Extrapolate a 2D array using Fourier Transform
You can use the Fourier transform for a lot of things – it can be part of a particular step in a scaling algorithm. Still, what you're looking for are scaling algorithms.

In this particular case, your original data size is (74 × 128) – what that means in wavelengths or any physical unit doesn't matter to algorithms – and you want to scale it to (512 × 512). 

So what you'd need to get 128 -> 512 would be simply inserting 3 zero rows after every image row and then low-pass filtering the result. A 128-FFT, followed by a zero-padding the 512-128 higher bins and then doing a 512-IFFT would do that – but not like you'd want it to, but with circular effects at the border; what you'd be doing mathematically would be a cyclic convolution with a 1/4 band-wide filter.

Same goes for the 74 -> 512 dimension – but here you'd need to interpolate higher first (to 37·512, to be exact, i.e. by a factor of 37·256) just to decimate down back to 512 again. And then you'd still be stuck with the circular convolution.

What you in fact want to do is apply resampling by a rational rate – the core filter for that could be applied by fast convolution, i.e. with the help of overlapping multiple FFTs, but it's not really the point of the algorithm (just like "using a screwdriver" isn't the central part in "building an airplane").

So, look for any of the existing image scaling algorithms.