Abstract

The functional relationship between the total illuminance and transfer function is obtained for systems having rotationally symmetric aberrations. It is shown that the behavior of the transfer function at zero spatial frequency determines the asymptotic behavior of the total illuminance. In addition, the moments of the transfer function determine the behavior of the total illuminance in the vicinity of the origin. Typical numerical results are presented.

© 1963 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. R. Barakat and M. V. Morello, “Computation of the Total Illuminance of an Optical System from the Design Data for Rotationally Symmetric Aberrations” (to be published).
  4. E. L. O’Neill, Selected Topics in Optics and Communication Theory (Boston University Physical Research Laboratory, Boston, 1959).
  5. H. S. Carelaw, Introduction to the Theory of Fourier’s Series and Integrals (Dover Publications, Inc., New York, 1956), 3rd ed., p. 219.
  6. B. van der Pol and H. Bremmer, Operational Calculus Based on the Two-Sided Laplace Transform (Cambridge University Press, Cambridge, England, 1950), Chap. 7.
  7. H. F. Willis, Phil. Mag. 39, 455 (1948). We had derived these relations independently although in a somewhat less satisfactory manner than Willis.
  8. R. Barakat, J. Opt. Soc. Am. 51, 152 (1961).
    [Crossref]
  9. Rayleigh (J. W. Strutt), Phil. Mag. 11, 214 (1881).
  10. T. J. Bromwich, An Introduction to the Theory of Infinite Series (Macmillan and Company Ltd., New York, 1955), 2nd ed., p. 338.
  11. R. K. Luneberg, Mathematical Theory of Optics (Brown University, Providence, Rhode Island, 1944).
  12. R. Barakat, J. Opt. Soc. Am. 52, 264 (1962).
    [Crossref]
  13. R. Barakat and L. Riseberg, “On the Theory of Aberration Balancing” (to be published).
  14. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959), Chap. 9.

1962 (3)

1961 (1)

1948 (1)

H. F. Willis, Phil. Mag. 39, 455 (1948). We had derived these relations independently although in a somewhat less satisfactory manner than Willis.

1881 (1)

Rayleigh (J. W. Strutt), Phil. Mag. 11, 214 (1881).

Rayleigh (J. W. Strutt), Phil. Mag. 11, 214 (1881).

Barakat, R.

R. Barakat, J. Opt. Soc. Am. 52, 985 (1962).
[Crossref]

R. Barakat and M. V. Morello, J. Opt. Soc. Am. 52, 992 (1962).
[Crossref]

R. Barakat, J. Opt. Soc. Am. 52, 264 (1962).
[Crossref]

R. Barakat, J. Opt. Soc. Am. 51, 152 (1961).
[Crossref]

R. Barakat and L. Riseberg, “On the Theory of Aberration Balancing” (to be published).

R. Barakat and M. V. Morello, “Computation of the Total Illuminance of an Optical System from the Design Data for Rotationally Symmetric Aberrations” (to be published).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959), Chap. 9.

Bremmer, H.

B. van der Pol and H. Bremmer, Operational Calculus Based on the Two-Sided Laplace Transform (Cambridge University Press, Cambridge, England, 1950), Chap. 7.

Bromwich, T. J.

T. J. Bromwich, An Introduction to the Theory of Infinite Series (Macmillan and Company Ltd., New York, 1955), 2nd ed., p. 338.

Carelaw, H. S.

H. S. Carelaw, Introduction to the Theory of Fourier’s Series and Integrals (Dover Publications, Inc., New York, 1956), 3rd ed., p. 219.

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (Brown University, Providence, Rhode Island, 1944).

Morello, M. V.

R. Barakat and M. V. Morello, J. Opt. Soc. Am. 52, 992 (1962).
[Crossref]

R. Barakat and M. V. Morello, “Computation of the Total Illuminance of an Optical System from the Design Data for Rotationally Symmetric Aberrations” (to be published).

O’Neill, E. L.

E. L. O’Neill, Selected Topics in Optics and Communication Theory (Boston University Physical Research Laboratory, Boston, 1959).

Rayleigh,

Rayleigh (J. W. Strutt), Phil. Mag. 11, 214 (1881).

Riseberg, L.

R. Barakat and L. Riseberg, “On the Theory of Aberration Balancing” (to be published).

Strutt, J. W.

Rayleigh (J. W. Strutt), Phil. Mag. 11, 214 (1881).

van der Pol, B.

B. van der Pol and H. Bremmer, Operational Calculus Based on the Two-Sided Laplace Transform (Cambridge University Press, Cambridge, England, 1950), Chap. 7.

Willis, H. F.

H. F. Willis, Phil. Mag. 39, 455 (1948). We had derived these relations independently although in a somewhat less satisfactory manner than Willis.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959), Chap. 9.

J. Opt. Soc. Am. (4)

Phil. Mag. (2)

H. F. Willis, Phil. Mag. 39, 455 (1948). We had derived these relations independently although in a somewhat less satisfactory manner than Willis.

Rayleigh (J. W. Strutt), Phil. Mag. 11, 214 (1881).

Other (8)

T. J. Bromwich, An Introduction to the Theory of Infinite Series (Macmillan and Company Ltd., New York, 1955), 2nd ed., p. 338.

R. K. Luneberg, Mathematical Theory of Optics (Brown University, Providence, Rhode Island, 1944).

R. Barakat and M. V. Morello, “Computation of the Total Illuminance of an Optical System from the Design Data for Rotationally Symmetric Aberrations” (to be published).

E. L. O’Neill, Selected Topics in Optics and Communication Theory (Boston University Physical Research Laboratory, Boston, 1959).

H. S. Carelaw, Introduction to the Theory of Fourier’s Series and Integrals (Dover Publications, Inc., New York, 1956), 3rd ed., p. 219.

B. van der Pol and H. Bremmer, Operational Calculus Based on the Two-Sided Laplace Transform (Cambridge University Press, Cambridge, England, 1950), Chap. 7.

R. Barakat and L. Riseberg, “On the Theory of Aberration Balancing” (to be published).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959), Chap. 9.

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Figures (8)

Fig. 1
Fig. 1

Total illuminance in various defocused receiving planes for aberration free slit aperture.

Fig. 2
Fig. 2

Slit-aperture transfer function for optimum balanced third-order spherical aberration.

Fig. 3
Fig. 3

Total illuminance corresponding to transfer-function data of Fig. 2.

Fig. 4
Fig. 4

Slit-aperture transfer function for optimum balanced fifth-order spherical aberration.

Fig. 5
Fig. 5

Total illuminance corresponding to transfer-function data of Fig. 4.

Fig. 6
Fig. 6

Circular-aperture transfer function for optimum balanced fifth-order spherical aberration.

Fig. 7
Fig. 7

Total illuminance corresponding to transfer-function data of Fig. 6.

Fig. 8
Fig. 8

Curves of constant total illuminance for slit aperture having W4=1 as a function of defocusing.

Tables (1)

Tables Icon

Table I Comparison of numerical values of L(v0) for aberration-free circular aperture as determined by exact solution (3.9) and the Tauberian representation (3.8).

Equations (41)

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I ( v x , v y ) = K - - T ( ω x , ω y ) e i ( v x ω x + v y ω y ) d ω x d ω y ,
L ( v x 0 , v y 0 ) = - v x 0 v x 0 - v y 0 v y 0 I ( v x , v y ) d v x d v y ,
L ( v x 0 , v y 0 ) = K - - T ( ω x , ω y ) d ω x d ω y × - v x 0 v x 0 - v y 0 v y 0 e i ( v x ω x + v y ω y ) d v x d v y = 4 K - - T ( ω x , ω y ) sin v x 0 ω x ω x sin v y 0 ω y ω y d ω x d ω y .
L ( v 0 ) = K 0 T ( ω ) sin v 0 ω ω d ω .
lim v 0 0 T ( ω ) sin v 0 ω ω d ω = π T ( 0 ) = π ,
lim v 0 L ( v 0 ) = 1 ,
L ( v 0 ) = 2 π 0 2 T ( ω ) sin v 0 ω ω d ω .
I ( v ) = K 0 T ( ω ) J 0 ( v ω ) ω d ω ,
L ( v 0 ) = 0 v 0 I ( v ) v d v = K 0 T ( ω ) ω d ω 0 v 0 J 0 ( v ω ) v d v = K v 0 0 T ( ω ) J 1 ( v 0 ω ) d ω .
lim v 0 v 0 0 T ( ω ) J 1 ( v 0 ω ) d ω = T ( 0 ) = 1 ,
L ( v 0 ) = v 0 0 2 T ( ω ) J 1 ( v 0 ω ) d ω .
T ( ω ) = 1 2 ( 2 - ω ) ,
L ( v 0 ) = 2 π 0 2 sin v 0 ω ω d ω - 1 π 0 2 sin v 0 ω d ω = 2 π S i ( 2 v 0 ) - 2 π v 0 sin 2 v 0 ,
L ( v 0 ) = K 0 v 0 sin 2 v v 2 d v = 2 π [ S i ( 2 v 0 ) - sin 2 v 0 v 0 ] .
S f ( x ) ~ ( π / 2 ) - ( cos x / x ) .
x 0 f ( s ) J 1 ( x s ) d s ~ f ( 0 ) + f ( 0 ) x - f ( 0 ) 2 x 3 + ,
2 π 0 f ( s ) sin x s s d s ~ f ( 0 ) + 2 π f ( 0 ) x - 2 f ( 0 ) 3 π x 3 + ,
T ( 0 ) = - 2 / π             ( circular aperture ) , T ( 0 ) = - 1 2             ( slit aperture ) .
L ( v 0 ) ~ 1 - ( 2 / π v 0 ) - [ T ( 0 ) / 2 v 0 3 ] +             ( circular aperture ) ,
L ( v 0 ) ~ 1 - ( 1 / π v 0 ) - [ 2 T ( 0 ) / 3 π v 0 3 ] +             ( slit aperture ) .
L ( v 0 ) ~ 1 - ( 2 / π v 0 )             ( circular aperture ) ,
L ( v 0 ) ~ 1 - ( 1 / π v 0 )             ( slit aperture ) .
T ( ω ) = ( 2 / π ) [ cos - 1 ( ω / 2 ) - ( ω / 4 ) ( 4 - ω 2 ) 1 2 ] .
L ( v 0 ) ~ 1 - ( 2 / π v 0 ) - ( 1 / 4 π v 0 3 ) + .
L ( v 0 ) = 2 0 v 0 J 1 2 ( v ) v 2 v d v = 1 - J 0 2 ( v 0 ) - J 1 2 ( v 0 ) .
L ( v 0 ) ~ 1 - ( 1 / π v 0 )
S i ( x ) ~ π 2 - cos x [ 1 x - 2 ! x 2 + ] - sin x [ 1 x 2 - 3 ! x 4 + ] ,
L ( v 0 ) ~ 1 - 1 π v 0 - sin 2 v 0 2 π v 0 2 + cos 2 v 0 2 π v 0 3 + .
T ( ) = L ( 0 ) = 0
L ( v 0 ) ~ 2 π v 0 0 2 T ( ω ) d ω - v 0 3 3 π 0 2 T ( ω ) ω 2 d ω + .
L ( v 0 ) ~ ( 2 v 0 / π ) I ( 0 )             ( slit aperture ) .
L ( v 0 ) ~ v 0 2 2 0 2 T ( ω ) ω d ω - v 0 4 16 0 2 T ( ω ) ω 4 d ω + .
L ( v 0 ) ~ ( v 0 2 / 2 ) I ( 0 )             ( circular aperture ) .
T ( ω ) = 1 2 ω - 1 1 e i k [ W ( ρ ) - W ( ω - ρ ) ] d ρ ,
W ( ρ ) = W 2 ρ 2 + W 4 ρ 4 + W 6 ρ 6 + ,
W 2 = - ( 30 / 35 ) W 4 ;
W 4 = - ( 315 / 231 ) W 6 ;             W 2 = ( 105 / 231 ) W 6 .
W 2 = - W 4 ,
W 4 = - 3 2 W 6 ;             W 2 = 3 5 W 6 .
L ( v 0 ) = 2 π n = 0 N H n T ( ω n ) sin v 0 ω n ω n ,
L ( v 0 ) = v 0 n = 0 N H n T ( ω n ) J 1 ( v 0 ω n ) ,