prof. Jose Sasian OPTI 517 Conclusion Assume no second order terms in the aberration functions of each surface Then for a system of surfaces the fourth- order coefficients are the sum of the coefficients contributed by each surface There are no fourth-order extrinsic terms from previous aberration in the system Prof. Jose Sasian OPTI 517 Prof. Jose Sasian OPTI 517 Prof. Jose Sasian OPTI 517 Example : Cooke triplet lens Prof. Jose Sasian OPTI 517 Wave coefficients In waves at 0.000587 mm W040 W131 W222 W220 W311 W020 W111 1.0000 5.8831 16.4912 11.5569 37.9424 61.2785 -10.4705 -14.6753 2.0000 4.6978 -50.6336 136.4338 -0.1227 -366.9639 -6.5847 35.4854 3.0000 -22.3709 117.1708 -153.4247 -36.2070 295.7159 18.4586 -48.3398 4.0000 -9.6490 -65.3484 -110.6438 -40.4402 -324.2767 14.8117 50.1564 5.0000 1.6894 24.1504 86.3110 10.3704 382.5921 -4.7481 -33.9387 6.0000 22.0849 -42.6064 20.5492 43.9016 -52.2587 -12.3359 11.8993 Totals 2.3352 -0.7760 -9.2175 15.4444 -3.9128 -0.8690 0.5872 Prof. Jose Sasian OPTI 517 Prescription F/4, f=50 mm; FOV +/- 20 degrees Stop at surface 3 SURFACE DATA SUMMARY: Surf Type Radius Thickness Glass Diameter Conic Comment OBJ STANDARD Infinity Infinity 0 0 1 STANDARD 23.713 4.831 LAK9 20.26679 0 2 STANDARD 7331.288 5.86 18.17704 0 3 STANDARD -24.456 0.975 SF5 9.598584 0 4 STANDARD 21.896 4.822 9.909458 0 5 STANDARD 86.759 3.127 LAK9 16.07715 0 6 STANDARD -20.4942 -10.0135 16.69285 0 STO STANDARD Infinity 51.25 12.90717 0 IMA STANDARD Infinity 36.46158 0 Prof. Jose Sasian OPTI 517 Summary • Spherical aberration • Stop shifting • Off-axis aberrations • Entrance/exit pupil concatenation • Seidel sums • Actual coefficients computation • Information acquired Prof. Jose Sasian OPTI 517.
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