Abstract

A compact phased-array antenna acousto-optic beam former with element-level analog phase (0–2π) and amplitude control using nematic-liquid-crystal display-type technology is experimentally demonstrated. Measurements indicate >6-bit phase control and 52.6 dB of amplitude-attenuation control. High-quality error calibration and antenna sidelobe-levelcontrol is possible with this low-control-power analog beam former. Optical system options using rf Bragg cells or wideband Bragg cells are discussed, with the rf design being the current preferred approach. Transmit–receive beam forming based on frequency upconversion–downconversion by electronic mixing is introduced for the rf Bragg-cell beam former, and comparisons with digital beam forming are highlighted. A millimeter-wave signal generation and control optical architecture is described.

© 1994 Optical Society of America

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References

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  1. See B. M. Hendrickson, S. Yao, eds., Optoelectronic Signal Processing for Phased-Array Antennas III, Proc. Soc. Photo-Opt. Instrum. Eng.1073 (1992).
  2. See Digest of the DARPA Symposium on Photonic Systems for Antenna Applications, M. L. VanBlaricum, ed. (Advanced Research Projects Agency, Arlington, Va., 1993), Vol. 3.
  3. N. A. Riza, “A compact high performance optical control system for phased array radars,” IEEE Photon. Tech. Lett. 4, 1073–1076 (1992).
    [CrossRef]
  4. N. A. Riza, “Liquid crystal-based optical control of phased array antennas,” J. Lightwave Technol. 10, 1974–1984 (1992).
    [CrossRef]
  5. N. A. Riza, “Phased array antenna control using liquid crystals,” in Analog Photonics, A. R. Pirich, P. Sierak, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1790, 65–75 (1993).
  6. N. A. Riza, “Optical control system for millimeter wave phased array antennas,” in Photonics for Processors, Neural Networks, and Memories, B. Javidi, J. L. Horner, W. J. Micelli, S. T. Kowel, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 344–351 (1993).
  7. N. A. Riza, “A compact electro-optic controller for microwave phased array antennas,” in Photonics for Processors, Neural Networks, and Memories, B. Javidi, J. L. Horner, W. J. Micelli, S. T. Kowel, eds., Proc. Soc. Soc. Photo-Opt. Instrum. Eng.2026, 286–296 (1993).
  8. N. A. Riza, “Liquid crystal-based optical time delay control system for wideband phased arrays,” in Analog Photonics, A. R. Pirich, P. Sierak, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1790, 171–183 (1993).
  9. N. A. Riza, “Three-dimensional optical time delay units for radar,” in Photonics for Processors, Neural Networks, and Memories, B. Javidi, J. L. Horner, W. J. Micelli, S. T. Kowel, eds., Proc. Soc. Opt. Instrum. Eng.2026, 227–237 (1993).
  10. F. G. Stremler, Introduction to Communication Systems, 2nd ed. (Addison-Wesley, Reading, Mass., 1982), pp. 241–244.
  11. G. J. Simonis, K. G. Purchase, “Optical generation, distribution, and control of microwaves using laser heterodyne,” IEEE Trans. Microwave Theory Tech. 38, 667–669 (1990).
    [CrossRef]
  12. T. Day, E. K. Gustafson, R. L. Byer, “Active frequency stabilization of a 1.062 μm, Nd:GGG, diode-laser-pumped nonplanar ring oscillator to less than 3 Hz of relative linewidth,” Opt. Lett. 15 (1990).
    [CrossRef] [PubMed]
  13. T. Day, E. K. Gustafson, R. L. Byer, “Subhertz relative frequency stabilization of two-diode-laser pumped Nd:YAG lasers locked to a Fabry-Perot interferometer,” IEEE J. Quantum Electron. 28, 1106–1117 (1992).
    [CrossRef]
  14. N. Uehara, K. Ueda, “193-mHz beat linewidth of frequency-stabilized laser-diode pumped Nd:YAG ring lasers,” Opt. Lett. 18 (1993).
    [CrossRef] [PubMed]
  15. T. Olson, T. J. Kane, F. J. Adams, “Single frequency diode-pumped Nd:YAG lasers used for microwave synthesis,” in Digest of the DARPA Symposium on Photonic Systems for Antenna Applications, M. L. VanBlaricum, ed. (Advanced Research Projects Agency, Arlington, Va., 1993), Vol. 3.
  16. U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18-GHz microwave carriers,” IEEE Photon. Tech. Lett. 4, 936–938 (1992).
    [CrossRef]
  17. “Liquid crystal optical controllers for phased array antennas,” in Optoelectronic Signal Processing for Phased Array Antennas IV, B. M. Hendrickson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2055 (to be published).
  18. Acousto-Optics Product Catalog (Brimrose Corporation of America, Baltimore, Md., 1993).
  19. Hamamatsu Corporation, Japan, silicon APD’s (e.g., Model S2381), 1993.
  20. 1993 Product Guide (Model QDEMW1 PIN detector) (Lasertron, Burlington, Mass., 1993).
  21. Radio Frequency/Intermediate Frequency Signal Processing Handbook (Mini-Circuits, Brooklyn, N.Y., 1993), Vols. 1 and 2.
  22. Filter Catalog (Tokyo America, Inc., Mt. Prospect, III., 1993).
  23. Lexel and Spectra Physics, air-cooled argon lasers.
  24. Specifications Sheet for 670-nm Laser Diode (Spectra Diode Labs, San Jose, Calif., 1993).
  25. Series 122 and 140 Diode-Pumped Ring Laser Specifications Sheets (Lightwave Electronics, Mountain View, Calif., 1993).
  26. Product Catalog, 1993 (SDL-3400 series linear arrays) (Spectra Diode Labs, San Jose, Calif., 1993).
  27. H. Steyskal, “Digital beamforming antennas: an introduction,” Microwave J. (1987), pp. 107–124.
  28. R. J. Mailloux, “Microwave and mm-wave array antennas,” Microwave J. (1990), pp. 21–32.

1993 (1)

N. Uehara, K. Ueda, “193-mHz beat linewidth of frequency-stabilized laser-diode pumped Nd:YAG ring lasers,” Opt. Lett. 18 (1993).
[CrossRef] [PubMed]

1992 (4)

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18-GHz microwave carriers,” IEEE Photon. Tech. Lett. 4, 936–938 (1992).
[CrossRef]

T. Day, E. K. Gustafson, R. L. Byer, “Subhertz relative frequency stabilization of two-diode-laser pumped Nd:YAG lasers locked to a Fabry-Perot interferometer,” IEEE J. Quantum Electron. 28, 1106–1117 (1992).
[CrossRef]

N. A. Riza, “A compact high performance optical control system for phased array radars,” IEEE Photon. Tech. Lett. 4, 1073–1076 (1992).
[CrossRef]

N. A. Riza, “Liquid crystal-based optical control of phased array antennas,” J. Lightwave Technol. 10, 1974–1984 (1992).
[CrossRef]

1990 (3)

G. J. Simonis, K. G. Purchase, “Optical generation, distribution, and control of microwaves using laser heterodyne,” IEEE Trans. Microwave Theory Tech. 38, 667–669 (1990).
[CrossRef]

T. Day, E. K. Gustafson, R. L. Byer, “Active frequency stabilization of a 1.062 μm, Nd:GGG, diode-laser-pumped nonplanar ring oscillator to less than 3 Hz of relative linewidth,” Opt. Lett. 15 (1990).
[CrossRef] [PubMed]

R. J. Mailloux, “Microwave and mm-wave array antennas,” Microwave J. (1990), pp. 21–32.

1987 (1)

H. Steyskal, “Digital beamforming antennas: an introduction,” Microwave J. (1987), pp. 107–124.

Adams, F. J.

T. Olson, T. J. Kane, F. J. Adams, “Single frequency diode-pumped Nd:YAG lasers used for microwave synthesis,” in Digest of the DARPA Symposium on Photonic Systems for Antenna Applications, M. L. VanBlaricum, ed. (Advanced Research Projects Agency, Arlington, Va., 1993), Vol. 3.

Broberg, B.

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18-GHz microwave carriers,” IEEE Photon. Tech. Lett. 4, 936–938 (1992).
[CrossRef]

Bruun, M.

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18-GHz microwave carriers,” IEEE Photon. Tech. Lett. 4, 936–938 (1992).
[CrossRef]

Byer, R. L.

T. Day, E. K. Gustafson, R. L. Byer, “Subhertz relative frequency stabilization of two-diode-laser pumped Nd:YAG lasers locked to a Fabry-Perot interferometer,” IEEE J. Quantum Electron. 28, 1106–1117 (1992).
[CrossRef]

T. Day, E. K. Gustafson, R. L. Byer, “Active frequency stabilization of a 1.062 μm, Nd:GGG, diode-laser-pumped nonplanar ring oscillator to less than 3 Hz of relative linewidth,” Opt. Lett. 15 (1990).
[CrossRef] [PubMed]

Christensen, E. L.

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18-GHz microwave carriers,” IEEE Photon. Tech. Lett. 4, 936–938 (1992).
[CrossRef]

Day, T.

T. Day, E. K. Gustafson, R. L. Byer, “Subhertz relative frequency stabilization of two-diode-laser pumped Nd:YAG lasers locked to a Fabry-Perot interferometer,” IEEE J. Quantum Electron. 28, 1106–1117 (1992).
[CrossRef]

T. Day, E. K. Gustafson, R. L. Byer, “Active frequency stabilization of a 1.062 μm, Nd:GGG, diode-laser-pumped nonplanar ring oscillator to less than 3 Hz of relative linewidth,” Opt. Lett. 15 (1990).
[CrossRef] [PubMed]

Gliese, U.

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18-GHz microwave carriers,” IEEE Photon. Tech. Lett. 4, 936–938 (1992).
[CrossRef]

Gustafson, E. K.

T. Day, E. K. Gustafson, R. L. Byer, “Subhertz relative frequency stabilization of two-diode-laser pumped Nd:YAG lasers locked to a Fabry-Perot interferometer,” IEEE J. Quantum Electron. 28, 1106–1117 (1992).
[CrossRef]

T. Day, E. K. Gustafson, R. L. Byer, “Active frequency stabilization of a 1.062 μm, Nd:GGG, diode-laser-pumped nonplanar ring oscillator to less than 3 Hz of relative linewidth,” Opt. Lett. 15 (1990).
[CrossRef] [PubMed]

Kane, T. J.

T. Olson, T. J. Kane, F. J. Adams, “Single frequency diode-pumped Nd:YAG lasers used for microwave synthesis,” in Digest of the DARPA Symposium on Photonic Systems for Antenna Applications, M. L. VanBlaricum, ed. (Advanced Research Projects Agency, Arlington, Va., 1993), Vol. 3.

Lindgren, S.

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18-GHz microwave carriers,” IEEE Photon. Tech. Lett. 4, 936–938 (1992).
[CrossRef]

Mailloux, R. J.

R. J. Mailloux, “Microwave and mm-wave array antennas,” Microwave J. (1990), pp. 21–32.

Nielsen, T. N.

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18-GHz microwave carriers,” IEEE Photon. Tech. Lett. 4, 936–938 (1992).
[CrossRef]

Olson, T.

T. Olson, T. J. Kane, F. J. Adams, “Single frequency diode-pumped Nd:YAG lasers used for microwave synthesis,” in Digest of the DARPA Symposium on Photonic Systems for Antenna Applications, M. L. VanBlaricum, ed. (Advanced Research Projects Agency, Arlington, Va., 1993), Vol. 3.

Purchase, K. G.

G. J. Simonis, K. G. Purchase, “Optical generation, distribution, and control of microwaves using laser heterodyne,” IEEE Trans. Microwave Theory Tech. 38, 667–669 (1990).
[CrossRef]

Riza, N. A.

N. A. Riza, “A compact high performance optical control system for phased array radars,” IEEE Photon. Tech. Lett. 4, 1073–1076 (1992).
[CrossRef]

N. A. Riza, “Liquid crystal-based optical control of phased array antennas,” J. Lightwave Technol. 10, 1974–1984 (1992).
[CrossRef]

N. A. Riza, “Phased array antenna control using liquid crystals,” in Analog Photonics, A. R. Pirich, P. Sierak, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1790, 65–75 (1993).

N. A. Riza, “Optical control system for millimeter wave phased array antennas,” in Photonics for Processors, Neural Networks, and Memories, B. Javidi, J. L. Horner, W. J. Micelli, S. T. Kowel, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 344–351 (1993).

N. A. Riza, “A compact electro-optic controller for microwave phased array antennas,” in Photonics for Processors, Neural Networks, and Memories, B. Javidi, J. L. Horner, W. J. Micelli, S. T. Kowel, eds., Proc. Soc. Soc. Photo-Opt. Instrum. Eng.2026, 286–296 (1993).

N. A. Riza, “Liquid crystal-based optical time delay control system for wideband phased arrays,” in Analog Photonics, A. R. Pirich, P. Sierak, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1790, 171–183 (1993).

N. A. Riza, “Three-dimensional optical time delay units for radar,” in Photonics for Processors, Neural Networks, and Memories, B. Javidi, J. L. Horner, W. J. Micelli, S. T. Kowel, eds., Proc. Soc. Opt. Instrum. Eng.2026, 227–237 (1993).

Simonis, G. J.

G. J. Simonis, K. G. Purchase, “Optical generation, distribution, and control of microwaves using laser heterodyne,” IEEE Trans. Microwave Theory Tech. 38, 667–669 (1990).
[CrossRef]

Steyskal, H.

H. Steyskal, “Digital beamforming antennas: an introduction,” Microwave J. (1987), pp. 107–124.

Stremler, F. G.

F. G. Stremler, Introduction to Communication Systems, 2nd ed. (Addison-Wesley, Reading, Mass., 1982), pp. 241–244.

Stubkjaer, K. E.

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18-GHz microwave carriers,” IEEE Photon. Tech. Lett. 4, 936–938 (1992).
[CrossRef]

Ueda, K.

N. Uehara, K. Ueda, “193-mHz beat linewidth of frequency-stabilized laser-diode pumped Nd:YAG ring lasers,” Opt. Lett. 18 (1993).
[CrossRef] [PubMed]

Uehara, N.

N. Uehara, K. Ueda, “193-mHz beat linewidth of frequency-stabilized laser-diode pumped Nd:YAG ring lasers,” Opt. Lett. 18 (1993).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

T. Day, E. K. Gustafson, R. L. Byer, “Subhertz relative frequency stabilization of two-diode-laser pumped Nd:YAG lasers locked to a Fabry-Perot interferometer,” IEEE J. Quantum Electron. 28, 1106–1117 (1992).
[CrossRef]

IEEE Photon. Tech. Lett. (2)

N. A. Riza, “A compact high performance optical control system for phased array radars,” IEEE Photon. Tech. Lett. 4, 1073–1076 (1992).
[CrossRef]

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18-GHz microwave carriers,” IEEE Photon. Tech. Lett. 4, 936–938 (1992).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

G. J. Simonis, K. G. Purchase, “Optical generation, distribution, and control of microwaves using laser heterodyne,” IEEE Trans. Microwave Theory Tech. 38, 667–669 (1990).
[CrossRef]

J. Lightwave Technol. (1)

N. A. Riza, “Liquid crystal-based optical control of phased array antennas,” J. Lightwave Technol. 10, 1974–1984 (1992).
[CrossRef]

Microwave J. (2)

H. Steyskal, “Digital beamforming antennas: an introduction,” Microwave J. (1987), pp. 107–124.

R. J. Mailloux, “Microwave and mm-wave array antennas,” Microwave J. (1990), pp. 21–32.

Opt. Lett. (2)

N. Uehara, K. Ueda, “193-mHz beat linewidth of frequency-stabilized laser-diode pumped Nd:YAG ring lasers,” Opt. Lett. 18 (1993).
[CrossRef] [PubMed]

T. Day, E. K. Gustafson, R. L. Byer, “Active frequency stabilization of a 1.062 μm, Nd:GGG, diode-laser-pumped nonplanar ring oscillator to less than 3 Hz of relative linewidth,” Opt. Lett. 15 (1990).
[CrossRef] [PubMed]

Other (19)

See B. M. Hendrickson, S. Yao, eds., Optoelectronic Signal Processing for Phased-Array Antennas III, Proc. Soc. Photo-Opt. Instrum. Eng.1073 (1992).

See Digest of the DARPA Symposium on Photonic Systems for Antenna Applications, M. L. VanBlaricum, ed. (Advanced Research Projects Agency, Arlington, Va., 1993), Vol. 3.

T. Olson, T. J. Kane, F. J. Adams, “Single frequency diode-pumped Nd:YAG lasers used for microwave synthesis,” in Digest of the DARPA Symposium on Photonic Systems for Antenna Applications, M. L. VanBlaricum, ed. (Advanced Research Projects Agency, Arlington, Va., 1993), Vol. 3.

N. A. Riza, “Phased array antenna control using liquid crystals,” in Analog Photonics, A. R. Pirich, P. Sierak, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1790, 65–75 (1993).

N. A. Riza, “Optical control system for millimeter wave phased array antennas,” in Photonics for Processors, Neural Networks, and Memories, B. Javidi, J. L. Horner, W. J. Micelli, S. T. Kowel, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 344–351 (1993).

N. A. Riza, “A compact electro-optic controller for microwave phased array antennas,” in Photonics for Processors, Neural Networks, and Memories, B. Javidi, J. L. Horner, W. J. Micelli, S. T. Kowel, eds., Proc. Soc. Soc. Photo-Opt. Instrum. Eng.2026, 286–296 (1993).

N. A. Riza, “Liquid crystal-based optical time delay control system for wideband phased arrays,” in Analog Photonics, A. R. Pirich, P. Sierak, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1790, 171–183 (1993).

N. A. Riza, “Three-dimensional optical time delay units for radar,” in Photonics for Processors, Neural Networks, and Memories, B. Javidi, J. L. Horner, W. J. Micelli, S. T. Kowel, eds., Proc. Soc. Opt. Instrum. Eng.2026, 227–237 (1993).

F. G. Stremler, Introduction to Communication Systems, 2nd ed. (Addison-Wesley, Reading, Mass., 1982), pp. 241–244.

“Liquid crystal optical controllers for phased array antennas,” in Optoelectronic Signal Processing for Phased Array Antennas IV, B. M. Hendrickson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2055 (to be published).

Acousto-Optics Product Catalog (Brimrose Corporation of America, Baltimore, Md., 1993).

Hamamatsu Corporation, Japan, silicon APD’s (e.g., Model S2381), 1993.

1993 Product Guide (Model QDEMW1 PIN detector) (Lasertron, Burlington, Mass., 1993).

Radio Frequency/Intermediate Frequency Signal Processing Handbook (Mini-Circuits, Brooklyn, N.Y., 1993), Vols. 1 and 2.

Filter Catalog (Tokyo America, Inc., Mt. Prospect, III., 1993).

Lexel and Spectra Physics, air-cooled argon lasers.

Specifications Sheet for 670-nm Laser Diode (Spectra Diode Labs, San Jose, Calif., 1993).

Series 122 and 140 Diode-Pumped Ring Laser Specifications Sheets (Lightwave Electronics, Mountain View, Calif., 1993).

Product Catalog, 1993 (SDL-3400 series linear arrays) (Spectra Diode Labs, San Jose, Calif., 1993).

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Figures (17)

Fig. 1
Fig. 1

Basic optical beam-former system diagram: LNA’s, low-noise amplifiers; PA’s, power amplifiers.

Fig. 2
Fig. 2

Acousto-optic (AO) technique that is based on the in-line AO interferometer.

Fig. 3
Fig. 3

Single-sideband quadrature phase-shift method approach to optical beat-signal generation.

Fig. 4
Fig. 4

Beat-frequency signal-generation method that uses optical mixing of two highly stable color-shifted lasers: BS, beam splitter.

Fig. 5
Fig. 5

NLC-based optical control system using the in-line AO interferometer for beat-frequency signal generation: TN-LC, twisted-nematic liquid crystal; T/R, transmit and receive.

Fig. 6
Fig. 6

1500-pixel NLC array using thin-film transistors for pixel control fabricated at General Electric Research and Development Center. This array is used as NLC Array 1 in the experimental controller.

Fig. 7
Fig. 7

Oscilloscope photographs showing NLC-based phase-shift control for 140-MHz output signals: top traces, signal from the optical system; bottom traces, signal from the external 140-MHz electronic reference source: (a) Near in-phase and (b) near out-of-phase signals.

Fig. 8
Fig. 8

Oscilloscope traces showing NLC-based amplitude control for 140-MHz output signals. Traces show (a) peak and (b) nulled signals.

Fig. 9
Fig. 9

Phase-shift data taken from the optical controller.

Fig. 10
Fig. 10

Amplitude-control data taken from the optical controller.

Fig. 11
Fig. 11

Phase ramp is programmed on the 1500-pixel NLC spatial light modulator and viewed between parallel polarizers to see the phase variation mapped to amplitude variation. The polarizers are at 45° to the nematic director.

Fig. 12
Fig. 12

Spectrum-analyzer traces showing 140-MHz signal amplitude control with (a) 10.7-dBm peak and (b) −41.9-dBm null, giving a 52.6-dB signal attenuation range.

Fig. 13
Fig. 13

Spectrum-analyzer trace showing 140-MHz signal dynamic range of 66.6 dB measured at a +20-kHz offset with a 3-kHz RBW.

Fig. 14
Fig. 14

Spectrum-analyzer trace showing 140-MHz signal carrier-to-noise ratio of 96.2 dB/Hz measured at a +10-kHz offset with a 3-kHz RBW.

Fig. 15
Fig. 15

Spectrum-analyzer trace showing 140-MHz signal carrier-to-noise ratio of 122.2 dB/Hz measured at a +1-MHz offset with a 300-kHz RBW.

Fig. 16
Fig. 16

Antenna-system diagram for an optically controlled phased-array antenna that uses the rf Bragg-cell-based optical controller and frequency upconversion. EOM, electro-optic modulator.

Fig. 17
Fig. 17

Two-channel time-multiplexed beam-scanning approach using a two-channel system for optical control of phased arrays. In this case, while channel 1 is active for antenna operation, channel 2 is resetting for the next antenna beam position and vice versa: TIR, total internal reflection; PBS’s, polarizing beam splitters; SLM1, SLM2, spatial light modulators.

Equations (3)

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E ( n , t ) a n ( v n a ) exp ( - j ω c t ) + a n ( v n a ) × exp ( + j ω c t ) exp [ - j ϕ n ( v n p ) ] ,
i n ( t ) E ( n , t ) 2 = C A n ( v n a ) + C A n ( v n a ) cos [ 2 ω c t - ϕ n ( v n p ) ] ,
i n ( t ) = C A n cos ( 2 ω c t - ϕ n ) .

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