1 Wave propagation in optical waveguides Giulio Ciraolo November, 005 Abstract We present a mathematical framework for studying the problem of electro...

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November 2, 2005

Abstract We present a mathematical framework for studying the problem of electromagnetic wave propagation in a 2-D or 3-D optical waveguide (optical fiber). We will consider both the case of a rectilinear waveguide and the one of a waveguide presenting imperfections, with applications to phenomenons of physical interest. Numerical examples will be given.

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Introduction

A typical optical fiber is made of silica glass or plastic. Its central region is called core, and it is surrounded by a cladding, which has a slightly lower index of refraction. The cladding is surrounded by a protective jacket. In optical waveguides, most of the electromagnetic radiation propagates without loss as a set of guided modes along the fiber axis. The electromagnetic field intensity of the guided modes in the cladding decays exponentially transversally to the waveguide’s axis. That is why the radius of the cladding, which is typically several times larger than the radius of the core, can be considered infinite. In the model we used, we study the following Helmholtz equation L0 u := ∆u + k 2 n(x)2 u = f,

x ∈ RN ,

(1)

N = 2, 3, where k is the wavenumber and n is a positive function representing the index of refraction. In Sections 2 and 3 we describe how to construct a Green’s function for the case, respectively, of a 2-D and a 3-D rectilinear waveguide. A mathematical framework for studying 2-D optical waveguides with small imperfections and related numerical experiments are shown in Sections 4 and 5, respectively. ∗ Dipartimento di matematica U. Dini, Universit` a di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy.

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2-D rectilinear waveguides

A rectilinear waveguide can be described by assuming that ( nco (x1 ), if |x1 | ≤ h, n(x) = n0 (x1 ) = ncl , if |x1 | > h, where nco is a bounded even function. Thanks to the symmetry of the problem, we can separate the variables and look for solutions of the homogeneous Helmholtz equation of the form u(x1 , x2 ) = v(x1 , λ)eikβx2 , with λ = k 2 (n2∗ − β 2 ) and n∗ = max n. This leads to consider the following eigenvalue equation v 00 (x1 , λ) + [λ − q(x1 )]v = 0,

x1 ∈ R,

(2)

where q(x1 ) = k 2 [n2∗ − n(x1 )2 ]. Bounded solutions of (2) are of the form φ0j (h,λ) x1 > h, φj (h, λ) cos Q(x1 − h) + Q sin Q(x1 − h), vj (x1 , λ) = φj (x1 , λ), |x1 | ≤ h, 0 φ (−h, λ) cos Q(x + h) + φj (−h,λ) sin Q(x + h), x < −h, 1 1 j 1 Q √ j = s, a, with Q = λ − d2 and d2 = k 2 (n2∗ − n2cl ). Here, vj is symmetric or anti-symmetric in x1 if j = s or j = a, respectively. Solutions can be classified as follows: • Guided modes. For 0 < λ < d2 , only a finite number of eigenvalues λjm are supported by (2). The solutions decay exponentially outside the core and they correspond to solutions of the Helmholtz equation which propagate most of their energy inside the core; • Radiation modes. For d2 < λ < k 2 n2∗ , vj (x1 , λ) are bounded and oscillatory. Thus, the corresponding solutions of the Helmholtz equation are bounded and oscillatory both in the x and the z directions. • Evanescent modes. For λ > k 2 n2∗ , vj are bounded and oscillatory, but the corresponding solutions of the Helmholtz equation decay exponentially in one direction along the x2 axis and increase along the other one. By using the theory of Titchmarsh for the eigenvalues problems of singular differential operators, it is possible to construct a resolution formula for (1): Z u(x) = G(x, y)f (y)dy, (3) R2

with

√ ∞ X Z ei|x2 −y2 | k2 n2∗ −λ p G(x, y) = vj (x1 , λ)vj (y1 , λ)dρj (λ), 2i k 2 n2∗ − λ j∈{s,a} 0

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where, for every η ∈ C0∞ ([0, +∞)), it holds that hdρj , ηi =

Mj X m=1

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rjm η(λm j )

√ 1 R∞ λ − d2 η(λ)dλ. + 2π d2 (λ − d2 )φj (h, λ)2 + φ0j (h, λ)2

3-D rectilinear waveguides

We study cylindrically symmetric optical fibers, i.e. when ( nco (r), if 0 < r ≤ R, n = n(r) = ncl , if r > R, where r is the distance from the fiber’s axis. In this case, we separate the variables by using cylindrical coordinates (r, ϑ, x3 ) and looking for solutions 1 of the homogeneous Helmholtz equation of the form u = eiβkx3 eimϑ w(r)r− 2 , m ∈ Z. Hence, the associated eigenvalue problem is m2 − 1/4 w00 + λ − q(r) − w = 0, r > 0. r2 The classification of the solutions is analogous to the one obtained in the 2-D case. In this case, we can still apply the theory of Titchmarsh in all its power 2 , the equation has (see [1]). Notice that in this case, due to the term m r−1/4 2 a singularity at r = 0 besides the one at r = +∞; this adds further technical difficulties. Numerical results in the 3-D case are shown in Fig.1, where we supposed n to be ncl , 0 < r < a, (4) n(r) = nco , a ≤ r < R, ncl , r ≥ R, with nco and ncl constants and such that nco > ncl .

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2-D waveguides with imperfections

Real-life waveguides are never perfect, since they might contain imperfections due to inhomogeneities or changes in the core’s width and shape. When a pure guided mode is excited inside a guide with imperfections, a sort of resonation takes place and the other modes supported by the fiber are excited. This effect causes a signal distortion, since every guided mode propagates at its own characteristic velocity, and a loss in the signal power, due to the transfer of power to radiation, evanescent and the other guided modes. These effects are not always to be avoided. It is possible to make optical devices which can “propagate the energy as desired”. We will show two examples in the last part of this paper. 3

Figure 1: The figures represent the real part of the Green’s function in the case of a coaxial cable, i.e. n given by (4). In the first one, the source is on the waveguide’s axis, and only the symmetric modes are excited. In the second one, the source is inside the waveguide but not on the axis and all the guided modes are excited. From the mathematical point of view, we consider the Helmholtz equation Lε u := ∆u + k 2 nε (x1 , x2 )2 u = f,

in R2 ,

(5)

where the index of refraction nε is supposed to be a small perturbation of n0 . We formally represent Lε and u := uε in terms of their Neumann series and find L0 u0 = f, L0 u1 = −L1 u0 , . . . , L0 uj = −

j−1 X

Lj−r ur , . . .

r=0

Each step of the above iterative method can be solved by using the resolution formula (3). It is possible to prove the existence of a solution by writing the equation Lε u = f as L0 u = f + (L0 − Lε )u and then as L0 − Lε −1 u = L−1 f + εL u. 0 0 ε 16 Consider a weight function µ(x) = (4+|x| 2 )2 . By using estimates on the solution (3), we are able to prove that the linear operators 2 2 −1 L−1 ) → H 2 (R2 , µ) and 0 : L (R , µ

L0 − Lε : H 2 (R2 , µ) → L2 (R2 , µ−1 ) ε

are continuous. Hence, by choosing ε small enough and using the contraction mapping theorem, we prove the existence of a solution of Lε u = f .

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Numerical results

In the following simulations, we will always suppose that the zeroth order term of the Neumann series of u is a pure guided mode and we will show the effect of the imperfections by computing the first term u1 . 4

Near-field In the figure on the right, the real part of u0 +εu1 in proximity of the waveguide is represented. In this case, we suppose that the profile of the perturbation is as the one in the small figure. Grating-couplers Grating couplers are optical devices where the radiation energy is directed along precise directions. The figure on the right represents the intensity of far-field of u1 due to a perturbation in the cladding of the same type as the one in the previous figure. We notice that the energy propagates mostly on some precise directions. The Mach-Zehnder coupler In this kind of optical devices, the perturbation in a waveguide excites the guided modes of other waveguides which are close to the perturbed one. An example is shown in the figures below, where we show the real part of u0 , u0 + εu1 and u1 , respectively.

References [1] O. Alexandrov – G. Ciraolo, Wave propagation in a 3-D optical waveguide, Math. Models Methods Appl. Sci. 14 (2004), no. 6, 819–852. [2] O. Alexandrov – G. Ciraolo, Wave propagation in a 3-D optical waveguide II. Numerical Results (preprint). [3] G. Ciraolo, PhD Thesis, work in progress... [4] R. Magnanini and F. Santosa, Wave propagation in a 2-D optical waveguide, SIAM J. Appl. Math., 61 (2001) 1237 – 1252.

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