We could also have shown the 3D FT of f; but, as you may have guessed, it would look exactly like the right plots: by doing a 2D FT followed by a 1D one in the third direction, we have effectively reconstructed the 3D version.
The same technique can be used to reconstruct higher-dimensional FTs. For instance, A 4D transform can be reconstructed as the combination of two 2D ones, a 5D transform as two 2D ones and a 1D one,… More generally, for any positive integer d, one can compute a d-dimensional FT by performing d//2 2D FTs (where // is the Euclidean division) plus a 1D transform if d is odd. In practice, however, the most important cases are d=1 (used, for instance, in signal or time series analysis and in cryptography), d=2 (for classical or deep-learning image processing), and d=3 (for computational fluid dynamics, wave propagation modelling and 3D deep learning CNNs).
Fourier transform at the speed of light
As we mentioned above, the most intensive operations in the Cooley-Tukey algorithm as well as its variation computing batched FTs is the computation of smaller transforms: even if each of them is much easier to do than the large 2D one, they still require, for large arrays, much more compute power than the other steps in these algorithms do. Or, at least, that is true when the algorithm runs on electronic hardware. Free-space optical computing completely changes the picture thanks to a peculiar property of light: the presence of interferences, due to its wave-like nature. We will not go into the details of light propagation here; but, to make a long story short, and under some approximations, a simple parabolic lens is equivalent to the 2D FT, in that the amplitude of the electro-magnetic field (the ‘essence’ of light) in one focal plane is equal to the FT of its amplitude in the other focal plane.
Moreover, the calculation happens literally at the speed of light, as electro-magnetic waves move from one focal plane to the other. Even better, no active component is required in the process, and power dissipation can in principle be reduced to virtually zero. We talked about this optical advantage in our previous articles, to which we refer for more details.
For this reason, an optical Fourier engine can be significantly more efficient than state-of-the-art electronic hardware for the 2D FT, as well as its 1D or higher-dimensional variants using the algorithms described above. Precisely how efficient it is will depend on many factors, like the quality of the optical components, sensitivity of light detectors, and type of interface. The specifications of the device we are developing will be given in a future article, along with quantitative estimates for the gain in efficiency. To give a glimpse of what is to come, let us simply state here that we expect the efficiency boost to be well over an order of magnitude vs the fastest and most efficient modern solutions.
Conclusion
While this article was quite long, the main message is clear and simple: the optical Fourier engine developed by Optalysys will be able to perform any kind of FTs of practical interest. To gather some examples, it can do:
small or large 2D FTs, with countless applications in image analysis and deep learning (a prominent example being convolutional neural networks, possibly including a Bayesian uncertainty estimate, and a more recent one is logic tensor networks; we have also recently proposed that they may be used in transformers),
large 1D FTs, which are at the core of signal analysis algorithms, error-correction codes, and cryptosystems (like the NTRU public-key system, and most other lattice-based systems like those used for post-quantum cryptography or in homomorphic encryption),
large 3D FTs, which can help in computer simulations of fluid dynamicsand wave propagation²,
and higher-dimensional Fourier transforms, if and when the need arises.
Thanks to the peculiar properties of light, the 2D FT can be computed without resorting to any active component and literally at the speed of light, so that optical Fourier engine can be vastly more efficient than state-of-the-art electronic systems.
In light of these arguments, and given the wealth of problems which can be accelerated using this mathematical operation, it seems certain that free-space optical computing will play a major role in the future of computing. The question is not if, but when. And, given the pace of the recent progress, it is increasingly evident that the optical computing revolution is just around the corner.
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¹ The values we used are: ax=0.02, ay=0.01, az=0.02, and x₀=y₀=z₀=60.
² The last two linked articles give examples in 2D to make visualisation easier; but more realistic examples would generally use 3D simulations.