Pupil exploration and wave-front-polynomial fitting algorithms are tools that are often employed in image-quality evaluation techniques, such as optical-transfer-function and point-spread-function calculations. These techniques require that aberration data be determined for a large number of points across the pupil. With optical systems increasing in complexity, it is necessary that these algorithms become more sophisticated to ensure that the proper pupil shapes and aberration maps are used to represent the wave fronts. Such algorithms are described. These algorithms can handle systems that not only lack the symmetry found with the more conventional lens systems but those that also have apertures with unusual shapes. As practical demonstrations the treatments employed in the pupil exploration and the wave-front-polynomial fitting have been applied to various lens arrangements and the results discussed.
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Example 1 is an f/5.0 triplet lens with a semifield of view of 12.5°.
The coefficients are in units of λ = 546.1 nm. Blank entries occur because no coefficient that has two associated digits adding up to a number higher than eight has been included in the table.
Table 3
Wave-Front-Aberration Polynomial Fitting for Type-ii Polynomial Coefficients Illustrating a Poor Fit for Example 1: Typical Rotationally Symmetrical Lensa
τ
Object-Field Azimuth
Type-ii Polynomial Coefficientsb and Their rms Errors
Wn0
Wn2
Wn4
Wn1
Wn3
rms Error
1.000
0.00°
W20 = −11.222
W22 = 2.241
W44 = −0.299
W11 = 0.000
W33 = 0.810
1.099
1.000
0.00°
W40 = 102.627
W42 = −1.511
W31 = −0.664
W53 = −0.470
1.099
1.000
0.00°
W60 = −194.339
W62 = 2.424
W51 = 0.679
1.099
1.000
0.00°
W80 = 105.750
W71 = −0.010
1.099
Example 1 is an f/5.0 triplet lens with a semifield of view of 12.5°.
The coefficients are in units of λ = 546.1 nm. Blank entries occur because no coefficient that has two associated digits adding up to a number higher than eight has been included in the table.
Table 4
Entrance-Pupil-Exploration Data for Example 2: Simple Maksutov Lens Arrangementa
Peripheral Level to Which Pupil Data Were Fitted and Pupil Shape
τ
Object-Field Azimuth
Pupil Parameters
Size of Axes
Pupil-Center Coordinates
Angle ϕ
Error Function
Major a
Minor b
x
y
Outer
Elliptical
0.000
0.00°
1.2740
1.2740
0.0000
0.0000
0.0°
0.7327
Rectangular
0.000
0.00°
1.0031
1.0031
0.0000
0.0000
0.0°
0.0012
Inner
Elliptical
0.000
0.00°
0.6747
0.6747
0.0000
0.0000
0.0°
0.7717
Rectangular
0.000
0.00°
0.5262
0.5262
0.0000
0.0000
0.0°
0.0001
Outer
Elliptical
1.000
0.00°
1.1230
1.0070
0.0000
0.0488
0.0°
0.3406
Rectangular
1.000
0.00°
1.0031
0.8562
0.0000
0.0969
0.0°
0.0164
Inner
Elliptical
1.000
0.00°
0.6123
0.5771
0.0000
0.2736
0.0°
0.1711
Rectangular
1.000
0.00°
0.5262
0.5278
0.0000
0.2766
0.0°
0.0000
Example 2 is an f/2.5 simple Maksutov lens setup that operated over a ±4 field of view.
Table 5
Wave-Front-Aberration Polynomial Fitting for Type-ii Polynomial Coefficients for Example 2: Simple Maksutov Lens Arrangementa
τ
Object-Field Azimuth
Type-ii Polynomial Coefficientsb and Their rms Errors
Wn0
Wn2
Wn4
Wn1
Wn3
rms Error
0.000
0.00°
W20 = −0.006
W22 = 0.000
W44 = 0.000
W11 = 0.000
W33 = 0.000
0.000
0.000
0.00°
W40 = 0.344
W42 = −0.001
W31 = 0.000
W53 = 0.000
0.000
0.000
0.00°
W60 = −0.415
W62 = 0.000
W51 = 0.000
0.000
0.000
0.00°
W80 = −0.057
W71 = 0.000
0.000
1.000
0.00°
W20 = −6.021
W22 = 3.643
W44 = 0.156
W11 = 0.000
W33 = 0.733
0.002
1.000
0.00°
W40 = 0.206
W42 = −0.680
W31 = −1.984
W53 = −0.535
0.002
1.000
0.00°
W60 = −0.799
W62 = 0.224
W51 = 1.167
0.002
1.000
0.00°
W80 = 0.131
W71 = 0.038
0.002
Example 2 is an f/2.5 simple Maksutov lens setup that operated over a ±4° field of view.
The coefficients are in units of λ = 546.1 nm. Blank entries occur because no coefficient that has two associated digits adding up to a number higher than eight has been included in the table.
Table 6
Entrance-Pupil-Exploration Data for Example 3: Off-Axis Paraboloidal Reflectora
Pupil Shape
Peripheral Level to Which Pupil Data Were Fitted
τ
Object-Field Azimuth
Pupil Parameters
Size of Axes
Pupil-Center Coordinates
Angle ϕ
Error Function
Major a
Minor b
x
y
Elliptical
Outer
0.000
0.00°
1.0000
1.0000
0.0000
0.0000
90.0°
0.0000
Rectangular
Outer
0.000
0.00°
1.0000
1.0000
0.0000
0.0000
0.0°
0.3431
Elliptical
Outer
1.000
0.00°
1.0000
1.0000
0.0000
0.0000
90.0°
0.0000
Rectangular
Outer
1.000
0.00°
1.0000
1.0000
0.0000
0.0000
0.0°
0.3431
Example 3 is an f/4.0 off-axis paraboloid reflector, decentered by 25 mm, with a field of view of ±1° at an azimuth of 90°.
Table 7
Wave-Front-Aberration Polynomial Fitting for Type-i, Type-ii, and Type-iii Polynomial Coefficients for Example 3: Off-Axis Paraboloidal Reflectora
Coefficient Type and Value of τ
Object- Field Azimuth
Wn0
Wn2
Wn4
Wn1
Wn3
wn2
wn4
wn1
wn3
rms Error
Type-i
Coefficients
0.000
0.00°
W20 = 0.000
0.000
0.000
0.00°
W40 = 0.000
0.000
0.000
0.00°
W60 = 0.000
0.000
0.000
0.00°
W80 = 0.000
0.000
Type- ii
Coefficients
1.000
90.00°
W20 = −0.422
W22 = −0.532
W44 = −0.105
W31 = −1.799
W33 = 1.032
1.372
1.000
90.00°
W40 = 1.825
W42 = 2.257
W51 = −0.068
W53 = −1.211
1.372
1.000
90.00°
W60 = −4.253
W62 = −1.639
W71 = 0.683
1.372
1.000
90.00°
W80 = 2.665
1.372
Type-iii
Coefficients
1.000
90.00°
W20 = −0.193
W22 = 0.002
W44 = 0.187
W11 = 0.000
W33 = −0.119
w22 = 5.500
w44 = 0.202
w11 = 0.000
w33 = 0.011
0.0060
1.000
90.00°
W40 = 0.025
W42 = −0.202
W31 = −1.299
W53 = 0.068
w42 = −0.228
w31 = − 0.014
w53 = −0.001
0.0060
1.000
90.00°
W60 = −0.041
W62 = 0.014
W51 = 0.137
w62 = 0.152
w51 = −0.020
0.0060
1.000
90.00°
W80 = 0.021
W71 = −0.142
w71 = 0.016
0.0060
Example 3 is an f/4.0 off-axis paraboloid reflector, decentered by 25 mm, with a field of view of ±1° at an azimuth of 90°.
The coefficients are in units of λ = 546.1 nm. Blank spaces occur because no coefficient that has two associated digits adding up to a number higher than eight has been included in the table.
Tables (7)
Table 1
Entrance-Pupil-Exploration Data for Example 1: Typical Rotationally Symmetrical Lensa
Pupil Shape
Peripheral Level to Which Pupil Data Were Fitted
τ
Object-Field Azimuth
Pupil Parameters
Size of Axes
Pupil-Center Coordinates
Angle ϕ
Error Function
Major a
Minor b
x
y
Elliptical
Outer
0.000
0.00°
1.0000
1.0000
0.0000
0.0000
0.0°
0.0000
Rectangular
Outer
0.000
0.00°
1.0000
1.0000
0.0000
0.0000
0.0°
0.3431
Elliptical
Outer
1.000
0.00°
1.0011
0.6888
0.0000
−0.0176
0.0°
0.0041
Rectangular
Outer
1.000
0.00°
1.0078
0.6984
0.0000
−0.0203
0.0°
0.0753
Example 1 is an f/5.0 triplet lens with a semifield of view of 12.5°.
Table 2
Wave-Front-Aberration Polynomial Fitting for Type-i and Type-ii Polynomial Coefficients for Example 1: Typical Rotationally Symmetrical Lensa
Example 1 is an f/5.0 triplet lens with a semifield of view of 12.5°.
The coefficients are in units of λ = 546.1 nm. Blank entries occur because no coefficient that has two associated digits adding up to a number higher than eight has been included in the table.
Table 3
Wave-Front-Aberration Polynomial Fitting for Type-ii Polynomial Coefficients Illustrating a Poor Fit for Example 1: Typical Rotationally Symmetrical Lensa
τ
Object-Field Azimuth
Type-ii Polynomial Coefficientsb and Their rms Errors
Wn0
Wn2
Wn4
Wn1
Wn3
rms Error
1.000
0.00°
W20 = −11.222
W22 = 2.241
W44 = −0.299
W11 = 0.000
W33 = 0.810
1.099
1.000
0.00°
W40 = 102.627
W42 = −1.511
W31 = −0.664
W53 = −0.470
1.099
1.000
0.00°
W60 = −194.339
W62 = 2.424
W51 = 0.679
1.099
1.000
0.00°
W80 = 105.750
W71 = −0.010
1.099
Example 1 is an f/5.0 triplet lens with a semifield of view of 12.5°.
The coefficients are in units of λ = 546.1 nm. Blank entries occur because no coefficient that has two associated digits adding up to a number higher than eight has been included in the table.
Table 4
Entrance-Pupil-Exploration Data for Example 2: Simple Maksutov Lens Arrangementa
Peripheral Level to Which Pupil Data Were Fitted and Pupil Shape
τ
Object-Field Azimuth
Pupil Parameters
Size of Axes
Pupil-Center Coordinates
Angle ϕ
Error Function
Major a
Minor b
x
y
Outer
Elliptical
0.000
0.00°
1.2740
1.2740
0.0000
0.0000
0.0°
0.7327
Rectangular
0.000
0.00°
1.0031
1.0031
0.0000
0.0000
0.0°
0.0012
Inner
Elliptical
0.000
0.00°
0.6747
0.6747
0.0000
0.0000
0.0°
0.7717
Rectangular
0.000
0.00°
0.5262
0.5262
0.0000
0.0000
0.0°
0.0001
Outer
Elliptical
1.000
0.00°
1.1230
1.0070
0.0000
0.0488
0.0°
0.3406
Rectangular
1.000
0.00°
1.0031
0.8562
0.0000
0.0969
0.0°
0.0164
Inner
Elliptical
1.000
0.00°
0.6123
0.5771
0.0000
0.2736
0.0°
0.1711
Rectangular
1.000
0.00°
0.5262
0.5278
0.0000
0.2766
0.0°
0.0000
Example 2 is an f/2.5 simple Maksutov lens setup that operated over a ±4 field of view.
Table 5
Wave-Front-Aberration Polynomial Fitting for Type-ii Polynomial Coefficients for Example 2: Simple Maksutov Lens Arrangementa
τ
Object-Field Azimuth
Type-ii Polynomial Coefficientsb and Their rms Errors
Wn0
Wn2
Wn4
Wn1
Wn3
rms Error
0.000
0.00°
W20 = −0.006
W22 = 0.000
W44 = 0.000
W11 = 0.000
W33 = 0.000
0.000
0.000
0.00°
W40 = 0.344
W42 = −0.001
W31 = 0.000
W53 = 0.000
0.000
0.000
0.00°
W60 = −0.415
W62 = 0.000
W51 = 0.000
0.000
0.000
0.00°
W80 = −0.057
W71 = 0.000
0.000
1.000
0.00°
W20 = −6.021
W22 = 3.643
W44 = 0.156
W11 = 0.000
W33 = 0.733
0.002
1.000
0.00°
W40 = 0.206
W42 = −0.680
W31 = −1.984
W53 = −0.535
0.002
1.000
0.00°
W60 = −0.799
W62 = 0.224
W51 = 1.167
0.002
1.000
0.00°
W80 = 0.131
W71 = 0.038
0.002
Example 2 is an f/2.5 simple Maksutov lens setup that operated over a ±4° field of view.
The coefficients are in units of λ = 546.1 nm. Blank entries occur because no coefficient that has two associated digits adding up to a number higher than eight has been included in the table.
Table 6
Entrance-Pupil-Exploration Data for Example 3: Off-Axis Paraboloidal Reflectora
Pupil Shape
Peripheral Level to Which Pupil Data Were Fitted
τ
Object-Field Azimuth
Pupil Parameters
Size of Axes
Pupil-Center Coordinates
Angle ϕ
Error Function
Major a
Minor b
x
y
Elliptical
Outer
0.000
0.00°
1.0000
1.0000
0.0000
0.0000
90.0°
0.0000
Rectangular
Outer
0.000
0.00°
1.0000
1.0000
0.0000
0.0000
0.0°
0.3431
Elliptical
Outer
1.000
0.00°
1.0000
1.0000
0.0000
0.0000
90.0°
0.0000
Rectangular
Outer
1.000
0.00°
1.0000
1.0000
0.0000
0.0000
0.0°
0.3431
Example 3 is an f/4.0 off-axis paraboloid reflector, decentered by 25 mm, with a field of view of ±1° at an azimuth of 90°.
Table 7
Wave-Front-Aberration Polynomial Fitting for Type-i, Type-ii, and Type-iii Polynomial Coefficients for Example 3: Off-Axis Paraboloidal Reflectora
Coefficient Type and Value of τ
Object- Field Azimuth
Wn0
Wn2
Wn4
Wn1
Wn3
wn2
wn4
wn1
wn3
rms Error
Type-i
Coefficients
0.000
0.00°
W20 = 0.000
0.000
0.000
0.00°
W40 = 0.000
0.000
0.000
0.00°
W60 = 0.000
0.000
0.000
0.00°
W80 = 0.000
0.000
Type- ii
Coefficients
1.000
90.00°
W20 = −0.422
W22 = −0.532
W44 = −0.105
W31 = −1.799
W33 = 1.032
1.372
1.000
90.00°
W40 = 1.825
W42 = 2.257
W51 = −0.068
W53 = −1.211
1.372
1.000
90.00°
W60 = −4.253
W62 = −1.639
W71 = 0.683
1.372
1.000
90.00°
W80 = 2.665
1.372
Type-iii
Coefficients
1.000
90.00°
W20 = −0.193
W22 = 0.002
W44 = 0.187
W11 = 0.000
W33 = −0.119
w22 = 5.500
w44 = 0.202
w11 = 0.000
w33 = 0.011
0.0060
1.000
90.00°
W40 = 0.025
W42 = −0.202
W31 = −1.299
W53 = 0.068
w42 = −0.228
w31 = − 0.014
w53 = −0.001
0.0060
1.000
90.00°
W60 = −0.041
W62 = 0.014
W51 = 0.137
w62 = 0.152
w51 = −0.020
0.0060
1.000
90.00°
W80 = 0.021
W71 = −0.142
w71 = 0.016
0.0060
Example 3 is an f/4.0 off-axis paraboloid reflector, decentered by 25 mm, with a field of view of ±1° at an azimuth of 90°.
The coefficients are in units of λ = 546.1 nm. Blank spaces occur because no coefficient that has two associated digits adding up to a number higher than eight has been included in the table.