Abstract

A matrix method for calculating the effective refractive index of guided modes on multilayer dielectric waveguides is developed and employed to calculate the effective electro-optic coefficient of such waveguide structures when one of the media composing them is electro-optic. In properly specified three-layer waveguides, enhancement of the effective electro-optic coefficient above the bulk value by as much as a factor equal to the square of the highest refractive index in the structure is possible. The maximum modulation enhancement for the three-layer guide is attainable with loosely, as well as with tightly, confined waveguide modes.

© 1976 Optical Society of America

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References

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  1. P. K. Tien, Appl. Opt. 10, 2395 (1971).
    [CrossRef] [PubMed]
  2. W. K. Burns and J. Warner, J. Opt. Soc. Am. 64, 441 (1974).
    [CrossRef]
  3. R. Ulrich and R. J. Martin, Appl. Opt. 10, 2077 (1971).
    [CrossRef] [PubMed]
  4. R. Shubert and J. H. Harris, J. Opt. Soc. Am. 61, 154 (1971).
    [CrossRef]
  5. J. F. Lotspeich, J. Opt. Soc. Am. 65, 797 (1975).
    [CrossRef]
  6. O. S. Heavens, Optical Properties of Thin Films (Dover, New York, 1965), pp. 69ff.
  7. S. Fukunushi, N. Uchida, S. Miyazawa, and J. Noda, Appl. Phys. Lett. 24, 424 (1974).
    [CrossRef]
  8. W. G. Oldham and A. Bahraman, IEEE J. Quant. Electron. QE-3, 278 (1967).
    [CrossRef]
  9. P. K. Tien, R. Ulrich, and R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
    [CrossRef]
  10. J. E. Midwinter, IEEE J. Quant. Electron. QE-8, 583 (1970).
    [CrossRef]
  11. N. Nakamura and A. Yariv, Opt. Commun. 11, 18 (1974).
    [CrossRef]

1975 (1)

1974 (3)

S. Fukunushi, N. Uchida, S. Miyazawa, and J. Noda, Appl. Phys. Lett. 24, 424 (1974).
[CrossRef]

W. K. Burns and J. Warner, J. Opt. Soc. Am. 64, 441 (1974).
[CrossRef]

N. Nakamura and A. Yariv, Opt. Commun. 11, 18 (1974).
[CrossRef]

1971 (3)

1970 (1)

J. E. Midwinter, IEEE J. Quant. Electron. QE-8, 583 (1970).
[CrossRef]

1969 (1)

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

1967 (1)

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

Bahraman, A.

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

Burns, W. K.

Fukunushi, S.

S. Fukunushi, N. Uchida, S. Miyazawa, and J. Noda, Appl. Phys. Lett. 24, 424 (1974).
[CrossRef]

Harris, J. H.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Films (Dover, New York, 1965), pp. 69ff.

Lotspeich, J. F.

Martin, R. J.

R. Ulrich and R. J. Martin, Appl. Opt. 10, 2077 (1971).
[CrossRef] [PubMed]

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

Midwinter, J. E.

J. E. Midwinter, IEEE J. Quant. Electron. QE-8, 583 (1970).
[CrossRef]

Miyazawa, S.

S. Fukunushi, N. Uchida, S. Miyazawa, and J. Noda, Appl. Phys. Lett. 24, 424 (1974).
[CrossRef]

Nakamura, N.

N. Nakamura and A. Yariv, Opt. Commun. 11, 18 (1974).
[CrossRef]

Noda, J.

S. Fukunushi, N. Uchida, S. Miyazawa, and J. Noda, Appl. Phys. Lett. 24, 424 (1974).
[CrossRef]

Oldham, W. G.

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

Shubert, R.

Tien, P. K.

P. K. Tien, Appl. Opt. 10, 2395 (1971).
[CrossRef] [PubMed]

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

Uchida, N.

S. Fukunushi, N. Uchida, S. Miyazawa, and J. Noda, Appl. Phys. Lett. 24, 424 (1974).
[CrossRef]

Ulrich, R.

R. Ulrich and R. J. Martin, Appl. Opt. 10, 2077 (1971).
[CrossRef] [PubMed]

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

Warner, J.

Yariv, A.

N. Nakamura and A. Yariv, Opt. Commun. 11, 18 (1974).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

S. Fukunushi, N. Uchida, S. Miyazawa, and J. Noda, Appl. Phys. Lett. 24, 424 (1974).
[CrossRef]

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

IEEE J. Quant. Electron. (2)

J. E. Midwinter, IEEE J. Quant. Electron. QE-8, 583 (1970).
[CrossRef]

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

J. Opt. Soc. Am. (3)

Opt. Commun. (1)

N. Nakamura and A. Yariv, Opt. Commun. 11, 18 (1974).
[CrossRef]

Other (1)

O. S. Heavens, Optical Properties of Thin Films (Dover, New York, 1965), pp. 69ff.

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Figures (5)

FIG. 1
FIG. 1

Multilayer dielectric waveguide structure. The effective reflection coefficients e1 and e2 for the upper and lower (primed and unprimed) stacks of films, respectively, are obtained from Eq. (1).

FIG. 2
FIG. 2

Three-layer, electro-optic dielectric waveguide structure with electro-optic film 3, showing refractive index profile and range of effective refractive index n2 n ¯n3 (arrows) to be considered.

FIG. 3
FIG. 3

ϕ2 vs q for the three-layer structure of Fig. 2. Curve a represents weak, and curve b stronger coupling.

FIG. 4
FIG. 4

Keff/K vs q0 for a resonant three-layer dielectric structure with na = 1.0, ng = 2.7, n2 = 1.56, n3 = 1.97, and ns = 1.52, for different mode number j and coupling-at-resonance, f. Note scale break at q0 = 0.720.

FIG. 5
FIG. 5

Keff/K vs q0 for a resonant three-layer structure with na = 1.0, ng =3.6, n2 = 3.45, n3 = 3.5, and ns = 3.45, for different mode number j. f = 1.5.

Equations (28)

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[ B C A D ] = [ 1 r g 1 r g 1 1 ] [ 1 0 0 exp ( - i γ 1 ) ] [ 1 r N B r N B 1 ] ,
γ k = 4 π d k ( n k 2 - n g 2 q 2 ) 1 / 2 / λ ,
ψ = 4 π n g d g ( 1 - q 2 ) 1 / 2 / λ - ϕ 1 - ϕ 2 - 2 m π = 0 ,
n = n ( Ē = 0 ) + K E ,
K eff = n ¯ / E .
K eff = [ q δ g e + n g ( q n e ) ] K e ,
q n e = - ψ / n e ψ / q
= ( ξ / n g ) δ g e - ξ / n e [ ξ q / ( 1 - q 2 ) ] + ξ / q ,
q / n g ( 1 - q c 2 ) / q c n g .
K eff = 1 / q c = n g / Max ( n a , n s ) ,
ϕ 1 = 2 tan - 1 ( n g 2 q 2 - n a 2 n g 2 ( 1 - q 2 ) ) 1 / 2 .
ϕ 2 = tan - 1 ( 1 - x 2 ) sin ϕ 12 2 x + ( 1 + x 2 ) cos ϕ 12 ,
ϕ 12 = 2 tan - 1 ( n g 2 q 2 - n 2 2 n g 2 ( 1 - q 2 ) ) 1 / 2 ,
x = cos ϕ 23 + cos ( ϕ 3 s - γ ) 1 + cos ( ϕ 23 + ϕ 3 s - γ ) exp ( - 4 π d 2 λ ( n g 2 q 2 - n 2 2 ) 1 / 2 )
ϕ 23 = - 2 tan - 1 ( n 3 2 - n g 2 q 2 n g 2 q 2 - n 2 2 ) 1 / 2 ,
ϕ 3 s = 2 tan - 1 ( n g 2 q 2 - n s 2 n 3 2 - n g 2 q 2 ) 1 / 2 ,
γ = ( 4 π d 3 / λ ) ( n 3 2 - n g 2 q 2 ) 1 / 2 .
4 π d 3 n 3 λ ( 1 - q 3 2 ) 1 / 2 = 2 tan - 1 ( n 3 2 q 3 2 - n 2 2 n 3 2 ( 1 - q 3 2 ) ) 1 / 2 + 2 tan - 1 ( n 3 2 q 3 2 - n s 2 n 3 2 ( 1 - q 3 2 ) ) 1 / 2 + 2 j π ,
4 π d 3 ( n 3 2 - n g 2 q 2 ) 1 / 2 λ = 2 tan - 1 ( n g 2 q 2 - n 2 2 n 3 2 - n g 2 q 2 ) 1 / 2 + 2 tan - 1 ( n g 2 q 2 - n s 2 n 3 2 - n g 2 q 2 ) 1 / 2 + 2 j π ,
γ = ϕ 23 + ϕ 3 s + ( 2 j + 1 ) π ,
( 4 π d 2 / λ ) ( n g 2 q 2 - n 2 2 ) 1 / 2 1 ,
ϕ 2 q | q 0 = sin ϕ 12 sin ϕ 23 q ( ϕ 23 + ϕ 3 s - γ ) × exp ( 4 π d 2 λ ( n g 2 q 0 2 - n 2 2 ) 1 / 2 ) - ϕ 12 q
ϕ 2 n 3 | q 0 = sin ϕ 12 sin ϕ 23 n 3 ( ϕ 23 + ϕ 3 s - γ ) × exp ( 4 π d 2 λ ( n g 2 q 0 2 - n 2 2 ) 1 / 2 ) .
( ϕ 23 + ϕ 3 s - γ ) q | q 0 = q 0 ( n 3 2 - n g 2 q 0 2 ) 1 / 2 ( 2 n g 2 [ ( n g 2 q 0 2 - n 2 2 ) - 1 / 2 + ( n g 2 q 0 2 - n s 2 ) - 1 / 2 ] + ϕ 23 + ϕ 3 s + ( 2 j + 1 ) π ( n 3 2 - n g 2 q 0 2 ) 1 / 2 ) ,
( ϕ 23 + ϕ 3 s - γ ) n 3 | q 0 = - n 3 ( n 3 2 - n g 2 q 0 2 ) 1 / 2 ( ( n g 2 q 0 2 - n 2 2 ) 1 / 2 n 3 2 - n 2 2 + ( n g 2 q 0 2 - n s 2 ) 1 / 2 n 3 2 - n s 2 + ϕ 23 + ϕ 3 s + ( 2 j + 1 ) π ( n 3 2 - n g 2 q 0 2 ) 1 / 2 ) ,
ϕ 12 q | q 0 = 2 n g q 0 [ ( n g 2 q 0 2 - n 2 2 ) ( 1 - q 0 2 ) n g 2 ] 1 / 2 ,
q n 3 | q 0 = - α n 3 ( n 3 2 - n g 2 q 0 2 ) 1 / 2 ( ( n g 2 q 0 2 - n 2 2 ) 1 / 2 n 3 2 - n 2 2 + ( n g 2 q 0 2 - n s 2 ) 1 / 2 n 3 2 - n s 2 + ϕ 23 + ϕ 3 s + ( 2 j + 1 ) π ( n 3 2 - n g 2 q 0 2 ) 1 / 2 ) / [ α q 0 ( n 3 2 - n g 2 q 0 2 ) 1 / 2 ( 2 n g 2 ( n g 2 q 0 2 - n 2 2 ) 1 / 2 + 2 n g 2 ( n g 2 q 0 2 - n s 2 ) 1 / 2 + ϕ 23 + ϕ 3 s + ( 2 j + 1 ) π ( n 3 2 - n g 2 q 0 2 ) 1 / 2 ) + ϕ a q | q 0 - ϕ 12 q | q 0 + ( ϕ a q 0 - Φ 12 q 0 + 2 m π ) 1 - q 0 2 q 0 ] .
α = ( sin ϕ 12 sin ϕ 23 ) | q 0 exp ( 4 π d 2 λ ( n g 2 q 0 2 - n 2 2 ) 1 / 2 ) .