## Slowly varying envelope approximation >[!Notice] >We adopt the notation used in [[Nonlinear Optics]] (Page 1), the $\tilde{}$ denotes a quantity varies rapidly in time. Constant quantities, slowly varying quantities, and Fourier amplitudes are written without the tilde. ### Approximation in wave equations In [[Sum frequency generation under wave equation description]], we applied $\left| \frac{\mathrm{d}^2 A_3}{\mathrm{d} z^2} \right| \ll \left| k_3 \frac{\mathrm{d} A_3}{\mathrm{d} z} \right|$ Here $A_{3}$ is the amplitude (independent with $t$, dependent on $z$). Although here we do not have the envelope, but the idea is similarly: the second derivative is much smaller than the first derivative times the wavevector, indicating that there is no rapid oscillation in $z$ direction, and the propagation along $z$ direction is relative smooth. For the optical (or electromagnetic) waves, this means in an optical wavelength $\lambda_{3}$, the variation of $A_{3}$ should be significantly smaller than $A_{3}$ itself. (In [[Nonlinear Optics]] [[nonlinear optics Robert.W.Boyd.pdf#page=84&selection=368,0,374,42|page 71]], it writes "This condition requires that the fractional change in A3 in a distance of the order of an optical wavelength must be much smaller than unity.", which is of the same meaning.)