## Abstract

In this work we present a numerical study of a three-level laser containing a polarizable saturable absorber inside the cavity. This model allows us to study the kinetics of solid-state lasers in a general form. The stability of Q-switching regime is analyzed by means of numerical solution of rate equations; and main results of our analysis let us suggest that under certain conditions, the laser may pass from unstable relaxation oscillations to a stable CW operation by changing the mutual orientation of the absorber and polarizer, or by choosing the pump level.

© 2001 Optical Society of America

In recent years, considerable efforts have been directed towards implementation of a Cr^{4+}:YAG crystal in practical Q-switching schemes of neodymium lasers [1–4]. This is due to both excellent bleaching and thermo-optics characteristics of the crystal and to the latent anisotropy of its absorption. Although the impact of this anisotropy on the Q-switching regime of a solid-state laser and even on type-II second-harmonic generation was considered recently [5–7], the stability of the CW regime of operation of the laser containing such a polarizing saturable absorber as Cr^{4+}:YAG crystal, has not been addressed so far.

The presence of a wide amplification bandwidth along with an intra-cavity polarizing absorption means a variety of operation regimes, which results from the balance between amplification and anisotropic absorption. The kinetics of the laser that incorporates the absorber with a self-induced anisotropy may be extremely rich, including self-organizing oscillations. Experiments that have been performed so far [1–3,8,9], allow us to assume that a solid-state laser containing a polarizing saturable absorber can be described within a three-level approximation. This approximation fits extremely well for such solid-state lasers as ruby and erbium ones as well as for a neodymium laser operating at a 940-nm laser transition. It may also be applied to the neodymium laser operating at 1.06 µm if we restrict ourselves to the pulsewidth longer than 1 ns; in this case, it is possible to neglect the effect of self-limiting of the lasing transition which is due to a four-level nature of this laser.

The model presented here is for the solid-state laser shown in Fig. 1, which is formed by a reflecting mirror (1) and output mirror (2). It also contains a three-level lasing medium and saturable absorber (SA) in the form of Cr^{4+}:YAG crystal which [001]-axis is assumed to be parallel to the longitudinal axis of the laser. The other two axes of the absorber, [100] and [010], lie in the transverse (*x*-*y*) plane of the cavity and are oriented with respect to the *x*-axis at the angles *θ* [100] and *θ*+*π*/*2* [010], correspondingly. In turn, the orientation of the *x* and *y* axes are chosen such that the losses which are due to the presence of the partial polarizer (PP) (5) inside the cavity has minimum along the *x*-axis and maximum - along *y*-axis. The PP has the form of a glass plate forming the angle *β* with the axis orthogonal to the longitudinal axis of the cavity. Therefore the linear anisotropy of the cavity is due to the PP specific orientation (angle *β*), whereas its nonlinear anisotropy is due to the latent anisotropy of the SA and so to be determined by the angle *θ*. We further assume that the laser output has an elliptic polarization, and the ellipse azimuth is denoted as the angle *φ.*

That is the laser is characterized by the three levels consisting of the ground state with a population density *N _{1}*, an upper laser level with population density

*N*, and a pump level with population density

_{2}*N*. In addition, the laser contains a two-level saturable absorber of density of ions

_{3}*n*=

_{0}*n*+

_{1}*n*, where

_{2}*n*and

_{1}*n*corresponds to the ground-state population of ion dipoles oriented along [100]- and [010]-axis, correspondingly. The total density of the lasing states is

_{2}The temporal behaviour of the laser is described by coupled-rate equations having the following form:

$$F\left(1n\frac{1}{r}+{\alpha}_{x}{\mathrm{cos}}^{2}\varphi +{\alpha}_{y}{\mathrm{sin}}^{2}\varphi \right),$$

for the density of the pump level

for the upper laser level

and for the absorber ground states

Here *γ* is the spontaneous emission capture ratio for photons trapped inside the cavity, *g _{a}* is the gain factor for the active medium,

*g*is the absorption factor for the polarizable absorber,

_{s}*τ*is the decay time of level

_{32}*N*to level

_{3}*N*,

_{2}*τ*is the spontaneous decay time of level

_{sp}*N*,

_{2}*τ*is the decay time of absorber state

_{s}*n*

_{1,2},

*τ*is the transit time for lasing atoms pumped from level

_{p}*N*to level

_{1}*N*,

_{3}*r*is the reflection coefficient of the output mirror,

*α*

_{x,y}are the losses caused by the PP along x- and y-axis of the cavity, correspondingly.

These rate equations have simple intuitive meanings. The first term on the right-hand side of Eq. (2) indicates that the photon density in the laser cavity grows due to spontaneous emission of photons, with a growth rate that is proportional to the population in the upper laser level. The next term accounts for stimulated emission, which is proportional to the photon density that is already present and to the excess of lasing atoms or ions in the upper laser state over the ground state. The remaining terms describe photon decay due to the loss on the mirrors and on the PP, and due to the presence of the saturable absorber.

In order to make the problem posed we need to describe the state of polarization of the laser. Assuming that this is always an eigenstate corresponding to the minimum of the total intracavity losses yields the following relationship for *φ*(*t*):

Where the PP partial losses difference of (*α _{y}*-

*α*) can be found by means of the Fresnel formulas for a tilted glass plate.

_{x}The rate equation (3) for the pump level contains the optical pump power *P* that is flowing into the laser cavity. This power is normalized with respect to the threshold power *P _{th}* so that only the relative power

*P*/

*P*is needed for practical calculations. The second term in (3) accounts for the decay process from level

_{th}*N*to level

_{3}*N*. No other decay path is is provided for the population

_{2}*N*. The remaining rate equations (4) and (5) should be self-explanatory.

_{3}Exact analytical solutions of the time-dependent rate equations can probably not be obtained. Fairly good insight into the behaviour of the laser, which is modelled by these equations, can be gained by the numerical integration of the set of coupled nonlinear equations (2) through (6). Examples of numerically obtained results are presented later.

With a saturable absorber in the cavity, the laser may not be able to settle down to a steady-state but may pulse indefinitely if it is continuously pumped above threshold. However, formally, a steady-state solution can always be obtained by setting all time derivatives equal to zero. A stability analysis must then decide whether the obtained steady-state solution is stable or unstable.

Contrary to its importance in, e.g., semiconductor lasers, the spontaneous emission factor *γ* in solid-state lasers is small and has practically no influence on the steady-state solution. For this reason, we set *γ*=*0* for the purpose of computing the steady-state. However, for the dynamical solutions this factor is vital to allow the photon population to build up from zero values.

The steady-state solution of Eqs. (2) through (5) is obtained by formally setting *d*/*dt*=*0*. In addition, we defined a normalized pump power as

The steady-state solution can then be expressed as follows:

$${\left(\frac{{g}_{s}}{{g}_{a}}{n}_{0}\frac{1+\xbd{g}_{s}{F}_{s}{\mathrm{sin}}^{2}2\left(\theta -\varphi \right)}{1+{g}_{s}{F}_{s}+\xbd{g}_{s}^{2}{F}_{s}^{2}{\mathrm{sin}}^{2}2\left(\theta -\varphi \right)}+\frac{1}{2{g}_{a}}1n\frac{1}{r}+\frac{{\alpha}_{x}{\mathrm{cos}}^{2}\varphi +{\alpha}_{y}{\mathrm{sin}}^{2}\varphi}{2{g}_{a}}\right)}^{-1},$$

$$\frac{1+2G}{2+3G}\left(\frac{{g}_{s}}{{g}_{a}}{n}_{0}\frac{1+\xbd{g}_{s}{F}_{s}{\mathrm{sin}}^{2}2\left(\theta -\varphi \right)}{1+{g}_{s}{F}_{s}+\xbd{{g}_{s}}^{2}{F}_{s}^{2}{\mathrm{sin}}^{2}2\left(\theta -\varphi \right)}+\frac{1}{2{g}_{s}}1n\frac{1}{r}+\frac{{\alpha}_{x}{\mathrm{cos}}^{2}\varphi +{\alpha}_{y}{\mathrm{sin}}^{2}\varphi}{2{g}_{s}}\right),$$

Except of (8), all other equations are explicit solutions that require only knowledge of steady-state quantities that precede them in order of appearance. The equation for the steady-state photon density is an implicit equation, since the unknown quantity F_{s} appears on its right-hand side. However, even though the dependence of the right-hand side of (8) on F_{s} is relatively weak and the method of, e.g., successive approximations seems to be applied, this procedure is cumbersome so that we prefer to find the solutions by a numerical one.

Therefore, we use numerical routine in order to find boundary between stable and unstable regions of the laser (Fig. 3). The laser parameter we use for this calculation are log_{10}g_{s}/g_{a} along the horizontal axis and log_{10}θ-β/θ on the vertical axis. The curve is computed for a relative pump power of P/P_{th}=3. It is interesting to note that there is an unstable region within which the laser will pulse, that is surrounded by stable regions. For very small values of the relative absorber concentration the laser is seen to be always stable. But for the larger values of this ratio it is readily seen a region of instability. For the density of the same value as the density of lasing three-level particles, atoms or ions, the devise does not raise. The diagram clearly shows that for larger angles of the relative absorber orientation, the laser is also stable. These can be explained by the fact that, in these cases, the absorber remains essentially in its ground state so that bleaching does not occur. For intermediate values of θ-β/θ, the absorber can be bleached by high photon concentration, but it is able to return to its ground state sufficiently quickly and to turn the laser off as it reaches the low point of its relaxation oscillations. For very small values of θ-β/θ it is again stable, because now the absorber remains bleached and does not have time to turn the laser off.

Figure 4 represents a stability diagram of a different kind. In this case we fixed the value g_{s}/g_{a}=0.1 and let the pump power vary along the horizontal axis. Again, there is an unstable region surrounded by stable regions. But this diagram shows that under certain conditions, the laser can pass from pulsing to stable CW oscillations simply by raising the pump power and by mutual orientation of the absorber and PP.

Figs. 5 and 6 represent typical pulsing behaviour of the output of the laser. It is readily seen the self-induced pulsations of the output power. In particular, it follows that the quasi-period of pulses generated is equal to the lifetime of the photon in the cavity, and the output pulsewidth is defined by the ratio of the length of the lasing element to that of the absorber. Fig. 5 shows the period doubling of the output train. The only difference between the parameter sets behind these two figures is the value of the self-induced anisotropy of the saturable absorption. The phase difference is equal to 1.5·10^{-3} in the first case and 10^{-3} in the second, correspondingly. It is worth noticing that if being taken at a longer time scale, the both trains look like a decaying sequence of pulses.

The appearance of such a complicated output may have the following interpretation. The spontaneous decay of the upper lasing level along with mirror reflection cause a gradual growth of the energy stored in the system as well as the anisotropy of losses. Then the SA bleaching triggers the instability process leading to the output pulse formation and subsequent draining away of the energy stored. However, the linear and nonlinear rotation of the vector of polarization of the pulse backreflected from the output mirror makes it fulfilled the threshold condition, but for the other direction of the polarization vector. As a result, the second pulse is emitted. Since the time of the generation is uncertain due to its spontaneous nature, the interference between the two pulses occurs and its character depends on the energy of either of individual pulses that, in turn, is defined by the losses brought by the PP and SA into the cavity.

In conclusion, we have shown theoretically that two basically different regimes may be observed in a solid-state laser Q-switched by a Cr^{4+}:YAG saturable absorber. We have developed a set of rate equations in order to describe the evolution of such a laser, in which the orientation-dependent interaction between the photon flux, saturable absorber, and partial polarizer have been taken into account. It is shown that inserting a polarizing saturable absorber may cause a unstable operation of the laser. The stationary solution is found and stability of the laser is analyzed by means of numerical solution of rate equations. It is also demonstrated that the laser may be turned from unstable relaxation oscillations into a stable CW ones either by means of mutual orientation of the absorber and partial polarizer or choosing the pump level. The coexistence of different types of behavior in the same nonlinear laser system is a remarkable feature that deserves further study. Our model may find application in other solid-state laser systems Q-switched by an anisotropic Cr^{4+}- doped crystal [10]. Finally, the new regimes of quasi-chaotic pulse train generation and period doubling may find also application in optical communications with chaotic signal encoding/decoding at a 100 MHz bit rate.

## Acknowledgements

This research was partially supported by the CONACyT grants 34684-E and I32892-U. The work of I.V.M. was also supported by the CONACyT through the Catedra Patrimonial program. We are pleased to acknowledge useful discussions with V.A. Vysloukh and technical assistance with the manuscript from A.V. Kir’yanov and B. Salazar-Hernandez.

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