Abstract

A general graphical study of the complex valued propagation constants, defining the various types of TE waves that are resonant on optical thin-film (slab) waveguides, is presented. The dielectric constants of both the slab and its surroundings are allowed to take on all possible complex values. The corresponding waves are classified in to two general types: surface waves, with conductive losses only; and leaky waves, with both conductive and radiative losses.

© 1970 Optical Society of America

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  1. A. Yariv, C. C. Leite, Appl. Phys. Lett. 2, 55 (1963).
    [CrossRef]
  2. A. Ashkin, M. Gershenzon, J. Appl. Phys. 34, 2116 (1963).
    [CrossRef]
  3. W. W. Anderson, IEEE J. Quantum Electron. QE-1, 228 (1965).
    [CrossRef]
  4. D. F. Nelson, J. McKenna, J. Appl. Phys. 38, 4057 (1967).
    [CrossRef]
  5. P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
    [CrossRef]
  6. J. Kane, H. Osterberg, J. Opt. Soc. Amer. 54, 347 (1964).
    [CrossRef]
  7. H. K. V. Lotsch, Optik 27, 239 (1968).
  8. E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).
  9. T. Sawatari, N. S. Kapany, J. Opt. Soc. Amer. 60, 132 (1970).
    [CrossRef]
  10. E. Snitzer, H. Osterberg, J. Opt. Soc. Amer. 51, 499 (1961).
    [CrossRef]
  11. N. S. Kapany, J. J. Burke, J. Opt. Soc. Amer. 51, 1067 (1961).
    [CrossRef]
  12. S. E. Miller, Bell Syst. Tech. J. 48, 2059 (1969).
  13. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).
  14. W. G. Oldham, A. Bahraman, IEEE J. Quantum Electron. QE-3, 278 (1967).
    [CrossRef]
  15. D. F. Nelson, F. K. Reinhart, Appl. Phys. Lett. 5, 148 (1964).
    [CrossRef]
  16. P. K. Tien, J. Opt. Soc. Amer. 60, 723A (1970).
  17. S. E. Miller, Bell Syst. Tech. J. 48, 2189 (1969).
  18. E. A. J. Marcatili, SPIE Seminar-in-Depth on Fiber Optics (Jan., 1970), Dallas.
  19. S. Barone, Microwave Res. Inst., Polytechnic Inst. of Brooklyn, N.Y., Report No. R-532-56 (November, 1956); S. Barone, A. Hessel, Report No. R-698-58 (Dec., 1958).
  20. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 11.
  21. T. Tamir, A. A. Oliner, Proc. IEEE 51, 317 (1963).
    [CrossRef]
  22. E. S. Cassedy, M. Cohn, IEEE Trans. Microwave Theory Tech. MTT-9, 243 (1961).
    [CrossRef]
  23. T. Tamir, A. A. Oliner, Proc. IEEE (London) 110, 310 (1963).
  24. H. M. Barlow, A. L. Cullen, Proc. IEEE (London) 100, 329 (1953).
  25. R. E. Collin, F. J. Zucker, Eds., Antenna Theory Part II (McGraw-Hill, New York, 1969).
  26. D. E. Norton, 1965 IEEE International Conv. Record, Part 5, p. 200.

1970 (2)

P. K. Tien, J. Opt. Soc. Amer. 60, 723A (1970).

T. Sawatari, N. S. Kapany, J. Opt. Soc. Amer. 60, 132 (1970).
[CrossRef]

1969 (4)

S. E. Miller, Bell Syst. Tech. J. 48, 2189 (1969).

S. E. Miller, Bell Syst. Tech. J. 48, 2059 (1969).

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

1968 (1)

H. K. V. Lotsch, Optik 27, 239 (1968).

1967 (2)

D. F. Nelson, J. McKenna, J. Appl. Phys. 38, 4057 (1967).
[CrossRef]

W. G. Oldham, A. Bahraman, IEEE J. Quantum Electron. QE-3, 278 (1967).
[CrossRef]

1965 (1)

W. W. Anderson, IEEE J. Quantum Electron. QE-1, 228 (1965).
[CrossRef]

1964 (3)

E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

J. Kane, H. Osterberg, J. Opt. Soc. Amer. 54, 347 (1964).
[CrossRef]

D. F. Nelson, F. K. Reinhart, Appl. Phys. Lett. 5, 148 (1964).
[CrossRef]

1963 (4)

T. Tamir, A. A. Oliner, Proc. IEEE 51, 317 (1963).
[CrossRef]

A. Yariv, C. C. Leite, Appl. Phys. Lett. 2, 55 (1963).
[CrossRef]

A. Ashkin, M. Gershenzon, J. Appl. Phys. 34, 2116 (1963).
[CrossRef]

T. Tamir, A. A. Oliner, Proc. IEEE (London) 110, 310 (1963).

1961 (3)

E. Snitzer, H. Osterberg, J. Opt. Soc. Amer. 51, 499 (1961).
[CrossRef]

N. S. Kapany, J. J. Burke, J. Opt. Soc. Amer. 51, 1067 (1961).
[CrossRef]

E. S. Cassedy, M. Cohn, IEEE Trans. Microwave Theory Tech. MTT-9, 243 (1961).
[CrossRef]

1953 (1)

H. M. Barlow, A. L. Cullen, Proc. IEEE (London) 100, 329 (1953).

Anderson, W. W.

W. W. Anderson, IEEE J. Quantum Electron. QE-1, 228 (1965).
[CrossRef]

Ashkin, A.

A. Ashkin, M. Gershenzon, J. Appl. Phys. 34, 2116 (1963).
[CrossRef]

Bahraman, A.

W. G. Oldham, A. Bahraman, IEEE J. Quantum Electron. QE-3, 278 (1967).
[CrossRef]

Barlow, H. M.

H. M. Barlow, A. L. Cullen, Proc. IEEE (London) 100, 329 (1953).

Barone, S.

S. Barone, Microwave Res. Inst., Polytechnic Inst. of Brooklyn, N.Y., Report No. R-532-56 (November, 1956); S. Barone, A. Hessel, Report No. R-698-58 (Dec., 1958).

Burke, J. J.

N. S. Kapany, J. J. Burke, J. Opt. Soc. Amer. 51, 1067 (1961).
[CrossRef]

Cassedy, E. S.

E. S. Cassedy, M. Cohn, IEEE Trans. Microwave Theory Tech. MTT-9, 243 (1961).
[CrossRef]

Cohn, M.

E. S. Cassedy, M. Cohn, IEEE Trans. Microwave Theory Tech. MTT-9, 243 (1961).
[CrossRef]

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 11.

Cullen, A. L.

H. M. Barlow, A. L. Cullen, Proc. IEEE (London) 100, 329 (1953).

Gershenzon, M.

A. Ashkin, M. Gershenzon, J. Appl. Phys. 34, 2116 (1963).
[CrossRef]

Kane, J.

J. Kane, H. Osterberg, J. Opt. Soc. Amer. 54, 347 (1964).
[CrossRef]

Kapany, N. S.

T. Sawatari, N. S. Kapany, J. Opt. Soc. Amer. 60, 132 (1970).
[CrossRef]

N. S. Kapany, J. J. Burke, J. Opt. Soc. Amer. 51, 1067 (1961).
[CrossRef]

Leite, C. C.

A. Yariv, C. C. Leite, Appl. Phys. Lett. 2, 55 (1963).
[CrossRef]

Lotsch, H. K. V.

H. K. V. Lotsch, Optik 27, 239 (1968).

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

E. A. J. Marcatili, SPIE Seminar-in-Depth on Fiber Optics (Jan., 1970), Dallas.

Martin, R. J.

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

McKenna, J.

D. F. Nelson, J. McKenna, J. Appl. Phys. 38, 4057 (1967).
[CrossRef]

Miller, S. E.

S. E. Miller, Bell Syst. Tech. J. 48, 2189 (1969).

S. E. Miller, Bell Syst. Tech. J. 48, 2059 (1969).

Nelson, D. F.

D. F. Nelson, J. McKenna, J. Appl. Phys. 38, 4057 (1967).
[CrossRef]

D. F. Nelson, F. K. Reinhart, Appl. Phys. Lett. 5, 148 (1964).
[CrossRef]

Norton, D. E.

D. E. Norton, 1965 IEEE International Conv. Record, Part 5, p. 200.

Oldham, W. G.

W. G. Oldham, A. Bahraman, IEEE J. Quantum Electron. QE-3, 278 (1967).
[CrossRef]

Oliner, A. A.

T. Tamir, A. A. Oliner, Proc. IEEE 51, 317 (1963).
[CrossRef]

T. Tamir, A. A. Oliner, Proc. IEEE (London) 110, 310 (1963).

Osterberg, H.

J. Kane, H. Osterberg, J. Opt. Soc. Amer. 54, 347 (1964).
[CrossRef]

E. Snitzer, H. Osterberg, J. Opt. Soc. Amer. 51, 499 (1961).
[CrossRef]

Reinhart, F. K.

D. F. Nelson, F. K. Reinhart, Appl. Phys. Lett. 5, 148 (1964).
[CrossRef]

Sawatari, T.

T. Sawatari, N. S. Kapany, J. Opt. Soc. Amer. 60, 132 (1970).
[CrossRef]

Schmeltzer, R. A.

E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

Snitzer, E.

E. Snitzer, H. Osterberg, J. Opt. Soc. Amer. 51, 499 (1961).
[CrossRef]

Tamir, T.

T. Tamir, A. A. Oliner, Proc. IEEE 51, 317 (1963).
[CrossRef]

T. Tamir, A. A. Oliner, Proc. IEEE (London) 110, 310 (1963).

Tien, P. K.

P. K. Tien, J. Opt. Soc. Amer. 60, 723A (1970).

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

Ulrich, R.

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

Yariv, A.

A. Yariv, C. C. Leite, Appl. Phys. Lett. 2, 55 (1963).
[CrossRef]

Appl. Phys. Lett. (3)

A. Yariv, C. C. Leite, Appl. Phys. Lett. 2, 55 (1963).
[CrossRef]

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

D. F. Nelson, F. K. Reinhart, Appl. Phys. Lett. 5, 148 (1964).
[CrossRef]

Bell Syst. Tech. J. (4)

S. E. Miller, Bell Syst. Tech. J. 48, 2059 (1969).

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

S. E. Miller, Bell Syst. Tech. J. 48, 2189 (1969).

E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

IEEE J. Quantum Electron. (2)

W. W. Anderson, IEEE J. Quantum Electron. QE-1, 228 (1965).
[CrossRef]

W. G. Oldham, A. Bahraman, IEEE J. Quantum Electron. QE-3, 278 (1967).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

E. S. Cassedy, M. Cohn, IEEE Trans. Microwave Theory Tech. MTT-9, 243 (1961).
[CrossRef]

J. Appl. Phys. (2)

D. F. Nelson, J. McKenna, J. Appl. Phys. 38, 4057 (1967).
[CrossRef]

A. Ashkin, M. Gershenzon, J. Appl. Phys. 34, 2116 (1963).
[CrossRef]

J. Opt. Soc. Amer. (5)

T. Sawatari, N. S. Kapany, J. Opt. Soc. Amer. 60, 132 (1970).
[CrossRef]

E. Snitzer, H. Osterberg, J. Opt. Soc. Amer. 51, 499 (1961).
[CrossRef]

N. S. Kapany, J. J. Burke, J. Opt. Soc. Amer. 51, 1067 (1961).
[CrossRef]

J. Kane, H. Osterberg, J. Opt. Soc. Amer. 54, 347 (1964).
[CrossRef]

P. K. Tien, J. Opt. Soc. Amer. 60, 723A (1970).

Optik (1)

H. K. V. Lotsch, Optik 27, 239 (1968).

Proc. IEEE (1)

T. Tamir, A. A. Oliner, Proc. IEEE 51, 317 (1963).
[CrossRef]

Proc. IEEE (London) (2)

T. Tamir, A. A. Oliner, Proc. IEEE (London) 110, 310 (1963).

H. M. Barlow, A. L. Cullen, Proc. IEEE (London) 100, 329 (1953).

Other (5)

R. E. Collin, F. J. Zucker, Eds., Antenna Theory Part II (McGraw-Hill, New York, 1969).

D. E. Norton, 1965 IEEE International Conv. Record, Part 5, p. 200.

E. A. J. Marcatili, SPIE Seminar-in-Depth on Fiber Optics (Jan., 1970), Dallas.

S. Barone, Microwave Res. Inst., Polytechnic Inst. of Brooklyn, N.Y., Report No. R-532-56 (November, 1956); S. Barone, A. Hessel, Report No. R-698-58 (Dec., 1958).

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 11.

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of thin film waveguide of thickness 2a embedded in a homogeneous medium. The permittivity, conductivity, and permeability of the media are as indicated.

Fig. 2
Fig. 2

A graphical solution of Eqs. (5a) and (9a) for real values of u and R2 = (2πa0)2 (n12n22) is shown. The intersections of the solid and dashed curve above the u axis yield the eigenvalues of the surface wave modes; those below the axis give negative values of q, so that the interior fields increase with distance from the slab. The integers identify the eigenvalues of the TEN mode.19

Fig. 3
Fig. 3

An alternate graphical solution of Eqs. (5), (6), and (9) for three values of the R2, with u and q real.

Fig. 4
Fig. 4

Solutions of Eqs. (5) and (6) (long dashes) and Eqs. (6) and (9) (short dashes) for imaginary values of u = i u″ and real values of q. These are plots of Eqs. (9c) and (5c) of text.

Fig. 5
Fig. 5

Root loci of Eqs. (5) and (6) on the right half of the complex u (= u′ + i u″) plane for complex values of R2Ms expiϕs [Eqs. (5d, 5e, 5f)]. ϕs = 0 on the u′ axis. The u′ and u″ axes are scaled above and to the left of the figure, respectively. The double lines divide the half plane into domains (branches) in each of which R2 takes on all possible physical values. On the first branch, lines of equal ϕs run from (u′,u″) = (0,0) where Ms = 0 to (π/2,0) where Ms = ∞. On other branches, they originate at u″ = ∞ where Ms = 0 and terminate at u′ = (2n + 1) π/2 where Ms = ∞. Roots lying inside the dashed cutoff curves, where Re(q) = 0, correspond to complex surface waves that are localized in and around the guide. Roots lying outside these curves yield leaky waves that radiate away from the guide.

Fig. 6
Fig. 6

Root loci of Eqs. (6) and (9) on the right half of the complex u plane for complex values of R2Maexpiϕa [Eqs. (9d, 9e, 9f)]. Lines of equal ϕa originate at u″ = ∞ where Ma = 0 and terminate at u = , where Ma = ∞ (n = 1,2,3⋯). As in Fig. 5, double lines separate successive branches, while the dashed cutoff curves, where Re(q) = 0, separate the complex surface wave roots from those of leaky waves.

Fig. 7
Fig. 7

Successive root loci of Eqs. (6) and (9) for a small value of arg (R2).

Fig. 8
Fig. 8

Successive root loci of Eqs. (5) and (6) for arg (R2) slightly less than π.

Fig. 9
Fig. 9

Successive root loci of Eqs. (5) and (6) for a small value of arg (R2).

Equations (35)

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E y = A cos ( u x / a ) exp ( i ω t - i h z ) , H x = ( - h / ω μ 0 ) E y , H z = ( - i u / ω μ 0 a ) tan ( u x / a ) E y , E x = E z = H y = 0 ,             x a
E y = A e ± q cos ( u ) exp ( q x / a + i ω t - i h z ) , H x = ( - h / ω μ 0 ) E y , H z = ( i q / ω μ 0 a ) E y , E x = E z = H y = 0 ,             a x
ω 2 1 μ 0 ( 1 - i σ 1 / ω 1 ) = k 1 2 = k 0 2 n 1 2 = ( 2 π n 1 / λ 0 ) 2 ω 2 2 μ 0 ( 1 - i σ 2 / ω 2 ) = k 2 2 = k 0 2 n 2 2 = ( 2 π n 2 / λ 0 ) 2
h 2 = k 1 2 - u 2 / a 2 = k 2 2 + q 2 / a 2
q = u tan ( u )
u 2 + q 2 = k 0 2 a 2 ( n 1 2 - n 2 2 ) = R 2 .
K = n 2 - n 2 ; σ / ω 0 = 2 n n , n n } = ( K / 2 ) 1 2 [ ( 1 + σ 2 / ω 2 2 ) 1 2 ± 1 ] 1 2 .
E y = A sin ( u x / a ) exp ( i ω t - i h z ) , H x = ( - h / ω μ 0 ) E y , H z = ( i u / ω μ 0 a ) cot ( u x / a ) E y , E x = E z = H y = 0 ,             x a
E y = ± A e ± q sin ( u ) exp ( q x / a + i ω t - i h z ) , H x = ( - h / ω μ 0 ) E y , H z = ( i q / ω μ 0 a ) E y , E x = E z = H y = 0 , a x
q = - u cot ( u ) .
cot ( u ) = u / ( R 2 - u 2 ) 1 2 ,
- tan ( u ) = u / ( R 2 - u 2 ) 1 2 .
u / a = k 1 cos ( θ N ) ; h = k 1 sin ( θ N ) .
q = - u tanh ( u ) ,
q 2 - u 2 = R 2 ,
q = - u coth ( u ) .
R 2 = u 2 / sinh 2 ( u ) .
h 2 = k 1 2 + u 2 / a 2 ,
R 2 = - u 2 / cosh 2 ( u ) .
R 2 = u 2 / cos 2 ( u ) .
M s = 2 ( u 2 + u 2 ) / ( cosh 2 u + cos 2 u )
ϕ s = 2 tan - 1 ( u / u ) + 2 tan - 1 ( tan u tanh u ) .
R 2 = u 2 / sin 2 ( u )
M a = 2 ( u 2 + u 2 ) / ( cosh 2 u - cos 2 u )
ϕ a = 2 tan - 1 ( u / u ) - 2 tan - 1 ( cot u tanh u ) .
ϕ a 2 u when M a 0.
u coth u = u cot u .
tan u N = u N .
u tanh u = - u tan u .
ϕ s - π + 2 u when M s 0.
E y = A exp ( - u x / a + h z ) exp ( i ω t + i u x / a - i h z ) + A exp ( u x / a + h z ) exp ( i ω t - i u x / a - i h z ) .
E y = C exp ( - q x / a + h z ) exp ( i ω t - i q x / a - i h z ) .
u sin 2 u = u sinh 2 u .
u sin 2 u = - u sinh 2 u ,
h 2 = k 0 2 n e 2 = ( n 1 2 / Δ n 2 ) R 2 / a 2 - u 2 / a 2 = ( n 2 2 / Δ n 2 ) R 2 / a 2 + q 2 / a 2 = ( k 2 2 u 2 + k 1 2 q 2 ) / R 2 ,

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