Abstract

We present a spatio-temporal operator formalism and beam propagation simulations that describe the broadband efficient adaptive method for a true-time-delay array processing (BEAMTAP) algorithm for an optical beamformer by use of a photorefractive crystal. The optical system consists of a tapped-delay line implemented with an acoustooptic Bragg cell, an accumulating scrolling time-delay detector achieved with a traveling-fringes detector, and a photorefractive crystal to store the adaptive spatio-temporal weights as volume holographic gratings. In this analysis, linear shift-invariant integral operators are used to describe the propagation, interference, grating accumulation, and volume holographic diffraction of the spatio-temporally modulated optical fields in the system to compute the adaptive array processing operation. In addition, it is shown that the random fluctuation in time and phase delays of the optically modulated and transmitted array signals produced by fiber perturbations (temperature fluctuations, vibrations, or bending) are dynamically compensated for through the process of holographic wavefront reconstruction as a byproduct of the adaptive beam-forming and jammer-excision operation. The complexity of the cascaded spatial-temporal integrals describing the holographic formation, and subsequent readout processes, is shown to collapse to a simple imaging condition through standard operator manipulation. We also present spatio-temporal beam propagation simulation results as an illustrative demonstration of our analysis and the operation of a BEAMTAP beamformer.

© 2003 Optical Society of America

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  3. D. Psaltis, J. Hong, “Adaptive acoustooptic filter,” Applied Optics 23, 3475–3481 (1984).
    [CrossRef] [PubMed]
  4. S.-C. Lin, J. Hong, R. Boughton, D. Psaltis, “Broadband beamforming via acoustooptics,” 936, Advances in Optical Information Processing IIID. R. Pape, ed. Proc. SPIE, 152–162 (1988).
    [CrossRef]
  5. N. A. Riza, D. Psaltis, “Acoustooptic signal processors for transmission and reception of phased-array antenna signals,” Applied Optics 30, 3294–3303 (1991).
    [CrossRef]
  6. R. M. Iodice, P. Rutterbusch, “Acoustooptic null steering processor (AONSP) hardware performance summary,” Proc. SPIE2489, (Orlando, Fla.), April1995.
  7. R. M. Montgomery, “Acousto-optic/photorefractive processor for adaptive antenna arrays,” in B. M. Hendrickson, G. A. Koepf, eds., 1217, Opto-electronic Signal Processing for Phased-Array Antennas II, Proc. SPIE, 207–217 (1990).
    [CrossRef]
  8. W. Ng, A. A. Walston, G. L. Tangonan, J. J. Lee, I. L. Newberg, N. Bernstein, “The first demonstration of an optically steered microwave phased array antenna using true-time-delay,” J. Lightwave Technol. 9, 1124–1131 (1991).
    [CrossRef]
  9. R. Soref, “Optical dispersion technique for time-delay beam steering,” Applied Optics 31, 7395–7397 (1992).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  12. D. R. Pape, “Multichannel Bragg cells: design, performance, and applications,” Opt. Eng. 31, 2148–2158 (1992).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  16. A. W. Sarto, “Adaptive Phased-Array Radar Signal Processing using Photorefractive Volume Holograms,” Ph.D. dissertation (University of Colorado, Boulder, Colo., 1996).
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    [CrossRef]
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    [CrossRef]
  19. G. Kriehn, A. M. Kiruluta, K. H. Wagner, D. Dolfi, J.-P. Huignard, “Detection of a broadband RF signal using a traveling fringes detector,” in Terahertz and Gigahertz Photonics, Proc. SPIE3795, 94–103 (1999).
    [CrossRef]
  20. A. W. Sarto, K. H. Wagner, R. T. Weverka, S. Weaver, E. K. Walge, “Wide angular aperture holograms in photorefractive crystals by the use of orthogonally polarized write and read beams,” Appl. Opt. 35, 5765–5775 (1996).
    [CrossRef] [PubMed]
  21. J. Shamir, Optical Systems and Processes (SPIE Press, Bellingham, Wash., 1999).
    [CrossRef]
  22. A. A. Zozulya, D. Z. Anderson, “Spatial structure of light and a nonlinear refractive index generated by fanning in photorefractive media,” Phys. Rev. A 52, 878–881 (1995).
    [CrossRef] [PubMed]
  23. D. Z. Anderson, R. Brockett, N. Nuttall, “Information dynamics of photorefractive two-beam coupling,” Phys. Rev. Lett. 82, 1418–1421 (1999).
    [CrossRef]
  24. S. M. Sze, Physics of Semiconductor Devices, 2nd ed. (Wiley, New York, 1981).
  25. G. Kriehn, G. S. Pati, P. E. X. Silveira, F. Schlottau, K. Wagner, D. Dolfi, “Demonstration of optical beam forming using BEAMTAP,” Microwave Photonics MWP-2000, 5–8 (2000).
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    [CrossRef]
  27. L. Thylen, D. Yevick, “Beam propagation method in anisotropic media,” Appl. Opt. 21, 2751–2754 (1982).
    [CrossRef] [PubMed]
  28. K. Wagner, T. M. Slagle, “Optical competitive learning with VLSI/liquid-crystal winner-take-all modulators,” Appl. Opt. 32, 1408–1435 (1993).
    [CrossRef] [PubMed]
  29. L. Thylen, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron., 15, 433–439 (1983).
    [CrossRef]
  30. J. Jarem, P. Banerjee, Computational Methods for Electromagnetic and Optical System (Marcel Dekker, New York, 2000).
  31. K. Wu, “Acoustooptic fiber crossbar switches,” Ph.D. dissertation (University of Colorado, Boulder, Colo., 1995).
  32. S. Blair, “Optical soliton-based logic gates,” Ph.D. dissertation (University of Colorado, Boulder, Colo., 1998).
  33. M. Cronin-Golomb, “Whole beam method for photorefractive nonlinear optics,” Opt. Commun. 89, 276–282 (1992).
    [CrossRef]
  34. P. Asthana, G. P. Tanguay, B. K. Jenkins, “Analysis of weighted fan-out fan-in volume holographic optical interconnections,” Appl. Opt. 32, 1441–1469 (1993).
    [CrossRef] [PubMed]

2002 (2)

P. E. X. Silveira, G. Pati, K. Wagner, “Optical finite impulse response neural networks,” Appl. Opt. 41, 4162–4180 (2002).
[CrossRef] [PubMed]

J. Shamir, K. H. Wagner, “Hologram recording in volume media: A generalized fourier optics analysis,” Appl. Opt.6773–6786 (2002).
[CrossRef]

2000 (2)

G. Kriehn, A. Kiruluta, P. E. X. Silveira, S. Weaver, S. Kraut, K. Wagner, R. T. Weverka, L. Griffiths, “Optical BEAMTAP beam-forming and jammer-nulling system for broadband phased-array antennas,” Appl. Opt. 39, 212–230 (2000).
[CrossRef]

G. Kriehn, G. S. Pati, P. E. X. Silveira, F. Schlottau, K. Wagner, D. Dolfi, “Demonstration of optical beam forming using BEAMTAP,” Microwave Photonics MWP-2000, 5–8 (2000).

1999 (1)

D. Z. Anderson, R. Brockett, N. Nuttall, “Information dynamics of photorefractive two-beam coupling,” Phys. Rev. Lett. 82, 1418–1421 (1999).
[CrossRef]

1996 (2)

1995 (1)

A. A. Zozulya, D. Z. Anderson, “Spatial structure of light and a nonlinear refractive index generated by fanning in photorefractive media,” Phys. Rev. A 52, 878–881 (1995).
[CrossRef] [PubMed]

1993 (3)

1992 (3)

M. Cronin-Golomb, “Whole beam method for photorefractive nonlinear optics,” Opt. Commun. 89, 276–282 (1992).
[CrossRef]

D. R. Pape, “Multichannel Bragg cells: design, performance, and applications,” Opt. Eng. 31, 2148–2158 (1992).
[CrossRef]

R. Soref, “Optical dispersion technique for time-delay beam steering,” Applied Optics 31, 7395–7397 (1992).
[CrossRef] [PubMed]

1991 (2)

N. A. Riza, D. Psaltis, “Acoustooptic signal processors for transmission and reception of phased-array antenna signals,” Applied Optics 30, 3294–3303 (1991).
[CrossRef]

W. Ng, A. A. Walston, G. L. Tangonan, J. J. Lee, I. L. Newberg, N. Bernstein, “The first demonstration of an optically steered microwave phased array antenna using true-time-delay,” J. Lightwave Technol. 9, 1124–1131 (1991).
[CrossRef]

1984 (1)

D. Psaltis, J. Hong, “Adaptive acoustooptic filter,” Applied Optics 23, 3475–3481 (1984).
[CrossRef] [PubMed]

1983 (1)

L. Thylen, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron., 15, 433–439 (1983).
[CrossRef]

1982 (1)

1981 (1)

1967 (1)

B. Widrow, P. Mantey, L. Griffiths, B. Goode, “Adaptive antenna systems,” Proc. IEEE 55, 2143–2161 (1967).
[CrossRef]

Anderson, D. Z.

D. Z. Anderson, R. Brockett, N. Nuttall, “Information dynamics of photorefractive two-beam coupling,” Phys. Rev. Lett. 82, 1418–1421 (1999).
[CrossRef]

A. A. Zozulya, D. Z. Anderson, “Spatial structure of light and a nonlinear refractive index generated by fanning in photorefractive media,” Phys. Rev. A 52, 878–881 (1995).
[CrossRef] [PubMed]

Asthana, P.

Banerjee, P.

J. Jarem, P. Banerjee, Computational Methods for Electromagnetic and Optical System (Marcel Dekker, New York, 2000).

Bernstein, N.

W. Ng, A. A. Walston, G. L. Tangonan, J. J. Lee, I. L. Newberg, N. Bernstein, “The first demonstration of an optically steered microwave phased array antenna using true-time-delay,” J. Lightwave Technol. 9, 1124–1131 (1991).
[CrossRef]

Blair, S.

S. Blair, “Optical soliton-based logic gates,” Ph.D. dissertation (University of Colorado, Boulder, Colo., 1998).

Boughton, R.

S.-C. Lin, J. Hong, R. Boughton, D. Psaltis, “Broadband beamforming via acoustooptics,” 936, Advances in Optical Information Processing IIID. R. Pape, ed. Proc. SPIE, 152–162 (1988).
[CrossRef]

Brock, J. C.

L. J. Lembo, T. Holcomb, M. Wickham, P. Wisseman, J. C. Brock, “Low-loss fiber optic time-delay element for phased-array antennas,” in Proc. SPIE, B. M. Hendrickson, ed., 2155Optoelectronic Signal Processing for Phased-Array Antennas IV, 13–23 (1994).
[CrossRef]

Brockett, R.

D. Z. Anderson, R. Brockett, N. Nuttall, “Information dynamics of photorefractive two-beam coupling,” Phys. Rev. Lett. 82, 1418–1421 (1999).
[CrossRef]

Compton, R.

R. Compton, Adaptive Antennas (Prentice-Hall, Englewood Cliffs, N.J., 1988).

Cooper, D. G.

R. D. Esman, M. Y. Frankel, J. L. Dexter, L. Goldberg, M. G. Parent, D. Stilwell, D. G. Cooper, “Fiber-optic prism true time-delay antenna feed,” IEEE Photon. Techn. Lett. 5, 1347–1349 (1993).
[CrossRef]

Cronin-Golomb, M.

M. Cronin-Golomb, “Whole beam method for photorefractive nonlinear optics,” Opt. Commun. 89, 276–282 (1992).
[CrossRef]

Dexter, J. L.

R. D. Esman, M. Y. Frankel, J. L. Dexter, L. Goldberg, M. G. Parent, D. Stilwell, D. G. Cooper, “Fiber-optic prism true time-delay antenna feed,” IEEE Photon. Techn. Lett. 5, 1347–1349 (1993).
[CrossRef]

Dolfi, D.

G. Kriehn, G. S. Pati, P. E. X. Silveira, F. Schlottau, K. Wagner, D. Dolfi, “Demonstration of optical beam forming using BEAMTAP,” Microwave Photonics MWP-2000, 5–8 (2000).

T. Merlet, D. Dolfi, J.-P. Huignard, “A traveling fringes photodetector for microwave signals,” IEEE J. Quantum Electron. 32, 778–783 (1996).
[CrossRef]

G. Kriehn, A. M. Kiruluta, K. H. Wagner, D. Dolfi, J.-P. Huignard, “Detection of a broadband RF signal using a traveling fringes detector,” in Terahertz and Gigahertz Photonics, Proc. SPIE3795, 94–103 (1999).
[CrossRef]

Esman, R. D.

R. D. Esman, M. Y. Frankel, J. L. Dexter, L. Goldberg, M. G. Parent, D. Stilwell, D. G. Cooper, “Fiber-optic prism true time-delay antenna feed,” IEEE Photon. Techn. Lett. 5, 1347–1349 (1993).
[CrossRef]

Frankel, M. Y.

R. D. Esman, M. Y. Frankel, J. L. Dexter, L. Goldberg, M. G. Parent, D. Stilwell, D. G. Cooper, “Fiber-optic prism true time-delay antenna feed,” IEEE Photon. Techn. Lett. 5, 1347–1349 (1993).
[CrossRef]

Goldberg, L.

R. D. Esman, M. Y. Frankel, J. L. Dexter, L. Goldberg, M. G. Parent, D. Stilwell, D. G. Cooper, “Fiber-optic prism true time-delay antenna feed,” IEEE Photon. Techn. Lett. 5, 1347–1349 (1993).
[CrossRef]

Goode, B.

B. Widrow, P. Mantey, L. Griffiths, B. Goode, “Adaptive antenna systems,” Proc. IEEE 55, 2143–2161 (1967).
[CrossRef]

Griffiths, L.

Holcomb, T.

L. J. Lembo, T. Holcomb, M. Wickham, P. Wisseman, J. C. Brock, “Low-loss fiber optic time-delay element for phased-array antennas,” in Proc. SPIE, B. M. Hendrickson, ed., 2155Optoelectronic Signal Processing for Phased-Array Antennas IV, 13–23 (1994).
[CrossRef]

Hong, J.

D. Psaltis, J. Hong, “Adaptive acoustooptic filter,” Applied Optics 23, 3475–3481 (1984).
[CrossRef] [PubMed]

S.-C. Lin, J. Hong, R. Boughton, D. Psaltis, “Broadband beamforming via acoustooptics,” 936, Advances in Optical Information Processing IIID. R. Pape, ed. Proc. SPIE, 152–162 (1988).
[CrossRef]

Huignard, J.-P.

T. Merlet, D. Dolfi, J.-P. Huignard, “A traveling fringes photodetector for microwave signals,” IEEE J. Quantum Electron. 32, 778–783 (1996).
[CrossRef]

G. Kriehn, A. M. Kiruluta, K. H. Wagner, D. Dolfi, J.-P. Huignard, “Detection of a broadband RF signal using a traveling fringes detector,” in Terahertz and Gigahertz Photonics, Proc. SPIE3795, 94–103 (1999).
[CrossRef]

Iodice, R. M.

R. M. Iodice, P. Rutterbusch, “Acoustooptic null steering processor (AONSP) hardware performance summary,” Proc. SPIE2489, (Orlando, Fla.), April1995.

Jarem, J.

J. Jarem, P. Banerjee, Computational Methods for Electromagnetic and Optical System (Marcel Dekker, New York, 2000).

Jenkins, B. K.

Kiruluta, A.

Kiruluta, A. M.

G. Kriehn, A. M. Kiruluta, K. H. Wagner, D. Dolfi, J.-P. Huignard, “Detection of a broadband RF signal using a traveling fringes detector,” in Terahertz and Gigahertz Photonics, Proc. SPIE3795, 94–103 (1999).
[CrossRef]

Kraut, S.

Kriehn, G.

G. Kriehn, A. Kiruluta, P. E. X. Silveira, S. Weaver, S. Kraut, K. Wagner, R. T. Weverka, L. Griffiths, “Optical BEAMTAP beam-forming and jammer-nulling system for broadband phased-array antennas,” Appl. Opt. 39, 212–230 (2000).
[CrossRef]

G. Kriehn, G. S. Pati, P. E. X. Silveira, F. Schlottau, K. Wagner, D. Dolfi, “Demonstration of optical beam forming using BEAMTAP,” Microwave Photonics MWP-2000, 5–8 (2000).

G. Kriehn, A. M. Kiruluta, K. H. Wagner, D. Dolfi, J.-P. Huignard, “Detection of a broadband RF signal using a traveling fringes detector,” in Terahertz and Gigahertz Photonics, Proc. SPIE3795, 94–103 (1999).
[CrossRef]

Lagasse, P. E.

Lee, J. J.

W. Ng, A. A. Walston, G. L. Tangonan, J. J. Lee, I. L. Newberg, N. Bernstein, “The first demonstration of an optically steered microwave phased array antenna using true-time-delay,” J. Lightwave Technol. 9, 1124–1131 (1991).
[CrossRef]

Lembo, L. J.

L. J. Lembo, T. Holcomb, M. Wickham, P. Wisseman, J. C. Brock, “Low-loss fiber optic time-delay element for phased-array antennas,” in Proc. SPIE, B. M. Hendrickson, ed., 2155Optoelectronic Signal Processing for Phased-Array Antennas IV, 13–23 (1994).
[CrossRef]

Lin, S.-C.

S.-C. Lin, J. Hong, R. Boughton, D. Psaltis, “Broadband beamforming via acoustooptics,” 936, Advances in Optical Information Processing IIID. R. Pape, ed. Proc. SPIE, 152–162 (1988).
[CrossRef]

Mantey, P.

B. Widrow, P. Mantey, L. Griffiths, B. Goode, “Adaptive antenna systems,” Proc. IEEE 55, 2143–2161 (1967).
[CrossRef]

Merlet, T.

T. Merlet, D. Dolfi, J.-P. Huignard, “A traveling fringes photodetector for microwave signals,” IEEE J. Quantum Electron. 32, 778–783 (1996).
[CrossRef]

Montgomery, R. M.

R. M. Montgomery, “Acousto-optic/photorefractive processor for adaptive antenna arrays,” in B. M. Hendrickson, G. A. Koepf, eds., 1217, Opto-electronic Signal Processing for Phased-Array Antennas II, Proc. SPIE, 207–217 (1990).
[CrossRef]

Newberg, I. L.

W. Ng, A. A. Walston, G. L. Tangonan, J. J. Lee, I. L. Newberg, N. Bernstein, “The first demonstration of an optically steered microwave phased array antenna using true-time-delay,” J. Lightwave Technol. 9, 1124–1131 (1991).
[CrossRef]

Ng, W.

W. Ng, A. A. Walston, G. L. Tangonan, J. J. Lee, I. L. Newberg, N. Bernstein, “The first demonstration of an optically steered microwave phased array antenna using true-time-delay,” J. Lightwave Technol. 9, 1124–1131 (1991).
[CrossRef]

Nuttall, N.

D. Z. Anderson, R. Brockett, N. Nuttall, “Information dynamics of photorefractive two-beam coupling,” Phys. Rev. Lett. 82, 1418–1421 (1999).
[CrossRef]

Pape, D. R.

D. R. Pape, “Multichannel Bragg cells: design, performance, and applications,” Opt. Eng. 31, 2148–2158 (1992).
[CrossRef]

Parent, M. G.

R. D. Esman, M. Y. Frankel, J. L. Dexter, L. Goldberg, M. G. Parent, D. Stilwell, D. G. Cooper, “Fiber-optic prism true time-delay antenna feed,” IEEE Photon. Techn. Lett. 5, 1347–1349 (1993).
[CrossRef]

Pati, G.

Pati, G. S.

G. Kriehn, G. S. Pati, P. E. X. Silveira, F. Schlottau, K. Wagner, D. Dolfi, “Demonstration of optical beam forming using BEAMTAP,” Microwave Photonics MWP-2000, 5–8 (2000).

Psaltis, D.

N. A. Riza, D. Psaltis, “Acoustooptic signal processors for transmission and reception of phased-array antenna signals,” Applied Optics 30, 3294–3303 (1991).
[CrossRef]

D. Psaltis, J. Hong, “Adaptive acoustooptic filter,” Applied Optics 23, 3475–3481 (1984).
[CrossRef] [PubMed]

S.-C. Lin, J. Hong, R. Boughton, D. Psaltis, “Broadband beamforming via acoustooptics,” 936, Advances in Optical Information Processing IIID. R. Pape, ed. Proc. SPIE, 152–162 (1988).
[CrossRef]

Riza, N. A.

N. A. Riza, D. Psaltis, “Acoustooptic signal processors for transmission and reception of phased-array antenna signals,” Applied Optics 30, 3294–3303 (1991).
[CrossRef]

Roey, J. V.

Rutterbusch, P.

R. M. Iodice, P. Rutterbusch, “Acoustooptic null steering processor (AONSP) hardware performance summary,” Proc. SPIE2489, (Orlando, Fla.), April1995.

Sarto, A. W.

A. W. Sarto, K. H. Wagner, R. T. Weverka, S. Weaver, E. K. Walge, “Wide angular aperture holograms in photorefractive crystals by the use of orthogonally polarized write and read beams,” Appl. Opt. 35, 5765–5775 (1996).
[CrossRef] [PubMed]

A. W. Sarto, “Adaptive Phased-Array Radar Signal Processing using Photorefractive Volume Holograms,” Ph.D. dissertation (University of Colorado, Boulder, Colo., 1996).

Schlottau, F.

G. Kriehn, G. S. Pati, P. E. X. Silveira, F. Schlottau, K. Wagner, D. Dolfi, “Demonstration of optical beam forming using BEAMTAP,” Microwave Photonics MWP-2000, 5–8 (2000).

Shamir, J.

J. Shamir, K. H. Wagner, “Hologram recording in volume media: A generalized fourier optics analysis,” Appl. Opt.6773–6786 (2002).
[CrossRef]

J. Shamir, Optical Systems and Processes (SPIE Press, Bellingham, Wash., 1999).
[CrossRef]

Silveira, P. E. X.

Slagle, T. M.

Soref, R.

R. Soref, “Optical dispersion technique for time-delay beam steering,” Applied Optics 31, 7395–7397 (1992).
[CrossRef] [PubMed]

Stearns, S.

B. Widrow, S. Stearns, Adaptive Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1985).

Stilwell, D.

R. D. Esman, M. Y. Frankel, J. L. Dexter, L. Goldberg, M. G. Parent, D. Stilwell, D. G. Cooper, “Fiber-optic prism true time-delay antenna feed,” IEEE Photon. Techn. Lett. 5, 1347–1349 (1993).
[CrossRef]

Sze, S. M.

S. M. Sze, Physics of Semiconductor Devices, 2nd ed. (Wiley, New York, 1981).

Tangonan, G. L.

W. Ng, A. A. Walston, G. L. Tangonan, J. J. Lee, I. L. Newberg, N. Bernstein, “The first demonstration of an optically steered microwave phased array antenna using true-time-delay,” J. Lightwave Technol. 9, 1124–1131 (1991).
[CrossRef]

Tanguay, G. P.

Thylen, L.

L. Thylen, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron., 15, 433–439 (1983).
[CrossRef]

L. Thylen, D. Yevick, “Beam propagation method in anisotropic media,” Appl. Opt. 21, 2751–2754 (1982).
[CrossRef] [PubMed]

van der Donk, J.

Wagner, K.

Wagner, K. H.

J. Shamir, K. H. Wagner, “Hologram recording in volume media: A generalized fourier optics analysis,” Appl. Opt.6773–6786 (2002).
[CrossRef]

A. W. Sarto, K. H. Wagner, R. T. Weverka, S. Weaver, E. K. Walge, “Wide angular aperture holograms in photorefractive crystals by the use of orthogonally polarized write and read beams,” Appl. Opt. 35, 5765–5775 (1996).
[CrossRef] [PubMed]

G. Kriehn, A. M. Kiruluta, K. H. Wagner, D. Dolfi, J.-P. Huignard, “Detection of a broadband RF signal using a traveling fringes detector,” in Terahertz and Gigahertz Photonics, Proc. SPIE3795, 94–103 (1999).
[CrossRef]

Walge, E. K.

Walston, A. A.

W. Ng, A. A. Walston, G. L. Tangonan, J. J. Lee, I. L. Newberg, N. Bernstein, “The first demonstration of an optically steered microwave phased array antenna using true-time-delay,” J. Lightwave Technol. 9, 1124–1131 (1991).
[CrossRef]

Weaver, S.

Weverka, R. T.

Wickham, M.

L. J. Lembo, T. Holcomb, M. Wickham, P. Wisseman, J. C. Brock, “Low-loss fiber optic time-delay element for phased-array antennas,” in Proc. SPIE, B. M. Hendrickson, ed., 2155Optoelectronic Signal Processing for Phased-Array Antennas IV, 13–23 (1994).
[CrossRef]

Widrow, B.

B. Widrow, P. Mantey, L. Griffiths, B. Goode, “Adaptive antenna systems,” Proc. IEEE 55, 2143–2161 (1967).
[CrossRef]

B. Widrow, S. Stearns, Adaptive Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1985).

Wisseman, P.

L. J. Lembo, T. Holcomb, M. Wickham, P. Wisseman, J. C. Brock, “Low-loss fiber optic time-delay element for phased-array antennas,” in Proc. SPIE, B. M. Hendrickson, ed., 2155Optoelectronic Signal Processing for Phased-Array Antennas IV, 13–23 (1994).
[CrossRef]

Wu, K.

K. Wu, “Acoustooptic fiber crossbar switches,” Ph.D. dissertation (University of Colorado, Boulder, Colo., 1995).

Yevick, D.

Zozulya, A. A.

A. A. Zozulya, D. Z. Anderson, “Spatial structure of light and a nonlinear refractive index generated by fanning in photorefractive media,” Phys. Rev. A 52, 878–881 (1995).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Optical architecture for implementation of the BEAMTAP algorithm with the PRC near an image plane of both the AOD and array fiber feed output. The system uses a photorefractive crystal to store the adaptive weights as index gratings that are formed as interference between the incident phased-array signals and the feedback-error signal calculated from the difference between the output and the reference signal.

Fig. 2
Fig. 2

Schematic diagram of the generalized BEAMTAP processor with appropriate coordinate systems, showing the transformation of the phased array and AOD fields onto the photorefractive crystal and subsequently to the TFD with an imaging lens system comprised of a combination of lens system 1 and 2.

Fig. 3
Fig. 3

Depicts how an electro-magnetic field incident on each antenna element is transformed into a corresponding set of nonoverlapping optical beams at the output of the remoting fiber bundle with the appropriate delays for each beam corresponding to the respective time-of-arrival. In this case, a broadband rf signal is incident from one angle and a jammer j from another angle onto a phased-array antenna with element spacing d, creating time delays t p and t j between the elements of the array for the signal and jammer, respectively. Hence the topology of the antenna array is permuted onto the fiber feed.

Fig. 4
Fig. 4

Simulated BEAMTAP phased-array optical processor used to illustrate the key points of the operator analysis. A simple 4f imaging system formed by lens 1 (focal length f 1 = 30 cm) and lens 2 (focal length f 2 = 20 cm) telescopically images the AOD onto the TFD through the photorefractive crystal. The rf antenna element signals from an equi-spaced linear array are single-sideband, suppressed-carrier, optically modulated onto an array of optical signals that are focussed into the PRC by lens 1 where they interfere with the AOD diffracted order and build up a grating. The writing beams are blocked (by a polarizer) at the output PRC face. The array signals diffract off the grating with a switched polarization, and are reimaged by lens 2 onto the TFD where they interfere with the DC beam from the AOD which produced constant velocity moving fringes that are resonantly accumulated by the drifting electrons in the TFD.

Fig. 5
Fig. 5

Simulation results showing spatio-temporal correlation gratings formed in the PR crystal for different AOA of the chirp signal, and the corresponding instantaneous readout of the correlation grating by the optical field distribution from the phased array. In (a), (f) and (k), tilted correlation gratings are shown corresponding to rf AOAs of -45°, 0° and +45° respectively with the grating in red (Δn > 0) and blue (Δn < 0). The tilt of the grating corresponds directly to its respective AOA. For each AOA, insets (b)-(e), (g)-(j), (l)-(o) show subsequent readouts of the correlation gratings at times t = 0 ns, 4 ns, 8 ns, and 12 ns by the focused beam from the array modulators (shown in green), and shows the beams diffracted by the grating at each of these times.

Fig. 6
Fig. 6

Simulated output of the TFD during holographic adaptive beamforming for the array processor for a repetitive chirp waveform. The interferometric fringe patterns illuminating the detector are shown in part (a) as a function of detector position and time, and the accumulating drifting electron density in the TFD is shown in (b) vs. detector position and time, while the temporal output of the detector produced at the collecting electrode is illustrated in (c).

Equations (30)

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ot=n=1N-TDv0- sntδt-t-xv×-t-xv sn*t2-Tft2+xv-Tdt2dtdx=n=1N-TDv0 snt-τ×-t-T sn*t2-τdt2-ot2dτdt2,Wnm*x
Eax0, z=zr; t=A0b02 exp-ikBx0-ωlt×1+ηAOf˜t-T+x0vA+cc.
Eax, z; t=D-zx:x11x1:x0Eax0; t.
Dzx:x1Ex1, z=0; t =- xEx1, z=0; t×expiωt-kxx+kzkxzdkx, expi2πz/λiλz- Ex1, z=0; t×expiπλzx1-x2dx1,
kzkx=ω2c2 ne2-ne2no2 kx2,
Dzx1:xD-zx:x1Fx1Fx1=Fx1
Dzx1:x*=D-zx:x1,
2x2:x11x1:x0Mx2:x0expibM2x22,
Utτz0:nSz0:nτnt:τexpiϕnP¯¯n:n
kxopt=kxrfDd.
θopt=sin-1λlλrfDdsin θrf.
Utτz0:nUtτz0:n,
Epz0; t=Utτz0:nCn:ξErfξ; τ.
Epx, z; t=Dxz:z13z1:z0Utτz0:nCn:ξErfξ; τ=Dxz:z13z1:z0Epz0; t,
gx, z; t=-t Ep*x, z; tEax, z; t×exp-t-tτprdt+c.c.,=-tDxz:z13z1:z0Epz0; t* D-zx:x1×1x1:x0Eax0; texp-t-tτprdt+c.c.=TτprttEp*x, z; tEax, z; t,
exp-ikx-iλx  3*z1:z0Ep*z0; texp-iπλxz-z12dz1 ×expikz-iλz  1x1:x0Eax0; texp-iπλzx-x12dx1=exp-ikx-iλxexpikziλz  3*z1:z0Ep*z0; texp-iπλxz-z12×1x1:x0Eax0; texp-iπλzx-x12dx1dz1=Dx*z:z13*z1:z0D-zx:x11x1:x0Ep*z0; t Eax0; t
gx, z; t=Dx*z:z13*z1:z0D-zx:x11x1:x0×-t Ep*z0; tEax0; t×exp-t-tτprdt+c.c.=Dx*z:z13*z1:z0D-zx:x11x1:x0×TτprttEp*z0; tEax0; t+c.c.
Edx2; t=2x2:x10L Dzx1:xDxz:z13z1:z0×Epz0; tgx, z; tdz=2x2:x10L Dzx1:xDxz:z13z1:z0×Epz0; tDx*z:z13*z1:z0D-zx:x1×1x1:x0TτprttEp*z0; tEax0; tdz.
Dxz:z13z1:z0Epz0; tDx*z:z13*z1:z0Ep*z0; t =Dxz:z13z1:z0Dx*z:z13*z1:z0Epz0; tEp*z0; t.
Edx2; t=0L 2x2:x1Dzx1:xDxz:z13z1:z0Dx*z:z1×3*z1:z0D-zx:x11x1:x0×Epz0; t-t Ep*z0; tEax0; t×expt-tτprdtdz,
Dxz:z13z1:z0Dxz:z13z1:z0*=Pz:z0P*z:z01z0,
2x2:x1Dzx1:x×Dxz:z13z1:z0Dx*z:z13*z1:z01z0 D-zx:x11x1:x0=2x2:x1Dzx1:xD-zx:x11x1:x0=21x2:x0Mx2:x0expibM2x22.
Edx2; t=0L 2x2:x11x1:x0TτprttEpz0; tEp*z0; tEax0; tdz0=Mx2:x0expibM2x220LTτprttEpz0; tEp*z0; tEax0; tdz0.
Ix2; t=|Edx2; t+Ar expibM2x22expikBMx2|2=Ir+AMx2:x00L TτprttEpz0; t×Ep*z0; tEax0; t2+2A*r AMx2:x0×0L TτprttEpz0; tEp*z0; t×Eax0; t+c.c.,
ot= -TDv0-t Ix2; t14πDt-t×exp-x2-vt-t24Dt-t-t-tτdtdx2,
ot -TDv0 Ix2; t-x2vdx2.
ot=A1-TDv00LEpz0; t--x2v×Ep*z0; tEax0; t×exp-t-tτprdtdz0dx2,
Epz0; t=A0arNn=1Nexpiωlt-tn×gz0-nD1+ηps˜n*t-tn+ηps˜n*t-tn,
ot=A1-TDv0n=1N s˜nt-tn-x2v×-t-x2v f˜t-T+x2vs˜n*t-tndtdx2=A1-TDv0n=1N s˜nt-tn-x2v×-t f˜t1-Ts˜n*t1-tn-x2vdt1dx2.
ot=A1n=1N-TDv0 s˜nt-tn-x2v-t-T×d˜t1-õt1s˜n*t1+T-tn-x2vdt1dx2.

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