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Nonlinearities in optical fibers : Derivation of the Nonlinear Schrödinger Equation

Background Summary From Maxwell’s equation we can derive the wave equation :

1 ∂2 E ∂2 P ∇ × ∇ × E = − 2 2 − µ0 2 ∂t c ∂t where

E ( r, t ) = Re {E ( r, t ) exp( −iω0t )} P ( r, t ) = Re {P ( r, t ) exp( −iω0t )}

Slowly varying amplitude

In a linear dispersive medium, P and E are related by: ∞

P ( r , t ) = PL ( r , t ) = ε 0 ∫ χ (1) ( r , t − τ ) E ( r ,τ ) dτ −∞

P ( r , ω ) = ε 0 χ (1) ( r , ω ) E ( r , ω )

χ(1) is the first order susceptibility of the medium which is responsible for the losses and the dispersion. Using the linear polarization we derive the frequency domain wave equation called Helmholtz equation:

∇2 E ( r, ω ) + ε ( r, ω )

ω2 2

E ( r, ω ) = 0

c ε ( r, ω ) = ε 0 (1 + χ (1) ( r, ω ) ) The electric field is:

E ( r , ω − ω0 ) =

+∞

∫ E ( r , t ) exp[i (ω − ω

−∞

F ( x , y ) A( z , ω − ω 0 ) exp( i β 0 z )

0

) t ]dt =

The time domain linear propagation equation developed in section 3.2 is called the linear Schrödinger equation. It describes the dispersion induced broadening of the electric field envelope :

∂A i ∂2 A 1 ∂3 A + β2 2 − β3 3 = 0 ∂z 2 ∂t 6 ∂t β 2 is the second order dispersion parameter expressed in [ps2/km] β3 is the third order dispersion parameter expressed in [ps3/km]

Non linear Polarization The response of any dielectric medium to an intense electromagnetic field is nonlinear. The nonlinearity manifests itself as a high order polarization, namely a polarization which depends on powers of E

P ( r, t ) = PL ( r, t ) + PNL ( r, t ) =

ε

(1) (2) 2 (3) 3 χ χ χ r , t • E r , t + r , t • • E r , t + r , t • • • E ( ) ( ) ( ) ( ) ( ) ( r, t ) + …) ( 0

Where χ(j) (j=1,2,3,…) is jth order susceptibility (a tensor of rank j+1). The linear susceptibility χ(1) represents the dominant contribution to P χ(2) is responsible for effects like second harmonic and sum frequency generation. χ(2) vanishes in optical fiber. χ(3) is responsible for all important nonlinear effects in optical fibers .

P ( r , t ) = PL ( r , t ) + PNL ( r , t ) PL ( r , t ) = ε 0

+∞

∫

χ (1) ( t − t1 ) E ( r , t1 ) dt1

−∞

PNL ( r , t ) = ε 0 ∫∫

+∞

∫

χ ( 3) ( t − t1 , t − t 2 , t − t 3 ) E ( r , t1 ) E ( r , t 2 ) E ( r , t 3 ) dt1 dt 2 dt 3

−∞

In general, PL>>PNL so that PNL is treated as a small perturbation to PL The nature of glass is such that the nonlinear response is essentially instantaneous. A typical response time is in the single fs time scale which is much shorter than any practical optical pulse width. The instantaneous response leads to considerable simplifications since the time dependence of χ(3) can be described by three delta function.

χ (3) (t − t1 , t − t2 , t − t3 ) = χ (3)δ (t − t1 )δ (t − t2 )δ (t − t3 )

Hence

PN L ( r , t ) = ε 0 χ

(3)

E (r , t)E (r , t)E (r , t)

PNL (r , t ) ≈ ε 0ε NL E (r , t )

ε NL

2 3 (3) = χ xxxx E ( r , t ) 4

εNL is the nonlinear component of the dielectric constant

To wave equation for the slowly varying amplitude, E(r,t), is obtained in the frequency domain. Since εNL is used as a perturbation parameter (as is PNL) it is assumed to be constant during the derivation of the propagation equation.

∇ E ( r , ω ) + ε (ω ) 2

εL

with

ω2 c

2

E (r ,ω ) = 0

ε(ω) =1+ χ (ω) +εNL =( n+∆n) ≈n2 +2n∆n 2

(1)

n is the linear refractive index

∆n =

iα 2 + nNL E 2k0

α is the fiber loss and nNL is the nonlinear refractive index

nNL

3 = Re(χ (3) ) 8n

The total refractive index is : 2

n ' = Re( ε ) = n + nNL E = n + nNL

P Aeff

P is the optical power and Aeff is the effective mode area

Aeff

2 ⎛ +∞ +∞ ⎞ ⎜ ∫ ∫ F ( x , y ) dxdy ⎟ ⎜ −∞ −∞ ⎟ ⎝ ⎠ = 4

2

+∞ +∞

∫ ∫

F ( x , y ) dxdy

−∞ −∞

F(x,y) the functional form of the fiber mode. For the fundamental mode Aeff~πw2 with w being the spot size (Aeff~20-100µm2 @ 1550nm) 2

n ' = Re( ε ) = n + nNL E = n + nNL

P Aeff

Kerr Effect.

This is the intensity dependence of the refractive index

The Nonlinear Schrödinger Equation (NLSE) Using variable separation for the slowly varying field amplitude :

E ( r , ω − ω 0 ) = F ( x , y ) A( z , ω − ω 0 ) ex p (iβ 0 z )

2i β 0

yields

∂A 2 + (β 2 − β0 ) A = 0 ∂z

In the first order perturbation theory, the field mode distribution, F(x,y) is assumed to not be dependent on ∆n. However the eigenvalue

β becomes : β (ω ) = β (ω ) + ∆β 2

+∞ +∞

k0 ∆β =

∫ ∫ ∆n F ( x, y )

dxdy

−∞ −∞

+∞ +∞

∫∫

−∞ −∞

2

F ( x , y ) dxdy

This leads to the Nonlinear Schrödinger Equation (NLSE) that completely describes the pulse propagation in the fiber :

∂A i ∂2A 1 ∂3A α 2 + β 2 2 + β 3 3 + A = iγ A A ∂z 2 ∂t 6 ∂t 2 n NLω 0 γ = cAeff

defines the non linear fiber parameter

Usually nNL~3 10-20 m2/W and γ has values in the range of 1 - 20 W-1/km

The propagation regimes Assume a pulse

A( z,τ ) = P0 exp( −α z / 2)U ( z,τ )

The optical power is P0, the pulse width is T0 and U is the normalized pulse envelope. NLSE for U :

LD = LNL =

T 02

β2

1 γ P0

∂U sgn( β 2 ) ∂ 2U exp( −α z ) 2 +i =i U U 2 ∂z 2 LD ∂ τ LNL Dispersion length

Nonlinear length

t τ= T0

There are two propagation regimes : L