Amplified spontaneous emission and its restraint in a terawatt Ti:sapphire amplifier

Amplified spontaneous emission (ASE) and its restraint in a femtosecond Ti: sapphire chirped-pulse amplifier were investigated. The noises arising from ASE were effectively filtered out in the spatial, temporal and spectral domain. Pulses as short as 38 fs were amplified to peak power of 1.4 MI. The power ratio between the amplified femtosecond pulse and the ASE was higher than 10⁶: 1.     

Experimental studies of high order soliton compression effect and gain characteristics in femtosecond laser pulses Er3+-doped fiber amplifier

Seed laser pulses with average power of 146 μW and pulse duration of 480 fs were amplified to 14.5 mW. The pulse duration was compressed to 260 fs using 6 m high concentration E³⁺ _r -doped fiber under forward pumping. The amplified signal pulse energy was 0.691 nJ (corresponding to a peak power of 2 657.7 W) and the repetition rate was 20.84 MHz. Spectrum breakup was observed simultaneously. The spectrum of pulses amplified by 3 m E³⁺ _r -doped fiber remains a single peak under different pump power. The amplified pulse duration was compressed abnormally with the increasing pump power using the backward pumping; that is, the amplified pulses were compressed with the increasing pump power under low pump power. When the pump power reached 38 mW, the shortest amplified pulse duration was 309 fs. With further increase in pump power, the amplified pulses began broadening, accompanied by a single peak spectrum under different pump power.     

Experimental studies of high order soliton compression effect and gain characteristics in femtosecond laser pulses Er 3+-doped fiber amplifier

Seed laser pulses with average power of 146 μW and pulse duration of 480 fs were amplified to 14.5 mW. The pulse duration was compressed to 260 fs using 6 m high concentration E³⁺ _r -doped fiber under forward pumping. The amplified signal pulse energy was 0.691 nJ (corresponding to a peak power of 2 657.7 W) and the repetition rate was 20.84 MHz. Spectrum breakup was observed simultaneously. The spectrum of pulses amplified by 3 m E³⁺ _r -doped fiber remains a single peak under different pump power. The amplified pulse duration was compressed abnormally with the increasing pump power using the backward pumping; that is, the amplified pulses were compressed with the increasing pump power under low pump power. When the pump power reached 38 mW, the shortest amplified pulse duration was 309 fs. With further increase in pump power, the amplified pulses began broadening, accompanied by a single peak spectrum under different pump power.     

Experimental studies of self-starting passive mode locking Er3+ -doped fiber ring soliton laser

The mechanism of femtosecond optical pulse generation in a self-starting Er³⁺-doped fiber ring soliton laser and experimental research results are discussed. Using the nonlinear polarization rotation effect of the fiber for sat-urable absorbers (and then self-amplitude modulation) which acts as the mode locking mechanism in an Er³⁺-doped fiber ring cavity laser, stable self-starting mode locking pulses have been generated. The shortest output pulse is 269 fs, with the central wavelength of 1,531 pm at the repetition rate of 21.37 MHz. The average output powen of the two terminators of the laser are 0.25 mW and 0.08 mW respectively. The threshold pump power which sustains the mode locking is 15 mW. Under high pump power, the laser works in a high order harmonic mode locking state. The mode locking pulse durations vs different cavity lengths are also studied. Project supported by Major Project of Chines: Academy of Sciences (No. KJ952-J1-705).     

Optical diagnostics of femtosecond laser plasmas

Optical diagnostics of evolution of plasmas produced by ultrashort laser pulses is carried out using a femtosecond probing beam. The time sequence of plasma shadowgrams and interferograms are obtained. The filamentation instability in high-density region induces the local density modification. Large-scale toroidal magnetic fields confine plasma expansion in the transverse direction, resulting in the formation of a plasma jet. The plasma expansion along the target normal direction is found to scale as t^1/2.     

Experimental studies of high order soliton compression effect and gain characteristics in femtosecond laser pulses E3+r-doped fiber amplifier

Seed laser pulses with average power of 146 μW and pulse duration of 480 fs were amplified to 14.5 mW. The pulse duration was compressed to 260 fs using 6 m high concentration E3+r-doped fiber under forward pumping. The amplified signal pulse energy was 0.691 nJ (corresponding to a peak power of 2 657.7 W) and the repetition rate was 20.84 MHz. Spectrum breakup was observed simultaneously. The spectrum of pulses amplified by 3 m E3+r-doped fiber remains a single peak under different pump power. The amplified pulse duration was compressed abnormally with the increasing pump power using the backward pumping; that is, the amplified pulses were compressed with the increasing pump power under low pump power. When the pump power reached 38 mW, the shortest amplified pulse duration was 309 fs. With further increase in pump power, the amplified pulses began broadening, accompanied by a single peak spectrum under different pump power.     

L p - L q estimates of solutions to the damped wave equation in 3-dimensional space and their application

 It has been asserted that the damped wave equation has the diffusive structure as t→∞. In this paper we consider the Cauchy problem in 3-dimensional space for the linear damped wave equation and the corresponding parabolic equation, and obtain the L ^p −L ^q estimates of the difference of each solution, which represent the assertion precisely. Explicit formulas of the solutions are analyzed for the proof. The second aim is to apply the L ^p −L ^q estimates to the semilinear damped wave equation with power nonlinearity. If the power is larger than the Fujita exponent, then the time global existence of small weak solution is proved and its optimal decay order is obtained. Received: 8 June 2001; in final form: 12 August 2002 / Published online: 1 April 2003 Mathematical Subject Classification (2000): 35L15.     

Stability and omitting types

We give a complete solution to the following question: when does a superstable theory have a model of powerκ omitting a partial typeq? In particular, for fixedq, if there is such a model of power ?₁ then there is one of power 2^? ₀; and if there is a model omittingq of power ?₁, then there are arbitrarily large ones. For stable theories, a model of power ? _ω ⁺ , omittingq implies one of power 2^? ₀, and this is sharp. Several improvements and some negative results are listed in the introduction.     

A projective analogue of schur's tensor power construction

For a prime ring R with σ ? Aut(R), we examine the extended center of the skew power series ring R[[x;σ]]. We prove the extended center of R[[x;σ]] is isomorphic to: (1) C^σ if no power of σ is X-inner where C^σ is the fixed field of a acting on the extended center C of R (2) C^σ((vx')) (Laurent series) if some positive power of a is X-inner and R is closed prime. In this case I is the least positive integer such that a¹ is X-inner with σ¹= inn(v). We provide an example showing this result fails if R is not closed prime.     

Finite convergence over a field of power series

In this paper, we define an analog of power series functions over R, when R is replaced by K = k((x))^τ , a field of generalized power series with coefficients in an ordered field k and exponents in an ordered abelian group τ. To this end for any power series S(Y)ε K[[Y]] and any y ε K, we define a notion of convergence of S(y). Thus to any power series S(Y) is associated a partial function S : K→ K. We show that these partial functions have a lot of similarities with analytic functions over R. Then we prove properties of zeros of such functions which extend properties of roots of polynomials over k((x))^τ.     

On universal graphs without instances of CH

We first prove the consistency of: there is a universal graph of power ?₁<2^?₀ = 2^?₁=?₂. The consistency of the non-existence of a universal graph of power ?₁ is trivial. Add ?₂ Cohen generic reals. We then show that we can have 2^?₀=?₂<2^?₁, and get similar results for other cardinals.     

Amorphe Potenzen kompakter Räume

A set is amorphous, if it is not a union of two disjoint infinite subsets. The following variants of the Tychonoff product theorem are investigated in the hierarchy of weak choice principles. TA1: An amorphous power of a compactT ₂ space is compact. TA2: An amorphous power of a compactT ₂ space which as a set is wellorderable is compact. In ZF⁰TA1 is equivalent to the assertion, that amorphous sets are finite. RT is Ramsey's theorem, that every finite colouring of the set ofn-element subsets of an infinite set has an infinite homogeneous subset and PW is Rubin's axiom, that the power set of an ordinal is wellorderable. In ZF⁰RT+PW implies TA2. Since RT+PW is compatible with the existence of infinite amorphous sets, TA2 does not imply TA1 in ZF⁰. But TA2 cannot be proved in ZF⁰ alone. As an application, we prove a theorem of Stone, using a weak wellordering axiomD ₃ (a set is wellorderable, if each of its infinite subsets is structured) together with RT.

Diese Arbeit ist Teil der Habilitationsschrift des Verfassers im Fachgebiet Mathematische Analysis an der Technischen Universität Wien.     

Four Wave Mixing of the Signal and ASE Noisein Long Distance Optical Communication System

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Multiplicative inequalities for the L 1 norm: Applications in analysis and number theory

The paper is devoted to multiplicative lower estimates for the L ₁ norm and their applications in analysis and number theory. Multiplicative inequalities of the following three types are considered: martingale (for the Haar system), complex trigonometric (for exponential sums), and real trigonometric. A new method for obtaining sharp bounds for the integral norm of trigonometric and power series is proposed; this method uses the number-theoretic and combinatorial characteristics of the spectrum. Applications of the method (both in H ¹ and L ₁) to an important class of power density spectra, including [n ^α] with 1 ≤ α < ∞, are developed. A new combinatorial theorem is proved that makes it possible to estimate the arithmetic characteristics of spectra under fairly general assumptions.     

Fractional Navier-Stokes equations and a Hölder-type inequality in a sum of singular spaces

In this paper we study the local well-posedness of the fractional Navier-Stokes system with initial data belonging to a sum of two pseudomeasure-type spaces denoted by PM^a,b:=PM^a+PM^b. The proof requires showing a Hölder-type inequality in PM^a,b, as well as establishing estimates of the semigroup generated by the fractional power of Laplacian (−Δ)^γ on these spaces.     

A power series expansion and its applications

Using the power series solution of a differential equation and the computation of a parametric integral, two elementary proofs are given for the power series expansion of (arcsin x)², as well as some applications of this expansion.     

The Nonlinear Schrödinger Equation on Hyperbolic Space

In this article we study some aspects of dispersive and concentration phenomena for the Schrödinger equation posed on hyperbolic space  ⁿ , in order to see if the negative curvature of the manifold gets the dynamics more stable than in the Euclidean case. It is indeed the case for the dispersive properties: we prove that the dispersion inequality is valid, in a stronger form than the one on ? ⁿ . However, the geometry does not have enough of an effect to avoid the concentration phenomena and the picture is actually worse than expected. The critical nonlinearity power for blow-up turns out to be the same as in the euclidean case, and we prove that there are more explosive solutions for critical and supercritical nonlinearities.     

Non-crossed products over function fields

Noncrossed product division algebras are constructed over all function fields and iterated power series fields over global fields, using Hilbert's Irreducibility Theorem and the construction of [B]. Minimum indexes obtained are p ² for odd p and 2³ otherwise. Examples are obtained with large index to exponent ratio. Received: 12 February 2001 / Revised version: 26 November 2001     

Logarithmic growth of the Bergman Kernel for weakly pseudoconvex domains in ℂ3 of finite type

If G???ⁿis pseudoconvex with smooth boundary and z_o∈bG is a point_of finite type, one can ask the question: Does the Bergman kernel K_G(z,¯z) grow like a rational power of dist(z,bG) when z approaches z_o nontangentially? This question is suggested by observations for the domains U_pg defined below. But the answer to this question is in general negative as is shown by a counterexample in ?³.