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 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.     

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).     

Amplification and compression of Ti:sapphire femtosecond pulses at high-repetition-rate

The experiment of the generation and amplification of femetosecond Ti:sapphire laser pulse at high repetition rate is reported. The laser pulses with minimum pulsewidth 15 fs, maximum spectrum width of 80 nm, average power of 200 mW are generated from a home-built self-mode-locked Ti:sapphire laser. As a seed pulse which is selected from the oscillator, the laser pulse is further amplified by using chirped-pulse-amplification technology in a Ti:sapphire amplifier from which a kind of pulses with single-pulse-energy of 100 uj, pulsewidth after compressing of 50 fs at 5 kHz repetition rate are produced. The system design and experimental results are discussed. Project supported by the National “Climbing Project” of China.     

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.     

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.     

Highly efficient TW multipass Ti:sapphire laser system

An efficient chirped-pulse amplification, Ti: sapphire laser system, has been developed using mainly domestic components. The gain-narrowing effect has been significantly overcome by shaping spectrum of seeding pulses. With a novel aberration-free stretcher and two stage multi-pass amplifiers, pulses with duration of 25 fs and 36-mJ energy have been obtained at 10 Hz repetition rate, using only less than 290 mJ green Nd:YAG pump energy. This corresponds to a 1.4 TW peak power and 32% main amplification efficiency. The energy stability of the laser systems is better than ± 3%.     

On exact order estimates of N-widths of classes of functions analytic in a simply connected domain

In the spaces E _q(Ω), 1 < q < ∞, introduced by Smirnov, we obtain exact order estimates of projective and spectral n-widths of the classes W ^r E _p(Ω) and W ^r E _p(Ω)Ф in the case where p and q are not equal. We also indicate extremal subspaces and operators for the approximative values under consideration.     

Lipschitz equivalence of generalized {1,3,5}-{1,4,5} self-similar sets

This paper investigates the Lipschitz equivalence of generalized {1,3,5}-{1,4,5} self-similar sets D=(r_1D)∪(r_2D (1 r_1-r_2-r_3)/2)∪(r_3D 1 r_3) and E=(r_1E)∪(r_2E 1-r_2- r_3)∪(r_3E 1-r_3),and proves that D and E are Lipschitz equivalent if and only if there are positive integers m and n such that r_1~m=r_3~n.     

Some remarks on the schemesW d r

Summary Let X be an irreducible smooth projective curve of genus g. Let _d ^r (g) be the Brill-Noether Number. In this paper we prove some results concerning the schemes W _d ^r of special divisors. 1) Suppose dim (W _d–1 ^r )= _{d– 1} ^r (g)0 and _d ^r (g) < g. If W _{d– 1} ^r is a reduced (resp. irreducible) scheme, then W _d ^r is a reduced (resp. irreducible) scheme. 2) Under certain conditions, if Z is a generically reduced irreducible component of W _d–1 ^r then Z W ₁ ⁰ is a generically reduced irreducible component of W _d ^r . For r=1, we obtain some further results in this direction. 3) As an application of it we are able to prove some dimension theorems for the schemes W _d ¹ .     

Imaging through flesh tissue using fs electronic holographic gating method

The experimental results of imaging through flesh tissue using fs electronic holographic gating method is reported. In the experiment, Ti: sapphire mode-locked laser is used as light source, of which the repetition rate is 100 MHz, central wavelength 800 nm, duration of pulse 20 fs, output power 80 mW. Tissue is a 7 mm thick chicken slice, and the imaged object is a metal wire with diameter of 0.5 mm. A general CCD is used to record holograms and a clear image of metal wire is obtained. Several relevant problems are discussed.     

Symplectic embeddings of ellipsoids

We study the rigidity and flexibility of symplectic embeddings in the model case in which the domain is a symplectic ellipsoid. It is first proved that under the conditionr _n ² ≤2r ₁ ² the symplectic ellipsoidE(r ₁,…,r _n)with radiir ₁≤…≤r _ndoes not symplectically embed into a ball of radius strictly smaller thanr _n.We then use symplectic folding to see that this condition is sharp. We finally sketch a proof of the fact that any connected symplectic 4-manifold of finite volume can be asymptotically filled with skinny ellipoids.     

W(R)-splines

In [3] Golomb describes, for 1 < p < ∞, the H^r,p(R)-extremal extension F_* of a function ƒ:E → R (i.e., the H^r,p-spline with knots in E) and studies the cone H_*E^r,p of all such splines. We study the problem of determining when F_* is in W^r,p ≡ H^r,p ∩ L^p. If F_* ε W^r,p, then F_* is called a W^r,p-spline, and we denote by W_*E^r,p the cone of all such splines. If E is quasiuniform, then F_* ε W^r,p if and only if {ƒ(t_i)}_tiεE ε l^p. The cone W_*E^r,p with E quasiuniform is shown to be homeomorphic to l^p. Similarly, H_*E^r,p is homeomorphic to h^r,p. Approximation properties of the W^r,p-splines are studied and error bounds in terms of the mesh size ¦ E ¦ are calculated. Restricting ourselves to the case p = 2 and to quasiuniform partitions E, the second integral relation is proved and better error bounds in terms of ¦ E ¦ are derived.     

A common property of R(E, F) and B(ℝ n , ℝ m ) and a new method for seeking a path to connect two operators

Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear operators from E into F, and R(E, F) the set of all operators in B(E, F) with finite rank. It is well-known that B(? ⁿ ) is a Banach space as well as an algebra, while B(? ⁿ , ? ^m ) for m ≠ n, is a Banach space but not an algebra; meanwhile, it is clear that R(E, F) is neither a Banach space nor an algebra. However, in this paper, it is proved that all of them have a common property in geometry and topology, i.e., they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces). Let Σ _r be the set of all operators of finite rank r in B(E, F) (or B(? ⁿ , ? ^m )). In fact, we have that 1) suppose Σ _r ∈ B(? ⁿ , ? ^m ), and then Σ _r is a smooth and path-connected submanifold of B(? ⁿ , ? ^m ) and dimΣ _r = (n + m)r ? r ², for each r ∈ [0, min{n,m}; if m ≠ n, the same conclusion for Σ _r and its dimension is valid for each r ∈ [0, min{n, m}]; 2) suppose Σ _r ∈ B(E, F), and dimF = ∞, and then Σ _r is a smooth and path-connected submanifold of B(E, F) with the tangent space T _A Σ _r = {B ∈ B(E, F): BN(A) ? R(A)} at each A ∈ Σ _r for 0 ? r ? ∞. The routine methods for seeking a path to connect two operators can hardly apply here. A new method and some fundamental theorems are introduced in this paper, which is development of elementary transformation of matrices in B(? ⁿ ), and more adapted and simple than the elementary transformation method. In addition to tensor analysis and application of Thom’s famous result for transversility, these will benefit the study of infinite geometry.     

Nonlocal problems for the equations of Kelvin-Voight fluids and their ɛ-approximations in classes of smooth functions

Existence theorems are proved for the solutions of the first and second initial boundary-value problems for the equations of Kelvin-Voight fluids and for the penalized equations of Kelvin-Voight fluids in the smoothness classes W _∞ ^r (ℝ⁺;W ₂ ^2+k (Ω)), W ₂ ^r (ℝ⁺;W ₂ ^2+k (Ω)) and S ₂ ^r (ℝ⁺;W ₂ ^2+k (Ω)) (r=1,2; k=0,1,2, …) under the condition that the right-hand sides f(x,t) belong to the classes W _∞ ^r-1 (ℝ⁺;W ₂ ^k (Ω)), W ₂ ^r-1 (ℝ⁺;W ₂ ^k (Ω)) and S ₂ ^r-1 (ℝ⁺;W ₂ ^k (Ω)), respectively, and for the solutions of the first and second T-periodic boundary-value problems for the same equations in the smoothness classes W _∞ ^r−1 (ℝ; W ₂ ^2+k (Ω)) and W ₂ ^r−1 (0, T; W ₂ ^2+k (Ω)) (r=1,2, k=0,1,2…) under the condition that f(x,t) are T-periodic and belong to the spaces W _∞ ^r−1 (ℝ⁺; W ₂ ^k (Ω)) and W ₂ ^r−1 (0,T; W ₂ ^k (Ω)), respectively. It is shown that as ɛ→0, the smooth solutions {v^ɛ} of the perturbed initial boundary-value and T-periodic boundary-value problems for the penalized equations of Kelvin-Voight fluids converge to the corresponding smooth solutions (v,p) of the initial boundary-value and T-periodic boundary-value problems for the equations of Kelvin-Voight fluids. Bibliography: 27 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 214–242. Translated by T. N. Surkova.     

Clifford structures on Riemannian manifolds

We introduce the notion of even Clifford structures on Riemannian manifolds, which for rank r=2 and r=3 reduce to almost Hermitian and quaternion-Hermitian structures respectively. We give the complete classification of manifolds carrying parallel rank r even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler manifolds for r=2,3 and 4 respectively, several classes of 8-dimensional manifolds (for 5?r?8), families of real, complex and quaternionic Grassmannians (for r=8,6 and 5 respectively), and Rosenfeld?s elliptic projective planes OP², (C⊗O)P², (H⊗O)P² and (O⊗O)P², which are symmetric spaces associated to the exceptional simple Lie groups F₄, E₆, E₇ and E₈ (for r=9,10,12 and 16 respectively). As an application, we classify all Riemannian manifolds whose metric is bundle-like along the curvature constancy distribution, generalizing well-known results in Sasakian and 3-Sasakian geometry.     

Problemi discontinui e di trasmissione per due equazioni ellittiche di ordine diverso a coefficienti variabili relativi ad angoli consecutivi

Summary In this paper we continue our investigations (see[5]) devoted to transmission problems in consecutive angles, for elliptic operators. In[5] the above mentioned boundary value problems in the case of homogeneous constant coefficient operators and within the Sobolev weight spaces W _s1,s2 ^r were investigated. This work is concerned with the problem for general operators, within the Sobolev weight spaces W _s1,s2 ^r and W _s1,s2 ^*r . Existence theorems and stability theorems for nullity, deficiency, index have been proved.

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Entrata in Redazione il 28 marzo 1975.     

On the integral of mean curvature of a flattened convex body in space forms

Under the assumptions that E _λ ⁿ is an n-dimensional, simply connected Riemannian manifold of constant sectional curvature λ and L _λ ^r is an r-dimensional, totally geodesic submanifold of E _λ ⁿ , the paper investigates the q-th integral of the mean curvature M _q ⁿ of a convex body K ^r in E _λ ⁿ and gives the expression of M _q ⁿ in the terms of M _p ^r , where M _p ^r is the p-th integral of the mean curvature of K ^r > in L _λ ^r . A result of L. A. Santaló [2] holds in particular.     

Tame Kernels of Quaternion Number Fields

Let E/F be a Galois extension of number fields with the quaternion Galois group Q ₈. In this paper, we prove some relations connecting orders of the odd part of the kernel of the transfer map of the tame kernel of E with the same orders of some of its subfields. Let E/? be a Galois extension of number fields with the Galois group Q ₈ and p an odd prime such that p ≡ 3 (mod 4). We prove that if there is at most one quadratic subfield such that the p-Sylow subgroup of the tame kernel is nontrivial, then p ^r -rank(K ₂(E/K)) is even, i.e., 2|p ^r -rank(K ₂(𝒪 _E )) ? p ^r -rank(K ₂(𝒪 _K )), where K is the quartic subfield of E.     

The uniqueness of optimal quadrature formula and optimal interpolation knots with multiplicities in L1

The optimal quadrature formula of type (r₁,..., r_n) for the class of w _∞ ^m is unique if 1<2[(r_i+1)/2]<m. And the set of optimal interpolation knots of type (r₁,...,r_n) for W _∞ ^m in the metric of L₁ is existent and unique in case all r_i being even.