The mode sources in FDTD Solutions can now build and inject a frequency dependent mode profile. This functionality was introduced in the 2017a release, and it is very useful for broadband simulations. The goal of this posting is to briefly outline when and how to use a frequency dependent mode profile.
A mode is a solution to the source free Maxwell’s equations obtained in the frequency domain. This solution is calculated over a 1-D cross section for 2-D simulations and over a 2-D cross section for 3-D simulations. Mode sources in FDTD Solutions calculate modes over the chosen waveguide cross section and construct a time domain field profile based on the chosen frequency domain mode. The constructed time domain fields are subsequently injected during the simulation. To construct the time domain fields, an inverse Fourier transform is needed. If we assume that the desired mode stays constant over frequency, then computing the needed inverse Fourier transform trivial. However, if the desired mode is assumed to vary over frequency, then the modal fields need to be computed over several frequencies and the inverse Fourier transform must be computed numerically. The mode sources in FDTD Solutions now offer the option to compute the inverse Fourier transform of a frequency dependent mode profile. In earlier versions, it was always assumed that the mode profile stays constant over frequency.
Broadband vs. Narrowband Injection
For narrowband simulations, it is usually fine to assume that the computed mode stays constant over frequency. For the purposes of this discussion, narrowband means that the simulation bandwidth does not span more than an octave in frequency or wavelength. If the simulation bandwidth is larger than an octave in frequency or wavelength, assuming that the mode stays constant over frequency can become a problem. To illustrate this point, a simple 2-D glass waveguide is employed here. The attached simulation file (bottom of the post) contains a mode source that injects a TE mode into the glass waveguide. The source bandwidth spans the entire visible spectrum (380-750nm), which is almost exactly one octave in wavelength. The attached script collects and plots the transmitted and reflected power as a function of wavelength for both mode calculation options. The option to assume that the mode profile is constant over frequency is hereby referred to as the narrowband injection method, and the option to calculate the mode over several frequencies is referred to as the narrowband injection method.
Glass Waveguide Example
The frequency dependence of the glass material generates a noticeable frequency dependence in the selected TE mode:
If the narrowband injection method is employed over such a large bandwidth, the mode will not inject cleanly. This will in turn introduce a noticeable power normalization error. The attached script (bottom of the post) compares both injection methods. The narrowband injection method is employed by default, and, to enable the broadband injection method, the following script command is used:
setnamed("mode-source","multifrequency mode calculation",true);
Since the glass waveguide is lossless, the transmitted power T is expected to be equal to one and the reflected power R is expected to be zero for the entire simulation bandwidth. However, this can only be achieved with the broadband injection method:
As the bandwidth of the simulation increases, or, alternatively, as the frequency dependence of the materials becomes more pronounced, then the differences between the results of two injection methods will become more pronounced.