Abstract

The transfer function of an optical system in the presence of off-axis aberrations is evaluated via Gauss quadrature. The transfer function in the presence of third-order coma is studied and the phase of the transfer function is shown to be a smooth function. The variance of the wavefront is obtained for all third- and fifth-order aberrations and its use is illustrated for an actual system along with the transfer function.

© 1965 Optical Society of America

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References

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  1. R. Barakat, J. Opt. Soc. Am. 52, 985 (1962).
    [CrossRef]
  2. R. Barakat and M. V. Morello, J. Opt. Soc. Am. 52, 992 (1962).
    [CrossRef]
  3. H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).
  4. R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1244 (1963).
    [CrossRef]
  5. R. Barakat and A. Houston, J. Opt. Soc. Am. 54, 768 (1964).
    [CrossRef]
  6. R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 538, (1965).
    [CrossRef]
  7. R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 881 (1965).
    [CrossRef]
  8. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964) Chap. 6. This is a corrected reprint of Luenburg’s original notes published by Brown University Press, 1944.
  9. The senior author wishes to thank M. Herzberger for a stimulating series of conversations in 1960 on the implications of Luneburg’s work with regard to the mixed characteristic and the usual treatment of wavefront aberrations.
  10. B. R. A. Nijboer, “The diffraction theory of aberrations” doctoral thesis, Groningen, 1942.
  11. H. H. Hopkins, Wave Theory of Aberrations (Oxford University Press, London, 1950).
  12. A. H. Bennett, J. Jupnik, H. Osterberg, and O. W. Richards, Phase Microscopy, Principles and Applications (John Wiley & Sons, Inc., New York, 1951), p. 241.
  13. See Ref. 8, p. 103.
  14. See Ref. 8, p. 270, as well as M. Herzberger, Modern Geometrical Optics (Interscience Publishers, Inc., New York, 1958) Chaps. 26, 27.
  15. The generalization to nonrotationally symmetric wavefronts has been treated by us and preliminary results communicated at the 1964 Fall Meeting of the Optical Society of America, J. Opt. Soc. Am.54, 1407A (1964). We hope to publish the final results shortly.
  16. There are a number of ways of obtaining W for an actual optical system. One common method is to trace rays and then curve fit via least squares: E. Marchand and R. Phillips, Appl. Opt. 2, 359 (1963);W. Brouwer, E. L. O’Neill, and A. Walther, Appl. Opt.,  2, 1239 (1963).Another procedure developed by us and as yet unpublished utilizes certain interpolating properties of the Zernike polynomials. The calculations in this paper for an actual system (see Sec. 6) employed the Buchdahl aberration coefficients. Our lens-design computing facilities directly calculate all the third- and fifth-order aberrations as well as seventh-order spherical aberration as a matter of routine using an expanded version of Buchdahl’s work developed by M. Rimmer of our design department. It was a simple matter to adapt these coefficients to our requirements.
    [CrossRef]
  17. Z. Kopal, Numerical Analysis (John Wiley & Sons, Inc., New York, 1955).
  18. V. I. Krylov, Approximate Calculation of Integrals (The Macmillan Co., New York, 1962).
  19. A. S. Marathay, Proc. Phys. Soc. (London) 74, 721 (1959).
    [CrossRef]
  20. H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).
  21. M. De and B. K. Nath, Optik 15, 739 (1958).
  22. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Inc., Reading, Massachusetts, 1963).
  23. The senior author wishes to thank E. L. O’Neill for a series of discussions concerning De and Nath’s work shortly after its publication. It is largely due to O’Neill that this problem has come to be re-examined by us and independently by DeVelis. DeVelis has kindly informed us (private communication) that he has located the error in their calculations.
  24. A. Maréchal, Rev. Opt. 26, 257 (1947).
  25. We are presently investigating the use of this merit factor in lens design.

1965 (2)

1964 (1)

1963 (2)

1962 (2)

1959 (1)

A. S. Marathay, Proc. Phys. Soc. (London) 74, 721 (1959).
[CrossRef]

1958 (1)

M. De and B. K. Nath, Optik 15, 739 (1958).

1957 (2)

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).

1947 (1)

A. Maréchal, Rev. Opt. 26, 257 (1947).

Barakat, R.

Bennett, A. H.

A. H. Bennett, J. Jupnik, H. Osterberg, and O. W. Richards, Phase Microscopy, Principles and Applications (John Wiley & Sons, Inc., New York, 1951), p. 241.

De, M.

M. De and B. K. Nath, Optik 15, 739 (1958).

Herzberger, M.

See Ref. 8, p. 270, as well as M. Herzberger, Modern Geometrical Optics (Interscience Publishers, Inc., New York, 1958) Chaps. 26, 27.

Hopkins, H. H.

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).

H. H. Hopkins, Wave Theory of Aberrations (Oxford University Press, London, 1950).

Houston, A.

Jupnik, J.

A. H. Bennett, J. Jupnik, H. Osterberg, and O. W. Richards, Phase Microscopy, Principles and Applications (John Wiley & Sons, Inc., New York, 1951), p. 241.

Kopal, Z.

Z. Kopal, Numerical Analysis (John Wiley & Sons, Inc., New York, 1955).

Krylov, V. I.

V. I. Krylov, Approximate Calculation of Integrals (The Macmillan Co., New York, 1962).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964) Chap. 6. This is a corrected reprint of Luenburg’s original notes published by Brown University Press, 1944.

Marathay, A. S.

A. S. Marathay, Proc. Phys. Soc. (London) 74, 721 (1959).
[CrossRef]

Marchand, E.

Maréchal, A.

A. Maréchal, Rev. Opt. 26, 257 (1947).

Morello, M. V.

Nath, B. K.

M. De and B. K. Nath, Optik 15, 739 (1958).

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of aberrations” doctoral thesis, Groningen, 1942.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Inc., Reading, Massachusetts, 1963).

Osterberg, H.

A. H. Bennett, J. Jupnik, H. Osterberg, and O. W. Richards, Phase Microscopy, Principles and Applications (John Wiley & Sons, Inc., New York, 1951), p. 241.

Phillips, R.

Richards, O. W.

A. H. Bennett, J. Jupnik, H. Osterberg, and O. W. Richards, Phase Microscopy, Principles and Applications (John Wiley & Sons, Inc., New York, 1951), p. 241.

Appl. Opt. (1)

J. Opt. Soc. Am. (6)

Optik (1)

M. De and B. K. Nath, Optik 15, 739 (1958).

Proc. Phys. Soc. (London) (3)

A. S. Marathay, Proc. Phys. Soc. (London) 74, 721 (1959).
[CrossRef]

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).

Rev. Opt. (1)

A. Maréchal, Rev. Opt. 26, 257 (1947).

Other (13)

We are presently investigating the use of this merit factor in lens design.

Z. Kopal, Numerical Analysis (John Wiley & Sons, Inc., New York, 1955).

V. I. Krylov, Approximate Calculation of Integrals (The Macmillan Co., New York, 1962).

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Inc., Reading, Massachusetts, 1963).

The senior author wishes to thank E. L. O’Neill for a series of discussions concerning De and Nath’s work shortly after its publication. It is largely due to O’Neill that this problem has come to be re-examined by us and independently by DeVelis. DeVelis has kindly informed us (private communication) that he has located the error in their calculations.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964) Chap. 6. This is a corrected reprint of Luenburg’s original notes published by Brown University Press, 1944.

The senior author wishes to thank M. Herzberger for a stimulating series of conversations in 1960 on the implications of Luneburg’s work with regard to the mixed characteristic and the usual treatment of wavefront aberrations.

B. R. A. Nijboer, “The diffraction theory of aberrations” doctoral thesis, Groningen, 1942.

H. H. Hopkins, Wave Theory of Aberrations (Oxford University Press, London, 1950).

A. H. Bennett, J. Jupnik, H. Osterberg, and O. W. Richards, Phase Microscopy, Principles and Applications (John Wiley & Sons, Inc., New York, 1951), p. 241.

See Ref. 8, p. 103.

See Ref. 8, p. 270, as well as M. Herzberger, Modern Geometrical Optics (Interscience Publishers, Inc., New York, 1958) Chaps. 26, 27.

The generalization to nonrotationally symmetric wavefronts has been treated by us and preliminary results communicated at the 1964 Fall Meeting of the Optical Society of America, J. Opt. Soc. Am.54, 1407A (1964). We hope to publish the final results shortly.

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Figures (10)

F. 1
F. 1

Region of integration (shaded area) in the transfer-function integral for a circular aperture.

F. 2
F. 2

Modulus of transfer function of circular aperture in the presence of third-order coma W131 = 1.0λ in the paraxial plane W020 = 0; – · – ϕ = 0°, – – – ϕ = 45°, ——— ϕ = 90°.

F. 3
F. 3

Modulus of transfer function of circular aperture in the presence of third-order coma W131 = 2.0λ in the paraxial plane W020 = 0:– · – ϕ = 0°, – – – ϕ = 45°, ——— ϕ = 90°.

F. 4
F. 4

Phase of transfer function of circular aperture in the presence of third-order coma W131 = 1.0λ in the paraxial plane W020 = 0. The phase vanishes identically in the azimuth ϕ = 0°. The curves labeled A are the computed curves:– – – ϕ = 45°, ——— ϕ = 90°. The curves labeled B are the curves A with a linear phase shift subtracted.

F. 5
F. 5

Phase of transfer function of circular aperture in the presence of third-order coma W131 = 2.0λ in the paraxial plane W020 = 0. The phase vanishes identically in the azimuth ϕ = 0°. The curves labeled A are the computed curves: – – – ϕ = 45°, ——— ϕ = 90°. The curves labeled B are the curves A with a linear phase shift subtracted.

F. 6
F. 6

E0 for f/3.5 lens operating at 6° off axis as a function of W020 for the wavelengths: (A) 7000 Å, (B) 6600 Å, (C) 6000 Å, (D) 5400 Å. The approximate minima of E0 are at: (A) W020 = 0.6λ, (B) W020 = 0.4λ, (C) W020 = 0, (D) W020 = −0.7λ.

F. 7
F. 7

Modulus of transfer function of f/3.5 lens operating 6° off axis at 7000 Å in plane of best focus W020 = 0.6 λ(minimum E0): – · – ϕ = 0°, – – – ϕ = 45°, ——— ϕ = 90°.

F. 8
F. 8

Modulus of transfer function of f/3.5 lens operating 6° off axis at 6600 Å in plane of best focus W020 = 0.4λ (minimum E0): – · – ϕ = 0°, – – – ϕ = 45°, ——— ϕ = 90°.

F. 9
F. 9

Modulus of transfer function of f/3.5 lens operating 6° off axis at 6000 Å in plane of best focus W020 = 0 (minimum E0): – · – ϕ = 0°,– – – ϕ = 45°, ——— ϕ = 90°.

F. 10
F. 10

Modulus of transfer function of f/3.5 lens operating 6° off axis at 5400 Å in plane of best focus W020 = 0.7λ (minimum E0): – · – ϕ = 0°, – – – ϕ = 45°, ——— ϕ = 90°.

Tables (1)

Tables Icon

Table I Values of the aberration coefficients in wavelength units for f/3.5 lens operating at 6° off axis in four colors. All data in paraxial plane, W020 = 0.

Equations (36)

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a ( x , y ) = aperture ϕ ( p , q ) e i k W ( x 0 , y 0 , p , q ) e i k ( p x + q y ) d p d q .
ϕ ( p , q ) = A 0 ( p , q ) / ( n 2 p 2 q 2 ) 1 2 ,
ϕ ( p , q ) ~ A 0 ( p , q ) .
x = W / p ; y = W / q .
W = W ( 0 ) + W ( 2 ) + W ( 4 ) + W ( 6 ) + ,
u 1 = x 0 2 + y 0 2 , u 2 = p 2 + q 2 , u 3 = x 0 p + y 0 q .
W 2 = P x 0 2 + W 020 ( p 2 + q 2 ) + M ( x 0 p + y 0 q ) ,
C 311 x 0 3 p = W 311 p ( primary distortion ) , C 040 ( P 2 + q 2 ) 2 = W 040 ( P 2 + q 2 ) 2 ( primary spherical aberration ) , C 131 x 0 p ( P 2 + q 2 ) = W 131 p ( P 2 + q 2 ) ( primary coma ) , C 222 x 0 2 p 2 = W 222 p 2 ( primary astigmatism ) , C 220 x 0 2 ( p 2 + q 2 ) = W 220 ( p 2 + q 2 ) ( primary field curvature ) .
C 420 x 0 4 ( p 2 + q 2 ) = W 420 ( p 2 + q 2 ) ( secondary field curvature ) , C 240 x 0 2 ( p 2 + q 2 ) 2 = W 240 ( p 2 + q 2 ) 2 ( secondary spherical aberration ) , C 060 ( p 2 + q 2 ) 3 = W 060 ( p 2 + q 2 ) 3 ( primary spherical aberration ) , C 511 x 0 5 p = W 511 p ( secondary distortion ) , C 331 x 0 3 p ( p 2 + q 2 ) = W 311 p ( p 2 + q 2 ) ( secondary coma ) , C 151 x 0 p ( p 2 + q 2 ) 2 = W 151 p ( p 2 + q 2 ) 2 ( primary coma ) , C 422 x 0 p 2 = C 422 p 2 ( primary astigmatism ) , C 242 x 0 2 p 2 ( p 2 + q 2 ) = W 242 p 2 ( p 2 + q 2 ) ( secondary astigmatism ) , C 333 x 0 3 p 3 = W 333 p 3 ( trefoil ) .
C 080 ( p 2 + q 2 ) 4 = W 080 ( p 2 + q 2 ) 4 ( special aberration ) ,
W = ( W 131 + W 151 ) q + ( W 222 + W 422 ) q 2 + W 333 q 3 + ( W 220 + W 420 + W 020 ) ( p 2 + q 2 ) + ( W 040 + W 210 ) × ( p 2 + q 2 ) 2 + W 060 ( p 2 + q 2 ) 3 + ( W 131 + W 331 ) × q ( p 2 + q 2 ) + W 151 q ( p 2 + q 2 ) 2 + W 242 q 2 ( p 2 + q 2 ) + W 080 ( p 2 + q 2 ) 4 .
k ( x M x 0 ) p + k ( y M y 0 ) q = υ p p + υ q q ,
a ( υ p , υ q ) = e i k W ( x 0 , y 0 , p , q ) e i ( υ p p + υ q q ) d p d q .
p = p / p max q = q / q max , υ p = υ p p max , υ q = υ q q max .
a ( υ p , υ q ) = aperture e i k W ( x 0 , y 0 , p , q ) × e i ( υ p p + υ q q ) d p d q .
t ( υ p , υ q ) = | a ( υ p , υ q ) / a ( 0 , 0 ) Ai | 2 ,
T ( ω p , ω q ) = [ T ( 0 , 0 ) ] 1 A ( p + 1 2 ω p , q + 1 2 ω q ) × A * ( p 1 2 ω p , q 1 2 ω q ) d p d q ,
A ( p , q ) = { 0 p 2 + q 2 > 1 , e i k W ( p , q ) p 2 + q 2 < 1 .
p = α cos ϕ β sin ϕ , q = α sin ϕ + β cos ϕ .
T ( ω , ϕ ) = [ T ( 0 , 0 ) ] 1 A ( α + 1 2 ω , β ) × A * ( α 1 2 ω , β ) d α d β ,
ω = ( ω p 2 + ω q 2 ) 1 2 , ϕ = arctan ( ω p / ω q ) .
Ω = ω / 2 λ F ,
T ( ω , ϕ ) = T r ( ω , ϕ ) + i T i ( ω , ϕ ) ,
T r ( ω , ϕ ) = [ T ( 0 , 0 ) ] 1 a a b b cos [ 2 π W ( α + 1 2 ω , β ) 2 π W ( α 1 2 ω , β ) ] d α d β , T i ( ω , ϕ ) = [ T ( 0 , 0 ) ] 1 b a b b sin [ 2 π W ( α + 1 2 ω , β ) 2 π W ( α 1 2 ω , β ) ] d α d β ,
a = ( 1 1 4 ω 2 ) 1 2 , ( 0 | ω | 2 ) b = ( ω β 2 ) 1 2 1 2 ω
b b sin cos [ ] d α = b b F ( ω , ϕ , α , β ) d α ,
b b F ( ω , ϕ , α , β ) d α = b 1 1 F ( ω , ϕ , s , β ) d s ,
b n = 1 N H n F ( ω , ϕ , α n , β ) .
a a [ ( ω β 2 ) 1 2 1 2 ω ] n = 1 20 H n F ( ω , ϕ , α n , β ) d β .
a a G ( ω , ϕ , β ) d β = a 1 1 G ( ω , ϕ , t ) d t ,
( 1 1 4 ω 2 ) 1 2 m = 1 N H m G ( ω , ϕ , β m ) .
SC [ 1 1 2 k 2 E 0 ] 2 .
E 0 = 1 π 0 2 π 0 1 [ W ( ρ , θ ) ] 2 ρ d ρ d θ 1 π 2 [ 0 2 π 0 1 W ( ρ , θ ) ρ d ρ d θ ] 2 ,
W 151 p ( p 2 + q 2 ) = W 151 ρ 3 cos θ .
E 0 = 1 / 4 ( W 311 + W 511 ) 2 + 1 / 16 ( W 222 + W 422 ) 2 + 5 / 64 ( W 333 ) 2 + 1 / 12 ( W 020 + W 220 + W 420 ) 2 + 4 / 45 ( W 040 + W 240 ) 2 + 9 / 112 ( W 060 ) 2 + 16 / 225 ( W 080 ) 2 + 1 / 8 ( W 131 + W 331 ) 2 + 1 / 12 ( W 151 ) 2 + 17 / 360 ( W 242 ) 2 + 1 / 4 W 333 ( W 311 + W 511 ) + 1 / 3 ( W 311 + W 511 ) ( W 131 + W 331 ) + 1 / 4 W 151 ( W 311 + W 511 ) + 1 / 12 ( W 222 + W 422 ) ( W 020 + W 220 + W 420 ) + 1 / 12 ( W 222 + W 422 ) ( W 040 + W 240 ) + 3 / 40 W 060 ( W 222 + W 422 ) + 1 / 15 W 080 ( W 222 + W 422 ) + 5 / 48 W 242 ( W 222 + W 422 ) + 3 / 16 W 333 ( W 131 + W 331 ) + 3 / 20 W 333 W 151 + 1 / 6 ( W 020 + W 220 + W 420 ) ( W 040 + W 240 ) + 3 / 20 W 060 ( W 020 + W 220 + W 420 ) + 2 / 15 W 080 ( W 020 + W 220 + W 420 ) + 1 / 12 W 242 ( W 020 + W 220 + W 420 ) + 1 / 6 W 060 ( W 040 + W 240 ) + 16 / 105 W 080 ( W 040 + W 240 ) + 4 / 45 W 242 ( W 040 + W 240 ) + 3 / 20 W 060 W 080 + 1 / 12 W 060 W 242 + 8 / 105 W 080 W 242 + 1 / 5 W 151 ( W 131 + W 331 ) .
M = E 0 ( x 0 ) Q ( x 0 ) d x 0 ,