Abstract

The ways in which eigenmode expansions describe nonbound fields are compared and used to understand leaky guiding in nontransparent waveguides. Accurate modeling of leaky behavior is seen to be quite different in the cases of absorption and gain.

© 1995 Optical Society of America

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References

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  1. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  2. V. V. Shevchenko, “The expansion of the fields of open waveguides in proper and improper modes,” Radio Phys. Quantum Electron. 14, 972–977 (1974).
    [Crossref]
  3. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, New York, 1991).
  4. A. K. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Electron. 17, 311–321 (1985).
    [Crossref]
  5. T. Tamir, A. A. Oliner, “The spectrum of electromagnetic waves guided by a plasma layer,” Proc. IEEE 110, 317–332 (1963).
    [Crossref]
  6. J. Gerdes, B. Lunitz, D. Benish, R. Pregla, “Analysis of slab waveguide discontinuities including radiation and absorption effects,” Electron. Lett.28, 1013–1015 (1992).
  7. E. C. M. Pennings, R. J. Deri, R. J. Hawkins, “Quenching of resonantly enhanced absorption in vertical coupled-waveguide photodetectors,” in Integrated Photonics Research, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 82–83.
  8. L. J. Mawst, D. Botez, C. Zmudzinski, C. Tu, “Antiresonant reflecting optical waveguide-type, single-mode diode lasers,” Appl. Phys. Lett.61, 503–505 (1992).
  9. F. Abeles, “Recherches sur le propagation des ondes electromagnétiques sinusoïdales dans les milieux stratifiés: application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–682.
  10. R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” J. Quantum Electron. 28, 1520–1526 (1992).
    [Crossref]
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. 1.
  12. R. E. Smith, “Modal expansions in transparent and nontransparent planar waveguides,” Ph.D. dissertation (College of Engineering and Applied Science, University of Rochester, Rochester, N.Y., 1993).
  13. R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Unfolding the multivalued planar waveguide dispersion relation,” J. Quantum Electron. 29, 1031–1034 (1993).
    [Crossref]
  14. R. E. Smith, S. N. Houde-Walter, “The migration of bound and leaky solutions to the waveguide dispersion relation,” J. Lightwave Technol. 11, 1760–1768 (1993).
    [Crossref]
  15. E. T. Whitaker, G. N. Watson, A Course in Modern Analysis (Cambridge U. Press, Cambridge, 1980).

1993 (2)

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Unfolding the multivalued planar waveguide dispersion relation,” J. Quantum Electron. 29, 1031–1034 (1993).
[Crossref]

R. E. Smith, S. N. Houde-Walter, “The migration of bound and leaky solutions to the waveguide dispersion relation,” J. Lightwave Technol. 11, 1760–1768 (1993).
[Crossref]

1992 (3)

J. Gerdes, B. Lunitz, D. Benish, R. Pregla, “Analysis of slab waveguide discontinuities including radiation and absorption effects,” Electron. Lett.28, 1013–1015 (1992).

L. J. Mawst, D. Botez, C. Zmudzinski, C. Tu, “Antiresonant reflecting optical waveguide-type, single-mode diode lasers,” Appl. Phys. Lett.61, 503–505 (1992).

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” J. Quantum Electron. 28, 1520–1526 (1992).
[Crossref]

1985 (1)

A. K. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Electron. 17, 311–321 (1985).
[Crossref]

1974 (1)

V. V. Shevchenko, “The expansion of the fields of open waveguides in proper and improper modes,” Radio Phys. Quantum Electron. 14, 972–977 (1974).
[Crossref]

1963 (1)

T. Tamir, A. A. Oliner, “The spectrum of electromagnetic waves guided by a plasma layer,” Proc. IEEE 110, 317–332 (1963).
[Crossref]

Abeles, F.

F. Abeles, “Recherches sur le propagation des ondes electromagnétiques sinusoïdales dans les milieux stratifiés: application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–682.

Benish, D.

J. Gerdes, B. Lunitz, D. Benish, R. Pregla, “Analysis of slab waveguide discontinuities including radiation and absorption effects,” Electron. Lett.28, 1013–1015 (1992).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. 1.

Botez, D.

L. J. Mawst, D. Botez, C. Zmudzinski, C. Tu, “Antiresonant reflecting optical waveguide-type, single-mode diode lasers,” Appl. Phys. Lett.61, 503–505 (1992).

Deri, R. J.

E. C. M. Pennings, R. J. Deri, R. J. Hawkins, “Quenching of resonantly enhanced absorption in vertical coupled-waveguide photodetectors,” in Integrated Photonics Research, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 82–83.

Forbes, G. W.

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Unfolding the multivalued planar waveguide dispersion relation,” J. Quantum Electron. 29, 1031–1034 (1993).
[Crossref]

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” J. Quantum Electron. 28, 1520–1526 (1992).
[Crossref]

Gerdes, J.

J. Gerdes, B. Lunitz, D. Benish, R. Pregla, “Analysis of slab waveguide discontinuities including radiation and absorption effects,” Electron. Lett.28, 1013–1015 (1992).

Ghatak, A. K.

A. K. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Electron. 17, 311–321 (1985).
[Crossref]

Hawkins, R. J.

E. C. M. Pennings, R. J. Deri, R. J. Hawkins, “Quenching of resonantly enhanced absorption in vertical coupled-waveguide photodetectors,” in Integrated Photonics Research, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 82–83.

Houde-Walter, S. N.

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Unfolding the multivalued planar waveguide dispersion relation,” J. Quantum Electron. 29, 1031–1034 (1993).
[Crossref]

R. E. Smith, S. N. Houde-Walter, “The migration of bound and leaky solutions to the waveguide dispersion relation,” J. Lightwave Technol. 11, 1760–1768 (1993).
[Crossref]

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” J. Quantum Electron. 28, 1520–1526 (1992).
[Crossref]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Lunitz, B.

J. Gerdes, B. Lunitz, D. Benish, R. Pregla, “Analysis of slab waveguide discontinuities including radiation and absorption effects,” Electron. Lett.28, 1013–1015 (1992).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, New York, 1991).

Mawst, L. J.

L. J. Mawst, D. Botez, C. Zmudzinski, C. Tu, “Antiresonant reflecting optical waveguide-type, single-mode diode lasers,” Appl. Phys. Lett.61, 503–505 (1992).

Oliner, A. A.

T. Tamir, A. A. Oliner, “The spectrum of electromagnetic waves guided by a plasma layer,” Proc. IEEE 110, 317–332 (1963).
[Crossref]

Pennings, E. C. M.

E. C. M. Pennings, R. J. Deri, R. J. Hawkins, “Quenching of resonantly enhanced absorption in vertical coupled-waveguide photodetectors,” in Integrated Photonics Research, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 82–83.

Pregla, R.

J. Gerdes, B. Lunitz, D. Benish, R. Pregla, “Analysis of slab waveguide discontinuities including radiation and absorption effects,” Electron. Lett.28, 1013–1015 (1992).

Shevchenko, V. V.

V. V. Shevchenko, “The expansion of the fields of open waveguides in proper and improper modes,” Radio Phys. Quantum Electron. 14, 972–977 (1974).
[Crossref]

Smith, R. E.

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Unfolding the multivalued planar waveguide dispersion relation,” J. Quantum Electron. 29, 1031–1034 (1993).
[Crossref]

R. E. Smith, S. N. Houde-Walter, “The migration of bound and leaky solutions to the waveguide dispersion relation,” J. Lightwave Technol. 11, 1760–1768 (1993).
[Crossref]

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” J. Quantum Electron. 28, 1520–1526 (1992).
[Crossref]

R. E. Smith, “Modal expansions in transparent and nontransparent planar waveguides,” Ph.D. dissertation (College of Engineering and Applied Science, University of Rochester, Rochester, N.Y., 1993).

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Tamir, T.

T. Tamir, A. A. Oliner, “The spectrum of electromagnetic waves guided by a plasma layer,” Proc. IEEE 110, 317–332 (1963).
[Crossref]

Tu, C.

L. J. Mawst, D. Botez, C. Zmudzinski, C. Tu, “Antiresonant reflecting optical waveguide-type, single-mode diode lasers,” Appl. Phys. Lett.61, 503–505 (1992).

Watson, G. N.

E. T. Whitaker, G. N. Watson, A Course in Modern Analysis (Cambridge U. Press, Cambridge, 1980).

Whitaker, E. T.

E. T. Whitaker, G. N. Watson, A Course in Modern Analysis (Cambridge U. Press, Cambridge, 1980).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. 1.

Zmudzinski, C.

L. J. Mawst, D. Botez, C. Zmudzinski, C. Tu, “Antiresonant reflecting optical waveguide-type, single-mode diode lasers,” Appl. Phys. Lett.61, 503–505 (1992).

Ann. Phys. (Paris) (1)

F. Abeles, “Recherches sur le propagation des ondes electromagnétiques sinusoïdales dans les milieux stratifiés: application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–682.

Appl. Phys. Lett. (1)

L. J. Mawst, D. Botez, C. Zmudzinski, C. Tu, “Antiresonant reflecting optical waveguide-type, single-mode diode lasers,” Appl. Phys. Lett.61, 503–505 (1992).

Electron. Lett. (1)

J. Gerdes, B. Lunitz, D. Benish, R. Pregla, “Analysis of slab waveguide discontinuities including radiation and absorption effects,” Electron. Lett.28, 1013–1015 (1992).

J. Lightwave Technol. (1)

R. E. Smith, S. N. Houde-Walter, “The migration of bound and leaky solutions to the waveguide dispersion relation,” J. Lightwave Technol. 11, 1760–1768 (1993).
[Crossref]

J. Quantum Electron. (2)

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Unfolding the multivalued planar waveguide dispersion relation,” J. Quantum Electron. 29, 1031–1034 (1993).
[Crossref]

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” J. Quantum Electron. 28, 1520–1526 (1992).
[Crossref]

Opt. Quantum Electron. (1)

A. K. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Electron. 17, 311–321 (1985).
[Crossref]

Proc. IEEE (1)

T. Tamir, A. A. Oliner, “The spectrum of electromagnetic waves guided by a plasma layer,” Proc. IEEE 110, 317–332 (1963).
[Crossref]

Radio Phys. Quantum Electron. (1)

V. V. Shevchenko, “The expansion of the fields of open waveguides in proper and improper modes,” Radio Phys. Quantum Electron. 14, 972–977 (1974).
[Crossref]

Other (6)

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, New York, 1991).

E. C. M. Pennings, R. J. Deri, R. J. Hawkins, “Quenching of resonantly enhanced absorption in vertical coupled-waveguide photodetectors,” in Integrated Photonics Research, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 82–83.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. 1.

R. E. Smith, “Modal expansions in transparent and nontransparent planar waveguides,” Ph.D. dissertation (College of Engineering and Applied Science, University of Rochester, Rochester, N.Y., 1993).

E. T. Whitaker, G. N. Watson, A Course in Modern Analysis (Cambridge U. Press, Cambridge, 1980).

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Figures (18)

Fig. 1
Fig. 1

Planar waveguide. Solid lines, layer interfaces; dashed lines, locations of imposed boundaries.

Fig. 2
Fig. 2

Schematic representation of eigenvalues of the (a) closed, (b) continuum, and (c) improper expansions. X’s, discrete eigenvalues; heavy solid line, continuum of eigenvalues.

Fig. 3
Fig. 3

Modal field profiles of the four types of mode found in the closed and the continuum bases (see text).

Fig. 4
Fig. 4

Three-section planar waveguide separated by abrupt longitudinal interfaces. Using this geometry, we consider the coupling efficiency between modes of the input and modes of the output waveguide.

Fig. 5
Fig. 5

Contour of integration used with Eq. (15).

Fig. 6
Fig. 6

Contours of integration used with Eq. (19) shown schematically with the associated poles of I(αN).

Fig. 7
Fig. 7

Three-section waveguide discussed in Section 4. The center section is leaky; both the input and the output guides support one bound mode of each polarization.

Fig. 8
Fig. 8

Amplitude coupling efficiency |t0/i0| between the input and the output waveguides of Fig. 7. Solid curve, four coincident traces generated by Eqs. (28)(30); dashed curve, spurious reflection that can occur as a result of placement of a perfectly reflecting boundary too close to the waveguide.

Fig. 9
Fig. 9

Highly localized peak in the excitation of the substrate modes compared with the Lorentzian peak predicted by Eq. (27).

Fig. 10
Fig. 10

Poles of I(αN) for the transparent case (n″ = 0) of the waveguide in Fig. 7. The two contours enclose a single pole corresponding to a single leaky mode. A detail of (a) is shown in (b). ×, T21 = 0; +, T11 = 0.

Fig. 11
Fig. 11

Amplitude coupling efficiency |t0/i0| between the input and the output mode as a function of the length of the center guide. Solid curves, different amounts of absorption; each solid curve represents coincident traces resulting from the evaluation of Eqs. (28)(30).

Fig. 12
Fig. 12

Excitation Θ 00 ( β ) of substrate modes corresponding to (a) n″ = 10−4 and n″ = 10−5, i.e., the two lower traces in Fig. 11.

Fig. 13
Fig. 13

Excitation of substrate modes shown in Fig. 12(a) compared with the approximation given by Eq. (31).

Fig. 14
Fig. 14

Amplitude coupling efficiency |t0/i0| between the input and the output modes as a function of the length of the center section D for the waveguide in Fig. 7 with n″ = 10−3. The three solid curves correspond to different amounts of absorption; they are actually three coincident traces generated by Eqs. (32)(34).

Fig. 15
Fig. 15

(a) Excitation of substrate modes Θ 00 ( β ) for the waveguide in Fig. 7 with n″ = 10−3; (b) a comparison of the amplitude coupling efficiency |t0/i0| between the input and the output modes predicted by use of the proper mode alone (long-dashed curve), the substrate modes alone (short-dashed curve), and the complete description, previously shown in Fig. 14 (solid curve).

Fig. 16
Fig. 16

Poles of I ( Ñ ) for which the addition of sufficient absorption (n″ = 10−3) has caused migration of a second pole into the first quadrant, [i.e., two residue terms in Eq. (21)].

Fig. 17
Fig. 17

Poles of I ( Ñ ) for which the addition of sufficient gain (n″ = −10−3) has caused the pole that originally contributed to Eq. (21) to migrate out of the first quadrant.

Fig. 18
Fig. 18

Amplitude coupling efficiency |t0/i0| between the input and the output modes as a function of the length of the center section D. The solid curve is actually three coincident traces generated by Eqs. (35)(37).

Tables (1)

Tables Icon

Table 1 Overlap Coefficients for the Various Values of n″

Equations (37)

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( TE ) Ē ( x , z ) = E 0 ŷ m ( x ) exp [ i ( β z ω t ) ] ,
( TM ) H ¯ ( x , z ) = H 0 ŷ m ( x ) exp [ i ( β z ω t ) ]
2 m ( x ) x 2 + ( k j 2 β 2 ) m ( x ) = 0 .
m ( x ) = A j ( x ) exp { α j ( x ) [ x d j ( x ) ] } + B j ( x ) exp { α j ( x ) [ x d j ( x ) ] } .
k j = k 0 n , with n = ( n Re + i n Im ) and k 0 = ω / c .
( A j B j ) = M j ( A j 1 B j 1 ) ,
M j = 1 2 ρ j [ ( ρ j + α j 1 α j ) e δ j ( ρ j α j 1 α j ) e δ j ( ρ j α j 1 α j ) e δ j ( ρ j + α j 1 α j ) e δ j ] ,
( A N B N ) = [ T 11 T 12 T 12 T 22 ] ( A 0 B 0 ) ,
T = M N M N 1 M 1 .
b 0 m ( d 0 w 0 ) + b 0 ( d 0 w 0 ) = 0 ,
b N m ( d N ) + b N ( d N ) = 0 ,
[ ( b N + b N α N ) ( b N b N α N ) [ T 11 T 12 T 21 T 22 ] [ e δ 0 ( b 0 b 0 / α 0 ) e δ 0 ( b 0 + b 0 / α 0 ) ] = 0 ,
t l i j = k bound O jk O kl O jj O kk exp ( i β k D ) + k nonbound O jk O kl O jj O kk exp ( i β k D ) ,
O ab = m a ( x ) m b ( x ) d x ,
t l i j = k bound O jk O kl O jj O kk exp ( i β D ) + C 1 Θ j l ( β ) exp ( i β D ) d β ,
Θ jl ( β ) = O j β O β l O jj O β β ( w Ñ π β k Ñ 2 β 2 ) .
Θ jl ( β ) = 1 2 π O j β O β l O j j Ã Ñ B Ñ ( β k Ñ 2 β 2 ) ,
P ( z = 0 ) = 1 2 E 0 2 μ 0 ω [ k = 0 bound | O jk | 2 β k + 0 k Ñ Θ jj ( β ) β d β ] ,
t l i j = k bound O jk O kl O jj O kk exp ( i β k D ) + 1 2 π i C 2 O j β O β l O jj Ã Ñ B Ñ exp ( i β D ) d Ñ ,
1 2 π i [ C 2 I ( Ñ ) exp ( i β D ) d Ñ C 2 I ( Ñ ) exp ( i β D ) d Ñ ] = Res [ I ( Ñ ) ] exp ( i β D ) ,
I ( Ñ ) = O j β O β l O jj Ã Ñ B Ñ .
t l i j = k bound O jk O kl O jj O kk exp ( i β k D ) + k Res Res [ O j β O β l O jj T 11 T 12 ] exp ( i β D ) + 1 2 π i C 2 O j β O β l O jj T 11 T 12 exp ( i β D ) d Ñ .
t l i j = k bound ( O jk ) 2 exp ( i β k D ) + k Res ( O ¯ jk ) 2 exp ( i β k D ) + ϑ ( D ) ,
( O ¯ jk ) 2 = ( O j β O β l O jj T 11 ) / d ( T 21 ) d ( Ñ )
( O ¯ jk ) 2 = ( O j β O β l O jj T 21 ) / d ( T 11 ) d ( Ñ ) ,
C 1 Θ 00 ( β ) exp ( i β D ) d β = ( O ¯ 0 L ) 2 exp ( i β L D ) ,
Θ 0 L ( β ) = ( O 0 L ) 2 1 π Im [ β L ] Im [ β L ] 2 + ( β Re [ β L ] ) 2 .
t 0 i 0 = k nonbound ( O 0 k ) 2 exp ( i β k D ) ,
t 0 i 0 = C 1 Θ 00 ( β ) exp ( i β D ) d β ,
t 0 i 0 = ( O ¯ 0 L ) 2 exp ( i β L D ) .
Θ 00 ( β ) = ( O ¯ 0 L ) 2 1 π × Im [ β L R ] Im [ β L R ] 2 + [ β ( Re [ β L ] + i Im [ β L A ] ) ] 2 ,
t 0 i 0 = ( O 0 P ) 2 exp ( i β P D ) + k nonbound ( O 0 k ) 2 exp ( i β D ) ,
t 0 i 0 = ( O 0 P ) 2 exp ( i β P D ) + C 1 Θ 00 ( β ) exp ( i β D ) d β ,
t 0 i 0 = ( O 0 P ) 2 exp ( i β P D ) + ( O ¯ 0 P ) 2 exp ( i β P D ) + ( O ¯ 0 L ) 2 exp ( i β L D ) ,
t 0 i 0 = ( O 0 P ) 2 exp ( i β P D ) + k substrate ( O 0 k ) 2 exp ( i β k D ) ,
t 0 i 0 = ( O 0 P ) 2 exp ( i β P D ) + C 1 Θ 00 ( β ) exp ( i β D ) d β ,
t 0 i 0 = ( O 0 P ) 2 exp ( i β P D ) .

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