which depends only on θ and not on φ. On the excitation plane, the incident field adopts a modulated Gaussian waveform and a complex phase delay along the periodicity direction with the following form:
:<math> H_x^{inc}(x,y,t) = -\frac{1}{\eta_0} \sin\phi \; \exp \left(-\frac{(t-t_0)^2}{\tau^2} \right) \exp(j2\pi f_0 t) \exp(-jk_x x) \exp(-jk_y y)</math>:<math> H_y^{inc}(x,y,t) = \frac{1}{\eta_0} \cos\phi \; \exp \left(-\frac{(t-t_0)^2}{\tau^2} \right) \exp(j2\pi f_0 t) \exp(-jk_x x) \exp(-jk_y y)</math><!--[[Image:FDTD87.png]]-->
for TM<sub>z</sub> polarization and
The modulated Gaussian waveform must be chosen such that its effective bandwidth avoids the horizontal resonant frequency. Otherwise, the temporal response of the structure starts to oscillate, and the time marching loop will not converge. To avoid this problem, the modulation frequency and bandwidth of the waveform are chosen to satisfy the following condition:
:<math> f_{mod} \ge f_{res} + \dfrac{1}{2}\Delta f = \dfrac{k_{l,fixed}\;c}{2\pi} + \dfrac{1}{2}\Delta f </math><!--[[Image:FDTD103.png]]-->
=Defining The Physical Structure=