Abstract

The mean-square fluctuations of log amplitude and phase are studied theoretically for guided beam modes propagating in an asymmetric slab waveguide having random inhomogeneities of the refractive index. The study is based on a method of the Rytov approximation, which has been mainly employed in analyzing the beam fluctuations in an atmosphere. The results show that the mean-square fluctuations of log amplitude and phase increase proportionally to the propagation distance for z/λ > 103, z and λ being the actual propagation distance and the wavelength of light, whereas the log-amplitude fluctuations increase obeying a square law of the distance for z/λ < 102. It is also found that an abrupt reduction of these fluctuations is observed near the center of the slab waveguide where the irradiance of the beam field has a maximum.

© 1976 Optical Society of America

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References

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  1. Integrated Optics, edited by D. Marcuse (IEEE, New York, 1973).
  2. H. F. Taylor and A. Yariv, Proc. IEEE 62, 1044 (1974).
    [Crossref]
  3. H. Kogelnik, IEEE Trans. MTT-23, 2 (1975).
  4. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 4.
  5. M. Imai and T. Asakura (unpublished work).
  6. M. Imai, T. Asakura, and Y. Kinoshita, Opt. Quant. Electron. 7, 95 (1975).
    [Crossref]
  7. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  8. M. Imai, S. Kikuchi, T. Matsumoto, and Y. Kinoshita, J. Opt. Soc. Am. 59, 904 (1969).

1975 (2)

H. Kogelnik, IEEE Trans. MTT-23, 2 (1975).

M. Imai, T. Asakura, and Y. Kinoshita, Opt. Quant. Electron. 7, 95 (1975).
[Crossref]

1974 (1)

H. F. Taylor and A. Yariv, Proc. IEEE 62, 1044 (1974).
[Crossref]

1969 (1)

Asakura, T.

M. Imai, T. Asakura, and Y. Kinoshita, Opt. Quant. Electron. 7, 95 (1975).
[Crossref]

M. Imai and T. Asakura (unpublished work).

Imai, M.

M. Imai, T. Asakura, and Y. Kinoshita, Opt. Quant. Electron. 7, 95 (1975).
[Crossref]

M. Imai, S. Kikuchi, T. Matsumoto, and Y. Kinoshita, J. Opt. Soc. Am. 59, 904 (1969).

M. Imai and T. Asakura (unpublished work).

Kikuchi, S.

Kinoshita, Y.

M. Imai, T. Asakura, and Y. Kinoshita, Opt. Quant. Electron. 7, 95 (1975).
[Crossref]

M. Imai, S. Kikuchi, T. Matsumoto, and Y. Kinoshita, J. Opt. Soc. Am. 59, 904 (1969).

Kogelnik, H.

H. Kogelnik, IEEE Trans. MTT-23, 2 (1975).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 4.

Matsumoto, T.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Taylor, H. F.

H. F. Taylor and A. Yariv, Proc. IEEE 62, 1044 (1974).
[Crossref]

Yariv, A.

H. F. Taylor and A. Yariv, Proc. IEEE 62, 1044 (1974).
[Crossref]

IEEE Trans. (1)

H. Kogelnik, IEEE Trans. MTT-23, 2 (1975).

J. Opt. Soc. Am. (1)

Opt. Quant. Electron. (1)

M. Imai, T. Asakura, and Y. Kinoshita, Opt. Quant. Electron. 7, 95 (1975).
[Crossref]

Proc. IEEE (1)

H. F. Taylor and A. Yariv, Proc. IEEE 62, 1044 (1974).
[Crossref]

Other (4)

Integrated Optics, edited by D. Marcuse (IEEE, New York, 1973).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 4.

M. Imai and T. Asakura (unpublished work).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

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Figures (9)

FIG. 1
FIG. 1

Schematic representation of a dielectric slab waveguide and the field distribution of a guided TE0 mode.

FIG. 2
FIG. 2

Log-amplitude fluctuations at the center of the waveguide (x/t = −0.5) vs the normalized propagation distance z/λ.

FIG. 3
FIG. 3

Phase fluctuations at the center of the waveguide (x/t = −0.5) vs the normalized propagation distance z/λ.

FIG. 4
FIG. 4

Log-amplitude fluctuations at the position of x/t = −0.5 vs the normalized scale of the refractive index inhomogeneity a/λ.

FIG. 5
FIG. 5

Phase fluctuations at the position of x/t = −0.5 vs the normalized scale of the refractive index inhomogeneity a/λ.

FIG. 6
FIG. 6

Distributions of the log-amplitude fluctuations across the waveguide for t/λ = 5.0

FIG. 7
FIG. 7

Distributions of the phase fluctuations across the waveguide for t/λ = 5.0

FIG. 8
FIG. 8

Distributions of the log-amplitude fluctuations across the waveguide for t/λ = 20.0.

FIG. 9
FIG. 9

Distributions of the phase fluctuations across the waveguide for t/λ = 20.0

Tables (1)

Tables Icon

TABLE I Sample calculations of the number of guided modes (TE mode).

Equations (13)

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n 2 ( x , z ) = n 2 [ 1 + δ n ( x , z ) ] ,             δ n ( x , z ) 1
δ n ( x 1 , z 1 ) δ n ( x 2 , z 2 ) = δ n 2 exp ( - x 1 - x 2 + z 1 - z 2 a ) ,
δ E y ( x , y ) = - j ( k n 2 ) 2 2 ω μ m 0 z d z 1 - t 0 d x 1 δ n ( x 1 , z 1 ) × E 0 ( x 1 , z 1 ) E m ( x , z ) E m * ( x 1 , z ) exp [ - j β m ( z - z 1 ) ] .
E m ( x , z ) = { C m exp ( - q m x ) exp ( - j β m z ) , 0 x < region 1 , C m ( cos ( h m x ) + q m h m sin ( h m x ) ) exp ( - j β m z ) , - t x < 0 region 2 , C m ( cos ( h m t ) + q m h m sin ( h m t ) ) × exp [ p m ( x + t ) ] exp ( - j β m z ) , - < x - t region 3 ,
C m = 2 h m ( ω μ β m ( t + 1 / p m + 1 / q m ) ( h m 2 + q m 2 ) ) 1 / 2 .
δ ψ = χ + j δ S δ E y ( x , y ) E 0 ( x , z ) ,             δ E y ( x , z ) E 0 ( x , z )
χ 2 δ S 2 } = 1 2 Re [ δ ψ 2 ± δ ψ 2 ] ,
χ 2 δ S 2 } = 1 2 ( k 2 n 2 2 2 ω μ ) 2 δ n 2 Re m n 0 z d z 1 d z 2 exp ( - z 1 - z 2 a + j ( β 0 - β m ) ( z - z 1 ) ) × { exp [ - j ( β 0 - β n ) ( z - z 2 ) ] exp [ j ( β 0 - β n ) ( z - z 2 ) ] } × m ( x ) n ( x ) 0 2 ( x ) - t 0 | d x 1 d x 2 exp ( - x 1 - x 2 a ) 0 ( x 1 ) m ( x 1 ) 0 ( x 2 ) n ( x 2 ) .
χ 2 δ S 2 } = ( k 2 n 2 2 2 ω μ ) 2 δ n 2 a m n ( sin ( β m - β n ) z ( β m - β n ) { 1 + a 2 [ β 0 - 1 2 ( β m + β n ) ] 2 } sin ( 2 β 0 - ( β m + β n ) ) z [ 2 β 0 - ( β m + β n ) ] { 1 + 1 4 [ a 2 ( β m - β n ) 2 ] } ) m ( x ) n ( x ) 0 2 ( x ) I m n .
I m n = - t 0 d x 1 d x 2 exp ( - x 1 - x 2 a ) 0 ( x 1 ) m ( x 1 ) 0 ( x 2 ) n ( x 2 ) = C 0 2 C m C n a 2 [ 1 + ( q 0 h 0 ) 2 ] { [ 1 + ( q m h m ) 2 ] [ 1 + ( q n h n ) 2 ] } 1 / 2 × [ 1 + 1 4 a 2 [ ( 2 h 0 ) 2 + ( h m + h n ) 2 ] ( h m - h n ) [ 1 + 1 4 a 2 ( 2 h 0 + h m + h n ) 2 ] [ 1 + 1 4 a 2 ( 2 h 0 - h m - h n ) 2 ] ( p m / h m - p n / h n { [ 1 + ( p m / h m ) 2 ] [ 1 + ( p n / h n ) 2 ] } 1 / 2 + q m / h m - q n / h n { [ 1 + ( q m / h m ) 2 ] [ 1 + ( q n / h n ) 2 ] } 1 / 2 ) + 1 + 1 4 a 2 [ ( 2 h 0 ) 2 + ( h m - h n ) 2 ] ( h m + h n ) [ 1 + 1 4 a 2 ( 2 h 0 + h m - h n ) 2 ] [ 1 + 1 4 a 2 ( 2 h 0 - h m + h n ) 2 ] ( p m / h m + p n / h n { [ 1 + ( p m / h m ) 2 ] [ 1 + ( p n / h n ) 2 ] } 1 / 2 + q m / h m + q n / h n { [ 1 + ( q m / h m ) 2 ] [ 1 + ( q n / h n ) 2 ] } 1 / 2 ) ] .
tan ( h m t ) = p m + q m h m ( 1 - p m q m / h m 2 )             for a TE m mode .
δ E y ( x , z ) = d r 1 2 k 2 n 2 2 δ n ( x 1 , z 1 ) E 0 ( x 1 , z 1 ) G ( r , r 1 ) ,
G ( r , r 1 ) = - j 4 ω μ m E m ( x , z ) E m * ( x 1 , z ) exp ( - j β m z - z 1 ) ,