Florian Bociort, Torben B. Andersen, and Leo H. J. F. Beckmann, "High-order optical aberration coefficients: extension to finite objects and to telecentricity in object space," Appl. Opt. 47, 5691-5700 (2008)
We extend the method for the automatic computation of high-order optical aberration coefficients to include (1) a finite object distance and (2) an infinite entrance pupil position (telecentricity in object space). We present coefficients of the power series expansion of the transverse aberration vector with respect to the normalized aperture and field coordinates. Aberration coefficients of very high order (e.g., 21) can be computed easily and—as shown by comparisons with trigonometric ray tracing—reliably.
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Relationships Between the Third and the Fifth Order Aberration Coefficients Used in [13] and Those Used in the Present Papera
Spherical aberration
bS2
bS2
Coma
bT2
Astigmatism
bS3
Petzval blur
bT3
Distortion
bS4
bT4
Spherical aberration
bS5
bS5
bT5
Coma
bS6
bT6
Sagittal oblique spherical aberration
bS6
bS7
bS10
bT7
bS8
bT8
Sagittal elliptical coma
bS9
bT9
bS8
bS10
Distortion
bT10
The σ coefficients are third order aberrations and the μ coefficients are of fifth order. Fifth order astigmatism and Petzval blur are given by ( and , respectively.
Table 2
Relationships Between the Third and Fifth Order Aberration Coefficients Used by Buchdahl [4] and Those Used in the Present Paper
Andersen
Buchdahl
Andersen
Buchdahl
n
i
j
k
#
n
i
j
k
#
1
1
0
0
bS2
0
1
1
0
0
bT2
_Ap
6
1
0
1
0
bS3
2
1
0
1
0
bT3
_Cp
8
1
0
0
1
bS4
Bp
1
1
0
0
1
bT4
7
2
2
0
0
bS5
12
2
2
0
0
bT5
_S1p
13
2
1
1
0
bS6
16
2
1
1
0
bT6
_S3p
17
2
1
0
1
bS7
S2p
14
2
1
0
1
bT7
15
2
0
2
0
bS8
22
2
0
2
0
bT8
_S6p
23
2
0
1
1
bS9
S5p
20
2
0
1
1
bT9
21
2
0
0
2
bS10
18
2
0
0
2
bT10
_S4p
19
Table 3
Set of Four Indices n, i, j, and k Appearing in and a
1
0000
15
3111
29
4112
43
5221
2
1100
16
3102
30
4103
44
5212
3
1010
17
3030
31
4040
45
5203
4
1001
18
3021
32
4031
46
5140
5
2200
19
3012
33
4022
47
5131
6
2110
20
3003
34
4013
48
5122
7
2101
21
4400
35
4004
49
5113
8
2020
22
4310
36
5500
50
5104
9
2011
23
4301
37
5410
51
5050
10
2002
24
4220
38
5401
52
5041
11
3300
25
4211
39
5320
53
5032
12
3210
26
4202
40
5311
54
5023
13
3201
27
4130
41
5302
55
5014
14
3120
28
4121
42
5230
56
5005
Corresponding to a given cumulative index in Figs. 2, 4, 6.
Using aberrations up to the order shown in the first column. For comparison, the last row gives the values of the x and y components computed by ray tracing. The last column lists the total number of aberration coefficients that were used for the reconstruction.
Gives the total number of aberration coefficients that have been used in the reconstruction.
Table 7
Reconstruction of Transverse Aberration for the System Shown in Fig. 3
Order
x
y
Coefficients
3
5
5
0.00011823761
11
7
0.00022135621
21
9
0.00023401047
36
11
0.00023543557
57
13
0.00023558595
85
15
0.00023560122
121
17
0.00023560275
166
19
0.00023560290
221
21
0.00023560292
287
Ray tracing
0.00023560292
Table 8
Reconstruction of the Transverse Aberration for the System Shown in Fig. 5.
Order
x
y
Coefficients
3
5
5
0.00001715466
0.00001119838
11
7
0.00000616644
0.00001691790
21
9
0.00000215993
0.00000884134
36
11
57
13
85
15
121
17
166
19
221
21
287
Ray tracing
Tables (8)
Table 1
Relationships Between the Third and the Fifth Order Aberration Coefficients Used in [13] and Those Used in the Present Papera
Spherical aberration
bS2
bS2
Coma
bT2
Astigmatism
bS3
Petzval blur
bT3
Distortion
bS4
bT4
Spherical aberration
bS5
bS5
bT5
Coma
bS6
bT6
Sagittal oblique spherical aberration
bS6
bS7
bS10
bT7
bS8
bT8
Sagittal elliptical coma
bS9
bT9
bS8
bS10
Distortion
bT10
The σ coefficients are third order aberrations and the μ coefficients are of fifth order. Fifth order astigmatism and Petzval blur are given by ( and , respectively.
Table 2
Relationships Between the Third and Fifth Order Aberration Coefficients Used by Buchdahl [4] and Those Used in the Present Paper
Andersen
Buchdahl
Andersen
Buchdahl
n
i
j
k
#
n
i
j
k
#
1
1
0
0
bS2
0
1
1
0
0
bT2
_Ap
6
1
0
1
0
bS3
2
1
0
1
0
bT3
_Cp
8
1
0
0
1
bS4
Bp
1
1
0
0
1
bT4
7
2
2
0
0
bS5
12
2
2
0
0
bT5
_S1p
13
2
1
1
0
bS6
16
2
1
1
0
bT6
_S3p
17
2
1
0
1
bS7
S2p
14
2
1
0
1
bT7
15
2
0
2
0
bS8
22
2
0
2
0
bT8
_S6p
23
2
0
1
1
bS9
S5p
20
2
0
1
1
bT9
21
2
0
0
2
bS10
18
2
0
0
2
bT10
_S4p
19
Table 3
Set of Four Indices n, i, j, and k Appearing in and a
1
0000
15
3111
29
4112
43
5221
2
1100
16
3102
30
4103
44
5212
3
1010
17
3030
31
4040
45
5203
4
1001
18
3021
32
4031
46
5140
5
2200
19
3012
33
4022
47
5131
6
2110
20
3003
34
4013
48
5122
7
2101
21
4400
35
4004
49
5113
8
2020
22
4310
36
5500
50
5104
9
2011
23
4301
37
5410
51
5050
10
2002
24
4220
38
5401
52
5041
11
3300
25
4211
39
5320
53
5032
12
3210
26
4202
40
5311
54
5023
13
3201
27
4130
41
5302
55
5014
14
3120
28
4121
42
5230
56
5005
Corresponding to a given cumulative index in Figs. 2, 4, 6.
Using aberrations up to the order shown in the first column. For comparison, the last row gives the values of the x and y components computed by ray tracing. The last column lists the total number of aberration coefficients that were used for the reconstruction.
Gives the total number of aberration coefficients that have been used in the reconstruction.
Table 7
Reconstruction of Transverse Aberration for the System Shown in Fig. 3
Order
x
y
Coefficients
3
5
5
0.00011823761
11
7
0.00022135621
21
9
0.00023401047
36
11
0.00023543557
57
13
0.00023558595
85
15
0.00023560122
121
17
0.00023560275
166
19
0.00023560290
221
21
0.00023560292
287
Ray tracing
0.00023560292
Table 8
Reconstruction of the Transverse Aberration for the System Shown in Fig. 5.