Abstract

Bounds on the intensity and the variation of the far field F(u,v) of a given object f(x,y) are developed in terms of the energy and the area of the object. An amplitude function f(x,y) is determined for maximizing F(u0,v0). The results are extended to objects with circular symmetry. The analysis is applied to the following apodization problem: Given a pupil of specified boundary R, a transmission function f(x,y) of energy E is sought such that the energy of its far field in a region S of the uv plane is maximum.

© 1967 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959).
  2. A. Papoulis, MRI Symposium on Generalized Networks (Polytechnic Inst. of Brooklyn Press, 1967).
  3. A. Papoulis, The Fourier Integral and its Applications (McGraw–Hill Book Co., New York, 1962).
  4. I. N. Sneddon, Fourier Transforms (McGraw–Hill Book Co., New York, 1951).
  5. Rayleigh, Phil. Mag. (5),  11, 214 (1881). See also Ref. 1, p. 397.
  6. G. Petiau, La Théorie des Fonctions de Bessel (Centre National de la Recherche Scientifique, Paris, 1955).
  7. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill Book Co., New York, 1951).
  8. A. Papoulis, IEEE Trans. Information Theory IT-11, 4 (1965).
  9. D. Slepian, Bell System Tech. J. 43, 3009 (1964).
    [Crossref]
  10. P. Jacquinot and B. Roizen-Dossier in Progress in Optics III, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1964).
  11. R. Barakat, J. Opt. Soc. Am.,  52, 264 (1962).
    [Crossref]
  12. The function 2πaD(w,w0) is called the directrix by Lansraux and Boivin.13
  13. G. Lansraux and G. Boivin, Can. J. Phys. 36, 1696 (1958).
    [Crossref]
  14. D. Slepian, H. O. Pollak, and H. J. Landau, Bell System Tech. J. 40, 43 (1961).
    [Crossref]

1965 (1)

A. Papoulis, IEEE Trans. Information Theory IT-11, 4 (1965).

1964 (1)

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

1962 (1)

1961 (1)

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

1958 (1)

G. Lansraux and G. Boivin, Can. J. Phys. 36, 1696 (1958).
[Crossref]

1881 (1)

Rayleigh, Phil. Mag. (5),  11, 214 (1881). See also Ref. 1, p. 397.

Barakat, R.

Boivin, G.

G. Lansraux and G. Boivin, Can. J. Phys. 36, 1696 (1958).
[Crossref]

Born, M.

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

Jacquinot, P.

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

Landau, H. J.

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

Lansraux, G.

G. Lansraux and G. Boivin, Can. J. Phys. 36, 1696 (1958).
[Crossref]

Papoulis, A.

A. Papoulis, IEEE Trans. Information Theory IT-11, 4 (1965).

A. Papoulis, MRI Symposium on Generalized Networks (Polytechnic Inst. of Brooklyn Press, 1967).

A. Papoulis, The Fourier Integral and its Applications (McGraw–Hill Book Co., New York, 1962).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill Book Co., New York, 1951).

Petiau, G.

G. Petiau, La Théorie des Fonctions de Bessel (Centre National de la Recherche Scientifique, Paris, 1955).

Pollak, H. O.

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

Rayleigh,

Rayleigh, Phil. Mag. (5),  11, 214 (1881). See also Ref. 1, p. 397.

Roizen-Dossier, B.

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

Slepian, D.

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

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

Sneddon, I. N.

I. N. Sneddon, Fourier Transforms (McGraw–Hill Book Co., New York, 1951).

Wolf, E.

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

Bell System Tech. J. (2)

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

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

Can. J. Phys. (1)

G. Lansraux and G. Boivin, Can. J. Phys. 36, 1696 (1958).
[Crossref]

IEEE Trans. Information Theory (1)

A. Papoulis, IEEE Trans. Information Theory IT-11, 4 (1965).

J. Opt. Soc. Am. (1)

Phil. Mag. (5) (1)

Rayleigh, Phil. Mag. (5),  11, 214 (1881). See also Ref. 1, p. 397.

Other (8)

G. Petiau, La Théorie des Fonctions de Bessel (Centre National de la Recherche Scientifique, Paris, 1955).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill Book Co., New York, 1951).

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

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

A. Papoulis, MRI Symposium on Generalized Networks (Polytechnic Inst. of Brooklyn Press, 1967).

A. Papoulis, The Fourier Integral and its Applications (McGraw–Hill Book Co., New York, 1962).

I. N. Sneddon, Fourier Transforms (McGraw–Hill Book Co., New York, 1951).

The function 2πaD(w,w0) is called the directrix by Lansraux and Boivin.13

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Figures (4)

Fig. 1
Fig. 1

Uniformly illuminated aperture R generating optimum far field Fm(u,v).

Fig. 2
Fig. 2

Upper bound πa2E[J02(wa)+J12(wa)] of far-field intensity |F(w)|2 of a circularly symmetrical object.

Fig. 3
Fig. 3

Optimum far field Fm(w) maximizing Fm(w0); corresponding object fm(r)=kJ0(w0r) for 0<r<a; uniformly illuminated ring (left) generating fm(r) as its far field.

Fig. 4
Fig. 4

lαβ: maximum projection of the points of region R in the 0A direction.

Equations (67)

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f ( x , y ) = 0             for             ( x , y )             not in R ,
F ( u , v ) = R f ( x , y ) e - j ( u x + v y ) d x d y .
E = R f ( x , y ) 2 d x d y = 1 4 π 2 - - F ( u , v ) 2 d u d v
F ( u , v ) 2 E A ,
| g h d x d y | 2 g 2 d x d y h 2 d x d y .
g ( x , y ) = k h * ( x , y )
g ( x , y ) = f ( x , y ) ,             h ( x , y ) = e - j ( u x + v y )
F ( u , v ) 2 R f ( x , y ) 2 d x d y R e - j ( u x + v y ) 2 d x d y
f ( x , y ) = f m ( x , y ) = k e - j ( u 0 x + v 0 y )
F m ( u , v ) = ( E A ) 1 2 R exp { - j [ ( u - u 0 ) x + ( v - v 0 ) y ] } d x d y .
F m ( u , v ) F m ( u 0 , v 0 ) = ( E A ) 1 2 .
F m ( u , v ) = 2 ( E / a b ) 1 2 sin α ( u - u 0 ) u - u 0 sin b ( v - v 0 ) v - v 0 .
F m ( u , v ) = 2 ( E π ) 1 2 J 1 { a [ ( u - u 0 ) 2 + ( v - v 0 ) 2 ] 1 2 } [ ( u - u 0 ) 2 + ( v - v 0 ) 2 ] 1 2 ,
f ( x ) = 0             for             x > a             and             E = - a a f ( x ) 2 d x .
F ( ω ) = - a a f ( x ) e - j ω x d x
F ( ω ) 2 2 a E .
f ( x ) = f m ( x ) = ( E / 2 a ) 1 2 e j ω 0 x             for             x < a
F m ( ω ) = ( 2 E / a ) 1 2 sin a ( ω - ω 0 ) / ( ω - ω 0 ) .
f ( x , y ) = f ( r )             r = ( x 2 + y 2 ) 1 2 ,
f ( r ) = 0             for             r > a .
f ( u , v ) = F ( w ) ,             w = ( u 2 + v 2 ) 1 2 ,
F ( w ) = 2 π 0 a r f ( r ) J 0 ( w r ) d r .
E = 2 π 0 a r f ( r ) 2 d r
F ( w ) 2 π a 2 E [ J 0 2 ( w a ) + J 1 2 ( w a ) ] .
g ( r ) = r 1 2 f ( r ) ,             h ( r ) = r 1 2 J 0 ( w r )
F ( w ) 2 4 π 2 0 a r f ( r ) 2 d r 0 a r J 0 2 ( w r ) d r .
0 a r J 0 2 ( w r ) d r = a 2 2 [ J 0 2 ( w a ) + J 1 2 ( w a ) ] ,
f m ( r ) = { k J 0 ( w 0 r ) ; r < a 0 r > a
k = ( E / π a 2 ) 1 2 [ J 0 2 ( a w 0 ) + J 1 2 ( a w 0 ) ] - 1 2 .
F m ( w ) = 2 π a k w 0 J 1 ( w 0 a ) J 0 ( w a ) - w J 0 ( w 0 a ) J 1 ( w a ) w 0 2 - w 2 .
F ( ω + ) - F ( ω ) a ( 2 a E ) 1 2 .
ρ ( ω ) = - a a f ( x ) 2 e - j ω x d x .
ρ ( ω ) = 1 2 π F ( ω ) * F * ( - ω ) = 1 2 π - F ( ξ ) F * ( ω + ξ ) d ξ .
1 - cos x = 2 sin 2 ( x / 2 ) x 2 / 2 ,
0 Re [ ρ ( 0 ) - ρ ( ω ) ] ( E / 2 ) a 2 ω 2 .
Re [ ρ ( 0 ) - ρ ( ω ) ] = - a a f ( x ) 2 ( 1 - cos ω x ) d x - a a f ( x ) 2 x 2 ω 2 2 d x a 2 ω 2 2 - a a f ( x ) 2 d x
f ( x ) ( e - j x - 1 ) .
1 2 π - F ( ω + ) - F ( ω ) 2 d ω = 2 Re [ ρ ( 0 ) - ρ ( ) ]
F ( ω + ) - F ( ω ) 2 4 a Re [ ρ ( 0 ) - ρ ( ) ]
ρ ( u , v ) = R f ( x , y ) 2 e - j ( u x + v y ) d x d y .
ρ ( u , v ) = 1 4 π 2 - - F ( ξ , η ) F * ( x + ξ , y + η ) d ξ d η .
ρ ( 0 , 0 ) = E .
( α x + β y ) 2 l 2 α β ( α 2 + β 2 )
0 Re [ ρ ( 0 , 0 ) - ρ ( α , β ) ] 1 2 E l 2 α β ( α 2 + β 2 ) .
Re [ ρ ( 0 , 0 ) - ρ ( α , β ) ] = R f ( x , y ) 2 × [ 1 - cos ( α x + β y ) ] d x d y R f ( x , y ) 2 × ( α x + β y ) 2 2 d x d y l α β ( α 2 + β 2 ) 2 R f ( x , y ) 2 d x d y ,
F ( u + α , v + β ) - F ( u , v ) 2 l 2 α β A E ( α 2 + β 2 ) ,
f ( x , y ) [ e - j ( α x + β y ) - 1 ] .
1 4 π 2 - - F ( u + α , v + β ) - F ( u , v ) 2 d u d v = 2 Re [ ρ ( 0 , 0 ) - ρ ( α , β ) ] .
F ( u + α , v + β ) - F ( u , v ) 2 2 A Re [ ρ ( 0 , 0 ) - ρ ( α , β ) ] ,
ρ ( w ) = 2 π 0 a r f ( r ) 2 J 0 ( w r ) d r ,
0 ρ ( 0 ) - ρ ( w ) ρ ( 0 ) a 2 w 2 4 .
ρ ( 0 ) - ρ ( w ) = 2 π 0 a r f ( r ) 2 [ 1 - J 0 ( w r ) ] d r .
0 1 - J 0 ( x ) x 2 / 4.
ρ ( 0 ) - ρ ( w ) 2 π 0 a r f ( r ) 2 w 2 r 2 4 d r π a 2 w 2 2 0 r f ( r ) 2 d r ,
ρ ( 0 ) - ρ ( w ) E a 2 w 2 / 2
r = S F ( u , v ) 2 d u d v / - - F ( u , v ) 2 d u d v .
r F ( u 0 , v 0 ) 2 B / 4 π 2 E ,
F ( u , v ) = F m ( u , v ) = ( E A ) 1 2 × R exp { - j [ ( u - u 0 ) x + ( v - v 0 ) y ] } d x d y
r m A B / 4 π 2 .
k R ( u , v ) = R e - j ( u x + v y ) d x d y .
S φ ( ξ , η ) k R ( u - ξ , v - η ) d ξ d η = λ φ ( u , v )
0 b w 0 φ 0 ( w 0 ) J 0 ( w 0 w ) d w 0 = ( b / a ) λ 0 1 2 φ 0 [ ( b / a ) w ]
0 b w 0 D ( w , w 0 ) φ 0 ( w 0 ) d w 0 = λ 0 φ 0 ( w ) ,
λ 0 a 2 b 2 / 4 ,             φ 0 ( w ) J 1 ( a w ) / w             for             a b 0.
r = 1 2 π E - b b F ( ω ) 2 d ω
r m ( 2 / π ) a b ,             F m ( ω ) ( 2 E / a ) 1 2 sin a ω / ω             for             a b 0.
- b b φ ( ξ ) sin a ( ω - ξ ) π ( ω - ξ ) d ξ = λ φ ( ω )