NON-STATIONARY OPERATION OF THE FAST-FLOW LASERS

AND NEW POSSIBILITIES OF CONTROLLING THE LASER OUTPUT CHARACTERISTICS

A.V. Mushenkov, A.Yu.Loskutov, A.I.Odintsov, A.I.Fedoseev and V.F.Sharkov

Physics department of M.V.Lomonosov Moscow State University, 119899, Vorob'evy gory, Moscow, Russia

Abstract

The dynamics of the lasing of the fast crossflow gas laser with the inhomogeneous steady state pumping in the unstable resonator by means of numerical modeling is investigated. Depending on the system parameters the various dynamical regimes of operation are observed. The detailed investigations of the chaotic lasing features as well as the chaos onset are performed. The possibilities of the control of the laser output characteristics are discussed.

**Keywords**: fast-flow gas laser, dynamics of lasing, inhomogeneous excitation, chaos, bifurcation

1. Introduction

Light-dynamic phenomena in fast-flow lasers (FFL) such as the optical instabilities, self-oscillations, chaotic lasing are of the great interest from the general point of view [1-3]. Practical applications of these phenomena are based on the fact that they can be used for the elaboration of new methods for the laser output temporal characteristic control [4,5]. The lasers, which use the motion of the active medium (note that the majority of the modern technological lasers are of the similar type) demonstrate the specific mechanisms of the continuous lasing instability. The mechanisms are caused by the non-local mode of the medium and field interaction and by the origination of the flow created feedback between the various spatial parts of the cavity [6,7]. In the case the dynamics of lasing is defined in a high degree by the spatial gradients of the system parameters in the flow direction [7,8]. As pointed out in Ref. [8], the suitable choice of the spatial pumping profile may cause the various operating regimes of the laser with fast crossflow of the active medium through an unstable cavity. Not only cw lasing, but also various self-pulsing regimes and chaotic lasing were observed.

The results of calculations of temporal and energy characteristics of the laser are presented in our report. The main attention is paid on the chaotic lasing features and chaos onset. The detailed investigations of the mentioned problems for the similar types of laser systems were performed for the first time.

2. Calculation model

Our calculations were carried out in the frameworks of the model given in Ref. [8]. The cylindrical unstable cavity supplied with the same mirrors was considered. The geometric-optics approximation was adopted for the description of the field in the cavity. We assumed that the losses in the active medium and in the mirrors, and the losses q
* = lnM/2L*, associated with the beam broadening in the cavity are distributed uniformly over the cavity length *L* (*M* is the total round-trip magnification). As known, if the magnitude of *M* is not too large (*Ì* £
2) the model can be simplified by ignoring the variation of the field along the cavity length. By this reason a one-dimensional distributions of the intensity *I(x,t)* and the gain a
*(x,t)* along a transverse coordinate can be considered. Since the decay time t
* _{r}=(cq
)^{-1} *of

, (1)

where the coordinate x is measured from the cavity axis opposite to the direction of the flow. We described the active medium by the simplest equation for the gain with one relaxation constant

, (2)

where s
is the cross section of an optical transition; t
is the inversion relaxation time; *S* is the pumping rate. In the majority of our calculations the spatial pumping profile was assumed to be

, (3)

where *h*_{0} is the pumping inhomogeneity zone near the cavity axis (*h _{0}<< h*),

3. Lasing regimes

Numerical simulations of the lasing dynamics of the system described above yielded transient profiles of the field intensity *I(x,t)* and the gain a
*(x,t), *corresponding to* *different parameters of the system. An analysis of these profiles enabled us to identify four main regions with different values of parameters each, of them characterized by a specific dynamic behavior of the system. These regions are plotted in Fig.1 in terms of the coordinates (t
/t
* _{f}*, q

The continuous curve shows the boundaries of the cw lasing (region I) The regular self-pulsations occur in regions II and IV. The chaotic lasing was observed in region III. The dashed curve represents the threshold cavity losses q
* *above which lasing is impossible*.*

Fig.2 shows typical time dependencies of the self-pulsing oscillations intensity *I(h,t)* at the cavity output (the data presented in Fig.2 as well as in following figures were obtained for the relative losses of the cavity q
*/a
_{m}*=0,26).

In the region II (t
*/t
_{f}<*1) there are regular self-pulsations with a relatively high pulse repetition frequency (period

The existence of two regions of self-modulated lasing with different pulsation frequencies identifies two different self-modulation mechanisms. The first occurs in region IV (Fig.1) for t
*>>t
_{f}*. The inversion recovery is the result of replacement of the active medium in the cavity. The scale of the spatial inhomogeneity of the field intensity and the gain is in this case governed by the cavity aperture

4. Chaotic lasing

Along with study of time dependencies of the intensity and Fourier spectra we use Poincare maps constructing and the determining of the attractor dimensions to point out the features of the chaotic lasing. The detailed study of the chaos onset and its evolution by the change of the governing parameter were performed.

When Poincare maps were constructed the value of the output intensity was assumed constant *I _{out}=x_{0}=const* as a rule. The intensity at the cavity axis and the gain in the mean point of the cavity aperture were taken as the dynamical variables

Further increase of the parameter to the value t
*/t
_{f} *=1,693 leads to next bifurcation which corresponds to the transformation of the limit cycle into the torus with the close winding (Fig.3a). The fact indicates quasi-periodical dynamic of the system at this step. Fourier spectrum of the intensity shows two incommensurable frequencies n

While the parameter t
*/t
_{f} * increases beginning from t

This fact is known from Gavrilov-Shilnikov chaos scenario as pleating of the torus [9]. The process of the deformation is accompanied with the transformations of Fourier spectrum of the intensity. Main harmonics become broadening and combined harmonics originate. In the range of the parameter t
*/t
_{f}* =1.696-1.71 the time dependencies of the intensity demonstrate the evolution from ordinary beatings to very complex dynamical regime.

The process of torus distruction and the conversion into chaotic dynamics completes at t
*/t
_{f} *=1,71. Further increasing of t

The structure of the vibrations simplifies near the right boundary of the chaotic lasing at t
*/t
_{f} @
*2,3. Points of Poincare section cluster into groups (it looks like separated spots in Fig 3f) while the Fourier spectrum and time structure of the intensity remain relatively complex (Fig.5c). The simplification of Poincare map indicates the tendency of transformation of chaos into regular regime. If t

The embedding dimension of an attractor along with the set of Luapunov numbers considered to be used as quantitative characteristic of the disordering of a system [2]. To find the embedding dimension is the important task for distributed system governed by equations similar to (1)-(2) because the dimension value makes clear the number of effective degrees of freedom for the system. This number in turn shows the number of ordinary differential equations needed for full description of the system [2]. In addition the correlative dimension of the attractor was chosen to characterize quantitatively the dynamical regimes. Both the embedding and correlative dimensions were calculated by use of known mathematical procedure invented by P.Grassberger and I.Proccaccia (look, for ins.[2]). Calculated value of the embedding dimension was found minimum at the boundaries of the chaotic lasing region at t
*/t
_{f} *=1,69 and 2,5 where the dynamics of the system is described with the limit cycle (

5. Conclusive remarks

Our calculations show that pumping inhomogeneity doesn't reduce essentially the energy efficiency in this system. The quantum efficiency h
* *of utilization of excitation energy in self-pulsation and chaotic mode of lasing can be quite high for a small dimension of inhomogeneity zone (*h _{0}<<h*) and may exceed the value of h
~
0,6. The similar systems may be applied in practice. The above mentioned method can be used for designing the technological FFL with various lasing regimes. The required control of pumping profile may be technically quite easy performed in case of electric discharge laser with sectioned electrodes

Note that our task was to make clear the general qualitative features of the dynamical behavior of considered systems and calculations were based on highly simplified model. The developed kinetic models of active medium should be applied to predict accurately the performances of real lasers such as ÑÎ_{2} and ÑÎ FFL.

This work was carried out with the support of the Russian Ministry of Science (project No 1.60, Laser Physics)

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