An iterative model based on the LambertW function was developed for estimating the fundamental mode parameters of resonators with saturable gain guiding. The process of pulse buildup in passively Q-switched, end-pumped lasers was analyzed. The effective ABCD cavity matrix for consecutive round-trips was calculated, taking into account spatially variable saturated gain in an active medium and absorption bleaching in a saturable absorber. The twofold decrease in beam width, as compared with the fundamental mode of the bare cavity, was demonstrated. The application of such a model for resonators with other nonlinear elements is feasible.
©2003 Optical Society of America
Determining the fundamental mode parameters of the laser cavity under real pumping conditions is one of the oldest and still vital problems in laser physics (see, e.g., Ref. ). The radially variable gain profiles [2–4] as well as heat-source densities [5,6] occur as a result of focusing of the pump beam in the majority of diode-pumped lasers. In analyzing this type of laser in the framework of the space-dependent rate-equation model [2,3,7], we assume that laser mode parameters are known a priori. However, it is well known that the simplest microchip lasers  have a fundamental mode width that is dependent on the thermal guiding effect of the pump beam width and absorbed pump power. The thermal guiding effect responsible for mode structure in lasers operating near threshold seems to be an unsatisfactory mechanism for explaining the properties of longitudinally pumped active media with high gain. Thus we developed complementary models that account for gain guiding , the Kerr effect , and gain-related effects [10–13].
Our aim in this paper is to analyze the influence of gain guiding and absorption bleaching effects on the properties of passively Q-switched lasers. In Section 2 an iterative model is developed, with the gain saturation taken into account, and applied to the analysis of a laser operating in a free-running regime . In Section 3, a similar procedure for cavities with variable gain and saturable losses is derived and applied for the analysis of passively Q-switched lasers. Conclusions are drawn in Section 4.
2. Iterative model of a cavity with radially variable saturated gain
The main limitation of the space-dependent rate-equation model is the near-threshold approximation that results in neglecting gain-saturation effects [2–3]. In the model including spatially variable gain saturation developed by Kapoor et al.  it is assumed that the fundamental laser mode is not changed. The ABCD model of a cavity with gain guiding was first proposed by Salin and Squier . The novel model for high intracavity intensities assuming the transverse Gaussian distribution of a saturated gain profile was proposed recently by Denchev et al.  and applied to the analysis of unstable resonators. For our purposes we decided to use a different approach, recently proposed Grace et al. . The main concept consists in application of the analytical solution of a saturable gain (or absorption) equation for a homogeneously broadened medium as follows,
where g is the small-signal logarithmic gain coefficient and I r=I/I sat is the relative intensity. The general integral of this type of first-order ordinary differential equation is known as the LambertW function W (see, e.g., Ref. ) and is defined as
Following the results of Ref.  we can express the relative intensity I r,1 after passage through an active medium of length l as an explicit function of the incident intensity I r,0 and the small-signal logarithmic gain gl as follows:
Applying Eq. (2), we have the output intensity expressed as
Knowing the output intensity at any point for the given incident one and small-signal gain, we can calculate the output profile for any given incident beam and gain profile. Further, Grace’s concept consists of approximating the output beam by the Gaussian one and calculating the parameters of the virtual “gain diaphragm” ABCD matrix M SG as follows:
where λ is the wavelength, w SG is the effective radius of the gain diaphragm,
w inp is the radius of the incident Gaussian beam, and w out,sg is the radius of the output approximated Gaussian beam. The value of w out,sg can be determined according to Siegman’s definition  with the second moments of the output intensity profile:
At the starting point we calculate the bare cavity matrix M BC including the effective thermal lensing power of a gain medium to determine the incident Gaussian beam profile at the point of entrance. We take the small value of the incident intensity magnitude compared with the saturation one. In each step of the iterative procedure we calculate the output intensity profile after passage through the gain medium according to Eqs. (3) and (4) and determine the effective gain matrix M SG; next, we pass the beam through the cavity, applying the product of both matrices M SG M BC. At each step we introduce the logarithmic passive losses of cavity δpas, multiplying the intensity profile at the output mirror by the factor exp(-Γpas). Because of the saturated gain profile and passive losses, the peak intensity converges with the number of round-trips to finite value, for which the procedure stops (see Figs. 2 and 3).
Let us note that such a procedure is similar to the Fox–Li approach (see, e.g., Ref. ); however, instead of performing an exact calculation of diffraction integrals for each round-trip, we pass the Gaussian beam modified by the gain medium through the cavity by applying Kogelnik’s ABCD rule. As a result, we dealt with the “effective” Gaussian beam of a cavity whose parameters change in each round-trip, converging to a stationary value. The solution for a laser operating in the free-running regime depends both on parameters of bare cavity M BC and on the magnitude of gain and its profile and existing linear losses.
3. Iterative model of a passively Q-switched laser
Equation (1) for g=αabs<0 also describes the change of intensity in the saturable absorber of the small-signal absorption coefficient αabs. Thus a similar iterative procedure was developed (see Fig. 4) with the additional components describing the transformation of the beam in the saturable absorber. The passive Q switch is placed at the opposite end of a cavity; thus the round-trip matrix M BC was divided into two components, M BC1 forward from the gain medium to the absorber and M BC2 backward to the gain medium. In the procedure, for the free-running regime we were looking for a stationary solution, assuming that the shape of gain profile does not change (i.e., radially variable gain depletion was neglected). Now we intend to describe the process of pulse formation in a passively Q-switched laser, and so both radially variable depletion of gain and bleaching of saturable absorbers should be taken into account.
We have found that application of the integral definition of the beam radius Eq. (7) leads to some numerical problems for depleted gain profiles and wide integrals limits. Thus we decided to use the 1/e 2 definition of beam width in the model, calculating relative intensity in two points and assuming a Gaussian approximation. Let us note that saturation intensities of gain and absorber media can differ significantly. The condition for efficient passive Q switching (see Refs.  and ) states that the ratio of saturation intensities between gain and absorbing media α=σa/σe should be much greater than 1. Thus, to build the iterative procedure for modeling the passively Q-switched laser, we should modify the function of the gain/absorption profile, adding the depletion/bleaching effect, and scale the relative intensity before and after the passive Q switch by the ratio of saturation intensities of the gain medium and absorber.
Examples of pulse formation and changes of beam width are shown in Fig. 5. As a result of rapid increase in instantaneous intensity during pulse formation, the saturable absorber acts as an instantaneous “soft” diaphragm with a profile and width changing in each round-trip. The effective width 2W SA of the passive Q-switch diaphragm rapidly decreases (see Fig. 5, dashed curve) causing narrowing of the pulse width during the pulse-formation process. Thus the effective width of the pulsed mode is narrower compared with the value of the bare cavity fundamental mode width 2W 00 (see Fig. 5, dotted curve). The change of instantaneous beam width in the process of pulse formation depends on the parameters of the Q switch (initial transmission) and parameters of the bare cavity. Examples of quasi–two-dimensional intensity maps of pulse formation for two passively Q-switched lasers having the same resonators but different initial saturable losses are shown in Fig. 6.
Our calculation shows that twofold narrowing of a pulse beam width (see upper plot of Fig. 6) compared with the value calculated for the bare cavity is possible. Asymmetry of the pulse intensity profile depends on the level of initial saturable loss. Moreover, the pulse beam width can change as a result of additional effects (e.g., thermal lensing, nonlinear losses). It was found that for near-confocal resonators (i.e., cavities with the trace of their ABCD matrix near 0) the changes in effective pulse beam width are the lowest and they depend mainly on initial losses of the Q switch. In contrast, for a passively Q-switched laser operating near the stability limit, the effective pulse width significantly varies with bare cavity parameters.
Such an effect was numerically investigated for a flat–flat cavity for which the thermal lensing in the active medium (proportional to pump power) shifts it across the stability diagram. The passive Q switch was placed near the gain medium. Such a scheme is typical for passively Q-switched microchip lasers. The calculation results are shown in Fig. 7.
We can see that a decrease in beam width is nearly constant in the almost whole range of pumping rate as a result of beam narrowing in the saturable absorber. That means that, as a rule, average peak power density is higher (~50% for the case shown in Fig. 7) compared with estimates performed for the bare cavity data.
A model of the saturated gain guiding effect has been derived and applied for passively Q-switched lasers. The simple iterative procedure was proposed to calculate effective fundamental mode parameters of a cavity under gain guiding for given bare cavity ABCD matrix and pumping parameters, including gain saturation and passive cavity losses. Application of such a method for resonators of passively Q-switched lasers and cavities with other nonlinear elements such as OPO or Raman crystals is possible. For free-running and passively Q-switched lasers, the change of beam width compared with fundamental mode width is in the range of 50%. The gain guiding effect should be taken into consideration in designing cavities destined for high-gain, high-power lasers, especially with decreased thermal load. The iterative model based on the LambertW function can be applied for low-and medium-power passively Q-switched microchips, lasers with intracavity conversion, and the like.
This study was financed in part by the Polish Committee for Scientific Research under projects 0T00A06519 and 4T11B02724.
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