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This paper is based on our previous works in the past decade and is a summary of the theory of the amplitude-phase retrieval problem in any linear transform system and its applications. We describe the general statement on the amplitude-phase retrieval problem in an imaging system and derive a set of equations governing the amplitude-phase distributions in terms of the rigorous mathematical derivation. By using these equations and an iterative algorithm, a variety of amplitude-phase problems can be successfully handled. We carry out systematical investigations and comprehensive numerical calculations to demonstrate the utilization of this new algorithm in various transform systems. We also indicated the relationship and distinction between our algorithm and the original Gerchberg-Saxton algorithm. This new algorithm possesses considerable capability to deal with the various phase-retrieval problems in the arbitrary linear transform system and some inverse problems in solid state physics.

A scheme for the design of diffractive phase elements (DPE’s) that integrates several optical functions is presented in a consistent sense based on the general theory of amplitude–phase retrieval and the Yang–Gu algorithm [Appl. Opt. 33, 209 (1994)]. We extend the original Yang–Gu algorithm to treat a system illuminated by a beam of incident light whose components are at different wavelengths, and a set of equations for determining the phase distribution of the DPE is derived. The profile of a surface-relief DPE can be designed with an iterative algorithm. Numerical simulations are carried out for the design of one-dimensional DPE’s capable of both demultiplexing different wavelength components and focusing each partial wave at predetermined positions. The influence of the extension of sampling points in the DPE’s from ideal geometric points to physical spots on design results is also investigated. The numerical simulation results show that the new algorithm can be used successfully to design the desired DPE’s. It is therefore expected to be useful in the design of DPE’s for micro-optical systems.

An iterative algorithm for the determination of the phonon density of states (DOS) from the specific heat is presented based on the general theory of the amplitude-phase retrieval problem. From the model calculation, it is found that a convergent solution can always be obtained. However, the profile of the recovered DOS depends on the initial frequency distribution used in the algorithm since the inverse problem is inherently ill-posed. In order to obtain the ideal reconstruction of the DOS, it is necessary to introduce additional information about the character of the DOS to guide the choice of the initial distribution. The algorithm can also give a robust reconstruction of the DOS from specific heat data contaminated with random noise. The recovery of the DOS based on specific heat data in a confined temperature range has also been examined in detail.

The optical bistability in a dye-ring cavity has been investigated by using the derived Maxwell-Bloch equations based on the band model. It is shown that the bistable conditions and behavior are closely related to the band structure of the dye molecules.

We report a new design for a polarization-selective laser cavity with birefringent diffractive phase elements. This laser cavity can create two modes with different polarizations and profiles launched separately from two end mirrors. The numerical simulation results show that the constructed laser cavity can successfully generate two orthogonal polarization modes with a uniform circular shaped pattern output from one end mirror and a uniform square-shaped pattern output from another end mirror.

The design of diffractive phase elements (DPE's) for generating color pseudo-nondiffracting beams (PNDB's) in a multiple-wave illuminating system by the conjugate-gradient method is described. The axial-intensity distributions for dual-color PNDB's generated by the DPE's are shown. The color PNDB's behave as segmented axial-intensity distributions; in each of segments only one color persists. The sequence of wavelength components in the color PNDB's can be arbitrarily preset. Three-dimensional plots of a dual-color PNDB indicate the characteristics of a beam with high transverse resolution.

We investigate the focusing action of refractive microlens based on the rigorous electromagnetic theory by boundary element method. We numerically simulate total electric-field patterns, the electric-field intensity distributions on the focal plane, and their diffractive efficiencies at the focal spots for describing the focusing behaviours of these microlenses with continuous and multilevel surface-envelopes. Focusing action of incident beam with a certain angle of inclination is indagated as well. The present numerical and graphical results may provide an useful information for the analysis and the design of refractive elements in micro-optics.

Green's tensors for photonic crystal (PC) slabs are numerically solved by the coupled-dipole approximation (CDA) technique. The obtained components of Green's tensors satisfy discontinuous or continuous conditions at interfaces of scatterers. This shows that the CDA technique is very applicable to studying the properties of PC slabs. Green's tensors exhibit obviously periodic oscillation with the increase of the number of scatterers; furthermore, the effect of each scatterer on Green's tensors displays a localization feature in the sample containing one row of scatterers; on the contrary, this localized effect disappears in the sample consisting of multiple rows of scatterers.

A coherent optical system for performing an arbitrary linear transform is described. The system consists of a holographic mask and two Fourier lenses. A set of equations for determining the amplitude-phase distribution of the mask is given, and the mask is generated by combination of a computer-generated hologram and optical holography. As an example, a Walsh-Hadamard transform of order 32 is realized. >

We first present nonparaxial designs for a microcylindrical axilens with different long focal depths and rigorously analyze electromagnetic field distributions of the axilens using integral equations and the boundary-element method. Numerical results show that the designed axilenses indeed have the special feature of attaining a long focal depth while keeping high transverse resolution for numerical apertures of 2.4, 2.0, and 1.0. The ratio between the extended focal depth of the designed axilens and the focal depth of the conventional focal lens is 1.41, the corresponding maximal extended focal depth of the axilens can reach 28 microm, and the spot size of the focal beam is approximately 10 microm over the focal range.

Spontaneous decay behaviors from an assembly of atoms (or molecules) in 3D photonic crystals (PC's) with pseudogaps are investigated. Theoretically, a lifetime distribution function for these atoms or molecules is defined to reveal decay kinetics. Our calculations show that quite wide or narrow lifetime distributions can occur for different spread configurations of the atoms (or molecules). The pure PC effect may lead to coexistence of both accelerated and inhibited decay processes. These results provide theoretical clarification for substantial discrepancies in the recent reported experiments.

Within the framework of mean field theory, we study the effects of multi-surface modification on Curie temperature of ferroelectric films using the transverse Ising model. The general nonlinear equations for Curie temperature of multi-surface ferroelectric films with arbitrary exchange constants and transverse fields are derived by the transfer matrix method. As an example, we consider a film consisting of top surface layers, bulk layers and bottom surface layers. Two types of surface modifications, modifications of a surface exchange constant and a surface transverse field are taken into account. The dependence of Curie temperature on the surface layer numbers, bulk layer numbers, surface exchange constants, surface transverse fields and bulk transverse fields is discussed., Comment: 11 pages, 5 figures

Within the framework of mean field theory, we examine the phase transitions in Ising magnetic films and superlattices. By transfer matrix method, we derive two general nonlinear equations for phase transition temperatures of Ising magnetic films and superlattices, respectively. The equations can be applied to the films and superlattices with arbitrary exchange interaction constants and arbitrary layer number. Numerical results for phase transition temperatures as a function of exchange interaction constants are presented., 10 pages, 3 figures, accepted by Solid State Communications

We study the nonclassical properties and algebraic characteristics of the negative binomial states introduced by Barnett recently. The ladder operator formalism and displacement operator formalism of the negative binomial states are found and the algebra involved turns out to be the SU(1,1) Lie algebra via the generalized Holstein-Primarkoff realization. These states are essentially Peremolov's SU(1,1) coherent states. We reveal their connection with the geometric states and find that they are excited geometric states. As intermediate states, they interpolate between the number states and geometric states. We also point out that they can be recognized as the nonlinear coherent states. Their nonclassical properties, such as sub-Poissonian distribution and squeezing effect are discussed. The quasiprobability distributions in phase space, namely the Q and Wigner functions, are studied in detail. We also propose two methods of generation of the negative binomial states., Comment: 17 pages, 5 figures, Accepted in EPJ D

We consider a finite ferroelectric superlattice in which the elementary unit cell is made up of l atomic layers of type A and n atomic layers of type B. Based on the transverse Ising model we examine the phase transition properties of the ferroelectric superlattice. Using the transfer matrix method we derive the equation for the Curie temperature of the superlattice. Numerical results are given for the dependence of the Curie temperature on the thickness and exchange constants of the superlattice., Comment: 8 pages, 3 figures, Accepted by Australian J.Phys

Nondiffracting beam has extensive applications such as optical alignment, surveying, industrial inspection, and optical interconnection. Recently, pseudo-nondiffracting beam (PNDB) has been proposed. All the PNDB’s, a single and two segments, are almost generated in the monochromatic light illuminating system.1-2

Optical correlation is one of important operations for many applications, for example, pattern recognition, feature extraction, and so on. With the development of optical information processing, many techniques have been applied to the optical correlation to meet different requirements. The conventional correlation is based on the Fourier transform,1,2 therefore, it is space-invariant. However, when one pays attention to the correct location of the object and does not care for whether it has been rotated at some angle. In this case, not only the space-variant but also rotation-invariant correlation is needed. In this presentation, we combine the fractional correlation with the method of the circular harmonics decomposition, and present a method to achieve the locating rotation-invariant correlation.

As integrated circuit (IC) technology continues to push further into the submicrometer regime, considerable effort has been devoted to finding new approaches for extending the resolution limits of optical lithographic systems. The idea of using phase–shifting masks in optical lithography is one of such resolution–enhancing techniques and is commonly attributed to Levenson.1 The problem of the design of phase–shifting mask is how to determine the phase of the mask that produces a predesignated image. There are several approaches to deal with this problem such as simulated annealing algorithm2 and optimal coherent approximations.3 In this paper we present an approach of the design of the phase–shifting mask for the enhancement of optical resolution in lithography based on general theory of amplitude–phase retrieval in optical system and an iteration algorithm. For several model objects the numerical investigating results are given.

We discuss magnetostatic modes in semi-infinite magnetic-nonmagnetic superlattices in an arbitrary-angle magnetization geometry. In this case, some very interesting nonlinearity features appear. We also discuss the consistency condition for the existence of surface waves.

A different set of the Maxwell-Bloch equations describing optical bistability in a dye ring cavity has been derived. A complete linear stability analysis including propagation effects is given. The corresponding numerical simulations show that the bistable regions are functions of both the frequency detuning and the band-structure parameters such as the bandwidth and the distribution of the dipole moments.

This paper is a summary of the theory of the amplitude-phase retrieval problem in any linear transform system and its applications based on our previous works in the past decade. We describe the general statement on the amplitude-phase retrieval problem in an imaging system and derive a set of equations governing the amplitude-phase distribution in terms of the rigorous mathematical derivation. We then show that, by using these equations and an iterative algorithm, a variety of amplitude-phase problems can be successfully handled. We carry out the systematic investigations and comprehensive numerical calculations to demonstrate the utilization of this new algorithm in various transform systems. For instance, we have achieved the phase retrieval from two intensity measurements in an imaging system with diffraction loss (non-unitary transform), both theoretically and experimentally, and the recovery of model real image from its Hartley-transform modulus only in one and two dimensional cases. We discuss the achievement of the phase retrieval problem from a single intensity only based on the sampling theorem and our algorithm. We also apply this algorithm to provide an optimal design of the phase-adjusted plate for a phase-adjustment focusing laser accelerator and a design approach of single phase-only element for implementing optical interconnect. In order to closely simulate the really measured data, we examine the reconstruction of image from its spectral modulus corrupted by a random noise in detail. The results show that the convergent solution can always be obtained and the quality of the recovered image is satisfactory. We also indicated the relationship and distinction between our algorithm and the original Gerchberg- Saxton algorithm. From these studies, we conclude that our algorithm shows great capability to deal with the comprehensive phase-retrieval problems in the imaging system and the inverse problem in solid state physics. It may open a new way to solve important inverse source problems extensively appearing in physics.