Two-Mirror Systems with Spherical Reflecting Surfaces

Orestes N. Stavroudis

doi:10.1364/JOSA.57.000741

Journal of the Optical Society of America
Vol. 57,
Issue 6,
pp. 741-748
(1967)
•https://doi.org/10.1364/JOSA.57.000741

Two-Mirror Systems with Spherical Reflecting Surfaces

Orestes N. Stavroudis

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Orestes N. Stavroudis, "Two-Mirror Systems with Spherical Reflecting Surfaces," J. Opt. Soc. Am. 57, 741-748 (1967)

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Abstract

An analysis is made of two-mirror systems consisting of spherical reflecting surfaces. Solutions are found for those systems having zero third-order spherical aberration. It is shown that no practical solution exists for the configuration resembling the Cassegrainian telescope; there are three one-parameter families of solutions. These are given by

\begin{array}{l} c_{1} = (q - 1) / 2 t_{0} & \begin{matrix} t_{1} = (t_{0} - f) / q & c_{2} = q / 2 f \end{matrix} \\ t_{0} = (27 f / 32) {sec}^{2} θ & q = - 3 {[1 + 4 cos \frac{2}{3} (θ + π r)]}^{- 1}, \end{array}

where c₁ and c₂ are the two curvatures; t₁ the axial separation of the two reflecting surfaces; t₀ the distance from a focus to the corresponding surface; and f the focal length. The free parameter is θ and r = 0, 1, −1.

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θ	Q	T₀	T₁	N₂	C₁	C₂	R₁	R₂
0.0	−0.600000	0.843750	+0.260417	+0.493827	−0.948148	−0.300000	−1.05469	−3.33333
0.1	−0.601068	0.852244	+0.245822	+0.461S14	−0.939325	−0.300534	−1.06459	−3.32741
0.2	−0.604291	0.878421	+0.201193	+0.367446	−0.913168	−0.302145	−1.09509	−3.30966
0.3	−0.609723	0.924488	+0.123847	+0.215643	−0.870603	−0.304862	−1.14863	−3.28018
0.4	−0.617459	0.994574	+0.008788	+0.014292	−0.813142	−0.308730	−1.22980	−3.23908
0.5	−0.627638	1.09556	−0.152260	−0.226207	−0.742831	−0.313819	−1.34620	−3.18655
0.6	−0.640445	1.23866	−0.372650	−0.493526	−0.662184	−0.320222	−1.51015	−3.12283
0.7	−0.656126	1.44235	−0.674182	−0.774106	−0.574108	−0.328063	−1.74183	−3.04819
0.8	−0.674996	1.73826	−1.09372	−1.05392	−0.481804	−0.337498	−2.07553	−2.96298
0.9	−0.697457	2.18362	−1.69706	−1.31922	−0.388679	−0.348728	−2.57282	−2.86756
1.0	−0.724017	2.89028	−2.61083	−1.55732	−0.298244	−0.362009	−3.35296	−2.76237
1.1	−0.755327	4.10086	−4.10532	−1.75724	−0.214019	−0.377664	−4.67248	−2.64786
1.2	−0.792220	6.42597	−6.84907	−1.91022	−0.139451	−0.396110	−7.17096	−2.52455
1.3	−0.835775	11.7915	−12.9120	−2.01022	−0.077843	−0.417887	−12.8464	−2.39299
1.4	−0.887407	29.2068	−31.7856	−2.05406	−0.032311	−0.443704	−30.9491	−2.25376
1.5	−0.949004	168.623	−176.631	−2.04156	−0.005779	−0.474502	−173.035	−2.10747
1.6	−1.02313	989.608	−966.259	−1.97540	−0.001022	−0.511564	−978.294	−1.95479

θ	Q	T₀	T₁	N₂	C₁	C₂	R₁	R₂
0.0	+3.00000	0.843750	−0.052083	+0.123457	+1.18518	+1.50000	+0.843750	+0.666667
0.1	+2.44633	0.852244	−0.060399	+0.102502	+0.848543	+1.22317	+1.17849	+0.817551
0.2	+2.07935	0.878421	−0.058470	+0.071844	+0.614367	+1.03967	+1.62769	+0.961841
0.3	+1.82001	0.924488	−0.041490	+0.036801	+0.443494	+0.910004	+2.25482	+1.09890
0.4	+1.62852	0.994574	−0.003332	+0.002106	+0.315976	+0.814261	+3.16480	+1.22811
0.5	+1.48269	1.09556	+0.064453	−0.028397	+0.220293	+0.741345	+4.53941	+1.34890
0.6	+1.36917	1.23866	+0.174311	−0.051952	+0.149020	+0.684586	+6.71050	+1.46074
0.7	+1.27949	1.44253	+0.345723	−0.066992	+0.096887	+0.639744	+10.3213	+1.56312
0.8	+1.20802	1.73826	+0.611130	−0.073134	+0.059835	+0.604009	+16.7125	+1.65560
0.9	+1.15090	2.18362	+1.02843	−0.071071	+0.034553	+0.575451	+28.9409	+1.73777
1.0	+1.10543	2.89028	+1.70999	−0.062378	+0.018239	+0.552716	+54.8272	+1.80925
1.1	1.06967	4.10086	+2.89888	−0.049253	+0.008495	+0.534837	+117.714	+1.86973
1.2	1.04224	6.42597	+5.20605	−0.034223	+0.003287	+0.521121	+304.239	+ 1.91894

θ	Q	T₀	T₁	N₂	C₁	C₂	R₁	R₂
0.0	+3.00000	0.843750	−0.052083	+0.123457	+1.18518	+1.50000	+0.843750	+0.666667
0.1	+3.92265	0.852244	−0.037667	+0.129175	+1.71468	+1.96133	+0.583199	+0.509859
0.2	+5.75004	0.878421	−0.021144	+0.114336	+2.70374	+2.87502	+0.369859	+0.347824
0.3	+11.0326	0.924487	−0.006845	+0.074277	+5.42601	+5.51628	+0.184298	+0.181282
0.4	+182.277	0.994574	−0.000030	+0.005426	+91.1329	+91.1384	+0.010973	+0.010972
0.5	−12.3192	1.09556	−0.007757	−0.094309	−6.07871	−6.15962	−0.164509	−0.162348
0.6	−5.91876	1.23866	−0.040323	−0.225230	−2.79284	−2.95938	−0.358059	−0.337908
0.7	−3.88403	1.44235	−0.113889	−0.385647	−1.69308	−1.94201	−0.590639	−0.514930
0.8	−2.88757	1.73826	−0.255667	−0.571793	−1.11824	−1.44378	−0.894265	−0.692625
0.9	−2.29831	2.18362	−0.514998	−0.777891	−0.755238	−1.14915	−1.32409	−0.870205
1.0	−1.91044	2.89028	−0.989449	−0.996350	−0.503487	−0.955219	−1.98615	−1.04688
1.1	−1.63684	4.10086	−1.89442	−1.21811	−0.321498	−0.818420	−3.11044	−1.22187
1.2	−1.43432	6.42597	−3.78295	−1.43308	−0.189413	−0.717161	−5.27947	−1.39439
1.3	−1.27904	11.7915	−8.43721	−1.63073	−0.096639	−0.639519	−10.3478	−1.56367
1.4	−1.15675	29.2068	−24.3844	−1.80065	−0.036922	−0.578377	−27.0840	−1.72898
1.5	−1.05845	168.623	−158.367	−1.93325	−0.006104	−0.529224	−163.835	−1.88956

φ	sinφ′
1°	−1.353×10⁻⁷
2°	−3.107
3°	−5.233
4°	−7.121
5°	−6.363
6°	−3.897
10°	3.350×10⁻⁶
20°	1.362×10⁻⁴
30°	1.041×10⁻³

φ	sinφ′
−30°	−4.9264×10⁻²
−20°	−5.0724
−10°	−5.1979
−5°	−5.2335
0°	−5.2482
5°	−5.2417
10°	−5.2133
20°	−5.0760
30°	−4.7605

Abstract

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Figures (10)

Tables (5)

Equations (61)

Journal of the Optical Society of America