# Second Harmonic Revisited: An Analytic Quantum Approach
## 2 Results  
### Classical Regime  
Given an electric field \(\mathbf{E}(\mathbf{x},t)\) propagating in a second-order nonlinear material, the SHG process can be described by solving [PERSON]'s equations in the absence of free charges and magnetic fields. The bound charges in the material are sources of a polarization vector \(\mathbf{P}_{\mathrm{NL}}\), which, in the particular case of SHG, is given by [2]:  
\[\mathbf{P}_{\mathrm{NL}}(\omega_{\mathrm{SH}})=\epsilon_{0}\chi^{(2)}(\omega_ {\mathrm{SH}}=2\omega_{\mathrm{F}}):\mathbf{E}(\omega_{\mathrm{F}})\mathbf{E}( \omega_{\mathrm{F}}) \tag{1}\]  
being \(\epsilon_{0}\) the permittivity of free space and \(\chi^{(2)}(\omega_{\mathrm{SH}}=2\omega_{\mathrm{F}})\) the nonlinear susceptibility tensor related to this process. From now on, we fix the geometry and the symmetries of the nonlinear crystal so that \(\chi^{(2)}\) can be assumed to be a real scalar factor [13]. In particular, we conform to the contracted notation and rename \(d_{\mathrm{eff}}\equiv\chi^{(2)}/2\). [PERSON]'s equations provide a wave equation driven by this nonlinear polarization, i.e., [2; 13]:  
\[\
abla^{2}\mathbf{E}-\frac{\epsilon}{c^{2}}\partial_{t}^{2}\mathbf{E}=\frac{ 1}{\epsilon_{0}c^{2}}\partial_{t}^{2}\mathbf{P}_{\mathrm{NL}}, \tag{2}\]  
where \(\epsilon\) is the dimensionless permittivity of the material and \(c\) is the speed of light in free space. We assume that the propagation of the field is along the \(z\) axis and that the angular wave number is \(k_{j}=n_{j}\omega_{j}/c\), where \(n_{j}\) is the refractive index of the material corresponding to frequency \(\omega_{j}\), being \(j=\mathrm{F}\), SH. We express the fundamental field \(\mathbf{E}_{\mathrm{F}}\) and the second-harmonic field \(\mathbf{E}_{\mathrm{SH}}\) as [13]:  
\[E_{j}(z,t)=A_{j}(z)e^{ik_{j}z-i\omega_{j}t} \tag{3}\]  
and the nonlinear polarization as:  
\[P_{\mathrm{NL}}(z,t)=P_{\mathrm{NL}}^{(1)}(z,t)+P_{\mathrm{NL}}^{(2)}(z,t), \tag{4}\]  
where [13]  
\[P_{\mathrm{NL}}^{(1)}(z,t) = 4\epsilon_{0}d_{\mathrm{eff}}A_{\mathrm{SH}}A_{\mathrm{F}}^{*}e^ {i(k_{\mathrm{SH}}-k_{\mathrm{F}})z-i\omega_{\mathrm{F}}t} \tag{5}\] \[P_{\mathrm{NL}}^{(2)}(z,t) = 2\epsilon_{0}d_{\mathrm{eff}}A_{\mathrm{F}}^{2}e^{ik_{\mathrm{ SH}}z-i\omega_{\mathrm{SH}}t}. \tag{6}\]  
It is common to assume the variation of the field amplitude small with respect to the field wavelength (slowly-varying-amplitude approximation), i.e.,  
\[|\partial_{z}^{2}A_{\mathrm{SH}}|\ll|k_{\mathrm{SH}}\partial_{z}A_{\mathrm{SH}}|. \tag{7}\]  
This assumption leads to the coupled-amplitude equations:  
\[\frac{dA_{\mathrm{F}}}{dz} = \frac{2i\omega_{\mathrm{F}}d_{\mathrm{eff}}}{n_{\mathrm{F}}c}A_{ \mathrm{SH}}A_{\mathrm{F}}^{*}\exp[-i\Delta kz] \tag{8}\] \[\frac{dA_{\mathrm{SH}}}{dz} = \frac{i\omega_{\mathrm{SH}}d_{\mathrm{eff}}}{n_{\mathrm{SH}}c}A_{ \mathrm{F}}^{2}\exp[i\Delta kz] \tag{9}\]where \(\Delta k=2k_{\rm F}-k_{\rm SH}\) is the phase mismatch. In the so-called undepleted-pump regime, we further assume that \(A_{\rm SH}(z)\ll A_{\rm F}(z)\ \forall z\), so that \(A_{\rm F}\) can be assumed to be constant with \(z\), and the only equation to be solved in the system (8)-(9) is the second one. Hence, it is possible to retrieve the output intensity \(I_{\rm SH}=2n_{\rm SH}\epsilon_{0}c|A_{\rm SH}|^{2}\) as [13]:  
\[I_{\rm SH}(L,\omega_{\rm SH})=\frac{\omega_{\rm SH}^{2}d_{\rm eff}^{2}L^{2}}{2 n_{\rm SH}n_{\rm F}^{2}c^{3}\epsilon_{0}}\text{sinc}^{2}\left(\frac{\Delta KL}{2} \right)\ I_{\rm F}^{2}(\omega_{\rm F}), \tag{10}\]  
where \(L\) is the crystal length.