Theoretical Resolution Angles for Point and Line Test Objects in the Presence of a Luminous Background

Howard S. Coleman; Madeline F. Coleman

doi:10.1364/JOSA.37.000572

Journal of the Optical Society of America
Vol. 37,
Issue 7,
pp. 572-576
(1947)
•https://doi.org/10.1364/JOSA.37.000572

Theoretical Resolution Angles for Point and Line Test Objects in the Presence of a Luminous Background

Howard S. Coleman and Madeline F. Coleman

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Howard S. Coleman and Madeline F. Coleman, "Theoretical Resolution Angles for Point and Line Test Objects in the Presence of a Luminous Background," J. Opt. Soc. Am. 37, 572-576 (1947)

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Abstract

A theoretical study has been made of the diffraction patterns for a circular aperture for point and line test objects in the presence of a luminous background. The study involves the superposition of the diffraction pattern elements of the test object and the image of the luminous background. From this superposition it is possible to relate the contrast of the image of the test object to the angle subtended by the test object elements and their apparent contrast at the entrance pupil of the optical system under consideration. This relation has been found useful in studying the resolving power of optical systems. For example, it is possible to determine the contrast of the image of a test object at the limit of resolution under a wide variety of conditions. It is also possible to determine the loss in resolution caused by a given reduction in image contrast regardless of the cause.

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Angle radians ×10⁵	Brightness for a point	Brightness for a line	Angle radians ×10⁵	Brightness for a point	Brightness for a line
0	100.00	100.00	105.996	.85	1.64
3.533	99.00	97.69	109.529	.52	1.37
7.066	96.06	95.62	113.062	.32	1.09
10.6000	91.34	90.15	116.596	.14	.91
14.133	85.03	85.34	120.129	.04	.86
17.666	77.46	76.75	123.662	.0002	.90
21.199	68.97	67.64	127.195	.02	1.04
24.732	59.94	58.43	130.728	.09	1.09
28.266	50.75	49.10	134.262	.18	1.14
31.799	41.74	39.77	137.795	.27	1.18
35.332	33.26	31.76	141.328	.34	1.24
38.865	25.54	23.86	144.861	.40	1.15
42.398	18.79	17.92	148.394	.42	1.12
45.932	13.12	12.90	151.928	.40	.91
49.465	8.57	8.35	155.461	.36	.73
52.998	5.11	5.76	158.994	.30	.65
56.531	2.67	3.76	162.527	.22	.41
60.064	1.11	3.00	166.060	.15	.26
63.598	.28	2.99	169.594	.08	.24
67.131	.004	3.05	173.127	.04	.26
70.664	.11	3.44	176.662	.008	.28
74.197	.44	3.73	180.193	.0002	.33
77.730	.85	4.01	183.726	.01	.36
81.264	1.25	4.16	187.260	.04	.42
84.797	1.55	4.17	190.793	.07	.45
88.330	1.72	4.04	194.326	.10	.48
91.863	1.74	3.70	197.859	.13	.47
95.396	1.64	3.09	201.392	.15	.45
98.930	1.43	2.56	204.926	.16	.43
102.463	1.15	2.24	208.459	.155	.32

α	B_max	B_min	Difference (B_max − B_min)	α	B_max	B_min	Difference (B_max − B_min)
49	1.199	1.198	.001	81	1.011	.448	.563
50	1.174	1.172	.002	82	1.013	.430	.583
51	1.150	1.147	.003	83	1.014	.410	.604
52	1.126	1.122	.004	84	1.015	.392	.623
53	1.108	1.098	.010	85	1.016	.375	.641
54	1.085	1.070	.015	86	1.017	.358	.659
55	1.070	1.048	.022	87	1.018	.338	.680
56	1.056	1.022	.034	88	1.019	.322	.697
57	1.040	1.000	.040	89	1.019	.305	.714
58	1.030	.975	.055	90	1.020	.292	.728
59	1.022	.952	.070	91	1.020	.278	.742
60	1.012	.926	.086	92	1.019	.264	.755
61	1.010	.900	.110	93	1.019	.249	.770
62	1.009	.880	.129	94	1.018	.234	.784
63	1.006	.856	.150	95	1.017	.220	.797
64	1.004	.830	.174	96	1.016	.209	.807
65	1.003	.802	.201	97	1.015	.195	.820
66	1.002	.775	.227	98	1.014	.182	.832
67	1.001	.752	.249	99	1.013	.170	.843
68	1.000	.730	.270	100	1.012	.160	.852
69	1.000	.707	.293	110	1.005	.070	.935
70	1.000	.680	.320	120	1.001	.024	.977
71	1.001	.658	.343	130	1.000	.002	.998
72	1.002	.632	.370	140	1.002	.006	.996
73	1.003	.610	.393	150	1.005	.008	.997
74	1.004	.590	.414	160	1.003	.022	.981
75	1.005	.570	.435	170	1.000	.012	.988
76	1.006	.546	.460	180	1.000	.006	.994
77	1.007	.526	.481	190	1.001	.005	.996
78	1.008	.502	.506	200	1.004	.008	.996
79	1.009	.488	.521	210	1.004	.006	.998
80	1.010	.468	.542

α	B_max	B_min	Difference (B_max −B_min)	Ca%	α	B_max	B_min	Difference (B_min − B_min)	Ca.%
49.0	—	—			75	1.040	.528	.512	49.23
49.5	1.160	1.159	.001		76	1.040	.510	.530	50.96
50.0	1.150	1.142	.008		77	1.040	.490	.550	52.88
50.5	1.150	1.140	.010		78	1.041	.470	.571	54.85
51.0	1.125	1.120	.005	.444	79	1.042	.452	.590	56.62
51.5	1.113	1.105	.008	.712	80	1.043	.430	.613	58.77
52.0	1.103	1.092	.011	.997	81	1.044	.413	.631	60.44
52.5	1.091	1.081	.010	.917	82	1.044	.396	.648	62.07
53.0	1.081	1.065	.016	1.48	83	1.045	.380	.665	63.64
53.5	1.073	1.054	.019	1.77	84	1.044	.364	.680	65.13
54	1.057	1.031	.026	.9524	85	1.044	.350	.694	66.48
55	1.045	1.007	.038	2.97	86	1.044	.333	.711	68.10
56	1.040	.985	.055	5.29	87	1.043	.320	.723	69.32
57	1.045	.958	.087	8.33	88	1.042	.300	.742	71.21
58	1.033	.933	.100	9.68	89	1.041	.288	.753	72.33
59	1.031	.905	.126	12.22	90	1.040	.276	.764	73.46
60	1.030	.880	.150	14.56	95	1.033	.210	.823	79.67
61	1.030	.850	.180	17.48	100	1.025	.160	.865	84.39
62	1.030	.830	.200	19.42	110	1.011	.090	.921	91.09
63	1.030	.800	.230	22.33	120	1.008	.060	.948	94.05
64	1.030	.780	.250	24.27	130	1.011	.060	.951	94.07
65	1.030	.757	.273	26.50	140	1.013	.070	.943	93.09
66	1.030	.732	.298	28.93	150	1.010	.080	.930	92.08
67	1.030	.708	.322	31.26	160	1.008	.087	.921	91.37
68	1.031	.680	.351	34.04	170	1.003	.050	.953	95.01
69	1.030	.660	.371	35.98	180	1.003	.040	.963	96.01
70	1.035	.638	.397	38.36	190	1.004	.040	.964	96.02
71	1.035	.612	.423	40.87	200	1.003	.050	.953	95.01
72	1.037	.593	.444	42.82	210	1.000	.035	.965	96.50
73	1.038	.568	.470	45.28	220	1.000	.025	.975	97.50
74	1.038	.550	.488	47.01	230	1.000	.020	.980	98.00

Apparent contrast (percent)	Coefficient of specific resolution (mm − rad × 10⁵)	Image contrast at limit of resolution (percent)
100	54.6	3.0
63	55.6	3.0
44	55.8	2.5
31	56.2	2.2
23	56.7	2.0
21	57.4	2.0
12	58.1	1.5
9	60.0	1.5
6	64.3	1.5
4	76.5	2.1
3	99.0	2.5

Angle radians ×10⁵	Brightness for a point	Brightness for a line	Angle radians ×10⁵	Brightness for a point	Brightness for a line
0	100.00	100.00	105.996	.85	1.64
3.533	99.00	97.69	109.529	.52	1.37
7.066	96.06	95.62	113.062	.32	1.09
10.6000	91.34	90.15	116.596	.14	.91
14.133	85.03	85.34	120.129	.04	.86
17.666	77.46	76.75	123.662	.0002	.90
21.199	68.97	67.64	127.195	.02	1.04
24.732	59.94	58.43	130.728	.09	1.09
28.266	50.75	49.10	134.262	.18	1.14
31.799	41.74	39.77	137.795	.27	1.18
35.332	33.26	31.76	141.328	.34	1.24
38.865	25.54	23.86	144.861	.40	1.15
42.398	18.79	17.92	148.394	.42	1.12
45.932	13.12	12.90	151.928	.40	.91
49.465	8.57	8.35	155.461	.36	.73
52.998	5.11	5.76	158.994	.30	.65
56.531	2.67	3.76	162.527	.22	.41
60.064	1.11	3.00	166.060	.15	.26
63.598	.28	2.99	169.594	.08	.24
67.131	.004	3.05	173.127	.04	.26
70.664	.11	3.44	176.662	.008	.28
74.197	.44	3.73	180.193	.0002	.33
77.730	.85	4.01	183.726	.01	.36
81.264	1.25	4.16	187.260	.04	.42
84.797	1.55	4.17	190.793	.07	.45
88.330	1.72	4.04	194.326	.10	.48
91.863	1.74	3.70	197.859	.13	.47
95.396	1.64	3.09	201.392	.15	.45
98.930	1.43	2.56	204.926	.16	.43
102.463	1.15	2.24	208.459	.155	.32

Abstract

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Figures (8)

Tables (4)

Equations (5)

Journal of the Optical Society of America