By use of matrix-based techniques it is shown how the space-bandwidth product (SBP) of a signal, as indicated by the location of the signal energy in the Wigner distribution function, can be tracked through any quadraticphase optical system whose operation is described by the linear canonical transform. Then, applying the regular uniform sampling criteria imposed by the SBP and linking the criteria explicitly to a decomposition of the optical matrix of the system, it is shown how numerical algorithms (employing interpolation and decimation), which exhibit both invertibility and additivity, can be implemented. Algorithms appearing in the literature for a variety of transforms (Fresnel, fractional Fourier) are shown to be special cases of our general approach. The method is shown to allow the existing algorithms to be optimized and is also shown to permit the invention of many new algorithms.

A number of methods have recently been proposed in the literature for the encryption of two-dimensional information by use of optical systems based on the fractional Fourier transform. Typically, these methods require random phase screen keys for decrypting the data, which must be stored at the receiver and must be carefully aligned with the received encrypted data. A new technique based on a random shifting, or jigsaw, algorithm is proposed. This method does not require the use of phase keys. The image is encrypted by juxtaposition of sections of the image in fractional Fourier domains. The new method has been compared with existing methods and shows comparable or superior robustness to blind decryption. Optical implementation is discussed, and the sensitivity of the various encryption keys to blind decryption is examined. It can be shown that if the phases in these screens can be accurately described as statistically independent white noise, then the resulting encrypted image is also a white-noise distribution. 2The first random phase plane serves to make the input image white but nonstationary and not encrypted. The second serves to make the image stationary and encoded. Thus, the random phase key located at the Fourier plane of this system serves as the only key in this encryption scheme.The fractional Fourier transform (FRT) was introduced to the optical community by Ozaktas and Mendolovic. 3,4 The transform was used to describe wave propagation in graded index media. Lohmann described the relationship between the FRT and the Wigner distribution function 5 and gave two possible optical implementations, one of which, like the optical implementation of the Fourier transform, uses a single lens and free space. The FRT has an order associated with it, indicating the domain into which it transforms; i.e., the FRT of order a 1 is simply the Fourier transform. The FRT is a linear transformation that is separable in both the x and y directions, and optical systems have been proposed that allow for implementation with different continuously variable orders in both the x and y directions. 6 A number of algorithms have been proposed to compute the FRT numerically 7 -9 with order N log͑N ͒ calculations that make use of the fast Fourier transform algorithm. We have implemented and compared these algorithms, and the algorithm outlined in Ref. 7 is used to produce the results presented here.Several techniques have been proposed in the literature to optically encrypt images by use of the FRT. 2,10 -14We have examined all these encryption methods numerically, using all three of the fast algorithms 7 -9 discussed above, and tested and compared their robustness to blind decryption. In Ref. 2 the method f irst presented in Ref. 1 is modif ied, with the two Fourier transform operations being replaced with two FRT operations; the phase key is therefore applied in some fractional domain. In this way, four new FRT order keys have been introduced, i.e., two in each direction. This work was developed in Refs. 10 and 11, in which the...

The linear canonical transform (LCT) describes the effect of any quadratic phase system (QPS) on an input optical wave field. Special cases of the LCT include the fractional Fourier transform (FRT), the Fourier transform (FT), and the Fresnel transform (FST) describing free-space propagation. Currently there are numerous efficient algorithms used (for purposes of numerical simulation in the area of optical signal processing) to calculate the discrete FT, FRT, and FST. All of these algorithms are based on the use of the fast Fourier transform (FFT). In this paper we develop theory for the discrete linear canonical transform (DLCT), which is to the LCT what the discrete Fourier transform (DFT) is to the FT. We then derive the fast linear canonical transform (FLCT), an N log N algorithm for its numerical implementation by an approach similar to that used in deriving the FFT from the DFT. Our algorithm is significantly different from the FFT, is based purely on the properties of the LCT, and can be used for FFT, FRT, and FST calculations and, in the most general case, for the rapid calculation of the effect of any QPS.

Comparison of Raman spectra recorded from fresh cheek cells across ten sample substrates and five source wavelengths.

Abstract. We examine some fundamental theoretical limits on the ability of practical digital holography ͑DH͒ systems to resolve detail in an image. Unlike conventional diffraction-limited imaging systems, where a projected image of the limiting aperture is used to define the system performance, there are at least three major effects that determine the performance of a DH system: ͑i͒ The spacing between adjacent pixels on the CCD, ͑ii͒ an averaging effect introduced by the finite size of these pixels, and ͑iii͒ the finite extent of the camera face itself. Using a theoretical model, we define a single expression that accounts for all these physical effects. With this model, we explore several different DH recording techniques: off-axis and inline, considering both the dc terms, as well as the real and twin images that are features of the holographic recording process. Our analysis shows that the imaging operation is shift variant and we demonstrate this using a simple example. We examine how our theoretical model can be used to optimize CCD design for lensless DH capture. We present a series of experimental results to confirm the validity of our theoretical model, demonstrating recovery of superNyquist frequencies for the first time. © 2009 Society of Photo-Optical Instrumentation Engineers.