Abstract

Imaging systems with noncentrally obscured circular pupils are considered, and expressions for the point-spread function, encircled energy, and the optical transfer function are obtained. It is shown that at image points which lie within the Airy circle, a noncentral obscuration yields higher or equal irradiance compared to a central obscuration, and consequently a higher concentration of energy near the image center. In general, a noncentral obscuration, compared to a central one, increases the transfer function at low intermediate spatial frequencies and decreases it at high intermediate frequencies; at very low and very high frequencies the transfer function is the same in both cases. Numerical results for pupils with linear obscuration ratios of 0.25, 0.50, and 0.75 are presented.

© 1978 Optical Society of America

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References

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  1. E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145–149(1953).
    [Crossref]
  2. C. A. Taylor and B. J. Thompson, “Attempt to investigate experimentally the intensity distribution near the focus in the error-free diffraction patterns of circular and annular apertures,” J. Opt. Soc. Am. 38, 844–850(1958).
    [Crossref]
  3. W. T. Welford, “Use of annular apertures to increase focal depth,” J. Opt. Soc. Am. 50, 749–753(1960).
    [Crossref]
  4. H. F. A. Tschunko, “Imaging performance of annular apertures,” Appl. Opt. 13, 1820–1823(1974).
    [Crossref] [PubMed]
  5. V. N. Mahajan, “Imaging with obscured pupils,” Opt. Lett. 1, 128–129(1977).
    [Crossref] [PubMed]
  6. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 482.
  7. Reference 6, p. 398.
  8. E. L. O’Neill, “Transfer function for an annular aperture,” J. Opt. Soc. Am. 46, 285–288(1956); note that a term of −2η2 is missing in the second of O’Neill’s Eq. (26).
    [Crossref]
  9. G. W. Sutton, M. M. Weiner, and S. A. Mani, “Fraunhofer diffraction patterns from uniformly illuminated square output apertures with noncentered square obscurations,” Appl. Opt. 15, 2228–2232(1976).
    [Crossref] [PubMed]

1977 (1)

1976 (1)

1974 (1)

1960 (1)

1958 (1)

C. A. Taylor and B. J. Thompson, “Attempt to investigate experimentally the intensity distribution near the focus in the error-free diffraction patterns of circular and annular apertures,” J. Opt. Soc. Am. 38, 844–850(1958).
[Crossref]

1956 (1)

1953 (1)

E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145–149(1953).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 482.

Linfoot, E. H.

E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145–149(1953).
[Crossref]

Mahajan, V. N.

Mani, S. A.

O’Neill, E. L.

Sutton, G. W.

Taylor, C. A.

C. A. Taylor and B. J. Thompson, “Attempt to investigate experimentally the intensity distribution near the focus in the error-free diffraction patterns of circular and annular apertures,” J. Opt. Soc. Am. 38, 844–850(1958).
[Crossref]

Thompson, B. J.

C. A. Taylor and B. J. Thompson, “Attempt to investigate experimentally the intensity distribution near the focus in the error-free diffraction patterns of circular and annular apertures,” J. Opt. Soc. Am. 38, 844–850(1958).
[Crossref]

Tschunko, H. F. A.

Weiner, M. M.

Welford, W. T.

Wolf, E.

E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145–149(1953).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 482.

Appl. Opt. (2)

J. Opt. Soc. Am. (3)

C. A. Taylor and B. J. Thompson, “Attempt to investigate experimentally the intensity distribution near the focus in the error-free diffraction patterns of circular and annular apertures,” J. Opt. Soc. Am. 38, 844–850(1958).
[Crossref]

W. T. Welford, “Use of annular apertures to increase focal depth,” J. Opt. Soc. Am. 50, 749–753(1960).
[Crossref]

E. L. O’Neill, “Transfer function for an annular aperture,” J. Opt. Soc. Am. 46, 285–288(1956); note that a term of −2η2 is missing in the second of O’Neill’s Eq. (26).
[Crossref]

Opt. Lett. (1)

Proc. Phys. Soc. B (1)

E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145–149(1953).
[Crossref]

Other (2)

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 482.

Reference 6, p. 398.

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Figures (14)

FIG. 1
FIG. 1

Geometry of the imaging system. (a) Circular pupil with a noncentral obscuration. (b) Image plane, where the point-spread function, encircled energy, and the optical transfer function are measured.

FIG. 2
FIG. 2

Point-spread function for centrally obscured circular pupils with ɛ = 0.25, 0.50, and 0.75.

FIG. 3
FIG. 3

Change in point-spread function when ɛ = 0.25 and the distance between the centers of the pupil and the obscuration is (a) r0 = (1 − ɛ)/4, (b) r0 = 1 − ɛ)/3,(c) r0 = (1 − ɛ)/2. Δθ = θθ0 is the angle formed by the line along which the change in the point-spread function is measured and the line joining the centers of the pupil and the obscuration.

FIG. 4
FIG. 4

Same as Fig.3 except that ɛ = 0.50.

FIG. 5
FIG. 5

Same as Fig. 3 except that ɛ = 0.75.

FIG. 6
FIG. 6

Encircled energy for centrally obscured circular pupils with ɛ = 0.25, 0.50, and 0.75.

FIG. 7
FIG. 7

Change in encircled energy when ɛ = 0.25 and the distance between the centers of the pupil and the obscuration is r0 = (1 − ɛ)/4, (1 − e)/3, and (1 − ɛ)/2.

FIG. 8
FIG. 8

Same as Fig. 7 except that ɛ = 0.50.

FIG. 9
FIG. 9

Same as Fig. 7 except that ɛ = 0.75.

FIG. 10
FIG. 10

Optical transfer function for centrally obscured circular pupils with ɛ = 0.25, 0.50, and 0.75.

FIG. 11
FIG. 11

Variation of τ12 with ρ.

FIG. 12
FIG. 12

Change in optical transfer function when ɛ = 0.25 and (a) r0 = (1 − ɛ)/4, (b) r0 = (1 − ɛ)/3, (c) r0 = (1 − ɛ)/2. Δϕ = ϕϕ0 is the angle formed by the line along which the change in optical transfer function is measured and the line joining the centers of the pupil and the obscuration.

FIG. 13
FIG. 13

Same as Fig. 12 except that ɛ = 0.50.

FIG. 14
FIG. 14

Same as Fig. 12 except that ɛ = 0.75.

Tables (1)

Tables Icon

TABLE I Percent energy in an Airy circle, E(a = 1.22;r0).

Equations (35)

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P ( x p , y p ; ɛ ; x 0 , y 0 ) = cyl ( r p / D ) - cyl { [ ( x p - x 0 ) 2 + ( y p - y 0 ) 2 ] 1 / 2 ɛ D } ,
r p = ( x p 2 + y p 2 ) 1 / 2
cyl ( r p / D ) = 1 ,             r p D / 2 = 0 ,             otherwise .
I ( x , y ; ɛ ; x 0 y 0 ) = I 0 λ 2 R 2 | - P ( x p , y p ; ɛ ; x 0 , y 0 ) × exp [ 2 π λ R i ( x x p + y y p ) ] d x p d y p | 2 ,
I ( x , y ; ɛ ; x 0 , y 0 ) = U 2 ( r ) + ɛ 4 U 2 ( ɛ r ) - 2 ɛ 2 U ( r ) U ( ɛ r ) cos [ 2 π ( x x 0 + y y 0 ) ] .
r = ( x 2 + y 2 ) 1 / 2
U ( r ) = ( π D / 4 ) [ 2 J 1 ( π r ) / π r ] I 0 ,
I ( 0 , 0 ; ɛ ; x 0 , y 0 ) = ( π D / 4 ) 2 ( 1 - ɛ 2 ) 2 I 0 .
Δ I ( x , y ; ɛ ; x 0 , y 0 ) = I ( x , y ; ɛ ; x 0 , y 0 ) - I ( x , y ; ɛ ; x 0 = y 0 = 0 ) = 2 ɛ 2 U ( r ) U ( ɛ r ) { 1 - cos [ 2 π ( x x 0 + y y 0 ) ] } .
x = r cos θ ,             y = r sin θ
x 0 = r 0 cos θ 0 ,             y 0 = r 0 sin θ 0 ,
Δ I ( r , θ ; ɛ ; r 0 , θ 0 ) = 2 ɛ 2 U ( r ) U ( ɛ r ) × { 1 - cos [ 2 π r r 0 cos ( θ - θ 0 ) ] } .
Δ I 0 , r 1.220 , 0 , 1.220 r smaller of 2.233 and 1.220 / ɛ , 0 , smaller of 2.233 and 1.220 / ɛ r larger of 2.233 and 1.220 / ɛ but smaller than 3.238 , 0 , larger of 2.233 and 1.220 / ɛ r smaller of 2.233 / ɛ and 3.238 ,
E ( a ; ɛ ; r 0 , θ 0 ) = ( 0 a 0 2 π I ( r , θ ; ɛ ; r 0 , θ 0 ) r d r d θ ) / ( 0 0 2 π I ( r , θ ; ɛ ; r 0 , θ 0 ) r d r d θ ) .
0 2 π cos ( z cos x ) d x = 2 π J 0 ( z ) ,
E ( a , ɛ , r 0 ) = 8 D 2 ( 1 - ɛ 2 ) I 0 0 a [ U 2 ( r ) + ɛ 4 U 2 ( ɛ r ) - 2 ɛ 2 U ( r ) U ( ɛ r ) J 0 ( 2 π r r 0 ) ] r d r = [ 1 / ( 1 - ɛ 2 ) ] [ E ( a ) + ɛ 2 E ( ɛ a ) - 4 ɛ 0 a J 1 ( π r ) J 1 ( π ɛ r ) J 0 ( 2 π r r 0 ) d r r ] .
E ( a ) = 8 D 2 I 0 0 a U 2 ( r ) r d r = 1 - J 0 2 ( π a ) - J 1 2 ( π a ) .
Δ E ( a , ɛ , r 0 ) = E ( a , ɛ , r 0 ) - E ( a , ɛ , r 0 = 0 ) = 4 ɛ 1 - ɛ 2 0 a J 1 ( π r ) J 1 ( π ɛ r ) [ 1 - J 0 ( 2 π r r 0 ) ] d r r .
τ ( ζ , η ; ɛ ; x 0 , y 0 ) = - I ( x , y ; ɛ ; x 0 , y 0 ) exp [ 2 π i ( ζ x + η y ) ] d x d y / - I ( x , y ; ɛ ; x 0 , y 0 ) d x d y = τ ( ρ ) + ɛ 2 τ ( ρ / ɛ ) - ½ [ τ 12 ( ζ + ζ 0 , η + η 0 ; ) + τ 12 ( ζ - ζ 0 , η - η 0 ; ɛ ) ] ,
ρ = ( ζ 2 + η 2 ) 1 / 2 ,
ζ 0 = x 0 ,             η 0 = y 0 ,
τ ( ρ ) = [ 2 / π ( 1 - ɛ 2 ) ] [ cos - 1 ρ - ρ ( 1 - ρ 2 ) 1 / 2 ] ,             0 ρ 1 = 0 ,             otherwise ,
τ 12 ( ζ , η ; ɛ ) = 2 ɛ 2 / ( 1 - ɛ 2 ) ,             0 ρ ( 1 - ɛ ) / 2 = 2 ɛ 2 1 - ɛ 2 [ 1 - 1 + ɛ 2 2 π ɛ 2 ψ - 1 π ɛ sin ψ + 1 - ɛ 2 π ɛ 2 tan - 1 ( 1 + ɛ 1 - ɛ tan ψ 2 ) ] , ( 1 - ɛ ) / 2 ρ ( 1 + ɛ ) / 2 = 0 ,             otherwise .
ψ = cos - 1 [ ( 1 + ɛ 2 - 4 ρ 2 ) / 2 ɛ ] .
Δ τ ( ζ , η ; ɛ ; x 0 , y 0 ) = τ ( ζ , η ; ɛ ; x 0 , y 0 ) - τ ( ζ , η ; ɛ ; x 0 = y 0 = 0 ) = τ 12 ( ζ , η ; ɛ ) - ½ [ τ 12 ( ζ + ζ 0 , η + η 0 ; ɛ ) + τ 12 ( ζ - ζ 0 , η - η 0 ; ɛ ) ] = τ 12 ( ρ ; ɛ ) - ½ [ τ 12 ( ρ + ; ɛ ) + τ 12 ( ρ - ; ɛ ) ] ,
ρ ± = [ ( ζ ± ζ 0 ) 2 + ( η ± η 0 ) 2 ] 1 / 2 .
ζ = ρ cos ϕ ,             η = ρ sin ϕ
ζ 0 = ρ 0 cos ϕ 0 ,             η 0 = ρ 0 sin ϕ 0 ,
ρ ± = [ ρ 2 + ρ 0 2 ± 2 ρ ρ 0 cos ( ϕ - ϕ 0 ) ] 1 / 2 .
τ 12 ( ρ ; ɛ ) = ½ [ τ 12 ( ρ + ; ɛ ) + τ 12 ( ρ - ; ɛ ) ] .
Δ τ ( ζ = η = 0 ; ɛ ; x 0 , y 0 ) = τ 12 ( ρ = 0 ; ɛ ) - τ 12 ( ρ 0 ; ɛ ) ,
ρ 0 = ( ζ 0 2 + η 0 2 ) 1 / 2 ( 1 - ɛ ) / 2.
I n ( r ; ɛ ) = ( 2 J 1 ( π r ) π r - ɛ 2 2 J 1 ( π ɛ r ) π ɛ r ) 2 .
Δ I n ( r , Δ θ ; ɛ ; r 0 ) = 8 ɛ [ J 1 ( π r ) J 1 ( π ɛ r ) / π 2 r 2 ] × [ 1 - cos ( 2 π r r 0 cos Δ θ ) ] ,
Δ θ = θ - θ 0 .