Abstract

An equation has been developed that predicts the average size of the spot diagram over the field of view of a lens. The equation is based on an analytic integration of Buchdahl’s aberration polynomials, and is used to supply the figure of merit in an automatic correction computer program. The equation results in aberration balancing through the seventh order, and combines the procedures of optimization and analysis for designers who utilize the geometric spot size as an image quality criterion.

© 1976 Optical Society of America

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References

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  1. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
  2. H. A. Buchdahl, “Optical Aberration Coefficients. III. The Computation of the Tertiary Coefficients,” J. Opt. Soc. Am. 48, 747–756 (1958).
    [Crossref]
  3. G. W. Hopkins, “Proximate Ray Tracing and Optical Aberration Coefficients,” J. Opt. Soc. Am. 66, 405–410 (1976).
    [Crossref]
  4. C. J. Woodruff, “A Comparison, Using Orthogonal Coefficients, of Two Forms of Aberration Balancing,” Opt. Acta 22, 933–941 (1975).
    [Crossref]
  5. F. D. Cruickshank and G. A. Hills, “Use of Optical Aberration Coefficients in Optical Design,” J. Opt. Soc. Am. 50, 379–387 (1960).
    [Crossref]
  6. H. A. Buchdahl, “Optical Aberration Coefficients. V. On the Qualtiy of Predicted Displacements,” J. Opt. Soc. Am. 49, 1113–1121 (1959).
    [Crossref]
  7. P. J. Sands, “Aberration Coefficients and Unusual Coordinates for Specifying Rays,” Appl. Opt. 9, 828–836 (1970).
    [Crossref] [PubMed]
  8. See Ref. 7, “… VI. Aperture Coordinates,” p. 832.
  9. R. B. Johnson, “Polynomial Ray Aberrations Computed in Various Lens Design Programs,” Appl. Opt. 12, 2079–2082 (1973).
    [Crossref] [PubMed]
  10. An accuracy of 823 significant figures means that the 9th place is a random quantity dependent on the machine configuration. In this case, accuracy to 9 places will be obtained 2 times out of 3 on the average, and to 8 places the rest of the time.
  11. A. Cox, A System of Optical Design (Focal, London, 1967).
  12. P. J. Sands, “Aberration Coefficients and Surfaces of Best Focus,” J. Opt. Soc. Am. 63, 582–588 (1973).
    [Crossref]

1976 (1)

1975 (1)

C. J. Woodruff, “A Comparison, Using Orthogonal Coefficients, of Two Forms of Aberration Balancing,” Opt. Acta 22, 933–941 (1975).
[Crossref]

1973 (2)

1970 (1)

1960 (1)

1959 (1)

1958 (1)

Buchdahl, H. A.

Cox, A.

A. Cox, A System of Optical Design (Focal, London, 1967).

Cruickshank, F. D.

Hills, G. A.

Hopkins, G. W.

Johnson, R. B.

Sands, P. J.

Woodruff, C. J.

C. J. Woodruff, “A Comparison, Using Orthogonal Coefficients, of Two Forms of Aberration Balancing,” Opt. Acta 22, 933–941 (1975).
[Crossref]

Appl. Opt. (2)

J. Opt. Soc. Am. (5)

Opt. Acta (1)

C. J. Woodruff, “A Comparison, Using Orthogonal Coefficients, of Two Forms of Aberration Balancing,” Opt. Acta 22, 933–941 (1975).
[Crossref]

Other (4)

An accuracy of 823 significant figures means that the 9th place is a random quantity dependent on the machine configuration. In this case, accuracy to 9 places will be obtained 2 times out of 3 on the average, and to 8 places the rest of the time.

A. Cox, A System of Optical Design (Focal, London, 1967).

See Ref. 7, “… VI. Aperture Coordinates,” p. 832.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

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Tables (4)

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TABLE I Equations for the 37 S(J) coefficients.

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TABLE II Constructional parameters for the lens described in the text. The height of the paraxial axial ray entering the lens was 0.08, and the half-field angle was 17.0°.

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TABLE III The aberration coefficients for the system having the constructional parameters of Table II.

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TABLE IV The 37 S(J) terms and rms spot size computed for the aberration coefficients of Table III.

Equations (27)

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h = h 0 + ( 3 ) + ( 5 ) + ( 7 ) ,
y ( 3 ) = σ 1 cos θ ρ 3 + σ 2 ( 2 + cos 2 θ ) ρ 2 H ¯ + ( 3 σ 3 + σ 4 ) cos θ ρ H ¯ 2 + σ 5 H ¯ 3 , z ( 3 ) = σ 1 sin θ ρ 3 + σ 2 sin 2 θ ρ 2 H ¯ + ( σ 3 + σ 4 ) sin θ ρ H ¯ 2 , y ( 5 ) = μ 1 cos θ ρ 5 + ( μ 2 + μ 3 cos 2 θ ) ρ 4 H ¯ + ( μ 4 + μ 6 cos 2 θ ) cos θ ρ 3 H ¯ 2 + ( μ 7 + μ 8 cos 2 θ ) ρ 2 H ¯ 3 + μ 10 cos θ ρ H ¯ 4 + μ 12 H ¯ 5 , z ( 5 ) = μ 1 sin θ ρ 5 + μ 3 sin 2 θ ρ 4 H ¯ + ( μ 5 + μ 6 cos 2 θ ) sin θ ρ 3 H ¯ 2 + μ 9 sin 2 θ ρ 2 H ¯ 3 + μ 11 sin θ ρ H ¯ 4 , y ( 7 ) = τ 1 cos θ ρ 7 + ( τ 2 + τ 3 cos 2 θ ) ρ 6 H ¯ + ( τ 4 + τ 6 cos 2 θ ) cos θ ρ 5 H ¯ 2 + ( τ 7 + τ 8 cos 2 θ + τ 10 cos 4 θ ) ρ 4 H ¯ 3 + ( τ 11 + τ 12 cos 2 θ ) cos θ ρ 3 H ¯ 4 + ( τ 15 + τ 16 cos 2 θ ) ρ 2 H ¯ 5 + τ 18 cos θ ρ H ¯ 6 + τ 20 H ¯ 7 , z ( 7 ) = τ 1 sin θ ρ 7 + τ 3 sin 2 θ ρ 6 H ¯ + ( τ 5 + τ 6 cos 2 θ ) sin θ ρ 5 H ¯ 2 + ( τ 9 sin 2 θ + τ 10 sin 4 θ ) ρ 4 H ¯ 3 + ( τ 13 + τ 14 cos 2 θ ) sin θ ρ 3 H ¯ 4 + τ 17 sin 2 θ ρ 2 H ¯ 5 + τ 19 sin θ ρ H ¯ 6 .
y ( 3 ) = α 0 + α 1 cos θ + α 2 cos 2 θ , z ( 3 ) = α 3 sin θ + α 2 sin 2 θ , y ( 5 ) = β 0 + β 1 cos θ + β 2 cos 2 θ + β 3 cos 3 θ , z ( 5 ) = β 4 sin θ + β 5 sin 2 θ + β 3 cos 2 θ sin θ , y ( 7 ) = γ 0 + γ 1 cos θ + γ 2 cos 2 θ + γ 3 cos 4 θ + γ 4 cos 3 θ , z ( 7 ) = γ 5 sin θ + γ 6 sin 2 θ + γ 7 sin 4 θ + γ 8 cos 2 θ sin θ ,
σ = ( 0 h ( σ y 2 + σ z 2 ) d H ¯ ) 1 / 2 ,
σ y 2 = 1 ρ 2 2 - ρ 1 2 1 π ρ = ρ 1 ρ 2 θ = 0 2 π ( h y - E [ h y ] ) 2 d θ ρ d ρ ,
σ z 2 = 1 ρ 2 2 - ρ 1 2 1 π ρ 1 ρ 2 0 2 π ( h z - E [ h z ] ) 2 d θ ρ d ρ ,
E [ f ] = 1 ρ 2 2 - ρ 1 2 1 π 0 2 π d θ ρ 1 ρ 2 f ( ρ , θ , H ¯ ) ρ d ρ .
E [ h z ] = h 0 z = 0.
E [ h y ] = 1 ρ 2 2 - ρ 1 2 ( ( h 0 y + σ 5 H ¯ 3 + μ 12 H ¯ 5 + τ 20 H ¯ 7 ) ( ρ 2 2 - ρ 1 2 ) + ( 2 σ 2 H ¯ + μ 7 H ¯ 3 + τ 15 H ¯ 5 ) ρ 2 4 - ρ 1 4 2 + ( μ 2 H ¯ + τ 7 H ¯ 3 ) ρ 2 6 - ρ 1 6 3 + τ 2 H ¯ ρ 2 8 - ρ 1 8 4 ) .
I y = 1 2 π 0 2 π ( h y - E [ h y ] ) 2 d θ = 1 2 π 0 2 π Δ y 2 d θ ,
σ y 2 = 2 ρ 2 2 - ρ 1 2 ρ 1 ρ 2 I y ρ d ρ ,
δ = E [ h y ] - h 0 y
Δ y = y ( 3 ) + y ( 5 ) + y ( 7 ) - δ , Δ z = z ( 3 ) + z ( 5 ) + z ( 7 ) .
Δ y = a 0 + a 1 cos θ + a 2 cos 2 θ + a 3 cos 4 θ + a 4 cos 3 θ - δ , Δ z = b 1 sin θ + b 2 sin 2 θ + b 3 sin 4 θ + b 4 cos 2 θ sin θ ,
a 0 = α 0 + β 0 + γ 0 , a 1 = α 1 + β 1 + γ 1 , a 2 = α 2 + β 2 + γ 2 , a 3 = γ 3 , a 4 = β 3 + γ 4 , b 1 = α 3 + β 4 + γ 5 , b 2 = α 2 + β 5 + γ 6 , b 3 = γ 7 , b 4 = β 3 + γ 8 .
I y = 1 2 π ( 2 a 0 2 - 4 a 0 δ + a 1 2 + 3 2 a 1 a 4 + a 2 2 + a 3 2 + 5 8 a 4 2 + 2 δ 2 )
I z = 1 2 π ( b 1 2 + 1 2 b 1 b 4 + b 2 2 + b 3 2 + 1 8 b 4 2 ) .
σ = W ( J = 1 37 S ( J ) ) 1 / 2 ,
h n = h n / n ,
R m = ( ρ 2 m - ρ 1 m ) / m ,
Q t = ( ρ 2 t + ρ 1 t ) / t ,
X = ( ρ 2 4 + ρ 2 2 ρ 1 2 + ρ 1 4 ) / 3.
U = σ 1 [ 2 R 8 h 1 σ 1 + 4 R 6 h 3 ( 2 σ 3 + σ 4 ) ] + σ 2 2 [ 2 h 3 ( 5 R 6 - 4 R 4 Q 2 ) ] + σ 3 [ 2 R 4 h 5 ( 5 σ 3 + 4 σ 4 ) ] + σ 4 2 [ 2 R 4 h 5 ]
σ = W U ,
U = σ 1 ( 1 4 σ 1 + 4 9 σ 3 + 2 9 σ 4 ) + 2 9 σ 2 2 + σ 3 ( 1 2 σ 3 + 2 5 σ 4 ) + 1 10 σ 4 2 .
U = σ 1 ( 2 R 8 h 1 σ 1 + 4 R 10 h 1 μ 1 + 4 R 12 h 1 τ 1 ) + μ 1 ( 2 R 12 h 1 μ 1 + 4 R 14 h 1 τ 1 ) + τ 1 ( 2 R 16 h 1 τ 1 ) .
U = 1 4 σ 1 2 + 1 6 μ 1 2 + 1 8 τ 1 2 .