Stimulated inelastic scattering effects in fiber :Stimulated inelastic scattering effects in fiber :Stimulated Stimulated BrillouinBrillouin Scattering and Scattering and

Stimulated Raman ScatteringStimulated Raman Scattering

The nonlinear effects governed by the third order suceptibility (SPM, XPM, FWM) are elastic in the sense that no energy is exchanged between the electromagnetic field and the dielectric medium.

A second class of nonlinear effects results from inelastic stimulated scattering in which the optical field transfers part of its energy to the nonlinear medium. Two such important effects in glass fiber are: Stimulated Brillouim Scattering(SBS) and Stimulated Raman Scattering (SRS). Both are related to the vibrational excitations mode of silica

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In the quantum mechanical approach, for both SBS and SRS, a photon of the incident field (called the pump wave) is scattered by a molecule of silica to a photon of lower energy (called a Stokes wave) while at the same time the residual energy is absorbed by the molecule via phonons (transition between vibrational states). These phonons are acoustic phonons for SBS and optical phonons for SRS. An anti Stokes wave whose frequency is up shifted is also generated in principle but it easy to show that it does not build up.

Stoke photonPump photonSi

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phononSi

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Stimulated Stimulated BrillouinBrillouin scatteringscattering

Brillouin scattering is the interaction between light and sound waves in the matter. It manifests itself through the generation of a backward propagating Stokes wave downshifted from the frequency of the incident pump wave by an amount determined by the nonlinear medium. The stoke waves carries most of the input energy, once the Brillouin threshold is reached.

The process of SBS can be viewed as a parametric interaction among the pump wave, the Stoke and anti Stoke waves and an acoustic wave. The pump field generates sound waves in the fiber which induce a periodic modulation of the refractive index due to the pressure.

The pump induced index grating scatters the pump light thought Bragg diffraction.

Scattered light which counter propagates with the pump is down shifted in frequency because of the Doppler shift associated with a grating moving at the acoustic velocity va.

A scattered wave which co propagates with the pump (anti Stokes) is up shifted by the same amount.

The down shifted wave adds energy to the acoustic wave increasing its intensity and therefore enhancing the effect by a positive feedback. The anti Stokes wave, on the other hand extracts energy from the acoustic wave so the effect dies down fast and only the backscattered wave matters.

In the quantum mechanical approach, the annihilation of a pump photon creates simultaneously a Stoke photon and an acoustic phonon. Since both the energy and the momentum must be conserved during the scattering event, the frequencies and the wave vectors of the three waves are related by :

( )2 sin 2A A A A pk v v k = =

Where p and s are the frequencies and kp and ks are the wave vectors of the pump and Stokes waves respectively. The frequency A and the wave vector kA of the acoustic wave satisfy the dispersion relation (using that |kp|~|ka|):

A p s

A p sk k k =

=

Where is the angle between the pump and the Stokes waves.

The frequency shift depends on the angle. In the forward direction there is no Brillouin scattering and the maximum scattering is obtained in the backward direction (~).

22

aAb

p

nvv = =

The bandwidth of this process is determined by the acoustic attenuation of the glass. For fiber the Brillouin bandwidth is DfB~20 MHz at 1550 nm

The gain peak of Brillouin scattering in silica fiber is gB~4 10-11 m/W

The frequency shift B in the backward direction is

With n being the refractive index and pthe pump wavelength.

The only relevant directions in single mode fiber are the forward and backward directions because of the well defined axis of propagation. Hence, the only possible diffraction direction is the backward direction.

The coupled equations governing the pump and the Stokes wave evolutions are

p Bp s p

s Bp s s

d I g I I Id z Kd I g I I Id z K

= =

Ip and Is are the pump and stoke intensities (Ps=IsAeff and Pp=Ip Aeff)

gB the Brillouin gain and the fiber losses. K refers to the degree of polarization (K=1 for co polarized pump and stoke waves, 2 for randomly polarized pump and stoke waves)

The sign refers to the backward propagation the Stokes wave

In the absence of fiber loss, because of the energy conservation during the SBS process: ( ) 0p sd I Idz =

Brillouin Threshold

SBS occurs when the incident light has a sufficiently high intensity so that the Stoke waves growth exponentially.

In the undepleted pump approximation, the stoke power at z=0 for a pump power Pp(0) at z=0 and a Stokes power Ps a z=L is given by

( 0 ) 1 - e x p ( - )( 0 ) ( ) e x p ( ) B p e f fs s e f fe f f

g P L zP P L L LK A

= + =

The threshold power 21

1effth sourceBB eff B

KA fPg L f

= + Dfsource = initial pump spectral width DfB = Brillouin gain bandwidth

The threshold power increases for CW beams whose spectral width is Dfsourcelarger than the Brillouin gain line width (DfB ~20 MHz). It also increases when short optical pulses propagate through the fiber because of their wide bandwidth.

Stimulated Raman scattering (SRS)

Because of SRS, in WDM systems, signals at long wavelengths are amplified by signals at shorter wavelength

1 2 3 4 1 2 3 4Fiber

SRS is a nonlinear parametric interaction between light and molecular vibrations. SRS is similar to SBS but occurs in either forward or backward direction. The Raman gain (gRmax~7 10-14 m/W) is about three order of magnitude lower than Brillouin gain but its bandwidth is much wider (~13 THz or 100 nm)

0 5 10 15 20 25 30 350

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Frequency shift [THz]

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~100 nm

Raman gain coefficient versus the frequency shift for fused silica with a pump at 1500 nm

The molecule absorbs temporarily and is then in a forbidden energy state, since in general Wn+hfp does not coincide with an allowed energy level.

The following different scenarios can occure :

W0

W1

2 s=s=

p=p=

p=p= p=

s=

Rayleigh Scattering (losses)

Spontaneous Stokes scattering

Stimulated Stokes scattering

Explanation based on particle view

Raman ThresholdThe coupled equations governing the pump and stoke waves in the co propagating case, is

p p Rp s p p

s

s Rp s s s

d I g I I Id z Kd I g I I Id z K

= =

In absence of loss, 0ps

s p

IIdd z

+ =

That means that the total number of photons in pump and Stokes remains the same during SRS.

The SRS threshold is defined as the input pump power at which Stokes wave power becomes equal to the pump power at the fiber output

( 0 ) 1 6 f o r f o r w a r d S to k e s ~

2 0 f o r b a c k w a r d S to k e se f f

pR p

e f f

g P LK A

In the undepleted pump approximation:

( ) ( ) (0) exp (0)eff

prs s eff s p s

eff

gP L I L A P P L zKA

= = ( )1 exp( )pp

effp

LL

=

Raman induced crosstalkIn a two channel system the short wavelength signal can act as a pump and transfer power to the longer wavelength channel : this leads to Raman induced crosstalk

1 1 11 2 1

2

21 2 2

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e f f

R

e f f

d P d P gd P P Pd z d t K Ad P g P P Pd z K A

+ = =

Assuming that both channel have the same losses, the Raman crosstalk is governed by the following equation (assuming that LD>>L and 1 < 2)

d is the walk off parameter between the two channels

In the two channel system, channel 1 acts as a pump and transfers a fraction of its energy to channels 2.

This leads to an amplification in channel 2 and to an attenuation in channel 1 when there is temporal overlap of pulses from two channel pulses.

The cross talk level is strongly affected by the channel detuning because of the walk off parameter.

To illustrate this, consider to channels separated by 100 nm with an equal input power of 10 mW. The fiber parameters are

L=50 Km, K=1, gr=1.3 10-3 W-1 km-1, Aeff=50 m2, =0.2[dB/km]

The two channels are modulated with NRZ format at 1 Gbit/s (the dispersion is then negligible for 50 km of regular fiber) and we examine the influence of different walk off parameter values.

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Time [ns]0 5 10 15 20 25 300

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0 5 10 15 20 25 300

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10 z=012> 1

z=50km

d=0 ps/km

z=50km

d=0.1 ps/km

z=50km

d=1 ps/km

The walk off reduces the impact of the Raman crosstalk thanks to the averaging effect : each pulse in channel 2 sees many pulses from channel 1 during the propagation