Space bandwidth considerations for optical encryption systems

There has been considerable attention paid in recent literature to the use of optical signal processing in the field of image encryption and security. In most cases this involves the use of optical transforms based on quadratic phase systems (to implement the optical Fourier Transform (FT), the optical fractional Fourier Transform (FRT), the Fresnel transform (FST) and the most general Linear Canonical Transform (LCT). Random phase screens or random shifting.stages are applied after optically transforming to a new domain. The process may be repeated for deeper encryption. Each stage of the encryption process may change the position and spatial extension of the complex distribution on the plane normal to the propagation axis. Each stage may also change the frequency distribution of the signal. Therefore, the space bandwidth product (SBP), which is equal to the number of discrete samples that are required to fully represent our signal, will also change. In general the encrypted image is complex and recording must be carried out using a holographic material or using digital holographic methods. In each case it is desirable to know the spatial extension of the signal to be recorded, its position, and its spatial frequency extension. In this way we can determine which holographic materials will meet the criteria or which cameras will have a suitable number of pixels, greater than or equal to the space bandwidth product, if digital holography is used. We show how the matrices associated with the effect of a LCT on the Wigner Distribution Function (WDF) provide us with an efficient method for finding the position, spatial extent, spatial frequency extent and space bandwidth product of the encrypted signal. We review a number of methods, which have recently been proposed in the literature for the encryption of two-dimensional information using optical systems based on the FT, FRT, FST and LCT. We apply the new matrix technique to some of them.

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