Abstract

Recursion relationships are developed which generate the characteristic mode equations for both free and prism-loaded optical waveguides composed of N arbitrary dielectric layers. The solutions of these equations allow the computation of the effective mode-propagation coefficients and, in the case of prism loading, leakage rates and coupling efficiencies for arbitrary gap shape and coupling strength. As an illustration four specific prism/waveguide couplers with linearly tapered gaps are numerically analyzed. Two of these are single-layer waveguides (N = 1), and the other two are multilayered waveguides (N = 19) with ni chosen to simulate exponential index gradients. Results obtained from the present theory in the weak coupling limit are shown to agree with those obtained in previous work.

© 1981 Optical Society of America

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  1. Y. K. Lee, S. Wang, Appl. Phys. Lett. 25, 164 (1974).
    [CrossRef]
  2. I. P. Kaminow, J. A. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
    [CrossRef]
  3. R. L. Holman, P. J. Cressman, J. F. Revelli, Appl. Phys. Lett. 32, 280 (1978).
    [CrossRef]
  4. R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
    [CrossRef]
  5. G. L. Tangonan, M. K. Barnoski, J. F. Lotspeich, A. Lee, Appl. Phys. Lett. 30, 238 (1977).
    [CrossRef]
  6. P. K. Tien, R. Ulrich, J. Opt. Soc. Am. 60, 1325 (1970).
    [CrossRef]
  7. R. Ulrich, J. Opt. Soc. Am. 60, 1337 (1970).
    [CrossRef]
  8. J. E. Midwinter, IEEE J. Quantum Electron. QE-6, 583 (1970).
    [CrossRef]
  9. R. Ulrich, J. Opt. Soc. Am. 61, 1467 (1971).
    [CrossRef]
  10. E. M. Conwell, Appl. Phys. Lett. 23, 328 (1973).
    [CrossRef]
  11. E. M. Conwell, IEEE J. Quantum Electron. QE-10, 608 (1974).
    [CrossRef]
  12. E. M. Conwell, Appl. Phys. Lett. 25, 40 (1974).
    [CrossRef]
  13. E. M. Conwell, J. Appl. Phys. 46, 1407 (1975).
    [CrossRef]
  14. P. K. Tien, S. Riva-Sanseverino, R. J. Martin, A. A. Ballman, H. Brown, Appl. Phys. Lett. 24, 503 (1974).
    [CrossRef]
  15. G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
    [CrossRef]
  16. D. Sarid, D. Kermisch, Appl. Phys. Lett. 33, 619 (1978).
    [CrossRef]
  17. N. Uchida, Appl. Opt. 15, 179 (1976).
    [CrossRef] [PubMed]
  18. N. Uchida, O. Mikami, S. Uehara, J. Noda, Appl. Opt. 15, 455 (1976).
    [CrossRef] [PubMed]
  19. J. Noda, S. Zembutsu, S. Fukunishi, N. Uchida, Appl. Opt. 17, 1953 (1978).
    [CrossRef] [PubMed]
  20. J. F. Revelli, D. Sarid, J. Appl. Phys. 51, 3566 (1980).
    [CrossRef]
  21. J. F. Revelli, J. Appl. Phys. 52, 3185 (1981).
    [CrossRef]
  22. See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).
  23. E. Conwell, Phys. Today 29, 48 (1976).
    [CrossRef]
  24. See, for example, L. D. Landau, E. M. Lifshits, Electrodynamics of Continuous Media (Pergamon, New York, 1960).
  25. D. Marcuse, IEEE J. Quantum Electron. QE-9, 1000 (1973).
    [CrossRef]
  26. G. L. Tangonan, M. K. Barnoski, A. Lee, Appl. Opt. 16, 1795 (1977).
    [CrossRef] [PubMed]
  27. Note that the sign in Eq. (22) is opposite to the one that appears in Eq. (39) of Ref. 9. There is apparently a typographical error in the reference.
  28. D. Sarid, D. Kermisch, J. F. Revelli, Appl. Phys. Lett. 51, 6105 (1980).
  29. R. L. Holman, P. J. Cressman, Ferroelectrics 27, 85 (1980).
    [CrossRef]

1981 (1)

J. F. Revelli, J. Appl. Phys. 52, 3185 (1981).
[CrossRef]

1980 (3)

J. F. Revelli, D. Sarid, J. Appl. Phys. 51, 3566 (1980).
[CrossRef]

D. Sarid, D. Kermisch, J. F. Revelli, Appl. Phys. Lett. 51, 6105 (1980).

R. L. Holman, P. J. Cressman, Ferroelectrics 27, 85 (1980).
[CrossRef]

1978 (3)

J. Noda, S. Zembutsu, S. Fukunishi, N. Uchida, Appl. Opt. 17, 1953 (1978).
[CrossRef] [PubMed]

R. L. Holman, P. J. Cressman, J. F. Revelli, Appl. Phys. Lett. 32, 280 (1978).
[CrossRef]

D. Sarid, D. Kermisch, Appl. Phys. Lett. 33, 619 (1978).
[CrossRef]

1977 (2)

G. L. Tangonan, M. K. Barnoski, J. F. Lotspeich, A. Lee, Appl. Phys. Lett. 30, 238 (1977).
[CrossRef]

G. L. Tangonan, M. K. Barnoski, A. Lee, Appl. Opt. 16, 1795 (1977).
[CrossRef] [PubMed]

1976 (3)

1975 (2)

E. M. Conwell, J. Appl. Phys. 46, 1407 (1975).
[CrossRef]

G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[CrossRef]

1974 (5)

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, A. A. Ballman, H. Brown, Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

E. M. Conwell, IEEE J. Quantum Electron. QE-10, 608 (1974).
[CrossRef]

E. M. Conwell, Appl. Phys. Lett. 25, 40 (1974).
[CrossRef]

R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
[CrossRef]

Y. K. Lee, S. Wang, Appl. Phys. Lett. 25, 164 (1974).
[CrossRef]

1973 (3)

I. P. Kaminow, J. A. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[CrossRef]

E. M. Conwell, Appl. Phys. Lett. 23, 328 (1973).
[CrossRef]

D. Marcuse, IEEE J. Quantum Electron. QE-9, 1000 (1973).
[CrossRef]

1971 (1)

1970 (3)

Ballman, A. A.

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, A. A. Ballman, H. Brown, Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

Barnoski, M. K.

G. L. Tangonan, M. K. Barnoski, J. F. Lotspeich, A. Lee, Appl. Phys. Lett. 30, 238 (1977).
[CrossRef]

G. L. Tangonan, M. K. Barnoski, A. Lee, Appl. Opt. 16, 1795 (1977).
[CrossRef] [PubMed]

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Brown, H.

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, A. A. Ballman, H. Brown, Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

Burns, W. K.

G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[CrossRef]

Carruthers, J. A.

I. P. Kaminow, J. A. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[CrossRef]

Conwell, E.

E. Conwell, Phys. Today 29, 48 (1976).
[CrossRef]

Conwell, E. M.

E. M. Conwell, J. Appl. Phys. 46, 1407 (1975).
[CrossRef]

E. M. Conwell, IEEE J. Quantum Electron. QE-10, 608 (1974).
[CrossRef]

E. M. Conwell, Appl. Phys. Lett. 25, 40 (1974).
[CrossRef]

E. M. Conwell, Appl. Phys. Lett. 23, 328 (1973).
[CrossRef]

Cressman, P. J.

R. L. Holman, P. J. Cressman, Ferroelectrics 27, 85 (1980).
[CrossRef]

R. L. Holman, P. J. Cressman, J. F. Revelli, Appl. Phys. Lett. 32, 280 (1978).
[CrossRef]

Fukunishi, S.

Hocker, G. B.

G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[CrossRef]

Holman, R. L.

R. L. Holman, P. J. Cressman, Ferroelectrics 27, 85 (1980).
[CrossRef]

R. L. Holman, P. J. Cressman, J. F. Revelli, Appl. Phys. Lett. 32, 280 (1978).
[CrossRef]

Kaminow, I. P.

R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
[CrossRef]

I. P. Kaminow, J. A. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[CrossRef]

Kermisch, D.

D. Sarid, D. Kermisch, J. F. Revelli, Appl. Phys. Lett. 51, 6105 (1980).

D. Sarid, D. Kermisch, Appl. Phys. Lett. 33, 619 (1978).
[CrossRef]

Landau, L. D.

See, for example, L. D. Landau, E. M. Lifshits, Electrodynamics of Continuous Media (Pergamon, New York, 1960).

Lee, A.

G. L. Tangonan, M. K. Barnoski, A. Lee, Appl. Opt. 16, 1795 (1977).
[CrossRef] [PubMed]

G. L. Tangonan, M. K. Barnoski, J. F. Lotspeich, A. Lee, Appl. Phys. Lett. 30, 238 (1977).
[CrossRef]

Lee, Y. K.

Y. K. Lee, S. Wang, Appl. Phys. Lett. 25, 164 (1974).
[CrossRef]

Lifshits, E. M.

See, for example, L. D. Landau, E. M. Lifshits, Electrodynamics of Continuous Media (Pergamon, New York, 1960).

Lotspeich, J. F.

G. L. Tangonan, M. K. Barnoski, J. F. Lotspeich, A. Lee, Appl. Phys. Lett. 30, 238 (1977).
[CrossRef]

Marcuse, D.

D. Marcuse, IEEE J. Quantum Electron. QE-9, 1000 (1973).
[CrossRef]

Martin, R. J.

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, A. A. Ballman, H. Brown, Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

Midwinter, J. E.

J. E. Midwinter, IEEE J. Quantum Electron. QE-6, 583 (1970).
[CrossRef]

Mikami, O.

Noda, J.

Revelli, J. F.

J. F. Revelli, J. Appl. Phys. 52, 3185 (1981).
[CrossRef]

J. F. Revelli, D. Sarid, J. Appl. Phys. 51, 3566 (1980).
[CrossRef]

D. Sarid, D. Kermisch, J. F. Revelli, Appl. Phys. Lett. 51, 6105 (1980).

R. L. Holman, P. J. Cressman, J. F. Revelli, Appl. Phys. Lett. 32, 280 (1978).
[CrossRef]

Riva-Sanseverino, S.

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, A. A. Ballman, H. Brown, Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

Sarid, D.

J. F. Revelli, D. Sarid, J. Appl. Phys. 51, 3566 (1980).
[CrossRef]

D. Sarid, D. Kermisch, J. F. Revelli, Appl. Phys. Lett. 51, 6105 (1980).

D. Sarid, D. Kermisch, Appl. Phys. Lett. 33, 619 (1978).
[CrossRef]

Schmidt, R. V.

R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
[CrossRef]

Tangonan, G. L.

G. L. Tangonan, M. K. Barnoski, J. F. Lotspeich, A. Lee, Appl. Phys. Lett. 30, 238 (1977).
[CrossRef]

G. L. Tangonan, M. K. Barnoski, A. Lee, Appl. Opt. 16, 1795 (1977).
[CrossRef] [PubMed]

Tien, P. K.

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, A. A. Ballman, H. Brown, Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

P. K. Tien, R. Ulrich, J. Opt. Soc. Am. 60, 1325 (1970).
[CrossRef]

Uchida, N.

Uehara, S.

Ulrich, R.

Wang, S.

Y. K. Lee, S. Wang, Appl. Phys. Lett. 25, 164 (1974).
[CrossRef]

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Zembutsu, S.

Appl. Opt. (4)

Appl. Phys. Lett. (10)

D. Sarid, D. Kermisch, J. F. Revelli, Appl. Phys. Lett. 51, 6105 (1980).

D. Sarid, D. Kermisch, Appl. Phys. Lett. 33, 619 (1978).
[CrossRef]

E. M. Conwell, Appl. Phys. Lett. 25, 40 (1974).
[CrossRef]

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, A. A. Ballman, H. Brown, Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

Y. K. Lee, S. Wang, Appl. Phys. Lett. 25, 164 (1974).
[CrossRef]

I. P. Kaminow, J. A. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[CrossRef]

R. L. Holman, P. J. Cressman, J. F. Revelli, Appl. Phys. Lett. 32, 280 (1978).
[CrossRef]

R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
[CrossRef]

G. L. Tangonan, M. K. Barnoski, J. F. Lotspeich, A. Lee, Appl. Phys. Lett. 30, 238 (1977).
[CrossRef]

E. M. Conwell, Appl. Phys. Lett. 23, 328 (1973).
[CrossRef]

Ferroelectrics (1)

R. L. Holman, P. J. Cressman, Ferroelectrics 27, 85 (1980).
[CrossRef]

IEEE J. Quantum Electron. (4)

D. Marcuse, IEEE J. Quantum Electron. QE-9, 1000 (1973).
[CrossRef]

E. M. Conwell, IEEE J. Quantum Electron. QE-10, 608 (1974).
[CrossRef]

J. E. Midwinter, IEEE J. Quantum Electron. QE-6, 583 (1970).
[CrossRef]

G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[CrossRef]

J. Appl. Phys. (3)

E. M. Conwell, J. Appl. Phys. 46, 1407 (1975).
[CrossRef]

J. F. Revelli, D. Sarid, J. Appl. Phys. 51, 3566 (1980).
[CrossRef]

J. F. Revelli, J. Appl. Phys. 52, 3185 (1981).
[CrossRef]

J. Opt. Soc. Am. (3)

Phys. Today (1)

E. Conwell, Phys. Today 29, 48 (1976).
[CrossRef]

Other (3)

See, for example, L. D. Landau, E. M. Lifshits, Electrodynamics of Continuous Media (Pergamon, New York, 1960).

Note that the sign in Eq. (22) is opposite to the one that appears in Eq. (39) of Ref. 9. There is apparently a typographical error in the reference.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

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Figures (7)

Fig. 1
Fig. 1

Schematic diagrams of N-layered free and prism-loaded waveguides analyzed in text. Layer designations are listed at left-hand side, indices are listed at right-hand side, and z-coordinates of the boundaries are listed in center of diagram.

Fig. 2
Fig. 2

Schematic diagram of prism/waveguide coupler with variable gap spacing. Circular inset at right represents magnified view of a specified region of the coupler in which it is assumed that the gap spacing can be taken as locally uniform.

Fig. 3
Fig. 3

Error in calculated value of β0 as function of N for successive multilayer waveguide approximations to an exponential index-gradient waveguide is shown plotted vertically at right. Index profiles of selected N-layer waveguides are shown at left plotted as functions of depth into the waveguide.

Fig. 4
Fig. 4

αU(x) (dotted curve) and leakage rates α0(x) for four example waveguides. In this and subsequent figures, solid curves represent multilayered waveguides with N = 19 and dashed curves represent single-layer waveguides.

Fig. 5
Fig. 5

k times relative waveguide effective index k[βIβ0(x)] for four example waveguides.

Fig. 6
Fig. 6

Magnitude (lower curve) and relative phase (upper curve) of the ideal beam profile Ω* as functions of x for four example waveguides.

Fig. 7
Fig. 7

Maximum coupling efficiency η as a function of radius ωI (cm) of Gaussian input beam.

Tables (3)

Tables Icon

Table I Comparison of β m and β m 20 for an Exponential Waveguide with Five Modes a

Tables Icon

Table II Waveguide Parameters

Tables Icon

Table III Parameters for Optimum Coupling Efficiency

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

2 σ j ± = - ( x , z ) ω 2 / c 2 σ j ± = - n 2 ( x , z ) k 2 σ j ± ,
2 σ j ± / z 2 + k 2 [ - γ 2 + n j 2 ] σ j ± = 0 ,
2 σ j ± / x 2 - k 2 γ 2 σ j ± = 0.
σ j ± ( x , z , t ) = e j ± ( γ ) exp ( i k γ x - i ω t ) [ exp ( ± i k ξ j z ) ( z in the prism or waveguide ) or exp ( ± k p j z ) ( z in the gap or substrate ) ,
e j ± ( γ m ) = C A s j ± ( γ m ) .
A 1 = 2 i k 2 exp ( - k p 0 W T ) [ ξ 1 ( p 0 cos 1 - ξ 1 sin 1 ) + p g ( ξ 1 cos 1 + p 0 sin 1 ) ] = C 1 ( ξ 1 Y 1 + p g X 1 ) ,
A 2 = 4 k 3 exp ( - k p 0 W T ) [ ξ 2 ( ξ 1 Y 1 cos 2 - ξ 2 X 1 sin 2 ) + p g ( ξ 2 X 1 cos 2 + ξ 1 Y 1 sin 2 ) ] = C 2 ( ξ 2 Y 2 + p g X 2 ) ,
A 3 = - 8 i k 4 exp ( - k p 0 W T ) [ ξ 3 Y 2 cos 3 - ξ 3 X 2 sin 3 ) + p g ( ξ 3 X 2 cos 3 + ξ 2 Y 2 sin 3 ) ] = C 3 ( ξ 3 Y 3 + p g X 3 ) ,
A N = C N ( ξ N Y N + p g X N ) ,
X N = ξ N X N - 1 cos N + ξ N - 1 Y N - 1 sin N ,
Y N = ξ N - 1 Y N - 1 cos N - ξ N X N - 1 sin N ,
A N = C N + 1 [ ξ N Y N B p + p g X N A p ] ,
A p = ξ p cosh S + i p g sinh S ,
B p = ξ p sinh S + i p g cosh S ,
Ω * ( x ) = e p ( γ m , x ) exp [ i k x 0 x γ m ( ζ ) d ζ ] ,
e p ( γ m , x ) = C ( ξ N Y N sinh S + p g X N cosh S ) = C ( ξ N Y N cosh S + p g X N sinh S ) ,
Ψ WKB = k 0 z t ξ ( z ) d z - φ 10 - φ 12 - m π = 0 ,
ξ ( z ) = [ n 2 ( z ) - β 2 ] 1 / 2 ,
Φ j = tan - 1 [ ξ j + 1 X j / ( ξ j Y j ) ] ,
X j = ( ξ j - 1 Y j - 1 / cos Φ J - 1 ) sin ( j + Φ j - 1 ) ,
Y j = ( ξ j - 1 Y j - 1 / cos Φ j - 1 ) cos ( j + Φ j - 1 ) .
X j = ( ξ j - 1 Y j - 1 / cos Φ j - 1 ) × sin { j + tan - 1 [ ξ j / ξ j - 1 tan { j - 1 + tan - 1 [ ξ j - 1 / ξ j - 2 tan ( j - 2 + + tan - 1 ( ξ 1 / p 0 ) ) ] } ] } ,
Y j = ( ξ j - 1 Y j - 1 / cos Φ j - 1 ) × cos { j + tan - 1 [ ξ j / ξ j - 1 tan { j - 1 + tan - 1 [ ξ j - 1 / ξ j - 2 tan ( j - 2 + + tan - 1 ( ξ 1 / p 0 ) ) ] } ] } .
X ( z ) = [ ξ ( z ) Y ( z ) / cos Φ ( z ) ] sin [ k 0 z t ξ ( z ) d z + π / 4 ] ,
X ( z ) = [ ξ ( z ) Y ( z ) / cos Φ ( z ) ] cos [ k 0 z t ξ ( z ) d z + π / 4 ] .
ξ ( 0 ) cos [ k 0 z t ξ ( z ) d z + π / 4 ] + p g sin [ k 0 z t ξ ( z ) d z + π / 4 ] = 0.
sin [ k 0 z t ξ ( z ) d z - π / 4 - φ 12 ] = 0 ,
n j = n 0 + ( j ) ( Δ n ) / ( N + 1 ) ,
W j = [ z N + ( z N - z N - 1 ) / 2 for j = N , ( z j + 1 - z j ) / 2 + ( z j - z j - 1 ) / 2 for N > j > 1 , ( z 2 - z 1 ) for j = 1 ,
η = ( E I Ω * 2 ) / ( E I E I Ω * Ω * / Γ ) ,
Ω * Ω * / Γ = Ω * Ω * + Ω * ( 0 ) 2 / [ 2 α m ( 0 ) ] ,
E I ( x ) = exp [ i k β I ( x - x C ) - ( x - x C ) 2 / w I 2 ] .
Φ Ω E ( x ) = arg [ E I ( x ) Ω * ( x ) ] + k β I x C = Φ e p ( x ) + x 0 x k Δ β 0 ( ζ ) d ζ ,
ω I , opt = 1.128 a ,
x C , opt = ( a / 2 ) log e ( a α opt / h opt 2 ) + 0.271 a ,
α opt = 0.582 / a = 0.656 / ω I , opt .
F j ± = exp ( i k ξ j i = j + 1 i = J W j )
A J = i C J ( - ξ J A 21 + i p g A 2 , 2 J + 1 ] = C J [ ξ J ( - A 21 ) + p g ( - A 2 , 2 J + 1 ) ] .
A J + 1 = C J + 1 [ - i ξ J A 21 ( ξ j + 1 cos J + 1 + p g sin J + 1 ) - ξ J + 1 A 2 , 2 J + 1 ( p g cos J + 1 - ξ J + 1 sin J + 1 ) ] ,
A J + 1 = C J + 1 [ ξ J + 1 Y J + 1 + p g X J + 1 ]

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