Abstract

The evolution of surface and leaky waves as a function of frequency is examined for perpendicular polarization [transverse-electric (TEN) modes] in configurations consisting of a dielectric slab bounded by media having different refractive indices. We find that, at any frequency, these waves occur in sets of four types for all higher-order (N > 1) mode numbers. The characteristics and the field patterns of these wave types are discussed, and the special behavior of the first two (N = 0, 1) modes is described in detail.

© 1984 Optical Society of America

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References

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  1. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Sec. 1.3.
  2. H. G. Unger, Planar Optical Waveguides and Fibers (Clarendon, Oxford, 1977), Chap. 2.
  3. T. Tamir, ed., Integrated Optics (Springer-Verlag, Berlin, 1983), Chap. 2.
  4. T. Tamir, ed., Integrated Optics (Springer-Verlag, Berlin, 1983), Chap. 3, Sec. 1.4.
  5. N. N. Ding, E. Garmire, “Measuring refractive index and thickness of thin films: a new technique,” Appl. Opt. 22, 3177–3181 (1983).
    [CrossRef] [PubMed]
  6. D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
    [CrossRef]
  7. G. I. Stegeman, J. J. Burke, D. G. Hall, “Surface-polaritonlike waves guided by thin, lossy metal films,” Opt. Lett. 8, 383–385 (1983).
    [CrossRef] [PubMed]
  8. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Sec. 1.5.
  9. K. Ogusu, M. Miyagi, S. Nishida, “Leaky TE modes on an asymmetric three-layered slab waveguide,” J. Opt. Soc. Am. 70, 48–52 (1980).
    [CrossRef]
  10. H. Blok, J. M. Van Splunter, H. G. Janssen, “Leaky-wave modes and their role in the numerical evaluation of the field excited by a line source in a nonsymmetric, inhomogeneously layered waveguide,” Appl. Sci. Res. (to be published).
  11. T. Tamir, A. A. Oliner, “Guided complex waves,” Proc. IEEE 110, 325–383 (1963).
  12. V. Shah, T. Tamir, “Anomalous absorption by multilayered media,” Opt. Commun. 37, 383–387 (1981).
    [CrossRef]
  13. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Sec. 5.3.
  14. R. E. Collin, F. J. Zucker, Antenna Theory (McGraw-Hill, New York, 1969), Vol. 1, Chap. 20.
  15. T. Tamir, ed., Integrated Optics (Springer-Verlag, Berlin, 1983), Chap. 3.

1983

1981

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
[CrossRef]

V. Shah, T. Tamir, “Anomalous absorption by multilayered media,” Opt. Commun. 37, 383–387 (1981).
[CrossRef]

1980

1963

T. Tamir, A. A. Oliner, “Guided complex waves,” Proc. IEEE 110, 325–383 (1963).

Blok, H.

H. Blok, J. M. Van Splunter, H. G. Janssen, “Leaky-wave modes and their role in the numerical evaluation of the field excited by a line source in a nonsymmetric, inhomogeneously layered waveguide,” Appl. Sci. Res. (to be published).

Burke, J. J.

Collin, R. E.

R. E. Collin, F. J. Zucker, Antenna Theory (McGraw-Hill, New York, 1969), Vol. 1, Chap. 20.

Ding, N. N.

Felsen, L. B.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Sec. 5.3.

Garmire, E.

Hall, D. G.

Janssen, H. G.

H. Blok, J. M. Van Splunter, H. G. Janssen, “Leaky-wave modes and their role in the numerical evaluation of the field excited by a line source in a nonsymmetric, inhomogeneously layered waveguide,” Appl. Sci. Res. (to be published).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Sec. 1.3.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Sec. 1.5.

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Sec. 5.3.

Miyagi, M.

Nishida, S.

Ogusu, K.

Oliner, A. A.

T. Tamir, A. A. Oliner, “Guided complex waves,” Proc. IEEE 110, 325–383 (1963).

Sarid, D.

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
[CrossRef]

Shah, V.

V. Shah, T. Tamir, “Anomalous absorption by multilayered media,” Opt. Commun. 37, 383–387 (1981).
[CrossRef]

Stegeman, G. I.

Tamir, T.

V. Shah, T. Tamir, “Anomalous absorption by multilayered media,” Opt. Commun. 37, 383–387 (1981).
[CrossRef]

T. Tamir, A. A. Oliner, “Guided complex waves,” Proc. IEEE 110, 325–383 (1963).

Unger, H. G.

H. G. Unger, Planar Optical Waveguides and Fibers (Clarendon, Oxford, 1977), Chap. 2.

Van Splunter, J. M.

H. Blok, J. M. Van Splunter, H. G. Janssen, “Leaky-wave modes and their role in the numerical evaluation of the field excited by a line source in a nonsymmetric, inhomogeneously layered waveguide,” Appl. Sci. Res. (to be published).

Zucker, F. J.

R. E. Collin, F. J. Zucker, Antenna Theory (McGraw-Hill, New York, 1969), Vol. 1, Chap. 20.

Appl. Opt.

J. Opt. Soc. Am.

Opt. Commun.

V. Shah, T. Tamir, “Anomalous absorption by multilayered media,” Opt. Commun. 37, 383–387 (1981).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
[CrossRef]

Proc. IEEE

T. Tamir, A. A. Oliner, “Guided complex waves,” Proc. IEEE 110, 325–383 (1963).

Other

H. Blok, J. M. Van Splunter, H. G. Janssen, “Leaky-wave modes and their role in the numerical evaluation of the field excited by a line source in a nonsymmetric, inhomogeneously layered waveguide,” Appl. Sci. Res. (to be published).

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Sec. 5.3.

R. E. Collin, F. J. Zucker, Antenna Theory (McGraw-Hill, New York, 1969), Vol. 1, Chap. 20.

T. Tamir, ed., Integrated Optics (Springer-Verlag, Berlin, 1983), Chap. 3.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Sec. 1.5.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Sec. 1.3.

H. G. Unger, Planar Optical Waveguides and Fibers (Clarendon, Oxford, 1977), Chap. 2.

T. Tamir, ed., Integrated Optics (Springer-Verlag, Berlin, 1983), Chap. 2.

T. Tamir, ed., Integrated Optics (Springer-Verlag, Berlin, 1983), Chap. 3, Sec. 1.4.

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Figures (10)

Fig. 1
Fig. 1

Geometry of the layered configuration.

Fig. 2
Fig. 2

Typical (n ≠ 0) surface-wave solutions: (a) variation of the field amplitude with z for the four wave varieties; (b) variation of p = 1 with K = 2πh/λ. Arrows indicate increasing values of K.

Fig. 3
Fig. 3

Typical (n ≠ 0) root loci for general solutions in (a) the complex τ0 plane and (b) the complex τ2 plane. For clarity, lines for loci on the imaginary axes are shown slightly away from those axes. Arrows indicate increasing values of K.

Fig. 4
Fig. 4

Root loci for n = 2 with 0 = 1.00 and 1 = 3.00: (a) 2 = 1.01, (b) 2 = 0. As K → ∞ the portions of the loci on the imaginary axes end at finite values of τ0″ and τ2″, as shown in Fig. 3; however, these points are too far away to be shown here.

Fig. 5
Fig. 5

Typical root loci in the complex p plane: (a) 20, (b) 2 = 0. As also shown in Fig. 2, the small circles refer to the branch points τ2 = 0.

Fig. 6
Fig. 6

Typical root loci in the complex τ0 plane for (a) N = 0 and (b) N = 1.

Fig. 7
Fig. 7

Typical (N > 1) root loci in the complex κ plan. Note symmetry with respect the origin and to the κ axis.

Fig. 8
Fig. 8

Varieties of leaky-wave fields. Solid and dashed lines correspond to equiamplitude and equiphase contours, respectively. Direction of energy flow is indicated by the arrows. The field amplitude variation is suggested by the striped exponential profiles.

Fig. 9
Fig. 9

Graphic construction for surface-wave solutions with 0 = 1.0, 1 = 3.0, and 2 = 1.5. Typical (n = 2) roots are shown by the points A, B, A′, and B′. For consistency with the notation in Eqs. (3) and with Fig. 2(a), point A refers to a (++) solution, whereas B, A′, and B′ denote (+−) solutions. Solutions of the type (−−) and (−+) are given by other intersections, e.g., points R and S, respectively.

Fig. 10
Fig. 10

Graphic construction for surface-wave solutions with 0 = 1.0, 1 = 3.0, and 2 = 1.5 in the n = 0 interval: (a) N = 0 mode, (b) part of N = 1 mode.

Equations (19)

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i τ 1 cot K τ 1 + τ 1 2 + τ 0 τ 2 τ 0 + τ 2 = 0 ,
τ j 2 + κ 2 = j .
( + + ) :             τ 0 = i τ 0 ,             τ 2 = i τ 2 ,
( + - ) :             τ 0 = i τ 0 ,             τ 2 = - i τ 2 ,
( - + ) :             τ 0 = - i τ 0 ,             τ 2 = i τ 2 ,
( - - ) :             τ 0 = - i τ 0 ,             τ 2 = - i τ 2 ,
p = K τ 1 ,
a = [ K 2 ( 1 - 0 ) - p 2 ] 1 / 2 ,
s = [ K 2 ( 1 - 2 ) - p 2 ] 1 / 2 ,
p cot p = { p 2 - a s a + s ( + + ) , p 2 + a s a - s + - , p 2 + a s - a + s ( - + ) , - p 2 + a s a + s ( - - ) .
[ i τ 1 tan ( p / 2 ) - τ 0 ] [ i τ 1 cot ( p / 2 ) + τ 0 ] = 0 ,
τ 2 = ± ( 2 - 0 + τ 0 2 ) 1 / 2 .
p n = n π ± i tanh - 1 ( 2 - 0 0 + 2 - 2 1 ) ,
p n = n π ± tan - 1 ( 2 - 0 1 - 2 ) 1 / 2 .
κ = ± ( 0 - τ 0 2 ) 1 / 2
κ = 0 sin θ 0 = 2 sin θ 2 ,
τ 0 = 0 cos θ 0 ,
τ 2 = 2 cos θ 2 ,
K τ 0 τ 2 = - i ( τ 0 + τ 2 ) ,

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