Abstract

A ray and wavefront analysis is the basis of an interpretation of the aberration balancing merit function in Grey’s lens design program. From each of several field angles, this program traces a coarse grid of rays through an axially symmetric optical system. A polynomial in a region around each ray describes the wave error. This polynomial includes terms (through second order) in aperture, field angle, and wavelength variables. The coefficients of the terms in the polynomial are calculated from real and differential ray trace data. The merit function is the variance of the wave error, obtained by squaring and integrating the polynomials over appropriate ranges of the variables. Distortion is removed from the wave error polynomials and is included in the merit function in a unique manner.

© 1974 Optical Society of America

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References

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  1. T. P. Vogl, L. C. Lintner, A. K. Rigler, The Lens II Optical Design Computer Program for IBM OS/360 (Westinghouse Research Labs., Pittsburgh, Pa., 1967).
  2. D. P. Feder, J. Res. Nat. Bur. Stand. 52, Res. Paper 2471 (1954).
  3. W. Brouwer, E. L. O’Neill, A. Walther, Appl. Opt. 2, 1239 (1963).
    [CrossRef]
  4. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. P., Cambridge1970), Chaps. 1 and 2.
  5. H. H. Hopkins, Opt. Acta 13, 343 (1966).
    [CrossRef]
  6. W. B. King, Appl. Opt. 7, 489 (1968).
    [CrossRef] [PubMed]
  7. J. Meiron, Appl. Opt. 7, 667 (1968).
    [CrossRef] [PubMed]

1968 (2)

1966 (1)

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[CrossRef]

1963 (1)

1954 (1)

D. P. Feder, J. Res. Nat. Bur. Stand. 52, Res. Paper 2471 (1954).

Brouwer, W.

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. P., Cambridge1970), Chaps. 1 and 2.

Feder, D. P.

D. P. Feder, J. Res. Nat. Bur. Stand. 52, Res. Paper 2471 (1954).

Hopkins, H. H.

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[CrossRef]

King, W. B.

Lintner, L. C.

T. P. Vogl, L. C. Lintner, A. K. Rigler, The Lens II Optical Design Computer Program for IBM OS/360 (Westinghouse Research Labs., Pittsburgh, Pa., 1967).

Meiron, J.

O’Neill, E. L.

Rigler, A. K.

T. P. Vogl, L. C. Lintner, A. K. Rigler, The Lens II Optical Design Computer Program for IBM OS/360 (Westinghouse Research Labs., Pittsburgh, Pa., 1967).

Vogl, T. P.

T. P. Vogl, L. C. Lintner, A. K. Rigler, The Lens II Optical Design Computer Program for IBM OS/360 (Westinghouse Research Labs., Pittsburgh, Pa., 1967).

Walther, A.

Appl. Opt. (3)

J. Res. Nat. Bur. Stand. (1)

D. P. Feder, J. Res. Nat. Bur. Stand. 52, Res. Paper 2471 (1954).

Opt. Acta (1)

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[CrossRef]

Other (2)

T. P. Vogl, L. C. Lintner, A. K. Rigler, The Lens II Optical Design Computer Program for IBM OS/360 (Westinghouse Research Labs., Pittsburgh, Pa., 1967).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. P., Cambridge1970), Chaps. 1 and 2.

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Figures (4)

Fig. 1
Fig. 1

Unbarred coordinate systems have z axis along the optical axis; barred ones have z ¯ axis along the base ray (solid line).

Fig. 2
Fig. 2

(a) Monochromatic image errors. Solid and dotted lines represent base ray and differential field ray, respectively. P′ is the Gaussian image point. (b) Chromatic errors. Dotted line represents a differential ray in wavelength.

Fig. 3
Fig. 3

(a) Lateral image shift of ψk from the kth image point P′. Optical path V* → V* + ψkβ′; transverse displacement y′ → ∊y′ − ψk. (b) Longitudinal shift χ of the image plane. Optical path V ¯ *V* + γχ; transverse displacement y′ → y′ + (β′/γ′)χ; the focus errors τ0τ0χ/γ′, σ0σ0χ/γ′.

Fig. 4
Fig. 4

(a) Optical path along the base ray. The distances Ā P ¯ and Ā R ¯ become equal when the wavefront is moved back to infinity. (b) Path change for field angle change. The vectors e, e′ are unit vectors along the ray. The vectors δs, δs′ are the distances P ¯ P ¯ 1 , P ¯ P ¯ 1 .

Tables (1)

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Table I Ray Traces and Associated Ray Trace Data

Equations (42)

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Φ = i k x y ( Δ i k ) 2 d x ¯ d y ¯ ,
Δ = ( a 0 + a 1 y ¯ + a 2 x ¯ + a 3 x ¯ 2 + a 4 x ¯ y ¯ + a 5 y ¯ 2 ) + ( b 0 + b 1 y ¯ + b 2 x ¯ ) δ ϕ + ( c 0 + c 1 y ¯ + c 2 x ¯ ) ω + ( d 0 ) ω δ ϕ + ( e 0 ) δ ϕ 2 + ( f 0 ) ω 2 .
Δ = ( a 0 + a 1 y ¯ + a 3 x ¯ 2 + a 4 y ¯ 2 ) Group 1 + ( b 0 + b 1 y ¯ ) δ ϕ Group 2 + ( c 0 + c 1 y ¯ ) ω Group 3 + ( d 0 ) ω δ ϕ . Group 4
Δ = ( a 0 + a 1 y ¯ + a 2 x ¯ ) + ( c 0 ) ω .
Δ = a 0 + a 1 y ¯ + a 3 x ¯ 2 + a 4 y ¯ 2 .
Δ = ( V * + V ¯ * ) + y y ¯ + 1 / 2 τ 0 y ¯ 2 + 1 / 2 σ 0 x ¯ 2 .
Φ i k = W 0 { [ ( V * - V ¯ * ) + W 1 τ 0 + W 2 σ 0 ] 2 + W 3 γ 2 y 2 + W 4 τ 0 2 + W 5 σ 0 2 } .
Φ = W 0 { [ ( V * - V ¯ * + B k ) + γ χ + β ψ k + W 1 ( τ 0 - χ / γ ) + W 2 ( σ 0 - χ / γ ) ] 2 + W 3 [ γ ( y - ψ k + β χ / γ ) ] 2 + W 4 ( τ 0 - χ / γ ) 2 + W 5 ( σ 0 - χ / γ ) 2 } .
Δ δ ϕ = ( b 0 + b 1 y ¯ ) δ ϕ .
Δ δ ϕ = ( γ y δ ϕ - γ y δ ϕ ) - γ y ϕ θ y .
Φ δ ϕ = W 6 ( { [ γ y - γ ( y - ψ k + χ β / γ ) δ ϕ ] + F k β / γ 2 + B k * } 2 + W 7 ( y ϕ - F k / γ 2 ) 2 ) .
Δ ω = ( c 0 + c 1 y ¯ ) ω .
Δ ω = ( Σ D Δ n - Σ D Δ n ¯ ) - ( γ y ω θ y ) .
Φ ω = W 8 [ ( Σ D Δ n - Σ D Δ n ) 2 ¯ + W 9 γ 2 y ω 2 ] .
Φ w δ ϕ = W 10 [ ( γ y ω δ ϕ ω ) - ( average value ) ] 2 .
a 0 = V * - V ¯ * = limit A R [ P A R - ( P A R ) average ] .
tilt angle = + γ y / A R ,
a 1 y ¯ = + ( γ y / A R ) y ¯ .
θ x = limit A R ( x ¯ / A R ) and θ y = limit A R ( y ¯ / A R ) .
a 1 y ¯ = + ( γ y ) θ y .
a 5 y ¯ 2 = 1 / 2 ( y ¯ ) 2 ( 1 / ( A R + τ 0 ) - 1 / A R ) .
a 5 y ¯ 2 = + ( 1 / 2 ) τ 0 θ y 2 .
Δ = ( V * - V ¯ * ) + γ y θ y + 1 / 2 ( τ 0 θ y 2 + σ 0 θ x 2 ) .
Φ = 4 θ x θ y { [ ( V * - V ¯ * ) + τ 0 θ y 2 / 6 + σ 0 θ x 2 / 6 ] 2 + τ 0 2 θ y 4 / 45 + σ 0 2 θ x 4 / 45 + γ 2 y 2 / 3 } = W 0 { [ ( V * - V ¯ * ) + W 1 τ 0 + W 2 σ 0 ] 2 + W 3 γ 2 y 2 + W 4 τ 0 2 + W 5 σ 0 2 } .
δ V * = V ( P 1 , P 1 ) - V ( P , R ) = e · δ s - e · δ s .
e · δ s = - γ y δ ϕ and e · δ s = - γ y δ ϕ ,
δ V * = γ y δ ϕ - γ y δ ϕ .
y y - ψ k + χ β / γ .
δ V * = γ y δ ϕ - ( y - ψ k + χ β / γ ) γ δ ϕ + F k ( δ ϕ / γ 2 ) β .
b 1 y ¯ δ ϕ = - γ ( y ϕ - F k δ ϕ / γ 2 ) θ y .
Δ δ ϕ = [ γ y δ ϕ - γ ( y - ψ k + χ β / γ ) δ ϕ + F k ( δ ϕ / γ 2 ) β + B k * ] - γ [ y ϕ - F k ( δ ϕ / γ 2 ) ] θ y .
d 0 ω = Σ D Δ n - Σ D Δ n ¯ .
d 1 y ¯ ω = γ y ω θ y .
Φ ω = W 8 [ Σ D Δ n - Σ D Δ n ] 2 ¯ + W 7 ( γ y ω θ y ) 2 .
d 0 ω δ ϕ = - γ y ω δ ϕ ω .
H k = f ( tan ϕ k ) = f β / γ .
distortion = H k + ψ k - f · tan ϕ k = ψ k .
δ H k = ( f / γ 2 ) δ ϕ .
( F k / γ 2 ) δ ϕ .
rate of change = ( F k - f ) / γ 2 .
distortion = ( H k + ψ k - f · tan ϕ k ) + ( F k - f ) δ ϕ / γ 2 .
calibrated distortion = ( H k + ψ k - F * tan ϕ k ) + ( F k - F * ) δ ϕ / γ 2 .

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