Multiscale theory of nonlinear wavepacket propagation in a planar optical waveguide

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Multiscale theory of nonlinear wavepacket propagation in a planar optical waveguide

View the table of contents for this issue, or go to the journal homepage for more 2002 J. Opt. A: Pure Appl. Opt. 4 514

(http://iopscience.iop.org/1464-4258/4/5/305)

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J. Opt. A: Pure Appl. Opt.4(2002) 514–520 PII: S1464-4258(02)33722-X

Multiscale theory of nonlinear wavepacket propagation in a planar optical waveguide

V Boucher, H Leblond and X Nguyen Phu

Laboratoire des Propri´et´es Optiques des Mat´eriaux et Applications, UMR CNRS 6136, Universit´e d’Angers, 2 Bd Lavoisier, 49045 Angers Cedex, France

Received 12 February 2002, in final form 10 June 2002 Published 6 August 2002

Online at stacks.iop.org/JOptA/4/514 Abstract

In this paper, the multiscale expansion formalism is applied for the first time, to our knowledge, in nonlinear planar optical waveguides. This formalism permits us to describe the linear and nonlinear propagation for both transverse electric and transverse magnetic modes. The modal field distributions and the nonlinear coefficient in the nonlinear Schr¨odinger equation are highlighted.

Keywords: Nonlinear waveguide, multiple scales, soliton, NLS

1. Introduction

Nonlinear optical guided modes are of great interest, owing to their potential applications to optical signal processing devices, especially when localized wavepackets propagating without being deformed by diffraction or dispersion, i.e. solitons, are formed [1]. However, they can rarely be described rigorously by any formalism. Along the interface between a nonlinear and a nonlinear material, nonlinear optical guided waves can propagate. References [2] give an exact study of the propagation of such waves in both the transverse electric (TE) and the transverse magnetic (TM) configurations, showing a rather important difference between these two situations. A wavepacket can also be guided between two interfaces. In such a situation, the description of the wavepacket evolution rests on nonlinear Schr¨odinger (NLS)-type equations [3–12]

that are often derived in a rather heuristic way. A quite rigorous and general derivation of the system of two coupled NLS equations describing the interaction between one TE and one TM mode is written in [13], but has not been published.

This approach introduced a priorithe guided mode profiles, which have the advantage of avoiding the large computational difficulties involved by their explicit expressions. It but has two drawbacks, from our viewpoint. First, the linear theory should arise naturally as the first step in the weakly nonlinear approximation considered. Second, explicit computation of the coefficients in the nonlinear evolution equation of NLS type is of major importance in order to compare the simulations of the derived model to the experiment. The multiscale expansion formalism or reductive perturbation method seems to be the most adequate mathematical tool to avoid this drawback. It was initiated in 1968 by Taniuti and Washimi [14] in plasma

physics, and recently in nonlinear optics [15, 16]. There are several methods commonly used in the derivation of the coefficients of a NLS equation, two of which are very rigorous:

the multiscale formalism and the Hamiltonian formalism [18].

Both procedures are equivalent from the mathematical point of view, but multiscale expansions have the advantage of leading straightforwardly to explicit formulae. The present paper uses a strict multiscale expansion formalism in which the variations of the mode profile in the transverse direction of the waveguide are determined explicitly without anya priori assumption about them. Such an approach has already been used to describe the evolution of localized solitary waves in a shallow viscous fluid [17]. In recent times it has been applied to wavepackets in the frame of magnetostatic backward volume waves propagating in magnetic thin films [19]. The present paper is its first application to nonlinear optics.

We consider a wave propagating in a Kerr-like nonlinear dielectric waveguide constituted by a nonlinear film bounded by two linear media, as represented in figure 1. The present paper is a first stage in the study of this problem: the (1 + 1)- dimensional situation of the so-called ‘temporal’ soliton for a single mode. This situation is indeed the most simple one for which the physical meaning of the three length or time scales involved by the multiscale expansion, wavelength λ, pulse length aboutLand propagation distance aboutD, clearly appear. The present computations give most mathematical tools needed for the accurate description of experimental situations, eventually more complicated than the present one, as the interaction of two modes with different polarizations in a stationary beam. The structure of this paper follows the perturbative scheme: the basis model is first presented, then the first order of the multiscale expansion, at which the linear

Multiscale theory of nonlinear wavepacket propagation in a planar optical waveguide

-a a

x

y z

n n n₂

3 2

λ λλ//εε

Figure 1.Space scales involved in waveguide geometry.

theory of the waveguide modes is found again. The second order gives the propagation at group velocity and the third one the nonlinear evolution equation. The derived nonlinear coefficients are discussed before we give a conclusion.

2. The scaling

Assuming the medium is non-magnetic, the Maxwell equations reduce to the following wave equation for the electric fieldE:

E−∇(∇ ·E)= 1

c²∂t²[_E+_P] (1) wherecis the light velocity in vacuum. The time variabletis rescaled ast=ct, so thatctakes the value 1. The primes are omitted below. We denote by∂tthe derivative operator_∂t^∂ with regard to the time variable tand so on, and by_∇the three- dimensional gradient operator relative to the space variables x,y andz. The dielectric polarization_P is described by the following standard model: it splits into the sum_P =PL+_PN L

of a linear polarization_PLsatisfying PL=χ⁽¹⁾∗E=

_t

−∞

dt1χ⁽¹⁾(t−t1)E(t1), (2) and a nonlinear polarization_PN Lsuch that

PN L=χ⁽³⁾∗(E,E,E)

= t

−∞dt1

t₁

−∞dt2

t₂

−∞dt3

×χ⁽³⁾(t−t1,t−t2,t−t3):_E(t1)E(t2)E(t3). (3) Experiments have been performed using a liquid medium such as CS2 [20]. Then, as in any centrosymmetrical material, the second nonlinear susceptibility tensor χ⁽²⁾is zero. This condition is assumed to be satisfied here. The fields _Eand P are expanded simultaneously in a power series of a small parameterε, and in a Fourier series with respect to some phase ϕ=kz−ωt. The propagation direction is thus chosen as the zaxis. This expansion is

E=

l1,p∈Z

ε^lEl^pe^ip(kz−ωt), (4)

with the reality condition _El^−p = El^p∗ for each l and p (∗ denotes complex conjugation). The polarization _P is expanded in the same way. In expansion (4), the dominant term isε(E1¹e^iϕ+_E1⁻¹e^−iϕ), i.e._E1^p =0 for all p= ±1. In

other words, only the fundamental wavepacket propagates. A consequence of this assumption is that the amplitudes of the harmonics remain small quantities of order O(ε³). This means physically that third harmonic generation (due to the cubic nonlinearity) is not resonant. So is any stimulated scattering.

For the sake of simplicity, we denote below the leading electric field amplitudes_E1¹,_E1⁻¹,_E2¹and_E⁻₂¹by_E1,_E1^∗,_E2and_E2^∗, respectively.

The amplitudes_El^pare functions ofyand of slow variables (ξ, ζ, τ) defined by

ξ =εx, ζ =ε²z, τ =ε(t−z/v).

(5)

The y dependence describes the transverse structure of the waveguide modes. y is a variable of orderε⁰, not a slow variable. This feature accounts for the following assumption:

the waveguide thickness has the same order of magnitude as the wavelengthλ. τdescribes the longitudinal or temporal shape of the pulse, in a frame moving at the wavepacket velocityv. It is a slow variable of order ε, which means that the pulse length has the same order of magnitude asL =λ/ε. ξ gives an account of the transverse shape of the beam in the same scale asτ.ζis the variable that describes the evolution of the pulse shape during the propagation. Its order isε², giving an account of propagation distances aboutD=λ/ε².

This scaling accounts for two important assumptions.

Firstly three length (or time) scales are involved: that of the wavelength λ, of the pulse length L = λ/ε and of the propagation distance D = λ/ε². Secondly the pulse length, about L, is very large with regard to the wavelengthλ, and very small with regard to the propagation length, aboutD. It is almost the same scaling as commonly used for the derivation of the NLS equation in bulk media [21]. The main difference lies in the existence of a transverse variable y, which is not considered as a slow variable, assuming as mentioned above that the guide thickness has the same order of magnitude as the wavelengthλ. The scaling can thus describe a pulse of some hundred microns width and length, propagating in a waveguide a few microns thick. For instance, the wavelength λbeing about 0.5µm and the perturbative parameterε 5×10⁻³, the order of magnitude of the pulse width and length is L = λ/ε 100µm, which corresponds to a characteristic duration of 0.3 ps, and the propagation distance is about D=λ/ε²2 cm.

For the sake of simplicity we will split the spatio-temporal study into two parts. In this paper we will study the case of a sufficiently wide beam in the x dimension. ‘Wide’ here means that the pulse width is very large with regard to its length. Taking into account the orders of magnitude of the pulse length and propagation distance specified above, the diffraction length (or Rayleigh length) becomes very large with regards to propagation length D. Diffraction can thus be neglected and it amounts to studying the propagation of a short pulse in a one-dimensional medium corresponding to the confinement direction. The propagation of a diffracting stationary beam in the nonlinear medium and its self-focusing will be considered in a future publication.

515

The derivatives with respect to the spatial coordinate x can thus be neglected in the vectorial wave equation (1), which reduces to

∂²_yE^x+∂_z²E^x =∂_t²[E^x +P^x],

∂y∂zE^z−∂z²E^y= −∂t²[E^y+P^y],

∂z∂yE^y−∂_y²E^z= −∂_t²[E^z+P^z].

(6)

In the expansion (4) of the fields_Eand_P, the amplitudes_El^p depend on yand on the slow variables(τ, ζ), expansions (4) and (5) are substituted into the basic equations (2), (3) and (6) and the coefficients of each power ofεare collected to obtain a set of equations, which we solve order by order.

3. The linear guided modes are found again

The equations obtained at order ε¹ yield the linear approximation. Equation (6) leads to the following ordinary differential equation of second order with respect to the variabley:

∂²yE₁^x +(ω²n²_i −k²)E1^x =0, (7a)

∂yE₁^y+ ik E₁^z=0, (7b)

∂²yE₁^z+(ω²n²_i −k²)E₁^z=0. (7c) niis the refraction index of the considered layer (n1=nin the waveguide,n2in superstrate andn3in substrate).niis related to the material linear susceptibility by

(1 +χˆ_i⁽¹⁾)=n²_i, (8) whereχˆ_i⁽¹⁾ is the Fourier transform of the material response functionχ_i⁽¹⁾defined by relation (2). The system (7) is thus valid in the three media. At the interfaces, the electromagnetic field satisfies the electromagnetic continuity conditions: the tangential component of the electric and magnetic fields and the normal component of the electric displacement and of the magnetic induction are continuous. This yields, for the electric part,

E₁^x|y=a⁺ =E₁^x|y=a⁻, (9a) n²₂E₁^y|y=a⁺ =n²E₁^y|y=a⁻, (9b) E₁^z|y=a⁺ =E₁^z|y=a⁻, (9c) and for the magnetic part,

∂yE₁^x|y=a⁺ =∂yE₁^x|y=a⁻, (10a) ik E₁^x|y=a⁺ =ik E₁^x|y=a⁻, (10b)

∂yE₁^z|y=a⁺−ik E₁^y|y=a⁺=∂yE₁^z|y=a⁻−ik E₁^y|y=a⁻. (10c) Equations (7a), (9a) and (10a) on the one hand and (7b), (7c), (9b), (9c), (10b) and (10c) on the other can be solved separately, giving respectively the x componentE₁^x of the electric field, and itsyandzcomponentsE₁^yandE^z₁.

3.1. Transverse electric modes

In order to describe a guided mode, the general solutionE₁^x(y) of equation (7a) must be finite asy→ ±∞. It must describe an oscillating wave inside the waveguide and evanescent waves outside, as

E₁^x =







A^x₁e^{iq y}+B₁^xe^{−iq y} if|y|a, C₁^xe^−k2y ifya, D^x₁e^k3y ify−a,

(11)

whereqandkiare defined by the dispersion relation of optical waves in bulk media:

q=

ω²n²−k² (12)

ki=

k²−ω²n²_i, i=2,3. (13) Since the wavevectorsqandkimust be real, the waveguiding conditions are deduced: n > n2,n3. The boundary conditions (9a) and (10a) impose that E₁^x and ∂yE₁^x are continuous across the interface. Thus the coefficientsA^x₁,B₁^x, C₁^x,D₁^x of the solution (11) must satisfy the linear system

M·





 A^x₁ B₁^x C₁^x D^x₁





=



 0 0 0 0



, (14)

with

M=





e^iqa e^−iqa −e^−k2a 0 iqe^iqa −iqe^−iqa k2e^−k2a 0

e^−iqa e^iqa 0 −e^−k3a iqe^−iqa −iqe^iqa 0 −k₃e^−k3a



. (15)

System (14) admits a nonzero solution only if the determinant ofMis equal to zero, which gives

tan(2qa)= q(k2+k3) q²−k2k3

. (16)

Equation (16) is the well known dispersion relation for the TE waveguide modes [21]. For eachω, this yields a finite family of wavevectorskallowed to propagate in the guide. Replacing thekibyki=

ω²(n²−n²_i)−q², equation (16) becomes an equation for the transverse wavevectorq, directly related tok through relation (12). The solution of system (14) can then be written as

B₁^x(τ, ζ)=(q−ik2)²e^2iqa

q²+k²₂ A^x₁(τ, ζ), (17) C₁^x(τ, ζ)=2q(q−ik₂)e^(iqa+k2a)

q²+k²₂ A^x₁(τ, ζ), (18) D₁^x(τ, ζ)= (q−ik2)e^(3iqa+k3a)+(q²+k²₂)e^{(−3iqa+k3a)}

q²+k₂²

×A^x₁(τ, ζ). (19)

At first order of the multiscale expansion, the linear theory of the waveguide is found again, with the expressions of the TE modes propagating in planar waveguides.

Multiscale theory of nonlinear wavepacket propagation in a planar optical waveguide

3.2. Transverse magnetic modes

The components E₁^y, E^z₁ of the electric field satisfy equations (7b) and (7c). Their resolution yields an expression ofE₁^zidentical to (11), replacing the superscriptx byz. The boundary conditions (9b), (9c), (10b) and (10c) yield the linear system

N ·





 A^z₁ B₁^z C₁^z D₁^z





=



 0 0 0 0



, (20)

where N =







e^iqa e^−iqa −e^−k2a 0

i^ω2_qⁿ²e^iqa −i^ω2_qⁿ²e^{−iqa ω}_k²ⁿ²²

2 e^−k2a 0

e^−iqa e^iqa 0 −e^−k3a

i^ω2_qⁿ²e^−iqa −i^ω2_qⁿ²e^iqa 0 −^ω_k²ⁿ₃²³e^−k3a





.

(21) Setting the determinant ofNto zero, we obtain the dispersion equation of the TM modes as

tan(2qa)= qn²(k2n²₃+k3n²₂)

(q²n²₂n²₃−k₂k₃n⁴). (22) The coefficients B₁^z,C₁^z, D^z₁of the guided mode profile componentE₁^z(y), defined as in (11), have rather complicated expressions: we give them for a symmetrical guide only, i.e.n3=n2, and hencek3=k2. Then

B₁^z(τ, ζ)= −e^2iqa ρ(1 +σ²)

×[µ−2iσ(k⁴+k²q²(1 +σ²)−q⁴σ²)]A₁^z(τ, ζ) (23) C₁^z(τ, ζ)= 2e^iqan²ω²σ(k²−iσq²)

ρ(σ−i) A₁^z(τ, ζ) (24) D₁^z(τ, ζ)= 2e^iqa(k²−iσq²)

ρ(σ−i)

×[n²ω²σcos 2qa−n²₂ω²sin 2qa]A^z₁(τ, ζ). (25) We use the following shortcuts:

σ= k2

q, (26)

ρ=(k⁴+σ²q⁴), (27) µ=[k²(1 +σ)+σq²(1−σ)][k²(1−σ)−σq²(1 +σ)]. (28) The expression of the other component E₁^y(y) of the electric field is then deduced from (7b), and is

E₁^y =

















−k

q(A^z₁e^{iq y}−B₁^ze^{−iq y})+Q if|y|a, ik

k2

C₁^ze^−k2y ify a,

−ik k3

D^z₁e^k3y ify −a.

(29)

The integration constantQis set to zero in order to satisfy the boundary conditions for the ycomponent given by (29).

It should be noticed that the modal dispersion relations, equation (16) for the TE modes and equation (22) for the TM modes, are different. According to the ansatz (4), we consider

a unique wavevectork, i.e. a single waveguide mode, which can be either a TE or a TM one. However, in experiments, the wavepacket almost always contains a linear superposition of several modes, and nonlinear coupling between them might occur. This is not considered in the present paper.

3.3. The wavepacket velocity

The equations obtained at orderε²are analogous to first order, but inhomogeneous:

∂_y²E₂^x+(ω²n²_i −k²)E₂^x = −αi∂_τE₁^x, (30a)

−ikz∂yE₂^z+(ω²n²_i −k²)E₂^y = −αi∂_τE₁^y−1

v∂_τ∂yE₁^y, (30b)

∂²yE₂^z+(ω²n²_i −k²)E₂^z = −αi∂τE₁^z, (30c) whereαi = 2iωn²_i + 2iω²nin_i− ^2ik_v. The dispersion of the medium isn_i = ^dn_dωⁱ and vis defined as the velocity of the wavepacket.

For a TE mode, E₁^y = E₁^z = 0, it is easily seen from equations (30b) and (30c) that E₂^y and E₂^z also vanish, and system (30) reduces to equation (30a). It involves the evolution of the amplitudeE₁^xwith respect to the variableτ =ε(t−z/v). The solutionE₂^xof equation (30a) has the same form as in (11), with the subscript 1 replaced by 2, and an inhomogeneous additional term. Making use of the boundary conditions, we get an inhomogeneous system with a right-hand-side member Fdepending on the first order:

M·





 A^x₂ B₂^x C₂^x D₂^x





=F(∂_τA^x₁). (31)

whereMis given by equation (15) andFis a four-component function of ∂_τA^x₁. Since the determinant of M is equal to zero, the system (31) must satisfy a compatibility condition, obtained by replacing one column ofMby the right-hand-side memberF, and setting the determinant of the obtained matrix to zero. This condition yields the expression of the wavepacket velocityv, according to

1

v =κ1n+κ2, (32) with

κ1= 2anω³k2k3

kω(k2+k3+ 2ak2k3)

1 + (k2+k3)(q²+k2k3) 2a(q²+k₂²)(q²+k²₃)

, (33) and

κ2= 2ak2k3(q²+k²)+k²(k2+k3)

kω(k2+k3+ 2ak2k3) . (34) On the other hand, taking the derivative of the dispersion relation, equations (16), (12) and (13), with respect to the pulsation ω, we compute ^d_dk^ω and we check that the usual expression

v=dω

dk (35)

of the group velocity holds.

517

The correction E₂^x(y) to the TE mode profile has an expression analogous to (11) in whichB₁^x,C₁^x,D₁^xare replaced by B₂^x,C₂^x,D₂^x, given in appendix A for a symmetrical waveguide.

If the considered mode is a TM one, E₁^x = E₂^x = 0 and system (30) reduces to equations (30b) and (30c).

An analogous computation leads to an expression of the wavepacket velocity v analogous to (32)–(34). It has been checked explicitly that the obtained expression of the wavepacket velocity coincides with the usual expression (35) of the group velocity.

4. The nonlinear evolution equation

4.1. Self-phase modulation of a TE mode

Let us consider first a TE mode. At this order, the equation derived from equation (6) is

∂²_yE₃^1,x +(ω²n²_i −k²)E¹₃^,x = −αi∂_τE₂^x+βi∂_τ²E^x₁

−2ik∂_ζE₁^x−ω²P_{N L}^1,x_,3, (36) where βi= −_v¹2 +n²_i+ω(ωn_i²+ωnin_i + 4nin_i).n_i denotes

d²ni

dω²and the nonlinear polarization amplitude componentP_{N L,3}^1,x is derived from equation (3) as

P_{N L,3}^1,x =

p1+p2 +p3=1 j,k,l=x,y,z

ˆ

χx j kl⁽³⁾(p1ω,p2ω,p3ω)E₁^p1,jE₁^p2,kE₁^p3,l.

(37) In an isotropic medium, becauseE₁^y = E₁^z =0 for a TE mode, and using the symmetry properties of theχ⁽³⁾tensor, equation (37) reduces to

P₃^1,x =3χˆx x x x⁽³⁾ (ω,−ω, ω)|E₁^x|²E₁^x, (38) whereχˆx x x x⁽³⁾ (ω,−ω, ω)is a real quantity, assuming a lossless medium. The differential equation (36) can be solved without difficulty since the right-hand-side member depends explicitly onE₂^xandE₁^x, computed at previous orders. In the same way as for the previous order, the boundary conditions lead to a linear system which admits a nonzero solution if some compatibility condition is satisfied. The latter can be put in the form

2ik∂ζA^x₁+∂_τ²A^x₁+e|A^x₁|²A^x₁=0 (39) whereis a real constant and the coefficienteof the nonlinear term is given by

e= ˆχ_{x x x x}⁽³⁾ 3ω² 8

k2(k²₂−q²)

q(k₂²+q²)²(1 +k₂a) (40) with

=(3q²−k₂²)sin 4qa−4k2qcos 4qa

+(9q²+ 7k²₂+ 24q²k2a)sin 2qa+ 2q(6(q²−k²₂)a+k2)

× cos 2qa+ 12q((q²+k²₂)a+k2). (41) In a lossless medium, the nonlinear susceptibility component

ˆ

χ_{x x x x}⁽³⁾ is real, and so also is the coefficient e. Then equation (39) is the NLS equation.

It can be shown using expressions (16), (12) and (13) that the dispersion coefficient is

= −kk= −kd²k

dω². (42)

The latter expression for this coefficient, well established for wave propagation in bulk media, is still valid in the present case. The coefficiente takes into account the waveguiding structure through its dependence with regard toq,k2anda.

Then, from the Maxwell equations and within the multiscale formalism, the nonlinear propagation of a laser pulse in a planar waveguide is described, taking into account the waveguide geometry and its nonlinear characteristics.

4.2. Self-phase modulation of a TM mode

Consider now a TM mode. The electric field owns, in this case, two nonzero components that satisfy the following equations:

∂yE₃^y+ ik E₃^z= −α∂τE₂^y+β∂_τ²E₁^y−2ik∂ζE₁^y

−∂τ∂yE₂^z

v +∂ζ∂yE^z₁− 1

n²ik P₃^1,z− 1 n²∂yP₃^1,y,

∂²yE₃^z+q²E₃^z= −α∂τE₂^z+β∂_τ²E₁^z−2ik∂ζE₁^z

− ik

n²∂yP₃^1,y−q² n²P₃^1,z.

(43)

In an isotropic medium, the nonlinear polarization amplitude components_P3^1,iare, fori=y,z,

P₃^1,i =(χˆx x y y⁽³⁾ +χˆx yx y⁽³⁾ )[|E¹₁^,y|²+|E₁^1,z|²]E₁^1,i

+χˆ_{x y yx}⁽³⁾ [(E₁^1,y)²+(E₁^1,z)²]E₁^1,i∗. (44) System (43), together with the boundary conditions, yields some compatibility condition, which reduces to

2ik∂ζA₁^y−kk∂_τ²A^y₁+m|A₁^y|²A^y₁=0. (45) The nonlinear coefficient m is a linear combination of two components of the third-order nonlinear susceptibility components:

m=(a₁χˆx yx y⁽³⁾ +a₂χˆx y yx⁽³⁾ ) (46) where the coefficientsa1anda2, that express the waveguide influence on the nonlinear coefficient are given by

a1= {c1(sin 6qa+ sin 2qa)+c2sin 4qa+c3cos 6qa +c4(cos 4qa+ 1)+c5cos 2qa}{c6sin 4qa+c7cos 4qa

+c₈}⁻¹, (47)

and

a₂= {d₁(sin 6qa+ sin 2qa)+d₂sin 4qa+d₃cos 6qa +d4(cos 4qa+ 1)+d5cos 2qa}{d6sin 4qa

+d7cos 4qa+d8}⁻¹. (48)

The coefficientsc1, . . . ,d8are given in appendix B. The sign ofm is related to those of componentsχx yx y andχx y yx(a1, a2are positive).

4.3. The nonlinear coefficients

An example of numerical computation of the nonlinear coefficientse andm of the NLS equations (39) and (45) is shown in figure 2. The considered waveguide is symmetrical, with a thickness varying from 0 to 20 µm, and refractive indices n = 1.63 and n2 = 1.37. The wavelength is λ = 532 nm, and the first TE mode (TE1) and the first TM mode (TM1) are considered. These data are very close to that of the experiments [20].

Multiscale theory of nonlinear wavepacket propagation in a planar optical waveguide

2a(µm) 2a(µm)

2 10 20

0.994 0.997

1 Γ

Γ

e

m

Figure 2.Nonlinear coefficient normalized value for TE1and TM1

modes versus waveguide thickness.

The comparison of the two coefficients shows that the light–matter interaction is stronger in the direction parallel to the waveguide. The waveguiding structure breaks the symmetry between the two polarizations, creating a privileged direction. Two important features have been observed experimentally. First, the input power required for spatial soliton formation is smaller for TE modes than for TM modes.

Second, the polarization perpendicular to the waveguide, corresponding to a TM mode, is unstable and gives its energy to the TE modes. The above obtained values of the nonlinear coefficientseandmare in good agreement with these experimental observations. It must be noticed that the relative difference between e and m is rather small, and cannot give a quantitative account of the important difference between the power required for soliton formation with the two polarizations. However, this small difference is sufficient to give an account of the observed polarization instability. More detail about these features will be given in a future publication.

5. Conclusion

From basic Maxwell equations we have derived the general wave propagation equations in a nonlinear waveguide for TE and TM modes. The multiscale expansion, at first and second orders, found the linear theory of the guided modes again, in perfect agreement with previous theories.

The modal field distributions and their own group velocities are determined. The following order of the perturbative scheme yields the nonlinear partial differential equation that describes the evolution of the pulse amplitude. It is a NLS equation, as expected. The dispersion coefficient of the latter, computed by means of multiscale expansion, coincides with the commonly admitted value, which had be derived using the linear dispersion theory. The discrepancy observed for magnetostatic backward volume waves in thin magnetic films [19] does not arise here. On the contrary, the nonlinear coefficient in the NLS equation depends on the considered mode and the waveguide geometry. It shows that the input power required for soliton formation for a given waveguide depends on the propagation mode. For reasonable numerical data, it is smaller for the TE1 mode than for the TM1 one, in qualitative agreement with experimental results. Using an expansion formed by a superposition of linear modes, it will be possible to describe the interaction that occurs during the

propagation, using a model with two coupled NLS equations, and numerical simulations.

Appendix A. The electric field at second order In this appendix, we give the solutionE₂^x(y)at the second order for TE modes, in a symmetrical waveguide,n3=n2:

B₂^x = (1−iσ)e^2iqa 1 + iσ

×

A₂^x+ iα σq²

σaq−1 + δ 1 +σ²

∂τA^x₁

(49) C₂^x = e^iqa

1 + iσ

2A₂^x− α σq²

(aq−i)(1−iσ) +δ

aq− i 1 + iσ

∂τA^x₁

, (50)

D₂^x = 2e^iqa

1 + iσ(cos 2qa+σsin 2qa)A^x₂ +

aα 2q²σ

1 + δ

1 + iσ

e^−iqa+ α 2σ²q²

aqσ(1−3iσ)

−2i +δ

aqσ− 2iσ 1 +σ²

e^3iqa

∂τA^x₁, (51) where

α=2iωn²+ 2iω²nn−2ik

v , (52)

α2=2iωn²₂−2ik

v and δ= α2−α

α . (53) Appendix B. Expression of the nonlinear coefficient for TM modes

The nonlinear coefficient of the NLS equation (45) that governs the evolution of the amplitude of the TM modes,m, expresses the action of the waveguide on this evolution. It is given by equations (46)–(48), with the following expressions of the coefficientsc1, . . . ,d8:

c1=2(k−q)q²[(k⁶−σ²q⁴(3k²+q²))(3−σ²)

+k⁴q²(1−11σ²)] (54)

c2=2q²[(k−q)(3k⁶+k⁴q²+(3k²q⁴+q⁶)σ⁴

×(13k⁶+ 16k⁵q+ 35k⁴q²+ 32k³q³+ 41k²q⁴+ 16kq⁵ + 15q⁶)σ²+ 8qaσn²n²₂ω⁴(k+q)(3n²ω²−2kq)], (55) c3= −8σq²n⁴n²₂ω⁶(k−q) (56) c4=4q²[σn²n²₂ω⁴(k−q)(7k²+ 8kq+ 5q²)

+ 2qa(k+q)(3n²ω²−2kq)µ], (57) c5=8q²[2aq(3(k³+q³)+kq(k+q))(1 +σ²)ρ

+σn²n²₂ω⁴(k−q)(5n²ω²+ 8kq)], (58) whereµandρare given by (28) and (27), respectively.

c6=2kn²q⁶(1 +σ²)ρ[k⁴(1 +σ⁴)

−2q⁴σ⁴+ 2qaσ³n²n²₂ω⁴], (59) c₇=2σn²kq⁶(1 +σ²)ρ[qaσµ−ρ(1−σ²)

−k²q²(1 +σ²)²], (60) 519

c8=2kn²q⁶σ(1 +σ²)

×[3qaσρµ+(1−σ²)(k⁸−10k⁴q⁴σ²−3q⁸σ⁴)

−k²q²(k⁴(15−σ²)(1 +σ²)

+ 3q⁴σ²(1 +σ²)²−16k⁴)]. (61) d1=4n²ω²q²[(k⁶+q⁶σ²)(3−σ²)−k²q²(k²(1 + 5σ²)

−q²σ²(7 + 3σ²))], (62)

d2=4q²(k²+q²)[8qaσ(3(k⁴+q⁴)−2k²q²)(k²−σ²q²) +k⁶(3 + 13σ²)−k⁴q²(1−5σ²)−3k²q⁴σ²(5−σ²)

−q⁶σ²(15 +σ²)], (63)

d3=2(k+q)c4 (64)

d4=8q²[σn⁴n²₂ω⁶(7k²−5q²)

+ 2qa(3(k⁴+q⁴)−2k²q²)µ], (65) d5=16q²[5σn⁴n²₂ω⁶(k²−q²)

+ 2qa(3(k⁴+q⁴)−2k²q²)(1 +σ²)ρ], (66) d6=4n²k²q⁶(1 +σ²)ρ[k⁴(1 +σ⁴)−2q⁴σ⁴

+ 2qaσ³n²ω²(k²−σ²q²)], (67) d7= −4n²k²q⁶σ(1 +σ²)ρ[(1−σ²)ρ

+k²q²(1 +σ²)²−aqσµ], (68) d8=4n²k²q⁶σ(1 +σ²)[(k⁸−10k⁴q⁴σ²−3q⁸σ⁴)(1−σ²)

−3k²q⁶σ²(1 +σ²)²+k⁶q²(1−14σ²+σ⁴)

+ 3aqσ(k⁴+q⁴σ²)µ]. (69)

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