Abstract

Orthogonal expansion of a partially coherent source with a given cross-spectral density <i>W</i> is used to define an effective number <i>N</i> of degrees of freedom and an effective number <i>N</i> of uncorrelated random variables characterizing the source. Relations <i>N</i> ≤ <i>N</i> ≋ Tr <i>W</i>/λ<sub>0</sub> ≤ (Tr <i>W</i>/‖<i>W</i>‖)<sup>2</sup> = <i>V</i><sub>e</sub><i>V</i><sub>ce</sub> are established and discussed. Here Tr <i>W</i>, ‖<i>W</i>‖, and λ<sub>0</sub> are, respectively, the trace, the norm, and the largest eigenvalue of <i>W</i> used as the kernel of a homogeneous Fredholm equation; <i>V</i><sub>e</sub> is an effective volume of the source; and <i>V</i><sub>ce</sub> is its effective coherence volume. The main results are illustrated by a Gaussian Schell-model source.

© 1982 Optical Society of America

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  1. E. T. Whittaker, "On the functions which are represented by the expansions of the interpolation-theory," Proc. R. Soc. Edinburgh, Sect. A 35, 181–194 (1915).
  2. V. A. Kotel'nikov, "On the transmission capacity of 'ether' and wire in electrocommunications," contribution to the First All-Union Conference on Technical Reconstruction of the Communication Network and Development of the Low-Current Industry (Izdatel'stvo Redactzii Upravl'eniya Svyaz'i RKKA, Moscow, 1933), (in Russian).
  3. C. E. Shannon, "Communication in the presence of noise," Proc. IRE 37, 10–19 (1949).
  4. D. Gabor, "Theory of communication," J. IEE 93, Part III, 429–457 (1946).
  5. L. Mandel, "Fluctuations of photon beams and their correlations," Proc. Phys. Soc. 72, 1037–1048 (1958); "Fluctuations of photon beams: the distribution of the photo-electrons," Proc. Phys. Soc. 74, 233–243 (1959).
  6. E. M. Purcell, Nature (London) t78, 1449–1450 (1956) (untitled).
  7. C. W. Helstrom, "The distribution of photoelectric counts from partially polarized Gaussian light," Proc. Phys. Soc. 83, 777–782 (1964); "Quantum detection theory," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1972), Vol. X, pp. 289–369.
  8. J. Bures, C. Delisle, and A. Zardecki, "Détermination de la surface de cohérence à partir d'une experience de photocomptage," Can. J. Phys. 50, 760–768 (1972).
  9. M. Kac and A. J. F. Siegert, "On the theory of noise in radio receivers with square law detectors," J. Appl. Phys. 18, 383–397 (1947).
  10. L. Mandel and E. Wolf, "Spectral coherence and the concept of cross-spectral purity," J. Opt. Soc. Am. 66, 529–535 (1976).
  11. E. Wolf, "New spectral representation of random sources and of the partially coherent fields that they generate," Opt. Commun. 38, 3–6 (1981).
  12. E. Wolf, "New theory of partial coherence in the space-frequency domain. Part I: Spectra and cross-spectra of steady-state sources," J. Opt. Soc. Am. 72, 343–351 (1982).
  13. H. Hochstadt, Integral Equations (Wiley, New York, 1973), p. 90.
  14. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), Secs. 15. 3-3a and 15. 2-5.
  15. This type of expansion is similar to the well-known Karhunen-Loève expansion of a random process.16 Some differences between them are pointed out in Sec. 5 of Ref. 12.
  16. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), pp. 453–464.
  17. Upper bounds on λ0 in terms of iterated kernels are given by expressions (122) and (127) of Ref. 13.
  18. Lower bounds on λ0 in terms of iterated kernels are given in F. G. Tricomi, Integral Equations (Interscience, New York, 1957), Sec. 3.8. Note the difference in notation, particularly that λn+1 in this reference means the same as I/λn in the present paper.
  19. L. Mandel, "Fluctuations of light beams," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1963), Vol. II, pp. 181–248.
  20. The word Gaussian refers here to the dependence of I(x) and µ(x) on x.
  21. E. Wolf and E. Collett, "Partially coherent sources that produce the same far-field intensity distribution as a laser," Opt. Commun. 25, 293–296 (1978).
  22. J. T. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297–300 (1978).
  23. F. Gori, "Collett-Wolf sources and multimode lasers," Opt. Commun. 34, 301–305 (1980).
  24. A. Starikov and E. Wolf, "Coherent mode representation of Gaussian Schell-model sources and of their radiation fields," J. Opt. Soc. Am. 72, 923–928 (1982).
  25. Even though the preceding theory has been formulated for the finite sources, it is possible to show that it is also valid for many infinite sources, including Gaussian Schell-model sources.
  26. G. Toraldo di Francia, "Some recent progress in classical optics," Riv. Nuovo Cimento Ser.I 1, 460–484 (1969).

1982 (2)

1981 (1)

E. Wolf, "New spectral representation of random sources and of the partially coherent fields that they generate," Opt. Commun. 38, 3–6 (1981).

1980 (1)

F. Gori, "Collett-Wolf sources and multimode lasers," Opt. Commun. 34, 301–305 (1980).

1978 (2)

E. Wolf and E. Collett, "Partially coherent sources that produce the same far-field intensity distribution as a laser," Opt. Commun. 25, 293–296 (1978).

J. T. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297–300 (1978).

1976 (1)

1972 (1)

J. Bures, C. Delisle, and A. Zardecki, "Détermination de la surface de cohérence à partir d'une experience de photocomptage," Can. J. Phys. 50, 760–768 (1972).

1969 (1)

G. Toraldo di Francia, "Some recent progress in classical optics," Riv. Nuovo Cimento Ser.I 1, 460–484 (1969).

1958 (1)

L. Mandel, "Fluctuations of photon beams and their correlations," Proc. Phys. Soc. 72, 1037–1048 (1958); "Fluctuations of photon beams: the distribution of the photo-electrons," Proc. Phys. Soc. 74, 233–243 (1959).

1949 (1)

C. E. Shannon, "Communication in the presence of noise," Proc. IRE 37, 10–19 (1949).

1947 (1)

M. Kac and A. J. F. Siegert, "On the theory of noise in radio receivers with square law detectors," J. Appl. Phys. 18, 383–397 (1947).

1946 (1)

D. Gabor, "Theory of communication," J. IEE 93, Part III, 429–457 (1946).

1915 (1)

E. T. Whittaker, "On the functions which are represented by the expansions of the interpolation-theory," Proc. R. Soc. Edinburgh, Sect. A 35, 181–194 (1915).

Bures, J.

J. Bures, C. Delisle, and A. Zardecki, "Détermination de la surface de cohérence à partir d'une experience de photocomptage," Can. J. Phys. 50, 760–768 (1972).

Collett, E.

E. Wolf and E. Collett, "Partially coherent sources that produce the same far-field intensity distribution as a laser," Opt. Commun. 25, 293–296 (1978).

Delisle, C.

J. Bures, C. Delisle, and A. Zardecki, "Détermination de la surface de cohérence à partir d'une experience de photocomptage," Can. J. Phys. 50, 760–768 (1972).

Foley, J. T.

J. T. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297–300 (1978).

Francia, G. Toraldo di

G. Toraldo di Francia, "Some recent progress in classical optics," Riv. Nuovo Cimento Ser.I 1, 460–484 (1969).

Gabor, D.

D. Gabor, "Theory of communication," J. IEE 93, Part III, 429–457 (1946).

Gori, F.

F. Gori, "Collett-Wolf sources and multimode lasers," Opt. Commun. 34, 301–305 (1980).

Helstrom, C. W.

C. W. Helstrom, "The distribution of photoelectric counts from partially polarized Gaussian light," Proc. Phys. Soc. 83, 777–782 (1964); "Quantum detection theory," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1972), Vol. X, pp. 289–369.

Hochstadt, H.

H. Hochstadt, Integral Equations (Wiley, New York, 1973), p. 90.

Kac, M.

M. Kac and A. J. F. Siegert, "On the theory of noise in radio receivers with square law detectors," J. Appl. Phys. 18, 383–397 (1947).

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), Secs. 15. 3-3a and 15. 2-5.

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), Secs. 15. 3-3a and 15. 2-5.

Kotel’nikov, V. A.

V. A. Kotel'nikov, "On the transmission capacity of 'ether' and wire in electrocommunications," contribution to the First All-Union Conference on Technical Reconstruction of the Communication Network and Development of the Low-Current Industry (Izdatel'stvo Redactzii Upravl'eniya Svyaz'i RKKA, Moscow, 1933), (in Russian).

Mandel, L.

L. Mandel and E. Wolf, "Spectral coherence and the concept of cross-spectral purity," J. Opt. Soc. Am. 66, 529–535 (1976).

L. Mandel, "Fluctuations of photon beams and their correlations," Proc. Phys. Soc. 72, 1037–1048 (1958); "Fluctuations of photon beams: the distribution of the photo-electrons," Proc. Phys. Soc. 74, 233–243 (1959).

L. Mandel, "Fluctuations of light beams," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1963), Vol. II, pp. 181–248.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), pp. 453–464.

Purcell, E. M.

E. M. Purcell, Nature (London) t78, 1449–1450 (1956) (untitled).

Shannon, C. E.

C. E. Shannon, "Communication in the presence of noise," Proc. IRE 37, 10–19 (1949).

Siegert, A. J. F.

M. Kac and A. J. F. Siegert, "On the theory of noise in radio receivers with square law detectors," J. Appl. Phys. 18, 383–397 (1947).

Starikov, A.

Whittaker, E. T.

E. T. Whittaker, "On the functions which are represented by the expansions of the interpolation-theory," Proc. R. Soc. Edinburgh, Sect. A 35, 181–194 (1915).

Wolf, E.

Zardecki, A.

J. Bures, C. Delisle, and A. Zardecki, "Détermination de la surface de cohérence à partir d'une experience de photocomptage," Can. J. Phys. 50, 760–768 (1972).

Zubairy, M. S.

J. T. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297–300 (1978).

Can. J. Phys. (1)

J. Bures, C. Delisle, and A. Zardecki, "Détermination de la surface de cohérence à partir d'une experience de photocomptage," Can. J. Phys. 50, 760–768 (1972).

J. Appl. Phys. (1)

M. Kac and A. J. F. Siegert, "On the theory of noise in radio receivers with square law detectors," J. Appl. Phys. 18, 383–397 (1947).

J. IEE (1)

D. Gabor, "Theory of communication," J. IEE 93, Part III, 429–457 (1946).

J. Opt. Soc. Am. (3)

Opt. Commun. (4)

E. Wolf and E. Collett, "Partially coherent sources that produce the same far-field intensity distribution as a laser," Opt. Commun. 25, 293–296 (1978).

J. T. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297–300 (1978).

F. Gori, "Collett-Wolf sources and multimode lasers," Opt. Commun. 34, 301–305 (1980).

E. Wolf, "New spectral representation of random sources and of the partially coherent fields that they generate," Opt. Commun. 38, 3–6 (1981).

Proc. IRE (1)

C. E. Shannon, "Communication in the presence of noise," Proc. IRE 37, 10–19 (1949).

Proc. Phys. Soc. (1)

L. Mandel, "Fluctuations of photon beams and their correlations," Proc. Phys. Soc. 72, 1037–1048 (1958); "Fluctuations of photon beams: the distribution of the photo-electrons," Proc. Phys. Soc. 74, 233–243 (1959).

Proc. R. Soc. Edinburgh, Sect. A (1)

E. T. Whittaker, "On the functions which are represented by the expansions of the interpolation-theory," Proc. R. Soc. Edinburgh, Sect. A 35, 181–194 (1915).

Riv. Nuovo Cimento Ser. (1)

G. Toraldo di Francia, "Some recent progress in classical optics," Riv. Nuovo Cimento Ser.I 1, 460–484 (1969).

Other (12)

V. A. Kotel'nikov, "On the transmission capacity of 'ether' and wire in electrocommunications," contribution to the First All-Union Conference on Technical Reconstruction of the Communication Network and Development of the Low-Current Industry (Izdatel'stvo Redactzii Upravl'eniya Svyaz'i RKKA, Moscow, 1933), (in Russian).

E. M. Purcell, Nature (London) t78, 1449–1450 (1956) (untitled).

C. W. Helstrom, "The distribution of photoelectric counts from partially polarized Gaussian light," Proc. Phys. Soc. 83, 777–782 (1964); "Quantum detection theory," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1972), Vol. X, pp. 289–369.

Even though the preceding theory has been formulated for the finite sources, it is possible to show that it is also valid for many infinite sources, including Gaussian Schell-model sources.

H. Hochstadt, Integral Equations (Wiley, New York, 1973), p. 90.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), Secs. 15. 3-3a and 15. 2-5.

This type of expansion is similar to the well-known Karhunen-Loève expansion of a random process.16 Some differences between them are pointed out in Sec. 5 of Ref. 12.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), pp. 453–464.

Upper bounds on λ0 in terms of iterated kernels are given by expressions (122) and (127) of Ref. 13.

Lower bounds on λ0 in terms of iterated kernels are given in F. G. Tricomi, Integral Equations (Interscience, New York, 1957), Sec. 3.8. Note the difference in notation, particularly that λn+1 in this reference means the same as I/λn in the present paper.

L. Mandel, "Fluctuations of light beams," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1963), Vol. II, pp. 181–248.

The word Gaussian refers here to the dependence of I(x) and µ(x) on x.

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