Irradiance Distribution, Resolution, and Size Estimates in Diffraction Limited Imagery of Extended Circular Targets

Yu. Mekler; J. Otterman; Z. Ravnoy

doi:10.1364/AO.14.000503

Applied Optics
Vol. 14,
Issue 2,
pp. 503-508
(1975)
•https://doi.org/10.1364/AO.14.000503

Irradiance Distribution, Resolution, and Size Estimates in Diffraction Limited Imagery of Extended Circular Targets

Yu. Mekler, J. Otterman, and Z. Ravnoy

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Yu. Mekler, J. Otterman, and Z. Ravnoy, "Irradiance Distribution, Resolution, and Size Estimates in Diffraction Limited Imagery of Extended Circular Targets," Appl. Opt. 14, 503-508 (1975)

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Abstract

Various aspects of diffraction effects on imagery from distributed sources are studied. First the irradiance distribution in the focal plane from distributed, uniform, circular sources (disks) for optical systems with various obscuration ratios, was computed and presented in tables. A resolution criterion, applicable both to point sources and distributed sources, is presented and discussed. The criterion is based on an assumption that detection requires a minimum contrast. The level of this contrast was selected to be 20%. The criterion is applied to the case of two neighboring disks, and the numerical results for resolution are tabulated. The problem of size estimate, i.e., apparent diameter of a single disk on a uniform background, is treated as a question of locating points the irradiance at which is a specified level higher than the background. The real (geometric optics) diameter of a source is given in graphs in terms of its apparent size.

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θ/∊	0.0	0.1	0.2	0.3	0.4	0.5
R = 0.5
0.0	1.000	1.000	1.000	1.000	1.000	1.000
0.1	0.975	0.975	0.974	0.973	0.972	0.971
0.2	0.903	0.903	0.900	0.897	0.893	0.889
0.3	0.795	0.793	0.789	0.782	0.774	0.766
0.4	0.662	0.660	0.654	0.644	0.633	0.621
0.5	0.521	0.519	0.510	0.499	0.486	0.475
0.6	0.386	0.383	0.375	0.364	0.352	0.343
0.7	0.268	0.265	0.258	0.249	0.241	0.239
0.8	0.174	0.172	0.167	0.162	0.161	0.166
0.9	0.105	0.104	0.102	0.103	0.110	0.124
R = 1.0
0.0	1.000	1.000	1.000	1.000	1.000	1.000
0.10	1.000	1.000	1.000	1.003	1.006	1.009
0.20	0.998	0.999	1.000	1.010	1.021	1.034
0.30	0.989	0.992	1.000	0.016	1.038	1.063
0.40	0.969	0.973	0.987	1.011	1.045	1.084
0.50	0.930	0.936	0.954	0.987	1.032	1.083
0.60	0.871	0.877	0.899	0.937	0.991	1.050
0.70	0.788	0.796	0.819	0.861	0.918	0.982
0.80	0.688	0.695	0.718	0.760	0.818	0.880
0.90	0.575	0.581	0.603	0.643	0.698	0.757
1.00	0.458	0.464	0.483	0.519	0.570	0.623
1.10	0.346	0.351	0.368	0.400	0.446	0.495
1.20	0.248	0.251	0.266	0.295	0.337	0.383
1.30	0.167	0.170	0.182	0.209	0.249	0.295
1.40	0.107	0.109	0.121	0.146	0.185	0.233
R = 1.5
0.00	1.000	1.000	1.000	1.000	1.000	1.000
0.10	0.998	0.998	0.997	0.999	0.995	0.993
0.20	0.994	0.993	0.991	0.987	0.981	0.974
0.30	0.989	0.987	0.982	0.974	0.962	0.948
0.40	0.984	0.982	0.974	0.961	0.942	0.919
0.50	0.981	0.978	0.968	0.951	0.925	0.894
0.60	0.979	0.975	0.964	0.944	0.912	0.875
0.70	0.976	0.972	0.960	0.937	0.903	0.862
0.80	0.967	0.963	0.951	0.928	0.893	0.853
0.90	0.948	0.944	0.932	0.911	0.878	0.841
1.00	0.912	0.908	0.898	0.880	0.851	0.819
1.10	0.856	0.853	0.845	0.830	0.807	0.782
1.20	0.779	0.777	0.771	0.761	0.743	0.726
1.30	0.685	0.683	0.679	0.674	0.662	0.652
1.40	0.577	0.576	0.574	0.573	0.567	0.565
1.50	0.465	0.464	0.465	0.467	0.467	0.473
1.60	0.356	0.356	0.359	0.364	0.370	0.382
1.70	0.259	0.259	0.264	0.273	0.284	0.302
1.80	0.178	0.179	0.186	0.198	0.213	0.237
1.90	0.117	0.118	0.127	0.142	0.161	0.190
R = 2.0
0.00	1.000	1.000	1.000	1.000	1.000	1.000
0.10	1.000	1.000	1.000	1.000	1.000	1.002
0.20	1.002	1.001	1.000	1.000	1.000	1.005
0.30	1.004	1.003	1.000	0.998	0.999	1.009
0.40	1.005	1.003	0.998	0.994	0.995	1.009
0.50	1.004	1.000	0.993	0.985	0.986	1.003
0.60	1.000	0.996	0.985	0.974	0.971	0.990
0.70	0.995	0.989	0.976	0.959	0.952	0.970
0.80	0.989	0.983	0.965	0.943	0.930	0.945
0.90	0.984	0.977	0.956	0.929	0.909	0.918
1.00	0.981	0.972	0.950	0.917	0.809	0.983
1.10	0.978	0.970	0.945	0.908	0.865	0.872
1.20	0.974	0.966	0.941	0.900	0.863	0.856
1.30	0.965	0.957	0.932	0.891	0.851	0.841
1.40	0.945	0.938	0.915	0.874	0.835	0.834
1.50	0.910	0.904	0.884	0.845	0.808	0.799
1.60	0.857	0.851	0.834	0.800	0.766	0.761
1.70	0.783	0.778	0.764	0.735	0.708	0.707
1.80	0.691	0.688	0.677	0.654	0.633	0.636
1.90	0.586	0.583	0.576	0.560	0.546	0.554
2.00	0.475	0.473	0.469	0.460	0.454	0.466
2.10	0.366	0.365	0.365	0.362	0.363	0.380
2.20	0.268	0.268	0.271	0.274	0.282	0.303
2.30	0.187	0.187	0.193	0.202	0.215	0.240
2.40	0.124	0.126	0.134	0.147	0.165	0.194
2.50	0.081	0.083	0.094	0.110	0.132	0.162
R = 2.5
0.00	1.000	1.000	1.000	1.000	1.000	1.000
0.10	0.999	0.999	1.000	1.000	1.000	1.999
0.20	0.998	0.998	1.000	1.000	0.998	0.995
0.30	0.996	0.997	1.000	1.000	0.997	0.990
0.40	0.994	0.996	1.000	1.000	0.995	0.984
0.50	0.992	0.996	1.001	1.001	0.994	0.980
0.60	0.992	0.996	1.002	1.002	0.994	0.977
0.70	0.991	0.996	1.004	1.003	0.993	0.875
0.80	0.991	0.996	1.005	1.002	0.991	0.973
0.90	0.990	0.996	1.004	0.999	0.987	0.968
1.00	0.988	0.994	1.000	0.993	0.978	0.960
1.10	0.984	0.989	0.994	0.983	0.966	0.946
1.20	0.980	0.984	0.986	0.970	0.949	0.927
1.30	0.947	0.977	0.977	0.956	0.930	0.904
1.40	0.969	0.971	0.968	0.942	0.910	0.879
1.50	0.965	0.967	0.960	0.931	0.894	0.856
1.60	0.962	0.963	0.954	0.921	0.880	0.836
1.70	0.958	0.958	0.948	0.913	0.868	0.820
1.80	0.948	0.949	0.939	0.903	0.856	0.805
1.90	0.930	0.930	0.921	0.886	0.840	0.789
2.00	0.896	0.897	0.890	0.858	0.814	0.765
2.10	0.845	0.846	0.841	0.812	0.773	0.729
2.20	0.774	0.776	0.772	0.749	0.716	0.679
2.30	0.685	0.687	0.686	0.668	0.642	0.614
2.40	0.582	0.585	0.586	0.573	0.556	0.537
2.50	0.474	0.476	0.479	0.473	0.464	0.454
2.60	0.367	0.370	0.375	0.374	0.374	0.373
2.70	0.271	0.273	0.280	0.285	0.292	0.300
2.80	0.190	0.193	0.202	0.211	0.225	0.240
2.90	0.128	0.131	0.142	0.156	0.174	0.196
3.00	0.085	0.088	0.100	0.118	0.140	0.166

∊	R
∊	0.5	1.0	1.5	2.0	2.5
0.0	12.5	20.0	16.7	14.3	14.3
0.1	12.5	20.0	16.7	16.7	14.3
0.2	14.3	16.7	16.7	16.7	14.3
0.3	16.7	14.3	16.7	16.7	14.3
0.4	25.0	11.1	16.7	20.0	14.3
0.5	50.0	10.0	14.3	14.3	16.7

θ/∊	0.0	0.1	0.2	0.3	0.4	0.5
R = 0.5
0.0	1.000	1.000	1.000	1.000	1.000	1.000
0.1	0.975	0.975	0.974	0.973	0.972	0.971
0.2	0.903	0.903	0.900	0.897	0.893	0.889
0.3	0.795	0.793	0.789	0.782	0.774	0.766
0.4	0.662	0.660	0.654	0.644	0.633	0.621
0.5	0.521	0.519	0.510	0.499	0.486	0.475
0.6	0.386	0.383	0.375	0.364	0.352	0.343
0.7	0.268	0.265	0.258	0.249	0.241	0.239
0.8	0.174	0.172	0.167	0.162	0.161	0.166
0.9	0.105	0.104	0.102	0.103	0.110	0.124
R = 1.0
0.0	1.000	1.000	1.000	1.000	1.000	1.000
0.10	1.000	1.000	1.000	1.003	1.006	1.009
0.20	0.998	0.999	1.000	1.010	1.021	1.034
0.30	0.989	0.992	1.000	0.016	1.038	1.063
0.40	0.969	0.973	0.987	1.011	1.045	1.084
0.50	0.930	0.936	0.954	0.987	1.032	1.083
0.60	0.871	0.877	0.899	0.937	0.991	1.050
0.70	0.788	0.796	0.819	0.861	0.918	0.982
0.80	0.688	0.695	0.718	0.760	0.818	0.880
0.90	0.575	0.581	0.603	0.643	0.698	0.757
1.00	0.458	0.464	0.483	0.519	0.570	0.623
1.10	0.346	0.351	0.368	0.400	0.446	0.495
1.20	0.248	0.251	0.266	0.295	0.337	0.383
1.30	0.167	0.170	0.182	0.209	0.249	0.295
1.40	0.107	0.109	0.121	0.146	0.185	0.233
R = 1.5
0.00	1.000	1.000	1.000	1.000	1.000	1.000
0.10	0.998	0.998	0.997	0.999	0.995	0.993
0.20	0.994	0.993	0.991	0.987	0.981	0.974
0.30	0.989	0.987	0.982	0.974	0.962	0.948
0.40	0.984	0.982	0.974	0.961	0.942	0.919
0.50	0.981	0.978	0.968	0.951	0.925	0.894
0.60	0.979	0.975	0.964	0.944	0.912	0.875
0.70	0.976	0.972	0.960	0.937	0.903	0.862
0.80	0.967	0.963	0.951	0.928	0.893	0.853
0.90	0.948	0.944	0.932	0.911	0.878	0.841
1.00	0.912	0.908	0.898	0.880	0.851	0.819
1.10	0.856	0.853	0.845	0.830	0.807	0.782
1.20	0.779	0.777	0.771	0.761	0.743	0.726
1.30	0.685	0.683	0.679	0.674	0.662	0.652
1.40	0.577	0.576	0.574	0.573	0.567	0.565
1.50	0.465	0.464	0.465	0.467	0.467	0.473
1.60	0.356	0.356	0.359	0.364	0.370	0.382
1.70	0.259	0.259	0.264	0.273	0.284	0.302
1.80	0.178	0.179	0.186	0.198	0.213	0.237
1.90	0.117	0.118	0.127	0.142	0.161	0.190
R = 2.0
0.00	1.000	1.000	1.000	1.000	1.000	1.000
0.10	1.000	1.000	1.000	1.000	1.000	1.002
0.20	1.002	1.001	1.000	1.000	1.000	1.005
0.30	1.004	1.003	1.000	0.998	0.999	1.009
0.40	1.005	1.003	0.998	0.994	0.995	1.009
0.50	1.004	1.000	0.993	0.985	0.986	1.003
0.60	1.000	0.996	0.985	0.974	0.971	0.990
0.70	0.995	0.989	0.976	0.959	0.952	0.970
0.80	0.989	0.983	0.965	0.943	0.930	0.945
0.90	0.984	0.977	0.956	0.929	0.909	0.918
1.00	0.981	0.972	0.950	0.917	0.809	0.983
1.10	0.978	0.970	0.945	0.908	0.865	0.872
1.20	0.974	0.966	0.941	0.900	0.863	0.856
1.30	0.965	0.957	0.932	0.891	0.851	0.841
1.40	0.945	0.938	0.915	0.874	0.835	0.834
1.50	0.910	0.904	0.884	0.845	0.808	0.799
1.60	0.857	0.851	0.834	0.800	0.766	0.761
1.70	0.783	0.778	0.764	0.735	0.708	0.707
1.80	0.691	0.688	0.677	0.654	0.633	0.636
1.90	0.586	0.583	0.576	0.560	0.546	0.554
2.00	0.475	0.473	0.469	0.460	0.454	0.466
2.10	0.366	0.365	0.365	0.362	0.363	0.380
2.20	0.268	0.268	0.271	0.274	0.282	0.303
2.30	0.187	0.187	0.193	0.202	0.215	0.240
2.40	0.124	0.126	0.134	0.147	0.165	0.194
2.50	0.081	0.083	0.094	0.110	0.132	0.162
R = 2.5
0.00	1.000	1.000	1.000	1.000	1.000	1.000
0.10	0.999	0.999	1.000	1.000	1.000	1.999
0.20	0.998	0.998	1.000	1.000	0.998	0.995
0.30	0.996	0.997	1.000	1.000	0.997	0.990
0.40	0.994	0.996	1.000	1.000	0.995	0.984
0.50	0.992	0.996	1.001	1.001	0.994	0.980
0.60	0.992	0.996	1.002	1.002	0.994	0.977
0.70	0.991	0.996	1.004	1.003	0.993	0.875
0.80	0.991	0.996	1.005	1.002	0.991	0.973
0.90	0.990	0.996	1.004	0.999	0.987	0.968
1.00	0.988	0.994	1.000	0.993	0.978	0.960
1.10	0.984	0.989	0.994	0.983	0.966	0.946
1.20	0.980	0.984	0.986	0.970	0.949	0.927
1.30	0.947	0.977	0.977	0.956	0.930	0.904
1.40	0.969	0.971	0.968	0.942	0.910	0.879
1.50	0.965	0.967	0.960	0.931	0.894	0.856
1.60	0.962	0.963	0.954	0.921	0.880	0.836
1.70	0.958	0.958	0.948	0.913	0.868	0.820
1.80	0.948	0.949	0.939	0.903	0.856	0.805
1.90	0.930	0.930	0.921	0.886	0.840	0.789
2.00	0.896	0.897	0.890	0.858	0.814	0.765
2.10	0.845	0.846	0.841	0.812	0.773	0.729
2.20	0.774	0.776	0.772	0.749	0.716	0.679
2.30	0.685	0.687	0.686	0.668	0.642	0.614
2.40	0.582	0.585	0.586	0.573	0.556	0.537
2.50	0.474	0.476	0.479	0.473	0.464	0.454
2.60	0.367	0.370	0.375	0.374	0.374	0.373
2.70	0.271	0.273	0.280	0.285	0.292	0.300
2.80	0.190	0.193	0.202	0.211	0.225	0.240
2.90	0.128	0.131	0.142	0.156	0.174	0.196
3.00	0.085	0.088	0.100	0.118	0.140	0.166

∊	R
∊	0.5	1.0	1.5	2.0	2.5
0.0	12.5	20.0	16.7	14.3	14.3
0.1	12.5	20.0	16.7	16.7	14.3
0.2	14.3	16.7	16.7	16.7	14.3
0.3	16.7	14.3	16.7	16.7	14.3
0.4	25.0	11.1	16.7	20.0	14.3
0.5	50.0	10.0	14.3	14.3	16.7

Abstract

Cited By

Figures (7)

Tables (2)

Equations (16)

Applied Optics