Abstract

The normalized transfer function of an elliptical annular aperture for incoherent illumination is derived. The relationship between the formulation developed here and a problem occurring in the diffraction from apertures illuminated with partially coherent radiation is shown. Results are discussed in terms of spatial-filtering concepts, with emphasis on the behavior of the cutoff frequency and the passband characteristics in the spatial-frequency plane.

The effect of the aperture ellipticity for arbitrary annulus width is to introduce an asymmetry in the behavior of the cutoff frequency. The presence of a central obstruction, for arbitrary ellipticity, emphasizes the high-frequency transmission, with the region of emphasis dependent upon the angular orientation in the frequency plane.

© 1967 Optical Society of America

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References

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  1. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Inc., Reading, Mass., 1963), Chs. 5 and 6.
  2. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, New York, 1964), Sees. 9.5 and 10.5.3.
  3. Ref. 1, p. 77.
  4. If u, υ are the rectangular coordinates defining a point in the aperture, then μ= ku/d, ν= kυ/d are the reduced coordinates, where k is the wave vector, and d is the distance from the aperture to the plane for which the transfer function is computed.
  5. Ref. 1, p. 76.
  6. Reference 1, references on pp. 95 and 103.
  7. R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 538, (1965).
    [Crossref]
  8. E. L. O’Neill, J. Opt. Soc. Am. 46, 285 (1956).
    [Crossref]
  9. J. V. Cornacchio and R. P. Soni, J. Opt. Soc. Am. 55, 107 (1965). The transformation involved is also used in the analysis of Fraunhofer diffraction from planar apertures, see Ref. 2, p. 399.
    [Crossref]
  10. J. V. Cornacchio, Phys. Letters 15, 306 (1965).
    [Crossref]
  11. J. V. Cornacchio and R. P. Soni, Nuovo Cimento 38, 1169 (1965).
    [Crossref]
  12. J. V. Cornacchio and K. A. Farnham, Nuovo Cimento 42, 108 (1966).
    [Crossref]
  13. There is an error in the result given in Ref. 8. Although the graphs are correct, the term C, as given in that reference, must have the factor −2η2added to it in the interval [1 − η, 1 + η]. In fact, C(Ω), and therefore τ˜(Ω), is discontinuous at Ω = 1 ∓ η; by this amount.

1966 (1)

J. V. Cornacchio and K. A. Farnham, Nuovo Cimento 42, 108 (1966).
[Crossref]

1965 (4)

1956 (1)

Barakat, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, New York, 1964), Sees. 9.5 and 10.5.3.

Cornacchio, J. V.

J. V. Cornacchio and K. A. Farnham, Nuovo Cimento 42, 108 (1966).
[Crossref]

J. V. Cornacchio, Phys. Letters 15, 306 (1965).
[Crossref]

J. V. Cornacchio and R. P. Soni, Nuovo Cimento 38, 1169 (1965).
[Crossref]

J. V. Cornacchio and R. P. Soni, J. Opt. Soc. Am. 55, 107 (1965). The transformation involved is also used in the analysis of Fraunhofer diffraction from planar apertures, see Ref. 2, p. 399.
[Crossref]

Farnham, K. A.

J. V. Cornacchio and K. A. Farnham, Nuovo Cimento 42, 108 (1966).
[Crossref]

Houston, A.

O’Neill, E. L.

E. L. O’Neill, J. Opt. Soc. Am. 46, 285 (1956).
[Crossref]

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Inc., Reading, Mass., 1963), Chs. 5 and 6.

Soni, R. P.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, New York, 1964), Sees. 9.5 and 10.5.3.

J. Opt. Soc. Am. (3)

Nuovo Cimento (2)

J. V. Cornacchio and R. P. Soni, Nuovo Cimento 38, 1169 (1965).
[Crossref]

J. V. Cornacchio and K. A. Farnham, Nuovo Cimento 42, 108 (1966).
[Crossref]

Phys. Letters (1)

J. V. Cornacchio, Phys. Letters 15, 306 (1965).
[Crossref]

Other (7)

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Inc., Reading, Mass., 1963), Chs. 5 and 6.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, New York, 1964), Sees. 9.5 and 10.5.3.

Ref. 1, p. 77.

If u, υ are the rectangular coordinates defining a point in the aperture, then μ= ku/d, ν= kυ/d are the reduced coordinates, where k is the wave vector, and d is the distance from the aperture to the plane for which the transfer function is computed.

Ref. 1, p. 76.

Reference 1, references on pp. 95 and 103.

There is an error in the result given in Ref. 8. Although the graphs are correct, the term C, as given in that reference, must have the factor −2η2added to it in the interval [1 − η, 1 + η]. In fact, C(Ω), and therefore τ˜(Ω), is discontinuous at Ω = 1 ∓ η; by this amount.

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Figures (9)

Fig. 1
Fig. 1

Effect of the transformation T on the elliptical annulus. The ellipses C1 and C2 with common center having coordinates (μ,ν) are transformed into the circles C ˜ 1 and C ˜ 2, respectively, centered at the point having coordinates ( μ ˜ , ν ˜ ), with μ ˜ = μ , ν ˜ = ν.

Fig. 2
Fig. 2

Normalized cutoff frequency ωc/2A vs ray angle ψ for fixed ratio .

Fig. 3
Fig. 3

Normalized cutoff frequency ωc/2A vs for fixed ray angle ψ.

Fig. 4
Fig. 4

Normalized transfer function vs ω/2A for = 3 and η = 0.

Fig. 5
Fig. 5

Normalized transfer function vs ω/2A for = 3 and η = 0. ω0/2A is shown for ψ = π/4 and π/2.

Fig. 6
Fig. 6

Normalized transfer function vs ω/2A for = 3 and η = 1 2. Ωc/2A is shown for ψ = π/4 and π/2.

Fig. 7
Fig. 7

Normalized transfer function vs ω/2A for = 3 and η = 3 4. ωc/2A is shown for ψ = π/4: and π/2.

Fig. 8
Fig. 8

Normalized transfer function vs ω/2A for = 3 2 and η = 0. ωc/2A is shown for ψ = π/4 and π/2.

Fig. 9
Fig. 9

Normalized transfer function vs ω/2A for = 3 2 and η = 1 2. ωc/2A is shown for ψ = π/4: and π/2.

Equations (34)

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τ ( μ , ν ) = + F ( μ , ν ) F * ( μ μ , ν ν ) d μ d ν / + | F ( μ , ν ) | 2 d μ d ν ,
F ( μ , ν ) = 0 for ( μ , ν ) A r .
| F ( μ , ν ) | = 1 , ( μ , ν ) A r .
F ( μ , ν ) = { e i k Δ ( μ , ν ) ( μ , ν ) A r 0 , ( μ , ν ) A r .
C 1 : μ 2 ( k a / d ) 2 + ν 2 ( k b / d ) 2 = 1 C 2 : μ 2 ( k A / d ) 2 + ν 2 ( k B / d ) 2 = 1 ,
τ ( μ , ν ) = Δ A r ( μ , ν ) / Δ A r ( 0 , 0 ) ,
Δ A r ( μ , ν ) = A r [ 0 , 0 ] A r [ μ , ν ] d μ d ν ,
T : { μ = μ ν = ν ,
C ˜ 1 : μ 2 + ν 2 = ( k a / d ) 2 C ˜ 2 : μ 2 + ν 2 = ( k A / d ) 2 ,
μ ˜ = μ ν ˜ = ν .
( μ μ ˜ ) 2 + ( ν ν ˜ ) 2 = ( k a / d ) 2 ( μ μ ˜ ) 2 + ( ν ν ˜ ) 2 = ( k A / d ) 2 .
Δ A r ( μ , ν ) = J A ˜ r [ 0 , 0 ] A ˜ r [ μ ˜ , ν ˜ ] d μ d ν = J Δ A r [ μ ˜ , ν ˜ ] ,
J | ( μ , ν ) ( μ , ν ) | = 1
τ ( μ , ν ) = Δ A r ( μ , ν ) Δ A r ( 0 , 0 ) = Δ A r ( μ ˜ , ν ˜ ) Δ A ˜ r ( 0 , 0 ) = τ ˜ ( μ ˜ , ν ˜ ) ,
τ ( μ , ν ) = τ ˜ ( μ , ν ) ,
τ ( μ , ν ) = τ ˜ ( μ , ν ) ,
τ ˜ ( μ ˜ , ν ˜ ) = ( 1 η 2 ) 1 [ A ( Ω ˜ ) + B ( Ω ˜ ) + C ( Ω ˜ ) ] ,
A ( Ω ˜ ) = { ( 2 / π ) [ cos 1 ( Ω ˜ / 2 ) ( Ω ˜ / 2 ) ( 1 ( Ω ˜ / 2 ) 2 ) 1 2 ] , 0 Ω ˜ / 2 1 0 , Ω ˜ / 2 > 1 ,
B ( Ω ˜ ) = A ( Ω ˜ / η ) , [ where A ( Ω ˜ / η ) 0 for η = 0 ] ,
C ( Ω ˜ ) = { 2 η 2 , Ω ˜ < 1 η 2 η 2 + 2 η π sin ϕ ˜ + ( 1 + η 2 ) π ϕ ˜ 2 ( 1 η 2 ) π tan 1 [ ( 1 + η 1 η ) tan ϕ ˜ / 2 ] , 1 η Ω ˜ 1 + η 0 , Ω ˜ > 1 + η
η = a / A Ω ˜ ( μ ˜ , ν ˜ ) = A 1 ( μ ˜ 2 + ν ˜ 2 ) 1 2 ϕ ˜ ( μ ˜ , ν ˜ ) = cos 1 [ ( 1 + η 2 Ω ˜ 2 ) / 2 η ] .
τ ( μ , ν ) = ( 1 η 2 ) 1 [ A ( Ω ) + B ( Ω ) + C ( Ω ) ] ,
Ω ( μ , ν ) = Ω ˜ ( μ , ν ) = ( 1 / A ) ( μ 2 + 2 ν 2 ) 1 2 ϕ ( μ , ν ) = ϕ ˜ ( μ , ν ) = cos 1 [ 1 + η 2 Ω 2 ( μ , ν ) 2 η ] .
ω 2 = μ 2 + ν 2 , ψ = tan 1 ( ν / μ ) .
Ω = ω A ( cos 2 ψ + 2 sin 2 ψ ) 1 2 ,
B = ( b / a ) A = ( 1 / ) A .
τ ( ω , ψ ) = { 2 π [ cos 1 ( Ω 2 ) ( Ω 2 ) ( 1 ( Ω 2 ) 2 ) 1 2 ] , 0 Ω 2 1 0 , Ω / 2 > 1 ,
1 2 Ω > max [ 1 , η , 1 2 ( 1 + η ) ] .
ω / 2 A > ( cos 2 ψ + 2 sin 2 ψ ) 1 2 .
ω c / 2 A = ( cos 2 ψ + 2 sin 2 ψ ) 1 2
ω c / 2 A = [ 1 + sin 2 ψ ( 2 1 ) ] 1 2 ,
ρ = A / [ 1 + sin 2 ψ ( A 2 / B 2 1 ) ] 1 2 .
ω c / 2 A = ρ / A = [ 1 + sin 2 ψ ( 2 1 ) ] 1 2 ,
ω c ( ψ = 0 ) / ω c ( ψ = π / 2 ) = ,