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Computes the 1D, 2D or 3D Fast Fourier Transform (FFT) of a matrix. In the 1D case the transform is given by

The FFT, inverse FFT and all associated functions have an option (option 1 below) that controls the format used to store the frequency domain data. When working with spectral data it is not possible to switch between formats; there are no functions to convert between formats. This implies that if you use option 1=n to produce a spectrum with fft, then you must also use option 1=n if you want to pass that same spectral data to invfft. Similarly, if you use option 1=n for fft, then you also need to use option 1=n with fftw to get the proper frequency vector corresponding to your spectrum. invfft and fftk work in the same way.

 Supported Product: FDTD, MODE, DEVICE, INTERCONNECT

 Syntax Description out = fft(Ex); Returns the fast Fourier transform of Ex. Ex can be 1D, 2D or 3D. out = fft(Ex,option1,option2); option1 This option controls the format used to store the frequency domain data. The options are: •1 : the standard FFT (zero frequency is at the first element of the matrix). This is the default option.•2 : zero frequency is the first element, but only data up to and including the Nyquist frequency is stored. This option is only useful for real valued, 1D time/spatial signals.•3 : the FFT is shifted so zero frequency is the central element of the spectrum (more precisely, this means the zero frequency point is at element floor(N/2 + 1), where N is the number of samples).  option2 This option is either a 1, 2 or 3 element vector depending on whether Ex is 1D, 2D or 3D. For each dimension, specify a value of either 0, 1 or N to obtain the desired 0 padding options. •0: no zero padding.•1: zero padding up to the next power of 2 longer than the length of Ex (default).•N: zero pad up to length N if N > length(Ex), where length of Ex is the length in a specific dimension. If N <= length(Ex), it will zero pad up to the next power of 2 longer than the length of Ex. For the fastest results, N should be a power of 2 and can be entered, for example, as 2^12.

Note: FFT Conventions

There are different, but equivalent conventions for defining Fourier transforms. Lumerical defines the forward FFT using a positive sign in the exponential term, and the inverse FFT using a negative sign in the exponential term. However, some other packages (e.g. MATLAB) use the opposite convention, with a negative sign in the exponential for the forward FFT and a positive sign in the exponential for the inverse FFT. To convert between the different FFT conventions, switch the invfft and fft and rescale the results. For a signal y with N elements this can be done as follows:

 Lumerical MATLAB fft(y,1,0) invfft(y,1,0) ifft(y)*N fft(y)/N

Example

This example transforms a time signal with 60 and 100 rad/s angular frequency components into the frequency domain. The function fftw is used to get the correct frequency vector. If this was a spatial signal, then fftk should be used in place of fftw.

t=linspace(0,1,1000);    # time signal

x=0.5*(sin(w1*t)+sin(w2*t)); # the signal

plot(t,x,"time","signal");

o1=2;            # option 1

o2=1;            # option 2

y=fft(x,o1,o2);       # fft

w=fftw(t,o1,o2);       # frequency signal

The following figure shows the output of the the example code.

This example shows how to calculate the FFT of the electric field intensity (i.e., combining all three field components). If you want to filter the high frequency data, set option1 to 2.

# get data from point time monitor

m = "monitor1";

t = getdata(m,"t");

Ex = getdata(m,"Ex");

Ey = getdata(m,"Ey");

Ez = getdata(m,"Ez");

# option1 = 1 -> standard fft

# option1 = 2 -> remove high frequency data

option1 = 2;

# do fft of each component

f  = fftw(t, option1)/2/pi;

Exw = fft(Ex, option1);  # fft each component separately

Eyw = fft(Ey, option1);

Ezw = fft(Ez, option1);

E2w = abs(Exw)^2+abs(Eyw)^2+abs(Ezw)^2; # combine field components

plot(f/1e12,E2w,"f (THz)","|E(f)|^2","fft E^2 intensity");

The following figures show the resulting plot for both the standard transform and the option to remove the high frequency data (option1 = 1 or 2).