Abstract

The theoretical irradiance is presented for the images of sharp-edged objects as a function of the coherence of the illumination of the object. These distributions are presented for optical systems possessing both square apertures and circular apertures. The irradiance for degraded edged objects is also determined for optical systems with square apertures. The influence of the degree of coherence upon contrast and edge gradient is considered in detail. Experimental results are given which show good agreement with the theoretical irradiance. In all cases the systems are assumed free from aberration and the quasimonochromatic assumption is employed.

© 1965 Optical Society of America

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References

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  1. M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).
  2. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), Chap. X.
  3. D. Canals-Frau and M. Rousseau, Opt. Acta 5, 15 (1958).
    [Crossref]
  4. M. De and S. C. Som, Opt. Acta 4, 17 (1962).
    [Crossref]
  5. W. N. Charman, J. Opt. Soc. Am. 53, 410 (1963).
    [Crossref]
  6. W. T. Welford, Optics in Meteorology (Pergamon Press, Inc., New York, 1960), pp. 85–91.
  7. H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).
  8. The mutual intensity is found by setting τ equal to zero in the expression for the mutual-coherence function which is useful when investigating the imaging process employing quasimonochromatic light.
  9. One of the restrictions imposed by Hopkins, namely that the mutual-intensity function in the object plane is spatially stationary, can be removed by employing other expressions for the propagation of the mutual-intensity function. These areJ(x1,x2)=K∫∫J(ξ1,ξ2)exp[i(x2⋅ξ2−x1⋅ξ1)]dξ1dξ2for the propagation of the mutual intensity between planes sufficiently separated to allow the far-field approximation andJ′(x1,x2)=J(x1,x2)T(x1)T*(x2)for propagation through diffracting objects or apertures having a transmittance function T(x). We note in the first expression that the “far-field approximation may be valid” in the near field as shown by the case when J(ξ1,ξ2) represents an incoherent source and the first expression reduces to the Van Cittert–Zernike theorem, which Hopkins uses to describe the effective source.
  10. D. Canals-Frau and M. Rousseau, Ref. 3, report a similar analysis employing the Fourier transforms of the functions which we use. However, the normalization criteria are different.
  11. In this case the normalized mutual-intensity function or complex degree of coherence measured perpendicular to the edged object is given by Γ0(u1−u2) = [sin(u2−u1)∊]/[(u2−u1)∊].
  12. By coherence interval, we mean the minimum separation between points in the object plane (spatial separation) for which the mutual-intensity function vanishes.
  13. See Fig. 8 of Ref. 5. Private communication with W. N. Charman revealed that a typographical error resulted in the mislabeling of the “s” value at the low-intensity region of the curves in this figure.
  14. That is, the width of the image of the square effective source in the plane of the entrance aperture normalized by the width of the square entrance aperture.

1963 (1)

1962 (1)

M. De and S. C. Som, Opt. Acta 4, 17 (1962).
[Crossref]

1958 (1)

D. Canals-Frau and M. Rousseau, Opt. Acta 5, 15 (1958).
[Crossref]

1953 (1)

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

Beran, M. J.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), Chap. X.

Canals-Frau, D.

D. Canals-Frau and M. Rousseau, Opt. Acta 5, 15 (1958).
[Crossref]

D. Canals-Frau and M. Rousseau, Ref. 3, report a similar analysis employing the Fourier transforms of the functions which we use. However, the normalization criteria are different.

Charman, W. N.

De, M.

M. De and S. C. Som, Opt. Acta 4, 17 (1962).
[Crossref]

Hopkins, H. H.

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

Parrent, G. B.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).

Rousseau, M.

D. Canals-Frau and M. Rousseau, Opt. Acta 5, 15 (1958).
[Crossref]

D. Canals-Frau and M. Rousseau, Ref. 3, report a similar analysis employing the Fourier transforms of the functions which we use. However, the normalization criteria are different.

Som, S. C.

M. De and S. C. Som, Opt. Acta 4, 17 (1962).
[Crossref]

Welford, W. T.

W. T. Welford, Optics in Meteorology (Pergamon Press, Inc., New York, 1960), pp. 85–91.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), Chap. X.

J. Opt. Soc. Am. (1)

Opt. Acta (2)

D. Canals-Frau and M. Rousseau, Opt. Acta 5, 15 (1958).
[Crossref]

M. De and S. C. Som, Opt. Acta 4, 17 (1962).
[Crossref]

Proc. Roy. Soc. (London) (1)

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

Other (10)

The mutual intensity is found by setting τ equal to zero in the expression for the mutual-coherence function which is useful when investigating the imaging process employing quasimonochromatic light.

One of the restrictions imposed by Hopkins, namely that the mutual-intensity function in the object plane is spatially stationary, can be removed by employing other expressions for the propagation of the mutual-intensity function. These areJ(x1,x2)=K∫∫J(ξ1,ξ2)exp[i(x2⋅ξ2−x1⋅ξ1)]dξ1dξ2for the propagation of the mutual intensity between planes sufficiently separated to allow the far-field approximation andJ′(x1,x2)=J(x1,x2)T(x1)T*(x2)for propagation through diffracting objects or apertures having a transmittance function T(x). We note in the first expression that the “far-field approximation may be valid” in the near field as shown by the case when J(ξ1,ξ2) represents an incoherent source and the first expression reduces to the Van Cittert–Zernike theorem, which Hopkins uses to describe the effective source.

D. Canals-Frau and M. Rousseau, Ref. 3, report a similar analysis employing the Fourier transforms of the functions which we use. However, the normalization criteria are different.

In this case the normalized mutual-intensity function or complex degree of coherence measured perpendicular to the edged object is given by Γ0(u1−u2) = [sin(u2−u1)∊]/[(u2−u1)∊].

By coherence interval, we mean the minimum separation between points in the object plane (spatial separation) for which the mutual-intensity function vanishes.

See Fig. 8 of Ref. 5. Private communication with W. N. Charman revealed that a typographical error resulted in the mislabeling of the “s” value at the low-intensity region of the curves in this figure.

That is, the width of the image of the square effective source in the plane of the entrance aperture normalized by the width of the square entrance aperture.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), Chap. X.

W. T. Welford, Optics in Meteorology (Pergamon Press, Inc., New York, 1960), pp. 85–91.

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Figures (10)

F. 1
F. 1

Optical system showing the entrance and exit pupils of the imaging system. The effective source need not exist in an actual optical system and the field lens images the effective source onto the entrance pupil of the imaging system. The space surrounding the system is assumed to be filled with air.

F. 2
F. 2

Irradiance in an image of a sharp edge for varying degrees of coherence. A square entrance pupil is assumed. The dashed curve is the irradiance in the image of an incoherently illuminated edge.

F. 3
F. 3

Same as Fig. 2, except a circular entrance pupil and effective sources are assumed.

F. 4
F. 4

Intensity transmission for the trapezoidal object.

F. 5
F. 5

Irradiance in the image of a degraded edge for various degrees of coherence. The slant parameter s of trapezoidal object is 1.0 reduced units.

F. 6
F. 6

Same as Fig. 5, except s = 10.0.

F. 7
F. 7

The variation of the edge gradients with coherence for the various objects. (A) sharp edge, system -with square apertures; (B) degraded edge with s = 1.0; (C) sharp edge, system with circular apertures; (D) degraded edge with s = 4.25; and (E) degraded edge with s = 10.0.

F.8
F.8

Comparison between theoretical irradiance and Char-man’s experimental results (see Ref. 13). (a) = 0.6; (b) = 0.8; and (c) = 1.0.

F. 9
F. 9

Experimental optical arrangement.

F. 10
F. 10

Experimental irradiance plotted against the theoretical distributions for a degraded edge with a slant s = 14 μ. (a) = 0.23; (b) = 0.54; (c) = 0.92; and (d) = 1.03.

Equations (15)

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Γ 12 ( τ ) = V 1 ( t + τ ) V 2 * ( t ) .
ψ ( x , y , u , υ ) = 1 2 π 0 ( m , n ) × f ( x + m , y + n ) e i ( m u + n υ ) d m d n ,
I ( u , υ ) = 2 π I s ( x , y ) | ψ ( x , y , u , υ ) | 2 d x d y .
T ( u , υ ) = { A 0 < u < w , | υ | < 0 otherwise .
0 ( m , n ) = 2 A δ ( n ) [ sin ( m w / 2 ) / m ] e i ( m w / 2 ) .
f ( x , y ) = { 1 | x | 1 , | y | 1 0 otherwise .
| ψ ( x , y , u ) | 2 = { I 0 / 4 π 2 { [ Si ( 1 + x ) u + Si ( 1 x ) u Si ( 1 + x ) ( u w ) Si ( 1 x ) ( u w ) ] 2 + [ Ci ( 1 + x ) u Ci ( 1 x ) u + Ci ( 1 x ) ( u w ) Ci ( 1 x ) ( u w ) ] 2 | y | < 1 0 for | y | > 1
I s ( x , y ) = { 1 / 2 π | x | , | y | 0 otherwise .
f ( r , θ ) = { 1 r 1 0 elsewhere .
I s ( r , θ ) = { 1 / 2 π r 0 otherwise .
T ( u ) = { 0 u 0 ( A / S ) u 0 u < S A S u w + S ( A / S ) ( w + 2 S u ) W + S u w + 2 S 0 W + 2 S < u .
0 ( m , n ) = ( 2 A / S ) δ ( n ) × { [ ( cos β m cos α m ) / m 2 ] } e i m α ,
β = w / 2 α = ( w / 2 ) + S .
J(x1,x2)=KJ(ξ1,ξ2)exp[i(x2ξ2x1ξ1)]dξ1dξ2
J(x1,x2)=J(x1,x2)T(x1)T*(x2)