The language of Einstein spoken by optical instruments

The mathematics of Lorentz transformations, called the Lorentz group, continues to play an important role in optical sciences. It is the basic mathematical language for coherent and squeezed states. It is noted that the six-parameter Lorentz group can be represented by two-by-two matrices. Since the beam transfer matrices in ray optics are largely based on two-by-two matrices or ABCD matrices, the Lorentz group is bound to be the basic language for ray optics, including polarization optics, interferometers, lens optics, multilayer optics, and the Poincaré sphere. Because the group of Lorentz transformations and ray optics are based on the same two-by-two matrix formalism, ray optics can perform mathematical operations that correspond to transformations in special relativity. It is shown, in particular, that one-lens optics provides a mathematical basis for unifying the internal space-time symmetries of massive and massless particles in the Lorentz-covariant world.     

一维大气边界层光学折射率结构常数数值模式的实验检验

提出了一种大气光学折射率结构常数的数值模式。一次输入日期、时间、经纬度、温度、湿度、风速、粗糙度、土壤参数，云量等相关参数，可得到24h内的温度、湿度、风场、折射率结构常数等量。通过与实测对比表明，无论是月平均折射率结构常数还是某天的折射率结构常数，模式计算与实测数据都符合得很好。模式计算的向下总辐射通量与实测数据基本一致。     

Wigner rotations and Iwasawa decompositions in polarization optics

Wigner rotations and Iwasawa decompositions are manifestations of the internal space-time symmetries of massive and massless particles, respectively. It is shown to be possible to produce combinations of optical filters which exhibit transformations corresponding to Wigner rotations and Iwasawa decompositions. This is possible because the combined effects of rotation, phase-shift, and attenuation filters lead to transformation matrices of the six-parameter Lorentz group applicable to Jones vectors and Stokes parameters for polarized light waves. The symmetry transformations in special relativity lead to a set of experiments which can be performed in optics laboratories.     

光学形式量子化算符法与矩阵法的对应关系

本文从光学位置坐标算符和正则动量算符的定义以及线性光学系统光学算符的幺正性假设出发,由线性系统的普遍性质建立光学系统算符表示和矩阵表示的关联方程,进而利用该方程实现了光学算符和光学矩阵的互为导出,因此证明了光学算符法和矩阵法的等效对应关系。最后,本文说明了光学算符的物理意义。 关键词：     

The acoustic simulation and analysis of complicated reciprocating compressor piping systems,I: Analysis technique and parameter matrices of acoustic elements

This paper describes the mathematical formulation, equations, and procedures employed in the development of a comprehensive digital computer program for acoustic simulation and analysis of large and complicated piping systems. The analysis technique used is the transfer matrix method in which the piping system, with or without multiple inputs and outputs, is represented by a combination of discrete acoustic elements interconnected to one another at two stations such that the acoustic pressure and volume velocity at one station are uniquely related to those at the other by a two-by-two parameter matrix. Parameter matrices of 19 acoustic elements are included in this paper. By making use of these parameter matrices and the analysis technique, any complicated practical reciprocating compressor piping system can be modelled or analyzed.     

A problem relating to positive linear maps on matrix algebras

In this paper we investigate conditions under which two finite-dimensional density matrices can be transformed simultaneously into two other ones by means of a positive linear map which maps density matrices into density matrices. The result of this paper provides a complete answer in case of the matrix algebra of two-by-two matrices.     

High temperature thermal behaviour modeling of large-scale fused silica optics for laser facility

High temperature annealing is often used for the stress control of optical materials.However,weight and viscosity at high temperature may destroy the surface morphology,especially for the large-scale,thin and heavy optics used for large laser facilities.It is necessary to understand the thermal behaviour and design proper support systems for large-scale optics at high temperature.In this work,three support systems for fused silica optics are designed and simulated with the finite element method.After the analysis of the thermal behaviours of different support systems,some advantages and disadvantages can be revealed.The results show that the support with the optical surface vertical is optimal because both pollution and deformation of optics could be well controlled during annealing at high temperature.Annealing process of the optics irradiated by CO2 laser is also simulated.It can be concluded that high temperature annealing can effectively reduce the residual stress.However,the effects of annealing on surface morphology of the optics are complex.Annealing creep is closely related to the residual stress and strain distribution.In the region with large residual stress,the creep is too large and probably increases the deformation gradient which may affect the laser beam propagation.     

Two Lagrange-like optical invariants and some applications

Geometric optics can be completely derived from Fermat's principle, as classical mechanics can be obtained by the application of the Hamilton principle. In Lagrangian optics, for optical systems with rotational symmetry, is known the invariant L?, the Lagrange optical invariant. For systems built only with spherical lenses, we demonstrate there are two other optical invariants, L? and L?, analogous to L?. A proof based on Snell's law, the Weierstrass-Erdman jump condition, and the expression of the ray between two optical surfaces in the Hamiltonian formalism is reported. The presence of a conserved vector, L, allows us to write the equation of an emerging ray without any approximation.     

Separating the variables in a massless Dirac equation in Minkowski space

Exact integration of the Dirac equation is a classical topic in mathematical physics, which has been researched for several decades. A basic method is complete segregation of the variables. Such separation can be attained in a Dirac equation containing an external electromagnetic field in Minkowski space by means of complete sets of first-order symmetry matrix operators. The purpose of this paper is to solve an analogous case for a free massless Dirac equation. That task has a special feature because external fields are absent and the massless equation is reduced to a D'Alambert equation by squaring. Nevertheless, interest attaches to states defined by the first-order symmetry-operator matrices that cannot be obtained by setting the mass to zero in systems containing a mass Dirac equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 105–110, January, 1995.     

发射光学系统分析

 采用矩阵光学方法导出了发射光学系统的传输变换矩阵和具有两级扩束系统的发射系统的调焦公式,研究了输入激光不是平行光时远场焦点位置的变化和一级扩束系统焦距变化对系统焦距的影响。     

Matrix methods in treating decentred optical systems

A set of matrix methods treating decentred optical systems in the paraxial approximation is reviewed. Misalignment phenomena of optical systems can generally be described by an augmented 4×4 matrix; propagation of optical rays in an asymmetric, inhomogeneous medium by a 3×3 matrix. In order to simplify the operations with these matrices, ray transfer flow graphs are introduced.A lot of optical problems can be solved in a clear and simple manner, including optical arrays. Some examples are given.     

Reflection and transmission of light by cholesteric liquid crystal-glass-cholesteric liquid crystal and cholesteric liquid crystal(1)-cholesteric crystal(2) systems

Reflection and transmission of light by cholesteric liquid crystal-glass-cholesteric liquid crystal and cholesteric liquid crystal(1)-cholesteric liquid crystal(2) systems were studied. The classical Ambartsumyan method of adding layers and the concept of a sewing function were used. This approach was developed earlier in astrophysics for the theory of radiation transfer. Here, we used a version of this method adapted to wave optics. The Jones matrices are constructed for these systems. The features of the reflection and transmission spectra, optical rotation and ellipticity of polarization were studied for these systems. It is proposed to use these systems as tunable narrow-band filters and mirrors. These systems can be used, for example, to develop a variety of optical elements for lasers and of polarimetric elements in ellipsometry. The specific features of eigenpolarization are also discussed. It is shown that optical rotation of the two layers of cholesteric liquid crystals, which differ from each other only by the sign of the helix, is nonzero, and it becomes substantial in the diffraction reflection region. A unique property of these systems is the degeneracy (coincidence) of eigenpolarizations.     

一阶光学系统分数傅里叶变换的相空间分析

在维格纳相空间中,通过将一阶光学系统的传输矩阵分解为坐标旋转、比例缩放和啁啾矩阵的组合,得到了一阶光学系统在空域的分数傅里叶表示.结果表明:任意一阶光学系统均可表示为经过比例缩放和二次相位调制的分数傅里叶变换.通过将输入输出光场在相空间中作π/2角旋转,得到了一阶光学系统在频域的传输矩阵和衍射积分公式,进而得到了一阶光学系统在频域的分数傅里叶表示.比较空域和频域一阶光学系统的相空间变换矩阵,说明2个系统本质上属同一变换在不同基坐标下的表示,并推导出了光学系统在空域和频域具有相同分数傅里叶变换的条件.     

Ring resonators with N × M couplers

Multiple-zero multiple-pole optical filter transfer functions may be implemented more efficiently in an integrated optics architecture if higher order N × M optical couplers are utilized. For example, a coherent ring resonator made from two 3×3 couplers offers some advantages over the three mirror Fabry-Perot etalon, which is its analog. To this end we develop the formalism for obtaining the transfer functions and scattering matrices of ring resonators made from two N × M couplers. We then present a methodology for analyzing serial and parallel systems of N × M optical coupler ring resonators.     

Use of Dirac matrices in polarization optics

Several matrix methods have been developed for studying polarization properties of light. Jones was the first to apply the matrix method to the study of polarization optics. In Jones matrix formalism the polarized wave field is represented by 2-element column matrix known as Jones Vector and the polarization device encountered by light is represented by a 2×2 matrix, known as the characteristic Jones matrix of the device. Mueller introduced a new matrix method where the wave field is represented by a 4-dimensional vector. The elements of the vector are the Stokes parameters of the beam. In Mueller matrix formalism the optical device is represented by a 4×4 real matrix known as ‘Mueller Matrix’ of the device. The use of coherency matrix also proves to the useful in the study of partially polarized light. Pauli spin matrices have been used to unify the different matrix treatments of polarization optical phenomena. The present article is an attempt to unify the analysis of polarization phenomena using Dirac matrices used by Dirac in quantum mechanics. We have however redefined the set of Dirac matrices in terms of the Kronecher product of Pauli spin matrices.     

Simple trace criterion for classification of multilayers

The action of any lossless multilayer is described by a transfer matrix that can be factorized in terms of three basic matrices. We introduce a simple trace criterion that sorts the multilayers into three classes, each of whose properties are closely related to one (and only one) of the three basic matrices.     

Application of the polar decomposition to light scattering particle systems

We have applied the polar decomposition theorem (PDT) to the scattering matrix of simple systems such as an isolated sphere, either metallic or dielectric. The discrete dipole approximation (DDA) method has been used as an intermediate tool to calculate these matrices. We show how the parameters introduced by the PDT for describing the scattering by a single sphere can be easily interpreted. It is also shown that the PDT provides us with an alternative frame to conventional Mueller matrix analysis, by characterizing the systems by means of a group of independent parameters representing magnitudes of simple (virtual) elements. This step is necessary in order to translate other complex system matrices into manageable information.     

Complex ABCD transformations for optical beams with complex astigmatism

The method of the ray 4 × 4 ABCD matrices is extended to the three-dimensional optical systems of first order possessing complex astigmatism and containing selective elements with loss and field gain. These systems are found to have complex matrix elements. Properties of such matrices are studied, and a number of examples are considered that are of practical interest. It is shown that in the systems under consideration, like in the systems without loss and gain, such matrices retain the symplecticity property, on which the use of the method applied to specific models of optical-wave propagation is based.     

Orbit Classification of Qutrit via the Gram Matrix

We classify the orbits generated by unitary transformation on the density matrices of the three-state quantum systems （qutrits） via the Gram matrix. The Gram matrix is a real symmetric matrix formed from the Hilbert- Schmidt scalar products of the vectors lying in the tangent space to the orbits. The rank of the Gram matrix determines the dimensions of the orbits, which fall into three classes for qutrits.     

Characterizing entanglement via uncertainty relations

We derive a family of necessary separability criteria for finite-dimensional systems based on inequalities for variances of observables. We show that every pure bipartite entangled state violates some of these inequalities. Furthermore, a family of bound entangled states and true multipartite entangled states can be detected. The inequalities also allow us to distinguish between different classes of true tripartite entanglement for qubits. We formulate an equivalent criterion in terms of covariance matrices. This allows us to apply criteria known from the regime of continuous variables to finite-dimensional systems.