Abstract

A confocal Fabry-Perot interferometer is used to produce feedback within an optical processor. The resulting confocal feedback system (CFS) has a complex-valued feedback transfer function and performs analog solution of partial differential equations (PDE’s). The CFS is interfaced to a microcomputer via a laser scanner and video electronics to create a hybrid processor, thus taking a major step toward the practical use of the CFS for the high-speed solution of PDE’s. Solutions to the three types of second-order linear PDE’s in two dimensions—elliptic, hyperbolic, and parabolic—were obtained experimentally and analyzed.

© 1980 Optical Society of America

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  1. J. Cederquist and S. H. Lee, "The Use of Feedback in Optical Information Processing," Appl. Phys. 18, 311–319 (1979).
  2. R. Akins, R. Athale, and S. H. Lee, "Feedback in Analog and Digital Optical Image Processing," Opt. Eng. (to be published).
  3. M. Hercher, "The Spherical Mirror Fabry-Perot Interferometer," Appl. Opt. 7, 951–966 (1968); 7, 1336 (E) (1968).
  4. H. K. V. Lotsch, "The Confocal Resonator System," Optik 30, 1–14, 181–201, 217–233, and 563–576 (1969).
  5. J. A. Clark, "Imaging Through a Confocal Optical Interference Filter," Opt. Commun. 5, 163167 (1972).
  6. A. Riazi, O. P. Gandhi, and D. A. Christensen, "Imaging Characteristics of a Confocal Cavity," Opt. Commun. 28, 163–165 (1979).
  7. J. R. Leger and S. H. Lee, "Coherent Optical Implementation of Generalized Two-Dimensional Transforms," Opt. Eng. 18, 518–523 (1979).
  8. G. A. Bekey and W. J. Karplus, Hybrid Computation (Wiley, New York, 1968).
  9. S. P. McGrew, in New Concepts and Technologies in Parallel Information Processing, edited by E. R. Caianiello (Noordhoff, Leyden, 1975), pp. 55–73.
  10. N. G. Nakhodkin, A. V. Kuzmenko, V. N. Kurashov, and I. M. Pochernyaev, "Operator Method for Solving Differential Equations in a Digital Computer-Coherent Optical Device Hybrid System," Opt. Spectrosc. 40, 420–422 (1976).
  11. J. Götz, G. Häusler, A. W. Lohmann, and M. Simon, "Solving Differential Equations with TV-Optical Feedback," Proc. Int. Opt. Comput. Conf., London, 1978, pp. 179–180, IEEE Order No. CH1305-2/79/0000-0179.
  12. D. Jablonowski and S. H. Lee, "A Coherent Optical Feedback System for Optical Information Processing," Appl. Phys. 8, 51–58 (1975).
  13. J. Cederquist and S. H. Lee, "The Solution of Partial Differential Equations with Time-Varying Inhomogeneous Terms Using Coherent Optical Feedback," Proc. Int. Opt. Comput. Conf., London, 1978, pp. 181–184, IEEE Order No. CH1305-2/79/0000-0181.
  14. G. Dahlquist and Å. Björck, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 389–392.
  15. R. L. Burden, J. D. Faires and A. C. Reynolds, Numerical Analysis (Prindle, Weber and Schmidt, Boston, 1978), p. 525.
  16. R. Akins and S. H. Lee, "Coherent Optical Image Amplification by an Injection-Locked Dye Amplifier at 632.8 nm," Appl. Phys. Lett. 35, 660–663 (1979).
  17. Reference 15, pp. 508–509.
  18. R. W. Hockney, Technical Report 53, Institute for Plasma Research, Stanford University, 1966 (unpublished).

1979 (4)

A. Riazi, O. P. Gandhi, and D. A. Christensen, "Imaging Characteristics of a Confocal Cavity," Opt. Commun. 28, 163–165 (1979).

J. R. Leger and S. H. Lee, "Coherent Optical Implementation of Generalized Two-Dimensional Transforms," Opt. Eng. 18, 518–523 (1979).

R. Akins and S. H. Lee, "Coherent Optical Image Amplification by an Injection-Locked Dye Amplifier at 632.8 nm," Appl. Phys. Lett. 35, 660–663 (1979).

J. Cederquist and S. H. Lee, "The Use of Feedback in Optical Information Processing," Appl. Phys. 18, 311–319 (1979).

1976 (1)

N. G. Nakhodkin, A. V. Kuzmenko, V. N. Kurashov, and I. M. Pochernyaev, "Operator Method for Solving Differential Equations in a Digital Computer-Coherent Optical Device Hybrid System," Opt. Spectrosc. 40, 420–422 (1976).

1975 (1)

D. Jablonowski and S. H. Lee, "A Coherent Optical Feedback System for Optical Information Processing," Appl. Phys. 8, 51–58 (1975).

1972 (1)

J. A. Clark, "Imaging Through a Confocal Optical Interference Filter," Opt. Commun. 5, 163167 (1972).

1969 (1)

H. K. V. Lotsch, "The Confocal Resonator System," Optik 30, 1–14, 181–201, 217–233, and 563–576 (1969).

1968 (1)

Akins, R.

R. Akins and S. H. Lee, "Coherent Optical Image Amplification by an Injection-Locked Dye Amplifier at 632.8 nm," Appl. Phys. Lett. 35, 660–663 (1979).

R. Akins, R. Athale, and S. H. Lee, "Feedback in Analog and Digital Optical Image Processing," Opt. Eng. (to be published).

Athale, R.

R. Akins, R. Athale, and S. H. Lee, "Feedback in Analog and Digital Optical Image Processing," Opt. Eng. (to be published).

Bekey, G. A.

G. A. Bekey and W. J. Karplus, Hybrid Computation (Wiley, New York, 1968).

Björck, Å.

G. Dahlquist and Å. Björck, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 389–392.

Burden, R. L.

R. L. Burden, J. D. Faires and A. C. Reynolds, Numerical Analysis (Prindle, Weber and Schmidt, Boston, 1978), p. 525.

Cederquist, J.

J. Cederquist and S. H. Lee, "The Use of Feedback in Optical Information Processing," Appl. Phys. 18, 311–319 (1979).

J. Cederquist and S. H. Lee, "The Solution of Partial Differential Equations with Time-Varying Inhomogeneous Terms Using Coherent Optical Feedback," Proc. Int. Opt. Comput. Conf., London, 1978, pp. 181–184, IEEE Order No. CH1305-2/79/0000-0181.

Christensen, D. A.

A. Riazi, O. P. Gandhi, and D. A. Christensen, "Imaging Characteristics of a Confocal Cavity," Opt. Commun. 28, 163–165 (1979).

Clark, J. A.

J. A. Clark, "Imaging Through a Confocal Optical Interference Filter," Opt. Commun. 5, 163167 (1972).

Dahlquist, G.

G. Dahlquist and Å. Björck, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 389–392.

Faires, J. D.

R. L. Burden, J. D. Faires and A. C. Reynolds, Numerical Analysis (Prindle, Weber and Schmidt, Boston, 1978), p. 525.

Gandhi, O. P.

A. Riazi, O. P. Gandhi, and D. A. Christensen, "Imaging Characteristics of a Confocal Cavity," Opt. Commun. 28, 163–165 (1979).

Götz, J.

J. Götz, G. Häusler, A. W. Lohmann, and M. Simon, "Solving Differential Equations with TV-Optical Feedback," Proc. Int. Opt. Comput. Conf., London, 1978, pp. 179–180, IEEE Order No. CH1305-2/79/0000-0179.

Häusler, G.

J. Götz, G. Häusler, A. W. Lohmann, and M. Simon, "Solving Differential Equations with TV-Optical Feedback," Proc. Int. Opt. Comput. Conf., London, 1978, pp. 179–180, IEEE Order No. CH1305-2/79/0000-0179.

Hercher, M.

Hockney, R. W.

R. W. Hockney, Technical Report 53, Institute for Plasma Research, Stanford University, 1966 (unpublished).

Jablonowski, D.

D. Jablonowski and S. H. Lee, "A Coherent Optical Feedback System for Optical Information Processing," Appl. Phys. 8, 51–58 (1975).

Karplus, W. J.

G. A. Bekey and W. J. Karplus, Hybrid Computation (Wiley, New York, 1968).

Kurashov, V. N.

N. G. Nakhodkin, A. V. Kuzmenko, V. N. Kurashov, and I. M. Pochernyaev, "Operator Method for Solving Differential Equations in a Digital Computer-Coherent Optical Device Hybrid System," Opt. Spectrosc. 40, 420–422 (1976).

Kuzmenko, A. V.

N. G. Nakhodkin, A. V. Kuzmenko, V. N. Kurashov, and I. M. Pochernyaev, "Operator Method for Solving Differential Equations in a Digital Computer-Coherent Optical Device Hybrid System," Opt. Spectrosc. 40, 420–422 (1976).

Lee, S. H.

J. Cederquist and S. H. Lee, "The Use of Feedback in Optical Information Processing," Appl. Phys. 18, 311–319 (1979).

R. Akins and S. H. Lee, "Coherent Optical Image Amplification by an Injection-Locked Dye Amplifier at 632.8 nm," Appl. Phys. Lett. 35, 660–663 (1979).

J. R. Leger and S. H. Lee, "Coherent Optical Implementation of Generalized Two-Dimensional Transforms," Opt. Eng. 18, 518–523 (1979).

D. Jablonowski and S. H. Lee, "A Coherent Optical Feedback System for Optical Information Processing," Appl. Phys. 8, 51–58 (1975).

R. Akins, R. Athale, and S. H. Lee, "Feedback in Analog and Digital Optical Image Processing," Opt. Eng. (to be published).

J. Cederquist and S. H. Lee, "The Solution of Partial Differential Equations with Time-Varying Inhomogeneous Terms Using Coherent Optical Feedback," Proc. Int. Opt. Comput. Conf., London, 1978, pp. 181–184, IEEE Order No. CH1305-2/79/0000-0181.

Leger, J. R.

J. R. Leger and S. H. Lee, "Coherent Optical Implementation of Generalized Two-Dimensional Transforms," Opt. Eng. 18, 518–523 (1979).

Lohmann, A. W.

J. Götz, G. Häusler, A. W. Lohmann, and M. Simon, "Solving Differential Equations with TV-Optical Feedback," Proc. Int. Opt. Comput. Conf., London, 1978, pp. 179–180, IEEE Order No. CH1305-2/79/0000-0179.

Lotsch, H. K. V.

H. K. V. Lotsch, "The Confocal Resonator System," Optik 30, 1–14, 181–201, 217–233, and 563–576 (1969).

McGrew, S. P.

S. P. McGrew, in New Concepts and Technologies in Parallel Information Processing, edited by E. R. Caianiello (Noordhoff, Leyden, 1975), pp. 55–73.

Nakhodkin, N. G.

N. G. Nakhodkin, A. V. Kuzmenko, V. N. Kurashov, and I. M. Pochernyaev, "Operator Method for Solving Differential Equations in a Digital Computer-Coherent Optical Device Hybrid System," Opt. Spectrosc. 40, 420–422 (1976).

Pochernyaev, I. M.

N. G. Nakhodkin, A. V. Kuzmenko, V. N. Kurashov, and I. M. Pochernyaev, "Operator Method for Solving Differential Equations in a Digital Computer-Coherent Optical Device Hybrid System," Opt. Spectrosc. 40, 420–422 (1976).

Reynolds, A. C.

R. L. Burden, J. D. Faires and A. C. Reynolds, Numerical Analysis (Prindle, Weber and Schmidt, Boston, 1978), p. 525.

Riazi, A.

A. Riazi, O. P. Gandhi, and D. A. Christensen, "Imaging Characteristics of a Confocal Cavity," Opt. Commun. 28, 163–165 (1979).

Simon, M.

J. Götz, G. Häusler, A. W. Lohmann, and M. Simon, "Solving Differential Equations with TV-Optical Feedback," Proc. Int. Opt. Comput. Conf., London, 1978, pp. 179–180, IEEE Order No. CH1305-2/79/0000-0179.

Appl. Opt. (1)

Appl. Phys. (2)

J. Cederquist and S. H. Lee, "The Use of Feedback in Optical Information Processing," Appl. Phys. 18, 311–319 (1979).

D. Jablonowski and S. H. Lee, "A Coherent Optical Feedback System for Optical Information Processing," Appl. Phys. 8, 51–58 (1975).

Appl. Phys. Lett. (1)

R. Akins and S. H. Lee, "Coherent Optical Image Amplification by an Injection-Locked Dye Amplifier at 632.8 nm," Appl. Phys. Lett. 35, 660–663 (1979).

Opt. Commun. (2)

J. A. Clark, "Imaging Through a Confocal Optical Interference Filter," Opt. Commun. 5, 163167 (1972).

A. Riazi, O. P. Gandhi, and D. A. Christensen, "Imaging Characteristics of a Confocal Cavity," Opt. Commun. 28, 163–165 (1979).

Opt. Eng. (1)

J. R. Leger and S. H. Lee, "Coherent Optical Implementation of Generalized Two-Dimensional Transforms," Opt. Eng. 18, 518–523 (1979).

Opt. Spectrosc. (1)

N. G. Nakhodkin, A. V. Kuzmenko, V. N. Kurashov, and I. M. Pochernyaev, "Operator Method for Solving Differential Equations in a Digital Computer-Coherent Optical Device Hybrid System," Opt. Spectrosc. 40, 420–422 (1976).

Optik (1)

H. K. V. Lotsch, "The Confocal Resonator System," Optik 30, 1–14, 181–201, 217–233, and 563–576 (1969).

Other (9)

R. Akins, R. Athale, and S. H. Lee, "Feedback in Analog and Digital Optical Image Processing," Opt. Eng. (to be published).

J. Götz, G. Häusler, A. W. Lohmann, and M. Simon, "Solving Differential Equations with TV-Optical Feedback," Proc. Int. Opt. Comput. Conf., London, 1978, pp. 179–180, IEEE Order No. CH1305-2/79/0000-0179.

Reference 15, pp. 508–509.

R. W. Hockney, Technical Report 53, Institute for Plasma Research, Stanford University, 1966 (unpublished).

G. A. Bekey and W. J. Karplus, Hybrid Computation (Wiley, New York, 1968).

S. P. McGrew, in New Concepts and Technologies in Parallel Information Processing, edited by E. R. Caianiello (Noordhoff, Leyden, 1975), pp. 55–73.

J. Cederquist and S. H. Lee, "The Solution of Partial Differential Equations with Time-Varying Inhomogeneous Terms Using Coherent Optical Feedback," Proc. Int. Opt. Comput. Conf., London, 1978, pp. 181–184, IEEE Order No. CH1305-2/79/0000-0181.

G. Dahlquist and Å. Björck, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 389–392.

R. L. Burden, J. D. Faires and A. C. Reynolds, Numerical Analysis (Prindle, Weber and Schmidt, Boston, 1978), p. 525.

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