Abstract

Compressive sensing (CS) combines data acquisition with compression coding to reduce the number of measurements required to reconstruct a sparse signal. In optics, this usually takes the form of projecting the field onto sequences of random spatial patterns that are selected from an appropriate random ensemble. We show here that CS can be exploited in ‘native’ optics hardware without introducing added components. Specifically, we show that random sub-Nyquist sampling of an interferogram suffices to reconstruct the field modal structure despite the structural constraints of the measurement system set by its limited degrees of freedom. The distribution of the reduced (and structurally constrained) sensing matrices corresponding to random measurements is provably incoherent and isotropic, which helps us carry out CS successfully. We implement compressive interferometry using a generalized Mach-Zehnder interferometer in which the traditional temporal delay is replaced with a linear transformation corresponding to a fractional transform. By randomly sampling the order of the fractional transform, we efficiently reconstruct the modal content of the input beam in the Hermite-Gaussian and Laguerre-Gaussian bases.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (3)

2016 (4)

2015 (9)

W. Yu, X. Yao, X. Liu, L. Li, and G. Zhai, “Three-dimensional single-pixel compressive reflectivity imaging based on complementary modulation,” Appl. Opt. 54, 363–367 (2015).
[Crossref]

M. Akhlaghi and A. Dogariu, “Compressive correlation imaging with random illumination,” Opt. Lett. 40, 4464–4467 (2015).
[Crossref] [PubMed]

R. Yao, Q. Pian, and X. Intes, “Wide-field fluorescence molecular tomography with compressive sensing based preconditioning,” Biomed. Opt. Express 6, 4887–4898 (2015).
[Crossref] [PubMed]

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[Crossref] [PubMed]

N. Rawat, B. Kim, I. Muniraj, G. Situ, and B.-G. Lee, “Compressive sensing based robust multispectral double-image encryption,” Appl. Opt. 54, 1782–1793 (2015).
[Crossref]

D. Mardani, A. F. Abouraddy, and G. K. Atia, “Efficient modal analysis using compressive optical interferometry,” Opt. Express 23, 28449–28458 (2015).
[Crossref] [PubMed]

A. Kalev, R. L. Kosut, and I. H. Deutsch, “Quantum tomography protocols with positivity are compressed sensing protocols,” npj Quantum Inf. 1, 15018 (2015).
[Crossref]

V. Durán, F. Soldevila, E. Irles, P. Clemente, E. Tajahuerce, P. Andrés, and J. Lancis, “Compressive imaging in scattering media,” Opt. Express 23, 14424–14433 (2015).
[Crossref] [PubMed]

C. G. Graff and E. Y. Sidky, “Compressive sensing in medical imaging,” Appl. Opt. 54, C23–C44 (2015).
[Crossref] [PubMed]

2014 (7)

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
[Crossref] [PubMed]

B. Deepan, C. Quan, and C. J. Tay, “Compressive sensing for digital holographic interferometry,” Proc. SPIE 9234, 923419 (2014).
[Crossref]

E. Tajahuerce, V. Durán, P. Clemente, E. Irles, F. Soldevila, P. Andrés, and J. Lancis, “Image transmission through dynamic scattering media by single-pixel photodetection,” Opt. Express 22, 16945–16955 (2014).
[Crossref] [PubMed]

Z. Wang and Z. Yu, “Spectral analysis based on compressive sensing in nanophotonic structures,” Opt. Express 22, 25608–25614 (2014).
[Crossref] [PubMed]

G. A. Howland, J. Schneeloch, D. J. Lum, and J. C. Howell, “Simultaneous measurement of complementary observables with compressive sensing,” Phys. Rev. Lett. 112, 253602 (2014).
[Crossref] [PubMed]

M. Mirhosseini, O. S. Magaña-Loaiza, S. M. H. Rafsanjani, and R. W. Boyd, “Compressive Direct Measurement of the Quantum Wave Function,” Phys. Rev. Lett. 113, 090402 (2014).
[Crossref] [PubMed]

D. Xu, Y. Huang, and J. Kang, “Real-time compressive sensing spectral domain optical coherence tomography,” Opt. Lett. 39, 76–79 (2014).
[Crossref]

2013 (7)

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

E. Thiebaut, “Principles of image reconstruction in interferometry,” New Concepts in Imaging: Optical and Statistical Models, EAS Publications Series 59, 157–187 (2013).

B. Lawrie and R. Pooser, “Toward real-time quantum imaging with a single pixel camera,” Opt. Express 21, 7549–7559 (2013).
[Crossref] [PubMed]

Y. Rivenson, A. Stern, and J. Rosen, “Reconstruction guarantees for compressive tomographic holography,” Opt. Lett. 38, 2509–2511 (2013).
[Crossref] [PubMed]

P. Clemente, V. Durán, E. Tajahuerce, P. Andrés, V. Climent, and J. Lancis, “Compressive holography with a single-pixel detector,” Opt. Lett. 38, 2524–2527 (2013).
[Crossref] [PubMed]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photon. 7, 354–362 (2013).
[Crossref]

2012 (4)

2011 (5)

2010 (5)

2009 (1)

2008 (2)

E. J. Candes, “The restricted isometry property and its implications for compressed sensing,” C. R. Acad. Sci. 346, 589–592 (2008).
[Crossref]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

2007 (2)

E. Candes and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Problems 23, 969–985 (2007).
[Crossref]

E. Candes and T. Tao, “The dantzig selector: Statistical estimation when p is much larger than n,” The Annals of Statistics 35, 2313–2351 (2007).
[Crossref]

2006 (1)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

2005 (1)

1980 (2)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” IMA J. Appl. Math. 25, 241–256 (1980).
[Crossref]

V. Namias, “Fractionalization of Hankel transforms,” IMA J. Appl. Math. 26, 187–197 (1980).
[Crossref]

Abolbashari, M.

Abouraddy, A. F.

Ahmed, N.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Akhlaghi, M.

Alieva, T.

Andrés, P.

Araújo, F. M.

Arce, G.

Atia, G. K.

L. Martin, D. Mardani, H. E. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral Hilbert-space analyzers,” Sci. Rep. 7, 44995 (2017).
[Crossref] [PubMed]

D. Mardani, A. F. Abouraddy, and G. K. Atia, “Efficient modal analysis using compressive optical interferometry,” Opt. Express 23, 28449–28458 (2015).
[Crossref] [PubMed]

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

Bie, H.

Bizheva, K.

Boyd, R. W.

M. Mirhosseini, O. S. Magaña-Loaiza, S. M. H. Rafsanjani, and R. W. Boyd, “Compressive Direct Measurement of the Quantum Wave Function,” Phys. Rev. Lett. 113, 090402 (2014).
[Crossref] [PubMed]

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Calvo, M. L.

Candes, E.

E. Candes and T. Tao, “The dantzig selector: Statistical estimation when p is much larger than n,” The Annals of Statistics 35, 2313–2351 (2007).
[Crossref]

E. Candes and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Problems 23, 969–985 (2007).
[Crossref]

Candes, E. J.

E. J. Candes and Y. Plan, “A probabilistic and RIPless theory of compressed sensing,” IEEE Trans. Inf. Theory 57, 7235–7254 (2011).
[Crossref]

E. J. Candes, “The restricted isometry property and its implications for compressed sensing,” C. R. Acad. Sci. 346, 589–592 (2008).
[Crossref]

E. J. Candes, “Compressed sensing,” Proc. Int. Congress of Mathematicians (2006).

Carrasco, Silvia

Carron, I.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
[Crossref] [PubMed]

Chardon, G.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
[Crossref] [PubMed]

Chen, Y.

Clemente, P.

Climent, V.

Correia, M. V.

Daudet, L.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
[Crossref] [PubMed]

Davenport, M. A.

M. A. Davenport and M. B. Wakin, “Analysis of orthogonal matching pursuit using the restricted isometry property,” IEEE Trans. Inf. Theory 56, 4395–4401 (2010).
[Crossref]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

Deepan, B.

B. Deepan, C. Quan, and C. J. Tay, “Compressive sensing for digital holographic interferometry,” Proc. SPIE 9234, 923419 (2014).
[Crossref]

Deutsch, I. H.

A. Kalev, R. L. Kosut, and I. H. Deutsch, “Quantum tomography protocols with positivity are compressed sensing protocols,” npj Quantum Inf. 1, 15018 (2015).
[Crossref]

Dixon, P.

Dogariu, A.

Dolinar, S.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

Durán, V.

Durduran, T.

Edgar, M.

Farahi, F.

Fazal, I. M.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Fieguth, P.

Fini, J. M.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photon. 7, 354–362 (2013).
[Crossref]

Gao, L.

Giannoula, A.

Gibson, G.

Gigan, S.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
[Crossref] [PubMed]

Giuseppe, G. Di

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

Graff, C. G.

Hempler, N.

Howell, J.

Howell, J. C.

G. A. Howland, S. H. Knarr, J. Schneeloch, D. J. Lum, and J. C. Howell, “Compressively characterizing high-dimensional entangled states with complementary, random filtering,” Phys. Rev. X 6, 021018 (2016).

G. A. Howland, J. Schneeloch, D. J. Lum, and J. C. Howell, “Simultaneous measurement of complementary observables with compressive sensing,” Phys. Rev. Lett. 112, 253602 (2014).
[Crossref] [PubMed]

Howland, G.

Howland, G. A.

G. A. Howland, S. H. Knarr, J. Schneeloch, D. J. Lum, and J. C. Howell, “Compressively characterizing high-dimensional entangled states with complementary, random filtering,” Phys. Rev. X 6, 021018 (2016).

G. A. Howland, J. Schneeloch, D. J. Lum, and J. C. Howell, “Simultaneous measurement of complementary observables with compressive sensing,” Phys. Rev. Lett. 112, 253602 (2014).
[Crossref] [PubMed]

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Huang, Y.

Intes, X.

Irles, E.

Jahromi, A. K.

L. Martin, D. Mardani, H. E. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral Hilbert-space analyzers,” Sci. Rep. 7, 44995 (2017).
[Crossref] [PubMed]

Javidi, B.

Kagalwala, K. H.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

Kalev, A.

A. Kalev, R. L. Kosut, and I. H. Deutsch, “Quantum tomography protocols with positivity are compressed sensing protocols,” npj Quantum Inf. 1, 15018 (2015).
[Crossref]

Kang, J.

Karl, W. C.

N. Mohan, I. Stojanovic, W. C. Karl, B. E. A. Saleh, and M. C. Teich, “Compressed sensing in optical coherence tomography,” Proc. SPIE 7570, 75700L (2010).
[Crossref]

Katz, O.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
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Kelly, K. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

Kim, B.

Knarr, S. H.

G. A. Howland, S. H. Knarr, J. Schneeloch, D. J. Lum, and J. C. Howell, “Compressively characterizing high-dimensional entangled states with complementary, random filtering,” Phys. Rev. X 6, 021018 (2016).

Kondakci, H. E.

L. Martin, D. Mardani, H. E. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral Hilbert-space analyzers,” Sci. Rep. 7, 44995 (2017).
[Crossref] [PubMed]

Kosut, R. L.

A. Kalev, R. L. Kosut, and I. H. Deutsch, “Quantum tomography protocols with positivity are compressed sensing protocols,” npj Quantum Inf. 1, 15018 (2015).
[Crossref]

Kristensen, P.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Lancis, J.

Larson, W. D.

L. Martin, D. Mardani, H. E. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral Hilbert-space analyzers,” Sci. Rep. 7, 44995 (2017).
[Crossref] [PubMed]

Laska, J. N.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

Lawrie, B.

Lee, B.-G.

Lerosey, G.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
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Li, J.

J. Li, J. S. Li, Y. Y. Pan, and R. Li, “Compressive optical image encryption,” Sci. Rep. 5, 10374 (2015).
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Li, J. S.

J. Li, J. S. Li, Y. Y. Pan, and R. Li, “Compressive optical image encryption,” Sci. Rep. 5, 10374 (2015).
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Li, L.

Li, R.

J. Li, J. S. Li, Y. Y. Pan, and R. Li, “Compressive optical image encryption,” Sci. Rep. 5, 10374 (2015).
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Liang, J.

Liu, C.

Liu, X.

Liutkus, A.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
[Crossref] [PubMed]

Lum, D. J.

G. A. Howland, S. H. Knarr, J. Schneeloch, D. J. Lum, and J. C. Howell, “Compressively characterizing high-dimensional entangled states with complementary, random filtering,” Phys. Rev. X 6, 021018 (2016).

G. A. Howland, J. Schneeloch, D. J. Lum, and J. C. Howell, “Simultaneous measurement of complementary observables with compressive sensing,” Phys. Rev. Lett. 112, 253602 (2014).
[Crossref] [PubMed]

Ma, C.

Magalhães, F.

Magaña-Loaiza, O. S.

M. Mirhosseini, O. S. Magaña-Loaiza, S. M. H. Rafsanjani, and R. W. Boyd, “Compressive Direct Measurement of the Quantum Wave Function,” Phys. Rev. Lett. 113, 090402 (2014).
[Crossref] [PubMed]

Maker, G.

Malcolm, G.

Malhotra, T.

L. Martin, D. Mardani, H. E. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral Hilbert-space analyzers,” Sci. Rep. 7, 44995 (2017).
[Crossref] [PubMed]

Mardani, D.

L. Martin, D. Mardani, H. E. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral Hilbert-space analyzers,” Sci. Rep. 7, 44995 (2017).
[Crossref] [PubMed]

D. Mardani, A. F. Abouraddy, and G. K. Atia, “Efficient modal analysis using compressive optical interferometry,” Opt. Express 23, 28449–28458 (2015).
[Crossref] [PubMed]

Martin, L.

L. Martin, D. Mardani, H. E. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral Hilbert-space analyzers,” Sci. Rep. 7, 44995 (2017).
[Crossref] [PubMed]

Martina, D.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
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Martínez-León, L.

Mirhosseini, M.

M. Mirhosseini, O. S. Magaña-Loaiza, S. M. H. Rafsanjani, and R. W. Boyd, “Compressive Direct Measurement of the Quantum Wave Function,” Phys. Rev. Lett. 113, 090402 (2014).
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Mirza, I.

Mohan, N.

N. Mohan, I. Stojanovic, W. C. Karl, B. E. A. Saleh, and M. C. Teich, “Compressed sensing in optical coherence tomography,” Proc. SPIE 7570, 75700L (2010).
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Mori, Y.

Muniraj, I.

Namias, V.

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” IMA J. Appl. Math. 25, 241–256 (1980).
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V. Namias, “Fractionalization of Hankel transforms,” IMA J. Appl. Math. 26, 187–197 (1980).
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Nelson, L. E.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photon. 7, 354–362 (2013).
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Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Padgett, M.

Pan, Y. Y.

J. Li, J. S. Li, Y. Y. Pan, and R. Li, “Compressive optical image encryption,” Sci. Rep. 5, 10374 (2015).
[Crossref] [PubMed]

Phillips, D.

Pian, Q.

Plan, Y.

E. J. Candes and Y. Plan, “A probabilistic and RIPless theory of compressed sensing,” IEEE Trans. Inf. Theory 57, 7235–7254 (2011).
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Pooser, R.

Popoff, S.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
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Prather, D.

Quan, C.

B. Deepan, C. Quan, and C. J. Tay, “Compressive sensing for digital holographic interferometry,” Proc. SPIE 9234, 923419 (2014).
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Rafsanjani, S. M. H.

M. Mirhosseini, O. S. Magaña-Loaiza, S. M. H. Rafsanjani, and R. W. Boyd, “Compressive Direct Measurement of the Quantum Wave Function,” Phys. Rev. Lett. 113, 090402 (2014).
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Ramachandran, S.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
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Rawat, N.

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
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J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
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D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photon. 7, 354–362 (2013).
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Rodrigo, J. A.

Romberg, J.

E. Candes and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Problems 23, 969–985 (2007).
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Rosen, J.

Saleh, B. E. A.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
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A. F. Abouraddy, T. M. Yarnall, and B. E. A. Saleh, “Generalized optical interferometry for modal analysis in arbitrary degrees of freedom,” Opt. Lett. 37, 2889–2891 (2012).
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A. F. Abouraddy, T. M. Yarnall, and B. E. A. Saleh, “Angular and radial mode analyzer for optical beams,” Opt. Lett. 36, 4683–4685 (2011).
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N. Mohan, I. Stojanovic, W. C. Karl, B. E. A. Saleh, and M. C. Teich, “Compressed sensing in optical coherence tomography,” Proc. SPIE 7570, 75700L (2010).
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Schneeloch, J.

G. A. Howland, S. H. Knarr, J. Schneeloch, D. J. Lum, and J. C. Howell, “Compressively characterizing high-dimensional entangled states with complementary, random filtering,” Phys. Rev. X 6, 021018 (2016).

G. A. Howland, J. Schneeloch, D. J. Lum, and J. C. Howell, “Simultaneous measurement of complementary observables with compressive sensing,” Phys. Rev. Lett. 112, 253602 (2014).
[Crossref] [PubMed]

Shabahang, S.

L. Martin, D. Mardani, H. E. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral Hilbert-space analyzers,” Sci. Rep. 7, 44995 (2017).
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Sidky, E. Y.

Situ, G.

Soldevila, F.

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N. Mohan, I. Stojanovic, W. C. Karl, B. E. A. Saleh, and M. C. Teich, “Compressed sensing in optical coherence tomography,” Proc. SPIE 7570, 75700L (2010).
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Sun, B.

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M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
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Süzen, M.

Tajahuerce, E.

Takhar, D.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
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E. Candes and T. Tao, “The dantzig selector: Statistical estimation when p is much larger than n,” The Annals of Statistics 35, 2313–2351 (2007).
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B. Deepan, C. Quan, and C. J. Tay, “Compressive sensing for digital holographic interferometry,” Proc. SPIE 9234, 923419 (2014).
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Teich, M. C.

N. Mohan, I. Stojanovic, W. C. Karl, B. E. A. Saleh, and M. C. Teich, “Compressed sensing in optical coherence tomography,” Proc. SPIE 7570, 75700L (2010).
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B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

Thiebaut, E.

E. Thiebaut, “Principles of image reconstruction in interferometry,” New Concepts in Imaging: Optical and Statistical Models, EAS Publications Series 59, 157–187 (2013).

Torner, Lluis

Torres, Juan P.

Tur, M.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
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J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
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Vamivakas, A. N.

L. Martin, D. Mardani, H. E. Kondakci, W. D. Larson, S. Shabahang, A. K. Jahromi, T. Malhotra, A. N. Vamivakas, G. K. Atia, and A. F. Abouraddy, “Basis-neutral Hilbert-space analyzers,” Sci. Rep. 7, 44995 (2017).
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M. A. Davenport and M. B. Wakin, “Analysis of orthogonal matching pursuit using the restricted isometry property,” IEEE Trans. Inf. Theory 56, 4395–4401 (2010).
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J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
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N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
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J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
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N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
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M. A. Davenport and M. B. Wakin, “Analysis of orthogonal matching pursuit using the restricted isometry property,” IEEE Trans. Inf. Theory 56, 4395–4401 (2010).
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E. J. Candes and Y. Plan, “A probabilistic and RIPless theory of compressed sensing,” IEEE Trans. Inf. Theory 57, 7235–7254 (2011).
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V. Namias, “Fractionalization of Hankel transforms,” IMA J. Appl. Math. 26, 187–197 (1980).
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Nat. Photon. (3)

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
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E. Thiebaut, “Principles of image reconstruction in interferometry,” New Concepts in Imaging: Optical and Statistical Models, EAS Publications Series 59, 157–187 (2013).

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Optica (1)

Phys. Rev. Lett. (2)

G. A. Howland, J. Schneeloch, D. J. Lum, and J. C. Howell, “Simultaneous measurement of complementary observables with compressive sensing,” Phys. Rev. Lett. 112, 253602 (2014).
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M. Mirhosseini, O. S. Magaña-Loaiza, S. M. H. Rafsanjani, and R. W. Boyd, “Compressive Direct Measurement of the Quantum Wave Function,” Phys. Rev. Lett. 113, 090402 (2014).
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Figures (5)

Fig. 1
Fig. 1 Concept of compressive optical interferometry. (a) Schematic of the traditional CS scheme in optics, where the field E(x) is subject to random projections before measurements. (b) The mathematical concept used in traditional CS. The field is represented by a M×1 vector x that is transformed by a M ×N sensing matrix Φ (with disordered entries corresponding to the random masks) to yield a M ×1 measurement vector y, from which x is reconstructed. (c) The generalized interferometry scheme. The input beam E (x) is directed to a Mach-Zhender interferometer in which a generalized delay α replaces the usual temporal delay. Two copies of E(x) are created at a beam splitter and the modes {ψn(x)} underlying the beam acquire phase shifts of the form einα after passing through the generalized delay to yield a new beam E(x; α). The original and ‘delayed’ beams combine at another beam splitter, and their superposition is integrated by a bucket detector to produce an interferogram. (d,e) Graphical depiction of the matrix form of a generalized interferogram. (d) With evenly spaced Nyquist-rate sampling of the interferogram M ≥ 2N, the interferometric sensing matrix Φint has a well-defined deterministic structure. (e) With sub-Nyquist randomly sampled points α, Φint now appears disordered and similar to Φ in (b).
Fig. 2
Fig. 2 (a) Calculated values of η ( x ) = ( 2 / M ) Φ int x 2 / x 2 1 for s-sparse (s =4) vectors x. (b) Histogram of η(x) for an ensemble of 106 realizations of s-sparse vectors. (c) The sensing matrix Φint from Eq. (2) with randomly sampled α. (d) Φint is incoherent with incoherence parameter µ=1 and satisfies the isotropy property E{ϕϕ} =0.5 I.
Fig. 3
Fig. 3 (a) Implementation of the sampling system as a common-path interferometer leveraging the polarization-selectivity of liquid-crystal-based SLMs. (b) Interferograms collected by assigning different values to the generalized delay parameter α when HG2 is the only active mode. The figure shows three interferograms collected by an ideal frFT, a simulation that accounts for the pixelation and quantization effects of the SLMs, and the actual experimental setup.
Fig. 4
Fig. 4 Reconstructed modal distributions using the FT and CS approaches. (a) The modal weights |cn|2 calculated for fields in the HG-basis by applying the FT to Nyquist-rate evenly sampled interferograms (blue) and CS to sub-Nyquist randomly sampled (yellow) interferograms. (b) Same as (a) for LG modes (LG0 and LG1) and (c) for superposition of HG modes (HG0+HG1 and HG1 + iHG2). For FT, we use 128 evenly spaced values of α from 0 to 2π. For CS, we use M =30 randomly selected α ~ U [ 0 , 2 π ]. The insets show the ideal (exact) and approximated modes that are implemented experimentally. For HG0 and LG0, the modes are produced without approximation.
Fig. 5
Fig. 5 (a) Reconstruction error versus number of measurements M. Each curve results from averaging over 100 runs of the experiment. The average curve (solid black) is the mean and the shaded area designate one standard deviation spread on either side of the required M for 1000 randomly generated examples of sparse vectors with support size s≤4. (b) Comparing the reconstruction error in the experiment to that of the ideal case with an approximate HG3 mode for SNR= 20 dB.

Tables (1)

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Table 1 Summary of existing approaches in which CS is exploited in optics along with properties of the corresponding sensing matrix. The rightmost column highlights distinguishing features when comparing these approaches to our work. The bottom row refers to our proposed approach studied here.

Equations (2)

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P ( α ) d x | E ( x ; α ) + E ( x ; 0 ) | 2 = 1 + n = 1 N | c n | 2 cos ( n α ) .
[ P ( α 1 ) 1 P ( α 2 ) 1 P ( α M ) 1 ] y = [ cos α 1 cos 2 α 1 cos N α 1 cos α 2 cos 2 α 2 cos N α 2 cos α M cos 2 α M cos N α M ] Φ int [ | c 1 | 2 | c 2 | 2 | c N | 2 ] x .

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