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Perhaps the most common task to which FDTD is applied is that of computing the transmission or scattering spectra from some finite structure, such as a resonant cavity, in response to some stimulus. One could, of course, compute the fields (and thus the transmitted flux) at each frequency ω separately, as described above. However, it is much more efficient to compute a broad-spectrum response via a single computation by Fourier-transforming the response to a short pulse.

For example, suppose we want the transmitted power through some structure. For fields at a given frequency ω, this is the integral of the Poynting vector (in the normal direction) over a plane on the far side of the structure:

Now, if we input a short pulse, it is tempting to compute the integral P(t) of the Poynting vector at each time, and then Fourier-transform this to find P(ω). That is incorrect, however, because what we want is the flux of the Fourier-transformed fields E and H, which is not the same as the transform of the flux (because the flux is not a linear function of the fields).

Instead, what one does is to accumulate the Fourier transforms E and H for every point in the flux plane via summation over the (discrete) time steps n:

and then, at the end of the time-stepping, computing P(ω) by the fluxes of these Fourier-transformed fields. Meep takes care of all of this for you automatically, of course — you simply specify the regions over which you want to integrate the flux, and the frequencies that you want to compute.

(There are other possible methods of time-series analysis, of course. One method that is sometimes very effective is construct a Padé approximant of the time series of field values at some point, from which one can often extrapolate a very accurate discrete-time Fourier transform, including sharp peaks and other resonant features, from a relatively short time series. Meep does not provide a Padé computation for you, but of course you can output the fields at a point over time, ideally in a single-mode waveguide for transmission spectra via a single point, and compute the Padé approximant yourself by standard methods.)

The power P(ω) by itself is not very useful — one needs to normalize, dividing by the incident power at each frequency, to get the transmission spectrum. Typically, this is done by running the simulation twice: once with only the incident wave and no scattering structure, and once with the scattering structure, where the first calculation is used for normalization.

It gets more tricky if one wants to compute the reflection spectrum as well as the transmission. You can't simply compute the flux in the backwards direction, because this would give you the sum of the reflected and the incident power. You also can't simply subtract the incident power from backwards flux to get the transmitted power, because in general there will be interference effects (between incident and reflected waves) that are not subtracted. Rather, you have to subtract the Fourier-transformed incident fields E0 and H0

to get the reflected/scattered power.

Again, you can do this easily in practice by running the simulation twice, once without and once with the scatterer, and telling Meep to subtract the Fourier transforms in the reflected plane before computing the flux. And again, after computing the reflected power you will normalize by the incident power to get the reflection spectrum.

Meep is designed to make these kinds of calculations easy, as long as you have some idea of what is going on.

For example, suppose we want the transmitted power through some structure. For fields at a given frequency ω, this is the integral of the Poynting vector (in the normal direction) over a plane on the far side of the structure:

Now, if we input a short pulse, it is tempting to compute the integral P(t) of the Poynting vector at each time, and then Fourier-transform this to find P(ω). That is incorrect, however, because what we want is the flux of the Fourier-transformed fields E and H, which is not the same as the transform of the flux (because the flux is not a linear function of the fields).

Instead, what one does is to accumulate the Fourier transforms E and H for every point in the flux plane via summation over the (discrete) time steps n:

and then, at the end of the time-stepping, computing P(ω) by the fluxes of these Fourier-transformed fields. Meep takes care of all of this for you automatically, of course — you simply specify the regions over which you want to integrate the flux, and the frequencies that you want to compute.

(There are other possible methods of time-series analysis, of course. One method that is sometimes very effective is construct a Padé approximant of the time series of field values at some point, from which one can often extrapolate a very accurate discrete-time Fourier transform, including sharp peaks and other resonant features, from a relatively short time series. Meep does not provide a Padé computation for you, but of course you can output the fields at a point over time, ideally in a single-mode waveguide for transmission spectra via a single point, and compute the Padé approximant yourself by standard methods.)

The power P(ω) by itself is not very useful — one needs to normalize, dividing by the incident power at each frequency, to get the transmission spectrum. Typically, this is done by running the simulation twice: once with only the incident wave and no scattering structure, and once with the scattering structure, where the first calculation is used for normalization.

It gets more tricky if one wants to compute the reflection spectrum as well as the transmission. You can't simply compute the flux in the backwards direction, because this would give you the sum of the reflected and the incident power. You also can't simply subtract the incident power from backwards flux to get the transmitted power, because in general there will be interference effects (between incident and reflected waves) that are not subtracted. Rather, you have to subtract the Fourier-transformed incident fields E0 and H0

to get the reflected/scattered power.

Again, you can do this easily in practice by running the simulation twice, once without and once with the scatterer, and telling Meep to subtract the Fourier transforms in the reflected plane before computing the flux. And again, after computing the reflected power you will normalize by the incident power to get the reflection spectrum.

Meep is designed to make these kinds of calculations easy, as long as you have some idea of what is going on.

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