Abstract

The problem is discussed of determining the distribution of the intensity and of the degree of spectral coherence of a planar, secondary quasi-homogeneous source from the cross-spectral density function of the field measured over any plane that is parallel to the source. A solution to this problem is presented under the assumption that the degree of spectral coherence g(ρ1ρ2, ω) of the quasi-homogeneous source does not vary appreciably across the source over distances |ρ1ρ2| that are of the order of or less than the wavelength λ corresponding to the frequency to. The results are illustrated by computational reconstruction of Gaussian-correlated quasi-homogeneous sources.

© 1985 Optical Society of America

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  1. We are aware of only two papers regarding problems of this kind, viz., A. J. Devaney, “The inverse problem for random sources,” J. Math. Phys. 20, 1687–1691 (1979);I. J. LaHaie, “Inverse source problem for three-dimensional partially coherent sources and fields,” J. Opt. Soc. Am. A 2, 35–45 (1985).
    [CrossRef]
  2. W. H. Carter, E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  3. J. C. Dainty, “An introduction to Gaussian speckle,” Proc. Soc. Photo-Opt. Instrum. Eng. 243, 2–8 (1980).
  4. See, for example, E. Collett, E. Wolf, “Is a complete spatial coherence necessary for the generation of highly directional beams?” Opt. Lett. 2, 27–29 (1978);P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979);J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 10.2.
  6. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976), Eq. (2.10).
    [CrossRef]
  7. E. Wolf, W. H. Carter, “Fields generated by homogeneous and by quasi-homogeneous planar secondary sources,” Opt. Commun. 50, 131–136 (1984).There is an error in Eq. (4.6) of this paper. The constant k should be omitted in the exponent of the integral in that equation. This error does not affect the subsequent equations.
    [CrossRef]
  8. To simplify the notation we do not display from now on the dependence of the various quantities on the frequency ω.
  9. The cross-spectral density may be determined by taking the temporal Fourier transform of the mutual coherence function Γ(ρ1, ρ2, z, τ). The mutual coherence function may be determined from Young’s interference experiments, with pinholes at the points P1(ρ1, z) and P2(ρ2, z) (Ref. 5, Sec. 10.3). Alternatively, the cross-spectral density may be determined more directly from such experiments, if narrow-band filters, with passbands centered at the frequency ω, are placed in front of the pinholes [cf. E. Wolf, “Young’s interference fringes with narrow-band light,” Opt. Lett. 8, 250–252 (1983)].
  10. Since strictly speaking the preceding analysis applies to fields produced by finite sources, it would be more appropriate to assume that Eq. (4.1) holds when ρ≤ a and that I(0)(ρ) = 0 when ρ> a, where a is the radius of the source. However, when a ≫ σI, as we will assume from now on, no significant error is introduced by taking I(0)(ρ) to be given by Eq. (4.1) for all values of ρ.
  11. E. Wolf, “Completeness of coherent-mode eigenfunctions of Schell-model sources,” Opt. Lett. 9, 387–389 (1984), Eqs. (2.9) and (2.15).
    [CrossRef] [PubMed]
  12. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1944), p. 20, Eq. (5) (with an obvious substitution).

1984

E. Wolf, W. H. Carter, “Fields generated by homogeneous and by quasi-homogeneous planar secondary sources,” Opt. Commun. 50, 131–136 (1984).There is an error in Eq. (4.6) of this paper. The constant k should be omitted in the exponent of the integral in that equation. This error does not affect the subsequent equations.
[CrossRef]

E. Wolf, “Completeness of coherent-mode eigenfunctions of Schell-model sources,” Opt. Lett. 9, 387–389 (1984), Eqs. (2.9) and (2.15).
[CrossRef] [PubMed]

1980

J. C. Dainty, “An introduction to Gaussian speckle,” Proc. Soc. Photo-Opt. Instrum. Eng. 243, 2–8 (1980).

1979

We are aware of only two papers regarding problems of this kind, viz., A. J. Devaney, “The inverse problem for random sources,” J. Math. Phys. 20, 1687–1691 (1979);I. J. LaHaie, “Inverse source problem for three-dimensional partially coherent sources and fields,” J. Opt. Soc. Am. A 2, 35–45 (1985).
[CrossRef]

1978

1977

1976

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 10.2.

Carter, W. H.

E. Wolf, W. H. Carter, “Fields generated by homogeneous and by quasi-homogeneous planar secondary sources,” Opt. Commun. 50, 131–136 (1984).There is an error in Eq. (4.6) of this paper. The constant k should be omitted in the exponent of the integral in that equation. This error does not affect the subsequent equations.
[CrossRef]

W. H. Carter, E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
[CrossRef]

Collett, E.

Dainty, J. C.

J. C. Dainty, “An introduction to Gaussian speckle,” Proc. Soc. Photo-Opt. Instrum. Eng. 243, 2–8 (1980).

Devaney, A. J.

We are aware of only two papers regarding problems of this kind, viz., A. J. Devaney, “The inverse problem for random sources,” J. Math. Phys. 20, 1687–1691 (1979);I. J. LaHaie, “Inverse source problem for three-dimensional partially coherent sources and fields,” J. Opt. Soc. Am. A 2, 35–45 (1985).
[CrossRef]

Mandel, L.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1944), p. 20, Eq. (5) (with an obvious substitution).

Wolf, E.

J. Math. Phys.

We are aware of only two papers regarding problems of this kind, viz., A. J. Devaney, “The inverse problem for random sources,” J. Math. Phys. 20, 1687–1691 (1979);I. J. LaHaie, “Inverse source problem for three-dimensional partially coherent sources and fields,” J. Opt. Soc. Am. A 2, 35–45 (1985).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

E. Wolf, W. H. Carter, “Fields generated by homogeneous and by quasi-homogeneous planar secondary sources,” Opt. Commun. 50, 131–136 (1984).There is an error in Eq. (4.6) of this paper. The constant k should be omitted in the exponent of the integral in that equation. This error does not affect the subsequent equations.
[CrossRef]

Opt. Lett.

Proc. Soc. Photo-Opt. Instrum. Eng.

J. C. Dainty, “An introduction to Gaussian speckle,” Proc. Soc. Photo-Opt. Instrum. Eng. 243, 2–8 (1980).

Other

To simplify the notation we do not display from now on the dependence of the various quantities on the frequency ω.

The cross-spectral density may be determined by taking the temporal Fourier transform of the mutual coherence function Γ(ρ1, ρ2, z, τ). The mutual coherence function may be determined from Young’s interference experiments, with pinholes at the points P1(ρ1, z) and P2(ρ2, z) (Ref. 5, Sec. 10.3). Alternatively, the cross-spectral density may be determined more directly from such experiments, if narrow-band filters, with passbands centered at the frequency ω, are placed in front of the pinholes [cf. E. Wolf, “Young’s interference fringes with narrow-band light,” Opt. Lett. 8, 250–252 (1983)].

Since strictly speaking the preceding analysis applies to fields produced by finite sources, it would be more appropriate to assume that Eq. (4.1) holds when ρ≤ a and that I(0)(ρ) = 0 when ρ> a, where a is the radius of the source. However, when a ≫ σI, as we will assume from now on, no significant error is introduced by taking I(0)(ρ) to be given by Eq. (4.1) for all values of ρ.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 10.2.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1944), p. 20, Eq. (5) (with an obvious substitution).

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Figures (3)

Fig. 1
Fig. 1

Illustrating the concept of a quasi-homogeneous source (at fixed frequency ω). The optical intensity distribution I(0)(ρ, ω) across the source varies much more slowly with ρ than the degree of spectral coherence g(0)(ρ′, ω) varies with ρ = |ρ1ρ2|. Thus the intensity remains sensibly constant over distances of the order of the spectral correlation length Δ across the source (the effective width of |g(0)(ρ′, ω)). For the purpose of illustration the source is considered here to be one dimensional.

Fig. 2
Fig. 2

Illustrating the notation relating to the determination of the intensity and of the degree of spectral coherence across the source from measurements of second-order field correlations in a plane 2 = constant parallel to the source and at a finite distance from it.

Fig. 3
Fig. 3

Computer simulation of reconstruction of a planar, secondary, quasi-homogeneous source, located in the plane z = 0, whose degree of spectral coherence and intensity are given by Eqs. (4.1) and (4.2), respectively, subject to the inequality (4.3). The solid curves show the reconstruction (a) of the degree of spectral coherence of the source and (b) of the source intensity, from knowledge of the cross-spectral density of the field in any plane z = constant > 0, for sources with selected correlation length σg. With the normalization indicated on the axes, the curves labeled “Exact” (shown in dashed lines) apply to all the sources. When σg ≳ 0.6λ, the reconstructed values agree with the exact ones to within negligible errors.

Equations (65)

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Γ ( 0 ) ( ρ 1 , ρ 2 , τ ) = V ( 0 ) ( ρ 1 , t + τ ) V ( 0 ) * ( ρ 2 , t ) .
W ( 0 ) ( ρ 1 , ρ 2 , ω ) = 1 2 π Γ ( 0 ) ( ρ 1 , ρ 2 , τ ) e i ω τ d τ ,
μ ( 0 ) ( ρ 1 , ρ 2 , ω ) = W ( 0 ) ( ρ 1 , ρ 2 , ω ) [ W ( 0 ) ( ρ 1 , ρ 1 , ω ) ] 1 / 2 [ W ( 0 ) ( ρ 2 , ρ 2 , ω ) ] 1 / 2 .
I ( 0 ) ( ρ , ω ) W ( 0 ) ( ρ , ρ , ω ) ,
W ( 0 ) ( ρ 1 , ρ 2 , ω ) = [ I ( 0 ) ( ρ 1 , ω ) ] 1 / 2 [ I ( 0 ) ( ρ 2 , ω ) ] 1 / 2 μ ( 0 ) ( ρ 1 , ρ 2 , ω ) .
μ ( 0 ) ( ρ 1 , ρ 2 , ω ) g ( 0 ) ( ρ 1 ρ 2 , ω ) .
W ( 0 ) ( ρ 1 , ρ 2 , ω ) I ( 0 ) ( ρ 1 + ρ 2 2 , ω ) g ( 0 ) ( ρ 1 ρ 2 , ω ) .
W ( ρ 1 , ρ 2 , z , ω ) = 1 2 π Γ ( ρ 1 , ρ 2 , z , τ ) e i ω τ d τ ,
Γ ( ρ 1 , ρ 2 , z , τ ) = V ( ρ 1 , z , t + τ ) V * ( ρ 2 , z , t ) .
W ( f 1 , f 2 , z , ω ) = 1 ( 2 π ) 4 W ( ρ 1 , ρ 2 , z , ω ) × exp [ i ( f 1 ρ 1 + f 2 ρ 2 ) ] d 2 ρ 1 d 2 ρ 2 .
W ( f 1 , f 2 , z ) = Ĩ ( 0 ) ( f 1 + f 2 ) g ( 0 ) [ ( f 1 f 2 ) / 2 ] exp [ i ( h 1 h 2 * ) z ] ,
Ĩ ( 0 ) ( f ) = 1 ( 2 π ) 2 I ( 0 ) ( ρ ) exp [ ( i f ρ ) d 2 ρ ] ,
g ( 0 ) ( f ) = 1 ( 2 π ) 2 g ( 0 ) ( ρ ) exp ( i f ρ ) d 2 ρ ,
h j = + ( k 2 f j 2 ) 1 / 2 when f j 2 k 2
= + i ( f j 2 k 2 ) 1 / 2 when f j 2 > k 2
k = ω / c ,
f 1 = f 2 = f .
W ( f , f , z ) = Ĩ ( 0 ) ( 2 f ) g ( 0 ) ( 0 ) exp [ i ( h h * ) z ] ,
h = + k 2 f 2 when f 2 k 2
= i + f 2 k 2 when f 2 k 2 .
f 1 = f 2 = f ,
W ( f , f , z ) = Ĩ ( 0 ) ( 0 ) g ( 0 ) ( f ) exp [ i ( h h * ) z ] ,
W ( f , f , z ) = I ( 0 ) ( 2 f ) g ( 0 ) ( 0 ) ( f 2 k 2 ) ,
W ( f , f , z ) = Ĩ ( 0 ) ( 2 f ) g ( 0 ) ( f ) ( f 2 k 2 ) .
Ĩ ( 0 ) ( f ) = 1 g ( 0 ) ( 0 ) W ( f 2 , f 2 , z ) ( f 2 k 2 ) ,
g ( 0 ) ( f ) = 1 Ĩ ( 0 ) ( 0 ) W ( f , f , z ) ( f 2 k 2 ) ,
I ( 0 ) ( ρ ) 1 g ( 0 ) ( 0 ) f 2 k 2 W ( f 2 , f 2 , z ) exp ( i f ρ ) d 2 f ,
g ( 0 ) ( ρ ) 1 Ĩ ( 0 ) ( 0 ) f 2 k 2 W ( f , f , z ) exp ( i f ρ ) d 2 f ,
g ( 0 ) ( 0 ) = 1 ,
Ĩ ( 0 ) ( 0 ) f 2 k 2 W ( f , f , z ) d 2 f .
Ĩ ( 0 ) ( 0 ) g ( 0 ) ( 0 ) = W ( 0 , 0 , z ) .
g ( 0 ) ( 0 ) W ( 0 , 0 , z ) f 2 k 2 W ( f , f , z ) d 2 f .
I ( 0 ) ( ρ ) = A exp ( ρ 2 / 2 σ I 2 ) ,
g ( 0 ) ( ρ ) = exp ( ρ 2 / 2 σ g 2 ) ,
σ I σ g .
Ĩ ( 0 ) ( f ) = ( A σ I 2 / 2 π ) exp ( σ I 2 f 2 / 2 ) ,
g ( 0 ) ( f ) = ( σ g 2 / 2 π ) exp ( σ g 2 f 2 / 2 ) .
W ( f , f , z ) = A ( σ I σ g / 2 π ) 2 exp ( 2 σ I 2 f 2 ) ( f 2 k 2 ) ,
W ( f , f , z ) = A ( σ I σ g / 2 π ) 2 exp ( σ g 2 f 2 / 2 ) ( f 2 k 2 ) ,
I ( 0 ) ( ρ ) A σ I 2 { 1 exp [ ( k σ g ) 2 / 2 ] } × 0 k exp ( σ I 2 f 2 / 2 ) J 0 ( f ρ ) f d f
g ( 0 ) ( ρ ) σ g 2 { 1 exp [ ( k σ g ) 2 / 2 ] } × 0 k exp ( σ g 2 f 2 / 2 ) J 0 ( f ρ ) f d f .
D D W ( 0 ) ( ρ 1 , ρ 2 ) f ( ρ 1 ) f * ( ρ 2 ) d 2 ρ , d 2 ρ 2 > 0 ,
f ( ρ ) = 1 when ρ D ,
D D W ( 0 ) ( ρ 1 , ρ 2 ) d 2 ρ , d 2 ρ 2 > 0 ,
W ( 0 ) ( 0 , 0 ) > 0 .
W ( 0 ) ( 0 , 0 ) = Ĩ ( 0 ) ( 0 ) g ( 0 ) ( 0 ) .
Ĩ ( 0 ) ( 0 ) g ( 0 ) ( 0 ) > 0 ,
Ĩ ( 0 ) ( 0 ) 0 , g ( 0 ) ( 0 ) 0 .
Ĩ ( 0 ) ( 0 ) = 1 ( 2 π ) 2 I ( 0 ) ( ρ ) d 2 ρ .
Ĩ ( 0 ) ( 0 ) > 0 .
g ( 0 ) ( 0 ) > 0 .
f 2 k 2 W ( f 2 , f 2 , z ) exp ( i f ρ ) d 2 f = A ( σ I σ g 2 π ) 2 f 2 k 2 exp ( σ I 2 f 2 / 2 ) exp ( i f ρ ) d 2 f .
f = ( f cos θ , f sin θ ) ,
ρ = ( ρ cos φ , ρ sin φ ) .
f 2 k 2 W ( f 2 , f 2 , z ) exp ( i f ρ ) d 2 f = A ( σ I σ g 2 π ) 2 × 0 2 π d θ 0 k exp ( σ I 2 f 2 / 2 ) exp [ if ρ cos ( θ φ ) ] f d f .
0 2 π exp [ if ρ cos ( θ φ ) ] d θ = 2 π J 0 ( f ρ ) ,
f 2 k 2 W ( f 2 , f 2 , z ) exp ( i f ρ ) d 2 f = A ( σ I σ g ) 2 2 π 0 k exp ( σ I 2 f 2 / 2 ) J 0 ( f ρ ) f d f .
f 2 k 2 W ( f , f , z ) exp ( i f ρ ) d 2 f = A ( σ I σ g 2 π ) 2 f 2 k 2 exp ( σ g 2 f 2 / 2 ) exp ( i f ρ ) d 2 f .
f 2 k 2 W ( f , f , z ) exp ( i f ρ ) d 2 f = A ( σ I σ g ) 2 2 π 0 k exp ( σ g 2 f 2 / 2 ) J 0 ( f ρ ) f d f .
I ( 0 ) ( 0 ) A ( σ I σ g ) 2 2 π 0 k exp ( σ g 2 f 2 / 2 ) f d f .
I ( 0 ) ( 0 ) A σ I 2 2 π { 1 exp [ ( k σ g ) 2 / 2 ] } .
W ( 0 , 0 , z ) = A ( σ I σ g / 2 π ) 2 ,
g ( 0 ) ( 0 ) σ g 2 2 π 1 { 1 exp [ ( k σ g ) 2 / 2 ] } .
I ( 0 ) ( ρ ) A σ I 2 { 1 exp [ ( k σ g ) 2 / 2 ] } × 0 k exp ( σ I 2 f 2 / 2 ) J 0 ( f ρ ) f d f .
g ( 0 ) ( ρ ) σ g 2 { 1 exp [ ( k σ g ) 2 / 2 ] } × 0 k exp ( σ g 2 f 2 / 2 ) J 0 ( f ρ ) f d f .

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