Asymptotic Characteristics of the Evolution of Polarization of Nonmonochromatic Radiation in Single-Mode Optical Fibers with a Random Twisting of Axes of Linear Birefringence. I. Limiting of Polarization in an Infinitely Long Fiber

We reduce the problem of finding the limiting value of the fiber-ensemble averaged degree of radiation polarization of in an infinitely long optical fiber to the problem of distributions (including joint distributions) of random complex amplitudes E(,z) of the electric field in an optical wave for different wavelengths and the fiber length z tending to infinity. We prove that the random complex vector E(,z) is uniformly distributed on a three-dimensional sphere if z. It is also proved that the random vectors E(₁,z) and E(₂,z) are independent if ₁₂ and z, whence it follows that their joint distribution is entirely determined by the distribution of each of them. The result obtained allows us to find the limiting average values of various quantities describing the radiation upon passing an optical fiber with a random twisting of the anisotropy axes. In particular, on the basis of this result, we show that the average degree of polarization of incoherent radiation upon passing a fiber with such random irregularities tends to zero as the optical-fiber length goes to infinity.     

The ground state energy of Sdhrödinger operators

We studye()=inf spec(-+V) and examine whene()<0 for all 0. We prove thatc ²e()–d ² for suitableV and all small ||.Research partially funded under NSF grant number DMS-9101716.     

Experimental modelling of groove effect on finline transitions at Ka-Band

The mounting groove of finline circuits is always the area of research for low loss E-Plane components. The normal practice says, the groove depth should be kept /4 or /2 according to the termination at the top surface for electrical short inside the waveguide. Here we have investigated the effect experimentally at Ka-Band for short ended circuit grooves. It has been found that if the groove depth is kept /10, just adequate to hold the card vertical rigidly, considerable improvement has been observed in the Insertion Loss of the line. An improvement of 0.6 dB has been observed with /4 to /10 groove depth for short ended circuit for Symmetric unilateral finline.     

Kac-Moody symmetry of generalized non-linear Schrödinger equations

The classical non-linear Schrödinger equation associated with a symmetric Lie algebra =km is known to possess a class of conserved quantities which from a realization of the algebrak []. The construction is now extended to provide a realization of the Kac-Moody algebrak[, ^–1] (with central extension). One can then define auxiliary quantities to obtain the full algebra [, ^–1]. This leads to the formal linearization of the system.     

On the existence of invariant,absolutely continuous measures

Let (, , ) be a measure space with normalized measure,f: a nonsingular transformation. We prove: there exists anf-invariant normalized measure which is absolutely continuous with respect to if and only if there exist >0, and , 0<<1, such that (E)< implies (f ^–k(E))< for allk0.     

Regularization in ellipsometry

The continuous variation of the refractive index with the depth in the vicinity of surfaces can be determined by ellipsometry without destroying the object of measurements. The presented method does not impress any given structure to the profile and is applicable to just the range of layer thicknesses interesting to optics (about /4 to /2). The theoretical approach to interpret the measured data leads to an integral equation that is numerically inverted by regularization to filter out the destabilizing effects of measurement errors. The regularizing operator and regularization parameter responsible for this filtering are founded on physical arguments and experiment, respectively. These results can be transferred to other regularization problems based on quantities related to volume (e.g. density, temperature).     

The central limit theorem for the spectrum of the random one-dimensional Schrödinger operator

Let H_L = –d²/dt²+q(t,) be an one-dimensional random Schrödinger operator in ²(–L, L) with the classical boundary conditions. The random potential q(t,) has a form q(t, )=F(x_t), where x_t is a Brownian motion on the Euclidean v-dimensional torus, FS^v R¹ is a smooth function with the nondegenerated critical points, min_s ^v F = 0. Let are the eigenvalues of H_L) be a spectral distribution function in the volume [– L,L] and N() = lim_L(1/2L)N_L() be a corresponding limit distribution function.Theorem 1. If L then the normalized difference N _L ^* ()=[N_L() -2L·N()]2L tends (in the sense of Levi-Prokhorov) to the limit Gaussian process N^*(); N^*()0, 0, and N^*() has nondegenerated finitedimensional distributions on the spectrum (i.e., > 0). Theorem 2. The limit process N^*() is a continuous process with the locally independent increments.     

Rigorous Spectral Analysis of the Metal–Insulator Transition in a Limit-Periodic Potential

We consider the limit-periodic Jacobi matrices associated with the real Julia sets of f (z)=z ²– for which [2, ) can be seen as the strength of the limit-periodic coefficients. The typical local spectral exponent of their spectral measures is shown to be a harmonic function in decreasing logarithmically from 1 to 0.     

Positron annihilation in synthetic zeolites (II) magnetic quenching effect

Positron lifetime spectra were re-measured for a series of synthetic zeolites using a large time window of observation. Magnetic quenching experiments were also performed for the zeolites, and it has been confirmed that both the ₄ and the ₃ components are due to o-Ps. The annihilation rate of the third component, ₃, showed a good correlation with the size of the largest voids, which is similar to the correlation reported for other compounds. However the annihilation rate of the longest-lived component, ₄, showed a poor correlation with the void size. The ₃ component has thus been assigned to o-Ps in the regular voids of the zeolites, and the ₄ component to that escaped to inter-particle open spaces.     

Analysis of the order parameter for uniform nearest particle system

The uniform nearest particle system (UNPS) is studied, which is a continuoustime Markov process with state space . The rigorous upper bound ^(mf) = ( – 1)/ for the order parameter ₂, is given by the correlation identity and the FKG inequality. Then an improvement of this bound ^(mf) is shown in a similar fashion; C( – 1)/|log( – 1) for >1. Recently, Mountford proved that the critical value _c=1. Combining his result and our improved bound implies that if the critical exponent exists, it is strictly greater than the mean-field value 1 in the weak sense.     

Theoretical investigation of noncollinear phase-matched parametric four-photon amplification of ultrashort light pulses in isotropic media

The amplification of light signals (angular frequency _S in some isotropic media (D₂O, fused silica, and Schott type SF10 glasses) by noncollinear phase-matched parametric four-photon interaction ₁+_2S+₁ is studied theoretically. Computer simulations are carried out for fundamental and second-harmonic pump pulses of a mode-locked Nd: glass laser. Degenerate interaction (wavelength ₁=₂=1054nm or 527 nm) and nondegenerate interaction (₁=1054nm, ₂=527 nm are considered. Characteristic phase-matching parameters and gain parameters versus wavelength are determined. Limitations by spectral bandwidth, optical absorption, optical damage, self-phase modulation, self-focusing and stimulated Raman scattering are analysed.     

Extremal invariant states with a spectrum condition and Borchers' theorem

If the energy spectrum of an extremal invariant state is not the whole real line, it is shown that is either pure or uniquely decomposed into mutually disjoint pure states in the way that =^-1 F _{0 t} dt where is a pure state satisfying = with >0. Next we give a slightly generalized version of Borchers' theorem [1] on the innerness of some automorphism group of a von Neumann algebra with a spectrum condition.     

Lyapunov Modes of Two-Dimensional Many-Body Systems; Soft Disks, Hard Disks, and Rotors

The dynamical instability of many-body systems can best be characterized through the local Lyapunov spectrum {}, its associated eigenvectors {}, and the time-averaged spectrum {}. Each local Lyapunov exponent describes the degree of instability associated with a well-defined direction—given by the associated unit vector —in the full many-body phase space. For a variety of hard-particle systems it is by now well-established that several of the vectors, all with relatively-small values of the time-averaged exponent , correspond to quite well-defined long-wavelength modes. We investigate soft particles from the same viewpoint here, and find no convincing evidence for corresponding modes. The situation is similar—no firm evidence for modes—in a simple two-dimensional lattice-rotor model. We believe that these differences are related to the form of the time-averaged Lyapunov spectrum near =0.     

On the existence of rotating stars in general relativity

The Newtonian equations of motion, and Newton's law of gravitation can be obtained by a limit of Einstein's equations. For a sufficiently small constant the existence of a set of solutions (0) of Einstein's equations of a stationary, axisymmetric star is proven. This existence is proven in weighted Sobolev spaces with the implicit function theorem. Since the value of the causality constant depends only on the units used to measure the velocity, the existence of a solution for any small is physically interesting.     

Theory of fluctuations in limit-cycles of second-order bifurcation type

The fluctuations in limit cycles of second-order bifurcation (transition from a stable to an unstable focus) are investigated near the bifurcation point _c, being an external control parameter. Two different methods are applied: a time- and space-dependent Fokker-Planck equation obtained from an ^1/2-expansion of the master equation ( being the volume) and a time- and space-dependent Langevin equation. Both methods give the same results. It is shown that the dependence of the radial correlation on ² = | – _c| and the time-behaviour of the phase correlation (ensemble dephasing) are determined by the dimensionality of space.     

The measurement of the phase constant of EH11 mode in the hollow dielectric waveguides

In this paper, we presented the results of experiments of propagation phase shift in the hollow dielectric waveguides of the radius a20 with farinfrared laser (wave length =0.337mm). The experimental results agree well with theoretical expecting values. The results also show that it is to exist a coupling length between TEM_oo mode and the funtamental hybrid waveguide EH₁₁ mode. The coupling length is obtained from our measurement.     

Infrared holographic recording in LiNbO3:Cu

Holograms may be recorded in photorefractive LiNbO₃:Cu with pulsed infrared light (wavelength =1064 nm, Q-switched Nd:YAG laser), if the crystals are previously or simultaneously illuminated with a green (=532 nm) light pulse. We study refractive index changes and time constants of as-grown and thermally treated crystals with different copper concentrations. A model explaining this effect is discussed.     

C λ-extended oscillator algebra and parasupersymmetric quantum mechanics

The C -extended oscillator algebra is generated by {1, a, a , N, T}, where T is the generator of the cyclic group C of order . It can be realized as a generalized deformed oscillator algebra (GDOA). Its unirreps can thus be easily exhibited using the representation theory of GDOAs and their carrier spaces show a Z-grading structure. Within its infinite-dimensional Fock space representation, this algebra provides a bosonization of parasupersymmetric quantum mechanics of order p = – 1.     

Calculation and optimization of parameters of radiating distributed feedback structures

A simple method is suggested for calculation of reflection, radiation and transmission coefficients for the distributed feedback structure in the second diffraction order. The method is based on a slight difference between coefficients of reflectionR and radiationI of the surface wave for = (where is the light wavelength corresponding to a precise resonance for the grating length I) and those for =_l (where _l is the light wavelength corresponding to the resonance for the finite grating length). The simplicity of the method makes it possible to use it for optimization of the distributed feedback structure by a number of parameters. The technique can be used in the case of thin-film and diffused waveguides for both TE and TM modes.     

The asymmetric contact process at its second critical value

The asymmetric contact process onZ has two distinct critical values ₁ > ₂ (at least with sufficient asymmetry). One can consider the process on {0,...,N} and analyze the time (which we call _N ) till complete vacany starting from complete occupation. Its behavior has already been resolved for all regions of except for =₂. For this value, Schinazi proved that lim _N log _N /logN=2 in probability and conjectured that _N /N ² converges in distribution. It is that result that we prove in this paper. We rely heavily on the Brownian motion behavior of the edge particle, which comes from Galves and Presutti and Kuczek.