Abstract

The architectures of classical analog coherent optical (ACO) spectrum analyzers and correlators are not designed to process the wave signal as a whole, i.e., simultaneously in three dimensions. In this paper, the theory of ACO three-dimensional direct spectrum-correlation processing of spatial–temporal optical replicas (copies) of wave signals is discussed. In the single-stage and two-stage ACO systems, the spatial power spectrum and spatial correlation function of the wave signal (envelope) are obtained on the basis of space–time integration. The geometry of the final compressed signal in the output plane of either optical system allows one to evaluate the angle of wave arrival. The wave signal to be processed can theoretically have any form (due to autocorrelation properties of the systems) and an unlimited duration (due to time integration of wave energy and possibility of electronic subtraction of the intermediate bias terms of the time integration).

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  2. A. VanderLugt, Optical Signal Processing (Wiley, 2005).
  3. C. S. Weaver and J. W. Goodman, “A technique for optical convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef]
  4. D. C. Beste and E. N. Leith, “An optical technique for simultaneous beamforming and cross-correlation,” IEEE Trans. Aerosp. Electron. Syst. AES-2, 376–381 (1966).
    [CrossRef]
  5. R. E. Williams and K. VonBieren, “Combined beam forming and cross-correlation of broadband signals from a multidimensional array using coherent optics,” Appl. Opt. 10, 1386–1392 (1971).
    [CrossRef]
  6. C. E. Thomas, “Optical spectrum analysis of large space bandwidth signals,” Appl. Opt. 5, 1782–1790 (1966).
    [CrossRef]
  7. R. A. Sprague and Ch. L. Koliopulos, “Time integrating acoustooptic correlator,” Appl. Opt. 15, 89–92 (1976).
    [CrossRef]
  8. T. M. Turpin, “Time integrating optical processors,” Proc. SPIE 154, 196–203 (1978).
  9. T. R. Bader, “Acoustooptic spectrum analysis: a high performance hybrid technique,” Appl. Opt. 18, 1668–1672 (1979).
    [CrossRef]
  10. D. Psaltis and D. Casasent, “Time- and space-integrating spectrum analyzer,” Appl. Opt. 18, 3203–3204 (1979).
    [CrossRef]
  11. V. Ezhov, “The three-dimensional coherent optical correlator with generalized hologram for implementing the time compression of a spatial–temporal spectrum of the wave process,” in Proceedings of the 7th International Conference HOLOEXPO-2010 (Bauman Moscow State Technical University, 2010), pp. 416–421, in Russian.
  12. The discrete aperture leads simply to a spatial multiplication of the spectrum, corresponding to a continuous aperture (without changing the shape of the spectrum).
  13. All necessary information is contained in the (+1)st diffraction order of the light, corresponding to the second term of the input optical signal 1+exp[i2π(…)]+exp[−i2π(…)], which describes a whole real-valued amplitude light modulation.
  14. S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
    [CrossRef]
  15. P. Q. Thai and A. Alphones, “Hybrid optical beam-former in receiver mode,” Opt. Photon. J. 1, 130–136 (2011).
    [CrossRef]
  16. T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. 80, 3247–3249 (2002).
    [CrossRef]
  17. V. Ezhov, “Coherent optical correlator with space–time integration for radio astronomy,” in Proceedings of the 4th All-Union School on Optical Information Processing (Holography Research Council of USSR Academy of Sciences, 1984), pp. 270–271, in Russian.

2011

P. Q. Thai and A. Alphones, “Hybrid optical beam-former in receiver mode,” Opt. Photon. J. 1, 130–136 (2011).
[CrossRef]

2008

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

2002

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. 80, 3247–3249 (2002).
[CrossRef]

1979

1978

T. M. Turpin, “Time integrating optical processors,” Proc. SPIE 154, 196–203 (1978).

1976

1971

1966

Alphones, A.

P. Q. Thai and A. Alphones, “Hybrid optical beam-former in receiver mode,” Opt. Photon. J. 1, 130–136 (2011).
[CrossRef]

Bader, T. R.

Beste, D. C.

D. C. Beste and E. N. Leith, “An optical technique for simultaneous beamforming and cross-correlation,” IEEE Trans. Aerosp. Electron. Syst. AES-2, 376–381 (1966).
[CrossRef]

Blanche, P.-A.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

Brauer, A.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. 80, 3247–3249 (2002).
[CrossRef]

Casasent, D.

Ezhov, V.

V. Ezhov, “Coherent optical correlator with space–time integration for radio astronomy,” in Proceedings of the 4th All-Union School on Optical Information Processing (Holography Research Council of USSR Academy of Sciences, 1984), pp. 270–271, in Russian.

V. Ezhov, “The three-dimensional coherent optical correlator with generalized hologram for implementing the time compression of a spatial–temporal spectrum of the wave process,” in Proceedings of the 7th International Conference HOLOEXPO-2010 (Bauman Moscow State Technical University, 2010), pp. 416–421, in Russian.

Flores, D.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

Goodman, J. W.

Gu, T.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

Hilaire, P.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

Koliopulos, Ch. L.

Lederer, F.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. 80, 3247–3249 (2002).
[CrossRef]

Leith, E. N.

D. C. Beste and E. N. Leith, “An optical technique for simultaneous beamforming and cross-correlation,” IEEE Trans. Aerosp. Electron. Syst. AES-2, 376–381 (1966).
[CrossRef]

Li, G.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

Lin, W.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

Norwood, R.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

Pertsch, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. 80, 3247–3249 (2002).
[CrossRef]

Peschel, U.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. 80, 3247–3249 (2002).
[CrossRef]

Peyghambarian, N.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

Psaltis, D.

Rokutanda, S.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

Sprague, R. A.

Tay, S.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

Thai, P. Q.

P. Q. Thai and A. Alphones, “Hybrid optical beam-former in receiver mode,” Opt. Photon. J. 1, 130–136 (2011).
[CrossRef]

Thomas, C. E.

Thomas, J.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

Tunc, A.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

Turpin, T. M.

T. M. Turpin, “Time integrating optical processors,” Proc. SPIE 154, 196–203 (1978).

VanderLugt, A.

A. VanderLugt, Optical Signal Processing (Wiley, 2005).

VonBieren, K.

Voorakaranam, R.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

Wang, P.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

Weaver, C. S.

Williams, R. E.

Yamamoto, M.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

Zentgraf, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. 80, 3247–3249 (2002).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. 80, 3247–3249 (2002).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst.

D. C. Beste and E. N. Leith, “An optical technique for simultaneous beamforming and cross-correlation,” IEEE Trans. Aerosp. Electron. Syst. AES-2, 376–381 (1966).
[CrossRef]

Nature

S. Tay, P.-A. Blanche, R. Voorakaranam, A. Tunc, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. Hilaire, J. Thomas, R. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic 3D display,” Nature 451, 694–698 (2008).
[CrossRef]

Opt. Photon. J.

P. Q. Thai and A. Alphones, “Hybrid optical beam-former in receiver mode,” Opt. Photon. J. 1, 130–136 (2011).
[CrossRef]

Proc. SPIE

T. M. Turpin, “Time integrating optical processors,” Proc. SPIE 154, 196–203 (1978).

Other

V. Ezhov, “The three-dimensional coherent optical correlator with generalized hologram for implementing the time compression of a spatial–temporal spectrum of the wave process,” in Proceedings of the 7th International Conference HOLOEXPO-2010 (Bauman Moscow State Technical University, 2010), pp. 416–421, in Russian.

The discrete aperture leads simply to a spatial multiplication of the spectrum, corresponding to a continuous aperture (without changing the shape of the spectrum).

All necessary information is contained in the (+1)st diffraction order of the light, corresponding to the second term of the input optical signal 1+exp[i2π(…)]+exp[−i2π(…)], which describes a whole real-valued amplitude light modulation.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

A. VanderLugt, Optical Signal Processing (Wiley, 2005).

V. Ezhov, “Coherent optical correlator with space–time integration for radio astronomy,” in Proceedings of the 4th All-Union School on Optical Information Processing (Holography Research Council of USSR Academy of Sciences, 1984), pp. 270–271, in Russian.

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 (7)

Fig. 1.
Fig. 1.

Reception of sinusoidal wave with plane wavefront by 1D array of receiving elements with individual informational outputs (n is the normal to the array X).

Fig. 2.
Fig. 2.

ACO channel for obtaining the spatial–temporal spectrum of sinusoidal wave signal: (a) set of time signals along the coordinate X in process of change of time, (b) instantaneous spatial scan of wave signal along the coordinate X at fixed time t, (c) conversion of the wave signal with changing the transverse scale M, and (d) ACO spectrum analyzer consisting of spherical lens L and two volumes of free space each with width f, where f is the focal length of the lens.

Fig. 3.
Fig. 3.

Optical spatial–temporal spectrum of nonsinusoidal wave signal.

Fig. 4.
Fig. 4.

Reception of wave along two spatial coordinates X and Y: (a) tilt angles θ and φ of the wavefront and (b) two mutually orthogonal 1D arrays in the reception plane.

Fig. 5.
Fig. 5.

One-stage ACO spectral-correlation processor with electronic filtering and electronic time integration of spatial–temporal spectrum of the wave signal: (a) system, (b) two mutually orthogonal 1D arrays of light-modulating elements with individual informational inputs and the common central element arranged in the input plane P1, (c) two instantaneous spectra arranged mutually orthogonally in the spectral plane P2, and (d) power spectrum of wave signal realized along the line χ in the output spectral plane P2EL after implementing the time electronic integration.

Fig. 6.
Fig. 6.

One-stage ACO spectral-correlation processor with power spectrum realized on a spatial carrier: (a) system, (b) two spatially separated mutually orthogonal 1D arrays of light-modulating elements, (c) power spectrum on a spatial carrier in the output spectral plane P2 after implementing the optical time integration, and (d) orientation of the spatial carrier.

Fig. 7.
Fig. 7.

Two-stage ACO spectral-correlation processor with spatial autocorrelation function: (a) coherent optical scheme, (b) power spectrum on a spatial carrier in the intermediate spectral plane P2 after implementing the time optical integration, (c) spatial autocorrelation function R in output plane P3, and (d) autocorrelation function R determines the transverse profile (along the direction of line χ) of 1D optical distribution, obtained on the line ψ; the maximum of R corresponds to the central axis ψo of the line ψ).

Equations (25)

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

ssin(X,t)=exp[i2πν0(t+sinθ0λ0ν0X)],
ssin(x,t)=exp[i2πν0(t+Msinθ0c0x)]rect(xD),
rect(xD)=1for|x|D;rect(xD)=0for the restx.
Ssin(ξ,t)Fx{ssin(x,t)}=D/2+D/2exp(i2πν0t+Msinθ0λ0x)exp(i2πξx)dx.
sinπDξπDξsinc(Dξ)=DD/2D/2exp(i2πξx)dx,
Ssin(ξ,t)=Dexp(i2πν0t)·sinc[D(ξξ0)],
ξ0=Msinθ0λ0=Msinθ0c0ν0,
x0=λlightfξ0=λlightMsinθ0c0fν0=λlightMsinθ0λ0f.
λν=λ0ν0=c0,
Fx,t{sa(x,t)}=Aθ0,t(ξξo).
sa(x,y,t)=ssin(x,t)·rect(yd)+ssin(y,t)·rect(xd),
ssin(y,t)=exp[i2πν0(t+Msinθ0c0y)]rect(yD).
sinc(dξ)1(ξ),sinc(dη)1(η),
S(ξ0,η0,t)Dexp(i2πν0t)·sinc[D(ξξ0)]·1(η)+Dexp(i2πν0t)·sinc[D(ηη0)]·1(ξ).
S(ξ,η,t)Aθ0,t(ξξo)·1(η)+Aφ0,t(ηηo)·1(ξ).
J(ξ,η,t)=|S(ξ,η,t)|2=|Aθ0,t(ξξo)|21(η)+|Aφ0,t(ηηo)|21(ξ)+Aθ0,t(ξξo)·Aφ0,t*(ηηo)·1(η)·1(ξ)+Aθ0,t*(ξξo)·Aφ0,t(ηηo)·1(η)·1(ξ).
tgαχ=sinθ0sinφ0,
|S(χθ0,φ0,t)|2|At(χθ0,φ0)|2,
|SΔν(χθ0,φ0,t)|2=|At(χ)|2cos[2πΔνt].
|ST(χ)|2=0T|At(χ)|2dt=|AT(χ)|2.
sa(x,y,t)=sa(x,t)rect(x+x0D)rect(yd)+sa(y,t)rect(yD)rect(xx0d).
|STcarrier(χ)|2=|AT(χ)|2cos(4πx0ξ)=|AT(χ)|2cos(4πx0fLλlaserx),
F|ST(χ)|2=Fχ{|AT(χ)|2}·Fψ{rect(ψρ)},
F|ST(χ)|2=R{sa}·sinc(ρψ),
F|ST(χ)|2R{sa}·1{ψ},

Metrics