Abstract

Wave-front distortion compensation using direct system performance metric optimization is studied both theoretically and experimentally. It is shown how different requirements for wave-front control can be incorporated, and how information from different wave-front sensor types can be fused, within a generalized gradient descent optimization paradigm. In our experiments a very-large-scale integration (VLSI) system implementing a simultaneous perturbation stochastic approximation optimization algorithm was applied for real-time adaptive control of multielement wave-front correctors. The custom-chip controller is used in two adaptive laser beam focusing systems, one with a 127-element liquid-crystal phase modulator and the other with beam steering and 37-control channel micromachined deformable mirrors. The submillisecond response time of the micromachined deformable mirror and the parallel nature of the analog VLSI control architecture provide for high-speed adaptive compensation of dynamical phase aberrations of a laser beam under conditions of strong intensity scintillations. Experimental results demonstrate improvement of laser beam quality at the receiver plane in the spectral band up to 60 Hz.

© 2000 Optical Society of America

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1999 (2)

1998 (4)

1997 (5)

M. C. Roggemann, V. M. Bright, S. R. Hick, W. D. Cowan, “Use of micro-electromechanical deformable mirrors to control aberrations in optical system,” Opt. Eng. 36, 1326–1338 (1997).
[CrossRef]

G. Cauwenberghs, “Analog VLSI stochastic perturbative learning architectures,” Int. J. Analog Integr. Circuits Signal Process. 13, 195–209 (1997).
[CrossRef]

M. A. Vorontsov, G. W. Carhart, J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22, 907–909 (1997).
[CrossRef] [PubMed]

M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, D. G. Voelz, “Adaptive imaging system for phase-distorted extended source/multiple distance objects,” Appl. Opt. 36, 3319–3328 (1997).
[CrossRef] [PubMed]

G. Vdovin, S. Middelhoek, P. M. Sarro, “Technology and applications of micromachined silicon adaptive mirrors,” Opt. Eng. 36, 1382–1390 (1997).
[CrossRef]

1996 (2)

1995 (1)

1994 (2)

R. Q. Fugate, “Laser beacon adaptive optics,” Opt. Photon. News 5(6), 14–19 (1994).

G. Cauwenberghs, “Analog VLSI recurrent neural network learning a continuous-time trajectory,” IEEE Trans. Neural Netw. 41, 827–829 (1994).

1992 (1)

J. C. Spall, “Multivariate stochastic approximation using a simultaneous perturbation gradient approximation,” IEEE Trans. Autom. Control 37, 332–341 (1992).
[CrossRef]

1983 (1)

1978 (1)

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

1977 (3)

1974 (1)

1971 (1)

Andreou, G.

G. Andreou, K. A. Boahen, “Translinear circuits in subthreshold MOS,” Analog Integr. Circuits Signal Process. 9, 141–166 (1996).
[CrossRef]

Baranova, N. B.

Barbier, P. R.

B. M. Levine, A. Wirth, H. DaSilva, F. M. Landers, S. Kahalas, T. L. Bruno, P. R. Barbier, D. W. Rush, P. Polak-Dingels, G. L. Burdge, D. P. Looze, “Active compensation for horizontal line-of-sight turbulence over near-ground paths,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3233, 221–232 (1998).
[CrossRef]

Barclay, H. T.

Bartsch, D.

Boahen, K. A.

G. Andreou, K. A. Boahen, “Translinear circuits in subthreshold MOS,” Analog Integr. Circuits Signal Process. 9, 141–166 (1996).
[CrossRef]

Bright, V. M.

M. C. Roggemann, V. M. Bright, S. R. Hick, W. D. Cowan, “Use of micro-electromechanical deformable mirrors to control aberrations in optical system,” Opt. Eng. 36, 1326–1338 (1997).
[CrossRef]

Bruno, T. L.

B. M. Levine, A. Wirth, H. DaSilva, F. M. Landers, S. Kahalas, T. L. Bruno, P. R. Barbier, D. W. Rush, P. Polak-Dingels, G. L. Burdge, D. P. Looze, “Active compensation for horizontal line-of-sight turbulence over near-ground paths,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3233, 221–232 (1998).
[CrossRef]

Buffington, A.

Burdge, G. L.

B. M. Levine, A. Wirth, H. DaSilva, F. M. Landers, S. Kahalas, T. L. Bruno, P. R. Barbier, D. W. Rush, P. Polak-Dingels, G. L. Burdge, D. P. Looze, “Active compensation for horizontal line-of-sight turbulence over near-ground paths,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3233, 221–232 (1998).
[CrossRef]

Carhart, G. W.

G. W. Carhart, M. A. Vorontsov, “Synthetic imaging: non-adaptive anisoplanatic image correction in atmospheric turbulence,” Opt. Lett. 23, 745–747 (1998).
[CrossRef]

M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, D. G. Voelz, “Adaptive imaging system for phase-distorted extended source/multiple distance objects,” Appl. Opt. 36, 3319–3328 (1997).
[CrossRef] [PubMed]

M. A. Vorontsov, G. W. Carhart, J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22, 907–909 (1997).
[CrossRef] [PubMed]

M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, D. G. Voelz, “Image quality criteria for an adaptive imaging system based on statistical analysis of the speckle field,” J. Opt. Soc. Am. A 13, 1456–1466 (1996).
[CrossRef]

G. W. Carhart, J. C. Ricklin, V. P. Sivokon, M. A. Vorontsov, “Parallel perturbation gradient descent in algorithm for adaptive wavefront correction,” in Adaptive Optics and Applications, R. Tyson, R. Fugate, eds., Proc. SPIE3126, 221–227 (1997).
[CrossRef]

R. T. Edwards, M. Cohen, G. Cauwenberghs, M. A. Vorontsov, G. W. Carhart, “Analog VLSI parallel stochastic optimization for adaptive optics,” in Learning on Silicon, G. Cauwenberghs, M. A. Bayoumi, eds. (Kluwer Academic, Boston, 1999), Chap. 1, pp. 359–382.

G. W. Carhart, M. A. Vorontsov, R. T. Edwards, M. Cohen, G. Cauwenberghs, “Adaptive wavefront control using a VLSI implementation of the parallel perturbation gradient descent algorithm,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. Gonglewski, M. Vorontsov, eds., Proc. SPIE3760, 61–66 (1999).
[CrossRef]

Cauwenberghs, G.

G. Cauwenberghs, “Analog VLSI stochastic perturbative learning architectures,” Int. J. Analog Integr. Circuits Signal Process. 13, 195–209 (1997).
[CrossRef]

G. Cauwenberghs, “Analog VLSI recurrent neural network learning a continuous-time trajectory,” IEEE Trans. Neural Netw. 41, 827–829 (1994).

G. Cauwenberghs, “A learning analog neural network chip with continuous-recurrent dynamics,” in Advances in Neural Information Processing Systems, S. J. Hanson, J. D. Cowan, C. L. Giles, eds., (Morgan Kaufmann, San Mateo, Calif.1994), Vol. 6, pp. 858–865.

R. T. Edwards, M. Cohen, G. Cauwenberghs, M. A. Vorontsov, G. W. Carhart, “Analog VLSI parallel stochastic optimization for adaptive optics,” in Learning on Silicon, G. Cauwenberghs, M. A. Bayoumi, eds. (Kluwer Academic, Boston, 1999), Chap. 1, pp. 359–382.

G. W. Carhart, M. A. Vorontsov, R. T. Edwards, M. Cohen, G. Cauwenberghs, “Adaptive wavefront control using a VLSI implementation of the parallel perturbation gradient descent algorithm,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. Gonglewski, M. Vorontsov, eds., Proc. SPIE3760, 61–66 (1999).
[CrossRef]

G. Cauwenberghs, “A fast stochastic error-descent algorithm for supervised learning and optimization,” in Advances in Neural Information Processing Systems, S. J. Hanson, J. D. Cowan, C. L. Giles, eds. (Morgan Kaufmann, San Mateo, Calif, 1993), Vol. 5, pp. 244–251.

Clark, D. S.

H. J. Kushner, D. S. Clark, Stochastic Approximation Methods for Constrained and Unconstrained Systems (Springer-Verlag, New York, 1978).

Cohen, M.

G. W. Carhart, M. A. Vorontsov, R. T. Edwards, M. Cohen, G. Cauwenberghs, “Adaptive wavefront control using a VLSI implementation of the parallel perturbation gradient descent algorithm,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. Gonglewski, M. Vorontsov, eds., Proc. SPIE3760, 61–66 (1999).
[CrossRef]

R. T. Edwards, M. Cohen, G. Cauwenberghs, M. A. Vorontsov, G. W. Carhart, “Analog VLSI parallel stochastic optimization for adaptive optics,” in Learning on Silicon, G. Cauwenberghs, M. A. Bayoumi, eds. (Kluwer Academic, Boston, 1999), Chap. 1, pp. 359–382.

Cowan, W. D.

M. C. Roggemann, V. M. Bright, S. R. Hick, W. D. Cowan, “Use of micro-electromechanical deformable mirrors to control aberrations in optical system,” Opt. Eng. 36, 1326–1338 (1997).
[CrossRef]

DaSilva, H.

B. M. Levine, A. Wirth, H. DaSilva, F. M. Landers, S. Kahalas, T. L. Bruno, P. R. Barbier, D. W. Rush, P. Polak-Dingels, G. L. Burdge, D. P. Looze, “Active compensation for horizontal line-of-sight turbulence over near-ground paths,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3233, 221–232 (1998).
[CrossRef]

Edwards, R. T.

R. T. Edwards, M. Cohen, G. Cauwenberghs, M. A. Vorontsov, G. W. Carhart, “Analog VLSI parallel stochastic optimization for adaptive optics,” in Learning on Silicon, G. Cauwenberghs, M. A. Bayoumi, eds. (Kluwer Academic, Boston, 1999), Chap. 1, pp. 359–382.

G. W. Carhart, M. A. Vorontsov, R. T. Edwards, M. Cohen, G. Cauwenberghs, “Adaptive wavefront control using a VLSI implementation of the parallel perturbation gradient descent algorithm,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. Gonglewski, M. Vorontsov, eds., Proc. SPIE3760, 61–66 (1999).
[CrossRef]

Fainman, Y.

Fender, J. S.

G. B. Love, J. S. Fender, S. R. Restaino, “Adaptive wavefront shaping with liquid crystals,” Opt. Photon. News (October1995), pp. 16–21.

Flower, B.

B. Flower, M. Jabri, “Summed weight neuron perturbation: an O(n)improvement over weight perturbation,” in Advances in Neural Information Processing Systems, S. J. Hanson, J. D. Cowan, C. L. Giles, eds. (Morgan Kaufmann, San Meteo, Calif., 1993), Vol. 5, pp. 212–219.

Freeman, W. R.

Fried, D. L.

Fugate, R. Q.

R. Q. Fugate, “Laser beacon adaptive optics,” Opt. Photon. News 5(6), 14–19 (1994).

Hansen, S.

Hardy, J. W.

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Herrmann, J.

Hick, S. R.

M. C. Roggemann, V. M. Bright, S. R. Hick, W. D. Cowan, “Use of micro-electromechanical deformable mirrors to control aberrations in optical system,” Opt. Eng. 36, 1326–1338 (1997).
[CrossRef]

Humphreys, R. A.

Jabri, M.

B. Flower, M. Jabri, “Summed weight neuron perturbation: an O(n)improvement over weight perturbation,” in Advances in Neural Information Processing Systems, S. J. Hanson, J. D. Cowan, C. L. Giles, eds. (Morgan Kaufmann, San Meteo, Calif., 1993), Vol. 5, pp. 212–219.

Kahalas, S.

B. M. Levine, A. Wirth, H. DaSilva, F. M. Landers, S. Kahalas, T. L. Bruno, P. R. Barbier, D. W. Rush, P. Polak-Dingels, G. L. Burdge, D. P. Looze, “Active compensation for horizontal line-of-sight turbulence over near-ground paths,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3233, 221–232 (1998).
[CrossRef]

Kokorowski, S. A.

Kushner, H. J.

H. J. Kushner, D. S. Clark, Stochastic Approximation Methods for Constrained and Unconstrained Systems (Springer-Verlag, New York, 1978).

Landers, F. M.

B. M. Levine, A. Wirth, H. DaSilva, F. M. Landers, S. Kahalas, T. L. Bruno, P. R. Barbier, D. W. Rush, P. Polak-Dingels, G. L. Burdge, D. P. Looze, “Active compensation for horizontal line-of-sight turbulence over near-ground paths,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3233, 221–232 (1998).
[CrossRef]

Levine, B. M.

B. M. Levine, A. Wirth, H. DaSilva, F. M. Landers, S. Kahalas, T. L. Bruno, P. R. Barbier, D. W. Rush, P. Polak-Dingels, G. L. Burdge, D. P. Looze, “Active compensation for horizontal line-of-sight turbulence over near-ground paths,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3233, 221–232 (1998).
[CrossRef]

Looze, D. P.

B. M. Levine, A. Wirth, H. DaSilva, F. M. Landers, S. Kahalas, T. L. Bruno, P. R. Barbier, D. W. Rush, P. Polak-Dingels, G. L. Burdge, D. P. Looze, “Active compensation for horizontal line-of-sight turbulence over near-ground paths,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3233, 221–232 (1998).
[CrossRef]

Love, G. B.

G. B. Love, J. S. Fender, S. R. Restaino, “Adaptive wavefront shaping with liquid crystals,” Opt. Photon. News (October1995), pp. 16–21.

Mamaev, A. V.

Middelhoek, S.

G. Vdovin, S. Middelhoek, P. M. Sarro, “Technology and applications of micromachined silicon adaptive mirrors,” Opt. Eng. 36, 1382–1390 (1997).
[CrossRef]

Muller, R. A.

O’Meara, T. R.

Pearson, J. E.

Pedinoff, M. E.

Pilipetsky, N. F.

Pilipetsky, N. V.

In the nonlinear-optics-based or dynamical-holography-based phase conjugation systems, we have true wave-front conjugation (both phase conjugation and amplitude correction); see, for example, B. Y. Zeldovich, N. V. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation, Vol. 42 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1985).

Polak-Dingels, P.

B. M. Levine, A. Wirth, H. DaSilva, F. M. Landers, S. Kahalas, T. L. Bruno, P. R. Barbier, D. W. Rush, P. Polak-Dingels, G. L. Burdge, D. P. Looze, “Active compensation for horizontal line-of-sight turbulence over near-ground paths,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3233, 221–232 (1998).
[CrossRef]

Polejaev, V. I.

V. I. Polejaev, M. A. Vorontsov, “Adaptive active imaging system based on radiation focusing for extended targets,” in Adaptive Optics and Applications, R. Tyson, R. Fugate, eds., Proc. SPIE3126, 216–220 (1997).
[CrossRef]

Price, T. R.

Primmerman, C. A.

Pruidze, D. V.

Restaino, S. R.

G. B. Love, J. S. Fender, S. R. Restaino, “Adaptive wavefront shaping with liquid crystals,” Opt. Photon. News (October1995), pp. 16–21.

Ricklin, J. C.

Roggemann, M. C.

M. C. Roggemann, V. M. Bright, S. R. Hick, W. D. Cowan, “Use of micro-electromechanical deformable mirrors to control aberrations in optical system,” Opt. Eng. 36, 1326–1338 (1997).
[CrossRef]

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Rush, D. W.

B. M. Levine, A. Wirth, H. DaSilva, F. M. Landers, S. Kahalas, T. L. Bruno, P. R. Barbier, D. W. Rush, P. Polak-Dingels, G. L. Burdge, D. P. Looze, “Active compensation for horizontal line-of-sight turbulence over near-ground paths,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3233, 221–232 (1998).
[CrossRef]

Sarro, P. M.

G. Vdovin, S. Middelhoek, P. M. Sarro, “Technology and applications of micromachined silicon adaptive mirrors,” Opt. Eng. 36, 1382–1390 (1997).
[CrossRef]

Shapiro, J. H.

Shkunov, V. V.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, B. Ya. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
[CrossRef]

In the nonlinear-optics-based or dynamical-holography-based phase conjugation systems, we have true wave-front conjugation (both phase conjugation and amplitude correction); see, for example, B. Y. Zeldovich, N. V. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation, Vol. 42 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1985).

Sivokon, V. P.

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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G. W. Carhart, M. A. Vorontsov, R. T. Edwards, M. Cohen, G. Cauwenberghs, “Adaptive wavefront control using a VLSI implementation of the parallel perturbation gradient descent algorithm,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. Gonglewski, M. Vorontsov, eds., Proc. SPIE3760, 61–66 (1999).
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[CrossRef]

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Zeldovich, B. Y.

In the nonlinear-optics-based or dynamical-holography-based phase conjugation systems, we have true wave-front conjugation (both phase conjugation and amplitude correction); see, for example, B. Y. Zeldovich, N. V. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation, Vol. 42 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1985).

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[CrossRef]

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[CrossRef]

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[CrossRef]

Other (19)

In the nonlinear-optics-based or dynamical-holography-based phase conjugation systems, we have true wave-front conjugation (both phase conjugation and amplitude correction); see, for example, B. Y. Zeldovich, N. V. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation, Vol. 42 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1985).

B. M. Levine, A. Wirth, H. DaSilva, F. M. Landers, S. Kahalas, T. L. Bruno, P. R. Barbier, D. W. Rush, P. Polak-Dingels, G. L. Burdge, D. P. Looze, “Active compensation for horizontal line-of-sight turbulence over near-ground paths,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3233, 221–232 (1998).
[CrossRef]

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, 1991).

J. C. Spall, “A stochastic approximation technique for generating maximum likelihood parameter estimates,” in Proceedings of the American Control Conference (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 1161–1167.

G. Cauwenberghs, “A fast stochastic error-descent algorithm for supervised learning and optimization,” in Advances in Neural Information Processing Systems, S. J. Hanson, J. D. Cowan, C. L. Giles, eds. (Morgan Kaufmann, San Mateo, Calif, 1993), Vol. 5, pp. 244–251.

J. C. Spall, “Adaptive stochastic approximation by the simultaneous perturbation method,” IEEE Trans. Autom. Control45 (to be published); in condensed form in Proceedings of the IEEE Conference on Decision and Control (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 3872–3879.

B. Flower, M. Jabri, “Summed weight neuron perturbation: an O(n)improvement over weight perturbation,” in Advances in Neural Information Processing Systems, S. J. Hanson, J. D. Cowan, C. L. Giles, eds. (Morgan Kaufmann, San Meteo, Calif., 1993), Vol. 5, pp. 212–219.

G. Cauwenberghs, “A learning analog neural network chip with continuous-recurrent dynamics,” in Advances in Neural Information Processing Systems, S. J. Hanson, J. D. Cowan, C. L. Giles, eds., (Morgan Kaufmann, San Mateo, Calif.1994), Vol. 6, pp. 858–865.

V. I. Polejaev, M. A. Vorontsov, “Adaptive active imaging system based on radiation focusing for extended targets,” in Adaptive Optics and Applications, R. Tyson, R. Fugate, eds., Proc. SPIE3126, 216–220 (1997).
[CrossRef]

B. M. Ter Haar Romey, ed., Geometry-Driven Diffusion in Computer Vision (Kluwer Academic, Dordrecht, The Netherlands, 1994).

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H. J. Kushner, D. S. Clark, Stochastic Approximation Methods for Constrained and Unconstrained Systems (Springer-Verlag, New York, 1978).

This is not necessarily true if the division form J˜j′=δJj/δujis used.

G. W. Carhart, J. C. Ricklin, V. P. Sivokon, M. A. Vorontsov, “Parallel perturbation gradient descent in algorithm for adaptive wavefront correction,” in Adaptive Optics and Applications, R. Tyson, R. Fugate, eds., Proc. SPIE3126, 221–227 (1997).
[CrossRef]

R. T. Edwards, M. Cohen, G. Cauwenberghs, M. A. Vorontsov, G. W. Carhart, “Analog VLSI parallel stochastic optimization for adaptive optics,” in Learning on Silicon, G. Cauwenberghs, M. A. Bayoumi, eds. (Kluwer Academic, Boston, 1999), Chap. 1, pp. 359–382.

G. W. Carhart, M. A. Vorontsov, R. T. Edwards, M. Cohen, G. Cauwenberghs, “Adaptive wavefront control using a VLSI implementation of the parallel perturbation gradient descent algorithm,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. Gonglewski, M. Vorontsov, eds., Proc. SPIE3760, 61–66 (1999).
[CrossRef]

Users Manual, Meadowlark Optics Inc. HEX127 phase modulator, 1997.

G. Cauwenberghs, M. A. Bayoumi, eds., Learning on Silicon (Kluwer Academic, Boston, 1999).

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Figures (12)

Fig. 1
Fig. 1

Wave-front control system architecture with block diagram of the multichannel mixed-mode VLSI stochastic gradient descent controller (AdOpt system) and (top left) micrograph of the 19-channel system, a 2.2-mm×2.25-mm chip fabricated by using 1.2-μm complementary metal-oxide semiconductor technology.

Fig. 2
Fig. 2

Schematic for adaptive laser beam focusing systems used in the experiments: (a) adaptive system with LC multielement spatial phase modulator (HEX127 phase SLM) and (b) system with 37-channel micromachined OKO mirror and beam steering mirror (tilt control). Pictured are the geometry of the HEX127 spatial light modulator electrodes, the OKO mirror phase profile with equal voltages (50 V) applied to the mirror’s two electrodes (peak value phase deviation of 2π rad), and a photograph of the OKO mirror. Focal lengths corresponding to lenses L1 and L2 are 14 in. (35.6 cm).

Fig. 3
Fig. 3

Experimental results of phase distortion compensation in the adaptive system with LC phase modulator: (a) averaged laser beam quality metric and (b) standard deviations for metric J and metric perturbation δJ obtained from averaging over 100 trials of beam quality metric maximization (m512) and minimization (m>512). The time scale is 30 s per 512 iteration steps, indicated by the dot in (a).

Fig. 4
Fig. 4

Phase (left column) and focal plane intensity (right column) patterns obtained in the adaptive system with LC phase modulator during two subsequent maximization–minimization trials. (a), (b) Pattern corresponding to metric maximization (m=512) for the first trial and (c), (d) that for the second trial. (e)–(h) Phase and intensity patterns corresponding to metric minimization: (e), (f) the first trial, and (g), (h) the second trial.

Fig. 5
Fig. 5

Self-induced phase distortion compensation in the adaptive system with beam steering and micromachined mirrors: evolution curves for (a) averaged beam quality metric and (b) standard deviation. Metric maximization corresponds to 0<m512, and metric minimization corresponds to m>512. The time scale is 0.34 s per 512 iteration steps, indicated by the dot in (a). The evolution curves 1–3 correspond to 1, wave-front tilt control only; 2, OKO mirror control only; and 3, control of both the beam steering and micromachined mirrors. The photographs show focal plane intensity distributions at the end of (top) maximization and (bottom) minimization trials.

Fig. 6
Fig. 6

Normalized adaptation evolution curves for beam quality metric maximization–minimization (averaged over 100 trials) in the system with LC phase modulator for different control channel numbers N. The photographs on the right show typical phase patterns for N=7, 19, and 127.

Fig. 7
Fig. 7

Schematics for adaptive systems used in the experiments with heater/fan-induced turbulence: (a) adaptive receiver and (b) adaptive transmitter configurations. The focal length corresponding to lens L is 70 in. (177.8 cm), and the pinhole size is 2 mm.

Fig. 8
Fig. 8

Experimental results for adaptive receiver system configuration with laser beam propagation through the turbulence created by (a), (b) heater and (c), (d) heater and fan: (a), (c) averaged adaptation evolution curves for beam quality metric and (b), (d) corresponding curves for standard deviation of the beam quality metric fluctuations. The time scale is 0.1 s per 155 iteration steps, indicated by the black dot in (a). The evolution curves 1–4 correspond to 1, disabled feedback control (no adaptation); 2, wave-front tilt control only (tilts); 3, OKO mirror control only (OKO); and 4, control of both the beam steering and micromachined mirrors (tilts+OKO).

Fig. 9
Fig. 9

Histograms for the beam quality metric optimization process in the adaptive receiver with dynamical phase distortions created by an electric heater. Curve labels are the same as those in Fig. 8. The photographs correspond to focal plane intensity distributions in the system (a), (b) without adaptation and (c) with adaptation.

Fig. 10
Fig. 10

Histograms for beam quality metric optimization in an adaptive-receiver-type system with dynamical phase distortions created by both heater and fan. The numbers in parentheses correspond to the number of the adaptive system states Mst with zero beam quality metric value. The photographs correspond to averaged focal plane intensity distributions in the system (a) without adaptation and (b) with adaptation.

Fig. 11
Fig. 11

Experimental results for adaptive transmitter configuration with laser beam propagation through the turbulence created by two heaters: (a) averaged adaptation evolution curves for beam quality metric and (b) corresponding histograms. The numbers in parentheses correspond to the standard deviation σJ for beam quality metric fluctuations.

Fig. 12
Fig. 12

Beam quality metric temporal spectral energy density averaged over 300 adaptation trials for the adaptive receiver system with feedback control off (without adaptation) and with control using both the beam steering and micromachined mirrors (tilts+OKO).

Equations (24)

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τ u(r, t)t=-δuJ,
u(r, t)=j=1Nuj(t)Sj(r),
τjduj(t)dt=-γJj(u1,, uN),j=1,, N,
dJ(t)dt=j=1NJujdujdt=-γj=1NJuj2τj-10
(forpositiveγ).
uj(n+1)=uj(n)-γJj(u1(n),, uN(n)),j=1,, N.
δJj=J(u1,, uj+δuj,, uN)-J(u1,, uj,, uN),j=1,, N.
δJ=αj=1NJujsin(ωjt)+α22j,kN2Jujuksin(ωjt)sin(ωkt)+ .
δJ=J(u1+δu1,, uj+δuj,, uN+δuN)-J(u1,, uj,, uN)
J˜j=δJδuj=Juj (δuj)2+kjNJuk δukδuj+ .
uj(m+1)=uj(m)-γδJ(m)πj(m),j=1,, N.
J1[u]=J[u]+ηJa[u],u¯(t)=s-1u(r, t)d2r.
τ u(r, t)t=-η[u¯(t)-u0]-δuJ,
τjduj(t)dt=-η[u¯N(t)-u0]-γJj(u1,, uN),
u¯N=N-1j=1Nαjuj(t),αj=(s/N)-1Sj(r)d2r.
uj(m+1)=uj(m)-η[u¯N(m)-u0]-γδJ(m)πj(m),
j=1,, N.
J1[u]=J[u]+ηJa[u]+χJg[u],
Jg[u]=s-1|u(r, t)|2d2r.
τ u(r, t)t=d2u(r, t)-η[u¯(t)-u0]-γδJ(t)δu(r, t),
J1[u]=J[u]+μJs[u],
Js[u]=12[Iδ(r, t)-Iδ0(r, t)]2d2r.
τ u(r, t)t=-μ[Iδ(r, t)-Iδ0(r)]×δIδ(r, t)δu(r, t)-γδJ(t)δu(r, t),
τ u(r, t)t=d2u(r, t)-η[u¯(t)-u0]-μ[Iδ(r, t)-Iδ0(r)]δIδ(r, t)δu(r, t)-γδJ(t)δu(r, t),

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