The optical see-saw diagram is a method that describes image correction to third-order approximation over a finite field of view in rotationally symmetric systems that employ aspheric surfaces. The aim of this paper is to describe the correction of aberrations caused by plane surfaces in all refracting optical systems in terms of the see-saw diagram. A lens correction algorithm based on the see-saw method is described to correct analytically the Seidel aberrations, primary spherical aberration, coma, astigmatism, and distortion, in such systems. We then apply this lens correction algorithm to the design of equivalent configurations by aspherizing different surfaces of the system, and the high-order aberrations of the equivalent configurations are evaluated by means of transverse-ray-aberration plots. Results indicate that this method gives information on what the contribution must be to the third-order aberrations that each component should provide to the system to give a better balance of high-order aberrations. Examples of the lens correction algorithm applied to lenses with six refracting surfaces and working for both finite and infinite object conjugates are given.
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Effective focal length, 45 mm. ne is the refractive index for the design wavelength, λ = 546.074 nm.
Table 2.
Case 1: Data of the Optical See-saw Diagram for the Triple t in Table 1 when the Object is Located at Infinity
Surface i
Strength of Missing Corrector Plate αi
Position of Missing Corrector Plate xi
Position of Surface νi
1
2.933832 × 10−6
9.86213
−11.93785
2
2.803547 × 10−6
−42.37853
−10.59955
3
−1.181577 × 10−5
−22.51869
−2.53823
4
−3.515516 × 10−6
25.96910
−1.70939
5
0
—
0
6
3.382659 × 10−7
72.13871
6.17491
7
1.054380 × 10−5
−10.49299
8.31590
Note that the strength of the asphericity on each surface is equal to zero because all surfaces are spherical, i.e., γi = 0, i = 1, 2, …, 7.
Table 3.
Case 1: Data for the Strengths of the Asphericities on the Surfaces in Star Space and Real Space for the System as in Table 2
Surface i
Strength of Asphericities on the Surface in Star Space γi
Strength of Asphericities on the Surface in Real Space ηi
Magnification when Imaging the Surface into Star Space Mi
2
−3.400457 × 10−6
−4.039057 × 10−6
1.043964
3
1.178428 × 10−5
3.457628 × 10−5
1.308787
6
−3.532404 × 10−5
−7.846188 × 10−5
1.220807
7
2.565206 × 10−5
5.388368 × 10−5
1.203882
Table 4.
Case 1: Data for the Triplet in Table 1 but Corrected of Seidel Spherical Aberration, Coma, Astigmatism, and Distortion by Aspherizing the Second, Third, Sixth, and Seventh Surfaces
ne is the refractive index for the design wavelength, λ = 564.074 nm.
Axial radius of curvature: z = 5.639959 × 10−6ρ4.
Axial radius of curvature: z = 5.199495 × 10−5ρ4.
Axial radius of curvature: z = −1.095607 × 10−4ρ4.
Axial radius of curvature: z = −7.524076 × 10−5ρ4.
Table 5.
Case 1: Data for the Triplet in Table 1 but Corrected of Seidel Spherical Aberration, Coma, Astigmatism, and Distortion by Aspherizing the Second, Third, Fourth, and Sixth Surfaces
ne is the refractive index for the design wavelength λ = 564.074 nm.
Axial radius of curvature: z = 1.388521 × 10−5ρ4.
Axial radius of curvature: z = 8.413307 × 10−4ρ4.
Axial radius of curvature: z = 8.808004 × 10−4ρ4.
Axial radius of curvature: z = 3.255072 × 10−5ρ4.
Effective focal length, 45 mm; object position, 79.4561 mm. ne is the refractive index for the design wavelength, λ = 546.074 nm.
Table 8.
Case 2: Triplet f/4 Designed for an Infinite Conjugate but Corrected for Primary Spherical Aberration, Coma, Astigmatism, Distortion, and Third-Order Petzval Curvaturea
Effective focal length, 64.6 mm. nd is the refractive index for the design wavelength, λ = 587.5618 nm.
Axial radius of curvature: z = −1.083044 × 10−5ρ4.
Axial radius of curvature: z = −8.681021 × 10−6ρ4.
Axial radius of curvature: z = −2.605423 × 10−6ρ4.
Axial radius of curvature: z = 9.486157 × 10−6ρ4.
Table 9.
Case 2: Data of the Optical See-saw Diagram in Star Space for the System Ideal Lens (Table 7) + Triplet (Table 8)a
Surface i
Strength of Missing Corrector Plate αi
Position of Missing Corrector Plate xi
Strength of Surface γi
Position of Surface νi
1
9.917612 × 10−8
−21.01261
−6.578193 × 10−6
−87.67928
2
−3.798535 × 10−6
−61.88528
4.699896 × 10−6
−84.4779
3
4.574953 × 10−6
−43.17343
−4.945031 × 10−6
−60.48184
4
8.356653 × 10−6
−74.07603
0
−58.68929
5
−2.310922 × 10−6
−57.09503
0
−13.87359
6
2.873048 × 10−7
−98.30444
−3.853019 × 10−7
−2.80958
7
1.904099 × 10−5
−170.91247
0
−182.21704
8
1.532466 × 10−6
−214.82598
0
−181.34727
9
−1.712233 × 10−5
−190.28338
0
−176.66258
10
−3.523677 × 10−5
−165.25779
0
−176.22862
11
0
—
0
0
12
9.186561 × 10−6
−155.22165
0
−172.46961
13
2.673661 × 10−5
−181.27928
0
−171.55143
The separation between the ideal lens and the triplet is 112.3031 mm.
Table 10.
Case 2: Data for the Strengths of the Asphericities on the Surfaces in Star Space and Real Space for the System in Table 9
Surface i
Strength of Asphericities on the Surface in Star-Space γi
Strength of Asphericities on the Surface in Real Space ηi
Magnification when Imaging the Surface into Star Space Mi
7
1.783559 × 10−4
7.788755 × 10−5
−0.812915
8
−2.301524 × 10−4
−1.11667 × 10−4
−0.8345979
10
6.136252 × 10−5
4.982052 × 10−5
−0.9492405
13
−1.370352 × 10−5
−5.068811 × 10−6
−0.7798632
Table 11.
Case 2: Data for the Triplet in Table 7 but Corrected of Seidel Spherical Aberration, Coma, Astigmatism, and Distortion by Aspherizing Four Surfaces
ne is the refractive index for the design wavelength, λ = 546.074 nm.
Axial radius of curvature: z = 1.087587 × 10−4ρ4.
Axial radius of curvature: z = 1.559268 × 10−4ρ4.
Axial radius of curvature: z = −7.491885 × 10−5ρ4.
Axial radius of curvature: z = 7.077861 × 10−6ρ4.
Effective focal length, 45 mm. ne is the refractive index for the design wavelength, λ = 546.074 nm.
Table 2.
Case 1: Data of the Optical See-saw Diagram for the Triple t in Table 1 when the Object is Located at Infinity
Surface i
Strength of Missing Corrector Plate αi
Position of Missing Corrector Plate xi
Position of Surface νi
1
2.933832 × 10−6
9.86213
−11.93785
2
2.803547 × 10−6
−42.37853
−10.59955
3
−1.181577 × 10−5
−22.51869
−2.53823
4
−3.515516 × 10−6
25.96910
−1.70939
5
0
—
0
6
3.382659 × 10−7
72.13871
6.17491
7
1.054380 × 10−5
−10.49299
8.31590
Note that the strength of the asphericity on each surface is equal to zero because all surfaces are spherical, i.e., γi = 0, i = 1, 2, …, 7.
Table 3.
Case 1: Data for the Strengths of the Asphericities on the Surfaces in Star Space and Real Space for the System as in Table 2
Surface i
Strength of Asphericities on the Surface in Star Space γi
Strength of Asphericities on the Surface in Real Space ηi
Magnification when Imaging the Surface into Star Space Mi
2
−3.400457 × 10−6
−4.039057 × 10−6
1.043964
3
1.178428 × 10−5
3.457628 × 10−5
1.308787
6
−3.532404 × 10−5
−7.846188 × 10−5
1.220807
7
2.565206 × 10−5
5.388368 × 10−5
1.203882
Table 4.
Case 1: Data for the Triplet in Table 1 but Corrected of Seidel Spherical Aberration, Coma, Astigmatism, and Distortion by Aspherizing the Second, Third, Sixth, and Seventh Surfaces
ne is the refractive index for the design wavelength, λ = 564.074 nm.
Axial radius of curvature: z = 5.639959 × 10−6ρ4.
Axial radius of curvature: z = 5.199495 × 10−5ρ4.
Axial radius of curvature: z = −1.095607 × 10−4ρ4.
Axial radius of curvature: z = −7.524076 × 10−5ρ4.
Table 5.
Case 1: Data for the Triplet in Table 1 but Corrected of Seidel Spherical Aberration, Coma, Astigmatism, and Distortion by Aspherizing the Second, Third, Fourth, and Sixth Surfaces
ne is the refractive index for the design wavelength λ = 564.074 nm.
Axial radius of curvature: z = 1.388521 × 10−5ρ4.
Axial radius of curvature: z = 8.413307 × 10−4ρ4.
Axial radius of curvature: z = 8.808004 × 10−4ρ4.
Axial radius of curvature: z = 3.255072 × 10−5ρ4.
Effective focal length, 45 mm; object position, 79.4561 mm. ne is the refractive index for the design wavelength, λ = 546.074 nm.
Table 8.
Case 2: Triplet f/4 Designed for an Infinite Conjugate but Corrected for Primary Spherical Aberration, Coma, Astigmatism, Distortion, and Third-Order Petzval Curvaturea
Effective focal length, 64.6 mm. nd is the refractive index for the design wavelength, λ = 587.5618 nm.
Axial radius of curvature: z = −1.083044 × 10−5ρ4.
Axial radius of curvature: z = −8.681021 × 10−6ρ4.
Axial radius of curvature: z = −2.605423 × 10−6ρ4.
Axial radius of curvature: z = 9.486157 × 10−6ρ4.
Table 9.
Case 2: Data of the Optical See-saw Diagram in Star Space for the System Ideal Lens (Table 7) + Triplet (Table 8)a
Surface i
Strength of Missing Corrector Plate αi
Position of Missing Corrector Plate xi
Strength of Surface γi
Position of Surface νi
1
9.917612 × 10−8
−21.01261
−6.578193 × 10−6
−87.67928
2
−3.798535 × 10−6
−61.88528
4.699896 × 10−6
−84.4779
3
4.574953 × 10−6
−43.17343
−4.945031 × 10−6
−60.48184
4
8.356653 × 10−6
−74.07603
0
−58.68929
5
−2.310922 × 10−6
−57.09503
0
−13.87359
6
2.873048 × 10−7
−98.30444
−3.853019 × 10−7
−2.80958
7
1.904099 × 10−5
−170.91247
0
−182.21704
8
1.532466 × 10−6
−214.82598
0
−181.34727
9
−1.712233 × 10−5
−190.28338
0
−176.66258
10
−3.523677 × 10−5
−165.25779
0
−176.22862
11
0
—
0
0
12
9.186561 × 10−6
−155.22165
0
−172.46961
13
2.673661 × 10−5
−181.27928
0
−171.55143
The separation between the ideal lens and the triplet is 112.3031 mm.
Table 10.
Case 2: Data for the Strengths of the Asphericities on the Surfaces in Star Space and Real Space for the System in Table 9
Surface i
Strength of Asphericities on the Surface in Star-Space γi
Strength of Asphericities on the Surface in Real Space ηi
Magnification when Imaging the Surface into Star Space Mi
7
1.783559 × 10−4
7.788755 × 10−5
−0.812915
8
−2.301524 × 10−4
−1.11667 × 10−4
−0.8345979
10
6.136252 × 10−5
4.982052 × 10−5
−0.9492405
13
−1.370352 × 10−5
−5.068811 × 10−6
−0.7798632
Table 11.
Case 2: Data for the Triplet in Table 7 but Corrected of Seidel Spherical Aberration, Coma, Astigmatism, and Distortion by Aspherizing Four Surfaces
ne is the refractive index for the design wavelength, λ = 546.074 nm.
Axial radius of curvature: z = 1.087587 × 10−4ρ4.
Axial radius of curvature: z = 1.559268 × 10−4ρ4.
Axial radius of curvature: z = −7.491885 × 10−5ρ4.
Axial radius of curvature: z = 7.077861 × 10−6ρ4.