In present research, the influence of higher-order aberrations on the correction of secondary axial color is under investigation. Analytical solutions have so far been restricted to special cases and simple optical systems. Common theories require the tracing of rays of different wavelengths. Such numerical approaches do not support the comprehension of the underlying physical effects. In this paper, a formula for second-order axial color contributions is derived which is based on paraxial ray data for the reference wavelength only. Therefore, it allows the determination of second-order axial color in early paraxial design stages without further numerical ray trace. For systems of thin lenses in air, three second-order effects are identified and discussed using simple examples. A quantitative comparison with intrinsic secondary axial color is given.
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System Data Combined with First- and Second-Order Lens Contributions to of Example A1 (NBK7 Version) for Axial Color between Blue (F-light) and Green (d-light) in Microns
No.
[mm]
Glass
d-Light [mm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
1
10.0
0.0
NBK7
50.00
−539.50
5.82
−533.68
50.0
Overall Sum
−539.50
5.82
−533.68
Paraxial Ray Trace
−533.74
−533.74
Error
−5.76
0.06
Table 2.
System Data Combined with First- and Second-Order Lens Contributions to of Example A1 (NSF6 Version) for Axial Color between Blue (F-Light) and Green (d-Light) in Microns
No.
[mm]
Glass
d-Light [mm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
1
10.0
0.0
NSF6
50.00
−1406.46
39.56
1366.89
50.0
Overall Sum
−1406.46
39.56
−1366.89
Paraxial Ray Trace
−1367.98
−1367.98
Error
−38.48
1.08
Table 3.
System Data Combined with First- and Second-Order Lens Contributions to of Example A2 (Achromatic Doublet) for Axial Color between Blue (F-Light) and Green (d-Light) in Microns
No.
[mm]
Glass
d-Light [mm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
1
10.0
0.0
NBK7
22.18
−1216.14
13.12
−1203.01
0.0
2
0.0
NF2
−39.87
1216.14
−13.12
1203.01
50.0
Overall Sum
0.00
0.00
0.00
Paraxial Ray Trace
0.00
0.00
Error
0.00
0.00
Table 4.
System Data Combined with First- and Second-Order Lens Contributions to of Example A3 (Split Doublet) for Axial Color between Blue (F-Light) and Green (d-Light) in Microns
No.
[mm]
Glass
d-Light [mm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
1
10.0
0.0
NBK7
75.00
−359.67
3.88
−355.78
0.0
2
0.0
NBK7
150.00
−179.83
1.94
−177.89
50.0
Overall Sum
−539.50
5.82
−533.68
Paraxial Ray Trace
−533.74
−533.74
Error
−5.76
0.06
Table 5.
System Data Combined with First- and Second-Order Lens Contributions to of Example A4 (Split Doublet) for Axial Color between Blue (F-Light) and Green (d-Light) in Microns
No.
[mm]
Glass
d-Light [mm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
1
10.0
0.0
NBK7
75.00
−359.67
3.88
−355.78
25.0
2
0.0
NBK7
44.36
−270.25
2.57
−267.68
0.0
3
0.0
NF2
−79.73
270.25
−4.51
265.74
33.3
Overall Sum
−359.67
1.94
−357.73
Paraxial Ray Trace
−357.74
−357.74
Error
−1.93
0.01
Table 6.
System Data of the Schupmann Achromat (B1) Combined with Lens Contributions to for First-Order Primary Axial Color between Blue and Red [, Eq. (1)], First-Order Primary Axial Color between Blue and Green [, Eq. (5)], First-Order Secondary Axial Color between Blue and Green [, Eq. (32)], Second-Order Axial Color Contributions between Blue and Green [, Eq. (27)], and Induced Axial Color According to Wynne between Blue and Green [, Eq. (29)]a
Lens No.
[mm]
Glass
d-Light [mm]
First Order [μm]
First Order [μm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
Induced [μm]
1
10.0
0.0
NBK7
50.00
−194.80
−134.87
6.00
1.46
−133.42
0.00
100.0
2
0.0
NBK7
−50.00
194.80
134.87
−6.00
1.46
136.33
2.91
−25.0
Overall Sum
0.00
0.00
0.00
2.91
2.91
2.91
Paraxial Ray Trace
2.27
2.85
2.85
Error
−2.27
−2.85
0.06
According to first-order theory, all contributions cancel out perfectly. However, paraxial ray trace shows a significant amount of axial color remaining that can only be explained with second-order terms.
Table 7.
System Data of the Schupmann Achromat (B2) Combined with Lens Contributions to Similar to Table 6
Lens No.
[mm]
Glass
d-Light [mm]
First Order [μm]
First Order [μm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
Induced [μm]
1
10.0
0.0
NBK7
50.00
−194.80
−134.87
6.00
1.46
−133.42
0.00
50.0
2
0.0
NBK7
25.00
0.00
0.00
0.00
−2.91
−2.91
0.00
50.0
3
0.0
NBK7
−50.00
194.80
134.87
−6.00
1.46
136.33
−0.03
−25.0
Overall Sum
0.00
0.00
0.00
0.00
0.00
−0.03
Paraxial Ray Trace
−0.03
−0.03
−0.03
Error
0.03
0.03
0.03
Table 8.
System Data of the System C1 Combined with Lens Contributions to Similar to Table 6
Lens No.
[mm]
Glass
d-Light [mm]
First Order [μm]
First Order [μm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
Induced [μm]
1
10.0
0.0
K7
29.85
−1386.59
−963.78
38.94
11.09
−952.69
0.00
0.0
2
0.0
F2
−49.58
1386.59
979.28
−23.44
−11.08
968.19
0.00
25.0
3
0.0
K7
39.80
−462.20
−321.26
12.98
1.85
−319.41
−0.07
0.0
4
0.0
F2
−66.10
462.20
326.43
−7.81
−1.85
324.57
0.08
33.3
Overall Sum
0.00
20.66
20.66
0.01
20.67
0.00
Paraxial Ray Trace
0.00
20.67
20.67
Error
0.00
−0.01
0.00
Table 9.
System Data of the System C2 with Glass K7 in the Front Lens, Combined with Lens Contributions to Similar to Table 6
Lens No.
[mm]
Glass
d-Light [mm]
First Order [μm]
First Order [μm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
Induced [μm]
1
10.0
0.0
K7
75.00
−551.82
−383.55
15.50
4.41
−379.14
0.00
25.0
2
0.0
K7
14.18
−1296.98
−901.49
36.42
9.80
−891.69
5.19
0.0
3
0.0
F2
−16.53
1848.80
1305.71
−31.26
−16.41
1289.30
−7.51
33.3
Overall Sum
0.00
20.66
20.66
−2.19
18.47
−2.33
Paraxial Ray Trace
−1.80
18.45
18.45
Error
1.80
2.21
0.02
Table 10.
System Data of the System C2 with Glass F2 in the Front Lens, Combined with Lens Contributions to Similar to Table 6
Lens No.
[mm]
Glass
d-Light [mm]
First Order [μm]
First Order [μm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
Induced [μm]
1
10.0
0.0
F2
75.00
−916.59
−647.34
15.50
12.57
−634.77
0.00
25.0
2
0.0
K7
9.95
−1848.80
−1285.04
51.92
9.03
−1276.01
12.48
0.0
3
0.0
F2
−11.05
2765.39
1953.05
−46.75
−27.87
1925.17
−18.96
33.3
Overall Sum
0.00
20.66
20.66
−6.27
14.39
−6.49
Paraxial Ray Trace
−5.27
14.32
14.32
Error
5.27
6.34
0.07
Tables (10)
Table 1.
System Data Combined with First- and Second-Order Lens Contributions to of Example A1 (NBK7 Version) for Axial Color between Blue (F-light) and Green (d-light) in Microns
No.
[mm]
Glass
d-Light [mm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
1
10.0
0.0
NBK7
50.00
−539.50
5.82
−533.68
50.0
Overall Sum
−539.50
5.82
−533.68
Paraxial Ray Trace
−533.74
−533.74
Error
−5.76
0.06
Table 2.
System Data Combined with First- and Second-Order Lens Contributions to of Example A1 (NSF6 Version) for Axial Color between Blue (F-Light) and Green (d-Light) in Microns
No.
[mm]
Glass
d-Light [mm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
1
10.0
0.0
NSF6
50.00
−1406.46
39.56
1366.89
50.0
Overall Sum
−1406.46
39.56
−1366.89
Paraxial Ray Trace
−1367.98
−1367.98
Error
−38.48
1.08
Table 3.
System Data Combined with First- and Second-Order Lens Contributions to of Example A2 (Achromatic Doublet) for Axial Color between Blue (F-Light) and Green (d-Light) in Microns
No.
[mm]
Glass
d-Light [mm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
1
10.0
0.0
NBK7
22.18
−1216.14
13.12
−1203.01
0.0
2
0.0
NF2
−39.87
1216.14
−13.12
1203.01
50.0
Overall Sum
0.00
0.00
0.00
Paraxial Ray Trace
0.00
0.00
Error
0.00
0.00
Table 4.
System Data Combined with First- and Second-Order Lens Contributions to of Example A3 (Split Doublet) for Axial Color between Blue (F-Light) and Green (d-Light) in Microns
No.
[mm]
Glass
d-Light [mm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
1
10.0
0.0
NBK7
75.00
−359.67
3.88
−355.78
0.0
2
0.0
NBK7
150.00
−179.83
1.94
−177.89
50.0
Overall Sum
−539.50
5.82
−533.68
Paraxial Ray Trace
−533.74
−533.74
Error
−5.76
0.06
Table 5.
System Data Combined with First- and Second-Order Lens Contributions to of Example A4 (Split Doublet) for Axial Color between Blue (F-Light) and Green (d-Light) in Microns
No.
[mm]
Glass
d-Light [mm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
1
10.0
0.0
NBK7
75.00
−359.67
3.88
−355.78
25.0
2
0.0
NBK7
44.36
−270.25
2.57
−267.68
0.0
3
0.0
NF2
−79.73
270.25
−4.51
265.74
33.3
Overall Sum
−359.67
1.94
−357.73
Paraxial Ray Trace
−357.74
−357.74
Error
−1.93
0.01
Table 6.
System Data of the Schupmann Achromat (B1) Combined with Lens Contributions to for First-Order Primary Axial Color between Blue and Red [, Eq. (1)], First-Order Primary Axial Color between Blue and Green [, Eq. (5)], First-Order Secondary Axial Color between Blue and Green [, Eq. (32)], Second-Order Axial Color Contributions between Blue and Green [, Eq. (27)], and Induced Axial Color According to Wynne between Blue and Green [, Eq. (29)]a
Lens No.
[mm]
Glass
d-Light [mm]
First Order [μm]
First Order [μm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
Induced [μm]
1
10.0
0.0
NBK7
50.00
−194.80
−134.87
6.00
1.46
−133.42
0.00
100.0
2
0.0
NBK7
−50.00
194.80
134.87
−6.00
1.46
136.33
2.91
−25.0
Overall Sum
0.00
0.00
0.00
2.91
2.91
2.91
Paraxial Ray Trace
2.27
2.85
2.85
Error
−2.27
−2.85
0.06
According to first-order theory, all contributions cancel out perfectly. However, paraxial ray trace shows a significant amount of axial color remaining that can only be explained with second-order terms.
Table 7.
System Data of the Schupmann Achromat (B2) Combined with Lens Contributions to Similar to Table 6
Lens No.
[mm]
Glass
d-Light [mm]
First Order [μm]
First Order [μm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
Induced [μm]
1
10.0
0.0
NBK7
50.00
−194.80
−134.87
6.00
1.46
−133.42
0.00
50.0
2
0.0
NBK7
25.00
0.00
0.00
0.00
−2.91
−2.91
0.00
50.0
3
0.0
NBK7
−50.00
194.80
134.87
−6.00
1.46
136.33
−0.03
−25.0
Overall Sum
0.00
0.00
0.00
0.00
0.00
−0.03
Paraxial Ray Trace
−0.03
−0.03
−0.03
Error
0.03
0.03
0.03
Table 8.
System Data of the System C1 Combined with Lens Contributions to Similar to Table 6
Lens No.
[mm]
Glass
d-Light [mm]
First Order [μm]
First Order [μm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
Induced [μm]
1
10.0
0.0
K7
29.85
−1386.59
−963.78
38.94
11.09
−952.69
0.00
0.0
2
0.0
F2
−49.58
1386.59
979.28
−23.44
−11.08
968.19
0.00
25.0
3
0.0
K7
39.80
−462.20
−321.26
12.98
1.85
−319.41
−0.07
0.0
4
0.0
F2
−66.10
462.20
326.43
−7.81
−1.85
324.57
0.08
33.3
Overall Sum
0.00
20.66
20.66
0.01
20.67
0.00
Paraxial Ray Trace
0.00
20.67
20.67
Error
0.00
−0.01
0.00
Table 9.
System Data of the System C2 with Glass K7 in the Front Lens, Combined with Lens Contributions to Similar to Table 6
Lens No.
[mm]
Glass
d-Light [mm]
First Order [μm]
First Order [μm]
First Order [μm]
Second Order [μm]
First + Second Order [μm]
Induced [μm]
1
10.0
0.0
K7
75.00
−551.82
−383.55
15.50
4.41
−379.14
0.00
25.0
2
0.0
K7
14.18
−1296.98
−901.49
36.42
9.80
−891.69
5.19
0.0
3
0.0
F2
−16.53
1848.80
1305.71
−31.26
−16.41
1289.30
−7.51
33.3
Overall Sum
0.00
20.66
20.66
−2.19
18.47
−2.33
Paraxial Ray Trace
−1.80
18.45
18.45
Error
1.80
2.21
0.02
Table 10.
System Data of the System C2 with Glass F2 in the Front Lens, Combined with Lens Contributions to Similar to Table 6