Ray-trace analysis and data reduction methods for the Ritchey-Common test

K. L. Shu

doi:10.1364/AO.22.001879

Applied Optics
Vol. 22,
Issue 12,
pp. 1879-1886
(1983)
•https://doi.org/10.1364/AO.22.001879

Ray-trace analysis and data reduction methods for the Ritchey-Common test

K. L. Shu

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K. L. Shu, "Ray-trace analysis and data reduction methods for the Ritchey-Common test," Appl. Opt. 22, 1879-1886 (1983)

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Abstract

The relationships between figure errors on an optical flat and wave-front aberrations when the flat is tested in the Ritchey-Common configuration are examined by the real ray-tracing lens analysis program polycon. A new data reduction method for the test is demonstrated. The test is simulated, and rays are traced using polycon. Unit magnitude Zernike term surface errors are added, one by one, to the flat under simulated test, and the effect on the wave front is derived by real ray tracing. This results in a table of influence functions for the particular test geometry. The simulation is repeated for one or more different field angles or azimuthal rotations of the flat to obtain other sets of influence functions. By using these influence functions it is possible to deduce what surface errors on the flat are responsible for any arbitrary test wave-front errors measured up to the order of the influence functions derived. The effectiveness of this method is compared with other existing methods by computer simulations.

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Surface figure of a flat
Δ_s(X_x,Y_s) = −0.342Z_2,0 −0.146Z_2,2 −0.2421Z_2,−2 + 0.0262Z_3,1 −0.0378Z_3,−1 + 0.0473Z_4,0
rms = 0.231, span = 0.961

	(A) F/10, θ = 11° tanθ/2(F/#) = 0.01		(B) F/31.25, θ = 32° tanθ/2(F/#) = 0.01		(C) F/2.53, θ = 32° tanθ/2(F/#) = 0.12

	Real ray	Analytic equations	Real ray	Analytic equations	Real ray	Analytic equations
W_2,0	−0.6806	−0.6812	−0.6570	−0.6567	−0.6422	−0.6567
W_2,2	−0.3120	−0.3129	−0.4504	−0.4480	−0.4307	−0.4480
W_2,−2	−0.4863	−0.4842	−0.4858	−0.4842	−0.5142	−0.4842
W_3,1	0.0542	0.0540	0.0668	0.0667	0.0923	0.0667
W_3,−1	−0.0759	−0.0748	−0.0712	−0.0704	−0.0842	−00704
W_4,0	0.0970	0.0966	0.1155	0.1160	0.1294	0.1160

	Surface term weight
Ritchey Angle (θ)	S_2,0 2 sin²θ/cosθ	S_2,2 (1 + cos²θ)/cosθ)	$S_{4, 0} \frac{3 cos θ}{2} (\frac{3}{{cos}^{4} θ} - \frac{4}{{cos}^{2} θ} + 1)$
0	0	2	0
10	0.061	2.000	0.96
20	0.249	2.004	0.448
30	0.577	2.021	1.299
40	1.079	2.071	3.327
50	1.826	2.199	8.574
60	3.000	2.500	24.750
70	5.164	3.266	95.446
80	11.170	5.932	825.121

Ritchey angle (θ) (deg)	W_2,2	W_2,−2	W_3,1	W_3,−1	W_4,0
12.5	0.5731	0.4425	0.5851	0.4151	0.4568
32.1	1.3379	0.4571	0.8122	0.3583	0.4489
32.2	1.3448	0.4571	0.8143	0.3580	0.4491
47.8	2.3148	0.4282	0.7485	0.2574	0.3223
48	2.3384	0.4313	0.7535	0.2565	0.3212

Surface error term	Induced wave front errors

	Ritchey angle θ = 32.1°	Ritchey angle θ = 48°
S_2,0	S_2,0(1.3639W_2,2 + 0.3427W_3,1 + 0.0286W_4,0)	S_2,0(3.2546W_2,2 + 0.4568W_3,1 + 0.0205W_4,0)
S_2,2	S_2,2(4.0392W_2,2 + 0.0271W_3,1 − 0.0103W_4,0)	S_2,2(4.1693W_2,2 − 0.1115W_3,1 − 0.0080W_4,0)
S_2,−2	S_2,−2(3.9832W_2,−2 + 0.0169W_3,1 + 0.0013W_4,0)	S_2,−2(3.9135W_2,−2 − 0.0253W_3,−1 + 0.0013W_4,0)
S_3,1	S_3,1(0.8209W_2,2 + 4.903W_3,1 + 0.1124W_4,0)	S_3,1 (0.3543W_2,2 + 4.6352W_3,1 + 0.0390W_4,0)
S_3,−1	S_3,−1(0.5874W_2,−2 + 3.5661W_3,−1 + 0.0014W_4,0)	S_3,−1(0.3989W_2,2 + 2.5901W_3,−1 + 0.0014W_4,0)
S_4,0	S_4,0(1.6993W_2,2 + 1.4779W_3,1 + 4.2413W_4,0)	S_4,0(2.5893W_2,2 + 0.7278W_3,1 + 3.0817W_4,0)

Exact figure of flat A
Δ_s = 0.5Z_2,0 + 0.1Z_2,2 + 0.1Z_2,−2 + 0.1Z_3,1 + 0.1Z_3,−1 + 0.1Z_4,0

Method	Codd. app.	Inver. map.	Codd. + inver. map.	Codd. + small θ	Figure design

Error
Root mean square	0.077	0.042	0.136	0.034	0.000
Span	0.307	0.268	0.624	0.152	0.000

Exact Ritchey angles: 32.1, 48°
Measured Ritchey angle: 32.1, 48°

	S_2,0	S_2,2	S_2,−2	S_3,1	S_3,−1	S_4,0
Avg ( $\bar{S}$ )	0.4997	0.1020	0.1000	0.1002	0.0998	0.1000
SDV (σ)	0.0116	0.0068	0.0028	0.0027	0.0033	0.0023

	Z_2,0	Z_2,2	Z_2,−2	Z_3,1	Z_3,−1	Z_4,0	Z_3,3	Z_3,−3	Z_4,2	Z_4,−2	Z_5,1	Z_5,−1	Z_6,0	*
Flat C	0.5	0.1	0.1	0.1	0.1	0.1	0.05	0.02	0.01	0.02	0.01	0	−0.03
Recovered figure for flat C	0.4583	0.1039	0.0928	0.0785	0.0904	0.0994								0.00637
Flat D	0.5	0.1	0.1	0.1	0.1	0.1	0.1	0.04	0.02	0.04	0.02	0	−0.03
Recovered figure for flat D	0.4203	0.1081	0.0857	0.0582	0.0808	0.0959								0.02383

Abstract

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Figures (8)

Tables (9)

Equations (28)

Applied Optics