Abstract

The phenomena of optical coupling of parallel fibers and scattering of light from a fiber due to rough walls are considered from a mode point of view. With the use of a Green’s function, the problems are cast in the form of integral equations. Coupled ordinary differential equations are obtained which are used to study the coupling of modes in parallel fibers, including the case when the diameters are slowly-varying functions of the axial distance. The analysis of the problem of propagation in an optical fiber having rough walls shows that the various modes in a fiber will couple and that the roughness will cause radiation through the walls of the fiber. The character of the radiation may be determined if the spatial spectral density of the surface roughness is known.

© 1965 Optical Society of America

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References

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  1. N. S. Kapany, J. Opt. Soc. Am. 47, 413 (1957).
    [Crossref]
  2. R. J. Potter, J. Opt. Soc. Am. 51, 1079 (1961).
    [Crossref]
  3. N. S. Kapany and J. J. Burke, J. Opt. Soc. Am. 51, 1067 (1961).
    [Crossref]
  4. E. Snitzer, J. Opt. Soc. Am. 51, 491 (1961).
    [Crossref]
  5. E. Snitzer and H. Osterberg, J. Opt. Soc. Am. 51, 499 (1961).
    [Crossref]
  6. E. Snitzer, Advances in Quantum Electronics (Columbia University Press, New York, 1961).
  7. M. F. Bracey, A. L. Cullen, E. F. T. Gillespie, and J. A. Staniforth, IRE Trans. Antennas Propagation AP-7, S219 (1959).
    [Crossref]
  8. J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941).
  9. G. Goubau, Proc. IRE 40, 865, (1952).
    [Crossref]
  10. S. E. Miller, Bell System Tech. J. 33, 661 (1954).
    [Crossref]
  11. T. Tamir and A. A. Oliner, Proc. IEEE 51, 317 (1963).
    [Crossref]
  12. W. R. MacLean, Quart. Appl. Math. 2, 329 (1945).
  13. W. Magnus and F. Oberhettinger, Special Functions of Mathematical Physics (Chelsea Publishing Company, New York, 1949).
  14. J. W. Miles, J. Acoust. Soc. Am. 26, 191 (1954).
    [Crossref]
  15. E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962), p. 70.

1963 (1)

T. Tamir and A. A. Oliner, Proc. IEEE 51, 317 (1963).
[Crossref]

1961 (4)

1959 (1)

M. F. Bracey, A. L. Cullen, E. F. T. Gillespie, and J. A. Staniforth, IRE Trans. Antennas Propagation AP-7, S219 (1959).
[Crossref]

1957 (1)

1954 (2)

S. E. Miller, Bell System Tech. J. 33, 661 (1954).
[Crossref]

J. W. Miles, J. Acoust. Soc. Am. 26, 191 (1954).
[Crossref]

1952 (1)

G. Goubau, Proc. IRE 40, 865, (1952).
[Crossref]

1945 (1)

W. R. MacLean, Quart. Appl. Math. 2, 329 (1945).

Bracey, M. F.

M. F. Bracey, A. L. Cullen, E. F. T. Gillespie, and J. A. Staniforth, IRE Trans. Antennas Propagation AP-7, S219 (1959).
[Crossref]

Burke, J. J.

Cullen, A. L.

M. F. Bracey, A. L. Cullen, E. F. T. Gillespie, and J. A. Staniforth, IRE Trans. Antennas Propagation AP-7, S219 (1959).
[Crossref]

Gillespie, E. F. T.

M. F. Bracey, A. L. Cullen, E. F. T. Gillespie, and J. A. Staniforth, IRE Trans. Antennas Propagation AP-7, S219 (1959).
[Crossref]

Goubau, G.

G. Goubau, Proc. IRE 40, 865, (1952).
[Crossref]

Kapany, N. S.

MacLean, W. R.

W. R. MacLean, Quart. Appl. Math. 2, 329 (1945).

Magnus, W.

W. Magnus and F. Oberhettinger, Special Functions of Mathematical Physics (Chelsea Publishing Company, New York, 1949).

Miles, J. W.

J. W. Miles, J. Acoust. Soc. Am. 26, 191 (1954).
[Crossref]

Miller, S. E.

S. E. Miller, Bell System Tech. J. 33, 661 (1954).
[Crossref]

Oberhettinger, F.

W. Magnus and F. Oberhettinger, Special Functions of Mathematical Physics (Chelsea Publishing Company, New York, 1949).

Oliner, A. A.

T. Tamir and A. A. Oliner, Proc. IEEE 51, 317 (1963).
[Crossref]

Osterberg, H.

Parzen, E.

E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962), p. 70.

Potter, R. J.

Snitzer, E.

E. Snitzer, J. Opt. Soc. Am. 51, 491 (1961).
[Crossref]

E. Snitzer and H. Osterberg, J. Opt. Soc. Am. 51, 499 (1961).
[Crossref]

E. Snitzer, Advances in Quantum Electronics (Columbia University Press, New York, 1961).

Staniforth, J. A.

M. F. Bracey, A. L. Cullen, E. F. T. Gillespie, and J. A. Staniforth, IRE Trans. Antennas Propagation AP-7, S219 (1959).
[Crossref]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941).

Tamir, T.

T. Tamir and A. A. Oliner, Proc. IEEE 51, 317 (1963).
[Crossref]

Bell System Tech. J. (1)

S. E. Miller, Bell System Tech. J. 33, 661 (1954).
[Crossref]

IRE Trans. Antennas Propagation (1)

M. F. Bracey, A. L. Cullen, E. F. T. Gillespie, and J. A. Staniforth, IRE Trans. Antennas Propagation AP-7, S219 (1959).
[Crossref]

J. Acoust. Soc. Am. (1)

J. W. Miles, J. Acoust. Soc. Am. 26, 191 (1954).
[Crossref]

J. Opt. Soc. Am. (5)

Proc. IEEE (1)

T. Tamir and A. A. Oliner, Proc. IEEE 51, 317 (1963).
[Crossref]

Proc. IRE (1)

G. Goubau, Proc. IRE 40, 865, (1952).
[Crossref]

Quart. Appl. Math. (1)

W. R. MacLean, Quart. Appl. Math. 2, 329 (1945).

Other (4)

W. Magnus and F. Oberhettinger, Special Functions of Mathematical Physics (Chelsea Publishing Company, New York, 1949).

E. Snitzer, Advances in Quantum Electronics (Columbia University Press, New York, 1961).

E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962), p. 70.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941).

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Figures (12)

Fig. 1
Fig. 1

Propagation constant vs radius for various modes, n1=1.8, n2=1.5.

Fig. 2
Fig. 2

Path of integration when z>z0.

Fig. 3
Fig. 3

Geometry of two-fiber problem.

Fig. 4
Fig. 4

Energy in two parallel coupled fibers.

Fig. 5
Fig. 5

Comparison of solution for coupling constant with experiment of Bracey. Open circles indicate experimental values of βc while solid circles indicate experimental values of βs. n1=1.6, n2=1.0, a0=0.2146.

Fig. 6
Fig. 6

Coupling constants for HE11 modes in two parallel fibers. n1=1.8, n2=1.5.

Fig. 7
Fig. 7

Energy in center fibers of an infinite line of fibers.

Fig. 8
Fig. 8

Transfer of energy in two fibers which have linearly varying propagation parameters.

Fig. 9
Fig. 9

Attenuation constant vs reciprocal of roughness wavelength, n1=1.8, n2=1.5, a00=1.0.

Fig. 10
Fig. 10

Expected angular energy distribution, band-limited white noise, n1=1.8, n2=1.5, a00=1.0, a00=4.0, arrow indicates direction of light in fiber.

Fig. 11
Fig. 11

Expected angular energy distribution, Gaussian spectral density, n1=1.8, n2=1.5, a00=1.0, a0c=1.0, arrow indicates direction of light in fiber.

Fig. 12
Fig. 12

Expected angular energy distribution, band-limited white noise, n1=1.8, n2=1.5, a00=2.0, a0c=8.0, arrow indicates direction of light in fiber.

Equations (35)

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g n ( γ ) = [ - 4 ( - 1 ) n / a 6 π 2 ] [ ( v J n K n + u J n K n ) × ( v k 1 2 J n K n + u k 2 2 J n K n ) - n 2 γ 2 J n 2 K n 2 R 4 ] = 0 ,
u = a ( k 1 2 - γ 2 ) 1 2 ,             v = a ( γ 2 - k 2 2 ) 1 2 ,             R 2 = u 2 + v 2 ,             k j 2 = ω 2 μ j .
n j = k j / k 0 ,
k 0 2 = ω 2 0 μ = ( 2 π / λ 0 ) 2 ,
2 E + k l 2 E = 0.
n 1 × E 1 S 1 = - M 1 + S 2 [ ( n 2 × E 2 ) · ( × G 1 ) + i ω μ ( n 2 × H 2 ) · G 1 ] d S 2 ,
n 2 × E 2 S 2 = - M 2 + S 1 [ ( n 1 × E 1 ) · ( × G 2 ) + i ω μ ( n 1 × H 1 ) · G 2 ] d S 1 .
E j ( a j , θ j , z ) = n , m u n m j ( + ) ( z ) E n m j ( a j , θ j ; γ n m j ) / N n m j + n , m u n m j ( - ) ( z ) E n m j ( a j , θ j ; - γ n m j ) / N n m j ,
H j ( a j , θ j , z ) = n , m u n m j ( + ) ( z ) H n m j ( a j , θ j ; γ n m j ) / N n m j + n , m u n m j ( - ) ( z ) H n m j ( a j , θ j ; - γ n m j ) / N n m j ,
E j ( z ) = ( u n m j ( + ) ( z ) 2 - u n m j ( - ) ( z ) 2 ) .
d u n m 1 ( + ) ( z ) d z = i γ n m 1 u n m 1 ( + ) ( z ) + i k , l [ β k l n m 2 ( + ) u k l 2 ( + ) ( z ) + β k l n m 2 ( - ) u k l 2 ( - ) ( z ) ] ,
d u n m 1 ( - ) ( z ) d z = - i γ n m 1 u n m 1 ( - ) ( z ) - i k , l [ β k l n m 2 ( - ) u k l 2 ( + ) ( z ) + β k l n m 2 ( + ) u k l 2 ( - ) ( z ) ] ,
u n m j ( ± ) ( z ) = A n m j ( ± ) exp ± i γ n m j z .
d u 1 / d z = i γ 1 u 1 + i β u 2
d u 2 / d z = i γ 2 u 2 + i β u 1 ,
d u j d z = i γ j u j + i k j β j k u k ,
E j ( z ) = u j ( z ) 2 = J j 2 ( 2 β z ) .
d u 1 / d z = i β u 2 + i γ 1 ( z ) u 1 , d u 2 / d z = i β u 1 + i γ 2 ( z ) u 2 ,
E 1 ( z ) = u 1 ( z ) 2 = e - π ν / 2 D i ν ( β z ν - 1 2 e 3 π i / 4 ) 2 , E 2 ( z ) = u 2 ( z ) 2 = ν e - π ν / 2 D i ν - 1 ( β z ν - 1 2 e 3 π i / 4 ) 2 ,
a ( z ) = a 0 [ 1 + f ( z ) ] ,
f 1 and that d a / d z 1.
u m ( + ) ( z ) = A N δ N m exp ( i γ N z ) + i - z f ( z 0 ) × { n = 1 M β n m [ u n ( + ) ( z 0 ) + u n ( - ) ( z 0 ) ] + 0 δ m ( α ) × [ v ( + ) ( z 0 ; α ) + v ( - ) ( z 0 ; α ) ] d α } e i γ m ( z - z 0 ) d z 0 .
u m ( + ) ( z ) = A m δ N m e i γ m z + i A N β N m e i γ N z - z f ( z 0 ) exp [ ( i γ N - γ m ) z 0 ] d z 0 .
f ( z ) = sin ( Ω z ) ,             Ω = 2 π / λ ,
Ω = γ N - γ m ,             Ω = γ N + γ m .
d u 1 / d z = i γ 1 u 1 + i f ( z ) β u 2 , d u 2 / d z = i γ 2 u 2 + i f ( z ) β u 1 .
κ = ( γ 1 - γ 2 - Ω ) ( β ) - 1 ,
E θ ( r , z ) = i A N - f ( z 0 ) exp ( i γ N z 0 ) C 1 ( r , z - z 0 ) d z 0 ,
θ = tan - 1 { ( γ N - Ω ) [ k 2 2 - ( γ N - Ω ) 2 ] - 1 2 } .
S F ( Ω ) = ( π / 2 ) 1 2 Ω c - 1 exp ( - Ω 2 / 2 Ω c 2 ) ,
β n c = b n ( d ) k 2 2 ,
β n s = b n ( d ) P 2 γ n 2 ,
b n ( d ) = 2 a n ( - 1 ) n u 2 e - v d / a g n ( γ n ) K n 2 ( v ) v 2 ,
a n g n ( γ n ) = - n J n 2 K n 2 R 2 P u 2 v 2 k 1 2 [ J n K n + D 1 ± D 2 ] d / d γ [ J n K n + D 1 D 2 ] .
d d γ [ J n K n + D 1 D 2 ] = γ a 2 [ 1 u 2 J n K n + 1 u ( 1 - n 2 u 2 ) J n K n + 1 v J n K n ] + D 1 D 2 ,