Abstract

Orthogonal expansion of a partially coherent source with a given cross-spectral density W is used to define an effective number Nof degrees of freedom and an effective number N of uncorrelated random variables characterizing the source. Relations NNTrW/λ0(TrW/W)2=Ve/Vce are established and discussed. Here Tr W, ‖W‖, and λ0 are, respectively, the trace, the norm, and the largest eigenvalue of W used as the kernel of a homogeneous Fredholm equation; Ve is an effective volume of the source; and Vce 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-Uniori 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).
    [CrossRef]
  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).
    [CrossRef]
  6. E. M. Purcell, Nature (London)178, 1449–1450 (1956) (untitled).
    [CrossRef]
  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.
    [CrossRef]
  8. J. Bures, C. Delisle, and A. Zardecki, “Détermination de la surface de cohérence á partir d’une expérience de photocomptage,” Can. J. Phys. 50, 760–768 (1972).
    [CrossRef]
  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).
    [CrossRef]
  10. L. Mandel and E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [CrossRef]
  11. E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
    [CrossRef]
  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).
    [CrossRef]
  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), Sees. 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 1/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.
    [CrossRef]
  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).
    [CrossRef]
  22. J. T. Foley and M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
    [CrossRef]
  23. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  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).
    [CrossRef]
  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).
[CrossRef]

1980 (1)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

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).
[CrossRef]

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

1976 (1)

1972 (1)

J. Bures, C. Delisle, and A. Zardecki, “Détermination de la surface de cohérence á partir d’une expérience de photocomptage,” Can. J. Phys. 50, 760–768 (1972).
[CrossRef]

1969 (1)

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

1964 (1)

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.
[CrossRef]

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).
[CrossRef]

1949 (1)

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–19 (1949).
[CrossRef]

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).
[CrossRef]

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 á partir d’une expérience de photocomptage,” Can. J. Phys. 50, 760–768 (1972).
[CrossRef]

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).
[CrossRef]

Delisle, C.

J. Bures, C. Delisle, and A. Zardecki, “Détermination de la surface de cohérence á partir d’une expérience de photocomptage,” Can. J. Phys. 50, 760–768 (1972).
[CrossRef]

Foley, J. T.

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

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).
[CrossRef]

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.
[CrossRef]

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).
[CrossRef]

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), Sees. 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), Sees. 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-Uniori 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).
[CrossRef]

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).
[CrossRef]

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

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)178, 1449–1450 (1956) (untitled).
[CrossRef]

Shannon, C. E.

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–19 (1949).
[CrossRef]

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).
[CrossRef]

Starikov, A.

Toraldo di Francia, G.

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

Tricomi, F. G.

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 1/λn in the present paper.

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 á partir d’une expérience de photocomptage,” Can. J. Phys. 50, 760–768 (1972).
[CrossRef]

Zubairy, M. S.

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

Can. J. Phys. (1)

J. Bures, C. Delisle, and A. Zardecki, “Détermination de la surface de cohérence á partir d’une expérience de photocomptage,” Can. J. Phys. 50, 760–768 (1972).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

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

Proc. IRE (1)

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–19 (1949).
[CrossRef]

Proc. Phys. Soc. (2)

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.
[CrossRef]

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).
[CrossRef]

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. I (1)

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

Other (11)

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.

V. A. Kotel’nikov, “On the transmission capacity of ‘ether’ and wire in electrocommunications,” contribution to the First All-Uniori 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)178, 1449–1450 (1956) (untitled).
[CrossRef]

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), Sees. 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 1/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.
[CrossRef]

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

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Figures (3)

Fig. 1
Fig. 1

(a) A typical distribution of the source-integrated strength λn = 〈|an|2〉 of the nth mode as a function of n and the definition N ≡ Σn=0n0) of the effective number N of uncorrelated random variables {an} in the representation of the source. (b) Definition of the effective number of statistically independent random variables (degrees of freedom) in the representation of the source. Circles denote the variances of the uncorrelated random variables an that have higher-order correlations with those of lower value of n. For the particular diagram presented, N = N 5.

Fig. 2
Fig. 2

The distribution of the source-integrated strength of the nth mode of the Gaussian Schell-model source as a function of n for various values of the parameter ασI/σμ characterizing the degree of homogeneity of the source. Note the abrupt transition between large λn’s and small λn’s as a function of n for a « 1 and also the slow change for a » 1. (This figure is, apart from notation, Fig. 1 of Ref. 24).

Fig. 3
Fig. 3

The dependence of the effective coherence volume Vce on the effective volume Ve of the Gaussian Schell-model source. Note that Vce/Ve → 1 for Ve/Vce → 0 and Vce/Vce → 1 for Ve/Vce → ∞, Vce being the limiting value of Vce for large Ve.

Equations (62)

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Γ ( r 1 , r 2 , τ ) = Q * ( r 1 , t ) Q ( r 2 , t + τ ) , r 1 D , r 2 D .
W ( r 1 , r 2 , ω ) 1 2 π Γ ( r 1 , r 2 , τ ) e i ω τ d τ .
μ ( r 1 , r 2 , ω ) W ( r 1 , r 2 , ω ) [ ( r 1 , r 1 , ω ) ] 1 / 2 W [ ( r 2 , r 2 , ω ) ] 1 / 2 .
I ( r , ω ) W ( r , r , ω )
W ( r 1 , r 2 , ω ) = [ I ( r 1 , ω ) ] 1 / 2 [ I ( r 2 , ω ) ] 1 / 2 μ ( r 1 , r 2 , ω ) .
Tr W D W ( r , r , ω ) d 3 r < .
W 2 D d 3 r 1 D d 3 r 2 | W ( r 1 , r 2 , ω ) | 2 < .
W ( r 1 , r 2 , ω ) = n λ n ( ω ) φ n * ( r 1 , ω ) φ n ( r 2 , ω ) ,
D W ( r 1 , r 2 , ω ) φ n ( r 1 ) d 3 r 1 = λ n φ n ( r 2 ) .
M λ 0 λ 1 λ 2 λ n 0 ,
D φ n * ( r ) φ m ( r ) d 3 r = δ n m ,
Tr W = n λ n <
W 2 = n λ n 2 < .
U ( r , ω ) = n a n ( ω ) φ n ( r , ω )
a n ( ω ) = 0 ,
a n * ( ω ) a m ( ω ) = λ n ( ω ) δ n m .
W ( r 1 , r 2 , ω ) = U * ( r 1 , ω ) U ( r 2 , ω ) .
λ n = | a n ( ω ) | 2 .
Tr W n = 0 N 1 λ 0 = λ 0 N
N Tr W λ 0 = n = 0 λ n λ 0 .
Tr W = n = 0 λ n = λ 0 + n = 1 λ n λ 0 ,
N Tr W λ 0 1 .
0 λ n λ 0 1
W 2 = λ 0 2 n = 0 ( λ n λ 0 ) λ 0 2 n = 0 λ n λ 0 = λ 0 n = 0 λ n = λ 0 Tr W .
λ 0 W 2 Tr W = D D | W ( r 1 , r 2 , ω ) | 2 d 3 r 1 d 3 r 2 D W ( r , r , ω ) d 3 r .
N ( Tr W W ) 2 = [ D W ( r , r , ω ) d 3 r ] 2 D D | W ( r 1 , r 2 , ω ) | 2 d 3 r 1 d 3 r 2 .
N N ( Tr W W ) 2 = [ D W ( r , r , ω ) d 3 r ] 2 D D | W ( r 1 , r 2 , ω ) | 2 d 3 r 1 d 3 r 2 .
W 2 = n = 0 λ n 2 = λ 0 2 = λ 0 Tr W ,
λ 0 = W 2 Tr W ,
N = ( Tr W W ) 2 = 1.
W 2 = n = 0 λ n 2 = λ 0 2 N = λ 0 Tr W ,
λ 0 = W 2 Tr W ,
N = ( Tr W W ) 2 .
N = T ξ ,
ξ ( T ) 1 Γ 2 ( 0 ) T T / 2 T / 2 d t 1 T / 2 T / 2 d t 2 | Γ ( t 1 t 2 ) | 2 .
ξ lim T 0 ξ ( T ) = 1 Γ 2 ( 0 ) | Γ ( τ ) | 2 d τ ,
lim T 0 T ξ ( T ) = 1.
V c e W 2 V e I max 2 ( ω ) = 1 V e I max 2 ( ω ) D d 3 r 1 D d 3 r 2 | W ( r 1 , r 2 , ω ) | 2
V e Tr W I max ( ω ) = 1 I max ( ω ) D d 3 r W ( r , r , ω ) ,
I max ( ω ) max r D W ( r , r , ω ) max r D I ( r , ω ) .
N N Tr W λ 0 ( Tr W W ) 2 = V e V c e .
W ( r 1 , r 2 , ω ) = W ( r 1 r 2 , ω ) = I ( ω ) μ ( r 1 r 2 , ω ) ,
I ( ω ) = W ( r , r , ω ) = W ( 0 , ω ) .
V c e = 1 V D d 3 r 1 D d 3 r 2 | μ ( r 1 , r 2 , ω ) | 2
V e = D d 3 r = V ,
V c = D d 3 r | μ ( r , ω ) | 2 .
W ( x 1 , x 2 , ω ) = I 0 ( ω ) exp [ x 1 2 + x 2 2 4 σ I 2 ( ω ) ( x 1 x 2 ) 2 2 σ μ 2 ( ω ) ] ,
λ n = I 0 ( π a + b + c ) 1 / 2 ( b a + b + c ) n
φ n ( x ) = ( 2 c π ) 1 / 4 1 ( 2 n n ! ) 1 / 2 H n ( x 2 c ) e c x 2 ,
a 1 4 σ I 2 , b 1 2 σ μ 2 , c ( a 2 + 2 a b ) 1 / 2 ,
α σ I ( ω ) σ μ ( ω )
Tr W = I 0 ( 2 π σ I 2 ) 1 / 2 = I 0 ( π 2 a ) 1 / 2 .
0 < ( b a + b + c ) 2 < 1 .
W 2 = n = 0 I 0 2 π a + b + c ( b a + b + c ) 2 n = I 0 2 π 2 c .
N Tr W λ 0 = [ 1 + 2 α 2 + ( 1 + 4 α 2 ) 1 / 2 2 ] 1 / 2 .
N = 1 + Q ( α ) .
N = α + Q ( 1 ) .
N ( Tr W W ) 2 = ( 1 + 4 α 2 ) 1 / 2
L e = Tr W I max ( ω ) = ( 2 π σ I 2 ) 1 / 2 .
L c e = W 2 L e I max 2 ( ω ) = ( 2 π σ I 2 ) 1 / 2 ( 1 + 4 α 2 ) 1 / 2 = ( 2 π σ μ 2 ) 1 / 2 ( 1 α 2 + 4 ) 1 / 2 .
L c e lim L e 0 L c e = ( π 2 ) 1 / 2 σ μ .
L c e 0 lim L e 0 L c e = L e .