deletions | additions       diff --git a/section_Four_wave_mixing_Bragg__.tex b/section_Four_wave_mixing_Bragg__.tex  index b3ab36c..584076d 100644  --- a/section_Four_wave_mixing_Bragg__.tex  +++ b/section_Four_wave_mixing_Bragg__.tex 
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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 Δω 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 $\beta(\omega) = \beta^{(3)} \beta^{(3)}\6  (\omega-\omega_{ZDW})^3 + \beta^{(3)}/6 \beta^{(4)}/24  \cdot (\omega-\beta^{(3)}/24 \cdot  \omega_{ZDW})^4$, and to introduce the average pump frequency $\Delta \Omega = ..$ 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 1b). The phasematching now reads $$ \kappa = ... $$  and simplifies to   $$ \kappa = \beta^{(3)}/6 \left[ 3 \delta\omega \epsilon (\epsilon-2\Delta\Omega) + ..  $$    In the straightforward case of = 0, the term (as well as all the other odd terms) cancels out, leaving only smaller contribution from higher order dispersion.