Abstract

We present a scheme for recovering the complex input field launched into a waveguide array, from partial measurements of its output intensity, given advance knowledge that the input is sparse. In spite of the fact that in general the inversion problem is ill-conditioned, we demonstrate experimentally and in simulations that the prior knowledge of sparsity helps overcome the loss of information. Our method is based on GESPAR, a recently proposed efficient phase retrieval algorithm. Possible applications include optical interconnects and quantum state tomography, and the ideas are extendable to other multiple input and multiple output (MIMO) communication schemes.

© 2013 OSA

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2013 (2)

W. Dong, L. Zhang, R. Lukac, and G. Shi, “Sparse Representation Based Image Interpolation With Nonlocal Autoregressive Modeling,” IEEE Trans. Image Process.22(4), 1382–1394 (2013).
[CrossRef] [PubMed]

Y. Shechtman, Y. C. Eldar, O. Cohen, and M. Segev, “Efficient coherent diffractive imaging for sparsely varying objects,” Opt. Express21(5), 6327–6338 (2013).
[CrossRef] [PubMed]

2012 (2)

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater.11(5), 455–459 (2012).
[CrossRef] [PubMed]

Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pugatch, and Y. Silberberg, “Quantum walk of two interacting bosons,” Phys. Rev. A86(1), 011603 (2012).
[CrossRef]

2011 (1)

2010 (2)

Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett.35(8), 1148–1150 (2010).
[CrossRef] [PubMed]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science329(5998), 1500–1503 (2010).
[CrossRef] [PubMed]

2009 (3)

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett.102(25), 253904 (2009).
[CrossRef] [PubMed]

D. A. B. Miller, “Device Requirements for Optical Interconnects to Silicon Chips,” Proc. IEEE97(7), 1166–1185 (2009).
[CrossRef]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express17(26), 23920–23946 (2009).
[CrossRef] [PubMed]

2006 (2)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory52(4), 1289–1306 (2006).
[CrossRef]

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52(2), 489–509 (2006).
[CrossRef]

2005 (1)

2004 (1)

M. A. Nielsen, “Optical Quantum Computation Using Cluster States,” Phys. Rev. Lett.93(4), 040503 (2004).
[CrossRef] [PubMed]

2001 (1)

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature409(6816), 46–52 (2001).
[CrossRef] [PubMed]

2000 (1)

D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE88(6), 728–749 (2000).
[CrossRef]

1965 (1)

Blömer, D.

Bromberg, Y.

Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pugatch, and Y. Silberberg, “Quantum walk of two interacting bosons,” Phys. Rev. A86(1), 011603 (2012).
[CrossRef]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science329(5998), 1500–1503 (2010).
[CrossRef] [PubMed]

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett.102(25), 253904 (2009).
[CrossRef] [PubMed]

Bullkich, E.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater.11(5), 455–459 (2012).
[CrossRef] [PubMed]

Burghoff, J.

Candes, E. J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52(2), 489–509 (2006).
[CrossRef]

Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, and M. Segev, “Efficient coherent diffractive imaging for sparsely varying objects,” Opt. Express21(5), 6327–6338 (2013).
[CrossRef] [PubMed]

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater.11(5), 455–459 (2012).
[CrossRef] [PubMed]

Cohen-Hyams, T.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater.11(5), 455–459 (2012).
[CrossRef] [PubMed]

Dana, H.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater.11(5), 455–459 (2012).
[CrossRef] [PubMed]

Dong, W.

W. Dong, L. Zhang, R. Lukac, and G. Shi, “Sparse Representation Based Image Interpolation With Nonlocal Autoregressive Modeling,” IEEE Trans. Image Process.22(4), 1382–1394 (2013).
[CrossRef] [PubMed]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory52(4), 1289–1306 (2006).
[CrossRef]

Eldar, Y. C.

Gazit, S.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater.11(5), 455–459 (2012).
[CrossRef] [PubMed]

Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett.35(8), 1148–1150 (2010).
[CrossRef] [PubMed]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express17(26), 23920–23946 (2009).
[CrossRef] [PubMed]

Huber, S. D.

Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pugatch, and Y. Silberberg, “Quantum walk of two interacting bosons,” Phys. Rev. A86(1), 011603 (2012).
[CrossRef]

Ismail, N.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science329(5998), 1500–1503 (2010).
[CrossRef] [PubMed]

Jones, A. L.

Kley, E. B.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater.11(5), 455–459 (2012).
[CrossRef] [PubMed]

Knill, E.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature409(6816), 46–52 (2001).
[CrossRef] [PubMed]

Laflamme, R.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature409(6816), 46–52 (2001).
[CrossRef] [PubMed]

Lahini, Y.

Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pugatch, and Y. Silberberg, “Quantum walk of two interacting bosons,” Phys. Rev. A86(1), 011603 (2012).
[CrossRef]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science329(5998), 1500–1503 (2010).
[CrossRef] [PubMed]

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett.102(25), 253904 (2009).
[CrossRef] [PubMed]

Lederer, F.

Lobino, M.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science329(5998), 1500–1503 (2010).
[CrossRef] [PubMed]

Lukac, R.

W. Dong, L. Zhang, R. Lukac, and G. Shi, “Sparse Representation Based Image Interpolation With Nonlocal Autoregressive Modeling,” IEEE Trans. Image Process.22(4), 1382–1394 (2013).
[CrossRef] [PubMed]

Matsuda, N.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science329(5998), 1500–1503 (2010).
[CrossRef] [PubMed]

Matthews, J. C. F.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science329(5998), 1500–1503 (2010).
[CrossRef] [PubMed]

Milburn, G. J.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature409(6816), 46–52 (2001).
[CrossRef] [PubMed]

Miller, D. A. B.

D. A. B. Miller, “Device Requirements for Optical Interconnects to Silicon Chips,” Proc. IEEE97(7), 1166–1185 (2009).
[CrossRef]

D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE88(6), 728–749 (2000).
[CrossRef]

Morandotti, R.

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett.102(25), 253904 (2009).
[CrossRef] [PubMed]

Nielsen, M. A.

M. A. Nielsen, “Optical Quantum Computation Using Cluster States,” Phys. Rev. Lett.93(4), 040503 (2004).
[CrossRef] [PubMed]

Nolte, S.

OBrien, J. L.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science329(5998), 1500–1503 (2010).
[CrossRef] [PubMed]

Osherovich, E.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater.11(5), 455–459 (2012).
[CrossRef] [PubMed]

Pertsch, T.

Peruzzo, A.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science329(5998), 1500–1503 (2010).
[CrossRef] [PubMed]

Politi, A.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science329(5998), 1500–1503 (2010).
[CrossRef] [PubMed]

Poulios, K.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science329(5998), 1500–1503 (2010).
[CrossRef] [PubMed]

Pugatch, R.

Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pugatch, and Y. Silberberg, “Quantum walk of two interacting bosons,” Phys. Rev. A86(1), 011603 (2012).
[CrossRef]

Romberg, J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52(2), 489–509 (2006).
[CrossRef]

Schreiber, T.

Segev, M.

Shechtman, Y.

Shi, G.

W. Dong, L. Zhang, R. Lukac, and G. Shi, “Sparse Representation Based Image Interpolation With Nonlocal Autoregressive Modeling,” IEEE Trans. Image Process.22(4), 1382–1394 (2013).
[CrossRef] [PubMed]

Shoham, S.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater.11(5), 455–459 (2012).
[CrossRef] [PubMed]

Sidorenko, P.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater.11(5), 455–459 (2012).
[CrossRef] [PubMed]

Silberberg, Y.

Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pugatch, and Y. Silberberg, “Quantum walk of two interacting bosons,” Phys. Rev. A86(1), 011603 (2012).
[CrossRef]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science329(5998), 1500–1503 (2010).
[CrossRef] [PubMed]

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett.102(25), 253904 (2009).
[CrossRef] [PubMed]

Steiner, S.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater.11(5), 455–459 (2012).
[CrossRef] [PubMed]

Szameit, A.

Tao, T.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52(2), 489–509 (2006).
[CrossRef]

Thompson, M. G.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science329(5998), 1500–1503 (2010).
[CrossRef] [PubMed]

Tünnermann, A.

Verbin, M.

Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pugatch, and Y. Silberberg, “Quantum walk of two interacting bosons,” Phys. Rev. A86(1), 011603 (2012).
[CrossRef]

Wörhoff, K.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science329(5998), 1500–1503 (2010).
[CrossRef] [PubMed]

Yavneh, I.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater.11(5), 455–459 (2012).
[CrossRef] [PubMed]

Zhang, L.

W. Dong, L. Zhang, R. Lukac, and G. Shi, “Sparse Representation Based Image Interpolation With Nonlocal Autoregressive Modeling,” IEEE Trans. Image Process.22(4), 1382–1394 (2013).
[CrossRef] [PubMed]

Zhou, X.-Q.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science329(5998), 1500–1503 (2010).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

(a) Impulse response for light propagating in a periodic waveguide array (intensity). The light is launched into the central waveguide (#0), and evolves such that the envelope corresponds to the absolute value squared of the expression in Eq. (1), from left to right. Each row represents an individual waveguide. (b) A different input signal would yield a correspondingly different propagation pattern.

Fig. 2
Fig. 2

Experimental scheme. The EM field emerging from the 1D SLM (working in amplitude mode between two crossed polarizers) is imaged onto the input facet of the waveguide array. The light undergoes discrete diffraction in the array. The output intensity is imaged onto a CCD camera.

Fig. 3
Fig. 3

Recovery from the complete set of measurements: (a)measured output intensity, (b) discretized measurements vector, and recovery results of (c) amplitude and (d) phase.

Fig. 4
Fig. 4

Recovery from partial measurements. (a) Point spread function of the system. (b) Input and recovered amplitudes. (c) Measured output intensities. (d) Input and recovered phases. (e) The 11 measurements used and the consistency of the recovered signal at the system's output. (f) Super-resolution: accurate recovery of the non-measured output intensities (blue) using half of the output intensities (dashed blue).

Fig. 5
Fig. 5

(a) Reconstruction error vs. sparsity level (k) for various SNR values. (b) Support location vs. sparsity levels for different values of m - number of evenly spaced intensity measurements

Equations (6)

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i z a k +c( a k1 + a k+1 )=0
h[ k ]= i k J k ( 2c z 0 )
y i = | h i x | 2 = x * H ˜ i x,iM
x ^ = argmin x i=1 m ( y i x * H ˜ i x ) 2 subjectto x 0 s
n c = q a ' k q p ' k ( m k )= q a ' k q p ' k m! k!( mk )!
α= n c n uc = q a ' k q p ' k m! q m k!( mk )! >1

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