John T. Sheridan

Optical image encryption by random shifting in fractional Fourier domains

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.

Nonlocal-response diffusion model of holographic recording in photopolymer

The standard one-dimensional diffusion equation is extended to include nonlocal temporal and spatial medium responses. How such nonlocal effects arise in a photopolymer is discussed. It is argued that assuming rapid polymer chain growth, any nonlocal temporal response can be dealt with so that the response can be completely understood in terms of a steady-state nonlocal spatial response. The resulting nonlocal diffusion equation is then solved numerically, in low-harmonic approximation, to describe grating formation. The effects of the diffusion rate, the rate of polymerization, and a new parameter, the nonlocal response length, are examined by using the predictions of the model. By applying the two-wave coupled-wave model, assuming a linear relationship between polymerized concentration and index modulation, the resulting variation of the grating diffraction efficiency is examined.

Photopolymer holographic recording material

Summary Photopolymers are promising materials for use in holography. They have many advantages, such as ease of preparation, and are capable of efficiencies of up to 100%. A disadvantage of these materials is their inability to record high spatial frequency gratings when compared to other materials such as dichromated gelatin and silver halide photographic emulsion. Until recently, the drop off at high spatial frequencies of the material response was not predicted by any of the diffusion based models available. It has recently been proposed that this effect is due to polymer chains growing away from their initiation point and causing a smeared profile to be recorded. This is termed a non-local material response. Simple analytic expressions have been derived using this model and fits to experimental data have allowed values to be estimated for material parameters such as the diffusion coefficient of monomer, the ratio of polymerisation rate to diffusion rate and the distance that the polymer chains spread during holographic recording. The model predicts that the spatial frequency response might be improved by decreasing the mean polymer chain lengths and/or by increasing the mobility of the molecules used in the material. The experimental work carried out to investigate these predictions is reported here. This work involved (a) the changing of the molecular weights of chemical components within the material (dyes and binders) and (b) the addition of a chemical retarder in order to shorten the polymer chains, thereby decreasing the extent of the non-local effect. Although no significant improvement in spatial frequency response was observed the model appears to offer an improved understanding of the operation of the material.

Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation

The special affine Fourier transformation (SAFT) is a generalization of the fractional Fourier transformation (FRT) and represents the most general lossless inhomogeneous linear mapping, in phase space, as the integral transformation of a wave function. Here we first summarize the most well-known optical operations on light-wave functions (i.e., the FRT, lens transformation, free-space propagation, and magnification), in a unified way, from the viewpoint of the one-parameter Abelian subgroups of the SAFT. Then we present a new operation, which is the Lorentz-type hyperbolic transformation in phase space and exhibits squeezing. We also show that the SAFT including these five operations can be generated from any two independent operations.

A known-plaintext heuristic attack on the Fourier plane encryption algorithm

The Fourier plane encryption algorithm is subjected to a known-plaintext attack. The simulated annealing heuristic algorithm is used to estimate the key, using a known plaintext-ciphertext pair, which decrypts the ciphertext with arbitrarily low error. The strength of the algorithm is tested by using this estimated key to decrypt a different ciphertext which was also encrypted using the same original key. We assume that the plaintext is amplitude-encoded real-valued image, and analyze only the mathematical algorithm rather than a real optical system that can be more secure. The Fourier plane encryption algorithm is found to be susceptible to a known-plaintext heuristic attack.

Fast numerical algorithm for the linear canonical transform.

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.

Image encryption and the fractional Fourier transform

Summary A number of method have been recently proposed in the literature for the encryption of 2-D information using optical systems based on the fractional Fourier fransform, FRT. In this paper a brief review of the methods proposed to date is presented. A measure of the strength/robustness of the level of encryption of the various techniques is proposed and a comparison is carried out between the methods. Optical implementations are discussed. Robustness of system with respect to misalignment and blind decryption are also discussed.

Sampling and discretization of the linear canonical transform

The numerical approximation of the linear canonical transform (LCT) is of importance in modeling first order optical systems and many signal processing applications. We have considered an approach based on discretizing the continuous LCT, making careful consideration of the consequences for the range and resolution of the output.

Photoresist reflow method of microlens production Part I: Background and experiments

Summary We present a two part study of melted microlens arrays. This first part concentrates on the production and measurement of microlens arrays while the second part examines attempts to model the microlens profiles. In this paper we first review some of the fabrication techniques used over the past twenty years to produce lens arrays. Some applications of microlens arrays are then discussed. Particular emphasis is placed on the photoresist reflow method of microlens production that was suggested by Popovic et al., as this was the method used to produce the microlens examined in this study. Lenses produced using this method can have large deviations from the spherical case, i.e. the profile that would be expected from a simple minimisation of the surface energy. These deviations have not been explained to date in the literature, however a number of possible causes for this deviation are given in this paper. Therefore the fundamental questions we wish to explore here are: (1) Why physically do dips occur? and (2) Can the resulting surface profile be predicted? Any model developed to quantitatively estimate the optical effects of surface shape will depend on the physical assumptions made regarding the surface formation mechanism. However as we shall indicate at this point only an informed guess regarding the relative importance of a number of possible mechanisms can be made.

Temporal analysis of grating formation in photopolymer using the nonlocal polymerization-driven diffusion model

The nonlocal polymerization-driven diffusion model (NPDD) has been shown to predict high spatial frequency cut-off in photopolymers and to accurately predict higher order grating components. We propose an extension to the NPDD model to account for the temporal response associated with polymer chain growth. An exponential response function is proposed to describe transient effects during the polymerization process. The extended model is then solved using a finite element technique and the nature of grating evolution examined in the case when illumination is stopped prior to the saturation of the grating recording process. Based on independently determined refractive index measurements we determine the temporal evolution of the refractive index modulation and the resulting diffraction efficiency using rigorous coupled wave theory. Material parameters are then extracted based on fits to experimental data for nonlinear and both ideal and non-ideal kinetic models.