Abstract

We experimentally implement an optical algorithm for integration of a real-valued bivariate function. A user-defined function is encoded in the position-dependent phase of one of the polarization components of an optical beam. The integral of this function is retrieved by measuring a Stokes parameter of the polarization. We analyze the performance of the system as an integration device.

© 2014 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, 1996).
  2. B. E. A. Saleh and M. C. Teich, Fundamental Photonics (Wiley, 1991).
  3. D. A. B. Miller, “Are optical transistors the logical next step?” Nat. Photonics 4, 3–5 (2010).
    [CrossRef]
  4. H. J. Caulfield and S. Dolev, “The role of optics in computing,” Nat. Photonics 4, 406–407 (2010).
    [CrossRef]
  5. R. S. Tucker, “The role of optics in computing,” Nat. Photonics 4, 405 (2010).
    [CrossRef]
  6. D. A. B. Miller, “The role of optics in computing,” Nat. Photonics 4, 406 (2010).
    [CrossRef]
  7. H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4, 261–263 (2010).
    [CrossRef]
  8. D. Woods and T. J. Naughton, “Optical computing: photonic neural networks,” Nat. Phys. 8, 257–259 (2012).
    [CrossRef]
  9. H. M. Ozaktas and D. A. B. Miller, “Digital Fourier optics,” Appl. Opt. 35, 1212–1219 (1996).
    [CrossRef]
  10. A. Rodan and P. Tino, “Minimum complexity echo state network,” IEEE Trans. Neural Netw. 22, 131–144 (2011).
  11. M. Suzuki and H. Uenohara, “Investigation of all-optical error detection circuit using SOA-MZI based XOR gates at 10  gbit/s,” Electron. Lett. 45, 224–225 (2009).
    [CrossRef]
  12. J. Wang, Q. Sun, and J. Sun, “Ultrafast all-optical logic and gate for CSRZ signals using periodically poled lithium niobate,” J. Opt. Soc. Am. B 26, 951–958 (2009).
    [CrossRef]
  13. T. Chattopadhyay and J. N. Roy, “An all-optical technique for a binary-to-quaternary encoder and a quaternary-to-binary decoder,” J. Opt. A 11, 075501 (2009).
  14. T. Chattopadhyay, “Optical programmable Boolean logic unit,” Appl. Opt. 50, 6049–6056 (2011).
    [CrossRef]
  15. M. Oltean, in Fifth International Conference on Unconventional Computation (UCÕ06), Vol. 4135 of LNCS (Springer, 2006), pp. 217–227.
  16. N. T. Shaked, S. Messika, S. Dolev, and J. Rosen, “Optical solution for bounded np-complete problems,” Appl. Opt. 46, 711–724 (2007).
    [CrossRef]
  17. T. Haist and W. Osten, “An optical solution for the traveling salesman problem,” Opt. Express 15, 10473–10482 (2007).
    [CrossRef]
  18. L. Appeltant, M. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2, 468 (2011).
    [CrossRef]
  19. Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
    [CrossRef]
  20. E. Hecht, Optics (Addison-Wesley, 1987).
  21. G. B. Lemos, J. O. de Almeida, S. P. Walborn, P. H. S. Ribeiro, and M. Hor-Meyll are preparing a manuscript to be called “Characterization of a spatial light modulator as a polarization quantum channel.”

2012

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

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

2011

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

T. Chattopadhyay, “Optical programmable Boolean logic unit,” Appl. Opt. 50, 6049–6056 (2011).
[CrossRef]

A. Rodan and P. Tino, “Minimum complexity echo state network,” IEEE Trans. Neural Netw. 22, 131–144 (2011).

2010

D. A. B. Miller, “Are optical transistors the logical next step?” Nat. Photonics 4, 3–5 (2010).
[CrossRef]

H. J. Caulfield and S. Dolev, “The role of optics in computing,” Nat. Photonics 4, 406–407 (2010).
[CrossRef]

R. S. Tucker, “The role of optics in computing,” Nat. Photonics 4, 405 (2010).
[CrossRef]

D. A. B. Miller, “The role of optics in computing,” Nat. Photonics 4, 406 (2010).
[CrossRef]

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

2009

M. Suzuki and H. Uenohara, “Investigation of all-optical error detection circuit using SOA-MZI based XOR gates at 10  gbit/s,” Electron. Lett. 45, 224–225 (2009).
[CrossRef]

T. Chattopadhyay and J. N. Roy, “An all-optical technique for a binary-to-quaternary encoder and a quaternary-to-binary decoder,” J. Opt. A 11, 075501 (2009).

J. Wang, Q. Sun, and J. Sun, “Ultrafast all-optical logic and gate for CSRZ signals using periodically poled lithium niobate,” J. Opt. Soc. Am. B 26, 951–958 (2009).
[CrossRef]

2007

1996

Appeltant, L.

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

Caulfield, H. J.

H. J. Caulfield and S. Dolev, “The role of optics in computing,” Nat. Photonics 4, 406–407 (2010).
[CrossRef]

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

Chattopadhyay, T.

T. Chattopadhyay, “Optical programmable Boolean logic unit,” Appl. Opt. 50, 6049–6056 (2011).
[CrossRef]

T. Chattopadhyay and J. N. Roy, “An all-optical technique for a binary-to-quaternary encoder and a quaternary-to-binary decoder,” J. Opt. A 11, 075501 (2009).

Dambre, J.

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

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

Danckaert, J.

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

de Almeida, J. O.

G. B. Lemos, J. O. de Almeida, S. P. Walborn, P. H. S. Ribeiro, and M. Hor-Meyll are preparing a manuscript to be called “Characterization of a spatial light modulator as a polarization quantum channel.”

Dolev, S.

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

H. J. Caulfield and S. Dolev, “The role of optics in computing,” Nat. Photonics 4, 406–407 (2010).
[CrossRef]

N. T. Shaked, S. Messika, S. Dolev, and J. Rosen, “Optical solution for bounded np-complete problems,” Appl. Opt. 46, 711–724 (2007).
[CrossRef]

Duport, F.

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

Fischer, I.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, 1996).

Haelterman, M.

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

Haist, T.

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 1987).

Hor-Meyll, M.

G. B. Lemos, J. O. de Almeida, S. P. Walborn, P. H. S. Ribeiro, and M. Hor-Meyll are preparing a manuscript to be called “Characterization of a spatial light modulator as a polarization quantum channel.”

Lemos, G. B.

G. B. Lemos, J. O. de Almeida, S. P. Walborn, P. H. S. Ribeiro, and M. Hor-Meyll are preparing a manuscript to be called “Characterization of a spatial light modulator as a polarization quantum channel.”

Massar, S.

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

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

Messika, S.

Miller, D. A. B.

D. A. B. Miller, “Are optical transistors the logical next step?” Nat. Photonics 4, 3–5 (2010).
[CrossRef]

D. A. B. Miller, “The role of optics in computing,” Nat. Photonics 4, 406 (2010).
[CrossRef]

H. M. Ozaktas and D. A. B. Miller, “Digital Fourier optics,” Appl. Opt. 35, 1212–1219 (1996).
[CrossRef]

Mirasso, C.

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

Naughton, T. J.

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

Oltean, M.

M. Oltean, in Fifth International Conference on Unconventional Computation (UCÕ06), Vol. 4135 of LNCS (Springer, 2006), pp. 217–227.

Osten, W.

Ozaktas, H. M.

Paquot, Y.

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

Ribeiro, P. H. S.

G. B. Lemos, J. O. de Almeida, S. P. Walborn, P. H. S. Ribeiro, and M. Hor-Meyll are preparing a manuscript to be called “Characterization of a spatial light modulator as a polarization quantum channel.”

Rodan, A.

A. Rodan and P. Tino, “Minimum complexity echo state network,” IEEE Trans. Neural Netw. 22, 131–144 (2011).

Rosen, J.

Roy, J. N.

T. Chattopadhyay and J. N. Roy, “An all-optical technique for a binary-to-quaternary encoder and a quaternary-to-binary decoder,” J. Opt. A 11, 075501 (2009).

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamental Photonics (Wiley, 1991).

Schrauwen, B.

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

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

Shaked, N. T.

Smerieri, A.

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

Soriano, M.

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

Sun, J.

Sun, Q.

Suzuki, M.

M. Suzuki and H. Uenohara, “Investigation of all-optical error detection circuit using SOA-MZI based XOR gates at 10  gbit/s,” Electron. Lett. 45, 224–225 (2009).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamental Photonics (Wiley, 1991).

Tino, P.

A. Rodan and P. Tino, “Minimum complexity echo state network,” IEEE Trans. Neural Netw. 22, 131–144 (2011).

Tucker, R. S.

R. S. Tucker, “The role of optics in computing,” Nat. Photonics 4, 405 (2010).
[CrossRef]

Uenohara, H.

M. Suzuki and H. Uenohara, “Investigation of all-optical error detection circuit using SOA-MZI based XOR gates at 10  gbit/s,” Electron. Lett. 45, 224–225 (2009).
[CrossRef]

Van der Sande, G.

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

Walborn, S. P.

G. B. Lemos, J. O. de Almeida, S. P. Walborn, P. H. S. Ribeiro, and M. Hor-Meyll are preparing a manuscript to be called “Characterization of a spatial light modulator as a polarization quantum channel.”

Wang, J.

Woods, D.

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

Appl. Opt.

Electron. Lett.

M. Suzuki and H. Uenohara, “Investigation of all-optical error detection circuit using SOA-MZI based XOR gates at 10  gbit/s,” Electron. Lett. 45, 224–225 (2009).
[CrossRef]

IEEE Trans. Neural Netw.

A. Rodan and P. Tino, “Minimum complexity echo state network,” IEEE Trans. Neural Netw. 22, 131–144 (2011).

J. Opt. A

T. Chattopadhyay and J. N. Roy, “An all-optical technique for a binary-to-quaternary encoder and a quaternary-to-binary decoder,” J. Opt. A 11, 075501 (2009).

J. Opt. Soc. Am. B

Nat. Commun.

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

Nat. Photonics

D. A. B. Miller, “Are optical transistors the logical next step?” Nat. Photonics 4, 3–5 (2010).
[CrossRef]

H. J. Caulfield and S. Dolev, “The role of optics in computing,” Nat. Photonics 4, 406–407 (2010).
[CrossRef]

R. S. Tucker, “The role of optics in computing,” Nat. Photonics 4, 405 (2010).
[CrossRef]

D. A. B. Miller, “The role of optics in computing,” Nat. Photonics 4, 406 (2010).
[CrossRef]

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

Nat. Phys.

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

Opt. Express

Sci. Rep.

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

Other

E. Hecht, Optics (Addison-Wesley, 1987).

G. B. Lemos, J. O. de Almeida, S. P. Walborn, P. H. S. Ribeiro, and M. Hor-Meyll are preparing a manuscript to be called “Characterization of a spatial light modulator as a polarization quantum channel.”

M. Oltean, in Fifth International Conference on Unconventional Computation (UCÕ06), Vol. 4135 of LNCS (Springer, 2006), pp. 217–227.

J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, 1996).

B. E. A. Saleh and M. C. Teich, Fundamental Photonics (Wiley, 1991).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Schematic diagram of the optical integration algorithm. (b) Experimental setup for the integration of the function f(x,y).

Fig. 2.
Fig. 2.

(a) Linear fit used to obtain the calibration parameters A and B for test function fG(x,y) in Eq. (10). (b) Quadratic fit used to obtain the calibration parameters A and B for test function fG2(x,y) Eq. (11).

Fig. 3.
Fig. 3.

Optical integration (blue) and analytical result (black) for the function h(x,y) given in Eq. (12). The figures along the top show the gray-scale image of the phase used on the SLM in each case.

Fig. 4.
Fig. 4.

Optical integration (blue) and value for the analytical integration (black) of different 2D functions h(x,y) and S(x,y) given in Eqs. (14) and (15). The figures along the top show the gray-scale image of the phase used on the SLM in each case.

Equations (15)

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

J=xxyyf(x,y)dxdy,
Eo(x,y)=E(x,y)2(eia(x,y)eH+eV),
Eo(x,y)=E(x,y)(1+eia(x,y)2e++1eia(x,y)2e),
I±=|E(x,y)|2{12±14(eia(x,y)+eia(x,y))}dxdy,
I±=|E(x,y)|212{1±cos[a(x,y)]}dxdy.
D=|E(x,y)|2f(x,y)dxdy.
D|E(x0,y0)|2J.
δJ(g/T)1/2(1+D/T)2(g/T)1/2.
D=I+I=B+AJ,
fG(x,y)=e(x/σ)2R(y,y),
fG2(x,y)=exp(x2+y2σ2)R(x,x,y,y).
hn(x,y)=[Hn(x/20)]2e(x/20)22n7(n1)!R(y,200),
xxyyHn(x,y)dxdy207400πn,
hn(x,y)=130n(n1)!Hn(xy20)exp(x220y230),
Sν(x,y)=sin(νxy1000)exp(x2+y2100),

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