Abstract

We discuss the relation between the coherence of light from nonstationary sources in the space–time domain and the coherence of the monochromatic components of the light. For this purpose we use Loéve’s theory of harmonizable stochastic processes, particularly the spectral representation of a nonstationary stochastic process and the relation between its correlation function and the two-frequency spectral function as a measure of coherence of the monochromatic components of the source.

© 1995 Optical Society of America

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  1. F. Zernike, Proc. Phys. Soc. London 61, 158 (1948).
    [CrossRef]
  2. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  3. R. Gase and M. Schubert, Opt. Acta 29, 1331 (1982).
    [CrossRef]
  4. M. Kàlal, in 16th Congress of the International Commission for Optics: Optics as a Key to High Technology, G. Akos, T. Lippenyi, G. Lupkovics, and A. Podmaniczky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 686 (1993).
  5. J. H. Eberly and K. Wòdkiewicz, J. Opt. Soc. Am. 67, 1252 (1977).
    [CrossRef]
  6. J. H. Eberly, C. V. Kunasz, and K. Wòdkiewicz, J. Phys. B 13, 217 (1980).
    [CrossRef]
  7. B. Cairns and E. Wolf, Opt. Commun. 62, 215 (1987).
    [CrossRef]
  8. L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
    [CrossRef]
  9. L. Mandel and E. Wolf, J. Opt.Soc. Am. 66, 529 (1976).
    [CrossRef]
  10. L. Mandel and E. Wolf, Opt. Commun. 36, 247 (1981).
    [CrossRef]
  11. E. Wolf, J. Opt. Soc. Am. A 72, 343 (1982).
  12. E. Wolf, J. Opt. Soc. Am. A 3, 76 (1986).
    [CrossRef]
  13. M. Loéve, Probability Theory (Van Nostrand, New York, 1955).
  14. Yu. V. Prokhorov and Yu. A. Rozanov, Probability Theory (Springer-Verlag, Berlin, 1969).
    [CrossRef]
  15. Applications of such lasers were recently reported: Congress of the Society of Photo-Optical and Instrumentation Engineers, Budapest, 1993:S. Yokoyama, T. Araki, and N. Suzuki, “Ultra-wideband optical heterodyne interferometer”;R. Dandliker, K. Hug, and J. Politch, “Optical distance measurement with mm accuracy.” Here the measurements of periodic light intensity were also presented.
  16. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  17. Throughout this paper angle brackets denote an ensemble average or a time average for ergodic stochastic processes, even nonstationary ones. In the latter case the time of the average must be small enough to avoid masking the intensity fluctuations.
  18. A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions (Prentice-Hall, Englewood Cliffs, N.J., 1962).
  19. M. S. Bartlett, An Introduction to Stochastic Processes (Cambridge U. Press, Cambridge, 1955).
  20. S. Bochner, Harmonic Analysis and the Theory of Probability (U. California Press, Berkeley, Calif, 1955).
  21. R. Barakat, J. Opt. Soc. Am. A 10, 180 (1993).
    [CrossRef]
  22. Here we use the formalism of the theory of measure and the Lebesgue–Stieltjes integral, rather than that of delta functions, to describe with the same formulas sources with both discrete and continuous spectra (see Refs. 13, 14, 18, and 21).
  23. D. James and E. Wolf, Opt. Commun. 81, 150 (1991).
    [CrossRef]
  24. D. James and E. Wolf, Phys. Lett. A 157, 6 (1991).
    [CrossRef]

1993 (1)

1991 (2)

D. James and E. Wolf, Opt. Commun. 81, 150 (1991).
[CrossRef]

D. James and E. Wolf, Phys. Lett. A 157, 6 (1991).
[CrossRef]

1987 (1)

B. Cairns and E. Wolf, Opt. Commun. 62, 215 (1987).
[CrossRef]

1986 (1)

1982 (2)

E. Wolf, J. Opt. Soc. Am. A 72, 343 (1982).

R. Gase and M. Schubert, Opt. Acta 29, 1331 (1982).
[CrossRef]

1981 (1)

L. Mandel and E. Wolf, Opt. Commun. 36, 247 (1981).
[CrossRef]

1980 (1)

J. H. Eberly, C. V. Kunasz, and K. Wòdkiewicz, J. Phys. B 13, 217 (1980).
[CrossRef]

1977 (1)

1976 (1)

L. Mandel and E. Wolf, J. Opt.Soc. Am. 66, 529 (1976).
[CrossRef]

1965 (1)

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

1948 (1)

F. Zernike, Proc. Phys. Soc. London 61, 158 (1948).
[CrossRef]

Barakat, R.

Bartlett, M. S.

M. S. Bartlett, An Introduction to Stochastic Processes (Cambridge U. Press, Cambridge, 1955).

Bochner, S.

S. Bochner, Harmonic Analysis and the Theory of Probability (U. California Press, Berkeley, Calif, 1955).

Born, M.

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

Cairns, B.

B. Cairns and E. Wolf, Opt. Commun. 62, 215 (1987).
[CrossRef]

Eberly, J. H.

J. H. Eberly, C. V. Kunasz, and K. Wòdkiewicz, J. Phys. B 13, 217 (1980).
[CrossRef]

J. H. Eberly and K. Wòdkiewicz, J. Opt. Soc. Am. 67, 1252 (1977).
[CrossRef]

Gase, R.

R. Gase and M. Schubert, Opt. Acta 29, 1331 (1982).
[CrossRef]

James, D.

D. James and E. Wolf, Phys. Lett. A 157, 6 (1991).
[CrossRef]

D. James and E. Wolf, Opt. Commun. 81, 150 (1991).
[CrossRef]

Kàlal, M.

M. Kàlal, in 16th Congress of the International Commission for Optics: Optics as a Key to High Technology, G. Akos, T. Lippenyi, G. Lupkovics, and A. Podmaniczky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 686 (1993).

Kunasz, C. V.

J. H. Eberly, C. V. Kunasz, and K. Wòdkiewicz, J. Phys. B 13, 217 (1980).
[CrossRef]

Loéve, M.

M. Loéve, Probability Theory (Van Nostrand, New York, 1955).

Mandel, L.

L. Mandel and E. Wolf, Opt. Commun. 36, 247 (1981).
[CrossRef]

L. Mandel and E. Wolf, J. Opt.Soc. Am. 66, 529 (1976).
[CrossRef]

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

Prokhorov, Yu. V.

Yu. V. Prokhorov and Yu. A. Rozanov, Probability Theory (Springer-Verlag, Berlin, 1969).
[CrossRef]

Rozanov, Yu. A.

Yu. V. Prokhorov and Yu. A. Rozanov, Probability Theory (Springer-Verlag, Berlin, 1969).
[CrossRef]

Schubert, M.

R. Gase and M. Schubert, Opt. Acta 29, 1331 (1982).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Wòdkiewicz, K.

J. H. Eberly, C. V. Kunasz, and K. Wòdkiewicz, J. Phys. B 13, 217 (1980).
[CrossRef]

J. H. Eberly and K. Wòdkiewicz, J. Opt. Soc. Am. 67, 1252 (1977).
[CrossRef]

Wolf, E.

D. James and E. Wolf, Phys. Lett. A 157, 6 (1991).
[CrossRef]

D. James and E. Wolf, Opt. Commun. 81, 150 (1991).
[CrossRef]

B. Cairns and E. Wolf, Opt. Commun. 62, 215 (1987).
[CrossRef]

E. Wolf, J. Opt. Soc. Am. A 3, 76 (1986).
[CrossRef]

E. Wolf, J. Opt. Soc. Am. A 72, 343 (1982).

L. Mandel and E. Wolf, Opt. Commun. 36, 247 (1981).
[CrossRef]

L. Mandel and E. Wolf, J. Opt.Soc. Am. 66, 529 (1976).
[CrossRef]

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

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

Yaglom, A. M.

A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions (Prentice-Hall, Englewood Cliffs, N.J., 1962).

Zernike, F.

F. Zernike, Proc. Phys. Soc. London 61, 158 (1948).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Opt.Soc. Am. (1)

L. Mandel and E. Wolf, J. Opt.Soc. Am. 66, 529 (1976).
[CrossRef]

J. Phys. B (1)

J. H. Eberly, C. V. Kunasz, and K. Wòdkiewicz, J. Phys. B 13, 217 (1980).
[CrossRef]

Opt. Acta (1)

R. Gase and M. Schubert, Opt. Acta 29, 1331 (1982).
[CrossRef]

Opt. Commun. (3)

B. Cairns and E. Wolf, Opt. Commun. 62, 215 (1987).
[CrossRef]

L. Mandel and E. Wolf, Opt. Commun. 36, 247 (1981).
[CrossRef]

D. James and E. Wolf, Opt. Commun. 81, 150 (1991).
[CrossRef]

Phys. Lett. A (1)

D. James and E. Wolf, Phys. Lett. A 157, 6 (1991).
[CrossRef]

Proc. Phys. Soc. London (1)

F. Zernike, Proc. Phys. Soc. London 61, 158 (1948).
[CrossRef]

Rev. Mod. Phys. (1)

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

Other (11)

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

M. Kàlal, in 16th Congress of the International Commission for Optics: Optics as a Key to High Technology, G. Akos, T. Lippenyi, G. Lupkovics, and A. Podmaniczky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 686 (1993).

M. Loéve, Probability Theory (Van Nostrand, New York, 1955).

Yu. V. Prokhorov and Yu. A. Rozanov, Probability Theory (Springer-Verlag, Berlin, 1969).
[CrossRef]

Applications of such lasers were recently reported: Congress of the Society of Photo-Optical and Instrumentation Engineers, Budapest, 1993:S. Yokoyama, T. Araki, and N. Suzuki, “Ultra-wideband optical heterodyne interferometer”;R. Dandliker, K. Hug, and J. Politch, “Optical distance measurement with mm accuracy.” Here the measurements of periodic light intensity were also presented.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Throughout this paper angle brackets denote an ensemble average or a time average for ergodic stochastic processes, even nonstationary ones. In the latter case the time of the average must be small enough to avoid masking the intensity fluctuations.

A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions (Prentice-Hall, Englewood Cliffs, N.J., 1962).

M. S. Bartlett, An Introduction to Stochastic Processes (Cambridge U. Press, Cambridge, 1955).

S. Bochner, Harmonic Analysis and the Theory of Probability (U. California Press, Berkeley, Calif, 1955).

Here we use the formalism of the theory of measure and the Lebesgue–Stieltjes integral, rather than that of delta functions, to describe with the same formulas sources with both discrete and continuous spectra (see Refs. 13, 14, 18, and 21).

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Figures (3)

Fig. 1
Fig. 1

Geometry of Young’s experiment.

Fig. 2
Fig. 2

Intensity of the interference pattern with a Gaussian-shaped pulse source, ω0τ = 4 (normalized to the maximum value for Δs = 0).

Fig. 3
Fig. 3

Spectral composition of the interference pattern with a Gaussian-shaped pulse source, ω0τ = 1. The intensity is normalized to its maximum value for Δs = 0.

Equations (64)

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Q ( r , t ) = U ( r ) [ η 1 exp ( i ω 01 t ) + η 2 exp ( i ω 02 t ) ] .
η 1 η 2 * / ( | η 1 | 2 | η 2 | 2 ) 1 / 2 1 .
I Q ( r , t ) = | Q ( r , t ) | 2 = U 2 ( r ) × [ I 1 + I 2 + 2 | S 12 | cos ( ω 01 ω 02 ) t ] ,
S 12 = η 1 η 2 *
I j = | η j | 2 , j = 1 , 2
Q ( r , t ) = U ( r ) η ( t ) exp ( i ω 0 t ) ,
I Q ( r , t ) = U 2 ( r ) | η ( t ) | 2 .
η ( t ) = η exp [ 1 2 ( t τ ) 2 ] ,
Q ( r , t ) = U ( r ) η exp [ i ω 0 t 1 2 ( t τ ) 2 ] ,
I Q ( r , t ) = U 2 ( r ) I exp [ ( t τ ) 2 ] ,
I = | η | 2 .
I Q ( r , t ) = | Q ( r , t ) | 2 < ,
Q ( r , t ) = exp ( i ω t ) d U Q ( r , ω ) ,
Q ( r , t ) = lim n , Ω k = n n 1 exp ( i ω k t ) Δ U Q ( r , ω k ) ,
U Q ( ω ) = U ( r ) [ η 1 u ( ω ω 01 ) + η 2 u ( ω ω 02 ) ] .
d U Q ( ω ) / d ω = U ( r ) z η ( ω ω 0 )
z η ( ω ) = η τ 2 π exp [ 1 2 ( ω τ ) 2 ] .
s ( ω 1 , ω 2 ) = z η ( ω 1 ω 0 ) z η * ( ω 2 ω 0 ) = I τ 2 2 π exp { 1 2 [ ( ω 1 ω 0 ) 2 + ( ω 2 ω 0 ) 2 ] τ 2 } ,
I ( ω ) = s ( ω , ω ) = I τ 2 2 π exp [ ( ω ω 0 ) 2 τ 2 ] .
S Q ( r ; ω 1 , ω 2 ) = U Q ( r , ω 1 ) U Q * ( r , ω 2 ) ,
Γ Q ( r ; t 1 , t 2 ) = Q ( r , t 1 ) Q * ( r , t 2 ) = exp [ i ( ω 1 t 1 ω 2 t 2 ) d S Q ( r ; ω 1 , ω 2 ) ,
S Q ( r ; ω 1 , ω 2 ) = 2 S Q ( r ; ω 1 , ω 2 ) ω 1 ω 2 ,
Γ Q ( r ; t 1 , t 2 ) = exp [ i ( ω 1 t 1 ω 2 t 2 ) × S Q ( r ; ω 1 , ω 2 ) d ω 1 d ω 2 .
S Q ( r ; ω 1 , ω 2 ) = ( 2 π ) 2 exp [ i ( ω 1 t 1 ω 2 t 2 ) ] × Γ Q ( r ; t 1 , t 2 ) d t 1 d t 2 .
Δ S Q ( r ; ω k , ω n ) = s Q ( r ; ω k , ω n ) Δ ω k Δ ω k = Δ U Q ( r , ω k ) Δ U Q * ( r , ω n ) .
S Q ( r ; ω 1 , ω 2 ) = U 2 ( r ) [ I 1 u ( ω 1 ω 01 ) u ( ω 2 ω 01 ) + S 12 u ( ω 1 ω 01 ) u ( ω 2 ω 02 ) + S 12 * u ( ω 1 ω 02 ) u ( ω 2 ω 01 ) + I 2 u ( ω 1 ω 02 ) u ( ω 2 ω 02 )
s Q ( r ; ω 1 , ω 2 ) = U 2 ( r ) s ( ω 1 , ω 2 ) = U 2 ( r ) I τ 2 2 π × exp { 1 2 [ ( ω 1 ω 0 ) 2 + ( ω 2 ω 0 ) 2 ] } τ 2 .
Δ S Q ( r ; ω k , ω k ) = s Q ( r ; ω k , ω k ) ( Δ ω k ) 2 = | Δ U Q ( r , ω k ) | 2 .
S Q ( r ; ω , ω ) = U 2 ( r ) [ I 1 u 2 ( ω ω 01 ) + I 2 u 2 ( ω ω 02 ) ] ,
S Q ( r ; ω , ω ) = U 2 ( r ) I exp [ ( ω ω 0 ) 2 τ 2 ] .
I Q ( r ; t ) = Γ Q ( r ; t , t ) = exp [ i ( ω 1 ω 2 ) t ] d S Q ( r ; ω 1 , ω 2 )
Δ S Q ( r ; ω k , ω n ) = 0 , ω k ω n ,
Γ Q ( r ; t 1 , t 2 ) = exp [ i ω ( t 1 t 2 ) ] × exp ( i ν t 2 ) d S Q ( r ; ω , ω ν ) .
W Q ( r ; ω ) = ν d S Q ( r ; ω , ω ν ) ,
Γ Q ( r ; τ ) = exp ( i ω τ ) d W Q ( r ; ω ) ,
I Q ( r ) = d W Q ( r ; ω ) = ν d S Q ( r ; ω , ω ν ) ,
R Q ( r ; t , τ ) = Γ Q ( r ; t + τ , t ) ,
R Q ( r ; t , τ ) = exp ( i ω τ ) d G Q ( r ; ω , t ) .
G Q ( r ; ω , τ ) = ν = exp ( i ν τ ) d S Q ( r ; ω , ω ν ) .
Γ Q ( r 1 , r 2 ; t 1 , t 2 ) = Q ( r 1 , t 1 ) Q * ( r 2 , t 2 )
S Q ( r 1 , r 2 ; ω 1 , ω 2 ) = U Q ( r 1 , ω 1 ) U Q * ( r 2 , ω 2 ) ,
Γ Q ( r 1 , r 2 ; t 1 , t 2 ) = exp [ i ( ω 1 t 1 ω 2 t 2 ) ] × d S Q ( r 1 , r 2 ; ω 1 , ω 2 ) ,
S Q ( r 1 , r 2 ; ω 1 , ω 2 ) = 2 S Q ( r 1 , r 2 ; ω 1 , ω 2 ) ω 1 ω 2 ,
Γ Q ( r 1 , r 2 ; t 1 , t 2 ) = exp [ i ( ω 1 t 1 ω 2 t 2 ) ] × s Q ( r 1 , r 2 ; ω 1 , ω 2 ) d ω 1 d ω 2 ,
s Q ( r 1 , r 2 ; ω 1 , ω 2 ) = ( 2 π ) 2 × exp [ i ( ω 1 t 1 ω 2 t 2 ) ] × Γ Q ( r 1 , r 2 ; t 1 , t 2 ) d t 1 d t 2 ,
Δ S Q ( r 1 , r 2 ; ω k , ω n ) = s Q ( r 1 , r 2 ; ω k , ω n ) Δ ω k Δ ω n = Δ U Q ( r 1 , ω k ) Δ U Q * ( r 2 , ω n ) .
W Q ( r 1 , r 2 ; ω ) = ν d S Q ( r 1 , r 2 ; ω , ω ν ) .
R Q ( r 1 , r 2 ; t , τ ) = Γ Q ( r 1 , r 2 ; t + τ , t ) ,
R Q ( r 1 , r 2 ; t , τ ) = exp ( i ω τ ) d G Q ( r 1 , r 2 ; ω , t ) ,
G Q ( r 1 , r 2 ; ω , t ) = exp ( i ν τ ) d S Q ( r 1 , r 2 ; ω , ω ν ) .
γ Q ( r 1 , r 2 ; t 1 , t 2 ) = Γ Q ( r 1 , r 2 ; t 1 , t 2 ) [ I Q ( r 1 ; t 1 ) I Q ( r 2 ; t 2 ) ] 1 / 2
μ Q ( r 1 , r 2 ; ω 1 , ω 2 ) = d S Q ( r 1 , r 2 ; ω 1 , ω 2 ) [ d S Q ( r 1 ; ω 1 , ω 1 ) d S Q ( r 2 ; ω 2 , ω 2 ) ] 1 / 2
Q ( r 0 , t ) = exp ( i ω t ) d U Q ( r 0 , ω ) ,
V 0 ( r , t ) = V 0 ( s , t ) = exp [ i ω ( t s / υ ) ] d U Q ( r 0 , ω ) = exp [ i ( ω t k s ) ] d U Q ( r 0 , ω ) ,
V 0 ( s , t ) = η exp [ i ω 0 ( t s / υ ) 1 2 ( t s / υ τ ) 2 ] .
V ( r , t ) = K 1 V 0 ( s 1 , t ) + K 2 V 0 ( s 2 , t ) ,
I V ( r , t ) = | V ( r , t ) | 2 = | K 1 | 2 I 0 ( s 1 , t ) + | K 2 | 2 I 0 ( s 2 , t ) + 2 | K 1 K 2 * | Re [ J ( s 1 , s 2 ; t ) ] ,
I 0 ( s , t ) = | V 0 ( s , t ) | 2
J ( s 1 , s 2 ; t ) = V 0 ( s 1 , t ) V 0 * ( s 2 , t )
I V ( r , t ) = I { | K 1 | 2 exp [ ( t s 1 / υ τ ) 2 ] + | K 2 | 2 exp [ ( t s 2 / υ τ ) 2 ] + 2 | K 1 K 2 * | cos ( k 0 Δ s ) exp [ 1 2 ( t s 1 / υ τ ) 2 1 2 ( t s 2 / υ τ ) 2 ] } ,
V ( r , t ) = exp ( i ω t ) d U V ( r , ω ) ,
U V ( r , ω ) = K 1 U Q ( r 0 , ω ) exp ( i k s 1 ) + K 2 U Q ( r 0 , ω ) exp ( i k s 2 ) .
S V ( r ; ω 1 , ω 2 ) = U V ( r , ω 1 ) U υ * ( r , ω 2 ) = S Q ( r 0 ; ω 1 , ω 2 ) { | K 1 | 2 exp [ i υ ( ω 1 ω 2 ) s 1 ] + | K 2 | 2 exp [ i υ ( ω 1 ω 2 ) s 2 ] + K 1 K 2 * exp [ i υ ( ω 1 s 1 ω 2 s 2 ) ] + K 1 * K 2 exp [ i υ ( ω 1 s 2 ω 2 s 1 ) ] } ,
I V ( r ; ω ) = s V ( r ; ω , ω ) = I τ 2 2 π exp [ ( ω ω 0 ) 2 τ 2 ] × [ | K 1 | 2 + | K 2 | 2 + 2 | K 1 K 2 * | cos ( k 0 Δ s ) ] .

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