Abstract

Coupled power equations for multimode fibers were originally derived by neglecting mode losses, loss was later introduced by adding a phenomenological loss factor to the equation system. In this paper the time independent coupled power equations are derived taking mode losses into account from the beginning. The form of the equations so obtained is identical to the well known coupled power equations, but the coupling coefficients are different. The mode losses alter the analytical expressions for the coupling coefficients and cause the matrix of coupling coefficients to be asymmetrical. Some consequences of the coupled power theory incorporating mode losses are discussed for the specific example of a fiber supporting only two guided modes. Finally, we compare the steady-state losses of a fiber with two randomly coupled modes with the corresponding case of a fiber resonator and find that the resonator losses are equal to the steady-state losses of randomly coupled modes in the limit of very low and very high differential mode losses, for all other cases the resonator losses are much higher than the losses of randomly coupled modes.

© 1978 Optical Society of America

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References

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  1. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 5.
  2. D. Marcuse, Bell Syst. Tech. J. 55, 1445 (1976).
  3. Ref. 1, p. 104.
  4. Ref. 1, p. 179, Eq. (5.2-17).
  5. A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions (Dover, New York, 1962).
  6. A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

1976 (1)

D. Marcuse, Bell Syst. Tech. J. 55, 1445 (1976).

1961 (1)

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Fox, A. G.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Li, T.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 55, 1445 (1976).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 5.

Yaglom, A. M.

A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions (Dover, New York, 1962).

Bell Syst. Tech. J. (2)

D. Marcuse, Bell Syst. Tech. J. 55, 1445 (1976).

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Other (4)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 5.

Ref. 1, p. 104.

Ref. 1, p. 179, Eq. (5.2-17).

A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions (Dover, New York, 1962).

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Figures (3)

Fig. 1
Fig. 1

Comparison of the power loss coefficient 2γD of the resonator (asterisks) and σ(1)D of the randomly coupled fiber modes (circles) as functions of the loss coefficient of one of the modes. The other mode is assumed to be lossless. The value of the coupling parameter is chosen to result in a K ˆ D = 0.001. Note that the loss for the random case is multiplied by the factor 1000 to make it visible on the scale of this figure.

Fig. 2
Fig. 2

Same as Fig. 1 with a K ˆ D = 0.01.

Fig. 3
Fig. 3

Same as Fig. 1 with a K ˆ D = 0.1.

Equations (56)

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b ν = β ν i α ν
d a μ d z = i b μ a μ + ν = 1 N K ˆ μ ν f ( z ) a ν .
R ( u ) = f ( z + u ) f ( z ) .
R ( u ) = R ( u )
K ˆ ν μ = K ˆ μ ν * ,
a μ = c μ exp ( i b μ z ) .
d c μ d z = ν = 1 N K ˆ μ ν f ( z ) c ν exp [ i ( b μ b ν ) z ] .
c μ ( z ) = c μ ( z ) + ν = 1 N K ˆ μ ν c ν ( z ) z z f ( x ) exp [ i ( b μ b ν ) x ] d x .
P μ = a μ a μ * = c μ c μ * exp ( 2 α μ z ) .
d P μ d z = 2 α μ P μ + [ d c μ d z c μ * exp ( 2 α μ z ) + c . c ] ,
d P μ d z = 2 α μ P μ + { exp ( 2 α μ z ) ν = 1 N K ˆ μ ν c ν ( z ) c μ * ( z ) f ( z ) × exp [ i ( b μ b ν ) z ] + c . c } ,
A ν μ = c ν ( z ) c μ * ( z ) f ( z )
B ν μ = c ν ( z ) c μ * ( z ) f ( z ) f ( x ) .
A ν μ = c ν ( z ) c μ * ( z ) f ( z ) = 0.
B ν μ = c ν ( z ) c μ * ( z ) f ( z ) f ( x ) = | c μ ( z ) | 2 δ ν μ R ( z x ) .
B ν μ = exp ( 2 α μ z ) P μ ( z ) δ ν μ R ( z x ) .
d P μ d z = 2 α P μ + ν = 1 N ( h μ ν P ν h ν μ P μ ) ,
h μ ν = | K ˆ μ ν | 2 R ( u ) × exp [ ( α μ α ν ) | u | ] exp [ i ( β μ β ν ) u ] d u .
d P 1 / d z = 2 α 1 P 1 + h 12 P 2 h 21 P 1 ,
d P 2 / d z = 2 α 2 P 2 + h 21 P 1 h 12 P 2 .
P μ = B μ exp ( σ z )
( σ 2 α 1 h 21 ) B 1 + h 12 B 2 = 0 ,
h 21 B 1 + ( σ 2 α 2 h 21 ) B 2 = 0.
( σ 2 α 1 h 21 ) ( σ 2 α 2 h 12 ) h 12 h 21 = 0 ,
σ ( 1,2 ) = 1 2 ( 2 α 1 + 2 α 2 + h 12 + h 21 ) 1 2 [ ( 2 α 2 2 α 1 + h 12 + h 21 ) 2 8 h 21 ( α 2 α 1 ) ] 1 / 2 .
B 2 ( j ) = 2 α 1 + h 21 σ ( j ) h 12 B 2 ( j ) j = 1,2 .
P 1 ( z ) = c ( 1 ) B 1 ( 1 ) exp [ σ ( 1 ) z ] + c ( 2 ) B 1 ( 2 ) exp [ σ ( 2 ) z ] ,
P 2 ( z ) = c ( 1 ) B 2 ( 1 ) exp [ σ ( 1 ) z ] + c ( 2 ) B 2 ( 2 ) exp [ σ ( 2 ) z ] .
c ( 1 ) = P 1 ( 0 ) B 2 ( 2 ) P 2 ( 0 ) B 1 ( 2 ) B 1 ( 1 ) B 2 ( 2 ) B 2 ( 1 ) B 1 ( 2 ) ,
c ( 2 ) = P 1 ( 0 ) B 2 ( 1 ) + P 2 ( 0 ) B 1 ( 1 ) B 1 ( 1 ) B 2 ( 2 ) B 2 ( 1 ) B 1 ( 2 ) ,
f ( z ) = a cos [ Ω z + ϕ ( z ) ] .
F ( θ ) = lim L 1 L 0 L f ( z ) exp ( i θ z ) d z .
| F ( θ ) | 2 = a 2 sin 2 [ ( Ω | θ | ) D / 2 ] ( Ω | θ | ) 2 D ,
R ( u ) = 1 2 π | F ( θ ) | 2 exp ( i θ u ) d θ = a 2 D | u | 2 D cos ( Ω u ) for | u | < D .
h 12 = ( a | K ˆ | ) 2 2 D { exp [ ( α 2 α 1 ) D ] 1 ( α 2 α 1 ) 2 D α 2 α 1 } ,
h 21 = ( a | K ˆ | ) 2 2 D { exp [ ( α 2 α 1 ) D ] 1 ( α 2 α 1 ) 2 + D α 2 α 1 } ,
h 12 D 1 ; h 21 D 1.
h = ( a | K ˆ | 2 ) 2 D .
σ ( 1 ) = α 2 .
σ ( 1 ) = h .
h 12 = ( a | K ˆ | ) 2 2 D exp ( α 2 D ) α 2 2 ,
h 21 = ( a | K ˆ | ) 2 2 α 2 ( 1 1 α 2 D ) ,
h 12 h 21 .
σ ( 1 ) = h 21 ,
α 2 D = ( 3 h D ) 1 / 2 ,
σ ( 1 ) = h .
A ( D ) = M A ( 0 ) .
M = { ( 1 h 21 D 2 ) exp ( a 1 D ) ; a K 2 1 exp [ ( α 2 α 1 ) D ] α 2 α 1 exp ( i ϕ ) a K 2 1 exp [ ( α 2 α 1 ) D ] α 2 α 1 exp ( i ϕ ) ; ( 1 h 12 D 2 ) exp ( a 2 D ) } ,
A ( D ) = exp [ γ D ] A ( 0 )
exp ( γ D ) = 1 2 [ 1 h 21 D 2 + ( 1 h 12 D 2 ) exp ( α 2 D ) ] + 1 2 { [ 1 h 21 D 2 ( 1 h 12 D 2 ) exp ( α 2 D ) ] 2 c 2 } 1 / 2 ,
c = a K ˆ 1 exp ( α 2 D ) α 2 ,
2 γ = α 2 .
2 γ = [ ( a K ˆ ) 2 ] / ( 2 α 2 ) ,
α 2 = a K ˆ
2 γ = a K ˆ .
( 2 γ ) / [ σ ( 1 ) ] = 4 / ( a K ˆ D ) .

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