( 2c) (to within a rotation, which represents an arbitrary global phase delay). If we could perform the Fourier transform optically, we would escape the \(\) accessible states in an optimised polariser configuration provide the required modulation scheme, as shown in Fig. Historical approaches include: iterative algorithms 12 which are too slow for a Fourier coprocessor direct methods based on including a reference in the input plane 13 and of course interferometry. While a camera sensor can straightforwardly measure the magnitude, determining the phase is a venerable problem. However, using this phenomenon to evaluate a complex-to-complex Fourier transform requires full complex control of the light at the input spatial light modulator (SLM), and measurement of the complex amplitude across the focal plane at the output. The Fourier transform at the core of coherent optics is straightforwardly implemented with a classic 2f system 11. Specifically, we consider the natural application of the Optical Fourier Transform (OFT): replacing the Discrete Fourier Transform (DFT), as normally implemented by a Fast Fourier Transform (FFT) algorithm, with a dedicated optical coprocessor. However, it still potentially offers significant advantages over electronic methods when used appropriately 8, 9, 10. Despite this rich and prodigious history, OIP has often failed to compete with the formidable progress of digital electronic computers 7. Historically, optical information processing (OIP) has manifested itself in many imaginative–and some successful–ways: from image processing 1, 2, 3 and pattern matching 4, to numerical equation solving 5 and even to implementing a general purpose digital computer 6. This method could unlock the potential of the optical Fourier transform to permit 2D complex-to-complex discrete Fourier transforms with a performance that is currently untenable, with applications across information processing and computational physics. Performing larger optical Fourier transforms requires higher resolution spatial light modulators, but the execution time remains unchanged. By appropriately decomposing the input and exploiting symmetries of the Fourier transform we are able to determine the phase directly from straightforward intensity measurements, creating an optical Fourier transform with O(n) apparent complexity. Efficiently extracting the phase from the well-known optical Fourier transform is challenging. By implementing the Fourier transform optically we can overcome the limiting O(nlogn) complexity of fast Fourier transform algorithms. We propose and demonstrate a practical method to optically evaluate a complex-to-complex discrete Fourier transform. The Fourier transform is a ubiquitous mathematical operation which arises naturally in optics.
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