Abstract

This is a theoretical study of a partially coherent light source incorporating a stochastic, diffuser-like component and a deterministic, beam-splitter-like component. The resulting degree of spectral coherence in the far zone exhibits two peaks, whose shapes are determined by the Fourier transform of the source-plane intensity profile. Experimental realizations of this compound source are proposed. Implications for information coding are discussed.

© 1981 Optical Society of America

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References

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  1. H. P. Baltes, J. Geist, and A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, Vol. 9 of Topics in Current Physics, H. P. Baltes, ed. (Springer Verlag, Berlin, 1978), pp. 119–154; E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978); H. P. Baltes and B. Steinle, “Radiometry with fields of large coherence area,” Nuovo Cimento B 41, 428–440 (1977); W. H. Carter and E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977); A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  2. E. Jakeman and P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen. I. Theory,” J. Phys. A 8, 369–391 (1975).
    [CrossRef]
  3. J. C. Leader, “Far-zone range criteria for quasi-homogeneous partially coherent sources,” J. Opt. Soc. Am. 68, 1332–1338 (1978).
    [CrossRef]
  4. H. P. Baltes, H. A. Ferwerda, A. S. Glass, and B. Steinle, “Retrieval of structural information from far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
    [CrossRef]
  5. This interpretation was first suggested by A. Huiser, Laboratoire de Physique Théorique, EPF Lausanne, Switzerland (personal communication).
  6. T. Asakura and H. Fujii, “Multiple slit interference with partially coherent light,” Optik (Stuttgart) 40, 217–224 (1974); H. Fujii and T. Asakura, “Partially coherent multiple-beam interference,” Appl. Phys. 3, 121–129 (1974).
    [CrossRef]
  7. J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
    [CrossRef]

1981 (1)

H. P. Baltes, H. A. Ferwerda, A. S. Glass, and B. Steinle, “Retrieval of structural information from far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

1980 (1)

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

1978 (1)

1975 (1)

E. Jakeman and P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen. I. Theory,” J. Phys. A 8, 369–391 (1975).
[CrossRef]

1974 (1)

T. Asakura and H. Fujii, “Multiple slit interference with partially coherent light,” Optik (Stuttgart) 40, 217–224 (1974); H. Fujii and T. Asakura, “Partially coherent multiple-beam interference,” Appl. Phys. 3, 121–129 (1974).
[CrossRef]

Asakura, T.

T. Asakura and H. Fujii, “Multiple slit interference with partially coherent light,” Optik (Stuttgart) 40, 217–224 (1974); H. Fujii and T. Asakura, “Partially coherent multiple-beam interference,” Appl. Phys. 3, 121–129 (1974).
[CrossRef]

Baltes, H. P.

H. P. Baltes, H. A. Ferwerda, A. S. Glass, and B. Steinle, “Retrieval of structural information from far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

H. P. Baltes, J. Geist, and A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, Vol. 9 of Topics in Current Physics, H. P. Baltes, ed. (Springer Verlag, Berlin, 1978), pp. 119–154; E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978); H. P. Baltes and B. Steinle, “Radiometry with fields of large coherence area,” Nuovo Cimento B 41, 428–440 (1977); W. H. Carter and E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977); A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
[CrossRef]

Collett, E.

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

Farina, J. D.

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

Ferwerda, H. A.

H. P. Baltes, H. A. Ferwerda, A. S. Glass, and B. Steinle, “Retrieval of structural information from far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

Fujii, H.

T. Asakura and H. Fujii, “Multiple slit interference with partially coherent light,” Optik (Stuttgart) 40, 217–224 (1974); H. Fujii and T. Asakura, “Partially coherent multiple-beam interference,” Appl. Phys. 3, 121–129 (1974).
[CrossRef]

Geist, J.

H. P. Baltes, J. Geist, and A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, Vol. 9 of Topics in Current Physics, H. P. Baltes, ed. (Springer Verlag, Berlin, 1978), pp. 119–154; E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978); H. P. Baltes and B. Steinle, “Radiometry with fields of large coherence area,” Nuovo Cimento B 41, 428–440 (1977); W. H. Carter and E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977); A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
[CrossRef]

Glass, A. S.

H. P. Baltes, H. A. Ferwerda, A. S. Glass, and B. Steinle, “Retrieval of structural information from far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

Huiser, A.

This interpretation was first suggested by A. Huiser, Laboratoire de Physique Théorique, EPF Lausanne, Switzerland (personal communication).

Jakeman, E.

E. Jakeman and P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen. I. Theory,” J. Phys. A 8, 369–391 (1975).
[CrossRef]

Leader, J. C.

Narducci, L. M.

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

Pusey, P. N.

E. Jakeman and P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen. I. Theory,” J. Phys. A 8, 369–391 (1975).
[CrossRef]

Steinle, B.

H. P. Baltes, H. A. Ferwerda, A. S. Glass, and B. Steinle, “Retrieval of structural information from far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

Walther, A.

H. P. Baltes, J. Geist, and A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, Vol. 9 of Topics in Current Physics, H. P. Baltes, ed. (Springer Verlag, Berlin, 1978), pp. 119–154; E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978); H. P. Baltes and B. Steinle, “Radiometry with fields of large coherence area,” Nuovo Cimento B 41, 428–440 (1977); W. H. Carter and E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977); A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. A (1)

E. Jakeman and P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen. I. Theory,” J. Phys. A 8, 369–391 (1975).
[CrossRef]

Opt. Acta (1)

H. P. Baltes, H. A. Ferwerda, A. S. Glass, and B. Steinle, “Retrieval of structural information from far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

Opt. Commun. (1)

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

Optik (Stuttgart) (1)

T. Asakura and H. Fujii, “Multiple slit interference with partially coherent light,” Optik (Stuttgart) 40, 217–224 (1974); H. Fujii and T. Asakura, “Partially coherent multiple-beam interference,” Appl. Phys. 3, 121–129 (1974).
[CrossRef]

Other (2)

This interpretation was first suggested by A. Huiser, Laboratoire de Physique Théorique, EPF Lausanne, Switzerland (personal communication).

H. P. Baltes, J. Geist, and A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, Vol. 9 of Topics in Current Physics, H. P. Baltes, ed. (Springer Verlag, Berlin, 1978), pp. 119–154; E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978); H. P. Baltes and B. Steinle, “Radiometry with fields of large coherence area,” Nuovo Cimento B 41, 428–440 (1977); W. H. Carter and E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977); A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Geometry and notation.

Fig. 2
Fig. 2

Far-zone degree of spectral coherence [Eq. (24)] of a quasi-homogeneous source with beam splitter (Θ = 45°) and Gaussian intensity profile (ka = 5, 10, 20, 40).

Fig. 3
Fig. 3

(a) Angular distribution of far-zone average intensity [Eq. (30)] of a gauss-correlated source (correlation length l) with beam splitter (Θ = 45°) and Gaussian intensity profile (ka = 20). (b) Same as (a) but with beam-splitter angle Θ = 10°.

Fig. 4
Fig. 4

Far-zone degree of spectral coherence [Eq. (32)] of a gauss-correlated source (correlation length l) with beam-splitter (Θ = 45°) and Gaussian intensity profile (width a). The transition from the multilplex Van Cittert–Zernike regime (ka = 10, kl ≪ 1) to the coherent limit (la) is shown.

Fig. 5
Fig. 5

Example of proposed experiment: 1, laser and beam expander; 2, rotating semitransparent mirror with rough surface; 3, mirrors; 4, smooth, semitransparent mirror; 5, detector.

Equations (38)

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w ( x 1 - x 2 ) exp ( - x 1 - x 2 2 / 2 l 2 ) ,
u ( x ) = exp ( i k x sin Θ ) + exp ( - i k x sin Θ )
W ( x 1 , x 2 ) = u * ( x 1 ) u ( x 2 ) w ( x 1 - x 2 ) .
W ( x 1 , x 2 ) = [ I ( x 1 ) I ( x 2 ) ] 1 / 2 u * ( x 1 ) u ( x 2 ) w ( x 1 - x 2 ) .
I ( x ) = exp ( - x 2 / 2 a 2 ) ,
W ( x 1 , x 2 ) I [ ( x 1 + x 2 ) / 2 ] u * ( x 1 ) u ( x 2 ) w ( x 1 - x 2 ) .
W ˆ ( s 1 , s 2 ) cos θ 1 cos θ 2 d x 1 d x 2 × exp [ i k ( x 1 s 1 = x 2 s 2 ) ] W ( x 1 , x 2 ) .
ξ = x 1 = x 2 , x = ( x 1 + x 2 ) / 2 and σ = s 1 - s 2 , s = ( s 1 + s 2 ) / 2
W ˆ ( s , σ ) cos θ 1 cos θ 2 d ξ d x exp ( - i k s ξ ) × exp ( - i k σ x ) W ( x + ξ / 2 , x - ξ / 2 )
W ˆ ( s , σ ) cos θ 1 cos θ 2 d ξ exp ( - i k s ξ ) d x exp ( - i k σ x ) × w ( ξ ) I ( x ) u * ( x + ξ / 2 ) u ( x - ξ / 2 ) .
W ˆ ( s , σ ) cos θ 1 cos θ 2 d ξ w ( ξ ) exp ( - i k s ξ ) d x I ( x ) × exp ( - i k σ x ) { exp [ - i k sin Θ ( x + ξ / 2 ) ] + exp [ i k sin Θ ( x + ξ / 2 ) ] } { exp [ i k sin Θ ( x - ξ / 2 ) ] + exp [ - i k sin Θ ( x - ξ / 2 ) ] } ,
W ˆ ( s , σ ) cos θ 1 cos θ 2 { d ξ w ( ξ ) × exp [ - i k ξ ( s + sin Θ ) ] d x I ( x ) exp ( - i k σ x ) + d ξ w ( ξ ) exp [ - i k ξ ( s - sin Θ ) ] d x I ( x ) exp ( - i k σ x ) + d ξ w ( ξ ) exp ( - i k ξ s ) d x I ( x ) × exp [ - i k ( σ + 2 sin Θ ) x ] + d ξ w ( ξ ) × exp ( - i k ξ s ) d x I ( x ) exp [ - i k ( σ - 2 sin Θ ) x ] } .
ŵ ( s ) = d ξ w ( ξ ) exp ( - i k s ξ ) ,
Î ( σ ) = d x I ( x ) exp ( - i k σ x ) .
W ˆ ( s , σ ) cos θ 1 cos θ 2 { [ ŵ ( s + sin Θ ) + ŵ ( s - sin Θ ) ] Î ( σ ) + ŵ ( s ) [ Î ( σ + 2 sin Θ ) + Î ( σ - 2 sin Θ ) ] } .
W ˆ ( s , σ ) sin Θ = 0 cos θ 1 cos θ 2 ŵ ( s ) Î ( σ ) ,
I ( s ) = W ( s , σ = 0 ) cos 2 θ { Î ( 0 ) [ ŵ ( s + sin Θ ) + ŵ ( s - sin Θ ) ] + 2 Î ( 2 sin Θ ) ŵ ( s ) } ,
2 I ( 2 sin Θ ) = 2 d x I ( x ) exp [ - i k ( 2 sin Θ ) x ]
I ( 2 sin Θ ) = exp ( - 2 k 2 a 2 sin 2 Θ ) 1
I ( s ) cos 2 θ [ ŵ ( s + sin Θ ) + ŵ ( s - sin Θ ) ] ,
I ( s ) cos 2 θ { w ( s ) [ δ ( s + sin Θ ) + δ ( s - sin Θ ) ] } .
μ ( s , σ ) = W ˆ ( s , σ ) [ I ( s + σ / 2 ) I ( s - σ / 2 ) ] - 1 / 2 .
μ ( s , σ ) = { [ ŵ ( s + sin Θ ) + ŵ ( s - sin Θ ) ] Î ( σ ) + ŵ ( s ) [ Î ( σ + 2 sin Θ ) + Î ( σ - 2 sin Θ ) ] } × { Î ( 0 ) [ ŵ ( s + σ / 2 + sin Θ ) + ŵ ( s + σ / 2 - sin Θ ) ] } - 1 / 2 × { Î ( 0 ) [ ŵ ( s - σ / 2 + sin Θ ) + ŵ ( s - σ / 2 - sin Θ ) ] } - 1 / 2 .
μ ( s , σ ) = [ Î ( 0 ) ] - 1 [ Î ( σ ) + ½ Î ( σ + 2 sin Θ ) + ½ Î ( σ - 2 sin Θ ) ] ,
μ ( s , σ ) = exp ( - ½ k 2 a 2 σ 2 ) + ½ exp [ - ½ k 2 a 2 ( σ + 2 sin Θ ) 2 ] + ½ exp [ - ½ k 2 a 2 ( σ - 2 sin Θ ) 2 ] .
μ ( s , σ ) = Î ( σ ) / Î ( 0 ) = exp ( - ½ k 2 a 2 σ 2 ) ,
W ˆ ( x 1 , x 2 ) = exp ( - x 1 2 / 4 a 2 ) exp ( - x 2 2 / 4 a 2 ) × exp [ - ( x 1 - x 2 ) 2 / 2 l 2 ] [ exp ( - i k x 1 sin Θ ) + exp ( i k x 1 sin Θ ) ] [ exp ( i k x 2 sin Θ ) + exp ( - i k x 2 sin Θ ) ] .
W ˆ ( s , σ ) cos θ 1 cos θ 2 { d ξ exp ( - ξ 2 / 2 l ¯ 2 ) × exp [ - i k ( s + sin Θ ) ] d x exp ( - x 2 / 2 a 2 ) exp ( - i k σ x ) + d ξ exp ( - ξ 2 / 2 l ¯ 2 ) exp [ - i k ( s - sin Θ ) ] d x × exp ( - x 2 / 2 a 2 ) exp ( - i k σ x ) + d ξ × exp ( - ξ 2 / 2 l ¯ 2 ) exp ( - i k s ξ ) d x exp ( - x 2 / 2 a 2 ) × exp [ - i k ( σ + 2 sin Θ ) x ] + d ξ exp ( - ξ 2 / 2 l ¯ 2 ) × exp ( - i k s ξ ) d x exp ( - x 2 / 2 a 2 ) × exp [ - i k ( σ - 2 sin Θ ) x ] } ,
l ¯ = l ( 1 + l 2 / 4 a 2 ) - 1 .
W ( s , σ ) cos θ 1 cos θ 2 ( { exp [ - ½ k 2 l ¯ 2 ( s + sin Θ ) 2 ] + exp [ - ½ k 2 l ¯ 2 ( s - sin Θ ) 2 ] } exp ( - ½ a 2 k 2 σ 2 ) + exp ( - ½ k 2 l ¯ 2 s 2 ) { exp [ - ½ k 2 a 2 ( σ + 2 sin Θ ) 2 ] + exp [ - ½ k 2 a 2 ( σ - sin Θ ) 2 ] } ) .
I ( s ) = W ( s , σ = 0 ) = cos 2 θ × { exp [ - ½ k 2 l ¯ 2 ( s + sin Θ ) 2 ] + exp ( - ½ k 2 l ¯ 2 ( s - sin Θ ) 2 ] + 2 exp ( - 2 k 2 a 2 sin 2 Θ ) exp ( - ½ k 2 l ¯ 2 s 2 ) } .
W ( s = 0 , σ ) cos θ 1 cos θ 2 { 2 exp ( - ½ k 2 l ¯ 2 sin 2 Θ ) × exp ( - ½ k 2 a 2 σ 2 ) + exp [ - ½ k 2 a 2 ( σ + sin Θ ) 2 ] + exp [ - ½ k 2 a 2 ( σ - sin Θ ) 2 ] } .
μ ( 0 , σ ) = W ( 0 , σ ) [ I ( σ / 2 ) I ( - σ / 2 ) ] - 1 / 2 = { 2 exp ( - ½ k 2 l ¯ 2 sin 2 Θ ) exp ( - ½ k 2 a 2 σ 2 ) + exp [ - ½ k 2 a 2 ( σ + 2 sin Θ ) 2 ] + exp [ - ½ k 2 a 2 ( σ - 2 sin Θ ) 2 ] } × { exp [ - ½ k 2 l ¯ 2 ( σ / 2 + sin Θ ) 2 ] + exp [ - ½ k 2 l ¯ 2 ( σ / 2 - sin Θ ) 2 ] + 2 exp ( - 2 k 2 l ¯ 2 σ 2 ) exp ( - 2 k 2 a 2 sin Θ ) } - 1 .
μ ( 0 , σ ) k l 0 = { 2 exp ( - ½ k 2 a 2 σ 2 ) + exp [ - ½ k 2 a 2 ( σ + 2 sin Θ ) 2 ] + exp [ - ½ k 2 a 2 ( σ - 2 sin Θ ) 2 ] } { 2 + 2 exp ( - 2 k 2 a 2 sin 2 Θ ) } - 1 .
μ ( 0 , 2 sin Θ ) k a sin Θ 1 = [ 1 + exp ( - 2 k 2 l ¯ 2 sin 2 Θ ) ] - 1 ,
μ ( θ 1 , θ 2 ) = u * ( θ 1 ) u ( θ 2 ) [ I ( θ 1 ) I ( θ 2 ) ] - 1 / 2
g ( 2 ) ( θ 1 , θ 2 ) l = u ( θ 1 ) 2 u ( θ 2 ) 2 [ I ( θ 1 ) I ( θ 2 ) ] - 1 .
g ( 2 ) ( θ 1 , θ 2 ) = 1 + μ ( θ 1 , θ 2 ) 2 .