Abstract

The transfer function can be determined by convolution of the pupil function over the aperture. The pupil function itself is a function of the design data of the lens system (i.e., refractive indices, radii of curvature, etc.) Of particular importance, both practically and theoretically, is the frequency response on-axis where only rotationally symmetric aberrations are present. The aberration function is obtained from an integration over the ray-trace data and is curve-fitted by Chebyschev interpolation. Unlike the least-squares method, the Chebyschev approach allows a uniform approximation over the interval. This data is substituted into the transfer function which is numerically evaluated by application of very high-order Gauss quadrature theory.

© 1962 Optical Society of America

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References

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  1. P. M. Duffieux, L’integrale de Fourier et ses applications a l’optique (Rennes, 1946).
  2. K. Miyamoto, “Wave Optics and Geometrical Optics in Optical Design,” Progress in Optics, Volume One (North-Holland Publishing Company, Amsterdam, 1961). This article contains an excellent discussion of the general problem as well as a large bibliography.
    [CrossRef]
  3. H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).
  4. M. De, Proc. Roy. Soc. (London) A233, 91 (1955).
  5. G. B. Parrent and C. J. Drane, Optica Acta (Paris) 3, 195 (1956).
    [CrossRef]
  6. G. Black and E. H. Linfoot, Proc. Roy. Soc. (London) A239, 522 (1957).
  7. M. De and B. K. Kath, Optik 15, 739 (1958).
  8. E. L. O’Neill, “Selected Topics in Optics and Communication Theory,” (Itek Corporation, Boston, 1959). Contains an extensive bibliography of papers on the frequency response function up to 1959.
  9. H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).
  10. R. Barakat and M. Morello, J. Opt. Soc. Am. 52, 992 (1962).
    [CrossRef]
  11. W. H. Steel, Rev. optique 32, 4, 143, 269 (1953).
  12. J. Focke, Ber. Verhandl. sächs. Akad. Wiss. Leipzig, Math.-naturw. Kl. 101, 3 (1954).
  13. A. N. Lowan, N. Davids, and L. Levinson, Bull. Am. Math. Soc. 48, 739 (1942).
    [CrossRef]
  14. P. Davis and P. Rabinowitz, J. Research Natl. Bur. Standards 56, 35 (1955).
    [CrossRef]
  15. Z. Kopal, Numerical Analysis (John Wiley & Sons, Inc., New York, 1955).
  16. C. Lanczos, Applied Analysis (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1956).
  17. E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Cambridge University Press, Cambridge, 1931).
  18. H. Mineur, Techniques de calcul numerique (Librarie Polytechnique, Paris, 1952).
  19. The number 20 is somewhat of a compromise chosen because the work was originally programmed for the LGP-30 computer, a relatively slow machine. The program is now being redone on the PDP−1, an extremely fast machine, using 24 points (242=576 points in convolved area) for general use and another using 32 points (322=1024 points) for extermely large aberrations.
  20. A. S. Householder, Principles of Numerical Analysis (McGraw-Hill Book Company, Inc., New York, 1953), p. 244.
  21. C. Lanczos, J. Math. Phys. 17, 123 (1938).
  22. See reference 19.

1962 (1)

1958 (1)

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

1957 (2)

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

G. Black and E. H. Linfoot, Proc. Roy. Soc. (London) A239, 522 (1957).

1956 (1)

G. B. Parrent and C. J. Drane, Optica Acta (Paris) 3, 195 (1956).
[CrossRef]

1955 (3)

H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).

M. De, Proc. Roy. Soc. (London) A233, 91 (1955).

P. Davis and P. Rabinowitz, J. Research Natl. Bur. Standards 56, 35 (1955).
[CrossRef]

1954 (1)

J. Focke, Ber. Verhandl. sächs. Akad. Wiss. Leipzig, Math.-naturw. Kl. 101, 3 (1954).

1953 (1)

W. H. Steel, Rev. optique 32, 4, 143, 269 (1953).

1942 (1)

A. N. Lowan, N. Davids, and L. Levinson, Bull. Am. Math. Soc. 48, 739 (1942).
[CrossRef]

1938 (1)

C. Lanczos, J. Math. Phys. 17, 123 (1938).

Barakat, R.

Black, G.

G. Black and E. H. Linfoot, Proc. Roy. Soc. (London) A239, 522 (1957).

Davids, N.

A. N. Lowan, N. Davids, and L. Levinson, Bull. Am. Math. Soc. 48, 739 (1942).
[CrossRef]

Davis, P.

P. Davis and P. Rabinowitz, J. Research Natl. Bur. Standards 56, 35 (1955).
[CrossRef]

De, M.

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

M. De, Proc. Roy. Soc. (London) A233, 91 (1955).

Drane, C. J.

G. B. Parrent and C. J. Drane, Optica Acta (Paris) 3, 195 (1956).
[CrossRef]

Duffieux, P. M.

P. M. Duffieux, L’integrale de Fourier et ses applications a l’optique (Rennes, 1946).

Focke, J.

J. Focke, Ber. Verhandl. sächs. Akad. Wiss. Leipzig, Math.-naturw. Kl. 101, 3 (1954).

Hobson, E. W.

E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Cambridge University Press, Cambridge, 1931).

Hopkins, H. H.

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

H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).

Householder, A. S.

A. S. Householder, Principles of Numerical Analysis (McGraw-Hill Book Company, Inc., New York, 1953), p. 244.

Kath, B. K.

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

Kopal, Z.

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

Lanczos, C.

C. Lanczos, J. Math. Phys. 17, 123 (1938).

C. Lanczos, Applied Analysis (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1956).

Levinson, L.

A. N. Lowan, N. Davids, and L. Levinson, Bull. Am. Math. Soc. 48, 739 (1942).
[CrossRef]

Linfoot, E. H.

G. Black and E. H. Linfoot, Proc. Roy. Soc. (London) A239, 522 (1957).

Lowan, A. N.

A. N. Lowan, N. Davids, and L. Levinson, Bull. Am. Math. Soc. 48, 739 (1942).
[CrossRef]

Mineur, H.

H. Mineur, Techniques de calcul numerique (Librarie Polytechnique, Paris, 1952).

Miyamoto, K.

K. Miyamoto, “Wave Optics and Geometrical Optics in Optical Design,” Progress in Optics, Volume One (North-Holland Publishing Company, Amsterdam, 1961). This article contains an excellent discussion of the general problem as well as a large bibliography.
[CrossRef]

Morello, M.

O’Neill, E. L.

E. L. O’Neill, “Selected Topics in Optics and Communication Theory,” (Itek Corporation, Boston, 1959). Contains an extensive bibliography of papers on the frequency response function up to 1959.

Parrent, G. B.

G. B. Parrent and C. J. Drane, Optica Acta (Paris) 3, 195 (1956).
[CrossRef]

Rabinowitz, P.

P. Davis and P. Rabinowitz, J. Research Natl. Bur. Standards 56, 35 (1955).
[CrossRef]

Steel, W. H.

W. H. Steel, Rev. optique 32, 4, 143, 269 (1953).

Ber. Verhandl. sächs. Akad. Wiss. Leipzig, Math.-naturw. Kl. (1)

J. Focke, Ber. Verhandl. sächs. Akad. Wiss. Leipzig, Math.-naturw. Kl. 101, 3 (1954).

Bull. Am. Math. Soc. (1)

A. N. Lowan, N. Davids, and L. Levinson, Bull. Am. Math. Soc. 48, 739 (1942).
[CrossRef]

J. Math. Phys. (1)

C. Lanczos, J. Math. Phys. 17, 123 (1938).

J. Opt. Soc. Am. (1)

J. Research Natl. Bur. Standards (1)

P. Davis and P. Rabinowitz, J. Research Natl. Bur. Standards 56, 35 (1955).
[CrossRef]

Optica Acta (Paris) (1)

G. B. Parrent and C. J. Drane, Optica Acta (Paris) 3, 195 (1956).
[CrossRef]

Optik (1)

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

Proc. Phys. Soc. (London) (1)

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

Proc. Roy. Soc. (London) (3)

G. Black and E. H. Linfoot, Proc. Roy. Soc. (London) A239, 522 (1957).

H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).

M. De, Proc. Roy. Soc. (London) A233, 91 (1955).

Rev. optique (1)

W. H. Steel, Rev. optique 32, 4, 143, 269 (1953).

Other (10)

See reference 19.

P. M. Duffieux, L’integrale de Fourier et ses applications a l’optique (Rennes, 1946).

K. Miyamoto, “Wave Optics and Geometrical Optics in Optical Design,” Progress in Optics, Volume One (North-Holland Publishing Company, Amsterdam, 1961). This article contains an excellent discussion of the general problem as well as a large bibliography.
[CrossRef]

E. L. O’Neill, “Selected Topics in Optics and Communication Theory,” (Itek Corporation, Boston, 1959). Contains an extensive bibliography of papers on the frequency response function up to 1959.

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

C. Lanczos, Applied Analysis (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1956).

E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Cambridge University Press, Cambridge, 1931).

H. Mineur, Techniques de calcul numerique (Librarie Polytechnique, Paris, 1952).

The number 20 is somewhat of a compromise chosen because the work was originally programmed for the LGP-30 computer, a relatively slow machine. The program is now being redone on the PDP−1, an extremely fast machine, using 24 points (242=576 points in convolved area) for general use and another using 32 points (322=1024 points) for extermely large aberrations.

A. S. Householder, Principles of Numerical Analysis (McGraw-Hill Book Company, Inc., New York, 1953), p. 244.

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Figures (2)

Fig. 1
Fig. 1

Region of integration for rotationally symmetric transfer function is given by convolution of two circles (shaded area). Because of symmetry only upper-right-hand quadrant (cross-hatched area) needed be computed.

Fig. 2
Fig. 2

Optical-path difference for a test triplet in d light.

Tables (3)

Tables Icon

Table I Gauss quadrature constants for inner integral.

Tables Icon

Table II Gauss quadrature constants for outer integral.

Tables Icon

Table III Comparison of the aberration function and its representation by Chebyschev polynomials.

Equations (57)

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i ( x , y ) = - - t ( x - ζ , y - η ) O ( ζ , η ) d ζ d η
I ( ω x , ω y ) = T ( ω x , ω y ) O ( ω x , ω y ) ;
i ( x , y ) = - - I ( ω x , ω y ) e i ( ω x x + ω y y ) d x d y ,
I ( ω x , ω y ) = 1 ( 2 π ) 2 - - i ( x , y ) e η i ( ω x x + ω y y ) d ω x d ω y ,
t ( x , y ) = a ( x , y ) 2 ,
A ( β , γ ) = A 0 ( β , γ ) e i 2 π W ( β , γ ) ,
β = λ f / γ π x ;             γ = λ f / 2 π y ,
u = a ρ cos φ v = a ρ sin φ             ( 0 ρ 1 ) ,
α 2 = β 2 + γ 2 = ( λ f / 2 π ) 2 a 2 ρ 2 .
a ( x , y ) = 1 B - - A ( β , γ ) e i ( β x + γ y ) d β d γ ,
A ( β , γ ) = 1 ( 2 π ) 2 - - a ( x , y ) e - i ( β x + γ y ) d x d y ,
B = - - A ( β , γ ) d β d γ
T ( ω x , ω y ) = 1 T ( 0 , 0 ) - - A ( β , γ ) × A * ( β - ω x , γ - ω y ) d β d γ ,
T ( 0 , 0 ) = - - A ( β , γ ) 2 d β d γ
T ( ω ) = M 1 2 ω α 0 0 ( α 0 2 - β 2 ) 1 2 A ( β , γ ) A * ( β ω , γ ) d γ d β .
β = α 0 r , γ = α 0 s , ω 0 = α 0 ω ( 0 ω 0 2 ) , Ω = 1 2 α 0 ( 0 Ω 1 ) ;
Ω = ω 0 / 2 λ F * ,
T ( Ω ) = M α 0 2 1 2 ω 1 0 ( 1 - r 2 ) 1 2 A ( r , s ) A * ( r - ω 0 , s ) d s d r .
T ( 0 ) = 1 ,
A ( r , s ) = e i 2 π W ( r , s ) , A * ( r - ω 0 , s ) = e - i 2 π W ( r - ω 0 , s ) .
W ( r , s ) = W 2 ( r 2 + s 2 ) + W 4 ( r 2 + s 2 ) 2 + W 6 ( r 2 + s 2 ) 3 + ,
W ( r , s ) = J = 1 7 W 2 j ( r 2 + s 2 ) 2 j
A ( r , s ) A * ( r - ω 0 , s ) = exp { - i 2 π [ W ( r , s ) + W ( r - ω 0 , s ) ] } = exp [ i 2 π Δ ( r , ω 0 , s ) ] ,
Δ ( r , ω 0 , s ) = W 2 { ( r 2 + s 2 ) - [ ( r - ω 0 ) 2 + s 2 ] } + W 4 { ( r 2 + s 2 ) 2 - [ ( r - ω 0 ) 2 + s 2 ] } + + W 14 { ( r 2 + s 2 ) 7 - [ ( r - ω 0 ) 2 + s 2 ] 7 } .
T ( Ω ) = K 1 2 ω 1 0 ( 1 - r 2 ) 1 2 e i 2 π Δ ( r , ω 0 , s ) d s d r .
T ( Ω ) = K 1 2 ω 0 1 0 ( 1 - r 2 ) 1 2 cos [ 2 π Δ ( r , ω 0 , s ) ] d s d r
T ( Ω ) = ( 2 / π ) [ cos - 1 Ω - Ω ( 1 - Ω 2 ) 1 2 ] .
- 1 1 f ( x ) d x = n = 1 N H n f ( x n ) .
H n = - 1 1 P n ( x ) d x P n ( x n ) ( x - x n ) ,
T ( r , ω 0 ) = 0 ( 1 - r 2 ) 1 2 cos [ 2 π Δ ( r , ω 0 , s ) ] d s .
a b f ( s ) d s = b - a 2 - 1 1 f ( x ) d x ,
s = 1 2 ( b + a ) + 1 2 ( b - a ) x .
T ( r , ω 0 ) = ( 1 - r 2 ) 1 2 2 n = 1 20 H n cos [ 2 π Δ ( r , ω 0 , s n ) ] ,
s n = 1 2 ( 1 + x n ) ( 1 - r 2 ) 1 2
T ( ω 0 ) = 1 2 ω 0 1 ( 1 - r 2 ) 1 2 n = 1 20 H n 2 cos [ 2 π Δ ( r , ω 0 , s n ) ] d r .
D ( r , ω 0 , s n ) = ( 1 - r 2 ) 1 2 n = 1 20 H n 2 cos [ 2 π Δ ( r , ω 0 , s n ) ]
T ( ω 0 ) = 1 2 ω 0 1 D ( r , ω 0 , s n ) d r .
T ( ω 0 ) = ( 2 - ω 0 4 ) m = 1 20 H m D ( r m , ω 0 , s n ) ,
r m = 1 2 ( 1 + x m ) + 1 4 ( 1 - x m ) ω 0 = c m + b m ω 0 .
W = α 2 λ F * 0 1 ( y k + δ l tan μ k ) d ρ ,
W ( β , γ ) = W ( α 2 ) = W ( ρ ) = W 0 + W 2 ρ 2 + W 4 ρ 4 + + W 2 n ρ 2 n + ,
0 1 [ W A ( ρ ) - W N ( ρ ) ] 2 d ρ = min .
W N ( ρ ) = n = 0 N C n T n ( ρ ) ,
W A ( ρ l ) = n = 1 N C n T n ( ρ l )             ( l = 0 , 1 , , N ) .
W A ( ρ ) = n = 0 C n T n ( ρ ) .
= n = N C n T n ( ρ ) .
T N ( ρ ) = 0.
T n ( ρ ) = cos n t ,
t = arc cos { [ 2 ρ - ( b + a ) ] / ( b - a ) } ;
T 0 ( ρ ) = 1 , T 1 ( ρ ) = ρ , T 2 ( ρ ) = 2 ρ 2 - 1 , T 3 ( ρ ) = 4 ρ 3 - 3 ρ , T 4 ( ρ ) = 8 ρ 4 - 8 ρ 2 + 1 , T 5 ( ρ ) = 16 ρ 5 - 20 ρ 3 + 5 ρ , T 6 ( ρ ) = 32 ρ 6 - 48 ρ 4 + 18 ρ 2 - 1 ,
T n + 1 ( ρ ) = 2 ( 1 - 2 ρ ) T n ( ρ ) - T n - 1 ( ρ ) .
W n ( ρ ) = 1 2 C 0 + C 0 T 1 ( ρ ) + + C n - 1 T n - 1 ( ρ ) + 1 2 C N T N ( ρ ) .
C k = 2 N α = 0 N W ( ρ α ) cos ( k α π N ) = 1 7 α = 0 14 W ( ρ α ) cos ( k α π 14 ) .
ρ α = cos ( α π / 14 )             ( α = 0 , 1 , , 14 ) .
W ( ρ ) = W 0 + W 2 ρ 2 + + W 14 ρ 14 ,
W 0 = 1 2 C 0 - C 2 + C 4 - C 6 + C 8 - C 10 + C 12 - 1 2 C 14 0 , W 2 = 2 C 2 - 8 C 4 + 18 C 6 - 32 C 8 + 50 C 10 - 72 C 12 + 49 C 14 , W 4 = 8 C 4 - 48 C 6 + 160 C 8 - 400 C 10 + 840 C 12 - 784 C 14 , W 6 = 32 C 6 - 256 C 8 + 1120 C 10 - 3584 C 12 + 4704 C 14 , W 8 = 128 C 8 - 1280 C 10 + 6912 C 12 - 1344 C 14 , W 10 = 512 C 10 - 6144 C 12 + 19712 C 14 , W 12 = 2048 C 12 - 14336 C 14 , W 14 = 4096 C 14 .
W 2 = 5.027044 , W 4 = - 16.39149 , W 6 = 8.351982 , W 8 = 6.345953 , W 10 = - 3.730595 , W 12 = 1.732880 , W 14 = - 0.1089477.