Abstract

Apodization theory is concerned with the determination of the distribution of light over the exit pupil of an optical system required in order to achieve a suppression of the side lobes of the diffraction pattern. Here analytic solutions are given to the problem of determining the distribution of light in the exit pupil to concentrate maximally the illuminance in a geometrically similar region of the image plane. Both slit and circular apertures are treated.

© 1965 Optical Society of America

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References

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  1. P. Jacquinot and B. Roizen-Dossier in Progress in Optics, E. Wolf, ed. (North-Holland Publishing Co., Amsterdam, 1964), Vol. III, p. 31.
  2. R. Straubel, Pieter Zeeman. Verhandelingen op 25 Mei 1935 Aangeboden aan Prof. Dr. P. Zeeman (Martinus Nijhoff, The Hague, Netherlands, 1935), p. 302.
  3. R. K. Luneberg, Mathematical Theory of Optics (University of California Press, Berkeley, California, 1964), p. 353.
  4. G. Lansraux and G. Boivin, Can. J. Phys. 39, 158 (1961).
    [Crossref]
  5. R. Barakat, J. Opt. Soc. Am. 52, 264 (1962).
    [Crossref]
  6. D. Slepian, Bell System Tech. J. 43, 3009 (1964).
    [Crossref]
  7. P. Jacquinot and B. Roizen-Dossier, in Progress in OpticsIII edited by E. Wolf (North Holland Publishing Co., Amsterdam, 1964), pp. 47, 78.
  8. D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).
    [Crossref]
  9. H. J. Landau and H. O. Pollak, Bell System Tech. J. 40, 65 (1961);Bell System Tech. J. 41, 1295 (1962).
    [Crossref]
  10. D. Slepian, IRE Trans. PGIT-3, 68 (1954).
  11. J. Meixner and F. W. Schäfke, Mathieusche Funklionen und Sphäroidfunklionen (Springer-Verlag, Berlin, 1954).
    [Crossref]
  12. C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, California, 1957).
  13. J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and P. J. Corbató, Spheroidal Wave Functions (John Wiley & Sons, Inc., New York, 1956).
  14. G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
    [Crossref]
  15. D. Slepian, J. Math. Phys. (MIT) 44, 99 (1965).
  16. W. Fuchs, J. Math. Anal. Appl. 9, 317 (1964).
    [Crossref]
  17. A. G. Fox and Tingye Li, Bell System Tech. J. 40, 453 (1961).
    [Crossref]
  18. J. C. Heurtley, in Proc. Symposium on Quasi-Optics (Polytechnic Press, Brooklyn, New York, 1964), p. 367.
  19. H. Kogelnik, in Advances in Lasers, A. K. Levine, ed. (Dekker Publishers, New York, 1965).
  20. M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959).
  21. G. Szegö, Orthogonal Polynomials (Am. Math. Soc. Colloquium Publications, Vol. XXIII, New York, 1959), Chap. V.

1965 (1)

D. Slepian, J. Math. Phys. (MIT) 44, 99 (1965).

1964 (2)

W. Fuchs, J. Math. Anal. Appl. 9, 317 (1964).
[Crossref]

D. Slepian, Bell System Tech. J. 43, 3009 (1964).
[Crossref]

1962 (1)

1961 (5)

G. Lansraux and G. Boivin, Can. J. Phys. 39, 158 (1961).
[Crossref]

A. G. Fox and Tingye Li, Bell System Tech. J. 40, 453 (1961).
[Crossref]

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).
[Crossref]

H. J. Landau and H. O. Pollak, Bell System Tech. J. 40, 65 (1961);Bell System Tech. J. 41, 1295 (1962).
[Crossref]

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
[Crossref]

1954 (1)

D. Slepian, IRE Trans. PGIT-3, 68 (1954).

Barakat, R.

Boivin, G.

G. Lansraux and G. Boivin, Can. J. Phys. 39, 158 (1961).
[Crossref]

Born, M.

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

Boyd, G. D.

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
[Crossref]

Chu, L. J.

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and P. J. Corbató, Spheroidal Wave Functions (John Wiley & Sons, Inc., New York, 1956).

Corbató, P. J.

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and P. J. Corbató, Spheroidal Wave Functions (John Wiley & Sons, Inc., New York, 1956).

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, California, 1957).

Fox, A. G.

A. G. Fox and Tingye Li, Bell System Tech. J. 40, 453 (1961).
[Crossref]

Fuchs, W.

W. Fuchs, J. Math. Anal. Appl. 9, 317 (1964).
[Crossref]

Gordon, J. P.

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
[Crossref]

Heurtley, J. C.

J. C. Heurtley, in Proc. Symposium on Quasi-Optics (Polytechnic Press, Brooklyn, New York, 1964), p. 367.

Jacquinot, P.

P. Jacquinot and B. Roizen-Dossier, in Progress in OpticsIII edited by E. Wolf (North Holland Publishing Co., Amsterdam, 1964), pp. 47, 78.

P. Jacquinot and B. Roizen-Dossier in Progress in Optics, E. Wolf, ed. (North-Holland Publishing Co., Amsterdam, 1964), Vol. III, p. 31.

Kogelnik, H.

H. Kogelnik, in Advances in Lasers, A. K. Levine, ed. (Dekker Publishers, New York, 1965).

Landau, H. J.

H. J. Landau and H. O. Pollak, Bell System Tech. J. 40, 65 (1961);Bell System Tech. J. 41, 1295 (1962).
[Crossref]

Lansraux, G.

G. Lansraux and G. Boivin, Can. J. Phys. 39, 158 (1961).
[Crossref]

Li, Tingye

A. G. Fox and Tingye Li, Bell System Tech. J. 40, 453 (1961).
[Crossref]

Little, J. D. C.

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and P. J. Corbató, Spheroidal Wave Functions (John Wiley & Sons, Inc., New York, 1956).

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (University of California Press, Berkeley, California, 1964), p. 353.

Meixner, J.

J. Meixner and F. W. Schäfke, Mathieusche Funklionen und Sphäroidfunklionen (Springer-Verlag, Berlin, 1954).
[Crossref]

Morse, P. M.

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and P. J. Corbató, Spheroidal Wave Functions (John Wiley & Sons, Inc., New York, 1956).

Pollak, H. O.

H. J. Landau and H. O. Pollak, Bell System Tech. J. 40, 65 (1961);Bell System Tech. J. 41, 1295 (1962).
[Crossref]

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).
[Crossref]

Roizen-Dossier, B.

P. Jacquinot and B. Roizen-Dossier in Progress in Optics, E. Wolf, ed. (North-Holland Publishing Co., Amsterdam, 1964), Vol. III, p. 31.

P. Jacquinot and B. Roizen-Dossier, in Progress in OpticsIII edited by E. Wolf (North Holland Publishing Co., Amsterdam, 1964), pp. 47, 78.

Schäfke, F. W.

J. Meixner and F. W. Schäfke, Mathieusche Funklionen und Sphäroidfunklionen (Springer-Verlag, Berlin, 1954).
[Crossref]

Slepian, D.

D. Slepian, J. Math. Phys. (MIT) 44, 99 (1965).

D. Slepian, Bell System Tech. J. 43, 3009 (1964).
[Crossref]

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).
[Crossref]

D. Slepian, IRE Trans. PGIT-3, 68 (1954).

Stratton, J. A.

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and P. J. Corbató, Spheroidal Wave Functions (John Wiley & Sons, Inc., New York, 1956).

Straubel, R.

R. Straubel, Pieter Zeeman. Verhandelingen op 25 Mei 1935 Aangeboden aan Prof. Dr. P. Zeeman (Martinus Nijhoff, The Hague, Netherlands, 1935), p. 302.

Szegö, G.

G. Szegö, Orthogonal Polynomials (Am. Math. Soc. Colloquium Publications, Vol. XXIII, New York, 1959), Chap. V.

Wolf, E.

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

Bell System Tech. J. (5)

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).
[Crossref]

H. J. Landau and H. O. Pollak, Bell System Tech. J. 40, 65 (1961);Bell System Tech. J. 41, 1295 (1962).
[Crossref]

D. Slepian, Bell System Tech. J. 43, 3009 (1964).
[Crossref]

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
[Crossref]

A. G. Fox and Tingye Li, Bell System Tech. J. 40, 453 (1961).
[Crossref]

Can. J. Phys. (1)

G. Lansraux and G. Boivin, Can. J. Phys. 39, 158 (1961).
[Crossref]

IRE Trans. (1)

D. Slepian, IRE Trans. PGIT-3, 68 (1954).

J. Math. Anal. Appl. (1)

W. Fuchs, J. Math. Anal. Appl. 9, 317 (1964).
[Crossref]

J. Math. Phys. (MIT) (1)

D. Slepian, J. Math. Phys. (MIT) 44, 99 (1965).

J. Opt. Soc. Am. (1)

Other (11)

P. Jacquinot and B. Roizen-Dossier in Progress in Optics, E. Wolf, ed. (North-Holland Publishing Co., Amsterdam, 1964), Vol. III, p. 31.

R. Straubel, Pieter Zeeman. Verhandelingen op 25 Mei 1935 Aangeboden aan Prof. Dr. P. Zeeman (Martinus Nijhoff, The Hague, Netherlands, 1935), p. 302.

R. K. Luneberg, Mathematical Theory of Optics (University of California Press, Berkeley, California, 1964), p. 353.

J. Meixner and F. W. Schäfke, Mathieusche Funklionen und Sphäroidfunklionen (Springer-Verlag, Berlin, 1954).
[Crossref]

C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, California, 1957).

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and P. J. Corbató, Spheroidal Wave Functions (John Wiley & Sons, Inc., New York, 1956).

P. Jacquinot and B. Roizen-Dossier, in Progress in OpticsIII edited by E. Wolf (North Holland Publishing Co., Amsterdam, 1964), pp. 47, 78.

J. C. Heurtley, in Proc. Symposium on Quasi-Optics (Polytechnic Press, Brooklyn, New York, 1964), p. 367.

H. Kogelnik, in Advances in Lasers, A. K. Levine, ed. (Dekker Publishers, New York, 1965).

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

G. Szegö, Orthogonal Polynomials (Am. Math. Soc. Colloquium Publications, Vol. XXIII, New York, 1959), Chap. V.

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Figures (4)

F. 1
F. 1

Relative illuminance of diffraction pattern for apodized slit aperture.

F. 2
F. 2

Upper bound on relative illuminance of diffraction pattern for apodized slit aperture.

F. 3
F. 3

Relative illuminance of diffraction pattern for apodized circular aperture.

F. 4
F. 4

Upper bound on relative illuminance of diffraction pattern for apodized circular aperture.

Tables (2)

Tables Icon

Table I Measures of apodization for the slit aperture.

Tables Icon

Table II Measures of apodization for the circular aperture.

Equations (48)

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| x | a e i ( k / p ) ξ · x T ( x ) d x 1 d x 2 ,
ξ = ( a k / π p ) ξ , x = ( 1 / 2 a ) x , A ( ξ ) = A ( ξ ) , T ( x ) = T ( x ) ,
A ( ξ ) = | x | 1 2 e 2 π i ξ · x T ( x ) d x 1 d x 2 .
λ = | ξ | ρ | A ( ξ ) | 2 d ξ 1 d ξ 2 / | A ( ξ ) | 2 d ξ 1 d ξ 2
ρ = ( a k / π p ) b
f ( x ) = ( 2 π ) D R e i x · y F ( y ) d y ,
λ = C | f ( x ) | 2 d x E D | f ( x ) | 2 d x
λ ψ ( x ) = R K D ( x y ) ψ ( y ) d y
K D ( x y ) = ( c 2 π ) D R e i c z · ( x y ) d z .
α ψ ( x ) = R e i c x · y ψ ( y ) d y .
λ = ( c / 2 π ) D | α | 2 .
R ψ m ( x ) ψ n ( x ) d x = δ m n = λ n E D ψ m ( x ) ψ n ( x ) d x .
T ( x ) = ψ 0 ( 2 x )
c = k a b / p = π ρ .
A ( ξ ) = ( α / 2 D ) ψ 0 [ ( π / c ) ξ ] .
( ρ ) = λ 0 ,
I ( 0 ) I 0 ( 0 ) = α 0 2 4 = π λ 0 2 c , τ τ 0 = 1 2 N ,
I ( 0 ) I 0 ( 0 ) = α 0 2 π 2 = 4 λ 0 c 2 , τ τ 0 = 1 π N ,
N = | x | 1 ψ 0 2 ( x ) d x .
λ ψ ( x ) = 1 1 sin c ( x y ) π ( x y ) ψ ( y ) d y .
d d x ( 1 x 2 ) d ψ d x + ( χ c 2 x 2 ) ψ = 0
ψ 0 ( x ) = r = 0 d r ( c ) P r ( x )
ψ 0 ( x ) = K r = 0 d r ( c ) j r ( c x )
λ 0 = 2 c π d 0 2 / [ j = 0 ( 1 ) j 2 2 j ( 2 j j ) d 2 j ] 2 ,
N = 2 j = 0 d 2 j 2 4 j + 1 .
λ 0 = ( 2 / π ) c [ 1 ( c 2 / 9 ) + 0 ( c 4 ) ] ,
1 λ 0 = 4 ( π c ) 1 2 e 2 c [ 1 ( 3 / 32 c ) + 0 ( c 2 ) ] .
ψ 0 ( x ) = 1 + ( c 2 / 18 ) ( 1 3 x 2 ) + 0 ( c 4 ) .
ψ 0 ( x ) { e c x 2 / 2 , 0 x c 1 3 2 e c e c ( 1 x 2 ) 1 2 ( 1 x 2 ) 1 4 [ 1 + ( 1 x 2 ) 1 2 ] 1 2 , c 1 3 x 1 c 1 2 ( c π ) 1 2 e c I 0 [ c ( 1 x 2 ) 1 2 ] , 1 c 1 x 1 2 ( c π ) 1 2 e c J 0 [ c ( x 2 1 ) 1 2 ] , 1 x 1 + ( 1 / c ) 2 3 2 e c cos [ c ( x 2 1 ) 1 2 1 2 arctan ( x 2 1 ) 1 2 π / 4 ] x 1 2 ( x 2 1 ) 1 4 , 1 + ( 1 / c ) x
( ρ ) = 2 ρ [ 1 ( π 2 / 9 ) ρ 2 + 0 ( ρ 4 ) ] , I ( 0 ) / I 0 ( 0 ) = 1 ( π 2 / 9 ) ρ 2 + 0 ( ρ 4 ) , τ / τ 0 = 1 + 0 ( ρ 4 ) ,
( ρ ) 1 4 π ρ 1 2 e 2 π ρ [ 1 + 0 ( ρ 1 ) ] , I ( 0 ) / I 0 ( 0 ) ( 1 / 2 ρ ) ( 2 π e 2 π ρ / ρ 1 2 ) [ 1 + 0 ( ρ 1 ) ] , τ / τ 0 ( 1 / 2 ρ 1 2 ) [ 1 + 0 ( ρ 2 ) ] .
ψ 0 , n ( x 1 , x 2 ) = R 0 , n ( r ) , α 0 , n = 2 π β 0 , n , cos N θ ψ N , n ( x 1 , x 2 ) = R N , n ( r ) α N , n = 2 π β N , n , sin N θ , N = 1 , 2 , , n = 0 , 1 , 2 , ,
β N , n R N , n ( r ) = 0 1 J N ( c r r ) R N , n ( r ) r d r , n , N = 0 , 1 , 2 , .
γ N , n φ N , n ( r ) = 0 1 J N ( c r r ) ( c r r ) 1 2 φ N , n ( r ) d r ,
γ N , n = c 1 2 β N , n , φ N , n ( r ) = r 1 2 R N , n ( r ) .
d d r ( 1 r 2 ) d φ d r + ( 1 4 N 2 r 2 c 2 r 2 + χ ) φ = 0
φ N , n ( r ) = j = 0 d j N , n ( c ) T N , j ( r ) ,
T N , n ( r ) = ( n + N n ) 1 r N + 1 2 P n ( N , 0 ) ( 1 2 r 2 )
φ N , n ( r ) = 1 γ N , n j = 0 d j N , n ( c ) J N + 2 j + 1 ( c r ) ( N + j j ) ( c r ) 1 2 .
α 0 , 0 = π d 0 00 / j d j 00
λ 0 , 0 = ( c / 2 π ) 2 α 0 , 0 2 .
N = π j = 0 ( d j 0 , 0 ) 2 2 j + 1 / [ j = 0 d j 00 ( c ) ] .
α 0 , 0 = π [ 1 ( c 2 / 16 ) + 0 ( c 4 ) ] , λ 0 , 0 = ( c 2 / 4 ) [ 1 ( c 2 / 8 ) + 0 ( c 4 ) ] , N = π [ 1 ( c 2 / 8 ) + 0 ( c 4 ) ] ,
α 0 , 0 ( 2 π / c ) { 1 4 π c e 2 c [ 1 + 0 ( c 1 ) ] } , 1 λ 0 , 0 8 π c e 2 c [ 1 + 0 ( c 1 ) ] , N ( π / c 1 2 ) [ 1 + 0 ( c 1 ) ] .
ψ 0 , 0 = 1 ( c 2 r 2 / 8 ) + 0 ( c 4 ) .
ψ 0 , 0 ( r ) { e ( c r 2 / 2 ) , 0 r c 1 3 2 e c e c ( 1 r 2 ) 1 2 ( 1 r 2 ) 1 4 [ 1 + ( 1 r 2 ) 1 2 ] c 1 3 r 1 c 1 2 3 2 ( π c ) 1 2 e c I 0 [ c ( 1 r 2 ) 1 2 ] , 1 c 1 r 1 2 3 2 ( π c ) 1 2 e c J 0 [ c ( r 2 1 ) 1 2 ] , 1 r 1 + c 1 4 e c cos [ c ( r 2 1 ) 1 2 arctan ( r 2 1 ) 1 2 π / 4 ] ( r 2 1 ) 1 4 r , 1 + c 1 r .
( ρ ) = ( π 2 ρ 2 / 4 ) [ 1 + 0 ( ρ 4 ) ] , I ( 0 ) / I 0 = 1 ( π 2 ρ 2 / 8 ) + 0 ( ρ 4 ) , τ / τ 0 = 1 ( π 2 ρ 2 / 8 ) + 0 ( ρ 4 ) ,
( ρ ) = 1 8 π 2 ρ e 2 π ρ [ 1 + 0 ( ρ 1 ) ] , I ( 0 ) / I 0 = ( 4 / π 2 ρ 2 ) ( 32 / ρ ) e 2 π ρ [ 1 + 0 ( ρ 1 ) ] , τ / τ 0 = ( 1 / ( π ρ ) 1 2 [ 1 + 0 ( ρ 1 ) ] .