Abstract

For light propagation in dielectric waveguides, interconnections are established between phase velocity, group velocity, stored energy, power flow, and a zigzag-ray model. The ray model takes account of the Goos-Haenchen shifts that occur at the film boundaries and of the associated time delays and predicts the correct group velocity. Relationships between group velocity, power flow, and stored energy are shown to be valid for guides of quite general cross section and in the presence of refractive-index dispersion. Except in cases of anomalous dispersion, the group velocity is generally smaller than the phase velocity.

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  1. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 305.
  2. N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 7.
  3. P. K. Tien, Appl. Opt. 10, 2395 (1971).
  4. J. E. Goell and R. D. Standley, Proc. IEEE 58, 1504 (1970).
  5. J. J. Burke, Opt. Sci. Newsletter (U. Ariz.) 5, 31 (1971); Opt. Sci. Newsletter (U. Ariz.)5, 66 (1971).
  6. Reference 2, p. 74.
  7. Reference 2, p. 79.
  8. Reference 2, p. 80.
  9. R. Ulrich and W. Prettl, Appl. Phys. 1, 55 (1973).
  10. H. Kogelnik, T. P. Sosnowski, and H. P. Weber, IEEE J. Quantum Electron. 9, 795 (1973).
  11. L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).
  12. D. Gloge, Appl. Opt. 10, 2442 (1971).
  13. Reference 11, p. 122.
  14. K. Artmann, Ann. Phys. 2, 87 (1948).
  15. Reference 11, p. 99.
  16. Reference 11, p. 146.
  17. W. P. Allis, S. J. Buchsbaum, and A. Bers, Waves in Anisotropic Plasmas (Wiley, New York, 1962), p. 126.
  18. Reference 17, p. 106.
  19. H. L. Bertoni and A. Hessel, IEEE Trans. Antennas Propag. 14, 344 (1966).
  20. H. Kurss, Quant. Appl. Math. 26, 373 (1968).
  21. We can use Eq. (50) to rewrite the total stored energy W of Eq. (42) in a simple form in terms of the electric field E alone. Using Eqs. (43) and (44), we obtain, after some algebra,W=2Wε+Wδε=2ε0dx dy n·m· E. E*, where n = n (x,y) and m = m(xy) are the profiles of the phase and group indices of the guide materials as defined in Sec. I.
  22. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), p. 255.
  23. Reference 17, p. 100.
  24. F. N. H. Robinson [Int. J. Electron. XIX 2, 149 (1965)] has shown that an equation of this form is a direct consequence of energy conservation.
  25. Reference 1, p. 323.

Allis, W. P.

W. P. Allis, S. J. Buchsbaum, and A. Bers, Waves in Anisotropic Plasmas (Wiley, New York, 1962), p. 126.

Artmann, K.

K. Artmann, Ann. Phys. 2, 87 (1948).

Bers, A.

W. P. Allis, S. J. Buchsbaum, and A. Bers, Waves in Anisotropic Plasmas (Wiley, New York, 1962), p. 126.

Bertoni, H. L.

H. L. Bertoni and A. Hessel, IEEE Trans. Antennas Propag. 14, 344 (1966).

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).

Buchsbaum, S. J.

W. P. Allis, S. J. Buchsbaum, and A. Bers, Waves in Anisotropic Plasmas (Wiley, New York, 1962), p. 126.

Burke, J. J.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 7.

J. J. Burke, Opt. Sci. Newsletter (U. Ariz.) 5, 31 (1971); Opt. Sci. Newsletter (U. Ariz.)5, 66 (1971).

Gloge, D.

D. Gloge, Appl. Opt. 10, 2442 (1971).

Goell, J. E.

J. E. Goell and R. D. Standley, Proc. IEEE 58, 1504 (1970).

Hessel, A.

H. L. Bertoni and A. Hessel, IEEE Trans. Antennas Propag. 14, 344 (1966).

Kapany, N. S.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 7.

Kogelnik, H.

H. Kogelnik, T. P. Sosnowski, and H. P. Weber, IEEE J. Quantum Electron. 9, 795 (1973).

Kurss, H.

H. Kurss, Quant. Appl. Math. 26, 373 (1968).

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), p. 255.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), p. 255.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 305.

Prettl, W.

R. Ulrich and W. Prettl, Appl. Phys. 1, 55 (1973).

Robinson, F. N. H.

F. N. H. Robinson [Int. J. Electron. XIX 2, 149 (1965)] has shown that an equation of this form is a direct consequence of energy conservation.

Sosnowski, T. P.

H. Kogelnik, T. P. Sosnowski, and H. P. Weber, IEEE J. Quantum Electron. 9, 795 (1973).

Standley, R. D.

J. E. Goell and R. D. Standley, Proc. IEEE 58, 1504 (1970).

Tien, P. K.

P. K. Tien, Appl. Opt. 10, 2395 (1971).

Ulrich, R.

R. Ulrich and W. Prettl, Appl. Phys. 1, 55 (1973).

Weber, H. P.

H. Kogelnik, T. P. Sosnowski, and H. P. Weber, IEEE J. Quantum Electron. 9, 795 (1973).

Other (25)

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 305.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 7.

P. K. Tien, Appl. Opt. 10, 2395 (1971).

J. E. Goell and R. D. Standley, Proc. IEEE 58, 1504 (1970).

J. J. Burke, Opt. Sci. Newsletter (U. Ariz.) 5, 31 (1971); Opt. Sci. Newsletter (U. Ariz.)5, 66 (1971).

Reference 2, p. 74.

Reference 2, p. 79.

Reference 2, p. 80.

R. Ulrich and W. Prettl, Appl. Phys. 1, 55 (1973).

H. Kogelnik, T. P. Sosnowski, and H. P. Weber, IEEE J. Quantum Electron. 9, 795 (1973).

L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).

D. Gloge, Appl. Opt. 10, 2442 (1971).

Reference 11, p. 122.

K. Artmann, Ann. Phys. 2, 87 (1948).

Reference 11, p. 99.

Reference 11, p. 146.

W. P. Allis, S. J. Buchsbaum, and A. Bers, Waves in Anisotropic Plasmas (Wiley, New York, 1962), p. 126.

Reference 17, p. 106.

H. L. Bertoni and A. Hessel, IEEE Trans. Antennas Propag. 14, 344 (1966).

H. Kurss, Quant. Appl. Math. 26, 373 (1968).

We can use Eq. (50) to rewrite the total stored energy W of Eq. (42) in a simple form in terms of the electric field E alone. Using Eqs. (43) and (44), we obtain, after some algebra,W=2Wε+Wδε=2ε0dx dy n·m· E. E*, where n = n (x,y) and m = m(xy) are the profiles of the phase and group indices of the guide materials as defined in Sec. I.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), p. 255.

Reference 17, p. 100.

F. N. H. Robinson [Int. J. Electron. XIX 2, 149 (1965)] has shown that an equation of this form is a direct consequence of energy conservation.

Reference 1, p. 323.

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