Abstract

An approach to the theory of partial coherence for nonstationary optical fields is presented. Starting with a spectral representation, a favorable decomposition of the optical signals is discussed that supports a natural extension of the mathematical formalism. The coherence functions are redefined, but still as temporal correlation functions, allowing the obtaining of a more general form of the interference law for partially coherent optical signals. The general theory is applied in some relevant particular cases of nonstationary interference, namely, with quasi-monochromatic beams of different frequencies and with phase-modulated quasi-monochromatic beams of similar frequency spectra. All the results of the general treatment are reducible to the ones given in the literature for the case of stationary interference.

© 2009 Optical Society of America

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  1. E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884-888 (1954).
    [CrossRef]
  2. E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. I. Fields with a narrow spectral range,” Proc. R. Soc. London 225, 96-111 (1954).
    [CrossRef]
  3. E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. II. Fields with a spectral range of arbitrary width,” Proc. R. Soc. London 230, 246-265 (1955).
    [CrossRef]
  4. A. Blanc-Lapierre and P. Dumontet, “La notion de la coherence en optique,” Rev. Opt., Theor. Instrum. 34, 1-21 (1955).
  5. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).
  6. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).
  7. P. Hariharan, Optical Interferometry, 2nd ed. (Academic, 2003).
  8. J. Zheng, Optical Frequency-Modulated Continuous-Wave (FMCW) Interferometry (Springer-Verlag, 2005).
  9. G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, 2002).
    [CrossRef]
  10. P. Corkum and Z. Chang, “The attosecond revolution,” Opt. Photonics News 19, 24-29 (2008).
    [CrossRef]
  11. F. Fischer, “Interferenz von Licht verschiedener Frequenz,” Zeitschrift für Physik 199, 541-557 (1967).
    [CrossRef]
  12. G. G. Brătescu and T. Tudor, “On the coherence of disturbances of different frequencies,” J. Opt. (Paris) 12, 59-64 (1981).
    [CrossRef]
  13. T. Tudor, “Intensity waves in multifrequency optical fields,” Optik (Stuttgart) 100, 15-20 (1995).
  14. T. Tudor, “Coherent multifrequency optical fields,” J. Phys. Soc. Jpn. 73, 76-85 (2004).
    [CrossRef]
  15. W. Mark, “Spectral analysis of the convolution and filtering of non-stationary stochastic processes,” J. Sound Vib. 11, 19-63 (1970).
    [CrossRef]
  16. J. H. Eberly and K. Wódkiewicz, “The time-dependent physical spectrum of light,” J. Opt. Soc. Am. 67, 1252-1261 (1977).
    [CrossRef]
  17. R. Gase and J. Schubert, “On the determination of spectral properties of non-stationary radiation,” J. Mod. Opt. 29, 1331-1347 (1982).
    [CrossRef]
  18. M. Bertolotti, A. Ferrari, and L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B 12, 341-347 (1995).
    [CrossRef]
  19. M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153-171 (1997).
    [CrossRef]
  20. L. Sereda, M. Bertolotti, and A. Ferrari, “Coherence properties of nonstationary light wave fields,” J. Opt. Soc. Am. A 15, 695-705 (1998).
    [CrossRef]
  21. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11, 1894-1899 (2003).
    [CrossRef] [PubMed]
  22. W. A. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: half a century of research,” Signal Process. 86, 639-697 (2006).
    [CrossRef]
  23. B. J. Davis, “Observable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843-1-14 (2007).
    [CrossRef]
  24. R. W. Schoonover, B. J. Davis, and P. S. Carney, “The generalized Wolf shift for cyclostationary fields,” Opt. Express 17, 4705-4711 (2009).
    [CrossRef] [PubMed]
  25. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14, 5007-5012 (2006).
    [CrossRef] [PubMed]
  26. P. Vahimaa and J. Turunen, “Finite-elementary-source model for partially coherent radiation,” Opt. Express 14, 1376-1381 (2006).
    [CrossRef] [PubMed]
  27. A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary coherence-function representation for partially coherent light pulses,” Opt. Express 15, 5160-5165 (2007).
    [CrossRef] [PubMed]
  28. S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, “Energy spectrum of a nonstationary ensemble of pulses,” Opt. Lett. 29, 394-396 (2004).
    [CrossRef] [PubMed]
  29. T. Tudor, “Polarization waves as observable phenomena,” J. Opt. Soc. Am. A 14, 2013-2020 (1997).
    [CrossRef]
  30. O. V. Angelsky, N. N. Dominikov, P. P. Maksimyak, and T. Tudor, “Experimental revealing of polarization waves,” Appl. Opt. 38, 3112-3117 (1999).
    [CrossRef]
  31. O. V. Angelsky, S. B. Yermolenko, C. Yu. Zenkova, and A. O. Angelskaya, “On polarization manifestations of correlation (intrinsic coherence) of optical fields,” Appl. Opt. 47, 5492-5499 (2008).
    [PubMed]
  32. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  33. T. Setala, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young's interference experiment,” Opt. Lett. 31, 2208-2210 (2006).
    [CrossRef] [PubMed]
  34. T. Setala, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young's interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669-2671 (2006).
    [CrossRef] [PubMed]
  35. R. A. Silverman, “Locally stationary random processes,” IRE Trans. Inf. Theory 3, 182-187 (1957).
    [CrossRef]

2009 (1)

2008 (2)

2007 (2)

2006 (5)

2004 (2)

2003 (2)

1999 (1)

1998 (1)

1997 (2)

M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153-171 (1997).
[CrossRef]

T. Tudor, “Polarization waves as observable phenomena,” J. Opt. Soc. Am. A 14, 2013-2020 (1997).
[CrossRef]

1995 (2)

1982 (1)

R. Gase and J. Schubert, “On the determination of spectral properties of non-stationary radiation,” J. Mod. Opt. 29, 1331-1347 (1982).
[CrossRef]

1981 (1)

G. G. Brătescu and T. Tudor, “On the coherence of disturbances of different frequencies,” J. Opt. (Paris) 12, 59-64 (1981).
[CrossRef]

1977 (1)

1970 (1)

W. Mark, “Spectral analysis of the convolution and filtering of non-stationary stochastic processes,” J. Sound Vib. 11, 19-63 (1970).
[CrossRef]

1967 (1)

F. Fischer, “Interferenz von Licht verschiedener Frequenz,” Zeitschrift für Physik 199, 541-557 (1967).
[CrossRef]

1957 (1)

R. A. Silverman, “Locally stationary random processes,” IRE Trans. Inf. Theory 3, 182-187 (1957).
[CrossRef]

1955 (2)

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. II. Fields with a spectral range of arbitrary width,” Proc. R. Soc. London 230, 246-265 (1955).
[CrossRef]

A. Blanc-Lapierre and P. Dumontet, “La notion de la coherence en optique,” Rev. Opt., Theor. Instrum. 34, 1-21 (1955).

1954 (2)

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884-888 (1954).
[CrossRef]

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. I. Fields with a narrow spectral range,” Proc. R. Soc. London 225, 96-111 (1954).
[CrossRef]

Agrawal, G. P.

Angelskaya, A. O.

Angelsky, O. V.

Bertolotti, M.

Blanc-Lapierre, A.

A. Blanc-Lapierre and P. Dumontet, “La notion de la coherence en optique,” Rev. Opt., Theor. Instrum. 34, 1-21 (1955).

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).

Bratescu, G. G.

G. G. Brătescu and T. Tudor, “On the coherence of disturbances of different frequencies,” J. Opt. (Paris) 12, 59-64 (1981).
[CrossRef]

Carney, P. S.

Chang, Z.

P. Corkum and Z. Chang, “The attosecond revolution,” Opt. Photonics News 19, 24-29 (2008).
[CrossRef]

Corkum, P.

P. Corkum and Z. Chang, “The attosecond revolution,” Opt. Photonics News 19, 24-29 (2008).
[CrossRef]

Davis, B. J.

R. W. Schoonover, B. J. Davis, and P. S. Carney, “The generalized Wolf shift for cyclostationary fields,” Opt. Express 17, 4705-4711 (2009).
[CrossRef] [PubMed]

B. J. Davis, “Observable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843-1-14 (2007).
[CrossRef]

Dominikov, N. N.

Dumontet, P.

A. Blanc-Lapierre and P. Dumontet, “La notion de la coherence en optique,” Rev. Opt., Theor. Instrum. 34, 1-21 (1955).

Eberly, J. H.

Ferrari, A.

Fischer, F.

F. Fischer, “Interferenz von Licht verschiedener Frequenz,” Zeitschrift für Physik 199, 541-557 (1967).
[CrossRef]

Friberg, A. T.

Gardner, W. A.

W. A. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: half a century of research,” Signal Process. 86, 639-697 (2006).
[CrossRef]

Gase, R.

R. Gase and J. Schubert, “On the determination of spectral properties of non-stationary radiation,” J. Mod. Opt. 29, 1331-1347 (1982).
[CrossRef]

Hariharan, P.

P. Hariharan, Optical Interferometry, 2nd ed. (Academic, 2003).

Lajunen, H.

Maksimyak, P. P.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

Mark, W.

W. Mark, “Spectral analysis of the convolution and filtering of non-stationary stochastic processes,” J. Sound Vib. 11, 19-63 (1970).
[CrossRef]

Napolitano, A.

W. A. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: half a century of research,” Signal Process. 86, 639-697 (2006).
[CrossRef]

Paura, L.

W. A. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: half a century of research,” Signal Process. 86, 639-697 (2006).
[CrossRef]

Ponomarenko, S. A.

Schoonover, R. W.

Schubert, J.

R. Gase and J. Schubert, “On the determination of spectral properties of non-stationary radiation,” J. Mod. Opt. 29, 1331-1347 (1982).
[CrossRef]

Sereda, L.

Setala, T.

Silverman, R. A.

R. A. Silverman, “Locally stationary random processes,” IRE Trans. Inf. Theory 3, 182-187 (1957).
[CrossRef]

Tervo, J.

Torres-Company, V.

Tudor, T.

T. Tudor, “Coherent multifrequency optical fields,” J. Phys. Soc. Jpn. 73, 76-85 (2004).
[CrossRef]

O. V. Angelsky, N. N. Dominikov, P. P. Maksimyak, and T. Tudor, “Experimental revealing of polarization waves,” Appl. Opt. 38, 3112-3117 (1999).
[CrossRef]

T. Tudor, “Polarization waves as observable phenomena,” J. Opt. Soc. Am. A 14, 2013-2020 (1997).
[CrossRef]

T. Tudor, “Intensity waves in multifrequency optical fields,” Optik (Stuttgart) 100, 15-20 (1995).

G. G. Brătescu and T. Tudor, “On the coherence of disturbances of different frequencies,” J. Opt. (Paris) 12, 59-64 (1981).
[CrossRef]

Turunen, J.

Vahimaa, P.

Wódkiewicz, K.

Wolf, E.

S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, “Energy spectrum of a nonstationary ensemble of pulses,” Opt. Lett. 29, 394-396 (2004).
[CrossRef] [PubMed]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. II. Fields with a spectral range of arbitrary width,” Proc. R. Soc. London 230, 246-265 (1955).
[CrossRef]

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884-888 (1954).
[CrossRef]

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. I. Fields with a narrow spectral range,” Proc. R. Soc. London 225, 96-111 (1954).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

Wyrowski, F.

Yermolenko, S. B.

Yu. Zenkova, C.

Zheng, J.

J. Zheng, Optical Frequency-Modulated Continuous-Wave (FMCW) Interferometry (Springer-Verlag, 2005).

Appl. Opt. (2)

IRE Trans. Inf. Theory (1)

R. A. Silverman, “Locally stationary random processes,” IRE Trans. Inf. Theory 3, 182-187 (1957).
[CrossRef]

J. Mod. Opt. (1)

R. Gase and J. Schubert, “On the determination of spectral properties of non-stationary radiation,” J. Mod. Opt. 29, 1331-1347 (1982).
[CrossRef]

J. Opt. (Paris) (1)

G. G. Brătescu and T. Tudor, “On the coherence of disturbances of different frequencies,” J. Opt. (Paris) 12, 59-64 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

J. Phys. Soc. Jpn. (1)

T. Tudor, “Coherent multifrequency optical fields,” J. Phys. Soc. Jpn. 73, 76-85 (2004).
[CrossRef]

J. Sound Vib. (1)

W. Mark, “Spectral analysis of the convolution and filtering of non-stationary stochastic processes,” J. Sound Vib. 11, 19-63 (1970).
[CrossRef]

Nuovo Cimento (1)

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884-888 (1954).
[CrossRef]

Opt. Express (5)

Opt. Lett. (3)

Opt. Photonics News (1)

P. Corkum and Z. Chang, “The attosecond revolution,” Opt. Photonics News 19, 24-29 (2008).
[CrossRef]

Optik (Stuttgart) (1)

T. Tudor, “Intensity waves in multifrequency optical fields,” Optik (Stuttgart) 100, 15-20 (1995).

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

Phys. Rev. A (1)

B. J. Davis, “Observable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843-1-14 (2007).
[CrossRef]

Proc. R. Soc. London (2)

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. I. Fields with a narrow spectral range,” Proc. R. Soc. London 225, 96-111 (1954).
[CrossRef]

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. II. Fields with a spectral range of arbitrary width,” Proc. R. Soc. London 230, 246-265 (1955).
[CrossRef]

Pure Appl. Opt. (1)

M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153-171 (1997).
[CrossRef]

Rev. Opt., Theor. Instrum. (1)

A. Blanc-Lapierre and P. Dumontet, “La notion de la coherence en optique,” Rev. Opt., Theor. Instrum. 34, 1-21 (1955).

Signal Process. (1)

W. A. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: half a century of research,” Signal Process. 86, 639-697 (2006).
[CrossRef]

Zeitschrift für Physik (1)

F. Fischer, “Interferenz von Licht verschiedener Frequenz,” Zeitschrift für Physik 199, 541-557 (1967).
[CrossRef]

Other (5)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).

P. Hariharan, Optical Interferometry, 2nd ed. (Academic, 2003).

J. Zheng, Optical Frequency-Modulated Continuous-Wave (FMCW) Interferometry (Springer-Verlag, 2005).

G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, 2002).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Schematic representation of the Young’s interference arrangement.

Equations (50)

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Γ i j ( t 1 , t 2 ) = V i ( t 1 ) V j * ( t 2 ) e = W i j ( ω 1 , ω 2 ) e i ( ω 1 t 1 ω 2 t 2 ) d ω 1 d ω 2 , i , j = 1 , 2 .
W i j ( ω 1 , ω 2 ) = P i j ( ω 1 ω , ω 2 ω ) Q i j ( ω ) d ω ,
Γ i j ( t 1 , t 2 ) = V i ( t 1 ) V j * ( t 2 ) e = F i j ( t 1 , t 2 ) G i j ( t 1 t 2 ) ,
G i j ( t 1 t 2 ) = Q i j ( ω ) e i ω ( t 1 t 2 ) d ω .
{ V i ( t 1 ) = U i ( t 1 ) W i ( t 1 ) V j ( t 2 ) = U j ( t 2 ) W j ( t 2 ) } ,
Γ i j ( t 1 , t 2 ) = U i ( t 1 ) U j * ( t 2 ) W i ( t 1 ) W j * ( t 2 ) e .
W i j ( ω 1 , ω 2 ) = U ̃ i ( ω 1 ω ) U ̃ j * ( ω 2 ω ) Q i j ( ω ) d ω .
i n j n a i a j * W i j ( ω i , ω j ) 0 ,
i n j n a i a j * W i j ( ω i , ω j ) = i n j n a i a j * U ̃ i ( ω i ω ) U ̃ j * ( ω j ω ) Q i j ( ω ) d ω = [ i n j n a i U ̃ i ( ω i ω ) a j * U ̃ j * ( ω j ω ) Q i j ( ω ) ] d ω 0 ,
I ( r , t ) = V ( r , t ) V * ( r , t ) .
V ( r , t ) = K 1 V 1 ( r 1 , t t 1 ) + K 2 V 2 ( r 2 , t t 2 ) ,
I e ( r , t ) = [ K 1 V 1 ( r 1 , t t 1 ) + K 2 V 2 ( r 2 , t t 2 ) ] [ K 1 * V 1 * ( r 1 , t t 1 ) + K 2 * V 2 * ( r 2 , t t 2 ) ] e = | K 1 | 2 V 1 ( r 1 , t t 1 ) V 1 * ( r 1 , t t 1 ) e + | K 2 | 2 V 2 ( r 2 , t t 2 ) V 2 * ( r 2 , t t 2 ) e + 2 Re [ K 1 K 2 * V 1 ( r 1 , t t 1 ) V 2 * ( r 2 , t t 2 ) e ] .
F ( t ) T = lim T 1 2 T T T F ( t ) d t ,
I ( r , t ) = | K 1 | 2 V 1 ( r 1 , t t 1 ) V 1 * ( r 1 , t t 1 ) T + | K 2 | 2 V 2 ( r 2 , t t 2 ) V 2 * ( r 2 , t t 2 ) T + 2 Re [ K 1 K 2 * V 1 ( r 1 , t t 1 ) V 2 * ( r 2 , t t 2 ) T ] .
I ( r ) = I 1 ( r ) + I 2 ( r ) + 2 Re [ K 1 K 2 * Γ 12 ( r 1 , r 2 , τ ) ] = I 1 ( r ) + I 2 ( r ) + 2 I 1 ( r ) I 2 ( r ) Re [ γ 12 ( r 1 , r 2 , τ ) ]
V 1 ( r , t ) = U 1 ( r , t ) W 1 ( r , t ) ,
V 2 ( r , t ) = U 2 ( r , t ) W 2 ( r , t ) .
I e ( r , t ) = | K 1 | 2 U 1 ( r 1 , t t 1 ) W 1 ( r 1 , t t 1 ) U 1 * ( r 1 , t t 1 ) W 1 * ( r 1 , t t 1 ) e + | K 2 | 2 U 2 ( r 2 , t t 2 ) W 2 ( r 2 , t t 2 ) U 2 * ( r 2 , t t 2 ) W 2 * ( r 2 , t t 2 ) e + 2 Re K 1 K 2 * U 1 ( r 1 , t t 1 ) W 1 ( r 1 , t t 1 ) U 2 * ( r 2 , t t 2 ) W 2 * ( r 2 , t t 2 ) e .
I e ( r , t ) = | K 1 | 2 | U 1 ( r 1 , t t 1 ) | 2 W 1 ( r 1 , t t 1 ) W 1 * ( r 1 , t t 1 ) e + | K 2 | 2 | U 2 ( r 2 , t t 2 ) | 2 W 2 ( r 2 , t t 2 ) W 2 * ( r 2 , t t 2 ) e + 2 Re [ K 1 K 2 * U 1 ( r 1 , t t 1 ) U 2 * ( r 2 , t t 2 ) W 1 ( r 1 , t t 1 ) W 2 * ( r 2 , t t 2 ) e ] .
I ( r , t ) = | K 1 | 2 | U 1 ( r 1 , t t 1 ) | 2 W 1 ( r 1 , t t 1 ) W 1 * ( r 1 , t t 1 ) T + | K 2 | 2 | U 2 ( r 2 , t t 2 ) | 2 W 2 ( r 2 , t t 2 ) W 2 * ( r 2 , t t 2 ) T + 2 Re [ K 1 K 2 * U 1 ( r 1 , t t 1 ) U 2 * ( r 2 , t t 2 ) W 1 ( r 1 , t t 1 ) W 2 * ( r 2 , t t 2 ) T ] .
G 12 ( r 1 , r 2 , τ ) = W 1 ( r 1 , t ) W 2 * ( r 2 , t + τ ) T .
I 1 S E ( r 1 ) = G 11 ( r 1 , r 1 , 0 ) = W 1 ( r 1 , t ) W 1 * ( r 1 , t ) T .
I 1 ( r , t ) = | K 1 | 2 | U 1 ( r 1 , t t 1 ) | 2 I 1 S E ( r 1 ) = | K 1 | 2 | U 1 ( r 1 , t t 1 ) | 2 G 11 ( r 1 , r 1 , 0 ) | K 1 | 2 I 1 ( r 1 , t t 1 ) ,
g 12 ( r 1 , r 2 , τ ) = G 12 ( r 1 , r 2 , τ ) G 11 ( r 1 , r 1 , 0 ) G 22 ( r 2 , r 2 , 0 ) .
I ( r , t ) = | K 1 | 2 | U 1 ( r 1 , t t 1 ) | 2 G 11 ( r 1 , r 1 , 0 ) + | K 2 | 2 | U 2 ( r 2 , t t 2 ) | 2 G 22 ( r 2 , r 2 , 0 ) + 2 Re [ K 1 K 2 * U 1 ( r 1 , t t 1 ) U 2 * ( r 2 , t t 2 ) G 12 ( r 1 , r 2 , τ ) ] = I 1 ( r , t ) + I 2 ( r , t ) + 2 Re [ K 1 K 2 * U 1 ( r 1 , t t 1 ) U 2 * ( r 2 , t t 2 ) G 12 ( r 1 , r 2 , τ ) ] ,
I ( r , t ) = I 1 ( r , t ) + I 2 ( r , t ) + 2 I 1 ( r , t ) I 2 ( r , t ) Re [ Φ ( r 1 , r 2 , t t 1 , t t 2 ) g 12 ( r 1 , r 2 , τ ) ] ,
Φ ( r 1 , r 2 , t 1 , t 2 ) = U 1 ( r 1 , t 1 ) U 2 * ( r 2 , t 2 ) | U 1 ( r 1 , t 1 ) | | U 2 ( r 2 , t 2 ) | .
V ( r , t ) = a ( t ) e i [ ω 0 t + ϕ ( t ) ] ,
V 1 ( r 1 , t ) = a 1 ( t ) e i [ ω 1 t + ϕ 1 ( t ) ] ,
V 2 ( r 2 , t ) = a 2 ( t ) e i [ ω 2 t + ϕ 2 ( t ) ] .
U 1 ( r 1 , t ) = e i ω 1 t , W 1 ( r 1 , t ) = a 1 ( t ) e i ϕ 1 ( t ) ,
U 2 ( r 2 , t ) = e i ω 2 t , W 2 ( r 2 , t ) = a 2 ( t ) e i ϕ 2 ( t ) .
G 12 ( r 1 , r 2 , τ ) = W 1 ( r 1 , t ) W 2 * ( r 2 , t + τ ) T = a 1 ( t ) a 2 ( t + τ ) e i [ ϕ 1 ( t ) ϕ 2 ( t + τ ) ] T .
I ( r , t ) = I 1 ( r ) + I 2 ( r ) + 2 Re [ K 1 K 2 * e i ω 1 ( t t 1 ) e i ω 2 ( t t 2 ) G 12 ( r 1 , r 2 , τ ) ] = I 1 ( r ) + I 2 ( r ) + 2 Re K 1 K 2 * e i [ ( ω 1 ω 2 ) t + k 2 s 2 k 1 s 1 ] G 12 ( r 1 , r 2 , τ ) ,
I ( r , t ) = I 1 ( r ) + I 2 ( r ) + 2 I 1 ( r ) I 2 ( r ) × Re [ e i [ ( ω 1 ω 2 ) t + k 2 s 2 k 1 s 1 ] g 12 ( r 1 , r 2 , τ ) ] = I 1 ( r ) + I 2 ( r ) + 2 I 1 ( r ) I 2 ( r ) | g 12 ( r 1 , r 2 , τ ) | × cos [ ( ω 1 ω 2 ) t + k 2 s 2 k 1 s 1 + β 12 ( r 1 , r 2 , τ ) ] ,
I ( r , t ) = 2 I 0 ( r ) { 1 + | g 12 ( r 1 , r 2 , τ ) | cos [ ( ω 1 ω 2 ) t + k 2 s 2 k 1 s 1 + β 12 ( r 1 , r 2 , τ ) ] } .
V = I max I min I max + I min ,
V = | g 12 ( r 1 , r 2 , τ ) | ,
V out = e i A sin Ω t V in = e i A sin Ω t a ( t ) e i [ ω 0 t + ϕ ( t ) ] ,
V 1 ( r 1 , t ) = e i A sin Ω t a 1 ( t ) e i [ ω 0 t + ϕ 1 ( t ) ] ,
V 2 ( r 2 , t ) = e i A sin Ω t a 2 ( t ) e i [ ω 0 t + ϕ 2 ( t ) ] .
U 1 ( r 1 , t ) = e i A sin Ω t e i ω 0 t , W 1 ( r 1 , t ) = a 1 ( t ) e i ϕ 1 ( t ) ,
U 2 ( r 2 , t ) = e i A sin Ω t e i ω 0 t , W 2 ( r 2 , t ) = a 2 ( t ) e i ϕ 2 ( t ) .
I ( r , t ) = I 1 ( r ) + I 2 ( r ) + 2 Re { K 1 K 2 * e i [ ω 0 ( t 2 t 1 ) + A sin Ω ( t t 1 ) A sin Ω ( t t 2 ) ] G 12 ( r 1 , r 2 , τ ) } = I 1 ( r ) + I 2 ( r ) + 2 Re { K 1 K 2 * e i ω 0 ( t 2 t 1 ) + 2 i A sin Ω [ ( t 2 t 1 ) 2 ] cos Ω [ t ( t 1 + t 2 ) 2 ] G 12 ( r 1 , r 2 , τ ) } .
I ( r , t ) = I 1 ( r ) + I 2 ( r ) + 2 I 1 ( r ) I 2 ( r ) | g 12 ( r 1 , r 2 , τ ) | × cos { ω 0 ( t 2 t 1 ) + β 12 ( r 1 , r 2 , τ ) + 2 A sin [ Ω ( t 2 t 1 ) 2 ] cos [ Ω ( t t 1 + t 2 2 ) ] } .
Q 12 ( r 1 , r 2 , ω ) = α e ω 2 2 σ 2 .
G 12 ( r 1 , r 2 , τ ) = α e ω 2 2 σ 2 e i ω τ d ω = 2 π α σ e τ 2 σ 2 2 = G max e τ 2 σ 2 2 .
g 12 ( r 1 , r 2 , τ ) = g max e τ 2 σ 2 2 ,
I ( r , t ) = 2 I 0 ( r ) { 1 + | g 12 ( r 1 , r 2 , τ ) | } × cos { ω 0 ( t 2 t 1 ) + β 12 ( r 1 , r 2 , τ ) + 2 A sin [ Ω ( t 2 t 1 ) 2 ] cos [ Ω ( t t 1 + t 2 2 ) ] } ,
V = | g 12 ( r 1 , r 2 , τ ) | .

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