Abstract

This paper considers a dielectric waveguide that is uniform in the z direction and composed of N homogeneous regions with = k and μ = μk (k = 1, 2, …, N). If the electromagnetic field of a specified mode is tightly confined in the vicinity of the k th region, the phase constant β would be mainly determined by k and μk. We present a few simple general relations between dispersion and power-flow distribution. For example, the sum of kμk’s weighted by PkPk, where Pk denotes the fractional power carried in the k th region, is equal to 1/νpνg. Another main result is that the partial derivative (β2)/ω2(kμk) is close to PkPk in a weakly guiding dielectric waveguide. Applications of them to analysis of a dielectric surface waveguide are discussed.

© 1975 Optical Society of America

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References

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  1. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).
    [Crossref]
  2. K. M. Case, J. Math. Phys. 13, 360 (1972).
    [Crossref]
  3. R. E. Eaves, J. Math. Phys. 14, 432 (1973).
    [Crossref]
  4. J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).
    [Crossref]
  5. E. A. J. Marcatili, Bell Syst. Tech. J. 53, 645 (1974).
    [Crossref]
  6. H. Noda, H. Furuta, and A. Ihaya, 1974 IEEE/OSA CLEA Conference.
  7. S. Kawakami and S. Nishida, Electron. Lett. 10, 38 (1974).
    [Crossref]
  8. S. Kawakami and S. Nishida, IEEE J. Quantum Electron. 10, 879 (1974).
    [Crossref]
  9. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Ch. 11, Sec. 5.
  10. K. Kurokawa, Introduction to the Theory of Microwave Circuits (Academic, New York, 1969).

1974 (3)

E. A. J. Marcatili, Bell Syst. Tech. J. 53, 645 (1974).
[Crossref]

S. Kawakami and S. Nishida, Electron. Lett. 10, 38 (1974).
[Crossref]

S. Kawakami and S. Nishida, IEEE J. Quantum Electron. 10, 879 (1974).
[Crossref]

1973 (1)

R. E. Eaves, J. Math. Phys. 14, 432 (1973).
[Crossref]

1972 (1)

K. M. Case, J. Math. Phys. 13, 360 (1972).
[Crossref]

1969 (2)

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

J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).
[Crossref]

Case, K. M.

K. M. Case, J. Math. Phys. 13, 360 (1972).
[Crossref]

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Ch. 11, Sec. 5.

Eaves, R. E.

R. E. Eaves, J. Math. Phys. 14, 432 (1973).
[Crossref]

Furuta, H.

H. Noda, H. Furuta, and A. Ihaya, 1974 IEEE/OSA CLEA Conference.

Goell, J. E.

J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).
[Crossref]

Ihaya, A.

H. Noda, H. Furuta, and A. Ihaya, 1974 IEEE/OSA CLEA Conference.

Kawakami, S.

S. Kawakami and S. Nishida, Electron. Lett. 10, 38 (1974).
[Crossref]

S. Kawakami and S. Nishida, IEEE J. Quantum Electron. 10, 879 (1974).
[Crossref]

Kurokawa, K.

K. Kurokawa, Introduction to the Theory of Microwave Circuits (Academic, New York, 1969).

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 53, 645 (1974).
[Crossref]

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

Nishida, S.

S. Kawakami and S. Nishida, IEEE J. Quantum Electron. 10, 879 (1974).
[Crossref]

S. Kawakami and S. Nishida, Electron. Lett. 10, 38 (1974).
[Crossref]

Noda, H.

H. Noda, H. Furuta, and A. Ihaya, 1974 IEEE/OSA CLEA Conference.

Bell Syst. Tech. J. (3)

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

J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).
[Crossref]

E. A. J. Marcatili, Bell Syst. Tech. J. 53, 645 (1974).
[Crossref]

Electron. Lett. (1)

S. Kawakami and S. Nishida, Electron. Lett. 10, 38 (1974).
[Crossref]

IEEE J. Quantum Electron. (1)

S. Kawakami and S. Nishida, IEEE J. Quantum Electron. 10, 879 (1974).
[Crossref]

J. Math. Phys. (2)

K. M. Case, J. Math. Phys. 13, 360 (1972).
[Crossref]

R. E. Eaves, J. Math. Phys. 14, 432 (1973).
[Crossref]

Other (3)

H. Noda, H. Furuta, and A. Ihaya, 1974 IEEE/OSA CLEA Conference.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Ch. 11, Sec. 5.

K. Kurokawa, Introduction to the Theory of Microwave Circuits (Academic, New York, 1969).

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Figures (3)

FIG. 1
FIG. 1

Dielectric waveguide composed of N regions with = k and μ = μk.

FIG. 2
FIG. 2

Dielectric slab with three layers.

FIG. 3
FIG. 3

Comparison of the three expressions associated with power confinement of the TM0 mode on a slab waveguide.

Equations (43)

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R = 1 / υ p 2 1 / u 2 2 1 / u 1 2 1 / u 2 2 = β 2 ω 2 2 μ 2 ω 2 ( 1 μ 1 2 μ 2 )
1 υ p υ g = k = 1 N ρ k k μ k = k = 1 N ρ k / ( plane-wave velocity in region k ) 2 ,
ρ k = σ k ( E × H ) z d S / σ T ( E × H ) z d S .
1 / υ p υ g 1 / u 2 2 1 / u 1 2 1 / u 2 2 = ρ 1 .
( β 2 ) ω 2 ( k μ k ) ρ k .
[ β m 2 ( ω m ) β l 2 ( ω l ) ] [ E ( l ) ( ω l ) × H ( m ) ( ω m ) ] z d S = ( ω m 2 ω l 2 ) k = 1 N k μ k σ k [ E ( l ) ( ω l ) × H ( m ) ( ω m ) ] z d S ,
[ β m 2 ( ω ) β l 2 ( ω ) ] σ T [ E ( l ) ( ω l ) × H ( m ) ( ω m ) ] z d S = 2 ω Δ ω k k μ k σ k [ E ( l ) ( ω l ) × H ( m ) ( ω m ) ] z d S .
β 2 = ω 2 1 μ 0 h 2 ,
ω 2 ( 1 2 ) μ 0 = h 2 [ 1 + ( 2 / 1 ) 2 tan 2 h T ] ,
1 / υ p υ g = ( β / ω ) d β / d ω = 1 μ 0 · 1 / 2 + tan 2 h T + h T tan h T / cos 2 h T ( 1 / 2 ) 2 + tan 2 h T + h T tan h T / cos 2 h T ,
1 / υ p υ g 1 / u 2 2 1 / u 1 2 1 / u 2 2 = sin 2 h T + h T tan h T sin 2 h T + h T tan h T + ( 1 / 2 ) 3 cos 2 h T .
( E × H ) z cos 2 h x , | x | < T ( 1 / 2 ) 2 cos 2 h T · exp { 2 ( | x | T ) × ( 2 / 1 ) h T tanh T / T } , | x | > T .
ρ 1 = 0 T ( E × H ) z d S / 0 ( E × H ) z d S = ( h T tanh T + sin 1 h T ) / [ sin 2 h T + h T tanh T + ( 1 / 2 ) 2 cos 2 h T ] .
( β 2 ) ω 2 μ 0 1 = sin 2 h T + h T tan h T + 2 ( 1 2 ) ( sin 2 h T / 1 ) / [ 1 + ( 2 / 1 ) 2 tan 2 h T ] sin 2 h T + h T tan h T + ( 1 / 2 ) 2 cos 2 h T .
× E = j ω μ ( x , y ) H ,
× H = j ω ( x , y ) E .
t 2 E ( l ) + ω 2 μ E ( l ) · E ( l ) + 1 μ μ × ( × E ( l ) ) = β l 2 E ( l ) ,
t 2 H ( l ) + ω 2 μ H ( l ) · H ( l ) + 1 × ( × H ( l ) ) = β l 2 H ( l ) ) ,
ω ω ˆ , ˆ , μ μ ˆ
t 2 E ˆ ( m ) + ω ˆ 2 ˆ μ ˆ E ˆ ( m ) · E ˆ ( m ) + 1 μ ˆ ( μ ˆ ) × ( × E ˆ ( m ) ) = β ˆ m 2 E ˆ ( m ) ,
t 2 H ˆ ( m ) + ω ˆ 2 ˆ μ ˆ H ˆ ( m ) · H ( m ) + 1 ˆ ( ˆ ) × ( × H ˆ ( m ) ) = β ˆ m 2 H ˆ ( m ) .
( H ˆ y t 2 E x H ˆ x t 2 E y ) + ω 2 μ ( E × H ˆ ) z + ( H ˆ × ) z ( · E ) + 1 μ ˆ ( H ˆ · μ ) ( × E ) z = β l 2 ( E × H ˆ ) z .
( E x t 2 H ˆ y E y t 2 H ˆ x ) + ω ˆ 2 ˆ μ ˆ ( E × H ˆ ) z ( E × ) z ( · H ˆ ) 1 ˆ ( E · ˆ ) ( × H ˆ ) z = β ˆ m 2 ( E × H ˆ ) z ,
σ T ( ω ˆ 2 ˆ μ ˆ ω 2 μ ) ( E × H ˆ ) z d S ( β ˆ m 2 β l 2 ) σ T ( E × H ˆ ) z d S = σ T ( H ˆ y t 2 E x E x t 2 H ˆ y + E y t 2 H ˆ x H ˆ x t 2 E y ) d S + σ T [ ( H ˆ × ) z ( · E ) + 1 ˆ ( E · ˆ ) ( × H ˆ ) z + ( E × ) z ( · H ˆ ) + 1 μ ( H ˆ · μ ) ( × E ) z ] d S .
1 E · = · E ,
1 H ˆ · μ = · H ˆ .
( V × ) z ϕ ϕ ( × V ) z = ( × ϕ V ) z ,
σ T { × [ H ˆ ( · E ) + E ( · H ˆ ) ] } z d S ,
( ω ˆ 2 ω 2 ) σ T ( x , y ) μ ( x , y ) [ E ( l ) ( ω ) × H ˆ ( m ) ( ω ˆ ) ] z d S = ( β ˆ m 2 β l 2 ) [ E ( l ) ( ω ) × H ˆ ( m ) ( ω ) ] z d S ,
β Δ β ω Δ ω = σ T ( x , y ) μ ( x , y ) [ E ( l ) × H ˆ ( l ) ] z d S / σ T [ E ( l ) × H ˆ ( l ) ] z d S ;
· H ˆ = 0 , μ = 0.
E · = E · ˆ = 0 ,
ω 2 μ 0 σ T [ ˆ ( x , y ) ( x , y ) ] ( E ( l ) × H ˆ ( m ) ) z d S = ( β ˆ m 2 β l 2 ) σ T [ E ( l ) × H ˆ ( m ) ] z d S
( max min ) / ( max + min ) 1 ,
I = σ T { ( H ˆ × ) z ( · E ) + 1 ( E · ) ( × H ˆ ) z d S + σ T ( 1 ˆ E · ˆ 1 E · ) ( × H ˆ ) z d S = σ T ( 1 ˆ E · ˆ 1 E · ) j ω ˆ E ˆ d S .
I = Δ · C k E n · j ω E ˆ z d s ,
ω 2 Δ μ 0 σ k ( E ( l ) × H ( l ) ) z d S Δ ( β 2 ) σ T ( E ( l ) × H ( l ) ) z d S = Δ C k j ω E n E ˆ z d S .
| E z / E t | < ( max min ) / ( max + min ) ,
| σ k j ω E z E t d s / ω 2 μ 0 σ k ( E × H ) z d S | < ( 2 λ / π L ) · k 1 / 2 ( max min ) / ( max + min ) .
Δ ( β 2 ) ω 2 Δ ( μ ) ρ k ,
H z / y = j ( ω β 2 / ω μ ) E x ,
H z / x = j ( ω β 2 / ω μ ) E y .
j ( ω β 2 / ω μ ) · E + j ω E · = 0.