Few-cycle optical soliton propagation in coupled waveguides

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Few-cycle optical soliton propagation in coupled waveguides

Hervé Leblond, Dumitru Mihalache, David Kremer, Said Terniche

To cite this version:

Hervé Leblond, Dumitru Mihalache, David Kremer, Said Terniche. Few-cycle optical soliton propagation in coupled waveguides. Nonlinear Nanophotonics meets Nanomagnetism 2016, 2016, Le Mans, France. �hal-02564158�

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Few-cycle optical soliton propagation in coupled waveguides

Herv´e Leblond¹, Dumitru Mihalache², David Kremer³, Said Terniche^1,4

1Laboratoire de Photonique d’Angers LϕA EA 4464, Universit´e d’Angers.

2Horia Hulubei National Institute for Physics and Nuclear Engineering, and Academy of Romanian Scientists, Bucharest.

3Laboratoire MOLTECH-Anjou, CNRS UMR 6200, Universit´e d’Angers.

4Laboratoire Electronique Quantique, USTHB, Alger.

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1 Models for few-cycle solitons The mKdV-sG equation

Nonlinear widening of the linear guided modes

2 Waveguide coupling in the few-cycle regime Derivation of the coupling terms

Examples of behavior of the linear non-dispersive coupling Few-cycle optical solitons in linearly coupled waveguides

3 Few cycle spatiotemporal solitons in waveguide arrays Formation of a solitons from a Gaussian pulse Two kind of solitons

Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d’Angers LϕA EA 4464, Universit´Few-cycle optical solitons in coupled waveguides e d’Angers., Horia Hulubei National Institute for Physics and Nuclear Engineering, and Academy of Romanian Scientists, Bucharest., Laboratoire MOLTECH-Anjou, CNRS UMR 6200, Universit´2 / 47 e d’Angers., Laboratoire Electronique Quantique, USTHB, Alger. )

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Introduction

The shortest laser pulses: a duration of a few optical cycles.

Autocorrelation trace, R. Ell et al., Optics Letters 26 (6), 373 (2001).

Pulse duration down to a few fs. Ex. above: 5fs=5×10⁻¹⁵s.

How to model the propagation of such pulses?

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Solitary wave vs envelope solitons

Envelope soliton: the usual optical soliton in the ps range

Pulse durationLλwavelength Typical model: NonLinear Schr¨odinger equation (NLS)

It is a soliton if it propagates without deformation on DL, due to nonlinearity.

In linear regime: spread out by dispersion.

Solitary wave soliton: the hydrodynamical soliton A single oscillation

Typical model: Korteweg-de Vries equation (KdV)

Few-cycle optical solitons:L∼λ

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Solitary wave vs envelope solitons

Pulse durationLλwavelength Typical model: NonLinear Schr¨odinger equation (NLS) It is a soliton if it propagates without deformation on DL, due to nonlinearity.

In linear regime: spread out by dispersion.

Solitary wave soliton: the hydrodynamical soliton A single oscillation

The slowly varying envelope approximation is not valid Generalized NLS equation

We seek a different approach based on KdV-type models

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Solitary wave vs envelope solitons

Pulse durationLλwavelength Typical model: NonLinear Schr¨odinger equation (NLS) Solitary wave soliton: the hydrodynamical soliton

A single oscillation

The slowly varying envelope approximation is not valid Generalized NLS equation

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A transparent medium

The general absorption spectrum of a transparent medium

1/_p

₁ ₂

A simple model: A two-component medium, each component is described by a two-level model

We assume that the transparency range is very large:

ω1 (1/τp)ω2 (τp: FCP duration).

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A transparent medium

1/_p

₁ ₂

1/_p

₁ ₂

ω1 (1/τp)ω2

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A transparent medium

1/_p

₁ ₂

1/ _p

₁ ₂

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A transparent medium

1/ _p

₁ ₂

ω₁ (1/τ_p)ω₂

(τp: FCP duration).

In a first stage, the two components are treated separately UV transition only, with (1/τp)ω2

=⇒ Long-wave approximation

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A transparent medium

1/ _p

₁ ₂

ω₁ (1/τ_p)ω₂

(τp: FCP duration).

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A transparent medium

1/ _p

₁ ₂

1/_p ₂

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A transparent medium

In a first stage, the two components are treated separately UV transition only, with (1/τ_p)ω₂

1/_p ₂

modified Korteweg-de Vries (mKdV) equation

∂E

∂ζ = 1 6

d³k dω³ ω=0

∂³E

∂τ³ − 6π

ncχ⁽³⁾(ω;ω,ω,−ω) ω=0

∂E³

∂τ

H. Leblond and F. Sanchez,Phys. Rev. A67, 013804 (2003)

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A transparent medium

In a first stage, the two components are treated separately IR transition only, with ω1 (1/τp)

=⇒ Short-wave approximation sine-Gordon (sG) equation: ∂²ψ

∂z∂t =c1sinψ with c₁ = w∞

w_r : normalized initial population difference and ∂ψ

∂t = E

E_r : normalized electric field

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A transparent medium

In a first stage, the two components are treated separately IR transition only, with ω₁ (1/τ_p)

1/_p

₁

∂z∂t =c₁sinψ with c1 = w∞

∂t = E Er

: normalized electric field

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A transparent medium

In a first stage, the two components are treated separately IR transition only, with ω₁ (1/τ_p)

1/_p

₁

∂z∂t =c₁sinψ with c1 = w∞

∂t = E Er

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A transparent medium

Then the two approximations are brought together to yield a general model:

The mKdV-sG equation

∂²ψ

∂z∂t +c1sinψ+c2

∂

∂t ∂ψ

∂t 3

+c3

∂⁴ψ

∂t⁴ = 0

∂u

∂z +c1sin Z t

u+c2

∂u³

∂t +c3

∂³u

∂t³ = 0 with u = ∂ψ

∂t = E

Integrable by inverse scattering transform in some cases:

,

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A transparent medium

∂²ψ

∂z∂t +c₁sinψ+c₂ ∂

∂t ∂ψ

∂t 3

+c₃∂⁴ψ

∂t⁴ = 0 Or

∂u

∂z +c₁sin Z t

u+c₂∂u³

∂t +c₃∂³u

∂t³ = 0 with u = ∂ψ

∂t = E

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A transparent medium

∂u

∂z +c1sin Z t

u+c2

∂u³

∂t +c3

∂³u

∂t³ = 0 with u = ∂ψ

∂t = E

,

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A transparent medium

∂u

∂z +c2

∂u³

∂t +c3

∂³u

∂t³ = 0 with u = ∂ψ

∂t = E Er

mKdV,

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A transparent medium

∂u

∂z +c₁sin Z t

u = 0

with u = ∂ψ

∂t = E Er

mKdV, sG,

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A transparent medium

∂u

∂z +c₁sin Z t

u+c₂∂u³

∂t + 2c₂∂³u

∂t³ = 0 with u = ∂ψ

∂t = E

mKdV, sG, andc₃ = 2c₂.

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The analytical breather solution

1 2

2 4 6

-4 -2

-6 0

-0.05 0.05

0



 E

-2 0

-1

A few cycle soliton:

Not spread out by dispersion Stable

However, oscillates (breather)

H. Leblond, S.V. Sazonov, I.V. Mel’nikov, D. Mihalache, and F. Sanchez, Phys. Rev. A74, 063815 (2006)

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Waveguide description

The evolution of the electric field E: In (1+1) dimensions:

−→ The modified Korteweg-de Vries (mKdV) equation

∂_ζE+β∂_τ³E +γ∂_τE³= 0 Nonlinear coefficient γ = 1

2ncχ⁽³⁾, Dispersion parameter β= (−n⁰⁰)

2c ,

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Waveguide description

The evolution of the electric field E: We generalize to (2+1) dimensions:

−→ The cubic generalized Kadomtsev-Petviashvili (CGKP) equation

∂ζE+β∂_τ³E +γ∂τE³−V 2

Z τ

∂_ξ²Edτ⁰ = 0 Nonlinear coefficient γ = 1

2ncχ⁽³⁾, Dispersion parameter β= (−n⁰⁰)

2c , Linear group velocity: V = c

n.

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Waveguide description

The evolution of the electric field E:

A waveguide:

c g c x

core

cladding cladding

∂_ζE+β_α∂_τ³E +γ_α∂_τE³ −V_α 2

Z τ

∂_ξ²Edτ⁰ = 0 with α=g in the core andα=c in the cladding.

Nonlinear coefficient γ_α= 1 2n_αcχ⁽³⁾_α , Dispersion parameter β_α = (−n⁰⁰_α)

2c , Linear group velocity: Vα= c

nα

.

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Waveguide description

A waveguide:

c g c x

core

cladding cladding

∂_ζE+β_α∂_τ³E +γ_α∂_τE³+ 1

V_α∂_τE−Vα

2 Z _τ

∂_ξ²Edτ⁰ = 0 with α=g in the core andα=c in the cladding.

Velocities : V_g <V_c

Nonlinear coefficient γα= 1 2n_αcχ⁽³⁾_α , Dispersion parameter βα = (−n⁰⁰_α)

2c ,

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Waveguide description

A waveguide:

c g c x

core

cladding cladding

−→ The cubic generalized Kadomtsev-Petviashvili (CGKP) equation In dimensionless form:

∂zu =Aα∂_t³u+Bα∂tu³+vα∂tu+Wα

2 Z t

∂_x²udt⁰ with α=g in the core andα=c in the cladding.

Relative velocities :v_g >v_c Nonlinear coefficient γα= 1

2n_αcχ⁽³⁾_α , Dispersion parameter βα = (−n⁰⁰_α)

2c , Linear group velocity: Vα= c

nα

.

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Linear guided modes

The dimensionless CGKP equation

∂_zu =A_α∂_t³u+B_α∂_tu³+v_α∂_tu+W_α 2

Z t

∂_x²udt⁰

We linearize and seek for solutions of the form u =f(x) exp[i(ωt−kzz)].

Waveguide dispersion relation: tan(k_xa) = κ kx

Mode profiles of the form

f =Rcos(kxx) in the core f =Ce^−κ|x| in the cladding

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Linear guided modes

∂zu =Aα∂³_tu+ vα∂tu+Wα

2 Z t

∂_x²udt⁰ We linearize and seek for solutions of the form

u =f(x) exp[i(ωt−kzz)].

Material dispersion relation: k_x,α² = 2ω

V_α Aαω³−kz−ωvα

Waveguide dispersion relation: tan(k_xa) = κ k_x Mode profiles of the form

f =Rcos(k_xx) in the core f =Ce^−κ|x| in the cladding

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Linear guided modes

∂zu =Aα∂³_tu+ vα∂tu+Wα

2 Z t

∂_x²udt⁰ We linearize and seek for solutions of the form

u =f(x) exp[i(ωt−kzz)].

Waveguide dispersion relation: tan(k_xa) = κ k_x Mode profiles of the form

f =Rcos(kxx) in the core f =Ce^−κ|x| in the cladding Guiding condition:A2ω³−ωv2<kz <A1ω³

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Nonlinear propagation in linear guide

We solve the CGKP equation starting from

u(x,t,z = 0) =Acos(ωt)f(x)e^−t²^/w², f(x) =

cos(kxx), for |x| ≤a,

Ce^−κ|x|, for |x|>a, is a linear mode profile Normalized coefficients A₁=A₂=B₁ =B₂ =W₁=W₂ = 1,

−→ we assume that

- Temporal compression occurs

- Spatial defocusing occurs, (else it collapses!)

- Dispersion and nonlinearity are identical in core and cladding.

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Guided wave profiles

(Normalized so that the total power is 1.v2= 3,w= 2.)

The pulse is less confined in nonlinear (blue, and red)

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Nonlineary may reduce confinement. Indeed:

Waveguiding is due to total internal reflection.

−→ when the wave propagates too fast in the cladding, it cannot match the field oscillations in the core.

In the few-cycle regime,

not only linear, but also nonlinear velocity (large and positive here).

The nonlinear velocity is larger in the core and creates a nonlinear variation of the index.

It may compensate the linear part, and reduce the confinement.

Important in very narrow guides, where linear confinement itself is low.

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Nonlinear waveguide

Wave guided and confined by using nonlinear velocity:

a higher nonlinear coefficient in the cladding than that in the core.

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Two-cycle soliton of the nonlinear waveguide

x

t

-2

-1

0

1

2

-150 -100 -50 0 50 100 150

B2−B1= 1.

H. Leblond and D. Mihalache,Phys. Rev. A88, 023840 (2013)

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2D waveguiding structure: two cores 1 and 2 and dielectric cladding

The generalized Kadomtsev-Petviashvili (GKP) equation (dimensionless)

∂_zu =A_α∂_t³u+B_α∂_tu³+V_α∂_tu+w_α 2

Z t

∂_x²udt, α=g in the cores 1 and 2,α=c in the cladding.

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Individual waveguides, in the linearized model (B_α = 0):

The field is u=Rfj(x)e^i(ωt−βz)=Rfj(x)e^iϕ, f_j, (j = 1, 2): guided mode profile, R: amplitude.

We seek for a solution as

u=R(z)f1(x)e^iϕ+S(z)f2(x)eⁱ^ϕ, i.e., two interacting modes.

We report it into the GKP equation and get – in the cladding ∂zRf1+∂zSf2 = 0 – in guide 1 ∂zRf1+∂zSf2= iw_g

2ω (Kc−Kg)Sf2

and so on.

Multiplying this byf₁ and integrating over allx removes the x-dependency of f1,f2 and we get

∂_zR=∂_zS = iw_g 2ω

(K_c−K_g)I₂ 1 +I1

(R+S), Involve overlap integrals I₁ =R∞

−∞f₁f₂dx,I₂ =R

g1f₁f₂dx =R

g2f₁f₂dx

(“R

g_j·dx” is the integral over the corej= 1 or 2.)

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2ω (Kc−Kg)Sf2

and so on.

(K_c−K_g)I₂ 1 +I1

(R+S),

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2ω (Kc−Kg)Sf2

and so on.

(K_c−K_g)I₂ 1 +I1

g1f₁f₂dx =R

g2f₁f₂dx

(“R

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2ω (Kc−Kg)Sf2

and so on.

(K_c−K_g)I₂ 1 +I1

(R+S),

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2ω (Kc−Kg)Sf2

and so on.

(K_c−K_g)I₂ 1 +I1

g1f₁f₂dx =R

g2f₁f₂dx

(“R

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Multiplying this byf1 and integrating over allx removes the x-dependency of f₁,f₂ and we get

(K_c−K_g)I₂ 1 +I1

(R+S),

The few-cycle pulse is expanded as a Fourier integral of such modes, u₁=R

Re^iϕdω,u₂ =R

Seⁱ^ϕdω.

We report∂_zR and∂_zS into ∂_zu₁, and get the linear coupling terms.

Finally, we get the system of two coupled modified Korteweg-de Vries (mKdV) equations

3 3 3

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(K_c−K_g)I₂ 1 +I1

(R+S),

Re^iϕdω,u₂ =R

Seⁱ^ϕdω.

∂zu1=A∂_t³u1+B∂tu₁³+V∂tu1+C∂tu2+D∂³_tu2,

∂_zu₂=A∂_t³u₂+B∂_tu₂³+V∂_tu₂+C∂_tu₁+D∂³_tu₁,

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(K_c−K_g)I₂ 1 +I1

(R+S),

Re^iϕdω,u₂ =R

Seⁱ^ϕdω.

3 3 3

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(K_c−K_g)I₂ 1 +I1

(R+S),

Re^iϕdω,u₂ =R

Seⁱ^ϕdω.

∂zu1=A∂_t³u1+B∂tu₁³+V∂tu1+C∂tu2+D∂³_tu2,

∂_zu₂=A∂_t³u₂+B∂_tu₂³+V∂_tu₂+C∂_tu₁+D∂³_tu₁,

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Nonlinear coupling

An analogous procedure, treating the nonlinear term as a perturbation, yields a evolution equation

∂zu1=L(u₁) +∂t

I3u₁³+I4 3u²₁u2+u₂³ . The integrals involved areI₃ =R∞

−∞B_αf₁⁴dx,I₄ =R∞

−∞B_αf₁³f₂dx. The complete final system is

∂_zu₁ = A∂³_tu₁+B∂_tu³₁+V∂_tu₁

+C∂_tu₂+D∂_t³u₂+E∂_t 3u₁²u₂+u₂³

∂_zu₂ = A∂³_tu₂+B∂_tu³₂+V∂_tu₂

+C∂_tu₁+D∂_t³u₁+E∂_t 3u₁u₂+u₁³

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Nonlinear coupling

The complete final system is

∂zu1 = A∂³_tu1+B∂tu³₁+V∂tu1

+C∂tu2+D∂_t³u2+E∂t 3u₁²u2+u₂³

∂zu2 = A∂³_tu2+B∂tu³₂+V∂tu2

+C∂tu1+D∂_t³u1+E∂t 3u1u2+u₁³ We evidence

a standardlinear couplingterm,

a linearcouplingtermbased on dispersion, anonlinear couplingterm

H. Leblond, and S. Terniche,Phys. Rev. A93, 043839 (2016)

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Nonlinear coupling

+C∂tu2+D∂_t³u2+E∂t 3u₁²u2+u₂³

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Nonlinear coupling

+C∂tu2+D∂_t³u2+E∂t 3u₁²u2+u₂³

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Nonlinear coupling

+C∂tu2+D∂_t³u2+E∂t 3u₁²u2+u₂³

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We assume a purely linear and non-dispersive coupling

∂_zu = −∂_t(u³)−∂_t³u−C∂_tv,

∂_zv = −∂_t(v³)−∂_t³v−C∂_tu, The initial data is

u =A_usin (ω_ut+ϕ_u)e^−(t−t^u⁾²^/τ^u², v=A_vsin (ω_vt+ϕ_v)e^−(t−t^v⁾²^/τ^v.

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Asoliton is launched in channelu and a smaller inputin channel v. Initial pulses:

u forms asoliton,v is spread out.

Output, uncoupled:

Output, coupled:

Avector solitonforms.

(Blue line:u, red line:v. Output atz= 2,couplingC=−1 or 0.Au= 2,Av = 0.2, λu=λv = 1,FWHMu=FWHMv= 3,ϕu=ϕv = 0,tu=tv= 0.

Some energy is transferred form one channel to the other.

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Mutual trapping of solitons:

It can occur if theircenter don’t coincide initially Initial pulses:

Output, uncoupled:

Output, coupled:

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Two solitons with different frequenciescan lock together.

Initial pulses:

Output, uncoupled:

Output, coupled:

Blue line:u, red line:v. Output atz= 2, couplingC=−1 or 0.Au=Av = 1.8,λu= 1, λv= 0.8,FWHMu=FWHMv = 3,ϕu=ϕv = 0,tu=tv = 0).

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We look for stationary states (vector solitons) in this model The ”stationary” states oscillate witht andz: .

few-cycle solitons arebreathers.

Atypical example of few-cycle vector soliton

(Dotted lines:u, solid lines:v. Left: atz= 0, right: atz= 60.<Au>= 1.837).

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Evolution of soliton’s maximum amplitude during propagation.

1.6 1.8 2

0 1 2 3 4 5 6 7 8

maxt(|u|)

z

0.4 0.5 0.6

0 1 2 3 4 5 6 7 8

maxt(|v|)

z

Soliton with<Au>= 1.789.

Two types of oscillations:

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Consider now the coupled equations in thelinearized case.

Themonochromaticsolutions are u

v

= A

B

e^−i(ωt+bω³^z),

With, due to coupling,

A=u0coscωz+iv0sincωz, B=v0coscωz+iu0sincωz.

The maximum amplitude and the power density of the wave oscillate with spatial frequencycω/π=σ₀ = 1.326.

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Consider now the coupled equations in thelinearized case.

Themonochromaticsolutions are u

v

= A

B

e^−i(ωt+bω³^z), With, due to coupling,

A=u0coscωz+iv0sincωz, B=v0coscωz+iu0sincωz.

The maximum amplitude and the power density of the wave oscillate with spatial frequencycω/π=σ₀ = 1.326.