Generalized ray tracing, caustic surfaces, generalized bending, and the construction of a novel merit function for optical design

Rong-Seng Chang; Orestes N. Stavroudis

doi:10.1364/JOSA.70.000976

Journal of the Optical Society of America
Vol. 70,
Issue 8,
pp. 976-985
(1980)
•https://doi.org/10.1364/JOSA.70.000976

Generalized ray tracing, caustic surfaces, generalized bending, and the construction of a novel merit function for optical design

Rong-Seng Chang and Orestes N. Stavroudis

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Rong-Seng Chang and Orestes N. Stavroudis, "Generalized ray tracing, caustic surfaces, generalized bending, and the construction of a novel merit function for optical design," J. Opt. Soc. Am. 70, 976-985 (1980)

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Abstract

Generalized ray tracing is an algorithm for calculating the geometrical parameters of a wave front in the neighborhood of a traced ray. These calculations are applied, surface by surface, for each traced ray, to an optical system being designed. These calculations determine the two points of contact of each traced ray with the two sheets of a caustic surface. The caustic surfaces are, in fact, aberrated three-dimensional images of object points and therefore contain all information on the geometrical aberrations of the subject lens. Generalized bending is a procedure in which the curvature of a pair of adjacent spherical refracting surfaces, their separation, and the distance to the next succeeding or next preceding surface may be changed so that any paraxial ray is left invariant except at the two affected surfaces. In this study we show that the displacement of a caustic point caused by a generalized bending is in essentially a straight line, that the direction of the displacement is determined by which refracting surfaces are selected, and that the magnitude of the displacement is proportional to the logarithm of the bending parameter. This suggests that caustic surfaces can be used as a merit function in the optical design process, that the merit functions can be calculated by means of generalized ray tracing, and that generalized bending provides an effective means of optimizing the design when included in a feedback loop.

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Curvature (mm⁻¹)		Index	Thickness (mm)
Entrance	0.0
pupil		1.0	−15.999 51
sphere	2.695418 × 10⁻²	1.59015	7.4
plane	0.0	1.0	9.9
sphere	−1.81818 ×10⁻²	1.623770	2.6
sphere	2.724795×10⁻²	1 0	8.7
sphere	7.874016 × 10⁻³	1.621760	6.4
sphere	−2.314815 × 10−2	1.0	−21.74805
Exit pupil	0.0	1.0	103.2481
Image plane	0.0

Ray	Entrance pupil coordinate y (mm)
1	7.0
2	4.0
3	10.0

Bending parameter	Displacement
	Ray 1		Ray 2
	Direction (rad)	Distance (mm)	Direction (rad)	Distance (mm)
7.400	−0.074 307 19	0.260 282 34	−0.037 007 90	0.065 671 23
6.727	−0.074 309 27	0.233 152 85	−0.037 008 04	0.058 799 93
6.116	−0.074 311 32	0.208 458 77	−0.037 008 19	0.052 550 88
5.560	−0.074 313 37	0.185 983 18	−0.037 008 34	0.046 867 72
5.054	−0.074 315 45	0.165 528 07	−0.037 008 51	0.041 699 16
4.595	−0.074 317 56	0.146 912 67	−0.037 008 68	0.036 998 51
4.177	−0.074 319 73	0.129 972 05	−0.037 008 87	0.032 723 29
3.797	−0.074 321 97	0.114 555 78	−0.037 009 08	0.028 834 85
3.452	−0.074 324 32	0.100 526 64	−0.037 009 30	0.025 298 00
3.138	−0.074 326 78	0.087 759 53	−0.037 009 54	0.022 080 73
2.853	−0.074 329 39	0.076 140 40	−0.037 009 80	0.019 153 91
2.594	−0.074 332 16	0.065 565 27	−0.037 010 08	0.016 491 03
2.358	−0.074 335 12	0.055 939 37	−0.037 010 39	0.014 067 95
2.144	−0.074 338 30	0.047 176 31	−0.037 010 73	0.011 862 71
1.949	−0.074 341 73	0.039 197 36	−0.037 011 11	0.009 855 31
1.772	−0.074 345 44	0.031 930 73	−0.037 011 51	0.008 027 54
1.611	−0.074 349 46	0.025 310 98	−0.037 011 96	0.006 362 80
1.464	−0.074 353 82	0.019 278 41	−0.037 012 45	0.004 845 99
1.331	−0.074 358 57	0.013 778 59	−0.037 012 99	0.003 463 33
1.210	−0.074 363 75	0.008 761 84	−0.037 013 58	0.002 202 25
1.100	−0.074 369 40	0.004 182 79	−0.037 01422	0.001 051 30
1.0
0.900	−0.074 38306	0.004 208 26	−0.037 015 80	0.001 057 68
0.810	−0.074 39133	0.008 025 14	−0.037 016 76	0.002 017 03
0.729	−0.074 40046	0.011 492 97	−0.037 017 82	0.002 888 72
0.656	−0.074 41053	0.014 650 24	−0.037 018 99	0.003 682 48
0.590	−0.074 42163	0.017 532 00	−0.037 020 29	0.004 407 14
0.531	−0.074 43386	0.020 170 21	−0.037 021 73	0.005 070 81
0.478	−0.074 44729	0.022 594 12	−0.037 023 30	0.005 680 86
0.430	−0.074 46202	0.024 830 47	−0.037 025 04	0.006244 09
0.387	−0.074 47811	0.026 903 78	−0.037 026 94	0.006 766 74
0.349	−0.074 49561	0.028 836 52	−0.037 029 01	0.007 254 55
0.314	−0.074 51453	0.030 649 16	−0.037 031 26	0.007 712 80
0.282	−0.074 53480	0.032 360 26	−0.037 033 68	0.008 146 35
0.254	−0.074 55629	0.033 986 34	−0.037 036 25	0.008 559 59
0.229	−0.074 57869	0.035 541 63	−0.037 038 96	0.008 956 46
0.206	−0.074 60148	0.037 037 56	−0.037 041 75	0.009 340 34
0.185	−0.074 62381	0.038 482 02	−0.037 044 54	0.009 713 94
0.167	−0.074 64431	0.039 878 04	−0.037 047 19	0.010 079 09
0.150	−0.074 66084	0.041 221 85	−0.037 049 49	0.010 436 42
0.135	−0.074 67004	0.042 499 96	−0.037 051 11	0.010 784 88
0.122	−0.074 66662	0.043 654 67	−0.037 051 57	0.011 121 05

Bending parameter	Displacement
	Ray 1		Ray 2
	Direction (rad)	Distance (mm)	Direction (rad)	Distance (mm)
7.400	−0.073 568 14	0.021 146 63	−0.036 923 47	0.005 494 02
6.727	−0.073 495 72	0.017 232 88	−0.036 915 65	0.004 498 49
6.116	−0.0733 943 6	0.013 707 49	−0.036 904 84	0.003 601 87
5.560	−0.073 244 56	0.010 539 77	−0.036 889 13	0.002 796 30
5.054	−0.0730 046 5	0.007 702 45	−0.036 864 61	0.002 074 74
4.595	−0.072 566 94	0.005 171 50	−0.036 821 72	0.001 431 05
4.177	−0.071 542 35	0.002 926 02	−0.036 729 67	0.000 859 82
3.797	−0.066 614 37	0.000 948 14	−0.036 403 69	0.000 356 47
3.452	−0.082 802 77	0.000 777 11	−0.039 356 52	0.000 082 87
3.138	−0.0769 628 3	0.002 261 44	−0.037 391 00	0.000 461 27
2.853	−0.0758 712 7	0.003 513 66	−0.037 214 51	0.000 781 04
2.594	−0.075 422 54	0.004 539 18	−0.037 150 02	0.001 043 65
2.358	−0.075 186 67	0.005 339 81	−0.037 117 73	0.001 249 70
2.144	−0.075 048 49	0.005 913 32	−0.037 099 27	0.001 398 78
1.949	−0.074 964 28	0.006 252 89	−0.037 088 15	0.001 489 37
1.772	−0.074 914 02	0.006 346 24	−0.037 081 51	0.001 518 63
1.611	−0.074 887 41	0.006 174 59	−0.037 077 92	0.001 482 17
1.464	−0.074 878 80	0.005 711 22	−0.037 076 62	0.001 373 71
1.331	−0.074 88501	0.004 919 51	−0.037 077 18	0.001 184 66
1.210	−0.074 904 35	0.003 750 40	−0.037 079 36	0.000 903 54
1.100	−0.074 936 02	0.002 138 99	−0.037 083 06	0.000 515 23
1.0
0.900	−0.075 043 16	0.003 114 22	−0.037 095 79	0.000 748 36
0.810	−0.075 123 01	0.007 211 68	−0.037 105 35	0.001 728 88
0.729	−0.075 221 28	0.012 564 23	−0.037 117 15	0.003 002 28
0.656	−0.075 340 41	0.019 540 97	−0.037 131 50.	0.004 649 55
0.590	−0.075 483 55	0.028 646 04	−0.037 148 78	0.006 778 93
0.531	−0.075 654 69	0.040 573 27	−0.037 169 49	0.009 535 62
0.478	−0.075 858 77	0.056 286 38	−0.037 194 25	0.013 115 10
0.430	−0.076 101 90	0.077 138 38	−0.037 223 82	0.017 781 67
0.387	−0.076 391 59	0.105 053 23	−0.037 259 14	0.023 894 47
0.349	−0.076 737 07	0.142 808 32	−0.037 301 38	0.031 944 16
0.314	−0.077 149 68	0.194 485 94	−0.037 351 96	0.042 605 29
0.282	−0.077 643 34	0.266 218 26	−0.037 412 64	0.056 811 74
0.254	−0.078 351 2	0.367 648 1	−0.037 485 58	0.075 867 10
0.229	−0.078 946 02	0.513 307 48	−0.037 573 45	0.101 608 65
0.206	−0.079 801 87	0.728 817 77	−0.037 679 51	0.136 655 65
0.185	−0.080 834 68	1.057 994 53	−0.037 807 75	0.184 794 29
0.167	−0.082 084 38	1.583 879 03	−0.037 963 09	0.251 591 02
0.150	−0.836 015 6	2.480 091 24	−0.038 151 54	0.345 402 84
0.135	−0.085 452 04	4.170 252 91	−0.038 380 51	0.479 110 09
0.122	−0.087 725 32	7.996 825 01	−0.038 659 03	0.673 238 69

Surface	Bending factor	Displacement
Surface	Bending factor	Ray 1	Ray 2	Ray 3
4	1.1	−1.57235 ×10⁻³	−3.804 129 × 10⁻⁴	−1.262 466 × 10⁻³
2	1.69	−1.997813 × 10⁻⁵	−4.083 832 ×10⁻⁶	−1.541 791 × 10⁻⁵
combined		−1.592324 × 10⁻³	−3.844 963 × 10⁻⁴	−1.277 880 × 10⁻³
difference		−4.516551 × 10⁻⁹	−4.092 726 × 10⁻¹⁰	−3.317 837 × 10⁻⁹
4	1.1	−1.572350 × 10⁻³	−3.804 129 × 10⁻⁴	−1.262 466 × 10⁻³
2	1.52	−1.505603 × 10⁻⁵	−3.077 672 × 10⁻⁶	−1.161 932 × 10⁻⁵
combined		−1.587403 ×10⁻³	−3.834 902 × 10⁻⁴	−1.274 082 × 10⁻³
difference		−3.401510 × 10⁻⁹	−3.083 187 ×10⁻¹⁰	−2.500 656 ×10⁻⁹
4	1.1	−1.572350 ×10⁻³	−3.804 129 × 10⁻⁴	−1.262 466 × 10⁻³
2	1.37	−1.071297 × 10⁻⁵	−2.189 884 ×10⁻⁶	−8.267 612 × 10⁻⁶
combined		−1.583061 × 10⁻³	−3.826 025 × 10⁻⁴	−1.270 731 × 10⁻³
difference		2.421530 ×10⁻⁹	−2.173 692 × 10⁻¹⁰	−1.778 517 × 10⁻⁹
4	1.1	−1.572350 ×10⁻³	−3.804 129 × 10⁻⁴	−1.262 466 × 10⁻³
2	1.23	−6.659429 × l0⁻⁶	−1.361 280 × 10⁻⁶	−5.139 339 ×10⁻⁶
combined		−1.579008 × 10⁻³	−3.817 740 ×10⁻⁴	−1.267 604 ×10⁻⁹
difference		−1.50759 × 10⁻⁹	−1.364 242 × 10⁻¹⁰	−1.107 765 × 10⁻⁹
4	1.1	−1.572350 × 10⁻³	−3.804 129 ×10⁻⁴	−1.264 466 × 10⁻³
2	0.59	1.187128 × 10⁻⁵	2.426 639 × 10⁻⁶	9.161 513 × 10⁻⁶
combined		−1.560482 ×10⁻³	−3.779 865 × 10⁻⁴	−1.253 306 × 10⁻³
difference		2.684828 × 10⁻⁹	2.428 351 × 10⁻¹⁰	1.969 511 × 10⁻⁹
4	1.1	−1.572350 × 10⁻³	−3.804 129 × 10⁻⁴	−1.262 466 × 10⁻³
2	0.81^a	2.126653 × 10⁻⁴	6.983 643 × 10⁻⁵	1.860 279 × 10⁻⁴
combined		−1.360300 × 10⁻³	−3.106 152 × 10⁻⁴	−1.076 842 × 10⁻³
difference		6.153368 × 10⁻⁷	3.880 041 × 10⁻⁸	4.043 368 × 10⁻⁷
4	1.1	−1.572350 × 10⁻³	−3.804 129 × 10⁻⁴	−1.262 466 × 10⁻³
2	0.9^a	5.83084 × 10⁻⁵	2.250 209 ×10⁻⁵	5.383 521 × 10⁻⁵
combined		−1.514210 × 10⁻³	−3.579 258 × 10⁻⁴	−1.208 788 × 10⁻³
difference		2.405982 × l0⁻⁷	1.500 621 × 10⁻⁸	1.576 313 × 10⁻⁷
4	1.1	−1.572350 × 10⁻³	−3.804 129 × 10⁻⁴	−1.262 466 × 10⁻³
2	1.37^a	1.738308 × 10⁻⁴	2.415 801 × 10⁻⁵	1.247 036 × 10⁻⁴
combined		−1.398314 × 10⁻³	−3.562 435 × 10⁻⁴	−1.137 631 ×10⁻³
difference		−2.057873 × 10⁻⁷	−1.133 003 × 10⁻⁸	−1.306 148 × 10⁻⁷
4	1.1	−1.572350 × 10⁻³	−3.804 129 × 10⁻⁴	−1.262 466 × 10⁻³
2	1.52^a	2.953510 × 10⁻⁴	4.936 364 × 10⁻⁵	2.187 902 × 10⁻⁴
combined		−1.276829 × 10⁻³	−3.310 408 × 10⁻⁴	−1.043 570 × 10⁻³
difference		−1.704948 × 10⁻⁷	−8.366 897 × 10⁻⁹	−1.053 695 × 10⁻⁷

Surface	Bending factor		Bending factor	Bending factor
2 and 3	1.003 754		0.999 99	0.999 999
4 and 5	0.909 332 5		1.009 527	0.987 1664
6 and 7	1.077 137		0.997 594 6	1.007 008
	Initial system	1st iteration	2nd iteration	3rd iteration

c₁	0
t₁	−15.995 07
c₂	0.026 954 1
t₂	7.4	7.427 7860	7.427 775
	0
	9.9
c₃t₃		9.882 526 1	9.882 532 6
c₄	−0.018 181 8	−0.015 543 382	−0.015 818 626	−0.015 443 05
t₄	2.6	2.364 784 6	2.387 315 2	2.356 677
c₅	0.027 247 9	0.029 958 19	0.029 675 45	0:030 061 24
t₅	8.7	8.841 019 68	8.827 511 8	8.845 880
c₆	0.007 874 0	0.009 631 56	0.009 576 624	0.009 735 58
t₆	6.4	6.893 681 25	6.877 099 2	6.925 297 23
c₇	−0.023 1481	−0.021 490 42	−0.021 542 24	−0.021 392
t₇	−21.748 0460 103.248 066	−22.070 787	−22.059 474 76	−22.091 45
c₈
c₉	0
		Caustic point coordinates
ray 1
y	0.240 350	0.015 935 5	0.015 345 6	0.014 999 70
z	−0.563 745 5	−0.389 598	−0.377 163 3	−0.369 597 6
ray 2
y	0.006 453	0.004 911 9	0.004 810 15	0.004 739 35
z	−0.240 198	−0.182 285	−0.178 480 4	−0.175 811 1
ray 3
y	0.010 781	−0.011 649 7	−0.013 639 2	−0.014 422 8
z	−0.435 543	−0.093 579	−0.065 060 4	−0.052 109 2

Surface	Bending factor		Bending factor	Bending factor	Bending factor
2 and 3	1.014665		1.048 844	0.998 168	1.000 041
4 and 5	0.988210		0.986 853 4	1.014 663	1.008 712
6 and 7	1.043044		1.0430 44	0.971 575	0.982 847
	Initial system	1st iteration	2nd iteration	3rd iteration	4th iteration

c₁	0
t₁	−15.999 507
c₂	0.026 954 1
t₂	7.4	7.508 525 5	7.875 276	7.860 855	7.861 184 3
c₃	0.
t₃	9.9	9.831 751 3	9.601 112	9.610 180	9.609 974
c₄	−0.018 181 8	0.017 865 3	−0.017 507 78	−0.017 900 8	−0.018 132 4
t₄	2.6	2.569 348 4	2.535 570	2.572 750	2.595 166 0
c₅	0.027 247 9	0.027 573 0	0.027 940 32	0.027 536 5	0.027 298 7
t₅	8.7	8.718 376 6	8.738 627	8.716 336	8.702 898
c₆	0.007 874 0	0.008 886 8	0.009 857 8	0.009 197 8	0.008 792 6
t₆	6.4	6.675 487 2	6.962 832	6.764 918	6.648 880
c₇	−0.023 1481	−0.022 192 8	−0.021 2769	−0.021 899 4	−0.022 281 66
t₇	−21.748 046	−21.928 144 5	−22.115 995	−21.986 609	−21.910 750 4
c₈	0
t₈	103.248 066
c₉	0
		Caustic point coordinates
ray 1
y	17.776 333	17.733 599	17.692 602	17.698 431	17.701 704
z	0.685 623	0.517 018	0.356 266	0.307 900	0.276 199 8
ray 2
y	17.738 889	17.684 907	17.634 684	17.654 985	17.667 382
z	0.355 003	0.087 198	−0.155 639	−0.077 484	−0.029 633 4
ray 3
y	17.887 597	17.857 513	17.826 628	17.817 497	17.811 281 3
z	1.995 421	1.980 511	1.945 488	1.719 601	1.575 909 4

Abstract

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Figures (10)

Tables (7)

Equations (9)

Journal of the Optical Society of America