Abstract

Interferometry is routinely used for spectral or modal analysis of optical signals. By posing interferometric modal analysis as a sparse recovery problem, we show that compressive sampling helps exploit the sparsity of typical optical signals in modal space and reduces the number of required measurements. Instead of collecting evenly spaced interferometric samples at the Nyquist rate followed by a Fourier transform as is common practice, we show that random sampling at sub-Nyquist rates followed by a sparse reconstruction algorithm suffices. We demonstrate our approach, which we call compressive interferometry, numerically in the context of modal analysis of spatial beams using a generalized interferometric configuration. Compressive interferometry applies to widely used optical modal sets and is robust with respect to noise, thus holding promise to enhance real-time processing in optical imaging and communications.

© 2015 Optical Society of America

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References

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  1. 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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    [Crossref]
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    [Crossref] [PubMed]
  4. J. Demas and S. Ramachandran, “Sub-second mode measurement of fibers using C2 imaging,” Opt. Express 22, 23043–23056 (2014).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  24. J. Tropp, “Greed is good: Algorithmic results for sparseapproximation, IEEE Trans. Inf. Theory 50, 2231–2242 (2004).
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  25. W. Dai and O. Milenkovic, “Subspace pursuit for compressive sensing signal reconstruction, IEEE Trans. Inf. Theory 55, 2230–2249 (2009).
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2014 (3)

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]

J. Demas and S. Ramachandran, “Sub-second mode measurement of fibers using C2 imaging,” Opt. Express 22, 23043–23056 (2014).
[Crossref] [PubMed]

2013 (1)

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]

2012 (2)

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]

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).
[Crossref] [PubMed]

2011 (1)

2010 (1)

M. A. Davenport, P. T. Boufounos, M. B. Wakin, and R. G. Baraniuk, “Signal processing with compressive measurements,” IEEE J. Sel. Top. Signal Process. 4, 445–460 (2010).
[Crossref]

2009 (2)

W. Dai and O. Milenkovic, “Subspace pursuit for compressive sensing signal reconstruction, IEEE Trans. Inf. Theory 55, 2230–2249 (2009).
[Crossref]

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Programmable two-dimensional optical fractional Fourier processor,” Opt. Express 17, 4976–4983 (2009).
[Crossref] [PubMed]

2008 (2)

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]

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

2007 (1)

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

2006 (3)

E. J. Candès, J. K. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

E. J. 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. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

2005 (1)

E. J. Candès and T. Tao, “Decoding by Linear Programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[Crossref]

2004 (1)

J. Tropp, “Greed is good: Algorithmic results for sparseapproximation, IEEE Trans. Inf. Theory 50, 2231–2242 (2004).
[Crossref]

1998 (2)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8190 (1992).
[Crossref] [PubMed]

1980 (2)

V. Namias, “The fractional order Fourier transform and its applications to quantum mechanics,” IMA J. Appl. Math. 25, 241–265 (1980).
[Crossref]

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

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]

Alieva, T.

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8190 (1992).
[Crossref] [PubMed]

Baraniuk, R. G.

M. A. Davenport, P. T. Boufounos, M. B. Wakin, and R. G. Baraniuk, “Signal processing with compressive measurements,” IEEE J. Sel. Top. Signal Process. 4, 445–460 (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]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8190 (1992).
[Crossref] [PubMed]

Boufounos, P. T.

M. A. Davenport, P. T. Boufounos, M. B. Wakin, and R. G. Baraniuk, “Signal processing with compressive measurements,” IEEE J. Sel. Top. Signal Process. 4, 445–460 (2010).
[Crossref]

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.

Candès, E.

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

Candès, E. J.

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

E. J. Candès, J. K. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

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

E. J. Candès and T. Tao, “Decoding by Linear Programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[Crossref]

E. J. Candès,“Compressive sensing,” in Proceedings of the International Congress of Mathematicians (2006), M. Sanz-Solé, J. Soria, J.L. Varona, and J. Verdera, eds. (Eur. Math. Soc., 2007), pp. 1433–1452.

Chen, M.

Chen, S. S.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci Comput. 20, 33–61 (1998).
[Crossref]

Dai, W.

W. Dai and O. Milenkovic, “Subspace pursuit for compressive sensing signal reconstruction, IEEE Trans. Inf. Theory 55, 2230–2249 (2009).
[Crossref]

Davenport, M. A.

M. A. Davenport, P. T. Boufounos, M. B. Wakin, and R. G. Baraniuk, “Signal processing with compressive measurements,” IEEE J. Sel. Top. Signal Process. 4, 445–460 (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]

Demas, J.

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]

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci Comput. 20, 33–61 (1998).
[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]

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]

Howell, J. C.

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. A.

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, M.

Huang, W.

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]

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(Wiley, 2001).

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]

Lu, Y.

Lum, D. J.

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]

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]

Milenkovic, O.

W. Dai and O. Milenkovic, “Subspace pursuit for compressive sensing signal reconstruction, IEEE Trans. Inf. Theory 55, 2230–2249 (2009).
[Crossref]

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).
[Crossref] [PubMed]

Namias, V.

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

V. Namias, “The fractional order Fourier transform and its applications to quantum mechanics,” IMA J. Appl. Math. 25, 241–265 (1980).
[Crossref]

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform(Wiley, 2001).

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).
[Crossref] [PubMed]

Ramachandran, S.

J. Demas and S. Ramachandran, “Sub-second mode measurement of fibers using C2 imaging,” Opt. Express 22, 23043–23056 (2014).
[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]

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).
[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]

Rodrigo, J. A.

Romberg, J.

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

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

Romberg, J. K.

E. J. Candès, J. K. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

Saleh, B. E. A.

Saunders, M. A.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci Comput. 20, 33–61 (1998).
[Crossref]

Schneeloch, J.

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]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8190 (1992).
[Crossref] [PubMed]

Sun, T.

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]

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).
[Crossref]

Tao, T.

E. J. Candès, J. K. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

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

E. J. Candès and T. Tao, “Decoding by Linear Programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[Crossref]

Tropp, J.

J. Tropp, “Greed is good: Algorithmic results for sparseapproximation, IEEE Trans. Inf. Theory 50, 2231–2242 (2004).
[Crossref]

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).
[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]

Wakin, M. B.

M. A. Davenport, P. T. Boufounos, M. B. Wakin, and R. G. Baraniuk, “Signal processing with compressive measurements,” IEEE J. Sel. Top. Signal Process. 4, 445–460 (2010).
[Crossref]

Wang, J.

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]

Willner, A. E.

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]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8190 (1992).
[Crossref] [PubMed]

Yan, Y.

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]

Yang, J.

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]

Yarnall, T. M.

Yu, L.

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Figures (4)

Fig. 1
Fig. 1 (a) Generalized interferometer for a 2D beam with two GPOs (Λ1 and Λ2; delay parameters α and β, respectively) corresponding to two modal sets. (b) Graphical depiction of the matrix form of a generalized interferogram, Eq. (4): the interferogram is sampled at evenly spaced points and the number of measurements M exceeds the ambient dimension N. (c) The linear CS model: x is sparse, the interferogram is sampled at random points, and M < N.
Fig. 2
Fig. 2 Reconstruction of 1D beams with N = 100. (a),(b) Superpositions of four HG modes with indices (1, 5, 7, 10) and (30, 40, 50, 60) produce 1D beams with intensity profiles (c) |E1(x)|2 and (d) |E2(x)|2, respectively. (e) The recovery error of compressive interferometry for the beams in (a) when SNR = 10 dB. (f) Comparing the performance of compressive interferometry, M = 35 for E1(x), to that of the FT of an interferogram sampled at the Nyquist rate while varying SNR.
Fig. 3
Fig. 3 Reconstruction of 1D beams with N = 100. (a) Superpositions of four HG modes with indices (1, 5, 7, 10) and unequal modal weights. (b) The beam intensity |E1(x)|2. (c) The recovery error of compressive interferometry for the beam in (a) when SNR = 10 dB and 20 dB. The figure also shows the performance with and without SVD.
Fig. 4:
Fig. 4: Reconstruction of 2D beams using compressive interferometrywith N = N1×N2 =100. The first row shows themodalweights |cnm|2; the second shows the intensity distribution I(x,y), and the third the reconstruction error. Themodal bases in (a)–(b) are HG modes along x and y, in (c)–(d) are LG-OAM modes.(e) Recovery error as a function of SNR at selected measurement samples M using the beam E3(x,y) in (a). (f) SNR versus M for a recovery error <2×102.

Equations (9)

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E ( x , y ) = n = 1 N 1 m = 1 N 2 c n m ψ n ( x ) ϕ m ( y ) .
Λ ( x , x ; α ) = n = 1 N e i n α ψ n ( x ) ψ n * ( x ) ,
P ( α , β ) = 1 + n = 1 N 1 m = 1 N 2 | c n m | 2 cos ( n α + m β ) ,
[ 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 ^ [ | c 1 | 2 | c 2 | 2 | c N | 2 ] x .
P ( α i ) 1 = Φ i , x ; i = 1 , , M ,
P ( α i , β j ) 1 = Φ i , j , x ; i = 1 , , M 1 ; j = 1 , , M 2 ,
min x 0 subject to y = Φ ^ x ,
min x 1 subject to y = Φ ^ x ,
( 1 δ s ) x 2 Φ ^ x 2 ( 1 + δ s ) x 2 , s = 1 , 2 , ,

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