Abstract

The dispersion properties of a thin-film optical waveguide for TE and TM modes are expressed in simple and general analytic forms. These formulas describe the variation of the effective refractive index with respect to any physical parameter with which the refractive index of any layer or the thickness of the guiding layer may vary. Universal curves for both TE and TM modes are given, and applications of the formulas are discussed.

© 1986 Optical Society of America

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References

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  1. G. A. Bennett, C.-L. Chen, “Wavelength Dispersion of Optical Waveguides,” Appl. Opt. 19, 1990 (1980).
    [CrossRef] [PubMed]
  2. S. L. Chen, J. T. Boyd, “Temperature-Independent Thin-Film Optical Waveguide,” Appl. Opt. 20, 2280 (1981).
    [CrossRef] [PubMed]
  3. E. Colombini, “Design of Thin-Film Luneburg Lenses for Maximum Focal Length Control,” Appl. Opt. 20, 3589 (1981).
    [CrossRef] [PubMed]
  4. T. Tamir, Ed., Integrated Optics (Springer-Verlag, New York, 1975), pp. 27–29.
  5. H. Kogelnik, V. Ramaswamy, “Scaling Rules for Thin-Film Optical Waveguides,” Appl. Opt. 13, 1857 (1974).
    [CrossRef] [PubMed]

1981 (2)

1980 (1)

1974 (1)

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Figures (3)

Fig. 1
Fig. 1

Structure of the planar thin-film optical waveguide.

Fig. 2
Fig. 2

Dispersion coefficients as a function of normalized thickness for various n1/n2 ratios for the case a = 1.

Fig. 3
Fig. 3

Variation of the effective refractive index with respect to thickness as a function of guide thickness for various waveguide materials (n1 = 1.46, n3 = 1.0, λ = 632.8 nm).

Equations (31)

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V ( 1 - b ) 1 / 2 = m π + tan - 1 ( b 1 - b ) 1 / 2 + tan - 1 ( a + b 1 - b ) 1 / 2 ,
V = k h ( n 2 2 - n 1 2 ) 1 / 2 ;
a = ( n 1 2 - n 3 2 ) / ( n 2 2 - n 1 2 ) ;
b = ( N 2 - n 1 2 ) / ( n 2 2 - n 1 2 ) ;
k = 2 π / λ .
N N ξ = n 1 n 1 ξ ( 1 - b ) + n 2 n 2 ξ b + 1 / 2 ( n 2 2 - n 1 2 ) b ξ .
N N ξ = Q g ( n 2 2 - n 1 2 ) ( λ / h ) ξ ( h / λ ) + i = 1 3 Q i n i ( n i / ξ ) .
Q 1 = ( 1 - b ) α b 1 / 2 ,
Q 2 = 1 α [ V + b 1 / 2 + ( a + b ) 1 / 2 ( 1 + a ) ] ,
Q 3 = ( 1 - b ) α ( 1 + a ) ( a + b ) 1 / 2 ,
Q g = V ( 1 - b ) / α ,
α = V + 1 / b 1 / 2 + 1 / ( a + b ) 1 / 2 .
V q s 1 / 2 ( n 1 / n 2 ) 1 / 2 = m π + tan - 1 ( b 1 - b ) 1 / 2 + tan - 1 [ b + a ( 1 - b c d ) 1 - b ] 1 / 2 .
V = k h ( n 2 2 - n 1 2 ) 1 / 2 ,
a = ( n 2 4 n 3 4 ) ( n 1 2 - n 3 2 ) / ( n 2 2 - n 1 2 ) ,
b = ( n 2 2 n 1 2 q s ) ( N 2 - n 1 2 ) / ( n 2 2 - n 1 2 ) ,
q s = ( n 1 2 / n 2 2 ) [ ( 1 - b ) + b ( n 1 4 / n 2 4 ) ] - 1
c = 1 - ( n 1 2 / n 2 2 ) ;
d = 1 - ( n 3 2 / n 2 2 ) .
a = a ( 1 - b c d ) ,
V = V q s 1 / 2 ( n 2 / n 1 ) .
V ( 1 - b ) 1 / 2 = m π + tan - 1 ( b 1 - b ) 1 / 2 + tan - 1 ( a + b 1 - b ) 1 / 2 ,
N N ξ = Q g ( n 2 2 - n 1 2 ) ( λ / h ) ξ ( h / λ ) + i = 1 3 Q i n i ( n i / ξ ) ,
Q 1 = ( 1 - b ) α b 1 / 2 [ 1 - b c 2 ( 1 - c ) 2 ] ,
Q 2 = 1 α { V ( 1 + 2 b c + b c 2 ) + [ b 1 / 2 + ( a + b ) 1 / 2 ( 1 + a ) ] ( 1 - 2 c + b c 2 ) } ,
Q 3 = ( 1 - b ) α ( 1 + a ) ( a + b ) 1 / 2 { [ ( 1 - d ) - 2 c ( 1 + b d ) + b c 2 ( 1 + d ) ] / ( 1 - d ) 3 } ,
Q g = V ( 1 - b ) / α
α = W q s 1 / 2 ( n 1 / n 2 ) .
W = V + ( n 2 n 1 q s 3 / 2 b 1 / 2 ) + [ 1 + a ( 1 - c d ) ] n 2 ( 1 + a ) ( a + b ) 1 / 2 q s 3 / 2 n 1 .
N N T = Q g ( n 2 2 - n 1 2 ) ( 1 / h ) h T + i = 1 3 Q i n i ( n i / ξ ) .
N N h = V ( 1 - b ) h α ( n 2 2 - n 1 2 ) .

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