Analytical formula for time signal of Gaussian source

[Superwbb1999]

Hello Professor. As shown in the following figure, the comment on class Gaussian source in source.py reads: “A Gaussian-pulse source roughly proportional to $\exp (-i\omega t-(t-t_0{)}^2/2w^2)$.Technically, the "Gaussian" sources in Meep are the (discrete-time) derivative of aGaussian, i.e. they are $(-i\omega {)}^{-1}\frac{\partial }{\partial t}\exp (-i\omega t-(t-t_0{)}^2/2w^2)$, but the difference between this and a true Gaussian is usually Irelevant. "according to this information, if I need to calculate the time signal of the Gaussian source in MEEP, the analytical formula of the signal is the derivative of exp (− i ω t − ((t − t0) ^2/2w^2) over time, multiplied by (-i ω) ^-1. Is that correct?

[stevengj]

Technically it is the finite difference approximation $(f(t+\mathrm{\Delta }t)-f(t))/\mathrm{\Delta }t$ for the derivative, but it goes to the derivative in the limit as you increase the resolution (decreasing $\mathrm{\Delta }t$).

If you need to know the exact formula for a source, however, the most reliable thing is to put in a custom source, in which case you can put in whatever formula you want. (Usually it doesn't matter because in a linear simulation the precise formula for the source is usually cancelled out when you normalize the results.)

[Superwbb1999]

Dear Professor：
Thank you for your reply! I see what you mean. I currently have a new question that I need to ask you. In a two-dimensional simulation (as shown in the figure below), I placed a get_field_point and dft_monitor at the same position (the red line is source, propagating in the negative direction of the Y axis, and the blue line is two monitors). Then the time-domain signal of the Ex field component obtained by get_field_point is CZT transformed to obtain a spectrum curve (red), which is compared with the frequency-domain data Ex (green) obtained by dft_monitor. It is found that the two curves are not the same (the horizontal axis of the two curves is the same frequency). In lumerical, the results I calculated through such steps show that the two curves are the same. Could you please explain why these two curves are different in Meep?Thank you very much!