Invariants, Lie-Bäcklund operators and Bäcklund transformations

This thesis is mainly concerned with the application of groups of transformations to differential equations and in particular with the connection between the group structure of a given equation and the existence of exact solutions and conservation laws. In this respect the Lie-Bäcklund groups of tangent transformations, particular cases of which are the Lie tangent and the Lie point groups, are extensively used.

In Chapter I we first review the classical results of Lie, Bäcklund and Bianchi as well as the more recent ones due mainly to Ovsjannikov. We then concentrate on the Lie-Bäcklund groups (or more precisely on the corresponding Lie-Bäcklund operators), as introduced by Ibragimov and Anderson, and prove some lemmas about them which are useful for the following chapters. Finally we introduce the concept of a conditionally admissible operator (as opposed to an admissible one) and show how this can be used to generate exact solutions.

In Chapter II we establish the group nature of all separable solutions and conserved quantities in classical mechanics by analyzing the group structure of the Hamilton-Jacobi equation. It is shown that consideration of only Lie point groups is insufficient. For this purpose a special type of Lie-Bäcklund groups, those equivalent to Lie tangent groups, is used. It is also shown how these generalized groups induce Lie point groups on Hamilton's equations. The generalization of the above results to any first order equation, where the dependent variable does not appear explicitly, is obvious. In the second part of this chapter we investigate admissible operators (or equivalently constants of motion) of the Hamilton-Jacobi equation with polynornial dependence on the momenta. The form of the most general constant of motion linear, quadratic and cubic in the momenta is explicitly found. Emphasis is given to the quadratic case, where the particular case of a fixed (say zero) energy state is also considered; it is shown that in the latter case additional symmetries may appear. Finally, some potentials of physical interest admitting higher symmetries are considered. These include potentials due to two centers and limiting cases thereof. The most general two-center potential admitting a quadratic constant of motion is obtained, as well as the corresponding invariant. Also some new cubic invariants are found.

In Chapter III we first establish the group nature of all separable solutions of any linear, homogeneous equation. We then concentrate on the Schrodinger equation and look for an algorithm which generates a quantum invariant from a classical one. The problem of an isomorphism between functions in classical observables and quantum observables is studied concretely and constructively. For functions at most quadratic in the momenta an isomorphism is possible which agrees with Weyl' s transform and which takes invariants into invariants. It is not possible to extend the isomorphism indefinitely. The requirement that an invariant goes into an invariant may necessitate variants of Weyl' s transform. This is illustrated for the case of cubic invariants. Finally, the case of a specific value of energy is considered; in this case Weyl's transform does not yield an isomorphism even for the quadratic case. However, for this case a correspondence mapping a classical invariant to a quantum orie is explicitly found.

Chapters IV and V are concerned with the general group structure of evolution equations. In Chapter IV we establish a one to one correspondence between admissible Lie-Bäcklund operators of evolution equations (derivable from a variational principle) and conservation laws of these equations. This correspondence takes the form of a simple algorithm.

In Chapter V we first establish the group nature of all Bäcklund transformations (BT) by proving that any solution generated by a BT is invariant under the action of some conditionally admissible operator. We then use an algorithm based on invariance criteria to rederive many known BT and to derive some new ones. Finally, we propose a generalization of BT which, among other advantages, clarifies the connection between the wave-train solution and a BT in the sense that, a BT may be thought of as a variation of parameters of some. special case of the wave-train solution (usually the solitary wave one). Some open problems are indicated.

Most of the material of Chapters II and III is contained in [I], [II], [III] and [IV] and the first part of Chapter V in [V].

Crystal Coulomb energies: I. Organic donor-acceptor complexes--a review. II. Classical Ewald calculations of the Coulomb binding energy of four donor-acceptor crystals and Wurster's blue perchlorate. III. A rapid-convergence quantum-mechanical formalism for crystal electronic energies

Chapter I

Theories for organic donor-acceptor (DA) complexes in solution and in the solid state are reviewed, and compared with the available experimental data. As shown by McConnell et al. (Proc. Natl. Acad. Sci. U.S., 53, 46-50 (1965)), the DA crystals fall into two classes, the holoionic class with a fully or almost fully ionic ground state, and the nonionic class with little or no ionic character. If the total lattice binding energy 2ε₁ (per DA pair) gained in ionizing a DA lattice exceeds the cost 2ε_o of ionizing each DA pair, ε₁ + ε_o less than 0, then the lattice is holoionic. The charge-transfer (CT) band in crystals and in solution can be explained, following Mulliken, by a second-order mixing of states, or by any theory that makes the CT transition strongly allowed, and yet due to a small change in the ground state of the non-interacting components D and A (or D⁺ and A^-). The magnetic properties of the DA crystals are discussed.

Chapter II

A computer program, EWALD, was written to calculate by the Ewald fast-convergence method the crystal Coulomb binding energy E_C due to classical monopole-monopole interactions for crystals of any symmetry. The precision of E_C values obtained is high: the uncertainties, estimated by the effect on E_C of changing the Ewald convergence parameter η, ranged from ± 0.00002 eV to ± 0.01 eV in the worst case. The charge distribution for organic ions was idealized as fractional point charges localized at the crystallographic atomic positions: these charges were chosen from available theoretical and experimental estimates. The uncertainty in E_C due to different charge distribution models is typically ± 0.1 eV (± 3%): thus, even the simple Hückel model can give decent results.

E_C for Wurster's Blue Perchl orate is -4.1 eV/molecule: the crystal is stable under the binding provided by direct Coulomb interactions. E_C for N-Methylphenazinium Tetracyanoquino- dimethanide is 0.1 eV: exchange Coulomb interactions, which cannot be estimated classically, must provide the necessary binding.

EWALD was also used to test the McConnell classification of DA crystals. For the holoionic (1:1)-(N,N,N',N'-Tetramethyl-para- phenylenediamine: 7,7,8,8-Tetracyanoquinodimethan) E_C = -4.0 eV while 2ε_o = 4.6₅ eV: clearly, exchange forces must provide the balance. For the holoionic (1:1)-(N,N,N',N'-Tetramethyl-para- phenylenediamine:para-Chloranil) E_C = -4.4 eV, while 2ε_o = 5.0 eV: again EC falls short of 2ε₁. As a Gedankenexperiment, two nonionic crystals were assumed to be ionized: for (1:1)-(Hexamethyl- benzene:para-Chloranil) E_C = -4.5 eV, 2ε_o = 6.6 eV; for (1:1)- (Napthalene:Tetracyanoethylene) E_C = -4.3 eV, 2ε_o = 6.5 eV. Thus, exchange energies in these nonionic crystals must not exceed 1 eV.

Chapter III

A rapid-convergence quantum-mechanical formalism is derived to calculate the electronic energy of an arbitrary molecular (or molecular-ion) crystal: this provides estimates of crystal binding energies which include the exchange Coulomb inter- actions. Previously obtained LCAO-MO wavefunctions for the isolated molecule(s) ("unit cell spin-orbitals") provide the starting-point. Bloch's theorem is used to construct "crystal spin-orbitals". Overlap between the unit cell orbitals localized in different unit cells is neglected, or is eliminated by Löwdin orthogonalization. Then simple formulas for the total kinetic energy Q^(XT)_λ, nuclear attraction [λ/λ]^XT, direct Coulomb [λλ/λ'λ']^XT and exchange Coulomb [λλ'/λ'λ]^XT integrals are obtained, and direct-space brute-force expansions in atomic wavefunctions are given. Fourier series are obtained for [λ/λ]^XT, [λλ/λ'λ']^XT, and [λλ/λ'λ]^XT with the help of the convolution theorem; the Fourier coefficients require the evaluation of Silverstone's two-center Fourier transform integrals. If the short-range interactions are calculated by brute-force integrations in direct space, and the long-range effects are summed in Fourier space, then rapid convergence is possible for [λ/λ]^XT, [λλ/λ'λ']^XT and [λλ'/λ'λ]^XT. This is achieved, as in the Ewald method, by modifying each atomic wavefunction by a "Gaussian convergence acceleration factor", and evaluating separately in direct and in Fourier space appropriate portions of [λ/λ]^XT, etc., where some of the portions contain the Gaussian factor.

The modulated grating hologram

A phase and amplitude, off-axis hologram has been synthesized from three computer-generated transmission masks, using a multiple-exposure holographic recording method. Each of the masks controls one fixed-phase component of the complex hologram transmittance. The basic grating is generated optically, relieving the computer of the burden of drawing details the size of each fringe. The maximum information capacity of the computer plotting device can then be applied to the generation of the grating modulation function. By this method large digital holograms (25 mm by 25 mm) have been synthesized in dichromated gelatin. The recording method is applicable to virtually any holographic medium.

The modulated grating hologram was designed primarily for the application of spatial filtering, in which the requirement is a hologram with large dynamic range and large free spectral range. Choice of a low-noise, high-efficiency medium such as dichromated gelatin will allow exceptionally large dynamic range. Independence of the optically-generated carrier grating from the computer-generated modulation functions allows arbitrarily large free spectral range.

The performance of a holographic spatial filter will be limited ultimately by noise originating from imperfections in the holographic medium. The characteristics of this noise are analyzed, and in the case of a high diffraction efficiency hologram are shown to differ significantly from previous analyses. The dominant noise source in holograms of high diffraction efficiency will be scattering of the first order or imaging wave by deformations in the hologram surface or other effects of low spatial frequency. Experimental measurements in various low-noise holographic media verify these predictions.

LiNbO_3∶Ce∶Cu晶体中绿光非挥发全息记录优化

为了对在LiNbO3∶Ce∶Cu晶体中绿光作为记录光的非挥发全息记录进行优化,联立求解了双中心物质方程和双光束耦合波方程,数值分析了平均空间电荷场(SCF)和衍射效率随晶体的氧化还原态、记录光与敏化光的光强比以及深浅中心的掺杂浓度的变化。结果表明,采用绿光作为记录光在LiNbO3∶Ce∶Cu晶体中进行非挥发全息记录,可以记录得到强光折变光栅,其空间电荷场高达107V/m;获得高达80%以上的固定衍射效率,各相关参量都有较大的优化空间。

Nonvolatile holograms in LiNbO3 : Fe : Cu by use of the bleaching effect

We report our observation of a bleaching effect under an ultraviolet exposure in LiNbO3:Fe:Cu crystals. Two three-step recording-transferring-fixing schemes are proposed to record nonvolatile photorefractive holograms in such crystals. In the schemes two red laser beams and an ultraviolet illumination are used selectively to write the charge grating in the shallow-level Fe centers, to develop the charge grating in the deep-level Cu centers by transferring the charge grating in the Fe centers, and to fix only the charge grating in the Cu centers for unerasable read-out. Experimental results, verifications, and an optimal recording scheme are given. A comparison of the lithium niobate crystals of the same double-doping system of Fe:Mn, Ce:Mn, Ce:Cu, and Fe:Cu is outlined. (C) 2002 Optical Society of America.

LiNbO3：Cu：Ce非挥发全息记录掺杂组份比的优化

实验研究了掺杂组份比对LiNbO3：Cu：Ce晶体非挥发全息记录性能的影响。结果表明．在全息记录过程中，掺杂组份比通过改变晶体的紫外光吸收特性而引起全息记录性能的改变。增加LiNbO3：Cu：Ce晶体中Cu和Ce的掺杂组份比会导致晶体对紫外光吸收的增强，进而提高了全息记录灵敏度和固定衍射效率。在弱氧化处理的掺有CuO和Ce2O4的质量分数分别为0.085％和0．011％的LiNbO3：Ce：Cu晶体中．得到了最高的固定衍射效率ηf=32％和记录灵敏度S=0．022cm／J。

Polarizing beam splitter of a deep-etched fused-silica grating

We described a highly efficient polarizing beam splitter (PBS) of a deep-etched binary-phase fused-silica grating, where TE- and TM-polarized waves are mainly diffracted in the -1st and 0th orders, respectively. Tb achieve a high extinction ratio and diffraction efficiency, the grating depth and period are optimized by using rigorous coupled-wave analysis, which can be well explained based on the modal method with effective indices of the modes for TE/TM polarization. Holographic recording technology and inductively coupled plasma etching are employed to fabricate the fused-silica PBS grating. Experimental results of diffraction efficiencies approaching 80% for a TE-polarized wave in the -1st order and more than 85% for a TM-polarized wave in the 0th order were obtained at a wavelength of 1550 nm. Because of its compact structure and simple fabrication process, which is suitable for mass reproduction, a deep-etched fused-silica grating as a PBS should be a useful device for practical applications. (C) 2007 Optical Society of America

Three-port beam splitter of a binary fused-silica grating

A deep-etched polarization-independent binary fused-silica phase grating as a three-port beam splitter is designed and manufactured. The grating profile is optimized by use of the rigorous coupled-wave analysis around the 785 nm wavelength. The physical explanation of the grating is illustrated by the modal method. Simple analytical expressions of the diffraction efficiencies and modal guidelines for the three-port beam splitter grating design are given. Holographic recording technology and inductively coupled plasma etching are used to manufacture the fused-silica grating. Experimental results are in good agreement with the theoretical values. (c) 2008 Optical Society of America.

Deep silicon grating as high-extinction-ratio polarizing beam splitter

A deep binary silicon grating as high-extinction-ratio reflective polarizing beam splitter (PBS) at the wavelength of 1550 nm is presented. The design is based on the phenomenon of total internal reflection (TIR) by using the rigorous coupled wave analysis (RCWA). The extinction ratio of the rectangular PBS grating can reach 2.5×105 with the optimum grating period of 397 nm and groove depth of 1.092 μm. The effciencies of TM-polarized wave in the 0th order and TE-polarized wave in the −1st order can both reach unity at the Littrow angle. Holographic recording technology and inductively coupled plasma (ICP) etching could be used to fabricate the silicon PBS grating.

Photobleaching process of xanthene dyes initiated by N-phenylglycine in the polyvinylalcohol film

The parameters such as quantum yield and molar absorption coefficients of the photoinitiator that are responsible for holographic sensitivity in photopolymer material are investigated with a single beam exposure experiment. The influence of exposure intensity, the concentrations of N-phenylglycine and dye on the photobleaching process of xanthenes dyes are presented. In addition, the effect of diphenyliodonium hexafluorophosphate salt on the quantum yield and molar absorption of xanthene dyes is studied. (c) 2005 Elsevier Ltd. All rights reserved.

沢吨染料的光化漂白动力学及全息特性研究

研究了以三乙醇胺作为引发剂,五种沢吨染料的光化漂白过程.实验结果表明五种染料的光化漂白速率大小顺序是藻红B（ErB）＞曙红Y（EY）＞孟加拉玫瑰红B（RsB）＞罗丹明B（RoB）＞荧光素（F）.并研究了曝光强度及染料浓度对漂白过程的影响,以及这五种染料的全息记录性能.

一种核黄素敏化的宽带全息记录材料

制备了一种核黄素敏化的以聚乙烯醇为基质的新型全息存储材料，并研究了材料的全息特性,用Ar^＋激光器激发的488nm的蓝光对材料曝光，结果表明该材料具有较高的衍射效率和曝光灵敏度，衍射效率高达53％，灵敏度为3．3cm^2／J，折射率调制度为4．5×10^-4研究了材料的透过率与随时间和入射角度变化的关系，说明在曝光记录过程中有噪音光栅生成．在存储介质膜中存储了全息图像，再现图像较为清晰，说明该材料适合用作高密度全息存储介质．

Reducing group-velocity-dispersion-dependent broadening of stimulated Brillouin scattering slow light in an optical fiber by use of a single pump laser

We propose a method of effectively extending the stimulated Brillouin scattering (SBS) gain bandwidth in a single-mode optical fiber to reduce group-velocity-dispersion (GVD)-dependent pulse spread of SBS slow light. This can be done by overlapping doublet SBS gain spectra synthesized from a single pump laser. Numerical calculations are performed to verify our proposed method. We find that there exists the optimum spectral separation between two center frequencies of the doublet SBS gain spectrum with respect to the inherent spectral width of the pump laser, which makes it possible to effectively reduce the signal pulse broadening due to GVD. We show that the maximum time delay of the amplified signal pulse can be approximately two times longer than that by a previously reported method using a single broadband pump laser. (c) 2008 Optical Society of America.

Shell effect and capture cross sections in the synthesis of superheavy nuclei

The shell effect is included in the improved isospin dependent quantum molecular dynamics model in which the shell correction energy of the system is calculated by using the deformed two-center shell model. A switch function is introduced to connect the shell correction energy of the projectile and the target with that of the compound nucleus during the dynamical fusion process. It is found that the calculated capture cross sections reproduce the experimental data quantitatively at the energy near the Coulomb barrier. The capture cross sections for reaction (35) (80) Br + (82) (208) Pb -> (117) (288) X are also calculated and discussed.

The relation between formal potential and structure of indophenols and molecular mechnism of electrooxidation

The molecular structural parameters of indophenol and its derivatives were calculated by semi-empirical molecular orbital quantum chemical method,The relation between molecular structural parameters and formal potentials was analyzed by principal factor analysis and multiple Linear regression method. It was found that the formal potential of indophenols has a good relation with two-center electron exchange energy, E-ex (2), resonance energy of O-C bond, E-ex (C-1-O), and molecular ionization potential, I-p, among 19 moleclular structural parameters. The regression equation is E-0' = 1. 47 x 10 (-3) E-ex (two) - 5. 74 x 10 (-2) E-ex (C-1 - O) - 1. 41 x 10 (-2) I-p with RC = 0. 9999 and SD = 0. 00424. It was confirmed by the relation between structure parameters and formal potentials, and the thermodynamic stability of its intermediate products that the H (+) ionization is prior to the electron transfer step in the oxidation mechanism.