Abstract

We consider wave propagation in longitudinally periodic media near the first- and higher-order Bragg resonances. An extended coupled-waves (ECW) approach is applied to media with single or multiharmonic periodicities and numerically compared to the Floquet theory. The ECW approach predicts coupling coefficients χ that vary as η<sup><i>N</i></sup>, where η is the magnitude of the perturbation and N is the Bragg order. Explicit expressions are also given for the band-gap width and band-gap shift for all Bragg orders. A number of applications are discussed.

© 1976 Optical Society of America

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  1. L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1956).
  2. R. del Kronigand W. G. Penney, Proc. R. Soc. London Ser. A 130, 499 (1930).
  3. G. Allen, Phys. Rev. 91, 531 (1953).
  4. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1966).
  5. A. Hessel, in Antenna Theory, edited by R. E. Collin and F. J. Zucker (McGraw-Hill, New York, 1969), Pt. 2.
  6. D. J. Mead, J. Sound Vib. 27, 235 (1973).
  7. R. M. Bevensee, Electromagnetic Slow Wave Systems (Wiley, New York, 1964).
  8. R. M. White, Proc. IEEE 58, 1238 (1970).
  9. C. Elachi and C. Yeh, J. Opt. Soc. Am. 63, 840 (1973).
  10. C. Elachi and C. Yeh, J. Appl. Phys. 44, 3146 (1973).
  11. A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
  12. W. S. C. Chang, IEEE Trans. Microwave Theory Tech. 21, 775 (1974).
  13. H. Kogelnik and C. V. Shank, J. Appl. Phys. 43, 2327 (1972).
  14. S. Wang, Wave Electron. 1, 31 (1974).
  15. S. Wang, IEEE J. Quantum Electron. 10, 413 (1974).
  16. C. Elachi, G. Evans, and C. Yeh, J. Opt. Soc. Am. 65, 404 (1975).
  17. R. Dingle, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 34, 1327 (1975).
  18. T. Tamir, H. C. Wang, and A. A. Oliver, IEEE Trans. Microwave Theory Tech. 12, 324 (1964).
  19. C. Yeh, K. F. Casey, and Z. Kaprielian, IEEE Trans. Microwave Theory Tech. 13, 297 (1965).
  20. J. R. Pierce, Almost All About Waves (MIT, Cambridge, 1974).
  21. R. S. Chu and T. Tamir, IEEE Trans. Microwave Theory Tech. 17, 487 (1970).
  22. S. F. Su and T. K. Gaylord, J. Opt. Soc. Am. 65, 59 (1975).
  23. E. T. Whitaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. P., Cambridge, 1927).
  24. D. L. Jaggard and G. Evans, Coupled Waves and Floquet Approach to Periodic Structures (Caltech Ant. Lab. Report No. 73, Aug. 1975).
  25. A. Yariv and A. Gover, Appl. Phys. Lett. 26, 537 (1975).
  26. B. Y. Tong, Phys. Rev. 175, 710 (1961).
  27. C. V. Shank and R. V. Schmidt, Appl. Phys. Lett. 23, 154 (1973).
  28. J. F. Bjorkholm and V. V. Shank, Appl. Phys. Lett. 20, 306 (1972).
  29. C. Elachi, D. L. Jaggard, and C. Yeh, IEEE Trans. Antennas Propag. 23, 352 (1975).

Allen, G.

G. Allen, Phys. Rev. 91, 531 (1953).

Bevensee, R. M.

R. M. Bevensee, Electromagnetic Slow Wave Systems (Wiley, New York, 1964).

Bjorkholm, J. F.

J. F. Bjorkholm and V. V. Shank, Appl. Phys. Lett. 20, 306 (1972).

Brillouin, L.

L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1956).

Casey, K. F.

C. Yeh, K. F. Casey, and Z. Kaprielian, IEEE Trans. Microwave Theory Tech. 13, 297 (1965).

Chang, W. S. C.

W. S. C. Chang, IEEE Trans. Microwave Theory Tech. 21, 775 (1974).

Chu, R. S.

R. S. Chu and T. Tamir, IEEE Trans. Microwave Theory Tech. 17, 487 (1970).

del Kronigand, R.

R. del Kronigand W. G. Penney, Proc. R. Soc. London Ser. A 130, 499 (1930).

Dingle, R.

R. Dingle, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 34, 1327 (1975).

Elachi, C.

C. Elachi, G. Evans, and C. Yeh, J. Opt. Soc. Am. 65, 404 (1975).

C. Elachi and C. Yeh, J. Opt. Soc. Am. 63, 840 (1973).

C. Elachi and C. Yeh, J. Appl. Phys. 44, 3146 (1973).

C. Elachi, D. L. Jaggard, and C. Yeh, IEEE Trans. Antennas Propag. 23, 352 (1975).

Evans, G.

D. L. Jaggard and G. Evans, Coupled Waves and Floquet Approach to Periodic Structures (Caltech Ant. Lab. Report No. 73, Aug. 1975).

C. Elachi, G. Evans, and C. Yeh, J. Opt. Soc. Am. 65, 404 (1975).

Gaylord, T. K.

S. F. Su and T. K. Gaylord, J. Opt. Soc. Am. 65, 59 (1975).

Gossard, A. C.

R. Dingle, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 34, 1327 (1975).

Gover, A.

A. Yariv and A. Gover, Appl. Phys. Lett. 26, 537 (1975).

Hessel, A.

A. Hessel, in Antenna Theory, edited by R. E. Collin and F. J. Zucker (McGraw-Hill, New York, 1969), Pt. 2.

Jaggard, D. L.

D. L. Jaggard and G. Evans, Coupled Waves and Floquet Approach to Periodic Structures (Caltech Ant. Lab. Report No. 73, Aug. 1975).

C. Elachi, D. L. Jaggard, and C. Yeh, IEEE Trans. Antennas Propag. 23, 352 (1975).

Kaprielian, Z.

C. Yeh, K. F. Casey, and Z. Kaprielian, IEEE Trans. Microwave Theory Tech. 13, 297 (1965).

Kittel, C.

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1966).

Kogelnik, H.

H. Kogelnik and C. V. Shank, J. Appl. Phys. 43, 2327 (1972).

Mead, D. J.

D. J. Mead, J. Sound Vib. 27, 235 (1973).

Oliver, A. A.

T. Tamir, H. C. Wang, and A. A. Oliver, IEEE Trans. Microwave Theory Tech. 12, 324 (1964).

Penney, W. G.

R. del Kronigand W. G. Penney, Proc. R. Soc. London Ser. A 130, 499 (1930).

Pierce, J. R.

J. R. Pierce, Almost All About Waves (MIT, Cambridge, 1974).

Schmidt, R. V.

C. V. Shank and R. V. Schmidt, Appl. Phys. Lett. 23, 154 (1973).

Shank, C. V.

C. V. Shank and R. V. Schmidt, Appl. Phys. Lett. 23, 154 (1973).

H. Kogelnik and C. V. Shank, J. Appl. Phys. 43, 2327 (1972).

Shank, V. V.

J. F. Bjorkholm and V. V. Shank, Appl. Phys. Lett. 20, 306 (1972).

Su, S. F.

S. F. Su and T. K. Gaylord, J. Opt. Soc. Am. 65, 59 (1975).

Tamir, T.

R. S. Chu and T. Tamir, IEEE Trans. Microwave Theory Tech. 17, 487 (1970).

T. Tamir, H. C. Wang, and A. A. Oliver, IEEE Trans. Microwave Theory Tech. 12, 324 (1964).

Tong, B. Y.

B. Y. Tong, Phys. Rev. 175, 710 (1961).

Wang, H. C.

T. Tamir, H. C. Wang, and A. A. Oliver, IEEE Trans. Microwave Theory Tech. 12, 324 (1964).

Wang, S.

S. Wang, Wave Electron. 1, 31 (1974).

S. Wang, IEEE J. Quantum Electron. 10, 413 (1974).

Watson, G. N.

E. T. Whitaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. P., Cambridge, 1927).

Whitaker, E. T.

E. T. Whitaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. P., Cambridge, 1927).

White, R. M.

R. M. White, Proc. IEEE 58, 1238 (1970).

Wiegmann, W.

R. Dingle, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 34, 1327 (1975).

Yariv, A.

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).

A. Yariv and A. Gover, Appl. Phys. Lett. 26, 537 (1975).

Yeh, C.

C. Elachi, D. L. Jaggard, and C. Yeh, IEEE Trans. Antennas Propag. 23, 352 (1975).

C. Elachi, G. Evans, and C. Yeh, J. Opt. Soc. Am. 65, 404 (1975).

C. Yeh, K. F. Casey, and Z. Kaprielian, IEEE Trans. Microwave Theory Tech. 13, 297 (1965).

C. Elachi and C. Yeh, J. Appl. Phys. 44, 3146 (1973).

C. Elachi and C. Yeh, J. Opt. Soc. Am. 63, 840 (1973).

Other (29)

L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1956).

R. del Kronigand W. G. Penney, Proc. R. Soc. London Ser. A 130, 499 (1930).

G. Allen, Phys. Rev. 91, 531 (1953).

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1966).

A. Hessel, in Antenna Theory, edited by R. E. Collin and F. J. Zucker (McGraw-Hill, New York, 1969), Pt. 2.

D. J. Mead, J. Sound Vib. 27, 235 (1973).

R. M. Bevensee, Electromagnetic Slow Wave Systems (Wiley, New York, 1964).

R. M. White, Proc. IEEE 58, 1238 (1970).

C. Elachi and C. Yeh, J. Opt. Soc. Am. 63, 840 (1973).

C. Elachi and C. Yeh, J. Appl. Phys. 44, 3146 (1973).

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).

W. S. C. Chang, IEEE Trans. Microwave Theory Tech. 21, 775 (1974).

H. Kogelnik and C. V. Shank, J. Appl. Phys. 43, 2327 (1972).

S. Wang, Wave Electron. 1, 31 (1974).

S. Wang, IEEE J. Quantum Electron. 10, 413 (1974).

C. Elachi, G. Evans, and C. Yeh, J. Opt. Soc. Am. 65, 404 (1975).

R. Dingle, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 34, 1327 (1975).

T. Tamir, H. C. Wang, and A. A. Oliver, IEEE Trans. Microwave Theory Tech. 12, 324 (1964).

C. Yeh, K. F. Casey, and Z. Kaprielian, IEEE Trans. Microwave Theory Tech. 13, 297 (1965).

J. R. Pierce, Almost All About Waves (MIT, Cambridge, 1974).

R. S. Chu and T. Tamir, IEEE Trans. Microwave Theory Tech. 17, 487 (1970).

S. F. Su and T. K. Gaylord, J. Opt. Soc. Am. 65, 59 (1975).

E. T. Whitaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. P., Cambridge, 1927).

D. L. Jaggard and G. Evans, Coupled Waves and Floquet Approach to Periodic Structures (Caltech Ant. Lab. Report No. 73, Aug. 1975).

A. Yariv and A. Gover, Appl. Phys. Lett. 26, 537 (1975).

B. Y. Tong, Phys. Rev. 175, 710 (1961).

C. V. Shank and R. V. Schmidt, Appl. Phys. Lett. 23, 154 (1973).

J. F. Bjorkholm and V. V. Shank, Appl. Phys. Lett. 20, 306 (1972).

C. Elachi, D. L. Jaggard, and C. Yeh, IEEE Trans. Antennas Propag. 23, 352 (1975).

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