Abstract

The two-point coherence of an electromagnetic field is represented completely by a 4×4 coherency matrix G that encodes the joint polarization–spatial-field correlations. Here, we describe a systematic sequence of cascaded spatial and polarization projective measurements that are sufficient to tomographically reconstruct G—a task that, to the best of our knowledge, has not yet been realized. Our approach benefits from the correspondence between this reconstruction problem in classical optics and that of quantum state tomography for two-photon states in quantum optics. Identifying G uniquely determines all the measurable correlation characteristics of the field and, thus, lifts ambiguities that arise from reliance on traditional scalar descriptors, especially when the field’s degrees of freedom are correlated or classically entangled.

© 2014 Optical Society of America

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  1. L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
    [CrossRef]
  2. F. Gori, Opt. Lett. 23, 241 (1998).
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  3. E. Wolf, Phys. Lett. A 312, 263 (2003).
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  4. P. Réfrégier and F. Goudail, Opt. Express 13, 6051 (2005).
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  5. K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, Nat. Photonics 7, 72 (2013).
    [CrossRef]
  6. J. Peřina, Coherence of Light (Van Nostrand, 1972).
  7. F. Gori, M. Santarsiero, and R. Borghi, Opt. Lett. 31, 858 (2006).
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  8. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Phys. Rev. A 64, 052312 (2001).
    [CrossRef]
  9. A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, Opt. Commun. 201, 93 (2002).
    [CrossRef]
  10. U. Fano, Rev. Mod. Phys. 55, 855 (1983).
    [CrossRef]
  11. W. H. McMaster, Rev. Mod. Phys. 33, 8 (1961).
    [CrossRef]
  12. W. K. Wootters, Complexity, Entropy, and the Physics of Information, W. H. Zurek, ed. (Addison-Wesley, 1990), pp. 39–46.
  13. S. Bergia, F. Cannata, A. Cornia, and R. Livi, Found. Phys. 10, 723 (1980).
    [CrossRef]
  14. H. Araki, Commun. Math. Phys. 75, 1 (1980).
    [CrossRef]
  15. A. F. Abouraddy, T. M. Yarnall, and B. E. A. Saleh, Opt. Lett. 36, 4683 (2011).
    [CrossRef]
  16. A. F. Abouraddy, T. M. Yarnall, and B. E. A. Saleh, Opt. Lett. 37, 2889 (2012).
    [CrossRef]
  17. R. J. C. Spreeuw, Found. Phys. 28, 361 (1998).
    [CrossRef]

2013 (1)

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, Nat. Photonics 7, 72 (2013).
[CrossRef]

2012 (1)

2011 (1)

2006 (1)

2005 (1)

2003 (1)

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

2002 (1)

A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, Opt. Commun. 201, 93 (2002).
[CrossRef]

2001 (1)

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Phys. Rev. A 64, 052312 (2001).
[CrossRef]

1998 (2)

R. J. C. Spreeuw, Found. Phys. 28, 361 (1998).
[CrossRef]

F. Gori, Opt. Lett. 23, 241 (1998).
[CrossRef]

1983 (1)

U. Fano, Rev. Mod. Phys. 55, 855 (1983).
[CrossRef]

1980 (2)

S. Bergia, F. Cannata, A. Cornia, and R. Livi, Found. Phys. 10, 723 (1980).
[CrossRef]

H. Araki, Commun. Math. Phys. 75, 1 (1980).
[CrossRef]

1965 (1)

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

1961 (1)

W. H. McMaster, Rev. Mod. Phys. 33, 8 (1961).
[CrossRef]

Abouraddy, A. F.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, Nat. Photonics 7, 72 (2013).
[CrossRef]

A. F. Abouraddy, T. M. Yarnall, and B. E. A. Saleh, Opt. Lett. 37, 2889 (2012).
[CrossRef]

A. F. Abouraddy, T. M. Yarnall, and B. E. A. Saleh, Opt. Lett. 36, 4683 (2011).
[CrossRef]

A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, Opt. Commun. 201, 93 (2002).
[CrossRef]

Araki, H.

H. Araki, Commun. Math. Phys. 75, 1 (1980).
[CrossRef]

Bergia, S.

S. Bergia, F. Cannata, A. Cornia, and R. Livi, Found. Phys. 10, 723 (1980).
[CrossRef]

Borghi, R.

Cannata, F.

S. Bergia, F. Cannata, A. Cornia, and R. Livi, Found. Phys. 10, 723 (1980).
[CrossRef]

Cornia, A.

S. Bergia, F. Cannata, A. Cornia, and R. Livi, Found. Phys. 10, 723 (1980).
[CrossRef]

Di Giuseppe, G.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, Nat. Photonics 7, 72 (2013).
[CrossRef]

Fano, U.

U. Fano, Rev. Mod. Phys. 55, 855 (1983).
[CrossRef]

Gori, F.

Goudail, F.

James, D. F. V.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Phys. Rev. A 64, 052312 (2001).
[CrossRef]

Kagalwala, K. H.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, Nat. Photonics 7, 72 (2013).
[CrossRef]

Kwiat, P. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Phys. Rev. A 64, 052312 (2001).
[CrossRef]

Livi, R.

S. Bergia, F. Cannata, A. Cornia, and R. Livi, Found. Phys. 10, 723 (1980).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

McMaster, W. H.

W. H. McMaster, Rev. Mod. Phys. 33, 8 (1961).
[CrossRef]

Munro, W. J.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Phys. Rev. A 64, 052312 (2001).
[CrossRef]

Perina, J.

J. Peřina, Coherence of Light (Van Nostrand, 1972).

Réfrégier, P.

Saleh, B. E. A.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, Nat. Photonics 7, 72 (2013).
[CrossRef]

A. F. Abouraddy, T. M. Yarnall, and B. E. A. Saleh, Opt. Lett. 37, 2889 (2012).
[CrossRef]

A. F. Abouraddy, T. M. Yarnall, and B. E. A. Saleh, Opt. Lett. 36, 4683 (2011).
[CrossRef]

A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, Opt. Commun. 201, 93 (2002).
[CrossRef]

Santarsiero, M.

Sergienko, A. V.

A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, Opt. Commun. 201, 93 (2002).
[CrossRef]

Spreeuw, R. J. C.

R. J. C. Spreeuw, Found. Phys. 28, 361 (1998).
[CrossRef]

Teich, M. C.

A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, Opt. Commun. 201, 93 (2002).
[CrossRef]

White, A. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Phys. Rev. A 64, 052312 (2001).
[CrossRef]

Wolf, E.

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

Wootters, W. K.

W. K. Wootters, Complexity, Entropy, and the Physics of Information, W. H. Zurek, ed. (Addison-Wesley, 1990), pp. 39–46.

Yarnall, T. M.

Commun. Math. Phys. (1)

H. Araki, Commun. Math. Phys. 75, 1 (1980).
[CrossRef]

Found. Phys. (2)

S. Bergia, F. Cannata, A. Cornia, and R. Livi, Found. Phys. 10, 723 (1980).
[CrossRef]

R. J. C. Spreeuw, Found. Phys. 28, 361 (1998).
[CrossRef]

Nat. Photonics (1)

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, Nat. Photonics 7, 72 (2013).
[CrossRef]

Opt. Commun. (1)

A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, Opt. Commun. 201, 93 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Lett. A (1)

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

Phys. Rev. A (1)

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Phys. Rev. A 64, 052312 (2001).
[CrossRef]

Rev. Mod. Phys. (3)

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

U. Fano, Rev. Mod. Phys. 55, 855 (1983).
[CrossRef]

W. H. McMaster, Rev. Mod. Phys. 33, 8 (1961).
[CrossRef]

Other (2)

W. K. Wootters, Complexity, Entropy, and the Physics of Information, W. H. Zurek, ed. (Addison-Wesley, 1990), pp. 39–46.

J. Peřina, Coherence of Light (Van Nostrand, 1972).

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Figures (3)

Fig. 1.
Fig. 1.

(a) Projective measurements to reconstruct Gp. PBS, polarizing beam splitter; HWP, half-wave plate to rotate polarization by 45°; QWP, quarter-wave plate that transforms H to RHC polarization. The empty box corresponds to measuring the total power. (b) projective measurements to reconstruct Gs at r⃗a and r⃗b. BC, symmetric beam combiner; PS, a π/2 phase shifter; D, detector. All components in (b) are polarization insensitive. (c) Sixteen projective measurements to reconstruct G for a vector field at r⃗a and r⃗b constructed by cascading measurements from (a) and (b).

Fig. 2.
Fig. 2.

Setup for measuring the two-photon polarization density matrix ρ^ through projective measurements on photons P1 and P2 emitted from a two-photon source S. Cjk is the probability of coincidence detection after polarization projections j and k (shown here is the particular measurement C12 out of 16 potential measurements). See Fig. 1 for a definition of the components.

Fig. 3.
Fig. 3.

(a)–(f) Pictorial depictions of the real part of the coherency matrices G for the fields described in the text; the imaginary parts of the elements of G in all these cases are zero. The labels correspond to the indices of the elements of G in Eq. (2).

Equations (6)

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Gp=(GHHGHVGVHGVV)=12=03Spσ^=12(S0p+S1pS2piS3pS2p+iS3pS0pS1p),
σ^0=(1001),σ^1=(1001),σ^2=(0110),σ^3=(0ii0).
G=(GHa,HaGHa,HbGHa,VaGHa,VbGHb,HaGHb,HbGHb,VaGHb,VbGVa,HaGVa,HbGVa,VaGVa,VbGVb,HaGVb,HbGVb,VaGVb,Vb);
Gs(r)=(GHa,Ha+GVa,VaGHa,Hb+GVa,VbGHb,Ha+GVb,VaGHb,Hb+GVb,Vb).
Gp(r)=(GHa,Ha+GHb,HbGHa,Va+GHb,VbGVa,Ha+GVb,HbGVa,Va+GVb,Vb),
G=14(S00+S01+S10+S11S02+S12i(S03+S13)S20+S21i(S30+S31)S22S33i(S23+S32)S02+S12+i(S03+S13)S00S01+S10S11S22+S33+i(S23S32)S20S21i(S30S31)S20+S21+i(S30+S31)S22+S33i(S23S32)S00+S01S10S11S02S12i(S03S13)S22S33+i(S23+S32)S20S21+i(S30S31)S02S12+i(S03S13)S00S01S10+S11),

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