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.

J. Upatnieks, A. Vander Lugt, and E. Leith Appl. Opt. 5(4) 589-593 (1966)

References

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Effective focal length, 45 mm. n_{
e
} 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 α_{
i
}

Position of Missing Corrector Plate x_{
i
}

Position of Surface ν_{
i
}

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., γ_{
i
} = 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 γ_{
i
}

Strength of Asphericities on the Surface in Real Space η_{
i
}

Magnification when Imaging the Surface into Star Space M_{
i
}

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

n_{
e
} is the refractive index for the design wavelength, λ = 564.074 nm.
Axial radius of curvature: z = 5.639959 × 10^{−6}ρ^{4}.
Axial radius of curvature: z = 5.199495 × 10^{−5}ρ^{4}.
Axial radius of curvature: z = −1.095607 × 10^{−4}ρ^{4}.
Axial radius of curvature: z = −7.524076 × 10^{−5}ρ^{4}.

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

n_{
e
} is the refractive index for the design wavelength λ = 564.074 nm.
Axial radius of curvature: z = 1.388521 × 10^{−5}ρ^{4}.
Axial radius of curvature: z = 8.413307 × 10^{−4}ρ^{4}.
Axial radius of curvature: z = 8.808004 × 10^{−4}ρ^{4}.
Axial radius of curvature: z = 3.255072 × 10^{−5}ρ^{4}.

Effective focal length, 45 mm; object position, 79.4561 mm. n_{
e
} 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 Curvature^{
a
}

Effective focal length, 64.6 mm. n_{
d
} is the refractive index for the design wavelength, λ = 587.5618 nm.
Axial radius of curvature: z = −1.083044 × 10^{−5}ρ^{4}.
Axial radius of curvature: z = −8.681021 × 10^{−6}ρ^{4}.
Axial radius of curvature: z = −2.605423 × 10^{−6}ρ^{4}.
Axial radius of curvature: z = 9.486157 × 10^{−6}ρ^{4}.

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 α_{
i
}

Position of Missing Corrector Plate x_{
i
}

Strength of Surface γ_{
i
}

Position of Surface ν_{
i
}

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 γ_{
i
}

Strength of Asphericities on the Surface in Real Space η_{
i
}

Magnification when Imaging the Surface into Star Space M_{
i
}

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

n_{
e
} is the refractive index for the design wavelength, λ = 546.074 nm.
Axial radius of curvature: z = 1.087587 × 10^{−4}ρ^{4}.
Axial radius of curvature: z = 1.559268 × 10^{−4}ρ^{4}.
Axial radius of curvature: z = −7.491885 × 10^{−5}ρ^{4}.
Axial radius of curvature: z = 7.077861 × 10^{−6}ρ^{4}.

Effective focal length, 45 mm. n_{
e
} 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 α_{
i
}

Position of Missing Corrector Plate x_{
i
}

Position of Surface ν_{
i
}

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., γ_{
i
} = 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 γ_{
i
}

Strength of Asphericities on the Surface in Real Space η_{
i
}

Magnification when Imaging the Surface into Star Space M_{
i
}

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

n_{
e
} is the refractive index for the design wavelength, λ = 564.074 nm.
Axial radius of curvature: z = 5.639959 × 10^{−6}ρ^{4}.
Axial radius of curvature: z = 5.199495 × 10^{−5}ρ^{4}.
Axial radius of curvature: z = −1.095607 × 10^{−4}ρ^{4}.
Axial radius of curvature: z = −7.524076 × 10^{−5}ρ^{4}.

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

n_{
e
} is the refractive index for the design wavelength λ = 564.074 nm.
Axial radius of curvature: z = 1.388521 × 10^{−5}ρ^{4}.
Axial radius of curvature: z = 8.413307 × 10^{−4}ρ^{4}.
Axial radius of curvature: z = 8.808004 × 10^{−4}ρ^{4}.
Axial radius of curvature: z = 3.255072 × 10^{−5}ρ^{4}.

Effective focal length, 45 mm; object position, 79.4561 mm. n_{
e
} 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 Curvature^{
a
}

Effective focal length, 64.6 mm. n_{
d
} is the refractive index for the design wavelength, λ = 587.5618 nm.
Axial radius of curvature: z = −1.083044 × 10^{−5}ρ^{4}.
Axial radius of curvature: z = −8.681021 × 10^{−6}ρ^{4}.
Axial radius of curvature: z = −2.605423 × 10^{−6}ρ^{4}.
Axial radius of curvature: z = 9.486157 × 10^{−6}ρ^{4}.

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 α_{
i
}

Position of Missing Corrector Plate x_{
i
}

Strength of Surface γ_{
i
}

Position of Surface ν_{
i
}

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 γ_{
i
}

Strength of Asphericities on the Surface in Real Space η_{
i
}

Magnification when Imaging the Surface into Star Space M_{
i
}

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

n_{
e
} is the refractive index for the design wavelength, λ = 546.074 nm.
Axial radius of curvature: z = 1.087587 × 10^{−4}ρ^{4}.
Axial radius of curvature: z = 1.559268 × 10^{−4}ρ^{4}.
Axial radius of curvature: z = −7.491885 × 10^{−5}ρ^{4}.
Axial radius of curvature: z = 7.077861 × 10^{−6}ρ^{4}.