Abstract

The van Cittert–Zernike theorem is extended to the vectorial regime based on spatial averaging over the observation plane, and experimental demonstrations are presented. The theorem connects complex vectorial source structure to the degree of coherence and polarization of the spatially fluctuating vectorial field in the far field. Experimentation is carried out by making use of the space averages as a replacement of ensemble averages for the Gaussian stochastic field. For quantitative comparison with the theorem, analytical and experimental results are presented for a rectangular aperture with different vectorial source structures.

© 2013 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principle of Optics, 6th ed. (Cambridge University, 1997).
  2. E. Wolf, Introduction to the Theory of Coherence and Polarization (Cambridge University, 2007).
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    [CrossRef]

2013 (2)

J. Tervo, T. Setälä, J. Turunen, and A. T. Friberg, Opt. Lett. 38, 2301 (2013).
[CrossRef]

R. K. Singh, D. N. Naik, H. Itou, M. M. Brundavanam, Y. Miyamoto, and M. Takeda, Proc. SPIE 8788, 87880O (2013).
[CrossRef]

2012 (2)

2011 (1)

2009 (2)

2003 (1)

2000 (1)

1998 (1)

Born, M.

M. Born and E. Wolf, Principle of Optics, 6th ed. (Cambridge University, 1997).

Brundavanam, M. M.

R. K. Singh, D. N. Naik, H. Itou, M. M. Brundavanam, Y. Miyamoto, and M. Takeda, Proc. SPIE 8788, 87880O (2013).
[CrossRef]

D. N. Naik, R. K. Singh, H. Itou, M. M. Brundavanam, Y. Miyamoto, and M. Takeda, Opt. Lett. 37, 3282 (2012).
[CrossRef]

Friberg, A. T.

Goodman, J. W.

J. W. Goodman, Statistical Optics, 1st ed. (Wiley-Interscience, 2000).

Gori, F.

Itou, H.

Martinez-Niconoff, G.

Martinez-Vara, P.

Miyamoto, Y.

Naik, D. N.

Olvera-Santamaria, M. A.

Ostrovsky, A. S.

Piquero, G.

Rodriguez-Herrera, O. G.

Santarsiero, M.

Setäla, T.

Setälä, T.

Shirai, T.

Singh, R. K.

Takeda, M.

Tervo, J.

Turunen, J.

Tyo, J.

Wolf, E.

M. Born and E. Wolf, Principle of Optics, 6th ed. (Cambridge University, 1997).

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

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Figures (3)

Fig. 1.
Fig. 1.

Experimental setup for the generation and detection of polarization speckle.

Fig. 2.
Fig. 2.

Sagnac geometry for creating a desired polarization source structure on the ground glass.

Fig. 3.
Fig. 3.

Degree of coherence distribution and its profile for a 7.4 mm square beam cross section with α=0.44 and the overlap of the orthogonal polarization components (a) full, (b) half, (c) zero, and with (d) α=0.96 and zero overlap. Solid lines denote theoretical predictions and dotted lines denote experimental results. Red and blue indicate profiles along x and y directions, respectively.

Tables (1)

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Table 1. Beams with a 7.4 mm Square Beam Cross Section

Equations (12)

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W(r1,r2;t1,t2)=[Ex*(r1,t1)Ex(r2,t2)Ex*(r1,t1)Ey(r2,t2)Ey*(r1,t1)Ex(r2,t2)Ey*(r1,t1)Ey(r2,t2)],
Ei(r)=Ei(r^)exp[iφi(r^)]exp(i2πr·r^λf)dr^.
Wij(r1,r2)=Ei*(r1)Ej(r2)=Ei*(r1)Ej(r1+Δr)R=Wij(Δr){E˜i*(r^1)E˜j(r^2)exp[i2πλf[(r1+Δr)·r^2r1·r^1]]dr^1dr^2}dr1Iij(r^2)exp(2πiΔr·r^2λf)dr^2.
exp(2πi(r^2r^1)·r1λf)dr1δ(r^2r^1).
γ(Δr)=(tr[W*(Δr)W(Δr)]trW(0)trW(0))1/2,
P(0)=(14detW(0)[trW(0)]2)1/2.
P(0)=[2γ2(0)1]1/2.
γ2(Δr)=[(α2+β2)|Aexp(2πiΔr·r^λf)dr^|2+αβ|OAexp[iϕxy(r^)]exp(2πiΔr·r^λf)dr^|2+αβ|OAexp[iϕxy(r^)]exp(2πiΔr·r^λf)dr^|2]/[|Adr^|2].
γ2(Δx,Δy)=(α2+β2)sinc2(πaΔxλf)sinc2(πbΔyλf)+αβ(1Δx^a)2(1Δy^b)2×{sinc2[π(Δxξ)(aΔx^)λf]sinc2[π(Δyη)(bΔy^)λf]+sinc2[π(Δx+ξ)(aΔx^)λf]sinc2[π(Δy+η)(bΔy^)λf]},
γ2(Δx,Δy)=sinc2(πaΔxλf)sinc2(πbΔyλf).
γ2(Δx,Δy)=[α2+(1α)2]sinc2(πaΔxλf)sinc2(πbΔyλf),
P(0)=[4α24α+1]1/2.

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