An expression for attenuation in deformed optical waveguides

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An expression for attenuation in deformed optical waveguides

François Bentosela, Paola Piccoli

To cite this version:

François Bentosela, Paola Piccoli. An expression for attenuation in deformed optical waveguides. Jour-

nal de Physique, 1988, 49 (12), pp.2001-2007. �10.1051/jphys:0198800490120200100�. �jpa-00210880�

An expression for attenuation in deformed optical waveguides

François ^Bentosela (**) ^{and Paola Piccoli} (*)

Centre de Physique Théorique (***), CNRS-Luminy, Case 907 F-13288 Marseille Cedex 09, ^France

(Requ ^{le 19} juillet 1988, accepté le 24 aotit 1988)

Résumé.

Nous étudions dans cet article les effets de micro-déformations aléatoires sur la propagation ^des

ondes électromagnétiques dans les fibres optiques. Utilisant les techniques ^mises en 0153uvre pour l’étude de la localisation en physique ^{du solide, nous donnons, en} particulier pour ^une fibre mono-mode, l’expression ^de

l’atténuation en fonction de la distribution des micro-courbures aléatoires.

Abstract.

In this paper ^we study the effect of random microdeformations on the radiation propagation ⁱⁿ optical waveguides. ^{We use} the localization ideas coming from solid state physics. ^In particular ^{we deduce a}

formula linking the attenuation in a single mode fiber to the autocorrelation function of random bends.

Classification

Physics ^Abstracts 5.40 - 42.81 - 71.55J

1. Introduction.

In the optical waveguides ^{used in} long ^distance

communications one is interested in estimating ^the

different contributions to the radiation attenuation.

In this paper ^{we want to estimate} the part ^{of the} attenuation due to geometrical ^random imperfec-

tions. Two main imperfections ^{are observed: the}

variation of the fiber core section and, probably

more important, the microbends dues to the cabling

process.

This problem has been studied in general by writing coupled ^mode equations [9] ^and trying ^to

solve them by perturbation ^{methods, after} having performed, ^{at an} early stage, averages ^on the random quantities. Attempts ^{were also done to treat}

the problem deterministically ^{as far as} possible, considering ^that experiments ^are usually ^{done on a} single ^{fiber and not on an ensemble} [1,11].

Our point ^{of view} integrates the localization ideas which come from solid state physics ^{and which have}

been applied successfully ^{to other} domains, in par-

(***) Laboratoire Propre N° 7061, Centre National de la Recherche Scientifique.

ticular to wave propagation in acoustics or hydrody-

namics [6]. (See [15] ^{for a} review).

As our aim is to provide ^{more a method than}

formulas applicable ^{to all} possible situations, ^{we will}

restrict ourselves to the complete description ^{of a} special ^{case : a} single-mode fiber whose radius is constant and which presents microbends. We will also give ^{some ideas on} the way ^{to treat} the case in which the axis is straight ^{and the core radius is} slightly varying.

In section 2 we shall derive from Maxwell

equations ^the coupled ^mode equations ^{and discuss} why ^{it is} possible, ^{in some} special ^{cases, to} simplify drastically ^the problem reducing ^{it to a} single

differential equation ^with random coefficients.

In section 3 we shall give analytic expressions ^for

the Lyapunov exponents ^which give ^{the decrease}

rate for the solutions of the decoupled equations.

In section 4 we will use these results in order to

estimate the attenuation in the case of a fiber

presenting microbends.

In section 5 we will compare ^our results and method with the ones by ^Marcuse [11] and Arnold- Allen [1].

2. Coupled ^mode equations.

The components of the electric and magnetic ^fields satisfy ^Maxwell’s equations. ^{As the} dielectric con- stant and permeability ^are supposed ^{to be constant}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198800490120200100

2002

in a region, ^{one can convert Maxwell’s} equations ⁱⁿ

Helmholtz type equations. Calling ip any component of the fields we get

where x = (Xl, X2, z )

(X, z ), ^z being the propa-

gation direction, w ^{the wave} pulsation ^and n (x ) ^the

refractive index function.

We don’t make the assumption ^{of an infinite} cladding radius and then we will have some boundary

condition at its frontier.

Call Sz ^{the fiber section at z,} Hz ^the restriction of H to Sz, acting ^{on the Hilbert space} L 2(Sz ) ^{and let us}

denote by cp /z)(Xl’ ^x2), E/z) respectively ^{its eigenvec-}

tors and corresponding eigenvalues.

If the core and cladding ^{sections are} circular and concentric the eigenfunctions 0 1( z )(Xl, ^{X2) can be}

expressed ^{in terms} of Bessel functions. A finite number of E/z) ^{will be} negative. Among ^{these some}

°

of them correspond ^to eigenvectors ^{which are essen-} tially located in the core excepted ^an exponential ^tail

located in the cladding (when ^{there is} only ^{one such}

an eigenfunction ^{we are in the} single-mode case, if not in the multimode one).

The cp/Z) ^{vectors form a basis for L 2 (Sz ) so we can}

express ^a function f/1 of the domain of operator ^{H on} this basis :

where

(where ^we write t/J (z)(XI’ X2) for t/J (Xl’ X2)).

Plugging ^this expression ⁱⁿ equation (2.1) ^and using equation (2.2), ^we get :

Denoting ^as

we get ^{an infinite} system ^of coupled equations ^for

the {aj (Z ) }

whose coefficients K(1) Kmj(2), ^{Em(z) are} random vari- ables which can be expressed ^{in terms} of the random

imperfections ^{as we shall see below.} (In the absence of imperfections Em(z) ^{would be constant and the} Kmi (z) ^{equal to zero).}

In the perfect ^case only ^a finite number of modes

corresponding ^to negatives Em propagate, ^{the others} decrease exponentially ^{as it is} easily ^{seen from} equation (2.4). ^{In the} imperfect ^{case it} seems reason---

able to disregard ^all the modes which do not

propagate ^{in the} perfect ^{case, that is all the} equations

with E}z) ^{positive, i. e . to} reduce the problem ^{to a}

finite system ^of coupled differentiable equations.

Moreover, ^if (Et) (.> denoting ^{the mean} value) ^is

well separated ^{from the} (Ej >, j #: 1, ^{we have}

numerically checked that it is enough ^{to consider} only ^one equation ^{in order to} get ^a good approxi-

mation for the attenuation.

This property ^is supported by ^the following ^nu-

merical simulation. We select the first N second order differential coupled equations. ^After choosing

a mesh At, they ^{can be} transformed in a transfer matrix equation ^{2 Nx2 N :}

( (Tj ) ) ^can be transformed in a 2 N x 2 N matrix,

((Tj)) ^{belonging to the} symplectic group (see [13]).

((Tj)) ^{can be written in the form}

(J*(jOl)D(j01)J(jOl) ₁ -10 ₀ ),

where D (j Al

is the N x N diagonal matrix, Dmn = [2 ⁺ Ol 2 Em ( j Al)] ð mn’ and the matrix elements of

J(j Al) ^are expressed ^{in terms of the} Kmn (j Al (In

the perfect ^{case J is the} identity matrix and the

Em ( j Ol ) ^are constant.)

One calculates the matrix product Mn

Tn . Tn -1 .... T1. ^{It has been} proven [12] that, ^under

some conditions on the random matrix elements that i

are fulfilled here, (M"* Mn)2n ^has eingenvalues, ^the

limits of which, ^{as n -+} oo, exist and are independent

I

of the sample. ^{These limits are} generally ^denoted by

exp y1, exp Y2 ... exp Y2N, yi i being ^called Lyapunov exponents. They satisfy

General theorems [7] ^show that yN is strictly positive

and that the decrease of the Green function of the operator ^{H is} governed by yN. We recall that the solution of the Helmholtz equation ^{with a} light

source is the convolution of the light intensity ^term by ^{the Green} function. As the light ^{source is}

localized in front of the fiber, the solution decrease inside the fiber in the same way ^as the Green function.

Remark 1: As direct computer calculations of the

eigenvalues ^of M) Mn ^are impossible ^{due to} overflow

statement errors, ^{we use} the now classical orthonor- malisation method (see [4], [14]) which deduces the

Lyapunov exponents from the evolution of some

volumes in R2 N-space.

Remark 2 : The Lyapunov exponents calculations

were done imposing unrealistic parameters, ^{as the} refractive index difference An, ^the cladding ^radius

and the pulsation ^{w, in order to} get ^{a reasonable} number of propagating ^modes (N

4), ^and only

one guided ^mode.

Remark 3 : We specialize the calculations to the

case of a fiber having constant _sections, then

Em ( j Al ) ^{do not} depend ^{on the} site j (the approxi-

mate values of Ol 2 Em ^are åZ2 E1 = - ^0.8, åZ2 E2 = - 0.7, åZ2 E3 = - 0.5, åZ2 E4 ^{= - 0.2,} åZ2 E5

⁺ 0.02), ^{and the matrices} J(j Al) depend only ^{on the center} position ^{of the} fiber. The random fluctuations of this position ^are taken much greater than real _ones, in order to obtain a more rapid

convergence (after ¹⁰⁸ iterations we reach the stabi-

lity for the third digit and YN + 1 yN with ^a 3 digit precision).

In figure ^{1 are} plotted ^{the first N + 1} Lyapunov exponents obtained for a given ^fiber increasing ^the

number N of the coupled equations under consider- ation. One can see that the fact of considering

additional equations ^{do not} change significantly ^the

values of the smallest Lyapunov exponent, ^which

slightly decreases, passing ^{from value 0.0007176 for}

N = 1 to value 0.0006123 for N

3, where it almost stabilizes.

In a second numerical experiment, ^to observe the effect of couplings, ^{we consider a} fixed number of

equations (N

3) ^and multiply ^the coupling par-

Fig. ^1.

^{For N}

^5, yi

0.341549 does not appear in the figure being larger ^{than the} others y.

ameters Kmn (z) (m :0 n ) by ^a constant C (varying ⁱⁿ (Fig. 2) ^{from 0 to} 8). ^{We notice that} Y3 (C )

decreases so slightly that the upper bound Y3 (0) ^{is a} quite good approximation ^{for the fiber} attenuation.

In the subsequent ^{section we will} give ^an analytic

estimate for the Lyapunov exponents, ^{when one} neglects ^the coupling ^between modes. We will

rename the Lyapunov exponents, ^{in such a} way that

ym denote the positive Lyapunov exponent ^corre-

sponding ^{to the m-th} decoupled equation,

’Ym - yN+1-m(0).

Fig. ^2.

3. Lyapunov estimates for decoupled equations.

The decoupled equations ^are

We can eliminate the first order derivatives, using

the following change of function

2004

The coefficient of bm (z) ^{is a function of the}

random variable g (z), ^{which describes} the fluctu- ations. Denoting by

we obtain the equation ^{for a} random oscillator

This problem ^{has been studied} by ^{several authors} (see [2], [7]). ^In Appendix ^{1 we} mainly reproduce

the arguments by Arnold, Papanicolaou ^and

Wihstutz [3] ^who give ^an expression for the series in

e of the positive Lyapunov exponent. ^{The first term} of this series is

where Cm(z ) is the correlation function of the random variable Fm (c(z ) ), ^or

if Cm (z ) denotes the correlation function of Wm (z ).

In the next section we will specialize ^{this formula}

to the case in which the fiber has constant sections while its axis is randomly ^bended.

4. Lyapunov estimates for the bended fiber.

Hypotheses : ^The coordinates of the center of the circular section Sz ^{are two} independent ^random

variables xoi(z), x02 (z), whose derivatives are

stationnary ^{zero mean Gaussian} processes with the

same correlation function.

We impose ^Dirichlet boundary conditions on the

cladding ^surface.

Let us calculate now from (3.1) ^the expression ^for F. (C (z)). By hypothesis

then

Since

we get

and

with

As Em(z) is constant, ^it is equal to ( Em ) ^{an then we}

get :

Using ^the independence ^of

lation function of Wm (z ) ^is

,

the corre-

Now, using ^{he fact that}

process, ^we have

is a zero mean Gaussian

Then

Plugging ^this expression ⁱⁿ (3.3)

We recall that our main interest is in the evaluation of the smallest Lyapunov exponent Yi, correspond- ing ^{to the} guided ^{mode. In} this case I (Xl’ X2) depends only ^{on the radius r}

xl ^{+ x2 and}

Using the fact that a crude

estimate gives ^for All :

and

Then finally ^we get

...

dx01

to be

Supposing ^now the correlation function of ;;1 ^{to be}

Co(u)

a2 exp

^{we have}

So

5. Discussion.

The method we develop ^here is, ⁱⁿ spirit, ^different

from those by Arnold-Allen and Marcuse. Once

they get ^the system ^of coupled equations, they

suppose that the amplitudes corresponding ^{to the} guided ^{modes at the} origin ^{are much} larger ^than

those of the radiation modes and that they ^remain

constant. Then they ^{deduce from the} differential

equations ^the amplitudes ^{of the} radiation modes and calculate the energy passing ^{from the} guided ^modes

to the radiation modes. In our method the amplitude

of the guided ^{mode is not} supposed a priori ^{to be}

constant, ^{and in} fact it decreases exponentially, ^even

without taking ^{into account the} coupling ^{with the}

other modes.

At the level of the results, ^{the formulas} give ^a

similar behaviour of the attenuation as a function of the correlation function of the bends.

On the other hand their formulas differ in what

concerns the behaviour of the attenuation with respect ^{to the wave} length, ^{but more} significantly ⁱⁿ

the fact that in their formulas enters the statistics of the position ^{of the fiber center, while in our formula}

enters the statistics of the derivative of the position

of the fiber center. The difference could be great ;

unfortunately ^we don’t have precise informations about the statistics of the position ^of the fiber axis.

Let us simply notice that what we called a in formula (4.1), could be much greater ^{than the r.m.s.}

of the position, ^{which is} supposed ^to be of the order of few nanometers. For such amplitudes ^{and defor-}

mations correlation length, smaller than the

wavelength, ^the attenuation could reach 1 db/km.

As measured attenuations are smaller, ^{we can infer} that their amplitudes ^{are smaller or} their correlation

length greater.

Finally ^{we believe} that these methods could also

be used to calculate acoustical or electromagnetic

2006

attenuations in stratified media with guided ^modes

and irregular boundaries.

Appendix ^1.

Derivation of the Lyapunov exponent ^{for the ran-} dom oscillator.

Suppose the oscillator pulsation ^{at time z be} given by (d2(z) = - ( E ) - cF(c(z)), ^{the oscillator} equation ^is

We introduce the action-angle ^variables

from which one deduces the equations

Supposing that 6 (z) ^{is described} by ^{a diffusion}

process ^{on a} manifold whose generator ^{is G} (we ^can think, ^for example, ^{of a} Brownian motion on the

circle), ^the pair ( 0 (z ), 6 (z ) ) ^{is also a diffusion}

process, whose generator ^{L is}

(For the derivation of (A.3) ^{see the} Fokker-Planck

equation ^{related to a} multidimensional stochastic

equation, ^for example ⁱⁿ [2] p. 156.)

We define now the Lyapunov exponent

From (A.2) ^{we get}

If the process ( B (z), e (z ) ) ^{has an invariant measure} p (0, e ) d § d 0, then, by ^the ergodic ^theorem

The invariant measure has to satisfy L * p

^{0. Let}

us write

Arnold, Papanicolaou and Wihstutz [3] ^{build up a}

formal expression for pE

so that

po is evidently ^{the constant function}

and

Using the fact that formally

and denoting by g (u, e, q ) the transition probability

of the process 6(z), ^we get

Plugging ^the expression for po ⁺ ep, ⁱⁿ the integral,

the Lyapunov exponent up ^to the second order in s

becomes

Calling è (u ) ^the correlation function

References

[1] ^{ARNOLD, J.} M., ALLEN, R., IEE Proc. 130 (1983)

331-339.

[2] ARNOLD, L., Stochastic Differential Equations (John Wiley, 1974).

[3] ARNOLD, L., PAPANICOLAOU, G., WIHSTUTZ, V., SIAM J. Appl. ^{Math. 46} (1986) ^427-450.

[4] BENETTIN, G., GALGANI, L., GIORGILLI, A., ^STREL- CYN, J. M., Meccacina : J. of the Italian Associa- tion of Theor. and Appl. ^Mechanics (March 1980).

[5] ^BOUGEROL, P., LACROIX, J., Products of random matrices with Applications ^to Schrödinger Operators (Birkhausser, 1985).

[6] BELZONS, M., GUAZZELLI, E., PARODI, O., ^{J. Fluid}

Mech. 186 (1988) ^539-558.

[7] ^JOHNSTON, R., KUNZ, M., ^J. Phys. ^{C 16} (1983) ^3895.

[8] KOTANI, S., USHIROYA, N., ^{Comm. in Math.} Physics

115 (1988) ^247-266.

[9] LACROIX, J., Ann. Inst. Henri Poincaré A 40 (1984)

97-116.

[10] MARCUSE, D., Theory ^of dielectric optical waveguides (Academic Press, 1974).

[11] MARCUSE, D., Applied Optics ²³ (1984) ^1082-1091.

[12] OSELEDEC, V., Trans. Moscow Math. Soc. 19 (1968)

197-231.

[13] PICCOLI, P., ^Master Thesis, International School for Advanced Studies, ^Trieste (1985).

[14] ^{PICHARD, J.} L., Thesis, University ^of Paris, Orsay,

No. 2858 (1984).

[15] SOUILLARD, B., Waves and electrons in in-

homogeneous ^media (Les Houches, 1986).