On the algebraic characterization of a Mueller matrix in polarization optics. II. Necessary and sufficient conditions for Jones-derived Mueller matrices

Abstract

We show that every Mueller matrix, that is a real 4 × 4 matrix M which transforms Stokes vectors into Stokes vectors, may be factored as M = L ₂ KL ₁ where L ₁ and L ₂ are orthochronous proper Lorentz matrices and K is a canonical Mueller matrix having only two different forms, namely a diagonal form for type-I Mueller matrices and a non-diagonal form (with only one non-zero off-diagonal element) for type-II Mueller matrices. Using the general forms of Mueller matrices so derived, we then obtain the necessary and sufficient conditions for a Mueller matrix M to be Jones derived. These conditions for Jones derivability, unlike the Cloude conditions which are expressed in terms of the eigenvalues of the Hermitian coherency matrix T associated with M, characterize a Jones-derived matrix M through the G eigenvalues and G eigenvectors of the real symmetric N matrix N = [Mtilde]GM associated with M. Appending the passivity conditions for a Mueller matrix onto these Jones-derivability conditions, we then arrive at an algebraic identification of the physically important class of passive Jones-derived Mueller matrices.     

Conditions for the Physical Realizability of Polarization Matrices Characterizing Passive Systems

Abstract

We derive conditions for the physical realizability of polarization matrices characterizing passive systems or scattering media. By physically realizable, we mean that 0  g  1 where g ≡ (output intensity/input intensity). Using the singular-value decomposition of an arbitrary 2 × 2 complex-valued matrix, we prove that a Jones matrix T _J is physically realizable if 0  det T _J ⁺ T _J  1. Consequently singular Jones matrices (i.e. det T _J = 0) completely extinguish the output intensity irrespective of the input intensity because g ≡ 0. Corresponding results are obtained for Mueller-Jones matrices (the 4 × 4 real-valued matrices which are the four-dimensional representations of the two-dimensional 2 × 2 complex-valued Jones matrices). We also study the problem for general Mueller matrices; however because of their phenomenological character they do not admit of such criteria as do the Jones and Mueller-Jones matrices. This is because g now depends upon the matrix elements of the Mueller matrix and the input Stokes parameters; whereas for the Jones and Mueller-Jones matrices, g only depends upon the matrix elements. Finally we study the problem of relating the input and output mean randomness.     

A Simple Necessary and Sufficient Condition on Physically Realizable Mueller Matrices

Abstract

Development of simple tools to test physical realizability of measured or computed Mueller matrices is the subject of this paper. In particular, the overpolarization problem, i.e., the problem of ensuring that the output degree of polarization does not exceed unity is solved by finding an easily implementable necessary and sufficient condition. With G being the Lorentz metric, it states that a given matrix M is not overpolarizing if and only if the spectrum of GM ^T GM is real and an eigenvector associated with the largest eigenvalue is a physical Stokes vector. This result is used to characterize some M classes of special interest, and is used to test several examples from recent literature.     

On extensions of subshifts by finite groups

λ-Graph systems are labelled Bratteli diagrams with shift operations. They present subshifts. Their matrix presentations are called symbolic matrix systems. We define skew products of λ-graph systems and study extensions of subshifts by finite groups. We prove that two canonical symbolic matrix systems are G-strong shift equivalent if and only if their presented subshifts are G-conjugate.     

Reduced form of a Mueller matrix

Abstract

Through a simple procedure based on the Lu–Chipman decomposition [S.-Y. Lu and R.C. Chipman, J. Opt. Soc. Am. A 13, 1106 (1996)], any depolarizing Mueller matrix can be transformed into a reduced form which accumulates the depolarization and polarizance properties into a set of six parameters. The simple structure of this reduced form provides straightforward ways for the general characterization of Mueller matrices as well as for the analysis of singular Mueller matrices.     

Generation and Properties of Superpositions of Displaced Fock States

Abstract

We study the statistical properties of superpositions of displaced Fock states. We find that for the superposition of the form ¦ψ₁〉 = 1/√2(¦α, n〉 + ¦α, k〉) the direction of the displacement (α positive or negative) plays an important role; also, if n = 1 and k = 0 a strong sub-Poissonian character is found for α ≥ 0. We also analyse the ways in which superpositions of displaced Fock states may be generated.     

On the inverse of generalized λ‐matrices with singular leading term

An algorithm is introduced for the inverse of a λ‐matrix given as the truncated series A ₀?iλ A ₁?λ² A ₂+iλ³ A ₃+λ⁴ A ₄+···+O(λⁿ⁺¹) with square coefficient matrices and singular leading term A ₀. Moreover, A ₁ may be conditionally singular and no restrictions are made for the remaining terms. The result is a λ‐matrix given as a unique, truncated series of the same error order. Motivation for this problem is the evaluation of the frequency‐dependent stiffness matrix of general boundary or macro‐finite elements in the frame of a hybrid variational formulation that is based on a flexibility matrix F expressed as a truncated power series of the circular frequency ω. Copyright © 2005 John Wiley & Sons, Ltd.     

Superfluidity of solid4He

Using a procedure suggested by Leggett, an upper bound to the superfluid fraction in ground state solid ⁴ He slightly above the melting density is obtained numerically. The value obtained is 0.3±0.1. To judge the usefulness of this upper bound, we examine the conditions under which a symmetrized product of single-particle functions times a Jastrow function exhibits ODLRO, a necessary and sufficient condition for superfluid flow. It is found that ifU _ij (U _ij=? φ _i (x)φ _j (x) dx, and φ _i (x) is a single-particle wave function centered on the pointi) satisfies σ′_i U _ij>x, wherex varies from unity for long rangeU _ij (i.e.,U _ij decreases slowly enough asR _i?R_j increases) to a value of 12/7 for nearest-neighbor overlap only in the hcp lattice, then there is ODLRO, but not otherwise. Therefore, if the accepted single-particle functions are the true ones, then there is no ODLRO in solid ⁴ He, since the overlap is too small. We have explored the possibility of adding a flat tail, of magnitude λ′(VN)^?1/2 to the accepted single-particle functions. It is shown that if λ → 1 [λ ² =(λ′) ² +2(vNV ^?1)^1/2, andv=(? φ_i(x)dx)²], the system wave function becomes a pure Jastrow function, whereas if λ²?1??2×10 ^?1 , we have in effect the case where λ′=0; furthermore, there is ODLRO if λ ² ?1～?2×10 ^?1 . It is also concluded that the superfluid fraction upper bound of 0.3±0.1 obtained here as well as one suggested by Leggett are not very useful. We have not attempted to establish if there is some value of λ satisfying the above inequality such that the ground-state energy is lower than the value it takes for λ′=0.     

A contact problem for a nonhomogeneous half-space

A solution of a contact problem for a non-homogeneous half-space is given. The analysis is based on the assumption that the shearing displacement is prescribed within the band $\text{y = 0, ¦x¦<1, ¦z¦< ∞}$ and shearing stress is zero in the outside area $\text{|x|> 1, y = 0, ¦z¦<∞}$. The shear modulus of the non-homogeneous half-space is assumed in the following form μ = μ₀(1 + α₁x)² (1 + β₁y)², where α₁, β₁ and μ₀ are constants. Here x, y and z are referred to a rectangular coordinates system.     

A Problem of Ranking and Estimation With Poisson Processes

There are given k Poisson processes with parameters (rates of occurrence) λ₁, …, λ _k . Let λ₍₁₎ ≤ λ₍₂₎ ≤ … ≤ λ_(k) denote the ordered set of values of the parameters. A procedure is given for selecting the process corresponding to λ_(k) and estimating its parameter (λ_(k)). The given procedure controls the joint risk of improper selection and of large error in the estimate. Let θ > 1 and 0 < α, β < 1 be given numbers, and let δ denote the estimate of λ_(k). The joint probability that a correct selection is made and that |(δ/λ_(k)) ? 1| ≤ α is at least as large as β, for (λ_(k)/λ_(k?1)) ≥ θ. Two cases are considered, that is, when the processes are observed continuously in time, and when they are observed at successive intervals of time. Both the cases lead to the same theoretical results.     

Submillimeter surface impedance and relaxation: A case study of quasi-two-dimensional YBa2Cu3Ox

Surface impedance measurements in the normal and superconducting state are an excellent method to study the conduction electron dynamics in metals. This holds especially in the relaxation range, i.e., for distances traveled in one r.f. periods=υ _F/ω(υ_f is the Fermi velocity) being smaller or of the order of the penetration depth λ and mean free pathl. For materials withυ _F<-10⁷ cm/sec the relaxation range is easily accessible forf>0.1 THz. Then, in the normal state, relaxation defines the surface impedance with a penetration depth approaching the London penetration depth λ_L, andR≈μ ₀λ_l/2τ as surface resistance allowing a measure of λ_L and relaxation time τ(T, ω). In the superconducting state the photon interaction scales withξ _F/λ_L=l/γ (ξ _f is the dimension of Cooper pairs for l→∞) and causes at low frequencies an absorption rate growing withγ, which is decreasing withξ _F/l. The rate increase proportional toγ turns to a decrease above 0.1 THz, being accompanied by a decrease ofγ with frequency which is stronger for largeγ and smallξ _F/l. These characteristic dependences allow a measurement of material parameters, anisotropy, and dynamics of electrons. To evaluate the consequences of theâ, b, and?-direction anisotropy, the integral kernels for normal and superconducting surface impedances in its nonintegrated, angle-dependent form are presented, analyzed, and compared with impedance measurements above 0.1 THz of YBa₂Cu₃O_x.     

Microstructure and mechanical properties of directionally solidified Al–Si eutectic alloys with and without antimony

Abstract

Hardness H, interjlake spacing λ, and tensile properties are reported for Al–12·7Si and Al–12·7Si–0·2Sb (all wt-%) eutectic alloys directionally solidified at growth velocities of up to 250 μm s^?1 and under temperature gradients in the liquid of up to 12·9 K mm^?1. The hardness is related to interflake spacing by the equation H=H^o+Kλ^?0·2, where H^o is the initial hardness of the alloy. This behaviour contradicts previous results, which suggest that a Hall–Petch relationship is followed. The tensile properties are shown to follow similar behaviour, confirming that hardness shows the same dependence as proof stress on interflake spacing. However, the nature of the relationship depends on the Si morphology and caution should be exercised in using hardness or interflake spacing to indicate proof stress.

MST/1585     

Instrumentation in Astronomy III

Two-wavelength holography and phase-shifting interferometry are combined to measure the phase contours of deep wavefronts and surfaces, such as those produced by aspherics, with a variable sensitivity. When interference fringes are very closely spaced, the phase data contain high frequencies where 2 ~ ambiguities cannot be resolved. In this technique, the surface is tested at a synthesized longer equivalent wavelength. The phase of the wavefront is calculated modulo 2φ using phase-shifting techniques at each of two visible wavelengths. The difference between these two phase sets is the phase of the wavefront as it would be measured at λ_eq=λ₁λ₂/|λ₁ ? λ₂ |, assuming that 2π ambiguities can be removed at λ_eq. This technique enables surfaces to be contoured to an accuracy of λ_eq/100.     

Microstructures formed by directional solidification of two ternary eutectic alloys of Bi–Cd–In system

Abstract

Microstructures of the two ternary eutectic alloys of the Bi–Cd–In system were studied using slow unidirectional solidification, followed by quenching to form a representative solid/liquid interface for subsequent observation. The eutectic reactions were found to take the form L?BiIn+BiIn₂+Cd at 77.5°C and L?BiIn₂+_?+Cd at 61.5°C. The 77.5°C eutectic was observed to be of the faceted (BiIn)–faceted (Cd)–non-faceted (BiIn₂) type, while all three phases of the 61.5°C eutectic showed faceting. The BiIn and BiIn₂ phases of the 77.5°C eutectic formed a quasiregular microstructure with the Cd phase growing relatively independently. The phases of the 61.5°C eutectic tended to form a lamellar microstructure with a BiIn₂–?–Cd–_?–BiIn₂ phase sequence. Both eutectics were observed to obey the usual phase spacing law, λ²R=constant, where λ is the phase spacing and R is the growth rate.     

Dependency of microstructure parameters and microhardness on the temperature gradient for directionally solidified Ti–49Al alloy

Intermetallics Ti–49Al (at.%) alloy was directionally solidified with different temperature gradients (G) ranged from 2.8 K mm⁻¹ to 12.5 K mm⁻¹ at a constant growth rate (V = 10 μm s⁻¹) by using a Bridgman-type directional solidification furnace. The microstructure of the directionally solidified specimen is constituted of α₂(Ti₃Al) and γ(TiAl) lamellar structures. The values of the primary dendritic spacing (λ), interlamellar spacing (λ_L) and microhardness (HV) were measured. Dependencies of λ, λ_L and HV on G were determined by using the linear regressing analysis. According to these results, the values of λ and λ_L decrease with the increase of G, and the values of HV increase with the increase of G and with the decrease of λ and λ_L. In addition, these results were compared with the previous similar experimental results of TiAl-based alloys.     

Mueller Matrices and Depolarization Criteria

Abstract

The question of whether a given Mueller matrix represents a deterministic or a non-deterministic system is analysed by means of a matrix condition. The possibility of replacing this matrix condition by a scalar condition is examined. It is shown that this is permissible only for those cases where a Hermitian matrix constructed from the Mueller matrix is positive semidefinite.     

Fast Algorithms for Structured Least Squares and Total Least Squares Problems

We consider the problem of solving least squares problems involving a matrix M of small displacement rank with respect to two matrices Z₁ and Z₂. We develop formulas for the generators of the matrix M ^HM in terms of the generators of M and show that the Cholesky factorization of the matrix M ^HM can be computed quickly if Z₁ is close to unitary and Z₂ is triangular and nilpotent. These conditions are satisfied for several classes of matrices, including Toeplitz, block Toeplitz, Hankel, and block Hankel, and for matrices whose blocks have such structure. Fast Cholesky factorization enables fast solution of least squares problems, total least squares problems, and regularized total least squares problems involving these classes of matrices.     

Study of the Magnetic Properties of Ni/Ag Superlattice

Thin multilayer films of alternating ultrathin Ni and Ag layers (L(Ni)=11,15,30 Å, bulk and \(L(\mathrm{Ag})=50~{\AA})\) have been prepared by evaporation in ultrahigh vacuum under controlled conditions and have been studied by the magnetic measurements. The critical temperature T _C is studied as a function of the surface exchange interaction (J _S). The dependence of T _C on the thickness L of the film has been investigated. A critical value of the surface exchange interaction in the film, above which the surface magnetism appears, is obtained. The shift of the critical temperature T _C(L) from the bulk value \([\frac{T_{\mathrm{C}}(\infty )}{T_{\mathrm{C}}(L)}-1]\) can be described by a power law L ^?λ , where \(\lambda =\frac{1}{\nu_{\mathrm{b}}}\) is the inverse of the correlation length’s exponent. The effective critical exponent associated with the magnetization M(β) is deduced for different thicknesses of Ni layers, and the thickness L(Ag) was being kept constant at 50 Å.     

The Effect of the Demagnetizing Field on the NMR Spectra for Liquids Enclosed in a Cylinder

The NMR spectrum of a simple liquid in a cell of size L depends on D, the diffusion constant, G, the applied field gradient, and M ₀, the z-component of magnetization before tipping of the spins. For small tipping angles the shape of the spectrum depends on which of the corresponding frequency scales—ω _D = D/L ²,ω _G = γ FL, and ω _M = γ μ₀ M ₀—is the largest. We explore the evolution of the spectrum between the inhomogeneous broadening regime (ω _G ? ω _D ,ω _M ) and the regime where the spin dynamics is dominated by magnetostatic modes (ω _M ? ω _D ,ω _G )for a liquid confined in a cylinder of length 2L, both for classical liquids, and for liquids which exhibit the Leggett–Rice effect.