Abstract

Phase space tomography estimates correlation functions entirely from snapshots in the evolution of the wave function along a time or space variable. In contrast, traditional interferometric methods require measurement of multiple two–point correlations. However, as in every tomographic formulation, undersampling poses a severe limitation. Here we present the first, to our knowledge, experimental demonstration of compressive reconstruction of the classical optical correlation function, i.e. the mutual intensity function. Our compressive algorithm makes explicit use of the physically justifiable assumption of a low–entropy source (or state.) Since the source was directly accessible in our classical experiment, we were able to compare the compressive estimate of the mutual intensity to an independent ground–truth estimate from the van Cittert–Zernike theorem and verify substantial quantitative improvements in the reconstruction.

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References

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2011

2010

E. J. Candès and T. Tao, “The power of convex relaxation: near-optimal matrix completion,” IEEE Trans. Inform. Theory56, 2053–2080 (2010).
[CrossRef]

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett.105, 150401 (2010).
[CrossRef]

2009

E. J. Candès and B. Recht, “Exact matrix completion via convex optimization,” Found. Comput. Math.9, 717–772 (2009).
[CrossRef]

2006

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory52, 489–509 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure Appl. Math.59, 1207–1223 (2006).
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory52, 1289–1306 (2006).
[CrossRef]

2005

1999

1997

C. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature (London)386, 150–153 (1997).
[CrossRef]

J. Tu, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E55, 1946–1949 (1997).
[CrossRef]

1995

U. Leonhardt, “Quantum–state tomography and discrete Wigner function,” Phys. Rev. Lett.74, 4101–4105 (1995).
[CrossRef] [PubMed]

1994

M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
[CrossRef] [PubMed]

1993

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett.70, 1244–1247 (1993).
[CrossRef] [PubMed]

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett.18, 2041–2043 (1993).
[CrossRef] [PubMed]

1992

K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett.68, 2261–2264 (1992).
[CrossRef] [PubMed]

1989

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A40, 2847–2849 (1989).
[CrossRef] [PubMed]

1986

1984

K.-H. Brenner and J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta.31, 213–223 (1984).
[CrossRef]

M. J. Bastiaans, “New class of uncertainty relations for partially coherent light,” J. Opt. Soc. Am. A1, 711–715 (1984).
[CrossRef]

1983

K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun.44, 323–326 (1983).
[CrossRef]

1982

1978

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun.25, 26–30 (1978).
[CrossRef]

Bastiaans, M. J.

Beck, M.

M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett.70, 1244–1247 (1993).
[CrossRef] [PubMed]

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett.18, 2041–2043 (1993).
[CrossRef] [PubMed]

Becker, S.

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett.105, 150401 (2010).
[CrossRef]

Blum, K.

K. Blum, Density Matrix Theory and Applications (Plenum Press, 1981).

Brady, D. J.

Brenner, K.-H.

K.-H. Brenner and J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta.31, 213–223 (1984).
[CrossRef]

K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun.44, 323–326 (1983).
[CrossRef]

Cai, J.-F.

J.-F. Cai, E. J. Candès, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” ArXiv: 0810.3286 (2008).

Candès, E.

E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure Appl. Math.59, 1207–1223 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory52, 489–509 (2006).
[CrossRef]

Candès, E. J.

E. J. Candès and T. Tao, “The power of convex relaxation: near-optimal matrix completion,” IEEE Trans. Inform. Theory56, 2053–2080 (2010).
[CrossRef]

E. J. Candès and B. Recht, “Exact matrix completion via convex optimization,” Found. Comput. Math.9, 717–772 (2009).
[CrossRef]

J.-F. Cai, E. J. Candès, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” ArXiv: 0810.3286 (2008).

E. J. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” ArXiv: 1109.0573 (2011).

E. J. Candès and Y. Plan, “Matrix completion with noise,” ArXiv: 0903.3131 (2009).

E. J. Candès, T. Strohmer, and V. Voroninski, “Phaselift: exact and stable signal recovery from magnitude measurements via convex programming,” ArXiv: 1109.4499v1 (2011).

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory52, 1289–1306 (2006).
[CrossRef]

Eisert, J.

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett.105, 150401 (2010).
[CrossRef]

Eldar, Y.

E. J. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” ArXiv: 1109.0573 (2011).

Eldar, Y. C.

Faridani, A.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett.70, 1244–1247 (1993).
[CrossRef] [PubMed]

Flammia, S. T.

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett.105, 150401 (2010).
[CrossRef]

Goodman, J. W.

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

Gross, D.

D. Gross, “Recovering low-rank matrices from few coefficients in any basis,” IEEE Trans. Inf. Theory57, 1548–1566 (2011).
[CrossRef]

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett.105, 150401 (2010).
[CrossRef]

Itoh, K.

Kak, A. C.

A. C. Kak and M. Slaney, Principle of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

Kurtsiefer, C.

C. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature (London)386, 150–153 (1997).
[CrossRef]

Leonhardt, U.

U. Leonhardt, “Quantum–state tomography and discrete Wigner function,” Phys. Rev. Lett.74, 4101–4105 (1995).
[CrossRef] [PubMed]

Liu, Y.-K.

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett.105, 150401 (2010).
[CrossRef]

Lohmann, A.

K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun.44, 323–326 (1983).
[CrossRef]

Mandel, L.

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

Marks, D. L.

McAlister, D.

M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
[CrossRef] [PubMed]

McNulty, I.

Mlynek, J.

C. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature (London)386, 150–153 (1997).
[CrossRef]

Moser, H. O.

Nikulin, A. Y.

Nugent, K. A.

Ohtsuka, Y.

Ojeda-Castañeda, J.

K.-H. Brenner and J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta.31, 213–223 (1984).
[CrossRef]

K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun.44, 323–326 (1983).
[CrossRef]

Paterson, D.

Peele, A. G.

Pelliccia, D.

Pfau, T.

C. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature (London)386, 150–153 (1997).
[CrossRef]

Plan, Y.

E. J. Candès and Y. Plan, “Matrix completion with noise,” ArXiv: 0903.3131 (2009).

Raymer, M. G.

M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett.70, 1244–1247 (1993).
[CrossRef] [PubMed]

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett.18, 2041–2043 (1993).
[CrossRef] [PubMed]

Recht, B.

E. J. Candès and B. Recht, “Exact matrix completion via convex optimization,” Found. Comput. Math.9, 717–772 (2009).
[CrossRef]

Risken, H.

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A40, 2847–2849 (1989).
[CrossRef] [PubMed]

Roberts, A.

Romberg, J.

E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure Appl. Math.59, 1207–1223 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory52, 489–509 (2006).
[CrossRef]

Segev, M.

Shechtman, Y.

Shen, Z.

J.-F. Cai, E. J. Candès, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” ArXiv: 0810.3286 (2008).

Slaney, M.

A. C. Kak and M. Slaney, Principle of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

Smithey, D. T.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett.70, 1244–1247 (1993).
[CrossRef] [PubMed]

Stack, R. A.

Starikov, A.

Strohmer, T.

E. J. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” ArXiv: 1109.0573 (2011).

E. J. Candès, T. Strohmer, and V. Voroninski, “Phaselift: exact and stable signal recovery from magnitude measurements via convex programming,” ArXiv: 1109.4499v1 (2011).

Szameit, A.

Tao, T.

E. J. Candès and T. Tao, “The power of convex relaxation: near-optimal matrix completion,” IEEE Trans. Inform. Theory56, 2053–2080 (2010).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory52, 489–509 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure Appl. Math.59, 1207–1223 (2006).
[CrossRef]

Tran, C. Q.

Tu, J.

J. Tu, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E55, 1946–1949 (1997).
[CrossRef]

Vogel, K.

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A40, 2847–2849 (1989).
[CrossRef] [PubMed]

Voroninski, V.

E. J. Candès, T. Strohmer, and V. Voroninski, “Phaselift: exact and stable signal recovery from magnitude measurements via convex programming,” ArXiv: 1109.4499v1 (2011).

E. J. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” ArXiv: 1109.0573 (2011).

Walmsley, I. A.

Wolf, E.

Wong, V.

Appl. Opt.

Comm. Pure Appl. Math.

E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure Appl. Math.59, 1207–1223 (2006).
[CrossRef]

Found. Comput. Math.

E. J. Candès and B. Recht, “Exact matrix completion via convex optimization,” Found. Comput. Math.9, 717–772 (2009).
[CrossRef]

IEEE Trans. Inf. Theory

D. Gross, “Recovering low-rank matrices from few coefficients in any basis,” IEEE Trans. Inf. Theory57, 1548–1566 (2011).
[CrossRef]

IEEE Trans. Inform. Theory

E. J. Candès and T. Tao, “The power of convex relaxation: near-optimal matrix completion,” IEEE Trans. Inform. Theory56, 2053–2080 (2010).
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory52, 1289–1306 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory52, 489–509 (2006).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nature (London)

C. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature (London)386, 150–153 (1997).
[CrossRef]

Opt. Acta.

K.-H. Brenner and J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta.31, 213–223 (1984).
[CrossRef]

Opt. Commun.

K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun.44, 323–326 (1983).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun.25, 26–30 (1978).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A40, 2847–2849 (1989).
[CrossRef] [PubMed]

Phys. Rev. E

J. Tu, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E55, 1946–1949 (1997).
[CrossRef]

Phys. Rev. Lett.

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett.105, 150401 (2010).
[CrossRef]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett.70, 1244–1247 (1993).
[CrossRef] [PubMed]

U. Leonhardt, “Quantum–state tomography and discrete Wigner function,” Phys. Rev. Lett.74, 4101–4105 (1995).
[CrossRef] [PubMed]

K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett.68, 2261–2264 (1992).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
[CrossRef] [PubMed]

Other

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

K. Blum, Density Matrix Theory and Applications (Plenum Press, 1981).

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

E. J. Candès and Y. Plan, “Matrix completion with noise,” ArXiv: 0903.3131 (2009).

J.-F. Cai, E. J. Candès, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” ArXiv: 0810.3286 (2008).

A. C. Kak and M. Slaney, Principle of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

E. J. Candès, T. Strohmer, and V. Voroninski, “Phaselift: exact and stable signal recovery from magnitude measurements via convex programming,” ArXiv: 1109.4499v1 (2011).

E. J. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” ArXiv: 1109.0573 (2011).

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Figures (11)

Fig. 1
Fig. 1

(a) Input mutual intensity of a GSMS with paramters σI = 17 and σc = 13, (b) data point locations in the Ambiguity space, mutual intensities estimated by (c) FBP and (d) LRMR methods.

Fig. 2
Fig. 2

The first nine coherent modes of the mutual intensity in Fig. 1(a). (a) Theoretical modes, and (b) LRMR estimates.

Fig. 3
Fig. 3

Eigenvalues of the mutual intensity in Fig. 1(a). (a) Theoretical values, (b) FBP estimates, (c) LRMR estimates, and (d) absolute errors in the LRMR estimates versus mode index.

Fig. 4
Fig. 4

Oversampling rate versus relative MSE of LRMR estimates. The input field is a GSMS with parameters σI = 36 and σc = 18. The noisy data is generated with different SNR from (a) an additive random Gaussian noise model, and (b) a Poisson noise model.

Fig. 5
Fig. 5

Experimental arrangement

Fig. 6
Fig. 6

Intensity measurements at several unequally spaced propagation distances.

Fig. 7
Fig. 7

(a) Real and (b) imaginary parts of the radial slices in Ambiguity space from Fourier transforming the vectors of intensities measured at corresponding propagation distances.

Fig. 8
Fig. 8

Real part of the reconstructed mutual intensity from (a) FBP; (b) LRMR method.

Fig. 9
Fig. 9

Eigenvalues estimated by (a) FBP, and (b) LRMR method.

Fig. 10
Fig. 10

(a) Intensity measured immediately to the right of the illumination slit; (b) real part of van Cittert–Zernike theorem estimated mutual intensity immediately to the right of the object slit; (c) eigenvalues of the mutual intensity in (b); (d) absolute error between the eigenvalues in Fig. 9(b) and 10(c) versus mode index.

Fig. 11
Fig. 11

(a) LRMR estimated coherent modes of the mutual intensities in Fig. 8(b), and (b) coherent modes of the mutual intensities in Fig. 10(b), calculated via use of the van Cittert–Zernike theorem, and assumption of incoherent illumination.

Equations (11)

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J ( x 1 , x 2 ) = g * ( x 1 ) g ( x 2 ) ,
I ( x o ; z ) = d x 1 d x 2 J ( x 1 , x 2 ) exp ( i π λ z ( x 1 2 x 2 2 ) ) exp ( i 2 π x 1 x 2 λ z x o ) .
I = tr ( P x o J ) ,
𝒜 ( u , x ) = J ( x + x 2 , x x 2 ) exp ( i 2 π u x ) d x .
I ˜ ( u ; z ) = 𝒜 ( u , λ z u ) ,
𝒜 = 𝒡 T J + e .
minimize rank ( J ^ ) subject to 𝒜 = 𝒡 T J ^ , λ i 0 , and i λ i = 1.
minimize J ^ * subject to 𝒜 = 𝒡 T J ^ , λ i 0 , and i λ i = 1 ,
J ( x 1 , x 2 ) = [ I ( x 1 ) ] 1 / 2 [ I ( x 2 ) ] 1 / 2 μ ( x 1 x 2 ) ,
I ( x ) = exp ( x 2 2 σ I 2 ) , μ ( x 1 x 2 ) = exp ( ( x 1 x 2 ) 2 2 σ c 2 ) ,
𝒲 ( x , u ) = J ( x + x 2 , x x 2 ) exp ( i 2 π u x ) d x .

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