Jorge Castro-Ramos,
Oscar de Ita Prieto,
and Gilberto Silva-Ortigoza
J. Castro-Ramos (jcastro@inaoep.mx) is with the Instituto Nacional de Astrofísica, Óptica y Electrónica, Apartado Postal 51 y 216, Tonantzintla, Cholula, Puebla 72000, México.
O. de Ita Prieto and G. Silva-Ortigoza are with the Facultad de Ciencias Físico Matemáticas de la Universidad Autónoma de Puebla, Apartado Postal 1152, Puebla, Puebla 72001, México.
Jorge Castro-Ramos, Oscar de Ita Prieto, and Gilberto Silva-Ortigoza, "Computation of the disk of least confusion for conic mirrors," Appl. Opt. 43, 6080-6089 (2004)
We use geometrical optics to compute, in an exact way and by using the third-order approximation, the disk of least confusion (DLC) or the best image produced by a conic reflector when the point source is located at any position on the optical axis. In the approximate case we obtain analytical formulas to compute the DLC. Furthermore, we apply our equations to particular examples to compare the exact and approximate results.
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Exact and Approximate Results for the Radius and the z Coordinate of the Center of the DLC as Functions of the Position of the Point Source for a Hyperbolic Reflectora
s3
R
Ra
C
Ca
1350
145.4730
111.0700
18,633.5854
16,625.5259
1523
73.8167
68.0696
7673.7130
7448.3922
1746
52.8709
51.1347
4795.2109
4732.2629
1969
43.7194
42.8802
3698.7549
3668.6703
2192
38.2524
37.6998
3120.5682
3102.2116
2415
34.4910
34.0406
2763.3994
2750.5435
22,832
11.2437
10.7856
1332.4752
1331.0523
43,249
10.1441
9.7050
1291.3869
1291.3869
63,666
9.7515
9.3203
1278.8722
1277.6683
84,083
9.5499
9.1230
1271.8873
1270.7118
104,500
9.4272
9.0030
1267.6647
1266.5064
∞
8.9225
8.5101
1250.5300
1249.4400
c = (1/2415)(1/mm), κ = -2, and ρm = D/2 = 735 mm.
Table 2
Exact and Approximate Results for the Radius and the z Coordinate of the Center of the DLC as Functions of the Position of the Point Source for a Parabolic Reflectora
s3
R
Ra
C
Ca
1350
29.8221
30.4475
12,882.7061
12,861.1184
1523
26.7847
26.9889
6483.7222
6471.0202
1746
23.6782
23.5418
4300.5268
4291.3266
1969
21.2187
20.8756
3395.7344
3388.2493
2192
19.2228
18.7518
2900.6874
2894.2858
2415
17.5705
17.0203
2588.4026
2582.7717
22,832
1.9833
1.8003
1284.8220
1284.2944
43,249
1.0510
0.9504
1247.2779
1247.0000
63,666
0.7150
0.6456
1234.2765
1234.0879
84,083
0.5417
0.4888
1227.6805
1227.5378
104,500
0.4361
0.3933
1223.6919
1223.5771
∞
0
0
1207.5000
1207.5000
c = (1/2415)(1/mm), κ = -1, and ρm = D/2 = 735 mm.
Table 3
Exact and Approximate Results for the Radius and the z Coordinate of the Center of the DLC as Functions of the Position of the Point Source for an Ellipsoidal Reflectora
s3
R
Ra
C
Ca
1350
8.6216
9.8638
11,026.9923
10,978.9147
1523
6.1561
6.4485
5977.6633
5982.3342
1746
9.6886
9.7454
4070.9547
3248.0389
1969
10.0356
9.8733
3250.0240
3248.0389
2192
9.5730
9.2779
2792.8894
2790.3229
2415
8.8796
8.5102
2501.5549
2498.8859
22,832
3.0548
2.6924
1259.8901
1260.9156
43,249
3.9054
3.4269
1223.5014
1224.8066
63,666
4.2140
3.6917
1210.8917
1212.2978
84,083
4.3734
3.8282
1204.4928
1205.9507
104,500
4.4707
3.9115
1200.6228
1202.1124
∞
4.8734
4.2550
1184.9100
1186.5300
c = (1/2415)(1/mm), κ = -0.5, and ρm = D/2 = 735 mm.
Table 4
Exact and Approximate Results for the Radius and the z Coordinate of the Center of the DLC as Functions of the Position of the Point Source for a Spherical Reflectora
s3
R
Ra
C
Ca
1350
39.7208
50.1751
9561.7356
9096.7110
1523
12.9837
14.0918
5518.9932
5493.6482
1746
3.9867
4.0510
3107.7688
3850.3903
1969
1.1488
1.1290
3107.7688
3107.8284
2192
0.2039
0.1961
2686.3173
2686.3597
2415
0
0
2415
2415
22,832
8.4076
7.1851
1234.1295
1237.5365
43,249
9.1785
7.8042
1198.9058
1202.6131
63,666
9.4602
8.0291
1186.6913
1190.5075
84,083
9.6061
8.1453
1180.4913
1184.3637
104,500
9.6953
8.2163
1176.7410
1180.6478
∞
10.0656
8.5101
1161.5100
1165.5600
c = (1/2415)(1/mm), κ = 0, and ρm = D/2 = 735 mm.
Table 5
Exact and Approximate Results for the Radius and the z Coordinate of the Center of the DLC as Functions of the Position of the Point Source for an Oblate Spheroidal Reflectora
s3
R
Ra
C
Ca
1350
124.7206
211.4203
5852.8671
1567.8961
1523
79.2989
96.2533
4037.3948
3538.9042
1746
56.8681
59.2366
3061.1533
2968.5176
1969
46.7303
45.1382
2567.2090
2546.9865
2192
41.3088
38.0920
2268.8732
2270.5083
2415
38.1156
34.0406
2069.1193
2079.4565
22,832
33.8826
25.1558
1120.9165
1144.0208
43,249
34.3675
25.3133
1090.4584
1113.8394
63,666
34.5535
25.3784
1079.8643
1103.3468
84,083
34.6516
25.4136
1074.4803
1098.0156
104,500
34.7122
25.4356
1071.2216
1094.7892
∞
34.9679
25.5305
1057.9700
1081.6700
c = (1/2415)(1/mm), κ = 2, and ρm = D/2 = 735 mm.
Tables (5)
Table 1
Exact and Approximate Results for the Radius and the z Coordinate of the Center of the DLC as Functions of the Position of the Point Source for a Hyperbolic Reflectora
s3
R
Ra
C
Ca
1350
145.4730
111.0700
18,633.5854
16,625.5259
1523
73.8167
68.0696
7673.7130
7448.3922
1746
52.8709
51.1347
4795.2109
4732.2629
1969
43.7194
42.8802
3698.7549
3668.6703
2192
38.2524
37.6998
3120.5682
3102.2116
2415
34.4910
34.0406
2763.3994
2750.5435
22,832
11.2437
10.7856
1332.4752
1331.0523
43,249
10.1441
9.7050
1291.3869
1291.3869
63,666
9.7515
9.3203
1278.8722
1277.6683
84,083
9.5499
9.1230
1271.8873
1270.7118
104,500
9.4272
9.0030
1267.6647
1266.5064
∞
8.9225
8.5101
1250.5300
1249.4400
c = (1/2415)(1/mm), κ = -2, and ρm = D/2 = 735 mm.
Table 2
Exact and Approximate Results for the Radius and the z Coordinate of the Center of the DLC as Functions of the Position of the Point Source for a Parabolic Reflectora
s3
R
Ra
C
Ca
1350
29.8221
30.4475
12,882.7061
12,861.1184
1523
26.7847
26.9889
6483.7222
6471.0202
1746
23.6782
23.5418
4300.5268
4291.3266
1969
21.2187
20.8756
3395.7344
3388.2493
2192
19.2228
18.7518
2900.6874
2894.2858
2415
17.5705
17.0203
2588.4026
2582.7717
22,832
1.9833
1.8003
1284.8220
1284.2944
43,249
1.0510
0.9504
1247.2779
1247.0000
63,666
0.7150
0.6456
1234.2765
1234.0879
84,083
0.5417
0.4888
1227.6805
1227.5378
104,500
0.4361
0.3933
1223.6919
1223.5771
∞
0
0
1207.5000
1207.5000
c = (1/2415)(1/mm), κ = -1, and ρm = D/2 = 735 mm.
Table 3
Exact and Approximate Results for the Radius and the z Coordinate of the Center of the DLC as Functions of the Position of the Point Source for an Ellipsoidal Reflectora
s3
R
Ra
C
Ca
1350
8.6216
9.8638
11,026.9923
10,978.9147
1523
6.1561
6.4485
5977.6633
5982.3342
1746
9.6886
9.7454
4070.9547
3248.0389
1969
10.0356
9.8733
3250.0240
3248.0389
2192
9.5730
9.2779
2792.8894
2790.3229
2415
8.8796
8.5102
2501.5549
2498.8859
22,832
3.0548
2.6924
1259.8901
1260.9156
43,249
3.9054
3.4269
1223.5014
1224.8066
63,666
4.2140
3.6917
1210.8917
1212.2978
84,083
4.3734
3.8282
1204.4928
1205.9507
104,500
4.4707
3.9115
1200.6228
1202.1124
∞
4.8734
4.2550
1184.9100
1186.5300
c = (1/2415)(1/mm), κ = -0.5, and ρm = D/2 = 735 mm.
Table 4
Exact and Approximate Results for the Radius and the z Coordinate of the Center of the DLC as Functions of the Position of the Point Source for a Spherical Reflectora
s3
R
Ra
C
Ca
1350
39.7208
50.1751
9561.7356
9096.7110
1523
12.9837
14.0918
5518.9932
5493.6482
1746
3.9867
4.0510
3107.7688
3850.3903
1969
1.1488
1.1290
3107.7688
3107.8284
2192
0.2039
0.1961
2686.3173
2686.3597
2415
0
0
2415
2415
22,832
8.4076
7.1851
1234.1295
1237.5365
43,249
9.1785
7.8042
1198.9058
1202.6131
63,666
9.4602
8.0291
1186.6913
1190.5075
84,083
9.6061
8.1453
1180.4913
1184.3637
104,500
9.6953
8.2163
1176.7410
1180.6478
∞
10.0656
8.5101
1161.5100
1165.5600
c = (1/2415)(1/mm), κ = 0, and ρm = D/2 = 735 mm.
Table 5
Exact and Approximate Results for the Radius and the z Coordinate of the Center of the DLC as Functions of the Position of the Point Source for an Oblate Spheroidal Reflectora
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