Abstract

The random-walk approach has been extended and applied to study the development of polarization speckle by taking the vector nature into account for stochastic electric fields. Based on the random polarization phasor sum, the first and second moments of the Stokes parameters of the resultant polarization speckle have been examined. Under certain assumptions about the statistics of the component polarization phasors that make up the sum, we present some of the details of the spatial derivation that lead to the expressions for the degree of polarization and the newly proposed Stokes contrast that are suitable for describing the polarization speckle development. This vectorial extension of the random walk will provide an intuitive explanation for the development of the polarization speckle.

© 2019 Optical Society of America

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References

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  2. G. F. Lawer and V. Limic, Random Walk: A Modern Introduction (Cambridge University, 2010).
  3. J. C. Dainty, Laser Speckle and Related Phenomena, 2nd ed. (Springer Verlag, 1984).
  4. J. W. Goodman, Statistical Optics (Wiley, 2000).
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    [Crossref]
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  10. R. Barkat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985).
    [Crossref]
  11. R. Barkat, “Statistics of the Stokes parameters,” J. Opt. Soc. Am. A 4, 1256–1263 (1987).
    [Crossref]
  12. D. Eliyahu, “Vector statistics of correlated Gaussian fields,” Phys. Rev. E 47, 2881–2892 (1993).
    [Crossref]
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  18. E. Ochoa and J. W. Goodman, “Statistical properties of ray directions in a monochromatic speckle pattern,” J. Opt. Soc. Am. 73, 943–949 (1983).
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  19. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1960).
  20. M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001).
    [Crossref]
  21. W. Wang, S. G. Hanson, and M. Takeda, “Statistics of polarization speckle: theory versus experiment,” Proc. SPIE 7388, 738803 (2009).
    [Crossref]
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2015 (1)

2014 (1)

2011 (1)

2010 (2)

2009 (2)

W. Wang, S. G. Hanson, and M. Takeda, “Statistics of polarization speckle: theory versus experiment,” Proc. SPIE 7388, 738803 (2009).
[Crossref]

J. Sorrentini, M. Zerrad, and C. Amra, “Statistical signatures of random media and their correlation to polarization properties,” Opt. Lett. 34, 2429–2431 (2009).
[Crossref]

2001 (1)

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001).
[Crossref]

1994 (1)

D. Eliyahu, “Statistics of Stokes variables for correlated Gaussian fields,” Phys. Rev. E 50, 2381–2384 (1994).
[Crossref]

1993 (1)

D. Eliyahu, “Vector statistics of correlated Gaussian fields,” Phys. Rev. E 47, 2881–2892 (1993).
[Crossref]

1987 (1)

1985 (1)

R. Barkat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985).
[Crossref]

1984 (1)

1983 (2)

1981 (1)

F. Fercher and P. F. Steeger, “First-order statistics of Stokes parameters in speckle fields,” Opt. Acta 28, 443–448 (1981).
[Crossref]

1957 (1)

M. S. Longruet-Higgins, “The statistical analysis of a random moving surface,” Philos. Trans. R. Soc. London A 249, 321–387 (1957).
[Crossref]

Amra, C.

Asakura, T.

Barkat, R.

R. Barkat, “Statistics of the Stokes parameters,” J. Opt. Soc. Am. A 4, 1256–1263 (1987).
[Crossref]

R. Barkat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985).
[Crossref]

Berry, M. V.

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001).
[Crossref]

Born, M.

M. Born and E. Wolf, Principle of Optics (Cambridge University, 1999).

Broky, J.

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena, 2nd ed. (Springer Verlag, 1984).

Dennis, M. R.

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001).
[Crossref]

Dogriu, A.

Dupont, J.

Eliyahu, D.

D. Eliyahu, “Statistics of Stokes variables for correlated Gaussian fields,” Phys. Rev. E 50, 2381–2384 (1994).
[Crossref]

D. Eliyahu, “Vector statistics of correlated Gaussian fields,” Phys. Rev. E 47, 2881–2892 (1993).
[Crossref]

Fercher, A. F.

Fercher, F.

F. Fercher and P. F. Steeger, “First-order statistics of Stokes parameters in speckle fields,” Opt. Acta 28, 443–448 (1981).
[Crossref]

Ghabbach, A.

Goodman, J. W.

E. Ochoa and J. W. Goodman, “Statistical properties of ray directions in a monochromatic speckle pattern,” J. Opt. Soc. Am. 73, 943–949 (1983).
[Crossref]

J. W. Goodman, Speckle Phenomena in Optics: Theory and Application (Roberts and Company, 2006).

J. W. Goodman, Statistical Optics (Wiley, 2000).

J. W. Goodman, Speckle Phenomena in Optics (Robert and Company, 2007).

Hanson, S. G.

Lawer, G. F.

G. F. Lawer and V. Limic, Random Walk: A Modern Introduction (Cambridge University, 2010).

Limic, V.

G. F. Lawer and V. Limic, Random Walk: A Modern Introduction (Cambridge University, 2010).

Longruet-Higgins, M. S.

M. S. Longruet-Higgins, “The statistical analysis of a random moving surface,” Philos. Trans. R. Soc. London A 249, 321–387 (1957).
[Crossref]

Ma, N.

Ochoa, E.

Orlik, X.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1960).

Soriano, G.

Sorrentini, J.

Steeger, P. F.

Takeda, M.

Wang, W.

Weiss, G. H.

G. H. Weiss, Aspects and Applications of the Random Walk(North-Holland, 1994).

Wolf, E.

M. Born and E. Wolf, Principle of Optics (Cambridge University, 1999).

Zerrad, M.

Zhang, S.

Zocha, K.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Acta (2)

R. Barkat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985).
[Crossref]

F. Fercher and P. F. Steeger, “First-order statistics of Stokes parameters in speckle fields,” Opt. Acta 28, 443–448 (1981).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Philos. Trans. R. Soc. London A (1)

M. S. Longruet-Higgins, “The statistical analysis of a random moving surface,” Philos. Trans. R. Soc. London A 249, 321–387 (1957).
[Crossref]

Phys. Rev. E (2)

D. Eliyahu, “Vector statistics of correlated Gaussian fields,” Phys. Rev. E 47, 2881–2892 (1993).
[Crossref]

D. Eliyahu, “Statistics of Stokes variables for correlated Gaussian fields,” Phys. Rev. E 50, 2381–2384 (1994).
[Crossref]

Proc. R. Soc. London A (1)

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001).
[Crossref]

Proc. SPIE (1)

W. Wang, S. G. Hanson, and M. Takeda, “Statistics of polarization speckle: theory versus experiment,” Proc. SPIE 7388, 738803 (2009).
[Crossref]

Other (8)

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1960).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Application (Roberts and Company, 2006).

G. H. Weiss, Aspects and Applications of the Random Walk(North-Holland, 1994).

G. F. Lawer and V. Limic, Random Walk: A Modern Introduction (Cambridge University, 2010).

J. C. Dainty, Laser Speckle and Related Phenomena, 2nd ed. (Springer Verlag, 1984).

J. W. Goodman, Statistical Optics (Wiley, 2000).

J. W. Goodman, Speckle Phenomena in Optics (Robert and Company, 2007).

M. Born and E. Wolf, Principle of Optics (Cambridge University, 1999).

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Figures (5)

Fig. 1.
Fig. 1. Schematic diagram of random polarization phasor sum.
Fig. 2.
Fig. 2. Spatial degree of polarization PS of partially developed polarization speckles versus phase standard deviation of σxϕ and σyϕ for various values of N: (a) N=2, (b) N=10, (c) N=100.
Fig. 3.
Fig. 3. Spatial degree of polarization PS of partially developed polarization speckles versus random-walk number N for various values of σyϕ/σxϕ.
Fig. 4.
Fig. 4. Stokes contrast CS of partially developed polarization speckle versus phase standard deviation σxϕ and σyϕ for various values of N: (a) N=2, (b) N=10, (c) N=100.
Fig. 5.
Fig. 5. Stokes contrast CS of partially developed polarization speckles versus random-walk number N for various values of σyϕ/σxϕ.

Equations (26)

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E=E˜xx^+E˜yy^=axeiϕxx^+ayeiϕyy^=1Nn=1NEn=1Nn=1N(axneiϕxnx^+ayneiϕyny^),
S0=E˜xE˜x*+E˜yE˜y*=1Nn=1Nm=1N[axnaxmei(ϕxnϕxm)+aynaymei(ϕynϕym)],
S1=E˜xE˜x*EyE˜y*=1Nn=1Nm=1N[axnaxmei(ϕxnϕxm)aynaymei(ϕynϕym)],
S2=E˜x*E˜y+E˜y*E˜x=1Nn=1Nm=1N[aynaxmei(ϕynϕxm)+axnaymei(ϕxnϕym)],
S3=i(E˜x*E˜yE˜y*E˜x)=iNn=1Nm=1N[aynaxmei(ϕynϕxm)axnaymei(ϕxnϕym)],
S0=1Nn=1Nm=1N[axnaxmei(ϕxnϕxm)+aynaymei(ϕynϕym)]=ax2+ay2+(N1)ax2Mxϕ(1)Mxϕ(1)+(N1)ay2Myϕ(1)Myϕ(1),
S1=1Nn=1Nm=1N[axnaxmei(ϕxnϕxm)aynaymei(ϕynϕym)]=ax2ay2+(N1)ax2Mxϕ(1)Mxϕ(1)(N1)ay2Myϕ(1)Myϕ(1),
S2=1Nn=1Nm=1N[axnaymei(ϕxnϕym)+aynaxmei(ϕynϕxm)]=Naxay[Mxϕ(1)Myϕ(1)+Mxϕ(1)Myϕ(1)],
S3=iNn=1Nm=1N[aynaxmei(ϕynϕxm)axnaymei(ϕxnϕym)]=iNaxay[Mxϕ(1)Myϕ(1)Mxϕ(1)Myϕ(1)].
S02=1N2n=1Nm=1Np=1Nq=1N[axnaxmei(ϕxnϕxm)+aynaymei(ϕynϕym)]×[axpaxqei(ϕxpϕxq)+aypayqei(ϕypϕyq)],
S12=1N2n=1Nm=1Np=1Nq=1N[axnaxmei(ϕxnϕxm)aynaymei(ϕynϕym)]×[axpaxqei(ϕxpϕxq)aypayqei(ϕypϕyq)],
S22=1N2n=1Nm=1Np=1Nq=1N[aynaxmei(ϕynϕxm)+axnaymei(ϕxnϕym)]×[aypaxqei(ϕypϕxq)+axpayqei(ϕxpϕyq)],
S32=1N2n=1Nm=1Np=1Nq=1N[aynaxmei(ϕynϕxm)axnaymei(ϕxnϕym)]×[aypaxqei(ϕypϕxq)axpayqei(ϕxpϕyq)],
S02=1N{ax4+ay4+2(N1)(ax22+ay22)+2Nax2ay2+(N36N2+11N6)[ax4(Mxϕ(1))4+ay4(Myϕ(1))4]+(Mxϕ(1))2[4(N1)axax3+2(N23N+2)ax2ax2(Mxϕ(2)+2)+2N(N1)ax2ay2]+(Myϕ(1))2[4(N1)ayay3+2(N23N+2)ay2ay2(Myϕ(2)+2)+2N(N1)ay2ax2]+(N1)[ax22(Mxϕ(2))2+ay22(Myϕ(2))2]+2N(N1)2ax2ay2(Mxϕ(1))2(Myϕ(1))2},
S12=1N{ax4+ay4+2(N1)(ax22+ay22)2Nax2ay2+(N36N2+11N6)[ax4(Mxϕ(1))4+ay4(Myϕ(1))4]+(Mxϕ(1))2[4(N1)axax3+2(N23N+2)ax2ax2(Mxϕ(2)+2)2N(N1)ax2ay2]+(Myϕ(1))2[4(N1)ayay3+2(N23N+2)ay2ay2(Myϕ(2)+2)2N(N1)ay2ax2]+(N1)[ax22(Mxϕ(2))2+ay22(Myϕ(2))2]2N(N1)2ax2ay2(Mxϕ(1))2(Myϕ(1))2},
S22=2(N1)ax2(Mxϕ(1))2[2(N1)ay2(Myϕ(1))2+ay2(1+Myϕ(2))]+ax2[2(N1)ay2(1+Mxϕ(2))(Myϕ(1))2+2ay2(1+Mxϕ(2)Myϕ(2))],
S32=2(N1)ax2ay2(Mxϕ(1))2(Myϕ(2)1)ax2[2(N1)ay2(Mxϕ(2)1)(Myϕ(1))2+2ay2(Mxϕ(2)Myϕ(2)1)].
ax4=ay4=ax3=ay3=ax2=ay2=ax=ay=1.
Pϕ(ϕk)=exp{ϕk2/[2(σkϕ)2]}/(2πσkϕ),
Mkϕ(ω)=exp[ω2(σkϕ)2/2].
PS=1[4Det(J)]/[Tr(J)]2,
J=12[S0+S1S2+iS3S2iS3S0S1],
CS=[S12+S22+S32]S02S02,
PS={(N1)2e2(σxϕ)2+(N1)2e2(σyϕ)2+2(N2+2N1)×e[(σxϕ)2+(σyϕ)2]}1/2×{2+(N1)[e(σxϕ)2+e(σyϕ)2]}1.
CS={8(11/N)e2(σxϕ)2[N1+cosh((σxϕ)2)]×sinh2((σxϕ)2/2)+8(11/N)e2(σyϕ)2[N1+cosh((σyϕ)2)]sinh2((σyϕ)2/2)}1/2×[2+(N1)×(e(σxϕ)2+e(σyϕ)2)]1.
C={8(N1)[N1+cosh((σϕ)2)]sinh2((σϕ)2/2)}1/2×{N[N1+e(σϕ)2]2}1/2,