Abstract

In this paper we deal with the use of feedback systems to generate Gaussian Schell-model sources. Owing to the approximation of quasi-homogeneity of the beam and assuming slow transmittance variations of diffracting elements, the coherence-propagation equation can be simplifed and leads after several transits to a multiple-convolution product. According to the central-limit theorem and whatever the pupil functions are, this product reduces in general to a Gaussian distribution. This result is applied to imaging and lensless feedback systems.

© 1983 Optical Society of America

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References

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  1. W. H. Carter and E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [Crossref]
  2. H. P. Baltes, J. Geist, and A. Walther, Radiometry and Coherence, Vol. 9 of Topics in Current Physics (Springer-Verlag, Berlin, 1978), pp. 135–139.
  3. E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27 (1978).
    [Crossref] [PubMed]
  4. J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
    [Crossref]
  5. P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [Crossref]
  6. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1978).
    [Crossref]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  8. J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
    [Crossref]
  9. J. W. Goodman, Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), pp. 61–63.
    [Crossref]
  10. Although the reflection and transmission coefficients strongly limit the available energy in the output plane, they have been dropped in the mathematical development since this paper deals more with the qualitative study of the propagation of a partially coherent field in the system than with energy considerations.
  11. See, for instance, R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), pp. 168–173;A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), pp. 227–239.
  12. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–588 (1961).
    [Crossref]
  13. G. Dujardin and P. Flamant, “Conversion d’énergie dans les amplificateurs à colorants en presénce de superfluorescence,” Opt. Acta 25, 273–283 (1978).
    [Crossref]
  14. V. Ganiel, A. Hardy, G. Neumann, and D. Treves, “Amplified spontaneous emission and signal amplification in dye-laser systems,” IEEE J. Quantum Electron. QE-11, 881–891 (1975).
    [Crossref]
  15. E. Wolf, “Recent researches on coherence properties of light,” presented at Third International Conference on Quantum Electronics, Paris, 1964.

1980 (1)

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[Crossref]

1979 (1)

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

1978 (3)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1978).
[Crossref]

E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27 (1978).
[Crossref] [PubMed]

G. Dujardin and P. Flamant, “Conversion d’énergie dans les amplificateurs à colorants en presénce de superfluorescence,” Opt. Acta 25, 273–283 (1978).
[Crossref]

1977 (1)

1975 (1)

V. Ganiel, A. Hardy, G. Neumann, and D. Treves, “Amplified spontaneous emission and signal amplification in dye-laser systems,” IEEE J. Quantum Electron. QE-11, 881–891 (1975).
[Crossref]

1965 (1)

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[Crossref]

1961 (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–588 (1961).
[Crossref]

Baltes, H. P.

H. P. Baltes, J. Geist, and A. Walther, Radiometry and Coherence, Vol. 9 of Topics in Current Physics (Springer-Verlag, Berlin, 1978), pp. 135–139.

Bracewell, R.

See, for instance, R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), pp. 168–173;A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), pp. 227–239.

Carter, W. H.

Collett, E.

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[Crossref]

E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27 (1978).
[Crossref] [PubMed]

de Santis, P.

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Dujardin, G.

G. Dujardin and P. Flamant, “Conversion d’énergie dans les amplificateurs à colorants en presénce de superfluorescence,” Opt. Acta 25, 273–283 (1978).
[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, 203–207 (1980).
[Crossref]

Flamant, P.

G. Dujardin and P. Flamant, “Conversion d’énergie dans les amplificateurs à colorants en presénce de superfluorescence,” Opt. Acta 25, 273–283 (1978).
[Crossref]

Fox, A. G.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–588 (1961).
[Crossref]

Ganiel, V.

V. Ganiel, A. Hardy, G. Neumann, and D. Treves, “Amplified spontaneous emission and signal amplification in dye-laser systems,” IEEE J. Quantum Electron. QE-11, 881–891 (1975).
[Crossref]

Geist, J.

H. P. Baltes, J. Geist, and A. Walther, Radiometry and Coherence, Vol. 9 of Topics in Current Physics (Springer-Verlag, Berlin, 1978), pp. 135–139.

Goodman, J. W.

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[Crossref]

J. W. Goodman, Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), pp. 61–63.
[Crossref]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gori, F.

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1978).
[Crossref]

Guattari, G.

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Hardy, A.

V. Ganiel, A. Hardy, G. Neumann, and D. Treves, “Amplified spontaneous emission and signal amplification in dye-laser systems,” IEEE J. Quantum Electron. QE-11, 881–891 (1975).
[Crossref]

Li, T.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–588 (1961).
[Crossref]

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, 203–207 (1980).
[Crossref]

Neumann, G.

V. Ganiel, A. Hardy, G. Neumann, and D. Treves, “Amplified spontaneous emission and signal amplification in dye-laser systems,” IEEE J. Quantum Electron. QE-11, 881–891 (1975).
[Crossref]

Palma, C.

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Treves, D.

V. Ganiel, A. Hardy, G. Neumann, and D. Treves, “Amplified spontaneous emission and signal amplification in dye-laser systems,” IEEE J. Quantum Electron. QE-11, 881–891 (1975).
[Crossref]

Walther, A.

H. P. Baltes, J. Geist, and A. Walther, Radiometry and Coherence, Vol. 9 of Topics in Current Physics (Springer-Verlag, Berlin, 1978), pp. 135–139.

Wolf, E.

Bell Syst. Tech. J. (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–588 (1961).
[Crossref]

IEEE J. Quantum Electron. (1)

V. Ganiel, A. Hardy, G. Neumann, and D. Treves, “Amplified spontaneous emission and signal amplification in dye-laser systems,” IEEE J. Quantum Electron. QE-11, 881–891 (1975).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

G. Dujardin and P. Flamant, “Conversion d’énergie dans les amplificateurs à colorants en presénce de superfluorescence,” Opt. Acta 25, 273–283 (1978).
[Crossref]

Opt. Commun. (3)

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[Crossref]

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1978).
[Crossref]

Opt. Lett. (1)

Proc. IEEE (1)

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[Crossref]

Other (6)

J. W. Goodman, Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), pp. 61–63.
[Crossref]

Although the reflection and transmission coefficients strongly limit the available energy in the output plane, they have been dropped in the mathematical development since this paper deals more with the qualitative study of the propagation of a partially coherent field in the system than with energy considerations.

See, for instance, R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), pp. 168–173;A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), pp. 227–239.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

E. Wolf, “Recent researches on coherence properties of light,” presented at Third International Conference on Quantum Electronics, Paris, 1964.

H. P. Baltes, J. Geist, and A. Walther, Radiometry and Coherence, Vol. 9 of Topics in Current Physics (Springer-Verlag, Berlin, 1978), pp. 135–139.

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the imaging feedback system.

Fig. 2
Fig. 2

Numerical example of multiple convolutions. In spite of the strong variations of the input function, the convolution product rapidly tends toward a Gaussian distribution.

Fig. 3
Fig. 3

Numerical example of multiple convolutions. In this case, the difference between the Gaussian curve and the convolution is already no longer perceptible after the first product.

Fig. 4
Fig. 4

Experimental setup. The spatially incoherent source is obtained by destroying the coherence of a laser beam by means of a rotating diffuser. In order to avoid the overlapping of images after 1… n transits, the optical system is slightly off axis. Diameter of object, ∼5 mm; diameter of Airy disk of the pupil P, ∼10 mm; distance of object to lens, ∼600 mm.

Fig. 5
Fig. 5

Experimental result: (a) cross section of image intensity, (b) degree of spatial coherence after 0, 1, 2, and 3 transits.

Fig. 6
Fig. 6

Lensless feedback system (model of Fox and Li).

Fig. 7
Fig. 7

(a) Amplified spontaneous emission and (b) laser emission in dye laser after 5 m of propagation. The area of the spot, distorted by aberrations, is almost the same in both cases.

Equations (33)

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W ( y 1 , y 2 ) = + W ( x 1 , x 2 ) K ( x 1 , y 1 ) K * ( x 2 , y 2 ) d 2 x 1 d 2 x 2 ,
W ( x 1 , x 2 ) I [ ½ ( x 1 + x 2 ) ] μ ( x 1 x 2 ) ,
K ( x , y ) = 1 λ 2 d 2 exp [ j k 2 d ( y 2 + x 2 ) ] × + P ( ξ ) exp [ j k d ( x y ) ξ ] d 2 ξ ,
W ( y 1 , y 2 ) 1 ( λ 2 d 2 ) 2 exp [ j k 2 d ( y 1 2 y 2 2 ) ] × + I [ ½ ( x 1 + x 2 ) ] μ ( x 1 x 2 ) P ( ξ 1 ) P * ( ξ 2 ) × exp [ j k 2 d ( x 1 2 x 2 2 ) ] exp { j k d [ ( x 1 y 1 ) ξ 1 ( x 2 y 2 ) ξ 2 ] } d 2 x 1 d 2 x 2 d 2 ξ 1 d 2 ξ 2 .
d > 2 S c λ , d > 2 ( S s S c ) 1 / 2 λ ,
P ( ξ 1 ) P * ( ξ 2 ) [ P ( ξ 1 + ξ 2 ) 2 ] 2 ,
W ( y 1 , y 2 ) 1 ( λ 2 d 2 ) 2 exp [ j k 2 d ( y 1 2 y 2 2 ) ] × + I [ ½ ( x 1 + x 2 ) ] μ ( x 1 x 2 ) { P [ ½ ( ξ 1 + ξ 2 ) ] } 2 × exp { j k d [ ( x 1 y 1 ) ξ 1 ( x 2 y 2 ) ξ 2 ] } × d 2 x 1 d 2 x 2 d 2 ξ 1 d 2 ξ 2 .
W ( y 1 , y 2 ) 4 λ 2 d 2 + I [ ½ ( x 1 + x 2 ) ] μ ( x 1 x 2 ) × + exp { j k d ( x 1 + x 2 y 1 y 2 ) ξ 1 } d 2 ξ 1 × + [ P ( λ η d ) ] 2 exp { j 4 π ( y 2 x 2 ) η } d 2 η d 2 x 1 d 2 x 2 .
W ( y 1 , y 2 ) 4 exp { j k 2 d ( y 1 2 y 2 2 ) } × + I [ ½ ( x 1 + x 2 ) ] μ ( x 1 x 2 ) h [ 2 ( y 2 x 2 ) ] × δ 2 ( x 1 + x 2 y 1 y 2 ) d 2 x 1 d 2 x 2 ,
h [ 2 ( y 2 x 2 ) ] = + [ P ( λ η d ) ] 2 exp [ j 2 π ( y 2 x 2 ) η ] d 2 η .
W ( y 1 , y 2 ) exp [ j k 2 d ( y 1 2 y 2 2 ) ] I [ ½ ( y 1 + y 2 ) ] × + μ ( α ) h [ α ( y 1 y 2 ) ] d 2 α ,
W s ( y 1 , y 2 ) I [ ½ ( y 1 + y 2 ) ] g ( y 1 y 2 ) ,
W s ( y 1 , y 2 ) g ( 0 ) I [ ½ ( y 1 + y 2 ) ] μ s ( y 1 y 2 ) ,
W s ( y 1 , y 2 ) I [ ½ ( y 1 + y 2 ) ] μ s ( y 1 y 2 ) ,
W 1 ( y 1 , y 2 ) 1 ( λ 2 d 2 ) 2 exp [ j k 2 d ( y 1 2 y 2 2 ) ] × + I [ ½ ( x 1 + x 2 ) ] g ( x 1 x 2 ) × exp [ j k 2 d 2 ( x 1 2 x 2 2 ) ] { P [ ½ ( ξ 1 + ξ 2 ) ] } 2 × exp { j k d [ ( x 1 y 1 ) ξ 1 ( x 2 y 2 ) ξ 2 ] } × d 2 x 1 d 2 x 2 d 2 ξ 1 d 2 ξ 2 .
W 1 ( y 1 , y 2 ) exp [ j k 2 d ( y 1 2 y 2 2 ) ] × I [ ½ ( y 1 + y 2 ) ] g 1 ( y 1 y 2 ) ,
g 1 = g * h = μ * h * h .
W n ( y 1 , y 2 ) 1 ( λ 2 d 2 ) 2 exp [ j k 2 d ( y 1 2 y 2 2 ) ] × + I [ ½ ( x 1 + x 2 ) ] g n 1 ( x 1 x 2 ) × exp [ j k 2 d 2 ( x 1 2 x 2 2 ) ] { P [ ½ ( ξ 1 + ξ 2 ) ] } 2 × exp { j k d [ ( x 1 y 1 ) ξ 1 ( x 2 y 2 ) ξ 2 ] } × d 2 x 1 d 2 x 2 d 2 ξ 1 d 2 ξ 2 .
W s n ( y 1 , y 2 ) I [ ½ ( y 1 + y 2 ) ] ( μ * h * h * h h ) n + 1 times
f ( x ) = f 1 ( x ) * f 2 ( x ) * . f n ( x ) ,
I ( r ) = A exp ( r 2 / 2 σ I 2 ) , μ ( r ) = exp ( r 2 / 2 σ μ 2 ) ,
J ( θ ) = J ( 0 ) cos 2 θ exp ( sin 2 θ / 2 Δ 2 ) ,
Δ = ( 1 / k σ μ ) [ 1 + ( σ μ / 2 σ I ) 2 ] 1 / 2
k σ I ½ Δ , k σ μ 1 / Δ .
W n ( y 1 , y 2 ) = I [ ½ ( y 1 + y 2 ) ] exp [ ( y 1 y 2 ) 2 / 2 σ μ 2 ] .
I [ ½ ( y 1 + y 2 ) ] = A exp [ ( y 1 + y 2 ) 2 2 / 2 σ I 2 ] ,
f ( x ) = rect x / a ( 1 + cos x 2 )
P ( ξ 1 ) P * ( ξ 2 ) P [ ½ ( ξ 1 + ξ 2 ) ] 2 .
K ( x , y ) = 1 λ 2 d 2 exp [ j k 2 d ( x 2 + y 2 ) ] × + exp { j k d [ ξ 2 ( x y ) ξ ] } rect ξ / a d 2 ξ .
P ( ξ ) = exp ( j k d ξ 2 ) rect ξ / a ,
W s ( y 1 , y 2 ) I [ ½ ( y 1 + y 2 ) ] g ( y 1 y 2 ) ,
g ( y 1 y 2 ) = + μ ( α ) h [ α ( y 1 y 2 ) ] d 2 α .
W s n ( y 1 , y 2 ) I [ ½ ( y 1 + y 2 ) ] ( μ * h * h * h * h ) n times