Abstract
The Fourier transform is a ubiquitous mathematical operation which arises naturally in optics. We propose and demonstrate a practical method to optically evaluate a complex-to-complex discrete Fourier transform. By implementing the Fourier transform optically we can overcome the limiting O(nlogn) complexity of fast Fourier transform algorithms. Efficiently extracting the phase from the well-known optical Fourier transform is challenging. 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. Performing larger optical Fourier transforms requires higher resolution spatial light modulators, but the execution time remains unchanged. 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.
Highlights
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]
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
It is well known that the Fourier transform of a coherent optical field at the front focal plane of a lens is rendered at the back focal plane, as shown in Fig. (1a)
Summary
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]. Despite this rich and prodigious history, OIP has often failed to compete with the formidable progress of digital electronic computers[7]. We discuss how this method overcomes the limitations of the FFT implemented on a digital computer and the different set of trade-offs it introduces
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