Abstract

Ubiquitous radar systems look everywhere at all times and require both parallel radar processors and parallel beamformers. Current systems operate with subgigahertz bandwidths and produce a handful of angle-of-arrival (AOA) beams. We present an electro-optic radar processor that combines the multigigahertz wideband capabilities of a spectral hole burning correlator with wideband Doppler processing and the thousands of parallel channels available from an electro-optical beamformer. Preliminary experiments demonstrate 150MHz bandwidth range correlations across 20 AOA beams.

© 2010 Optical Society of America

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2008 (3)

B. Braker, F. Schlottau, and K. Wagner, “Squint-free Fourier-optical rf beamforming using a spectral hole burning crystal as an imaging detector,” IEEE J. Sel. Top. Quantum Electron. 14, 952–962 (2008).
[CrossRef]

M. Colice, J. Xiong, and K. H. Wagner, “Frequency-doubled fiber lasers for RF spectrum analysis in spectral-hole-burning media,” IEEE J. Quantum Electron. 44, 587–594 (2008).
[CrossRef]

H. Hashemi, T.-S. Chu, and J. Roderick, “Integrated true-time-delay-based ultra-wideband array processing,” IEEE Commun. Mag. 46, 162–172 (2008).
[CrossRef]

2007 (5)

G. Gorju, A. Chauve, V. Crozatier, I. Lorgeré, J.-L. L. Gouët, and F. Bretanäker, “10ghz bandwidth rf spectral analyzer with megaherz resolution based on spectral-spatial holography in Tm3+:YAG experimental and theoretical study,” J. Opt. Soc. Am. B 24, 457–470 (2007).
[CrossRef]

R. K. Mohan, T. Chang, M. Tian, S. Bekker, A. Olson, C. Ostrandera, A. Khallaayouna, C. Dollingera, W. R. Babbitt, Z. Cole, R. R. Reibel, K. D. Merkel, Y. Sun, R. Cone, F. Schlottau, and K. H. Wagner, “Ultra-wideband spectral analysis using S2 technology,” J. Lumin. 127, 116–128 (2007).
[CrossRef]

Z. Cole, P. A. Roos, T. Berg, B. Kaylor, K. D. Merkel, W. R. Babbitt, and R. R. Reibel, “Unambiguous range-Doppler LADAR processing using 2 giga-sample-per-second noise waveforms,” J. Lumin. 127, 146–151 (2007).
[CrossRef]

M. Colice, J. Y. Xion, and K. Wagner, “Spectral hole burning for pulse repetition frequency analysis,” J. Lumin. 127, 129–134(2007).
[CrossRef]

F. Schlottau, Y. Li, and K. Wagner, “Demonstration of a spatial-spectral holographic LIDAR range-Doppler processor,” J. Lumin. 127, 135–145 (2007).
[CrossRef]

2006 (3)

2005 (7)

M. Colice, T. Weverka, G. Kriehn, F. Schlottau, and K. Wagner, “Holographic method of cohering fiber tapped delay lines,” Appl. Opt. 44, 5257–5272 (2005).
[CrossRef] [PubMed]

T. Chang, M. Tian, R. Mohan, C. Renner, K. Merkel, and W. Babbitt, “Recovery of spectral features readout with frequency-chirped laser fields,” Opt. Lett. 30, 1129–1131 (2005).
[CrossRef] [PubMed]

D.-Y. Zhang, N. Justis, and Y.-H. Lo, “Fluidic adaptive zoom lens with high zoom ratio and widely tunable field of view,” Opt. Commun. 249, 175 (2005).
[CrossRef]

F. Schlottau, M. Colice, and K. Wagner, “Spectral hole burning for wideband, high-resoultion radio-frequency spectrum analysis,” Opt. Lett. 30, 3003–3005 (2005).
[CrossRef] [PubMed]

M. Colice, T. Weverka, G. Kriehn, F. Schlottau, and K. Wagner, “Holographic method of cohering fiber tapped-delay-lines,” Appl. Opt. 44, 5257–5272 (2005).
[CrossRef] [PubMed]

G. Tavik, C. Hilterbrick, J. Evins, J. J. Alter, J. G. Crnkovich, Jr., J. de Graaf, W. Habicht II, G. Hrin, S. Lessin, D. Wu, and S. Hagewood, “The advanced multifunction rf concept,” IEEE Trans. Microwave Theory Tech. 53, 1009–1020 (2005).
[CrossRef]

C. Schuetz, J. Murakowski, G. Schneider, and D. Prather, “Radiometric millimeter-wave detection via optical upconversion and carrier suppression,” IEEE Trans. Microwave Theory Tech. 53, 1732–1738 (2005).
[CrossRef]

2004 (4)

V. Crozatier, V. Lavielle, F. Bretenaker, J.-L. L. Gouët, and I. Lorgeré, “High-resolution radio frequency spectral analysis with photon echo chirp transform in an Er:YSO crystal,” IEEE J. Quantum Electron. 40, 1450–1457 (2004).
[CrossRef]

J. Alter, R. White, F. Kresmer, I. D. Olin, and C. L. Temes, “Ubiquitous radar: an implementation concept,” IEEE Trans. Microwave Theory Tech. 52, 65–70 (2004).

L. Li, G. M. Heymsfield, P. E. Racette, L. Tian, and E. Zenker, “A 94gHz cloud radar system on a nasa high-altitude er-2 aircraft,” J. Atmos. Ocean. Technol. 21, 1378–1388 (2004).
[CrossRef]

K. Merkel, R. Mohan, Z. Cole, T. Chang, A. Olson, and W. Babbitt, “Multi-gigahertz radar range processing of baseband and RF carrier modulated signals in Tm : YAG,” J. Lumin. 107, 62–74 (2004).
[CrossRef]

2003 (1)

W. Ching, L. Ouyang, and Y. Xu, “Electronic and optical properties of Y2SiO5 and Y2SiO7 with comparisons to α−SiO2 and Y2O3,” Phys. Rev. B 67, 245108 (2003).
[CrossRef]

2002 (5)

Z. Cole, T. Böttger, R. K. Mohan, R. Reibel, W. R. Baggitt, and R. L. Cone, “Coherent integration of 0.5ghz spectral holograms at 1536nm using dynamic biphase codes,” Appl. Phys. Lett. 81, 3525–3527 (2002).
[CrossRef]

Y. Sun, C. W. Thiel, R. L. Cone, R. W. Equall, and R. L. Hutcheson, “Recent progress in developing new rare earth materials for hole burning and coherent transient applications,” J. Lumin. 98, 281–287 (2002).
[CrossRef]

M. Lee, H. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, “Broadband modulation of light by using an electro-optic polymer,” Science 298, 1401–1403(2002).
[CrossRef] [PubMed]

M. Skolnik, “Role of radar in microwaves,” IEEE Trans. Microwave Theory Tech. 50, 625–632 (2002).
[CrossRef]

O. Shibata, K. Inagaki, Y. Karasawa, and Y. Mizuguchi, “Spatial optical beam-forming network for receiving-mode multibeam array antenna—proposal and experiment,” IEEE Trans. Microwave Theory Tech. 50, 1425–1430 (2002).
[CrossRef]

2001 (1)

D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microwave Theory Tech. 49, 1581–1591 (2001).
[CrossRef]

2000 (1)

C. E. Muehe and M. Labitt, “Displaced-phase-center antenna technique,” Lincoln Lab. J. 12, 281–297 (2000).

1999 (1)

1998 (2)

M. Frankel, P. Matthews, R. Esman, and L. Goldberg, “Practical optical beamforming networks,” Opt. Quantum Electron. 30, 1033–1049 (1998).
[CrossRef]

A. Farina, P. Lombardo, and M. Pirri, “Nonlinear nonadaptive space-time processing for airborne early warning radar,” IEE Proc. Radar Sonar Navig. 145, 9–18 (1998).
[CrossRef]

1997 (1)

A. Goutzoulis and R. Gouse, “Comparison of conventional and fiberoptic manifolds for a dual band (UHF and S) phased-array antenna,” IEEE Trans. Antennas Propag. 45, 246–253 (1997).
[CrossRef]

1996 (1)

P. Blanchard, A. H. Greenaway, R. Anderton, and R. Appleby, “Phase calibration of arrays at optical and millimeter wavelengths,” J. Opt. Soc. Am. 13, 1593–1600 (1996).
[CrossRef]

1991 (1)

1982 (1)

1965 (1)

E. Kelly and R. Wishner, “Matched-filter theory for high-velocity accelerating targets,” IEEE Trans. Military Electron. MIL-9, 56 (1965).
[CrossRef]

Alter, J.

J. Alter, R. White, F. Kresmer, I. D. Olin, and C. L. Temes, “Ubiquitous radar: an implementation concept,” IEEE Trans. Microwave Theory Tech. 52, 65–70 (2004).

Alter, J. J.

G. Tavik, C. Hilterbrick, J. Evins, J. J. Alter, J. G. Crnkovich, Jr., J. de Graaf, W. Habicht II, G. Hrin, S. Lessin, D. Wu, and S. Hagewood, “The advanced multifunction rf concept,” IEEE Trans. Microwave Theory Tech. 53, 1009–1020 (2005).
[CrossRef]

Anderton, R.

P. Blanchard, A. H. Greenaway, R. Anderton, and R. Appleby, “Phase calibration of arrays at optical and millimeter wavelengths,” J. Opt. Soc. Am. 13, 1593–1600 (1996).
[CrossRef]

Ansari, H.

Appleby, R.

P. Blanchard, A. H. Greenaway, R. Anderton, and R. Appleby, “Phase calibration of arrays at optical and millimeter wavelengths,” J. Opt. Soc. Am. 13, 1593–1600 (1996).
[CrossRef]

R. Appleby, “Passive millimetre wave imaging and security,” European Radar Conference (2004).

Babbitt, W.

T. Chang, M. Tian, R. Mohan, C. Renner, K. Merkel, and W. Babbitt, “Recovery of spectral features readout with frequency-chirped laser fields,” Opt. Lett. 30, 1129–1131 (2005).
[CrossRef] [PubMed]

K. Merkel, R. Mohan, Z. Cole, T. Chang, A. Olson, and W. Babbitt, “Multi-gigahertz radar range processing of baseband and RF carrier modulated signals in Tm : YAG,” J. Lumin. 107, 62–74 (2004).
[CrossRef]

Babbitt, W. R.

Z. Cole, P. A. Roos, T. Berg, B. Kaylor, K. D. Merkel, W. R. Babbitt, and R. R. Reibel, “Unambiguous range-Doppler LADAR processing using 2 giga-sample-per-second noise waveforms,” J. Lumin. 127, 146–151 (2007).
[CrossRef]

R. K. Mohan, T. Chang, M. Tian, S. Bekker, A. Olson, C. Ostrandera, A. Khallaayouna, C. Dollingera, W. R. Babbitt, Z. Cole, R. R. Reibel, K. D. Merkel, Y. Sun, R. Cone, F. Schlottau, and K. H. Wagner, “Ultra-wideband spectral analysis using S2 technology,” J. Lumin. 127, 116–128 (2007).
[CrossRef]

T. L. Harris, K. D. Merkel, R. K. Mohan, T. Chang, Z. Cole, A. Olson, and W. R. Babbitt, “Multigigahertz radar-doppler correlative signal processing in optical memory crystals,” Appl. Opt. 45, 343–352 (2006).
[CrossRef] [PubMed]

Baggitt, W. R.

Z. Cole, T. Böttger, R. K. Mohan, R. Reibel, W. R. Baggitt, and R. L. Cone, “Coherent integration of 0.5ghz spectral holograms at 1536nm using dynamic biphase codes,” Appl. Phys. Lett. 81, 3525–3527 (2002).
[CrossRef]

Bekker, S.

R. K. Mohan, T. Chang, M. Tian, S. Bekker, A. Olson, C. Ostrandera, A. Khallaayouna, C. Dollingera, W. R. Babbitt, Z. Cole, R. R. Reibel, K. D. Merkel, Y. Sun, R. Cone, F. Schlottau, and K. H. Wagner, “Ultra-wideband spectral analysis using S2 technology,” J. Lumin. 127, 116–128 (2007).
[CrossRef]

Berg, T.

Z. Cole, P. A. Roos, T. Berg, B. Kaylor, K. D. Merkel, W. R. Babbitt, and R. R. Reibel, “Unambiguous range-Doppler LADAR processing using 2 giga-sample-per-second noise waveforms,” J. Lumin. 127, 146–151 (2007).
[CrossRef]

Blanchard, P.

P. Blanchard, A. H. Greenaway, R. Anderton, and R. Appleby, “Phase calibration of arrays at optical and millimeter wavelengths,” J. Opt. Soc. Am. 13, 1593–1600 (1996).
[CrossRef]

Blanchard, P. M.

Böttger, T.

T. Böttger, C. W. Thiel, Y. Sun, and R. L. Cone, “Optical decoherence and spectral diffusion at 1.5μm in Er3+:Y2SiO5 versus magnetic field, temperature, and Er3+ concentration,” Phys. Rev. B 73, 075101 (2006).
[CrossRef]

Z. Cole, T. Böttger, R. K. Mohan, R. Reibel, W. R. Baggitt, and R. L. Cone, “Coherent integration of 0.5ghz spectral holograms at 1536nm using dynamic biphase codes,” Appl. Phys. Lett. 81, 3525–3527 (2002).
[CrossRef]

T. Böttger, “Laser frequency stabilization to spectral hole burning frequency references in erbium-doped crystals: material and device optimization,” Ph.D. thesis (Montana State University, 2002).

T. Böttger, Y. Sun, C. W. Thiel, and R. L. Cone, “Material optimization of Er3+:Y2SiO5 at 1.5μm for optical processing, memory, and laser frequency stabilization applications,” in Photonics West 2003 (2003).

Braker, B.

B. Braker, F. Schlottau, and K. Wagner, “Squint-free Fourier-optical rf beamforming using a spectral hole burning crystal as an imaging detector,” IEEE J. Sel. Top. Quantum Electron. 14, 952–962 (2008).
[CrossRef]

K. Wagner, B. Braker, M. Colice, F. Schlottau, and R. T. Weverka, “Spectrally-compensated, squint-free, multiple-beam forming system for broadband rf antenna arrays,” in ICO Optics in Computing (ICO, 2004).

B. Braker, Y. Li, D. Gu, and K. Wagner, “Broadband microwave imaging with spectral hole burning for squint compensation,” in SPIE Defense and Security Symposium (SPIE, 2005).

K. Wagner, B. Braker, G. Lu, D. Gu, Y. Li, S. Herriot, and R. T. Weverka, “Photonic multiple beam forming systems for broadband rf antenna arrays,” in IEEE Summer Topical Meeting on Optical Signal Processing, Invited Paper (IEEE, 2005).

B. Braker, Y. Li, F. Schlottau, and K. Wagner, “Progress towards a wideband rf imager,” in IEEE Conference on Sensor Arrays and Multichannel Processing (IEEE, 2007), pp. 125–1288.

B. Braker, M. Colice, and K. Wagner, “Fiber array phase cohering: holographic versus numerical,” in OSA Coherent Optical Technologies and Applications (COTA) (OSA, 2006), paper CThD2.

B. Braker, “Spatial-spectral processing for imaging systems: multibeam rf imaging and radar systems using spectral hole burning materials,” Ph.D. dissertation (University of Colorado, Boulder, 2008).

B. Braker, M. Colice, and K. Wagner, “Fiber array phase cohering: holographic versus numerical,” in OSA Coherent Optical Technologies and Applications (COTA) (Optical Society of America, 2006), paper CThD2.

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K. H. Wagner, F. Schlottau, and J. Bregman, “Array imaging using spatial-spectral holography,” in Optics in Computing(2002).

F. Schlottau, K. Wagner, J. Bregman, and J.-L. L. Gouët, “Sparse antenna array multiple beamforming and spectral analysis using spatial-spectral holography,” in IEEE MWP03 (Microwave Photonics) (IEEE, 2003).

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Bretenaker, F.

V. Crozatier, V. Lavielle, F. Bretenaker, J.-L. L. Gouët, and I. Lorgeré, “High-resolution radio frequency spectral analysis with photon echo chirp transform in an Er:YSO crystal,” IEEE J. Quantum Electron. 40, 1450–1457 (2004).
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R. K. Mohan, T. Chang, M. Tian, S. Bekker, A. Olson, C. Ostrandera, A. Khallaayouna, C. Dollingera, W. R. Babbitt, Z. Cole, R. R. Reibel, K. D. Merkel, Y. Sun, R. Cone, F. Schlottau, and K. H. Wagner, “Ultra-wideband spectral analysis using S2 technology,” J. Lumin. 127, 116–128 (2007).
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T. L. Harris, K. D. Merkel, R. K. Mohan, T. Chang, Z. Cole, A. Olson, and W. R. Babbitt, “Multigigahertz radar-doppler correlative signal processing in optical memory crystals,” Appl. Opt. 45, 343–352 (2006).
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T. Chang, M. Tian, R. Mohan, C. Renner, K. Merkel, and W. Babbitt, “Recovery of spectral features readout with frequency-chirped laser fields,” Opt. Lett. 30, 1129–1131 (2005).
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K. Merkel, R. Mohan, Z. Cole, T. Chang, A. Olson, and W. Babbitt, “Multi-gigahertz radar range processing of baseband and RF carrier modulated signals in Tm : YAG,” J. Lumin. 107, 62–74 (2004).
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Chen, R. T.

R. T. Chen and Z. Fu, “Optical true-time-delay control systems for wideband phased array antennas,” in Progress in Optics, E.Wolf, ed. (North-Holland, 2000), Vol. 41, pp. 283–359.
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H. Hashemi, T.-S. Chu, and J. Roderick, “Integrated true-time-delay-based ultra-wideband array processing,” IEEE Commun. Mag. 46, 162–172 (2008).
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C. Martin and S. Clark, “Advances in millimeter-wave imaging technology for enhanced vision systems,” in IEEE Proceedings of Digital Avionics Systems Conference (IEEE, 2002).

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R. K. Mohan, T. Chang, M. Tian, S. Bekker, A. Olson, C. Ostrandera, A. Khallaayouna, C. Dollingera, W. R. Babbitt, Z. Cole, R. R. Reibel, K. D. Merkel, Y. Sun, R. Cone, F. Schlottau, and K. H. Wagner, “Ultra-wideband spectral analysis using S2 technology,” J. Lumin. 127, 116–128 (2007).
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Z. Cole, P. A. Roos, T. Berg, B. Kaylor, K. D. Merkel, W. R. Babbitt, and R. R. Reibel, “Unambiguous range-Doppler LADAR processing using 2 giga-sample-per-second noise waveforms,” J. Lumin. 127, 146–151 (2007).
[CrossRef]

T. L. Harris, K. D. Merkel, R. K. Mohan, T. Chang, Z. Cole, A. Olson, and W. R. Babbitt, “Multigigahertz radar-doppler correlative signal processing in optical memory crystals,” Appl. Opt. 45, 343–352 (2006).
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K. Merkel, R. Mohan, Z. Cole, T. Chang, A. Olson, and W. Babbitt, “Multi-gigahertz radar range processing of baseband and RF carrier modulated signals in Tm : YAG,” J. Lumin. 107, 62–74 (2004).
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Z. Cole, T. Böttger, R. K. Mohan, R. Reibel, W. R. Baggitt, and R. L. Cone, “Coherent integration of 0.5ghz spectral holograms at 1536nm using dynamic biphase codes,” Appl. Phys. Lett. 81, 3525–3527 (2002).
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M. Colice, J. Xiong, and K. H. Wagner, “Frequency-doubled fiber lasers for RF spectrum analysis in spectral-hole-burning media,” IEEE J. Quantum Electron. 44, 587–594 (2008).
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M. Colice, J. Y. Xion, and K. Wagner, “Spectral hole burning for pulse repetition frequency analysis,” J. Lumin. 127, 129–134(2007).
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M. Colice, F. Schlottau, and K. Wagner, “Broadband radio-frequency spectrum analysis in spectral-hole-burning media,” Appl. Opt. 45, 6393–6408 (2006).
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F. Schlottau, M. Colice, and K. Wagner, “Spectral hole burning for wideband, high-resoultion radio-frequency spectrum analysis,” Opt. Lett. 30, 3003–3005 (2005).
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M. Colice, T. Weverka, G. Kriehn, F. Schlottau, and K. Wagner, “Holographic method of cohering fiber tapped-delay-lines,” Appl. Opt. 44, 5257–5272 (2005).
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M. Colice, T. Weverka, G. Kriehn, F. Schlottau, and K. Wagner, “Holographic method of cohering fiber tapped delay lines,” Appl. Opt. 44, 5257–5272 (2005).
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K. Wagner, B. Braker, M. Colice, F. Schlottau, and R. T. Weverka, “Spectrally-compensated, squint-free, multiple-beam forming system for broadband rf antenna arrays,” in ICO Optics in Computing (ICO, 2004).

Cone, R.

R. K. Mohan, T. Chang, M. Tian, S. Bekker, A. Olson, C. Ostrandera, A. Khallaayouna, C. Dollingera, W. R. Babbitt, Z. Cole, R. R. Reibel, K. D. Merkel, Y. Sun, R. Cone, F. Schlottau, and K. H. Wagner, “Ultra-wideband spectral analysis using S2 technology,” J. Lumin. 127, 116–128 (2007).
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T. Böttger, C. W. Thiel, Y. Sun, and R. L. Cone, “Optical decoherence and spectral diffusion at 1.5μm in Er3+:Y2SiO5 versus magnetic field, temperature, and Er3+ concentration,” Phys. Rev. B 73, 075101 (2006).
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Y. Sun, C. W. Thiel, R. L. Cone, R. W. Equall, and R. L. Hutcheson, “Recent progress in developing new rare earth materials for hole burning and coherent transient applications,” J. Lumin. 98, 281–287 (2002).
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G. Gorju, A. Chauve, V. Crozatier, I. Lorgeré, J.-L. L. Gouët, and F. Bretanäker, “10ghz bandwidth rf spectral analyzer with megaherz resolution based on spectral-spatial holography in Tm3+:YAG experimental and theoretical study,” J. Opt. Soc. Am. B 24, 457–470 (2007).
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V. Crozatier, V. Lavielle, F. Bretenaker, J.-L. L. Gouët, and I. Lorgeré, “High-resolution radio frequency spectral analysis with photon echo chirp transform in an Er:YSO crystal,” IEEE J. Quantum Electron. 40, 1450–1457 (2004).
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G. Tavik, C. Hilterbrick, J. Evins, J. J. Alter, J. G. Crnkovich, Jr., J. de Graaf, W. Habicht II, G. Hrin, S. Lessin, D. Wu, and S. Hagewood, “The advanced multifunction rf concept,” IEEE Trans. Microwave Theory Tech. 53, 1009–1020 (2005).
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R. K. Mohan, T. Chang, M. Tian, S. Bekker, A. Olson, C. Ostrandera, A. Khallaayouna, C. Dollingera, W. R. Babbitt, Z. Cole, R. R. Reibel, K. D. Merkel, Y. Sun, R. Cone, F. Schlottau, and K. H. Wagner, “Ultra-wideband spectral analysis using S2 technology,” J. Lumin. 127, 116–128 (2007).
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Y. Sun, C. W. Thiel, R. L. Cone, R. W. Equall, and R. L. Hutcheson, “Recent progress in developing new rare earth materials for hole burning and coherent transient applications,” J. Lumin. 98, 281–287 (2002).
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R. T. Chen and Z. Fu, “Optical true-time-delay control systems for wideband phased array antennas,” in Progress in Optics, E.Wolf, ed. (North-Holland, 2000), Vol. 41, pp. 283–359.
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G. Krieger, N. Gebert, and A. Moreira, “Multidimensional waveform encoding: a new digital beamforming technique for synthetic aperture radar remote sensing,” IEEE Trans. Geosci. Remote Sens. (to be published) .

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M. Lee, H. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, “Broadband modulation of light by using an electro-optic polymer,” Science 298, 1401–1403(2002).
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M. Lee, H. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, “Broadband modulation of light by using an electro-optic polymer,” Science 298, 1401–1403(2002).
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Gouët, J.-L. L.

G. Gorju, A. Chauve, V. Crozatier, I. Lorgeré, J.-L. L. Gouët, and F. Bretanäker, “10ghz bandwidth rf spectral analyzer with megaherz resolution based on spectral-spatial holography in Tm3+:YAG experimental and theoretical study,” J. Opt. Soc. Am. B 24, 457–470 (2007).
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V. Crozatier, V. Lavielle, F. Bretenaker, J.-L. L. Gouët, and I. Lorgeré, “High-resolution radio frequency spectral analysis with photon echo chirp transform in an Er:YSO crystal,” IEEE J. Quantum Electron. 40, 1450–1457 (2004).
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F. Schlottau, K. Wagner, J. Bregman, and J.-L. L. Gouët, “Sparse antenna array multiple beamforming and spectral analysis using spatial-spectral holography,” in IEEE MWP03 (Microwave Photonics) (IEEE, 2003).

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G. Tavik, C. Hilterbrick, J. Evins, J. J. Alter, J. G. Crnkovich, Jr., J. de Graaf, W. Habicht II, G. Hrin, S. Lessin, D. Wu, and S. Hagewood, “The advanced multifunction rf concept,” IEEE Trans. Microwave Theory Tech. 53, 1009–1020 (2005).
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G. Tavik, C. Hilterbrick, J. Evins, J. J. Alter, J. G. Crnkovich, Jr., J. de Graaf, W. Habicht II, G. Hrin, S. Lessin, D. Wu, and S. Hagewood, “The advanced multifunction rf concept,” IEEE Trans. Microwave Theory Tech. 53, 1009–1020 (2005).
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D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microwave Theory Tech. 49, 1581–1591 (2001).
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H. Hashemi, T.-S. Chu, and J. Roderick, “Integrated true-time-delay-based ultra-wideband array processing,” IEEE Commun. Mag. 46, 162–172 (2008).
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M. Lee, H. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, “Broadband modulation of light by using an electro-optic polymer,” Science 298, 1401–1403(2002).
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K. Wagner, B. Braker, G. Lu, D. Gu, Y. Li, S. Herriot, and R. T. Weverka, “Photonic multiple beam forming systems for broadband rf antenna arrays,” in IEEE Summer Topical Meeting on Optical Signal Processing, Invited Paper (IEEE, 2005).

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L. Li, G. M. Heymsfield, P. E. Racette, L. Tian, and E. Zenker, “A 94gHz cloud radar system on a nasa high-altitude er-2 aircraft,” J. Atmos. Ocean. Technol. 21, 1378–1388 (2004).
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G. Tavik, C. Hilterbrick, J. Evins, J. J. Alter, J. G. Crnkovich, Jr., J. de Graaf, W. Habicht II, G. Hrin, S. Lessin, D. Wu, and S. Hagewood, “The advanced multifunction rf concept,” IEEE Trans. Microwave Theory Tech. 53, 1009–1020 (2005).
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G. Tavik, C. Hilterbrick, J. Evins, J. J. Alter, J. G. Crnkovich, Jr., J. de Graaf, W. Habicht II, G. Hrin, S. Lessin, D. Wu, and S. Hagewood, “The advanced multifunction rf concept,” IEEE Trans. Microwave Theory Tech. 53, 1009–1020 (2005).
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Hutcheson, R. L.

Y. Sun, C. W. Thiel, R. L. Cone, R. W. Equall, and R. L. Hutcheson, “Recent progress in developing new rare earth materials for hole burning and coherent transient applications,” J. Lumin. 98, 281–287 (2002).
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M. Lee, H. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, “Broadband modulation of light by using an electro-optic polymer,” Science 298, 1401–1403(2002).
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Z. Cole, P. A. Roos, T. Berg, B. Kaylor, K. D. Merkel, W. R. Babbitt, and R. R. Reibel, “Unambiguous range-Doppler LADAR processing using 2 giga-sample-per-second noise waveforms,” J. Lumin. 127, 146–151 (2007).
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G. Krieger, N. Gebert, and A. Moreira, “Multidimensional waveform encoding: a new digital beamforming technique for synthetic aperture radar remote sensing,” IEEE Trans. Geosci. Remote Sens. (to be published) .

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V. Crozatier, V. Lavielle, F. Bretenaker, J.-L. L. Gouët, and I. Lorgeré, “High-resolution radio frequency spectral analysis with photon echo chirp transform in an Er:YSO crystal,” IEEE J. Quantum Electron. 40, 1450–1457 (2004).
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M. Lee, H. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, “Broadband modulation of light by using an electro-optic polymer,” Science 298, 1401–1403(2002).
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G. Tavik, C. Hilterbrick, J. Evins, J. J. Alter, J. G. Crnkovich, Jr., J. de Graaf, W. Habicht II, G. Hrin, S. Lessin, D. Wu, and S. Hagewood, “The advanced multifunction rf concept,” IEEE Trans. Microwave Theory Tech. 53, 1009–1020 (2005).
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F. Schlottau, Y. Li, and K. Wagner, “Demonstration of a spatial-spectral holographic LIDAR range-Doppler processor,” J. Lumin. 127, 135–145 (2007).
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B. Braker, Y. Li, D. Gu, and K. Wagner, “Broadband microwave imaging with spectral hole burning for squint compensation,” in SPIE Defense and Security Symposium (SPIE, 2005).

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V. C. Chen and H. Lin, Time-Frequency Transforms for Radar Imaging and Signal Analysis (Artech, 2002).

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D.-Y. Zhang, N. Justis, and Y.-H. Lo, “Fluidic adaptive zoom lens with high zoom ratio and widely tunable field of view,” Opt. Commun. 249, 175 (2005).
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A. Farina, P. Lombardo, and M. Pirri, “Nonlinear nonadaptive space-time processing for airborne early warning radar,” IEE Proc. Radar Sonar Navig. 145, 9–18 (1998).
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G. Gorju, A. Chauve, V. Crozatier, I. Lorgeré, J.-L. L. Gouët, and F. Bretanäker, “10ghz bandwidth rf spectral analyzer with megaherz resolution based on spectral-spatial holography in Tm3+:YAG experimental and theoretical study,” J. Opt. Soc. Am. B 24, 457–470 (2007).
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V. Crozatier, V. Lavielle, F. Bretenaker, J.-L. L. Gouët, and I. Lorgeré, “High-resolution radio frequency spectral analysis with photon echo chirp transform in an Er:YSO crystal,” IEEE J. Quantum Electron. 40, 1450–1457 (2004).
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K. Wagner, B. Braker, G. Lu, D. Gu, Y. Li, S. Herriot, and R. T. Weverka, “Photonic multiple beam forming systems for broadband rf antenna arrays,” in IEEE Summer Topical Meeting on Optical Signal Processing, Invited Paper (IEEE, 2005).

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M. Lee, H. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, “Broadband modulation of light by using an electro-optic polymer,” Science 298, 1401–1403(2002).
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T. Chang, M. Tian, R. Mohan, C. Renner, K. Merkel, and W. Babbitt, “Recovery of spectral features readout with frequency-chirped laser fields,” Opt. Lett. 30, 1129–1131 (2005).
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R. K. Mohan, T. Chang, M. Tian, S. Bekker, A. Olson, C. Ostrandera, A. Khallaayouna, C. Dollingera, W. R. Babbitt, Z. Cole, R. R. Reibel, K. D. Merkel, Y. Sun, R. Cone, F. Schlottau, and K. H. Wagner, “Ultra-wideband spectral analysis using S2 technology,” J. Lumin. 127, 116–128 (2007).
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R. K. Mohan, T. Chang, M. Tian, S. Bekker, A. Olson, C. Ostrandera, A. Khallaayouna, C. Dollingera, W. R. Babbitt, Z. Cole, R. R. Reibel, K. D. Merkel, Y. Sun, R. Cone, F. Schlottau, and K. H. Wagner, “Ultra-wideband spectral analysis using S2 technology,” J. Lumin. 127, 116–128 (2007).
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R. K. Mohan, T. Chang, M. Tian, S. Bekker, A. Olson, C. Ostrandera, A. Khallaayouna, C. Dollingera, W. R. Babbitt, Z. Cole, R. R. Reibel, K. D. Merkel, Y. Sun, R. Cone, F. Schlottau, and K. H. Wagner, “Ultra-wideband spectral analysis using S2 technology,” J. Lumin. 127, 116–128 (2007).
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T. Böttger, C. W. Thiel, Y. Sun, and R. L. Cone, “Optical decoherence and spectral diffusion at 1.5μm in Er3+:Y2SiO5 versus magnetic field, temperature, and Er3+ concentration,” Phys. Rev. B 73, 075101 (2006).
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R. K. Mohan, T. Chang, M. Tian, S. Bekker, A. Olson, C. Ostrandera, A. Khallaayouna, C. Dollingera, W. R. Babbitt, Z. Cole, R. R. Reibel, K. D. Merkel, Y. Sun, R. Cone, F. Schlottau, and K. H. Wagner, “Ultra-wideband spectral analysis using S2 technology,” J. Lumin. 127, 116–128 (2007).
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B. Braker, Y. Li, F. Schlottau, and K. Wagner, “Progress towards a wideband rf imager,” in IEEE Conference on Sensor Arrays and Multichannel Processing (IEEE, 2007), pp. 125–1288.

F. Schlottau, K. Wagner, J. Bregman, and J.-L. L. Gouët, “Sparse antenna array multiple beamforming and spectral analysis using spatial-spectral holography,” in IEEE MWP03 (Microwave Photonics) (IEEE, 2003).

K. Wagner, B. Braker, M. Colice, F. Schlottau, and R. T. Weverka, “Spectrally-compensated, squint-free, multiple-beam forming system for broadband rf antenna arrays,” in ICO Optics in Computing (ICO, 2004).

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Appl. Opt. (5)

Appl. Phys. Lett. (1)

Z. Cole, T. Böttger, R. K. Mohan, R. Reibel, W. R. Baggitt, and R. L. Cone, “Coherent integration of 0.5ghz spectral holograms at 1536nm using dynamic biphase codes,” Appl. Phys. Lett. 81, 3525–3527 (2002).
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IEE Proc. Radar Sonar Navig. (1)

A. Farina, P. Lombardo, and M. Pirri, “Nonlinear nonadaptive space-time processing for airborne early warning radar,” IEE Proc. Radar Sonar Navig. 145, 9–18 (1998).
[CrossRef]

IEEE Commun. Mag. (1)

H. Hashemi, T.-S. Chu, and J. Roderick, “Integrated true-time-delay-based ultra-wideband array processing,” IEEE Commun. Mag. 46, 162–172 (2008).
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IEEE J. Quantum Electron. (2)

M. Colice, J. Xiong, and K. H. Wagner, “Frequency-doubled fiber lasers for RF spectrum analysis in spectral-hole-burning media,” IEEE J. Quantum Electron. 44, 587–594 (2008).
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V. Crozatier, V. Lavielle, F. Bretenaker, J.-L. L. Gouët, and I. Lorgeré, “High-resolution radio frequency spectral analysis with photon echo chirp transform in an Er:YSO crystal,” IEEE J. Quantum Electron. 40, 1450–1457 (2004).
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IEEE J. Sel. Top. Quantum Electron. (1)

B. Braker, F. Schlottau, and K. Wagner, “Squint-free Fourier-optical rf beamforming using a spectral hole burning crystal as an imaging detector,” IEEE J. Sel. Top. Quantum Electron. 14, 952–962 (2008).
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IEEE Trans. Antennas Propag. (1)

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IEEE Trans. Geosci. Remote Sens. (to be published) (1)

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IEEE Trans. Microwave Theory Tech. (6)

C. Schuetz, J. Murakowski, G. Schneider, and D. Prather, “Radiometric millimeter-wave detection via optical upconversion and carrier suppression,” IEEE Trans. Microwave Theory Tech. 53, 1732–1738 (2005).
[CrossRef]

O. Shibata, K. Inagaki, Y. Karasawa, and Y. Mizuguchi, “Spatial optical beam-forming network for receiving-mode multibeam array antenna—proposal and experiment,” IEEE Trans. Microwave Theory Tech. 50, 1425–1430 (2002).
[CrossRef]

J. Alter, R. White, F. Kresmer, I. D. Olin, and C. L. Temes, “Ubiquitous radar: an implementation concept,” IEEE Trans. Microwave Theory Tech. 52, 65–70 (2004).

G. Tavik, C. Hilterbrick, J. Evins, J. J. Alter, J. G. Crnkovich, Jr., J. de Graaf, W. Habicht II, G. Hrin, S. Lessin, D. Wu, and S. Hagewood, “The advanced multifunction rf concept,” IEEE Trans. Microwave Theory Tech. 53, 1009–1020 (2005).
[CrossRef]

M. Skolnik, “Role of radar in microwaves,” IEEE Trans. Microwave Theory Tech. 50, 625–632 (2002).
[CrossRef]

D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microwave Theory Tech. 49, 1581–1591 (2001).
[CrossRef]

IEEE Trans. Military Electron. (1)

E. Kelly and R. Wishner, “Matched-filter theory for high-velocity accelerating targets,” IEEE Trans. Military Electron. MIL-9, 56 (1965).
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Figures (12)

Fig. 1
Fig. 1

Fourier optical beamformer (FOBF) upconverts signals from an array of antennas onto an optical carrier so a single optical lens can take the spatial Fourier transform across the array to produce the far-field image. The spatial phase scales with frequency. As a consequence, the image scales with frequency. This effect is called beam squint.

Fig. 2
Fig. 2

Simulated wideband radar results for (a) range-AOA (angle-of-arrival) processing and (b) range-Doppler processing of a 1-D array. (a) Cross-spectral range-correlation interference-fringes (correlation fringes) shift apparent AOA position (top) and (b) apparent Doppler position (bottom) proportional to the frequency. The entire set of correlation fringes therefore scale linearly with the frequency, as seen in the top-left image of each set. If this frequency scaling is not compensated, the resolution in the range-AOA or range-Doppler processing is severely reduced, as shown in the top right of each set. Compensating this frequency scaling corrects for both effects.

Fig. 3
Fig. 3

Er:YSO crystals used in the SHB-FOBF experiments. Each erbium dopant substitutes for yttrium within the YSO crystal lattice (reprinted with permission from Ching [68]) and exhibits an inhomogeneous resonance shift. A magnetic field at 60 ° from the crystal’s D2 axis yields the optimum Zeeman shift for both the site 1 and the site 2 dopant locations [66]. When the crystal is cryogenically cooled, the collection of dopant resonances act as a bank of frequency-selective absorbers spread across a bandwidth of 1 GHz , which can be increased to 20 GHz by using a Eu codoped crystal.

Fig. 4
Fig. 4

Box geometry write and read of an SHB correlator. The reference signal and the return signal are electro- optically modulated onto an optical carrier, input at corners A and B of the input box geometry, and then interfered in an SHB crystal. Spatial-spectral interference fringes are stored in the SHB crystal for all frequencies across the upconverted spectral band. A readout laser placed at corner C of the input box geometry is passed through the SHB crystal where the write beams overlap. As the frequency of the readout laser is swept, it diffracts light into corner D of the output box geometry, where it is interfered with the readout laser to produce a detected intensity beat that is a time-domain replica of the spectral correlation fringe.

Fig. 5
Fig. 5

Geometry of an airborne side-looking multibeam array radar fed in to a range-Doppler-angle optical processor based on spatial-spectral holography with frequency dependent variable magnification for wideband Doppler and squint com pensation. The transmit array illuminates a large footprint, and the multibeam receive array forms a fan array of narrow beams spanning the width of the transmit footprint. Both can be electronically steered in elevation, and after vertical beamforming the 1D array of column summed signals are fiber-remoted to the optical processor.

Fig. 6
Fig. 6

(a) AOA-Doppler image for this side-looking scenario in the narrowband case showing one approaching airborne target and four ground moving targets near the tilted ground-clutter ridge. (b) In the wideband case, beam squint and wideband Doppler effects radially scales the AOA-Doppler image with frequency. Illustrative range correlation spectral gratings cause sinusoidal modulation across the color coded radar bandwidth. (c) The synchronous magnification during swept frequency readout of the multibeam SHB wideband range-Doppler-angle processor compensates for both beam squint and wideband Doppler effects and allows detection of the full radar bandwidth.

Fig. 7
Fig. 7

SHB-FOBF radar processor. Here we show a spaceborne scenario with a thin 2D rectangular receive array that uses true-time-delay analog beamforming along the small dimension of the array to produce a linear array of column summed signals that are applied to an array of EOMs. The 1D array of modulated signals are holographically cohered (not shown) and then imaged into an SHB crystal where they spectrally interfere with an elliptically expanded reference beam. Then the 1D array of recording spots are vertically scanned across the crystal height with each pulse of the radar. After recording a 2D array of spectral holograms, a swept frequency laser reads out the spatial-spectral holograms in the backward direction and is deflected downward using a Faraday isolator and polarizing beam splitter through a Fourier lens and into a synchronously scanned zoom lens system that compensates for wideband Doppler and beam squint. This diffracted, Fourier transformed, and zoom compensated array of spots needs to be combined with a reference plane wave from the chirped laser (not shown), and then differentially detected on a pair of megahertz rate detector arrays, whose cross-spectral interference grating outputs for each Doppler and angle represents the radar 3D data cube, illustrated with a focused clutter ridge, and a few ground moving targets. Doppler-angle bins of interest can be digitized and Fourier transformed to produced wideband coherent range profiles.

Fig. 8
Fig. 8

SHB spectral readouts can occur when the carrier is (a) within the spectral absorption profile or (b) outside the spectral absorption profile. Two delayed baseband RF signals in (a) are modulated directly onto an optical carrier and passed through the SHB crystal, so the spectral holes written into the SHB crystal contain the optical carrier as well as the cross-spectral correlation grating. When RF-upconverted signals in (b) are modulated onto an optical carrier that falls outside the spectral absorption profile of the SHB crystal, the cross-spectral correlation grating is still written into the SHB crystal. By tuning the laser to center the modulated RF band on the spectral absorption profile, the SHB systems can be used for any RF or millimeter-wave band that can be modulated onto an optical carrier.

Fig. 9
Fig. 9

Experimental implementation of the SHB-FOBF radar range processor. A wideband RF signal is formed by mixing a chirped waveform of an arbitrary waveform generator (AWG) with an RF carrier to create a DSB chirp about the RF carrier. The RF signal is split in two to create a reference signal and a set of antenna array signals. The reference signal is amplified, optically modulated, optically delayed and fed to a separate optical beam that passes through the SHB crystal. The array signal set passes through five linear RF delay lines before it is amplified and modulated onto an optical fiber array. The optical fiber array is phase cohered [28] and Fourier transformed to produce AOA beams that overlap with the reference beam in the SHB crystal. This writes correlation fringes into the SHB crystal at each AOA position of a far-field image. A swept frequency readout laser beam then reads out the correlation fringes from the SHB crystal. In the output Fourier plane of the SHB crystal, a spatial filter blocks the writing beams and the readout beam in order to pass only the diffracted light from the SHB crystal. The filtered image of the diffracted light within the SHB crystal is detected with a linescan CCD.

Fig. 10
Fig. 10

Model of the expected light distribution in the image planes and Fourier planes of the experimental system illustrated in Fig. 9.

Fig. 11
Fig. 11

Demonstrated correlation fringes for a 75 MHz double-sideband chirp ( 150 MHz of modulation). The raw linescan readout data in (a) exhibits beam squint as diagonal correlation fringes. The beam-squint-compensated correlation in (b) is a vertical correlation fringe at 17.5 ° scaled by the inverse of the readout frequency to compensate for beam squint. These images have been averaged across 26 successive readouts, and the white level has been adjusted to emphasize the correlation fringes.

Fig. 12
Fig. 12

The demonstrated AOA-range images for emulated objects at an AOA of 17.5 ° with range delays of 80 ns (bottom) and 130 ns (top). The AOA profile at 0 ns is the wideband image, while each delay has a corresponding image across all AOAs.

Equations (26)

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s ( t ) = k s p * δ ( t k T ) ,
r ( t ) = k a s p * δ ( t k T 2 R ( t ) / c ) ,
C ( f , k ) = C o | S p ( f ) | 2 { exp [ j 2 π f ( 2 R / c ) ] + c.c. } * γ 2 ( f ) ,
C ( f , k ) = C o 1 2 { exp [ j ( 4 π f / c ) ( R o + v p T k ) ] + c.c. } | S p ( f ) | 2 exp [ j 2 π f ( 2 R / c ) ] ,
C ( f , Ω k ) = C o | S p ( f ) | 2 exp [ j 2 π f ( 2 R o / c ) ] δ [ Ω k 2 π f ( 2 v p T / c ) ] .
s ( t ) = p Π ( t p T T p ) s p ( t p T ) = p s p ( t p T ) , = p f p ( t p T ) e i ω p t
Q T ( x , t ) = 2 Z P 0 L r L a λ r R sinc [ L r x λ r R tan Ψ ] sinc [ L a ( y v a t ) λ r R ] ,
k ^ x = r 2 ( t ) x | r 2 ( t ) x | ,
v r = v a · ( r 0 ( t ) x | r 0 ( t ) x | ) ,
r ( t ; x ) = a ( x ) γ s ( γ ( x ) t τ ( x ) ) = a γ p s p ( γ [ t τ p ( T Δ ) ] ) a γ p e i 2 π f p γ t f p ( t τ p ( T Δ ) ) = a γ p s p ( t τ p ( T Δ ) ) ,
E ( r , t ) V a ( x , t ) Q R ( x , t ) p s p ( t τ p ( T Δ ) k ^ x · r n m / c ) d 3 x .
E nm ( t ) = a ( x , t ) r ( t τ nm ( x ) ) d 3 x , = a ( x , t ) R ( f ) e i 2 π f ( t τ nm ( x ) ) d f d 3 x + c . c . ,
R ( f ) = p S p ( f ) e i 2 π f ( t + p ( T + Δ ( x ) ) ) .
Δ ( x ) = 2 v a · ( r 0 ( t ) x ) c + v a · ( r 0 ( t ) x ) T
E n ( t ) = m M E nm ( t m T Ψ ) m M E nm ( t ) e i 2 π m f 0 D y sin Ψ / c ,
a n ( t ) = P 0 N e i ω o t e i ϕ n r 1 2 [ ( 1 b ) e i ϕ n ( t ) ] ,
o ( x 0 , t ) = n e i ( ω 0 t k 0 L n n p ) e i ϕ n r [ b η s E n ( t L n n g / c ) ] g ( x 0 n d x x ^ 0 ) .
h c ( x 0 ) = η c d ( x ) * t | o ( x 0 , t ) e i k y y + r ( x 0 , t ) | 2 e ( t t ) / T p d t , η c T p b n g ( x 0 n d x , y 0 ) g ( x 0 / N , y 0 ) cos ( k y y 0 n p k 0 L n ϕ n r ) ,
o c ( x 0 , t ) = η c b T p η s e i ( ω 0 t k 0 z ) g ( x 0 / N , y 0 ) n g ( x 0 n d x , y 0 ) E n ( t t f ) = η s e i ( ω 0 t k 0 z ) g ( x 0 / N , y 0 ) n sensors g ( x 0 n d x x ^ 0 ) a ( x , t ) e i 2 π f ( t τ ( x ) t n ( x ) t f ) d 3 x ,
o r ( x 0 , t ) = η r e i ( ω 0 t k r · x 0 ) g ( x 0 / N , y 0 ) p s p ( t p T T min ) ,
h p ( x c , f ) = η h o c ( x c , f ) o r * ( x c , f ) * γ 2 ( f ) + c . c . , = η r η s g 0 ( x c N , y c p d y ) S p * ( f ) e i 2 π f p T n N g ( x c n d x , y c p d y ) a S p ( f ) e i 2 π f ( τ + p ( T + Δ ) + n D x sin θ / c ) * γ 2 ( f ) + c . c . ,
γ 2 ( f ) = 1 2 / T 2 + i 2 π f .
h ( x c , f ) = η h a n , p N , P | S p ( f ) | 2 e i 2 π f ( τ + p Δ + n D x sin θ / c ) * γ 2 ( f ) g ( x c n d x , y c p d y ) g ( x c N , y c p d y ) e ( P p ) T / T 1 + c.c.
H ( u , v , f ) = η h a G ( u , v ) e i 2 π f τ n , p N , P | S p ( f ) | 2 e i 2 π [ p ( f Δ d y v ) + n ( f D x sin θ / c d x u ) ] * γ 2 ( f ) , = η h G ( u , v ) a | S P ( f ) | 2 e i 2 π f τ * γ 2 ( f ) P sinc [ P d y ( v f Δ d y ) ] N sinc [ N d x ( u f D x sin θ d x c ) ] ,
H G ( u , v , f ) = η h a | S P ( f ) | 2 e i 2 π f τ G 0 ( u , v ) P sinc [ P d y ( v f 2 v a sin θ T c d y ) ] N sinc [ N d x ( u f D x sin θ c d x ) ] .
I ( u , t ) = I 0 | h e i ( ω 0 t + π b t 2 ) + 1 h 2 η h e i ( ω 0 t + π b t 2 ) δ ( u u β t ) β t H ( u , f ) d u | 2 A a | S P ( b t ) | 2 M 2 ( t ) [ 1 + cos ( 2 π b t τ ) ] G 0 ( u β t , v β t ) P sinc [ P d y β t ( v b β Δ d y ) ] N sinc [ N d x β t ( u b β D x cos θ d x c ) ] ,

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