Abstract

This paper deals with the propagation of coherent light through lens systems; it relates diffraction theory to ray optics. A diffraction integral is derived which relates the electromagnetic fields on the input plane of a lens system to those on its output plane. The kernel of the diffraction integral is written in terms of the elements of the ray matrix that describes the complete lens system; that kernel indicates a connection between ray optics and diffraction theory. It also provides a simple method for writing the diffraction integral for a lens system. The results are limited to the paraxial-ray approximation, but apply to symmetric and asymmetric lens systems. In the case of asymmetric systems, i.e., those containing rotated elliptical or cylindrical lenses, the ray-matrix formalism is extended so as to use a single fourth-order matrix. The diffraction integrals derived are applied to optical spatial filtering, optical-beam waveguides, optical resonators, and holography.

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  1. S. A. Collins, Jr., Appl. Opt. 3, 1263 (1964).
  2. L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. L. Porcello, IRE Trans. Inf. Theory IT6, 386 (1960).
  3. Edward L. O'Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).
  4. W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).
  5. G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
  6. G. D. Boyd and H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).
  7. W. K. Kahn, Appl. Opt. 4, 758 (1965).
  8. N. Kurauchi and W. Kahn, Appl. Opt. 5, 1023 (1966).
  9. Y. Suematsu and H. Fukinuki, J. Inst. Elec. Comm. Engrs. Japan 48, 1684 (1965).
  10. H. Kogelnik, Bell System Tech. J. 44, 455 (1965).
  11. W. Steier, Appl. Opt. 5, 1229 (1966).
  12. W. Brouwer, E. L. O'Neill, and A. Walther, Appl. Opt. 2, 1239 (1963).
  13. Y. Suematsu and H. Fukinuki, Bull. Tokyo Inst. Tech. 88, 33 (1968).
  14. A. Walther, J. Opt. Soc. Am. 59, 1325 (1969).
  15. P. Baus, Opto-Electron. 1, 37 (1969).
  16. A. Vander Lugt, Proc. IEEE 54, 1055 (1966).
  17. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), Ch. III.
  18. Reference 3, p. 48.
  19. Reference 17, Sec. 3.2.1, Ch. IV.
  20. F. B. Hildebrandt, Methods of Applied Mathematics (Prentice-Hall, New York, 1952), Sec. 1.6.
  21. To show complete agreement, it is necessary to change notation by putting A to C, B to D, C to A, and D to B to make our ray matrix [Eq. (16)] consistent with Kogelnik's Eq. (9).
  22. This stability condition can be brought into agreement with the stability condition 0<AD<1 predicted on the basis of lens-waveguide theory. To show the equivalence, it is necessary to form the equivalent lens-waveguide-unit lens system from the unit mentioned here, followed by this unit, inverted end for end.
  23. W. Lukosz, J. Opt. Soc. Am. 58, 1084 (1968).
  24. For more complicated cases in which LN possesses a linear term in x1 or in x2 it is necessary to use Hamilton's equations. [Equation], where i = 1, 2, to derive the complete ray-matrix equations.
  25. The case considered in Ref. 13 is slightly different, in that they start with the fields across the output plane and derive the input fields. They also incorporate the factor of (ik) in the exponent of Eq. (35) into the P's without significant change of results.
  26. W. Kahn and J. Nemit, in Proceedings of Symposium on Quasi-Optics, edited by J. Fox (Polytechnic Press, Brooklyn, N. Y., 1964), p. 501.

Baus, P.

P. Baus, Opto-Electron. 1, 37 (1969).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), Ch. III.

Boyd, G. D.

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

G. D. Boyd and H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).

Brouwer, W.

W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).

W. Brouwer, E. L. O'Neill, and A. Walther, Appl. Opt. 2, 1239 (1963).

Collins, Jr., S. A.

S. A. Collins, Jr., Appl. Opt. 3, 1263 (1964).

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. L. Porcello, IRE Trans. Inf. Theory IT6, 386 (1960).

Fukinuki, H.

Y. Suematsu and H. Fukinuki, Bull. Tokyo Inst. Tech. 88, 33 (1968).

Y. Suematsu and H. Fukinuki, J. Inst. Elec. Comm. Engrs. Japan 48, 1684 (1965).

Gordon, J. P.

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

Hildebrandt, F. B.

F. B. Hildebrandt, Methods of Applied Mathematics (Prentice-Hall, New York, 1952), Sec. 1.6.

Kahn, W.

W. Kahn and J. Nemit, in Proceedings of Symposium on Quasi-Optics, edited by J. Fox (Polytechnic Press, Brooklyn, N. Y., 1964), p. 501.

N. Kurauchi and W. Kahn, Appl. Opt. 5, 1023 (1966).

Kahn, W. K.

W. K. Kahn, Appl. Opt. 4, 758 (1965).

Kogelnik, H.

H. Kogelnik, Bell System Tech. J. 44, 455 (1965).

G. D. Boyd and H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).

Kurauchi, N.

N. Kurauchi and W. Kahn, Appl. Opt. 5, 1023 (1966).

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. L. Porcello, IRE Trans. Inf. Theory IT6, 386 (1960).

Lugt, A. Vander

A. Vander Lugt, Proc. IEEE 54, 1055 (1966).

Lukosz, W.

W. Lukosz, J. Opt. Soc. Am. 58, 1084 (1968).

Nemit, J.

W. Kahn and J. Nemit, in Proceedings of Symposium on Quasi-Optics, edited by J. Fox (Polytechnic Press, Brooklyn, N. Y., 1964), p. 501.

O’Neill, E. L.

W. Brouwer, E. L. O'Neill, and A. Walther, Appl. Opt. 2, 1239 (1963).

O’Neill, Edward L.

Edward L. O'Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. L. Porcello, IRE Trans. Inf. Theory IT6, 386 (1960).

Porcello, L. L.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. L. Porcello, IRE Trans. Inf. Theory IT6, 386 (1960).

Steier, W.

W. Steier, Appl. Opt. 5, 1229 (1966).

Suematsu, Y.

Y. Suematsu and H. Fukinuki, Bull. Tokyo Inst. Tech. 88, 33 (1968).

Y. Suematsu and H. Fukinuki, J. Inst. Elec. Comm. Engrs. Japan 48, 1684 (1965).

Walther, A.

A. Walther, J. Opt. Soc. Am. 59, 1325 (1969).

W. Brouwer, E. L. O'Neill, and A. Walther, Appl. Opt. 2, 1239 (1963).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), Ch. III.

Other (26)

S. A. Collins, Jr., Appl. Opt. 3, 1263 (1964).

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. L. Porcello, IRE Trans. Inf. Theory IT6, 386 (1960).

Edward L. O'Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

G. D. Boyd and H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).

W. K. Kahn, Appl. Opt. 4, 758 (1965).

N. Kurauchi and W. Kahn, Appl. Opt. 5, 1023 (1966).

Y. Suematsu and H. Fukinuki, J. Inst. Elec. Comm. Engrs. Japan 48, 1684 (1965).

H. Kogelnik, Bell System Tech. J. 44, 455 (1965).

W. Steier, Appl. Opt. 5, 1229 (1966).

W. Brouwer, E. L. O'Neill, and A. Walther, Appl. Opt. 2, 1239 (1963).

Y. Suematsu and H. Fukinuki, Bull. Tokyo Inst. Tech. 88, 33 (1968).

A. Walther, J. Opt. Soc. Am. 59, 1325 (1969).

P. Baus, Opto-Electron. 1, 37 (1969).

A. Vander Lugt, Proc. IEEE 54, 1055 (1966).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), Ch. III.

Reference 3, p. 48.

Reference 17, Sec. 3.2.1, Ch. IV.

F. B. Hildebrandt, Methods of Applied Mathematics (Prentice-Hall, New York, 1952), Sec. 1.6.

To show complete agreement, it is necessary to change notation by putting A to C, B to D, C to A, and D to B to make our ray matrix [Eq. (16)] consistent with Kogelnik's Eq. (9).

This stability condition can be brought into agreement with the stability condition 0<AD<1 predicted on the basis of lens-waveguide theory. To show the equivalence, it is necessary to form the equivalent lens-waveguide-unit lens system from the unit mentioned here, followed by this unit, inverted end for end.

W. Lukosz, J. Opt. Soc. Am. 58, 1084 (1968).

For more complicated cases in which LN possesses a linear term in x1 or in x2 it is necessary to use Hamilton's equations. [Equation], where i = 1, 2, to derive the complete ray-matrix equations.

The case considered in Ref. 13 is slightly different, in that they start with the fields across the output plane and derive the input fields. They also incorporate the factor of (ik) in the exponent of Eq. (35) into the P's without significant change of results.

W. Kahn and J. Nemit, in Proceedings of Symposium on Quasi-Optics, edited by J. Fox (Polytechnic Press, Brooklyn, N. Y., 1964), p. 501.

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