Let us compute the nonlinearly-generated signal amplitude in the case of
second harmonic generation, where a strong pump at \omega_1 produces a frequency-doubled signal at \omega_2=2\omega_1, assuming a constant pump amplitude (undepleted pump approximation). The signal wavelength can be expressed as a sum over the number of domains that exist in the crystal. In general the spatial rate of change of the signal amplitude is \frac{\partial A_2}{\partial z}=A_1^2 \chi e^{i \Delta k z}, where A_2 is the generated frequency amplitude and A_1 is the pump frequency amplitude and \Delta k is the phase mismatch between the two optical waves. The \chi refers to the nonlinear susceptibility of the crystal. In the case of a periodically poled crystal the crystal axis is flipped by 180 degrees in every other domain, which changes the sign of \chi. For the n^{th} domain \chi can be expressed as \chi=\chi_0 (-1)^n where n is the index of the poled domain. The total signal amplitude A_2 can be expressed as a sum A_2=A_1^2 \chi_0 \sum^{N-1}_{n=0} (-1)^n \int^{\Lambda (n+1)}_{\Lambda n} e^{i \Delta k z} \partial z where \Lambda is the spacing between poles in the crystal. The above equation integrates to A_2=-\frac{i A_1^2 \chi_0}{\Delta k} \sum^{N-1}_{n=0} (-1)^n (e^{i \Delta k \Lambda (n+1)}-e^{i \Delta k \Lambda n}) and reduces to A_2=-i A_1^2 \chi_0 \frac{e^{i \Delta k \Lambda}-1}{\Delta k} \sum^{N-1}_{n=0} (-1)^n e^{i \Delta k \Lambda n} The summation yields s=\sum^{N-1}_{n=0} (-1)^n e^{i \Delta k \Lambda n}=1-e^{i \Delta k \Lambda}+e^{i 2 \Delta k \Lambda}-e^{i 3 \Delta k \Lambda}+...+(-1)^N e^{i \Delta k \Lambda (N-2)}-(-1)^N e^{i \Delta k \Lambda (N-1)}. Multiplying both sides of the above equation by a factor of e^{i \Delta k \Lambda} leads to s e^{i \Delta k \Lambda}=e^{i \Delta k \Lambda} -e^{i 2 \Delta k \Lambda}+e^{i 3 \Delta k \Lambda}+...+(-1)^N e^{i \Delta k \Lambda (N-1)}-(-1)^N e^{i \Delta k \Lambda N}. Adding both equation leads to the relation s(1+e^{i \Delta k \Lambda})=1-(-1)^N e^{i \Delta k \Lambda N}. Solving for s gives s=\frac{1-(-1)^N e^{i \Delta k \Lambda N} }{1+e^{i \Delta k \Lambda}}, which leads to A_2=-i A_1^2 \chi_0 \left( \frac{e^{i \Delta k \Lambda}-1}{\Delta k} \right)\left(\frac{1-(-1)^N e^{i \Delta k \Lambda N}}{e^{i \Delta k \Lambda}+1}\right). The total SHG intensity can be expressed by I_2=A_2 A_2^*= \left|A_{1}\right|^{4} \chi_0^2 \Lambda^2 \mbox{sinc}^2(\Delta k \Lambda/2) \left(\frac{1-(-1)^N \cos(\Delta k \Lambda N)}{1+\cos(\Delta k \Lambda)} \right). For the case of \Lambda=\frac{\pi}{\Delta k} the right part of the above equation is undefined so the limit needs to be taken when \Delta k \Lambda \rightarrow \pi by invoking
L'Hôpital's rule. \lim_{\Delta k \Lambda\to\pi}\frac{1-(-1)^N \cos(\Delta k \Lambda N)}{1+\cos(\Delta k \Lambda)}=N^2 Which leads to the signal intensity I_2=\frac{4 \left|A_{1}\right|^{4} \chi_0^2 L^2}{\pi^2}. In order to allow different domain widths, i.e. \Lambda=\frac{m \pi}{\Delta k}, for m=1,3,5,..., the above equation becomes I_2=A_2 A_2^*= \left|A_{1}\right|^{4}\chi_0^2 \Lambda^2 \mbox{sinc}^2(m \Delta k \Lambda/2) \left(\frac{1-(-1)^N \cos(m \Delta k \Lambda N)}{1+\cos(m \Delta k \Lambda)} \right). With \Lambda = \frac{m \pi}{\Delta k} the intensity becomes I_2=\frac{4 \left|A_{1}\right|^{4} \chi_0^2 L^2}{m^2 \pi^2}. This allows quasi-phase-matching to exist at different domain widths \Lambda. From this equation it is apparent, however, that as the quasi-phase match order m increases, the efficiency decreases by m^2 . For example, for 3rd order quasi-phase matching only a third of the crystal is effectively used for the generation of signal frequency, as a consequence the amplitude of the signal wavelength only third of the amount of amplitude for same length crystal for 1st order quasi-phase match. ==Calculation of domain width==