Abstract

We present a number of results relating to the difference of two Gaussian Schell-model cross-spectral densities (CSDs). They allow us to specify conditions under which such a difference represents itself in a valid CSD. In particular, a sufficient condition is derived for the non-negative definiteness of the resulting CSD, for any admissible choice of the involved parameters, while a necessary and sufficient condition is obtained for the case of CSDs endowed with the property of being shape-invariant upon propagation.

© 2014 Optical Society of America

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References

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  1. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
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  12. Z. Mei and O. Korotkova, Opt. Lett. 38, 91 (2013).
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  14. C. R. Fernández-Pousa, Opt. Express 21, 9390 (2013).
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  15. Y. Zhang, L. Liu, C. Zhao, and Y. Cai, Phys. Lett. A 378, 750 (2014).
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  16. Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, Phys. Rev. A 89, 013801 (2014).
    [CrossRef]
  17. M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, Opt. Lett. 39, 1713 (2014).
    [CrossRef]
  18. F. Riesz and B. Szőkefalvi-Nagy, Functional Analysis (Blackie and Sons, 1956).
  19. F. Gori, V. Ramirez-Sanchez, M. Santarsiero, and T. Shirai, J. Opt. A 11, 085706 (2009).
    [CrossRef]
  20. F. Gori, Opt. Commun. 46, 149 (1983).
    [CrossRef]
  21. E. Wolf, J. Opt. Soc. Am. 72, 343 (1982).
    [CrossRef]
  22. F. Gori, Opt. Commun. 34, 301 (1980).
    [CrossRef]
  23. A. Starikov and E. Wolf, J. Opt. Soc. Am. 72, 923 (1982).
    [CrossRef]

2014 (3)

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, Phys. Lett. A 378, 750 (2014).
[CrossRef]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, Phys. Rev. A 89, 013801 (2014).
[CrossRef]

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, Opt. Lett. 39, 1713 (2014).
[CrossRef]

2013 (3)

2012 (2)

2011 (1)

2010 (2)

2009 (2)

V. Torres-Company, A. Valencia, and J. P. Torres, Opt. Lett. 34, 1177 (2009).
[CrossRef]

F. Gori, V. Ramirez-Sanchez, M. Santarsiero, and T. Shirai, J. Opt. A 11, 085706 (2009).
[CrossRef]

2008 (1)

2007 (1)

1983 (1)

F. Gori, Opt. Commun. 46, 149 (1983).
[CrossRef]

1982 (2)

1980 (1)

F. Gori, Opt. Commun. 34, 301 (1980).
[CrossRef]

Cai, Y.

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, Phys. Lett. A 378, 750 (2014).
[CrossRef]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, Phys. Rev. A 89, 013801 (2014).
[CrossRef]

Chen, Y.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, Phys. Rev. A 89, 013801 (2014).
[CrossRef]

Cheng, X.

de Sande, J. C. G.

Fernández-Pousa, C. R.

Friberg, A. T.

J. Turunen and A. T. Friberg, Progress in Optics, E. Wolf, ed. (Elsevier, 2009), Vol. 54, pp. 1–88.

Gori, F.

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, Opt. Lett. 39, 1713 (2014).
[CrossRef]

F. Gori, V. Ramirez-Sanchez, M. Santarsiero, and T. Shirai, J. Opt. A 11, 085706 (2009).
[CrossRef]

F. Gori and M. Santarsiero, Opt. Lett. 32, 3531 (2007).
[CrossRef]

F. Gori, Opt. Commun. 46, 149 (1983).
[CrossRef]

F. Gori, Opt. Commun. 34, 301 (1980).
[CrossRef]

Korotkova, O.

Lajunen, H.

Liu, L.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, Phys. Rev. A 89, 013801 (2014).
[CrossRef]

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, Phys. Lett. A 378, 750 (2014).
[CrossRef]

Mandel, L.

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

Mei, Z.

Piquero, G.

Qiu, Z.

Ramirez-Sanchez, V.

F. Gori, V. Ramirez-Sanchez, M. Santarsiero, and T. Shirai, J. Opt. A 11, 085706 (2009).
[CrossRef]

Riesz, F.

F. Riesz and B. Szőkefalvi-Nagy, Functional Analysis (Blackie and Sons, 1956).

Saastamoinen, T.

Sahin, S.

Santarsiero, M.

Shchepakina, E.

Shirai, T.

F. Gori, V. Ramirez-Sanchez, M. Santarsiero, and T. Shirai, J. Opt. A 11, 085706 (2009).
[CrossRef]

Starikov, A.

Szokefalvi-Nagy, B.

F. Riesz and B. Szőkefalvi-Nagy, Functional Analysis (Blackie and Sons, 1956).

Tervo, J.

Tong, Z.

Torres, J. P.

Torres-Company, V.

Turunen, J.

Vahimaa, P.

Valencia, A.

Wang, F.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, Phys. Rev. A 89, 013801 (2014).
[CrossRef]

Wang, X.

Wolf, E.

A. Starikov and E. Wolf, J. Opt. Soc. Am. 72, 923 (1982).
[CrossRef]

E. Wolf, J. Opt. Soc. Am. 72, 343 (1982).
[CrossRef]

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

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Wyrowski, F.

Zhang, R.

Zhang, Y.

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, Phys. Lett. A 378, 750 (2014).
[CrossRef]

Zhao, C.

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, Phys. Lett. A 378, 750 (2014).
[CrossRef]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, Phys. Rev. A 89, 013801 (2014).
[CrossRef]

J. Opt. A (1)

F. Gori, V. Ramirez-Sanchez, M. Santarsiero, and T. Shirai, J. Opt. A 11, 085706 (2009).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Commun. (2)

F. Gori, Opt. Commun. 46, 149 (1983).
[CrossRef]

F. Gori, Opt. Commun. 34, 301 (1980).
[CrossRef]

Opt. Express (3)

Opt. Lett. (6)

Phys. Lett. A (1)

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, Phys. Lett. A 378, 750 (2014).
[CrossRef]

Phys. Rev. A (1)

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, Phys. Rev. A 89, 013801 (2014).
[CrossRef]

Other (4)

F. Riesz and B. Szőkefalvi-Nagy, Functional Analysis (Blackie and Sons, 1956).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

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

J. Turunen and A. T. Friberg, Progress in Optics, E. Wolf, ed. (Elsevier, 2009), Vol. 54, pp. 1–88.

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Figures (1)

Fig. 1.
Fig. 1.

Alleged degree of coherence μ(r,r) as a function of r, associated with Eq. (12), with B1=2, B2=1.9, P1=0.5, P2=2, M1=7, M2=3.

Equations (40)

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W(r1,r2)=Bexp[r12+r224σ2(r1r2)22δ2],
W(r1,r2)=Bexp[P(r1+r2)2M(r1r2)2]=Bexp[2P(r12+r22)(MP)(r1r2)2],
P=18σ2;M=18σ2+12δ2.
W(r1,r2)=B1exp[P1(r1+r2)2M1(r1r2)2]B2exp[P2(r1+r2)2M2(r1r2)2].
I(r)=W(r,r)=B1exp(4P1r2)B2exp(4P2r2).
B1B2;P1P2.
W(r1,r2)=W˜0(r1,r2),
W(r1,r2)=πB1M1P1exp[π24M1(r1+r2)2π24P1(r1r2)2]πB2M2P2exp[π24M2(r1+r2)2π24P2(r1r2)2],
I(r)=πB1M1P1exp(π2r2M1)πB2M2P2exp(π2r2M2).
B1M1P1B2M2P2;M1M2.
MjPj,(j=1,2).
μ(r,r)=B1e4M1r2B2e4M2r2B1e4P1r2B2e4P2r2.
Q=W(r1,r2)f*(r1)f(r2)d2r1d2r2,
Q=Q1Q2.
B1B2M1PM2P.
W(r1,r2)=exp[2P(r12+r22)]×{B1exp[(M1P)(r1r2)2]B2exp[(M2P)(r1r2)2]},
B1M1Pexp(π2ν2M1P)B2M2Pexp(π2ν2M2P)0,
B1B2MP1MP2.
B1B2M1P1M2P2.
f(r)exp(2Pjr2)=gj(r),(j=1,2).
Qj=Bjexp[(MjPj)(r1r2)2]×gj*(r1)gj(r2)d2r1d2r2,(j=1,2).
exp[(MjPj)(r1r2)2]=πMjPj×exp(π2ν2MjPj)exp[2πiν·(r1r2)]d2ν,
Q1=πB1M1P1exp(π2ν2M1P1)|g˜1(ν)|2d2ν,
Q2=πB2M2P2exp(π2ν2M2P2)|g˜2(ν)|2d2ν.
Q1=πB1M1P2exp(π2ν2M1P2)|g˜1(ν)|2d2ν,
B1M1P2=B1M1P1.
B1M1P2B2M2P2,
B1M1P1B2M2P2.
B1B2B1M1P1M2P2B2,
P1P2M2M1,
P1M1=P2M2,
W(r1,r2)φn(r2)d2r2=ηnφn(r1);(n=0,1,2,),
W(r1,r2)=n=0ηnφn*(r1)φn(r2).
v02=14MP.
ηn=η1nη2n;(n=0,1,),
ηjn=ηj0qjn,ηj0=πBj(Mj+Pj)2;qj=MjPjMj+Pj,(n=0,1,;j=1,2).
B1B2(M1+P1M2+P2)2.
η10η20;q1q2.
B1(M1+P1)2B2(M2+P2)2;M1P1M1+P1M2P2M2+P2,
M1P1M2P2(M1+P1M2+P2)2,

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