deletions | additions       diff --git a/section_Four_wave_mixing_Bragg__.tex b/section_Four_wave_mixing_Bragg__.tex  index c51225d..b285238 100644  --- a/section_Four_wave_mixing_Bragg__.tex  +++ b/section_Four_wave_mixing_Bragg__.tex 
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\section{Four-wave mixing Bragg Scattering}  \label{fig:cartoon} Fig. 1. Cartoon of FWM-BS. A) Energy level depiction. B) Dispersion picture.  FWM-BS interactions are driven by two strong pump fields, whose frequency separation $\Delta\omega$ define the amount of frequency translation between signal and idler (see fig. 1). The efficiency of the translation process is $\eta=r^2/k_{bs}^2 \sin(k_{bs} L)$, where $r = 2\gamma P$ is the coupling strength term, $k_{bs} = \sqrt{\kappa^2+r^2} $ the scattering wavevector, and $\gamma$, $L$ and $P$ respectively the medium nonlinearity, the interaction length and the pump power.  $\kappa = \beta(\omega_s) - \beta(\omega_i) - \beta(\omega_1) + \beta(\omega_2)$ is the phasematching term: it is convenient to expand the wavevector $\beta(\omega)$ in respect of the zero dispersion frequency $\omega_{ZDW}$ of the nonlinear medium (i.e. $\beta^{(2)}(\omega_{ZDW}= 0$) so that $\beta(\omega) = \beta^{(3)}/6 (\omega-\omega_{ZDW})^3 + \beta^{(4)}/24 \cdot (\omega- \omega_{ZDW})^4$, and to introduce the average pump frequency $\Delta \Omega = (\omega_1 + \omega_2) /2 - \omega_{ZDW}$  and the signal offset $\epsilon$ so that the frequency and idler signal $\omega_s = \epsilon -\Delta\Omega + \Delta\omega/2$ and $\omega_s = \epsilon -\Delta\Omega - \Delta\omega/2$ (see figure \ref{fig:cartoon}b ).