Abstract

Random electromagnetic fields have a number of distinctive statistical properties that may depend on their origin. We show here that when two mutually coherent fields are overlapped, the individual characteristics are not completely lost. In particular, we demonstrate that if assumptions can be made regarding the coherence properties of one of the fields, both the relative average strength and the field correlation length of the second one can be retrieved using higher-order polarization properties of the combined field.

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References

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  1. J. W. Goodman, Speckle Phenomena in Optics, 1st ed. (Roberts & Co., 2007).
  2. P. Sebbah, O. Legrand, and A. Z. Genack, “Fluctuations in photon local delay time and their relation to phase spectra in random media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(2), 2406–2411 (1999).
    [CrossRef]
  3. M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987).
    [CrossRef] [PubMed]
  4. P. A. Lee and A. D. Stone, “Universal conductance fluctuations in metals,” Phys. Rev. Lett. 55(15), 1622–1625 (1985).
    [CrossRef] [PubMed]
  5. I. Freund and R. Berkovits, “Surface reflections and optical transport through random media: Coherent backscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B Condens. Matter 41(1), 496–503 (1990).
    [CrossRef] [PubMed]
  6. A. H. Gandjbakhche and G. H. Weiss, “Random walk and diffusion-like model of photon migration in turbid media,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, 1995), pp. 333–402.
  7. I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
    [CrossRef] [PubMed]
  8. I. Freund, M. Kaveh, R. Berkovits, and M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B Condens. Matter 42(4), 2613–2616 (1990).
    [CrossRef] [PubMed]
  9. F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter 40(13), 9342–9345 (1989).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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  14. G. G. Stokes, Trans. Cambridge Philos Soc. 9 (1852) 399, in Polarized Light, W. Swindell, ed., (Dowden, Hutchinson, and Ross, Inc., 1975).
  15. E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. 33(7), 642–644 (2008).
    [CrossRef] [PubMed]
  16. E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1993).
  17. B. Shapiro, “Large intensity fluctuations for wave propagation in random media,” Phys. Rev. Lett. 57(17), 2168–2171 (1986).
    [CrossRef] [PubMed]
  18. M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987).
    [CrossRef] [PubMed]
  19. H. Fuji, T. Asakura, and Y. Shindo, “Measurement of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16(1), 68–72 (1976).
    [CrossRef]
  20. J. Broky and A. Dogariu, “Complex degree of mutual polarization in randomly scattered fields,” Opt. Express 18(19), 20105–20113 (2010).
    [CrossRef] [PubMed]
  21. J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29(6), 536–538 (2004).
    [CrossRef] [PubMed]

2010

2008

2004

2003

A. Dogariu and E. Wolf, “Coherence theory of pairs of correlated wave fields,” J. Mod. Opt. 50(11), 1791–1796 (2003).
[CrossRef]

1999

P. Sebbah, O. Legrand, and A. Z. Genack, “Fluctuations in photon local delay time and their relation to phase spectra in random media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(2), 2406–2411 (1999).
[CrossRef]

1995

1990

I. Freund and R. Berkovits, “Surface reflections and optical transport through random media: Coherent backscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B Condens. Matter 41(1), 496–503 (1990).
[CrossRef] [PubMed]

I. Freund, M. Kaveh, R. Berkovits, and M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B Condens. Matter 42(4), 2613–2616 (1990).
[CrossRef] [PubMed]

1989

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter 40(13), 9342–9345 (1989).
[CrossRef] [PubMed]

1988

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
[CrossRef] [PubMed]

1987

M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987).
[CrossRef] [PubMed]

B. Ruth, “Superposition of Two Dynamic Speckle Patterns–An application to Non-contact Blood Flow Measurements,” J. Mod. Opt. 34(2), 257–273 (1987).
[CrossRef]

M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987).
[CrossRef] [PubMed]

1986

B. Shapiro, “Large intensity fluctuations for wave propagation in random media,” Phys. Rev. Lett. 57(17), 2168–2171 (1986).
[CrossRef] [PubMed]

1985

P. A. Lee and A. D. Stone, “Universal conductance fluctuations in metals,” Phys. Rev. Lett. 55(15), 1622–1625 (1985).
[CrossRef] [PubMed]

1976

H. Fuji, T. Asakura, and Y. Shindo, “Measurement of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16(1), 68–72 (1976).
[CrossRef]

Asakura, T.

H. Fuji, T. Asakura, and Y. Shindo, “Measurement of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16(1), 68–72 (1976).
[CrossRef]

Berkovits, R.

I. Freund, M. Kaveh, R. Berkovits, and M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B Condens. Matter 42(4), 2613–2616 (1990).
[CrossRef] [PubMed]

I. Freund and R. Berkovits, “Surface reflections and optical transport through random media: Coherent backscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B Condens. Matter 41(1), 496–503 (1990).
[CrossRef] [PubMed]

Broky, J.

Brueck, S. R. J.

Burckel, D.

Cwilich, G.

M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987).
[CrossRef] [PubMed]

M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987).
[CrossRef] [PubMed]

Dogariu, A.

Ellis, J.

Feng, S.

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
[CrossRef] [PubMed]

Frauenglass, A.

Freund, I.

I. Freund and R. Berkovits, “Surface reflections and optical transport through random media: Coherent backscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B Condens. Matter 41(1), 496–503 (1990).
[CrossRef] [PubMed]

I. Freund, M. Kaveh, R. Berkovits, and M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B Condens. Matter 42(4), 2613–2616 (1990).
[CrossRef] [PubMed]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
[CrossRef] [PubMed]

Fuji, H.

H. Fuji, T. Asakura, and Y. Shindo, “Measurement of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16(1), 68–72 (1976).
[CrossRef]

Genack, A. Z.

P. Sebbah, O. Legrand, and A. Z. Genack, “Fluctuations in photon local delay time and their relation to phase spectra in random media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(2), 2406–2411 (1999).
[CrossRef]

Kaveh, M.

I. Freund, M. Kaveh, R. Berkovits, and M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B Condens. Matter 42(4), 2613–2616 (1990).
[CrossRef] [PubMed]

Lang, M.

Lee, P. A.

P. A. Lee and A. D. Stone, “Universal conductance fluctuations in metals,” Phys. Rev. Lett. 55(15), 1622–1625 (1985).
[CrossRef] [PubMed]

Legrand, O.

P. Sebbah, O. Legrand, and A. Z. Genack, “Fluctuations in photon local delay time and their relation to phase spectra in random media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(2), 2406–2411 (1999).
[CrossRef]

MacKintosh, F. C.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter 40(13), 9342–9345 (1989).
[CrossRef] [PubMed]

Pine, D. J.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter 40(13), 9342–9345 (1989).
[CrossRef] [PubMed]

Rosenbluh, M.

I. Freund, M. Kaveh, R. Berkovits, and M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B Condens. Matter 42(4), 2613–2616 (1990).
[CrossRef] [PubMed]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
[CrossRef] [PubMed]

Ruth, B.

B. Ruth, “Superposition of Two Dynamic Speckle Patterns–An application to Non-contact Blood Flow Measurements,” J. Mod. Opt. 34(2), 257–273 (1987).
[CrossRef]

Sebbah, P.

P. Sebbah, O. Legrand, and A. Z. Genack, “Fluctuations in photon local delay time and their relation to phase spectra in random media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(2), 2406–2411 (1999).
[CrossRef]

Shapiro, B.

B. Shapiro, “Large intensity fluctuations for wave propagation in random media,” Phys. Rev. Lett. 57(17), 2168–2171 (1986).
[CrossRef] [PubMed]

Shindo, Y.

H. Fuji, T. Asakura, and Y. Shindo, “Measurement of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16(1), 68–72 (1976).
[CrossRef]

Stephen, M. J.

M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987).
[CrossRef] [PubMed]

M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987).
[CrossRef] [PubMed]

Stone, A. D.

P. A. Lee and A. D. Stone, “Universal conductance fluctuations in metals,” Phys. Rev. Lett. 55(15), 1622–1625 (1985).
[CrossRef] [PubMed]

Weitz, D. A.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter 40(13), 9342–9345 (1989).
[CrossRef] [PubMed]

Wolf, E.

E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. 33(7), 642–644 (2008).
[CrossRef] [PubMed]

A. Dogariu and E. Wolf, “Coherence theory of pairs of correlated wave fields,” J. Mod. Opt. 50(11), 1791–1796 (2003).
[CrossRef]

Zaidi, S. H.

Zhu, J. X.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter 40(13), 9342–9345 (1989).
[CrossRef] [PubMed]

J. Mod. Opt.

A. Dogariu and E. Wolf, “Coherence theory of pairs of correlated wave fields,” J. Mod. Opt. 50(11), 1791–1796 (2003).
[CrossRef]

B. Ruth, “Superposition of Two Dynamic Speckle Patterns–An application to Non-contact Blood Flow Measurements,” J. Mod. Opt. 34(2), 257–273 (1987).
[CrossRef]

Opt. Commun.

H. Fuji, T. Asakura, and Y. Shindo, “Measurement of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16(1), 68–72 (1976).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B Condens. Matter

I. Freund and R. Berkovits, “Surface reflections and optical transport through random media: Coherent backscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B Condens. Matter 41(1), 496–503 (1990).
[CrossRef] [PubMed]

I. Freund, M. Kaveh, R. Berkovits, and M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B Condens. Matter 42(4), 2613–2616 (1990).
[CrossRef] [PubMed]

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter 40(13), 9342–9345 (1989).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics

P. Sebbah, O. Legrand, and A. Z. Genack, “Fluctuations in photon local delay time and their relation to phase spectra in random media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(2), 2406–2411 (1999).
[CrossRef]

Phys. Rev. Lett.

M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987).
[CrossRef] [PubMed]

P. A. Lee and A. D. Stone, “Universal conductance fluctuations in metals,” Phys. Rev. Lett. 55(15), 1622–1625 (1985).
[CrossRef] [PubMed]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
[CrossRef] [PubMed]

B. Shapiro, “Large intensity fluctuations for wave propagation in random media,” Phys. Rev. Lett. 57(17), 2168–2171 (1986).
[CrossRef] [PubMed]

M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987).
[CrossRef] [PubMed]

Other

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1993).

G. G. Stokes, Trans. Cambridge Philos Soc. 9 (1852) 399, in Polarized Light, W. Swindell, ed., (Dowden, Hutchinson, and Ross, Inc., 1975).

J. W. Goodman, Speckle Phenomena in Optics, 1st ed. (Roberts & Co., 2007).

T. S. McKechnie, “Speckle reduction,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1984).

A. H. Gandjbakhche and G. H. Weiss, “Random walk and diffusion-like model of photon migration in turbid media,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, 1995), pp. 333–402.

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Figures (3)

Fig. 1
Fig. 1

Intensity speckle images of the superposition between an unpolarized field of coherence length δ U and a polarized field characterized by: a) β = 0.15 , δ P = 4 δ U b) β = 0.15 , δ P = δ U c) β = 0.45 , δ P = δ U and the corresponding CDMP maps for: d) β = 0.15 , δ P = 4 δ U e) β = 0.15 , δ P = δ U and f) β = 0.45 , δ P = δ U . Areas of blue and red correspond to CDMP values of 0 and 1, respectively. The values of β = 0.15 and β = 0.45 correspond to global degrees of polarization P ¯ = 0.11 and P ¯ = 0.31 , respectively.

Fig. 2
Fig. 2

The power spectral density of CDMP maps calculated for β = 0.45 and correlation lengths δ P equal to A) 2 δ U , B) 4 / 3 δ U , and C) δ U . Also shown with solid lines are the best fits with power spectrum dependence given in Eq. (9). The inset shows a log-log plot of the high spatial frequencies region.

Fig. 3
Fig. 3

The power spectral density of CDMP maps calculated for δ P = 4 δ U and field ratios β equal to A) 0.04, B), 0.19,and C).0.45. Also shown with solid lines are the best fits with power spectrum dependence given in Eq. (9). The inset shows a log-log plot of the high spatial frequencies region.

Equations (11)

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E μ U ( r ) = j a j exp ( i ( k j r + ϕ μ , j ) , μ = x , y ,
E x P ( r ) = j b j exp ( i ( k j r + ϕ x , j ) ,
P ¯ ( r ) = S 1 ( r ) 2 + S 2 ( r ) 2 + S 3 ( r ) 2 / I ( r ) ,
S 1 ( r ) = E x R * ( r ) E x R ( r ) E y R * ( r ) E y R ( r ) , S 2 ( r ) = E x R * ( r ) E y R ( r ) E y R * ( r ) E x R ( r ) , S 3 ( r ) = i ( E x R * ( r ) E y R ( r ) E y R * ( r ) E x R ( r ) ) .
E U ( r ) E U ( r + δ ) r = f ( δ U ) ,
E x P ( r ) E x P ( r + δ ) r = f ( δ P ) ,
| V 2 ( r ) | = ( E x R * ( r ) E x P + E y R * ( r ) E y P ) 2 ( | E x R ( r ) | 2 + | E y R ( r ) | 2 ) ( | E x P | 2 + | E y P | 2 ) .
P ( ω ) = F { | V 2 ( r ) | | V 2 ( r ) | } ,
P ( ω ) = I ¯ 1 2 p 1 ( ω ) + I ¯ 2 2 p 2 ( ω ) + I ¯ 1 I ¯ 2 p 12 ( ω ) ,
p j ( ω ) = F { a j * a j a j * a j } , j = 1 ,     2
p 12 ( ω ) = F { 2 a 1 * a 1 a 2 * a 2 + ( a 1 * a 2 + a 2 * a 1 ) ( a 1 * a 2 + a 2 * a 1 ) } ,

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