Optimum profiles for dispersive multimode fibres

A simple closed-form expression is given for the index profile of multimode fibres with arbitrary dispersion providing transmission capacities as large as 1.6/ ² Mbit/s km, wheren/n. Our result reduces to a previous result of Marcatili for the special case of circularly symmetric fibres. A transmission capacity of 150 Mbit/s over a 10 km long fibre appears possible with LED sources operating at optimum wavelengths for the medium.     

Approximations in the evaluation of the impulse response of nearly parabolic fibres

Formulae are derived giving the flight time of a ray impinging on a nearly-parabolic fibre whose core index profile includes a fourth order perturbation term in addition to the usual second order term of parabolic profiles. The method is applied to the case of parallel beam injection to calculate the impulse responses of nearly parabolic fibres characterized either by a fourth order term or by an value very close to 2. Numerical results are given for parameters typical of silica fibres made by the CVD technique.     

Modelling of attenuation properties of PCS fibres and its application to the backscattering measurement

The time-independent diffusion equation that describes an optical power flow in an axially perturbed multimode fibre with a lossy cladding is solved. On this basis the attenuation of plastic-clad silica (PCS) fibres depending on cladding loss, mode coupling due to fibre perturbation, launching conditions and fibre length is investigated by means of computer modelling. This involves not only attenuation properties of PCS fibres by themselves, but also how they are seen by the backscattering measurement technique.     

Phase conjugation in an optical fibre

We report experimental studies of phase conjugation by stimulated Mandelstam Brillouin scattering in multimode fibres. A dramatic decrease in the threshold power as compared to that (10⁴ to 10⁵ W) typically observed in conventional media was achieved. A ruby pulsed pump with 500-ns pulses was used. For the 7-m-long fibre, the threshold power was 50 W, the degree of phase conjugation was almost one, the pump-to-Stokes conversion efficiency was about 80%. For the 130-m-long fibre, the threshold power decreased to 10 W and the pump-to-Stokes conversion efficiency reached 100%. Polarization scrambling was observed in the 130-m-long fibre, but the Stokes wave with polarization coinciding with that of the pump wave was a phase-conjugate replica of the pump, as before.     

Pulse broadening in near-optimum graded-index fibres

Impulse responses of a wide class of multimode optical fibres with near-optimum index profiles are investigated. The index profile is described by three parameters: the power law exponent, the magnitude and the width of a dip at the centre of the core. It is found that for overcompensated profiles (< _opt) a moderate dip decreases the r.m.s. dispersion. Leaky modes are found to improve the dispersion characteristics for fibres with a dip in the index profile. Impulse response measurements are interpretated to give information about the index profile.     

Far-field techniques for the characterization of single-mode fibres

The characterization of single-mode fibres is important not only for determining the system parameters for a particular fibre, but also for the specification of fibres for manufacturing purposes. We present a detailed analysis of two new methods of characterizing single-mode fibres. The strip integrated far-field method involves measuring the light power passed by a thin slit scanned across the far-field pattern and the area integrated far-field method involves measuring the light power passed by a knife edge scanned across the far-field pattern. It is shown that, for either method, the measured data is related to the transverse offset transmission coefficient by the Fourier-cosine transform and hence that the far-field r.m.s. spot-size can be directly calculated. Noise analysis shows that the area integrated far-field method is superior to the strip integrated far-field method and similar to the far-field mask method when a broadband light source is used to measure the far-field r.m.s. spot-size. It is also shown that the area integrated far-field method has smaller systematic errors than the far-field mask method.     

Ray theory of the impulse response of randomly bent multimode fibres

A ray theory based on the time-independent Fokker-Planck equation and the integration of time along ray trajectories provides analytical expressions for the average arrival time and spread of optical pulses propagating in randomly distorted, multimode, optical fibres. A clear physical picture emerges from the theory. The analytical expressions obtained for t and t ² coincide with the ones obtained by Olshansky from coupled-mode theory. The t ³ and t ⁴ moments of the impulse response are also calculated. Simple closed-form formulae are given for the step-index slab. The coupling between all modes is effectively taken into account in our ray theory.On leave of absence at the Laboratoire des Signaux et Systèmes.     

More inequalities for critical exponents

A variety of rigorous inequalities for critical exponents is proved. Most notable is the low-temperature Josephson inequalitydv +2 2–. Others are 1 1 +v, 1 1 , 1,d 1 + 1/ (for d),dv, ₃ + (for d), ₄ , and _{2m 2m+2} (form 2). The hypotheses vary; all inequalities are true for the spin-1/2 Ising model with nearest-neighbor ferromagnetic pair interactions.NSF Predoctoral Fellow (1976–1979). Research supported in part by NSF Grant PHY 78-23952.     

A contribution to the theory of the triple crystal diffractometer

The first part of the paper gives a general equation for triple-crystal arrangement with perfect crystals on the assumption that the third crystal is rotated. It is shown that in the case of perfect crystals the shape of the reflection curve is practically independent of the vertical divergence. The case of mosaic crystals is also solved and the possibility of rotation by other than the third crystal is considered. A method is proposed for investigating the imperfection of a crystal which is different from methods used up to now. The paper is supplemented by some experimental results.

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Anderson localization for the almost Mathieu equation: A nonperturbative proof

We prove that for any diophantine rotation angle and a.e. phase the almost Mathieu operator (H()) _n = _n–1 + _n+1 +cos(2(+n)) _n has pure point spectrum with exponentially decaying eigenfunctions for 15. We also prove the existence of some pure point spectrum for any 5.4.     

On the Effect of Increasing the Radiative-Transition Rate Near a Nanosize Point

A short review of theoretical and experimental studies concerning the photoexcited florescence and Raman scattering of light for a substance in a space containing small material bodies is presented. Calculations of the radiativetransition probability for atoms (molecules) in the vicinity of bodies with a size much smaller than the light wavelength are performed. The probabilities of the singlephoton and doublephoton transitions are shown to increase by factors of 9 and 81 in the vicinity of a nanosize sphere with dielectric constant ||\ 1. The probability of a radiative transition in the vicinity of the vertex of a conic needle bearing up against a plane (both with || 1) increases by factors of (/R _in)² and (/R _in)⁴ for singlephoton and doublephoton transitions, respectively (R _in is the curvature radius for the needle vertex). This theoretical result is suggested as an explanation of the effect of increasing the radiation process intensity in the experiments carried out in the studies cited below.     

Kinetic equation in the kinetic region of the dilute and nonuniform electron plasma

Mori's scaling method is used to derive the kinetic equation for a dilute, nonuniform electron plasma in the kinetic region where the space-time cutoff (b, t _c) satisfies _Dbl _f , _D t _{c f} , with _D the Debye length, _D ^–1= _p the plasma frequency, andl _f and _f the mean free path and time, respectively. The kinetic equation takes account of the nonuniformity of the order ofl _f and _D for the single-and the two-particle distribution function, respectively. Thus the Vlasov term associated with the two-particle distribution function is retained. This kinetic equation is deduced from the kinetic equation in the coherent region obtained by Morita, Mori, and Tokuyama, where the space-time cutoff of the coherent region satisfies _Dbr ₀, _Dt _{c 0}, withr ₀ the Landau length and ₀ the corresponding time scale.     

A matrix method for estimating the Liapunov exponent of one-dimensional systems

Let : [0, 1][0, 1] be a piecewise monotonie expanding map. Then admits an absolutely continuous invariant measure. A result of Kosyakin and Sandler shows that can be approximated by a sequence of absolutely continuous measures _n invariant under piecewise linear Markov maps _itn. Each _itn is constructed on the inverse images of the turning points of . The easily computable measures _n are used to estimate the Liapunov exponent of . The idea of using Markov maps for estimating the Liapunov exponent is applied to both expanding and nonexpanding maps.     

The strength of Einstein's equations

The strength of Einstein's empty-space field equations is computed anew and shown to be equal to the amount of initial data required for a local solution of the equations. This same amount of initial data is shown to be precisely that required for a set of 16 unknown first-order differential equations containing 10 field variables and having six identities of second order. The 10 field variables must be functions of second order in the metric coefficients. The 16 field equationsC , = 0 whereC is Weyl's conformal tensor, are shown to have the same properties as those of the unknown equations, suggesting thatC = 0 is a satisfactory local first-order formulation of Einstein's second-order empty-space field equations.     

The effect of oxygen on the electric conductivity of a cuprous oxide single crystal

The effect of an excess of oxygen on the electric conductivity of a pre-illuminated and heated single crystal of Cu₂O is investigated. It is found that the influence of illumination on the electric conductivity, together with the concentration of impurities, increases with increasing oxygen pressure during annealing.

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In conclusion the author thanks E. Klier and J. Pastrak for valuable remarks and discussion.     

Parametric dependence of the properties of pulsed-laser-deposited diamond-like carbon films

The properties of Pulsed-Laser-Deposited Diamond-Like Carbon (PLD DLC) films are studied as functions of the power density and the wavelength of the laser beam, and the incident angle of the beam relative to the normal of the target surface. All the films have a similar structure consisting of graphite particulates embedded in a continuous matrix, so the macroscopic performance of the films is determined by the overall contributions of the particulates and the matrix. The use of higher, shorter, or larger leads to an enhancement of the diamond-like characteristics and a simultaneous increase of the particulate density. These two effects give opposite contributions to the electrical conductivity _R, leading to the following results. (i) _R drops with increasing in the low range (region I) due to the stronger diamond-like nature of the matrix, but increases sharply after has exceeded a threshold _min as a result of the rapid increase in particulate density. (ii) In region I, the use of shorter or larger leads to a more diamond-like matrix, and this overwhelms the degradation effect caused by the slight increase in particulate density. The samples thus become more insulating. In the high region (region II), however, the use of shorter or larger gives rise to higher particulate density, thereby increasing the electrical conductivity.     

The physical nature of the production of moving striations in a d-c discharge plasma

The physical nature of the stratification of the d-c discharge plasma is found and physically interpreted. The interpretation is based on the mathematical expression of the production of periodic structure in plasma after an aperiodic disturbance, derived from an extremely simplified system of equations. Only three basic phenomena occurring in the plasma of each d-c discharge are included: a) the dependence of the rate of ionization on the electron temperature and hence on the electric field, b) the production of space charges due to the different rates of diffusion of the electrons and ions, c) the creation of additional electric fields due to the creation of space charges. The interactions of these phenomena gives rise to a chain, expanded in time and space, which leads to the production of moving striations. In agreement with experiment this structure is developed only on the side towards the anode from the place where the equilibrium state is disturbed.

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In conclusion the authors thank J. Fousek, J. Kaczér and M. Novák for carefully reading this paper and for valuable remarks, and J. Holub for carrying out the numerical and graphical work.     

К ВОПРОСУ O КРИТИЧЕСКО Й ЧАСТОТЕ ПРИ ПРЕРЫВИСТОМ ОСВЕ ЩЕНИИ

c , c n , n AgBr. .     

Grassmann-Cartan algebra and Klein-Gordon equations

Given a Riemannian structure (M, g), a hypothesis is investigated that if= _p=0 ⁿ _p (M) is submitted to the differential condition (g++)=0, =mc/—which implies that each component of fulfills the Klein-Gordon equation (- ²) _p =0, ought to be interpreted as a natural complex of the bosonic fields. Then it is found that the complex admits the interpretation in the sense of first quantization with (M) being a convex set of states, with the structure of a Hilbert space over . The definite spin states of bosons are then pure states which are not conserved by the temporal evolution.     

On the Surface Pressure for the Edwards-Anderson Model

For the Edwards-Anderson model we introduce an integral representation for the surface pressure (per unit surface) in terms of a quenched moment of the bond-overlap on the surface. We consider free , periodic and antiperiodic ^* boundary conditions (by symmetry ⁽⁾=^(*)), and prove the bounds We show moreover that at high temperatures ⁽⁾ is close to ^2/4 and ⁽⁾ is close to ^2/4 uniformly in the volume .