Abstract

With this paper we bring about a discussion on the computing potential of complex optical networks and provide experimental demonstration that an optical fiber network can be used as an analog processor to calculate matrix inversion. A 3x3 matrix is inverted as a proof-of-concept demonstration using a fiber network containing three nodes and operating at telecomm wavelength. For an NxN matrix, the overall solving time (including setting time of the matrix elements and calculation time of inversion) scales as O(N2), whereas matrix inversion by most advanced computer algorithms requires ~O(N 2.37) computational time. For well-conditioned matrices, the error of the inversion performed optically is found to be around 3%, limited by the accuracy of measurement equipment.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. J. Caulfield, S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
    [CrossRef]
  2. S. Dolev, H. Fitoussi, “Masking traveling beams: Optical solutions for Np-complete problems, trading space for time,” Theor. Comput. Sci. 411(6), 837–853 (2010).
    [CrossRef]
  3. M. Oltean and O. Muntean, “Solving Np-complete problems with delayed signals: An overview of current research directions,” in Optical Supercomputing, S. Dolev, T. Haist, and M. Oltean, eds. (Springer, 2008), pp. 115–127.
  4. K. Wu, J. García de Abajo, C. Soci, P. P. Shum, N. I. Zheludev, “Fiber non-Turing all-optical computer for solving complex decision problems,” in Conference on Lasers and Electro-Optics / Europe (CLEO/Europe), (Munich, Germany, 2013), pp. CI-5.2.
  5. J. L. O’Brien, “Optical quantum computing,” Science 318(5856), 1567–1570 (2007).
    [CrossRef] [PubMed]
  6. M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, A. G. White, “Photonic Boson sampling in a tunable circuit,” Science 339(6121), 794–798 (2013).
    [CrossRef] [PubMed]
  7. J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
    [CrossRef] [PubMed]
  8. D. Woods, T. J. Naughton, “Optical computing: photonic neural networks,” Nat. Phys. 8(4), 257–259 (2012).
    [CrossRef]
  9. L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, I. Fischer, “Information processing using a single dynamical node as complex system,” Nat Commun. 2, 468 (2011).
    [CrossRef] [PubMed]
  10. Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
  11. D. Petrov, Y. Shkuratov, G. Videen, “Optimized matrix inversion technique for the T-matrix method,” Opt. Lett. 32(9), 1168–1170 (2007).
    [CrossRef] [PubMed]
  12. O. Arteaga, A. Canillas, “Analytic inversion of the Mueller-Jones polarization matrices for homogeneous media,” Opt. Lett. 35(4), 559–561 (2010).
    [CrossRef] [PubMed]
  13. V. V. Williams, “Multiplying matrices faster than Coppersmith-Winograd,” in 44th Symposium on Theory of Computing (ACM, 2012), 887–898.
  14. A. J. Stothers, On the Complexity of Matrix Multiplication (University of Edinburgh, 2010).
  15. H. Rajbenbach, Y. Fainman, S. H. Lee, “Optical implementation of an iterative algorithm formatrix inversion,” Appl. Opt. 26(6), 1024–1031 (1987).
    [CrossRef] [PubMed]
  16. D. Casasent, J. S. Smokelin, “New algorithm for analog optical matrix inversion,” Appl. Opt. 30(23), 3281–3287 (1991).
    [CrossRef] [PubMed]
  17. Q. Cao, J. W. Goodman, “Coherent optical techniques for diagonalization and inversion of circulant matrices and circulant approximations to Toeplitz matrices,” Appl. Opt. 23(6), 803–811 (1984).
    [CrossRef] [PubMed]
  18. E. Barnard, D. Casasent, “Optical neural net for matrix inversion,” Appl. Opt. 28(13), 2499–2504 (1989).
    [CrossRef] [PubMed]
  19. R. Paschotta, “Noise of mode-locked lasers (part I): Numerical model,” Appl. Phys. B 79(2), 153–162 (2004).
    [CrossRef]
  20. Z. L. Sámson, P. Horak, K. F. MacDonald, N. I. Zheludev, “Femtosecond surface plasmon pulse propagation,” Opt. Lett. 36(2), 250–252 (2011).
    [CrossRef] [PubMed]

2013 (2)

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, A. G. White, “Photonic Boson sampling in a tunable circuit,” Science 339(6121), 794–798 (2013).
[CrossRef] [PubMed]

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[CrossRef] [PubMed]

2012 (2)

D. Woods, T. J. Naughton, “Optical computing: photonic neural networks,” Nat. Phys. 8(4), 257–259 (2012).
[CrossRef]

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).

2011 (2)

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, I. Fischer, “Information processing using a single dynamical node as complex system,” Nat Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Z. L. Sámson, P. Horak, K. F. MacDonald, N. I. Zheludev, “Femtosecond surface plasmon pulse propagation,” Opt. Lett. 36(2), 250–252 (2011).
[CrossRef] [PubMed]

2010 (3)

O. Arteaga, A. Canillas, “Analytic inversion of the Mueller-Jones polarization matrices for homogeneous media,” Opt. Lett. 35(4), 559–561 (2010).
[CrossRef] [PubMed]

H. J. Caulfield, S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
[CrossRef]

S. Dolev, H. Fitoussi, “Masking traveling beams: Optical solutions for Np-complete problems, trading space for time,” Theor. Comput. Sci. 411(6), 837–853 (2010).
[CrossRef]

2007 (2)

2004 (1)

R. Paschotta, “Noise of mode-locked lasers (part I): Numerical model,” Appl. Phys. B 79(2), 153–162 (2004).
[CrossRef]

1991 (1)

1989 (1)

1987 (1)

1984 (1)

Aaronson, S.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, A. G. White, “Photonic Boson sampling in a tunable circuit,” Science 339(6121), 794–798 (2013).
[CrossRef] [PubMed]

Appeltant, L.

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, I. Fischer, “Information processing using a single dynamical node as complex system,” Nat Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Arteaga, O.

Barbieri, M.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[CrossRef] [PubMed]

Barnard, E.

Broome, M. A.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, A. G. White, “Photonic Boson sampling in a tunable circuit,” Science 339(6121), 794–798 (2013).
[CrossRef] [PubMed]

Canillas, A.

Cao, Q.

Casasent, D.

Caulfield, H. J.

H. J. Caulfield, S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
[CrossRef]

Dambre, J.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, I. Fischer, “Information processing using a single dynamical node as complex system,” Nat Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Danckaert, J.

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, I. Fischer, “Information processing using a single dynamical node as complex system,” Nat Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Datta, A.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[CrossRef] [PubMed]

Dolev, S.

H. J. Caulfield, S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
[CrossRef]

S. Dolev, H. Fitoussi, “Masking traveling beams: Optical solutions for Np-complete problems, trading space for time,” Theor. Comput. Sci. 411(6), 837–853 (2010).
[CrossRef]

Dove, J.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, A. G. White, “Photonic Boson sampling in a tunable circuit,” Science 339(6121), 794–798 (2013).
[CrossRef] [PubMed]

Duport, F.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).

Fainman, Y.

Fedrizzi, A.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, A. G. White, “Photonic Boson sampling in a tunable circuit,” Science 339(6121), 794–798 (2013).
[CrossRef] [PubMed]

Fischer, I.

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, I. Fischer, “Information processing using a single dynamical node as complex system,” Nat Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Fitoussi, H.

S. Dolev, H. Fitoussi, “Masking traveling beams: Optical solutions for Np-complete problems, trading space for time,” Theor. Comput. Sci. 411(6), 837–853 (2010).
[CrossRef]

García de Abajo, J.

K. Wu, J. García de Abajo, C. Soci, P. P. Shum, N. I. Zheludev, “Fiber non-Turing all-optical computer for solving complex decision problems,” in Conference on Lasers and Electro-Optics / Europe (CLEO/Europe), (Munich, Germany, 2013), pp. CI-5.2.

Gates, J. C.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[CrossRef] [PubMed]

Goodman, J. W.

Haelterman, M.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).

Horak, P.

Humphreys, P. C.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[CrossRef] [PubMed]

Jin, X.-M.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[CrossRef] [PubMed]

Kolthammer, W. S.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[CrossRef] [PubMed]

Kundys, D.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[CrossRef] [PubMed]

Langford, N. K.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[CrossRef] [PubMed]

Lee, S. H.

MacDonald, K. F.

Massar, S.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, I. Fischer, “Information processing using a single dynamical node as complex system,” Nat Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Metcalf, B. J.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[CrossRef] [PubMed]

Mirasso, C. R.

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, I. Fischer, “Information processing using a single dynamical node as complex system,” Nat Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Naughton, T. J.

D. Woods, T. J. Naughton, “Optical computing: photonic neural networks,” Nat. Phys. 8(4), 257–259 (2012).
[CrossRef]

O’Brien, J. L.

J. L. O’Brien, “Optical quantum computing,” Science 318(5856), 1567–1570 (2007).
[CrossRef] [PubMed]

Paquot, Y.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).

Paschotta, R.

R. Paschotta, “Noise of mode-locked lasers (part I): Numerical model,” Appl. Phys. B 79(2), 153–162 (2004).
[CrossRef]

Petrov, D.

Rahimi-Keshari, S.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, A. G. White, “Photonic Boson sampling in a tunable circuit,” Science 339(6121), 794–798 (2013).
[CrossRef] [PubMed]

Rajbenbach, H.

Ralph, T. C.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, A. G. White, “Photonic Boson sampling in a tunable circuit,” Science 339(6121), 794–798 (2013).
[CrossRef] [PubMed]

Sámson, Z. L.

Schrauwen, B.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, I. Fischer, “Information processing using a single dynamical node as complex system,” Nat Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Shkuratov, Y.

Shum, P. P.

K. Wu, J. García de Abajo, C. Soci, P. P. Shum, N. I. Zheludev, “Fiber non-Turing all-optical computer for solving complex decision problems,” in Conference on Lasers and Electro-Optics / Europe (CLEO/Europe), (Munich, Germany, 2013), pp. CI-5.2.

Smerieri, A.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).

Smith, B. J.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[CrossRef] [PubMed]

Smith, P. G. R.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[CrossRef] [PubMed]

Smokelin, J. S.

Soci, C.

K. Wu, J. García de Abajo, C. Soci, P. P. Shum, N. I. Zheludev, “Fiber non-Turing all-optical computer for solving complex decision problems,” in Conference on Lasers and Electro-Optics / Europe (CLEO/Europe), (Munich, Germany, 2013), pp. CI-5.2.

Soriano, M. C.

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, I. Fischer, “Information processing using a single dynamical node as complex system,” Nat Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Spring, J. B.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[CrossRef] [PubMed]

Thomas-Peter, N.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[CrossRef] [PubMed]

Van der Sande, G.

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, I. Fischer, “Information processing using a single dynamical node as complex system,” Nat Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Videen, G.

Walmsley, I. A.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[CrossRef] [PubMed]

White, A. G.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, A. G. White, “Photonic Boson sampling in a tunable circuit,” Science 339(6121), 794–798 (2013).
[CrossRef] [PubMed]

Williams, V. V.

V. V. Williams, “Multiplying matrices faster than Coppersmith-Winograd,” in 44th Symposium on Theory of Computing (ACM, 2012), 887–898.

Woods, D.

D. Woods, T. J. Naughton, “Optical computing: photonic neural networks,” Nat. Phys. 8(4), 257–259 (2012).
[CrossRef]

Wu, K.

K. Wu, J. García de Abajo, C. Soci, P. P. Shum, N. I. Zheludev, “Fiber non-Turing all-optical computer for solving complex decision problems,” in Conference on Lasers and Electro-Optics / Europe (CLEO/Europe), (Munich, Germany, 2013), pp. CI-5.2.

Zheludev, N. I.

Z. L. Sámson, P. Horak, K. F. MacDonald, N. I. Zheludev, “Femtosecond surface plasmon pulse propagation,” Opt. Lett. 36(2), 250–252 (2011).
[CrossRef] [PubMed]

K. Wu, J. García de Abajo, C. Soci, P. P. Shum, N. I. Zheludev, “Fiber non-Turing all-optical computer for solving complex decision problems,” in Conference on Lasers and Electro-Optics / Europe (CLEO/Europe), (Munich, Germany, 2013), pp. CI-5.2.

Appl. Opt. (4)

Appl. Phys. B (1)

R. Paschotta, “Noise of mode-locked lasers (part I): Numerical model,” Appl. Phys. B 79(2), 153–162 (2004).
[CrossRef]

Nat Commun. (1)

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, I. Fischer, “Information processing using a single dynamical node as complex system,” Nat Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Nat. Photonics (1)

H. J. Caulfield, S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
[CrossRef]

Nat. Phys. (1)

D. Woods, T. J. Naughton, “Optical computing: photonic neural networks,” Nat. Phys. 8(4), 257–259 (2012).
[CrossRef]

Opt. Lett. (3)

Sci. Rep. (1)

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).

Science (3)

J. L. O’Brien, “Optical quantum computing,” Science 318(5856), 1567–1570 (2007).
[CrossRef] [PubMed]

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, A. G. White, “Photonic Boson sampling in a tunable circuit,” Science 339(6121), 794–798 (2013).
[CrossRef] [PubMed]

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339(6121), 798–801 (2013).
[CrossRef] [PubMed]

Theor. Comput. Sci. (1)

S. Dolev, H. Fitoussi, “Masking traveling beams: Optical solutions for Np-complete problems, trading space for time,” Theor. Comput. Sci. 411(6), 837–853 (2010).
[CrossRef]

Other (4)

M. Oltean and O. Muntean, “Solving Np-complete problems with delayed signals: An overview of current research directions,” in Optical Supercomputing, S. Dolev, T. Haist, and M. Oltean, eds. (Springer, 2008), pp. 115–127.

K. Wu, J. García de Abajo, C. Soci, P. P. Shum, N. I. Zheludev, “Fiber non-Turing all-optical computer for solving complex decision problems,” in Conference on Lasers and Electro-Optics / Europe (CLEO/Europe), (Munich, Germany, 2013), pp. CI-5.2.

V. V. Williams, “Multiplying matrices faster than Coppersmith-Winograd,” in 44th Symposium on Theory of Computing (ACM, 2012), 887–898.

A. J. Stothers, On the Complexity of Matrix Multiplication (University of Edinburgh, 2010).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

(a) Schematic of an optical fiber network with three nodes where xi and yi (i = 1, 2, 3) are the input and output ports of each node; (b) Actual design of a node with optical fiber, couplers and attenuators and (c) Experimental setup of calculating matrix inversion.

Fig. 2
Fig. 2

Simulation results obtained for the inversion of 1000 3x3 matrices A = I-M with random elements (−0.125 < aij < 0 and aii = 1 for i, j = 1, 2, 3) and ~3% error in the elements. (a) Matrix error and (b) condition number of matrix A; (c) Inverse matrix error.

Fig. 3
Fig. 3

Calculated distributions of (a) condition numbers and (b) inverse matrix errors for different matrix sizes ranging from 3x3 to 100x100 and 1% initial (rms) error in matrix elements aij; (c) nominal error ε0 with respect to the matrix size. The solid squares are the values extracted from the simulation results in Fig. 3(b) with initial matrix error of 1%. The dashed line is a linear fit to the data according to log ε 0 0.59logN2.1 ; (d) inverse matrix errors for different initial (rms) errors ranging from 1% to 0.01% in matrix elements of 100x100 matrices. 1000 simulations were performed for each case.

Fig. 4
Fig. 4

General node design for NxN complex arbitrary matrices. Input signals are merged in a combiner after propagating through attenuators and phase shifters. The signals are then amplified to compensate for losses and split to generate the outputs. In the example shown for node 1, one of the outputs is fed back to the input corresponding to the diagonal element m11.

Fig. 5
Fig. 5

Time-domain waveform of (a) input pulse, (b) output at node 1 and (c) output at node 2. The black dashed lines represent the waveforms generated by a 300 ns pulse and the red solid lines waveforms generated by an 8 ns pulse.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

y i = x i + m ij y j + m ik y k
Y=X+MY
M=( 0 m 12 m 13 m 21 0 m 23 m 31 m 32 0 )
Y= (IM) 1 X A 1 X
M=( 0 0.121 0.122 0.120 0 0.121 0.0979 0.0912 0 )
A meas 1 =( 1.020 0.139 0.144 0.137 1.040 0.143 0.113 0.109 1.030 ), A calc 1 =( 1.030 0.138 0.143 0.138 1.030 0.142 0.113 0.107 1.030 )
(A+ΔA)( A 1 +Δ A 1 )=I
ε ||Δ A 1 || || A 1 +Δ A 1 || || A 1 ||||A|| ||ΔA|| ||A|| =κ ||ΔA|| ||A||
M =( 0 0.121 0.0763 0.120 0 0.121 0.0979 0.0345 0 )
B meas 1 =( 1.020 0.128 0.092 0.135 1.030 0.135 0.104 0.048 1.020 ), B calc 1 =( 1.030 0.127 0.094 0.136 1.020 0.134 0.105 0.048 1.010 )
y 1 = x 1 + j=1 N m 1j y j
Y= (IM) 1 X= k=0 M k X
(IM)Y=X
λ y 1 = j=1 N m 1j y j ;
λY=MY
(λIM)Y=0.

Metrics