Automatic computation of optical focal surfaces

T. Brender Andersen

doi:10.1364/AO.20.002754

Automatic computation of optical focal surfaces

T. Brender Andersen

Applied Optics
Vol. 20,
Issue 15,
pp. 2754-2760
(1981)
•https://doi.org/10.1364/AO.20.002754

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T. Brender Andersen, "Automatic computation of optical focal surfaces," Appl. Opt. 20, 2754-2760 (1981)

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History
- Original Manuscript: March 5, 1981
- Published: August 1, 1981

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Abstract

The use of automatically computed aberration coefficients of arbitrary order for an exact determination of the surface of best focus of a symmetrical system, by minimizing the gyration radius of the spot diagram, is demonstrated. For the two optical systems studied, a Schmidt camera and a Ritchey-Chrétien telescope, the results from the exact model are found to be significantly different from the paraxial approximation predictions.

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n i j	Sands Σ₁		Sands Σ₂

	Δ_min	M_min	Δ_min	M_min
2 2 0	0.0463	0.0169	0.00007	0.00046
2 1 1	0.0276	0.0444	−0.0033	−0.00021
2 0 2	−0.0065	0.0156	0.0053	0.0134
3 3 0	−0.0751	−0.0457	0.0122	0.0191
3 2 1	−0.4425	−0.6578	0.0737	0.0900
3 1 2	−0.2340	−0.2059	0.0237	0.0390
3 0 3	0.0661	0.1224	0.0612	0.0832
4 4 0	−0.0618	−0.0142	0.1834	0.1928
4 3 1	0.1967	0.0427	0.2838	0.3054
4 2 2	0.1413	0.1279	0.1309	0.1530
4 1 3	−0.3332	−0.0545	0.1710	0.2020
4 0 4	−1.3966	−1.3236	0.2776	0.3145
5 5 0	−0.0222	0.0336	0.3498	0.3607
5 4 1	0.3729	0.1852	0.5224	0.5458
5 3 2	0.4558	0.4798	0.2093	0.2358
5 2 3	0.0309	−2.6388	0.2958	0.3335
5 1 4	−0.3935	−0.3368	0.5506	0.6057
5 0 5	0.4630	0.3166	0.8236	0.8748
6 6 0	0.0145	0.0754	0.4779	0.4895
6 5 1	0.2572	0.1480	0.7645	0.7872
6 4 2	8.3928	−18.2449	0.1506	0.1661
6 3 3	0.0302	−0.1045	0.1906	0.2119
6 2 4	−0.0378	0.0710	0.7070	0.8238
6 1 5	0.1293	0.0790	1.6759	1.9693
6 0 6	0.2214	0.0107	2.3334	2.4946

n i j	c₁	C₂	σ⁻¹π₀	σ⁻¹π₁	σ⁻¹π₂	Δ_min	M_min	σ⁻¹k_g²	Δ_min,par.	M_min,par.
0 0 0	1.245737,+02	2.635665,+00	8.968310,−44	−3.399727,−24	3.221940,−05	5.275900,−20	1.245737,+02	0	5.275900,−20	1.245737,+02
1 1 0	−5.459957,−05	8.378662,−05	−1.596551,−28	3.026120,−09	1.466973,−09	−4.696115,−05	−1.783734,−04	−1.751623,−46	−4.696115,−05	−1.783734,−04
1 0 1	1.463244,+02	8.355896,+00	3.015673,−24	−5.550537,−06	2.317027,−04	8.613657,−02	1.465515,+02	2.722832,−24	8.613657,−02	1.465515,+02
2 2 0	−6.990160,−10	4.207338,−09	7.993693,−14	−1.070064,−12	5.087120,−14	1.874408,−08	4.476937,−08	8.881882,−15	1.780863,−08	4.623855,−08
2 1 1	4.503585,−03	8.120155,−04	1.726749,−09	−8.385405,−09	3.567896,−08	4.639245,−04	5.341149,−03	1.987409,−09	2.026162,−04	5.037613,−03
2 0 2	3.403928,+02	4.115548,+01	2.090575,−05	−1.634838,−02	1.708040,−03	2.530845,+02	1.008158,+03	2.066669,−05	2.548484,+02	1.012088,+03
3 3 0	1.485920,−12	2.332504,−13	−6.466926,−17	2.978264,−16	8.183409,−18	−5.401134,−12	−1.137673,−11	−1.118261,−17	−5.080293,−12	−1.190403,−11
3 2 1	1.426880,−07	8.146050,−08	−5.139490,−13	−8.109287,−12	3.598592,−12	2.179332,−08	3.578514,−07	2.867852,−13	6.893636,−08	3.243811,−07
3 1 2	3.994495,−02	6.761237,−03	−1.069119,−06	−7.919247,−07	5.021067,−07	−1.756765,−04	6.270076,−02	−3.039659,−07	1.494928,−02	7.934623,−02
3 0 3	9.276628,+02	2.255314,+02	−7.128444,−03	−1.608895,−01	1.248396,−02	6.721805,+02	4.817598,+03	−8.534918,−03	1.286715,+03	4.319011,+03
4 4 0	−8.198851,−17	1.795858,−17	3.284136,−20	−2.922054,−20	−4.562565,−23	6.817123,−16	1.330143,−15	7.647300,−21	5.928635,−16	1.480601,−15
4 3 1	1.786114,−11	7.875860,−12	−1.932391,−16	1.826490,−15	3.913655,−16	−6.761098,−12 −	−2.989719,−11	−2.688402,−16	−1.475132,−11	−2.101838,−11
4 2 2	2.134500,−06	1.018626,−06	2.365844,−10	−5.207013,−11	8.542292,−11	−5.254584,−07	2.819400,−06	−3.606738,−11	8.628370,−07	4.408649,−06
4 1 3	2.760709,−01	5.281299,−02	3.869722,−06	−1.845951,−05	5.568345,−06	−3.087554,−02	4.641379,−01	3.777616,−06	1.895833,−01	7.757490,−01
4 0 4	2.708320,+03	1.296636,+03	2.928588,+00	−1.185032,+00	8.934533,−02	1.060197,+02	1.903966,+04	8.510330,−01	5.471606,+03	1.712964,+04
5 5 0	−1.934562,−21	6.278842,−22	−1.063520,−23	−1.107488,−24	−3.205811,−26	−1.015166,−20	9.231734,−21 −	−3.236596,−24	3.150655,−22 −	1.104155,−21
5 4 1	−9.794609,−16	7.631881,−16	8.927875,−20	−7.882326,−20	3.312480,−20	9.344583,−16	3.585973,−15	3.708048,−20	1.432503,−15	2.796136,−15
5 3 2	1.804279,−10	1.428193,−10	−5.046851,−14	8.259640,−15	1.221815,−14	−7.790160,−11 −	1.543456,−10	2.259364,−14	3.848911,−11	2.818723,−10
5 2 3	2.874651,−05	1.067619,−05	3.047624,−09	−1.282073,−09	1.348097,−09	−1.181340,−05	1.980314,−05	1.787925,−10	1.427085,−05	6.635969,−05
5 1 4	1.548191,+00	3.962048,−01	4.337823,−04	−2.376300,−04	5.438144,−05	−8.386297,−01	1.386777,+00	3.221442,−04	1.610577,+00	5.793132,+00
5 0 5	8.308124,+03	7.663034,+03	3.207344,+01	−7.867670,+00	6.325428,−01	−1.260217,+04	6.083301,+04	6.168259,+00	2.168102,+04	6.545203,+04
6 6 0	−7.866014,−25	−4.171550,−26	2.318591,−27	4.453890,−29	−2.722901,−30	4.632807,−26	4.004388,−25	8.863409,−28	−1.620078,−25	−1.213600,−24
6 5 1	8.341725,−20	6.805964,−20	−4.572437,−23	−1.858826,−23	2.097950,−24	−6.254397,−21	2.707916,−19	−6.235140,−24	1.027724,−19	3.542909,−19
6 4 2	2.939951,−15	1.860506,−14	5.092769,−18	9.945402,−19	1.521145,−18	−1.900952,−15	5.432903,−15	−2.225830,−18	−7.890347,−16	8.603201,−16
6 3 3	2.456372,−09	1.958862,−09	−1.499020,−13	−1.705974,−14	2.531843,−13	−1.657483,−09	−2.349388,−09	5.096073,−13	1.227939,−09	5.692806,−09
6 2 4	2.925940,−04	1.012364,−04	5.284732,−08	−2.695853,−08	1.722863,−08	−2.049028,−04	−1.152887,−04	3.511742,−08	1.922186,−04	7.992178,−04
6 1 5	7.783067,+00	2.895288,+00	7.939518,−03	−2.335364,−03	4.925012,−04	−1.029094,+01	−1.044131,+01	5.486402,−03	1.104663,+01	3.689828,+01
6 0 6	2.669330,+04	4.610821,+04	2.282427,+02	−5.015204,+01	4.449849,+00	−1.006532,+05	1.408823,+05	2.304224,+01	8.277725,+04	2.448664,+05

i	μ_i (450 nm)	d_i	a_i₁	a_i₂	a_i₃
0	1.000000	0.00000	0	0	0
1	−1.000000	−3515.1	−4.69440,−5	1.75196,−14	−1.30767,−23
2	−1.000000	4308.91	−8.20996,−5	3.85102,−12	−3.61276,−19
3	0.655606	15.00	2.25893,−5	−1.4585,−9	0
4	1.525306	145.09	−2.25893,−5	1.4585,−9	0

n i j	Sands Σ₁		Sands Σ₂

	Δ_min	M_min	Δ_min	M_min
2 2 0	0.0463	0.0169	0.00007	0.00046
2 1 1	0.0276	0.0444	−0.0033	−0.00021
2 0 2	−0.0065	0.0156	0.0053	0.0134
3 3 0	−0.0751	−0.0457	0.0122	0.0191
3 2 1	−0.4425	−0.6578	0.0737	0.0900
3 1 2	−0.2340	−0.2059	0.0237	0.0390
3 0 3	0.0661	0.1224	0.0612	0.0832
4 4 0	−0.0618	−0.0142	0.1834	0.1928
4 3 1	0.1967	0.0427	0.2838	0.3054
4 2 2	0.1413	0.1279	0.1309	0.1530
4 1 3	−0.3332	−0.0545	0.1710	0.2020
4 0 4	−1.3966	−1.3236	0.2776	0.3145
5 5 0	−0.0222	0.0336	0.3498	0.3607
5 4 1	0.3729	0.1852	0.5224	0.5458
5 3 2	0.4558	0.4798	0.2093	0.2358
5 2 3	0.0309	−2.6388	0.2958	0.3335
5 1 4	−0.3935	−0.3368	0.5506	0.6057
5 0 5	0.4630	0.3166	0.8236	0.8748
6 6 0	0.0145	0.0754	0.4779	0.4895
6 5 1	0.2572	0.1480	0.7645	0.7872
6 4 2	8.3928	−18.2449	0.1506	0.1661
6 3 3	0.0302	−0.1045	0.1906	0.2119
6 2 4	−0.0378	0.0710	0.7070	0.8238
6 1 5	0.1293	0.0790	1.6759	1.9693
6 0 6	0.2214	0.0107	2.3334	2.4946

n i j	c₁	C₂	σ⁻¹π₀	σ⁻¹π₁	σ⁻¹π₂	Δ_min	M_min	σ⁻¹k_g²	Δ_min,par.	M_min,par.
0 0 0	1.245737,+02	2.635665,+00	8.968310,−44	−3.399727,−24	3.221940,−05	5.275900,−20	1.245737,+02	0	5.275900,−20	1.245737,+02
1 1 0	−5.459957,−05	8.378662,−05	−1.596551,−28	3.026120,−09	1.466973,−09	−4.696115,−05	−1.783734,−04	−1.751623,−46	−4.696115,−05	−1.783734,−04
1 0 1	1.463244,+02	8.355896,+00	3.015673,−24	−5.550537,−06	2.317027,−04	8.613657,−02	1.465515,+02	2.722832,−24	8.613657,−02	1.465515,+02
2 2 0	−6.990160,−10	4.207338,−09	7.993693,−14	−1.070064,−12	5.087120,−14	1.874408,−08	4.476937,−08	8.881882,−15	1.780863,−08	4.623855,−08
2 1 1	4.503585,−03	8.120155,−04	1.726749,−09	−8.385405,−09	3.567896,−08	4.639245,−04	5.341149,−03	1.987409,−09	2.026162,−04	5.037613,−03
2 0 2	3.403928,+02	4.115548,+01	2.090575,−05	−1.634838,−02	1.708040,−03	2.530845,+02	1.008158,+03	2.066669,−05	2.548484,+02	1.012088,+03
3 3 0	1.485920,−12	2.332504,−13	−6.466926,−17	2.978264,−16	8.183409,−18	−5.401134,−12	−1.137673,−11	−1.118261,−17	−5.080293,−12	−1.190403,−11
3 2 1	1.426880,−07	8.146050,−08	−5.139490,−13	−8.109287,−12	3.598592,−12	2.179332,−08	3.578514,−07	2.867852,−13	6.893636,−08	3.243811,−07
3 1 2	3.994495,−02	6.761237,−03	−1.069119,−06	−7.919247,−07	5.021067,−07	−1.756765,−04	6.270076,−02	−3.039659,−07	1.494928,−02	7.934623,−02
3 0 3	9.276628,+02	2.255314,+02	−7.128444,−03	−1.608895,−01	1.248396,−02	6.721805,+02	4.817598,+03	−8.534918,−03	1.286715,+03	4.319011,+03
4 4 0	−8.198851,−17	1.795858,−17	3.284136,−20	−2.922054,−20	−4.562565,−23	6.817123,−16	1.330143,−15	7.647300,−21	5.928635,−16	1.480601,−15
4 3 1	1.786114,−11	7.875860,−12	−1.932391,−16	1.826490,−15	3.913655,−16	−6.761098,−12 −	−2.989719,−11	−2.688402,−16	−1.475132,−11	−2.101838,−11
4 2 2	2.134500,−06	1.018626,−06	2.365844,−10	−5.207013,−11	8.542292,−11	−5.254584,−07	2.819400,−06	−3.606738,−11	8.628370,−07	4.408649,−06
4 1 3	2.760709,−01	5.281299,−02	3.869722,−06	−1.845951,−05	5.568345,−06	−3.087554,−02	4.641379,−01	3.777616,−06	1.895833,−01	7.757490,−01
4 0 4	2.708320,+03	1.296636,+03	2.928588,+00	−1.185032,+00	8.934533,−02	1.060197,+02	1.903966,+04	8.510330,−01	5.471606,+03	1.712964,+04
5 5 0	−1.934562,−21	6.278842,−22	−1.063520,−23	−1.107488,−24	−3.205811,−26	−1.015166,−20	9.231734,−21 −	−3.236596,−24	3.150655,−22 −	1.104155,−21
5 4 1	−9.794609,−16	7.631881,−16	8.927875,−20	−7.882326,−20	3.312480,−20	9.344583,−16	3.585973,−15	3.708048,−20	1.432503,−15	2.796136,−15
5 3 2	1.804279,−10	1.428193,−10	−5.046851,−14	8.259640,−15	1.221815,−14	−7.790160,−11 −	1.543456,−10	2.259364,−14	3.848911,−11	2.818723,−10
5 2 3	2.874651,−05	1.067619,−05	3.047624,−09	−1.282073,−09	1.348097,−09	−1.181340,−05	1.980314,−05	1.787925,−10	1.427085,−05	6.635969,−05
5 1 4	1.548191,+00	3.962048,−01	4.337823,−04	−2.376300,−04	5.438144,−05	−8.386297,−01	1.386777,+00	3.221442,−04	1.610577,+00	5.793132,+00
5 0 5	8.308124,+03	7.663034,+03	3.207344,+01	−7.867670,+00	6.325428,−01	−1.260217,+04	6.083301,+04	6.168259,+00	2.168102,+04	6.545203,+04
6 6 0	−7.866014,−25	−4.171550,−26	2.318591,−27	4.453890,−29	−2.722901,−30	4.632807,−26	4.004388,−25	8.863409,−28	−1.620078,−25	−1.213600,−24
6 5 1	8.341725,−20	6.805964,−20	−4.572437,−23	−1.858826,−23	2.097950,−24	−6.254397,−21	2.707916,−19	−6.235140,−24	1.027724,−19	3.542909,−19
6 4 2	2.939951,−15	1.860506,−14	5.092769,−18	9.945402,−19	1.521145,−18	−1.900952,−15	5.432903,−15	−2.225830,−18	−7.890347,−16	8.603201,−16
6 3 3	2.456372,−09	1.958862,−09	−1.499020,−13	−1.705974,−14	2.531843,−13	−1.657483,−09	−2.349388,−09	5.096073,−13	1.227939,−09	5.692806,−09
6 2 4	2.925940,−04	1.012364,−04	5.284732,−08	−2.695853,−08	1.722863,−08	−2.049028,−04	−1.152887,−04	3.511742,−08	1.922186,−04	7.992178,−04
6 1 5	7.783067,+00	2.895288,+00	7.939518,−03	−2.335364,−03	4.925012,−04	−1.029094,+01	−1.044131,+01	5.486402,−03	1.104663,+01	3.689828,+01
6 0 6	2.669330,+04	4.610821,+04	2.282427,+02	−5.015204,+01	4.449849,+00	−1.006532,+05	1.408823,+05	2.304224,+01	8.277725,+04	2.448664,+05

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