1. Introduction

During the past few years there has been an ever increasing interest in experimental studies of passively Q-switched(PQS) microchip lasers. A detail review of devices using solid state saturable absorbers is given in [1], [2]. However, the same trend is not evident in theoretical modeling[3, 4, 5, 6] studies. Only recently[7] has there been a comprehensive theoretical and numerical investigation of the pulse dynamics. This study included the limit-cycle analysis and presented a successful numerical analysis of the differential equations in the numerically challenging region where the pulse separation is six orders-of-magnitude larger than the pulse width. However, one area not thoroughly addressed was the pulse shape. In the following we derive simple design formulae for the peak power, pulse width, repetition rate, pulse energy and the pulse shape for microchip lasers which are derived with the experimentalists in mind. These approximate solutions are anchored to our previous simulations[7] which agree with experiment[8]. Finally, since our solutions are approximate we also discuss their limitations.

Initial studies of Q-switching were concerned with active[9] and then passive switching[10, 11, 12] of single pulses. These studies described numerically the pulse shape for various losses, and cross sections. A later paper[13] devoted a small section to pulse trains based on single pulse results. And, finally, a very recent paper[14] divided the pulse into four regions and deduced some design equations. Their development is for a semiconductor saturable absorber which has a decay rate much faster than the gain medium. Also, they restrict the saturable absorber unsaturated losses to be much less than the outcoupling losses, and finally they consider operation just above threshold. These assumptions lead to a symmetric pulse shape, for example. Subsequently, we showed[15] that their major results are contained within our earlier work[7]. In the following we relax these restrictions and develop design equations which include solid state saturable absorbers as well. Generally, the limit-cycle solution for passive Q-switching is described by a set of five coupled equations for the period, and the initial and final values of the laser gain and saturable absorber gain[7]. Our problem here is to reduce these equations to a reliable and managable form for solid state abosrbers.

The following development relies on our recent work[7]. Here, however, we develop the gain quick-pulse differential equation in more detail. In fact we are able to integrate it after introducing several reasonable approximations which results in the gain as an implicit function of time. The integration of the gain equation depends on the initial laser gain before the pulse. This, as just mentioned, is coupled to the period and the final gain after the pulse along with the initial and final saturable absorber gains. In this paper we show that the conditions under which these equations reduce to a single equation correspond to solid state saturable absorber parameters. We compare the approximate solution to the exact numerical simulations.

We close by restating the difference between previous singel pulse work in lasers and our studies. Pulsing in lasers is intrinsically a problem in laser instability and is physically and mathematically quite different from the single pulse situation. This difference is partly embodied in the period equation which is critical in determining the pulse characteristics.

2. Theory

We treat the gain medium as an ideal four-level system where the decay from the lower laser level to the ground state is nonradiative and very fast[7, 16, 17]. This process is described by the dimensionless gain G≡2L_gγeN ₂. N ₂ is the upper lasing level concentration, and N ₀ is the ground state concentration with N ₀+N ₂=N_T where N_T is the doping concentration. The gain length is L_g and the laser effective emission cross section is γ_e. On the other hand, we treat the saturable absorber as a two-level system described by the dimensionless gain G_s≡2L_s(σ_eN ₂-σ_aN ₁) where N ₂ is the upper level concentration and N ₁+N ₂=N ₀, where the total doping concentration of the saturable absorber is N ₀. The saturable absorber length is L_s and the laser flux is coupled through the effective emission cross section σ_e and the effective absorption cross section σ_a. Finally, the gain decay rate is A_g=1/τ_g and the saturable absorber decay rate is A_s=1/τ_s. The pump effective absorption cross section is γ′_a.

Using these notations, the upper level rate equation for the gain medium is

which yields

where Γ=γ′_aγ_e, R(t) is the laser flux, and P is the cw pump flux. The saturable absorber is a two-level atom with the rate equation for the upper lasing level given by Ṅ ₂=-Ṅ ₁=R(_γaN ₁-_γeN ₂)-N ₂=τ_g. Thus, the differential equation for the dimensionless saturable absorber gain G_s is

where σ ₊=σ_e+σ_a. Accompanying eqs. (2,3) is the differential equation for the sum of the forward and reverse laser fluxes R(t). In the mean field approximation this sum obeys

The cavity round trip time is τ_c=2L_c/v where v=c/n, c is the speed of light in vacuum, n is the index of refraction, and L_c=L_g+L_s. The laser outcoupling reflectivity is r and all other losses are included in L. Using the above equations we form the following three dimensionless differential equations

where I, D, and D̄ are the dimensionless laser intensity, dimensionless gain, and the dimensionless saturable absorber inversion defined by

The parameters in eq. (5) are related to the physical constants through

During the pulse when, I>>1, the three above normalized differential equations can be reduced to a single differential equation for the gain[7]. This is accomplished by dividing the second equation in eq.(5) by the third equation and integrating. This step allows elimination of D̄ when the first equation in eq. (5) is divided by the second equation. Integrating this last division gives

and thus the dD=ds equation becomes

D_b is the dimensionless gain inversion just before the pulse starts; we discuss this in more detail later.

Our objective is to develop approximate, yet reliable, analytic solutions for the microchip laser pulse intensity. To this end, we rely on the results of our numerical simulations of eqs. (9,10). These show that if m is set equal to unity then the pulse shape is very close to the actual pulse shape obtained when microchip constants are used. Note that for the microchip experiment the exponent m=αγ̄/γ=σ ₊/γ_a=3.2. The choice of m=1 allows us to solve eq. (10) analytically. The restriction m=1 requires the laser absorption cross sections of the gain to equal that of the saturable abosrber. This constraint, however, gives good results since the pulse shape is not strongly dependent on m, rather the shape depends on other parameters as we discuss near the end of the next section. Figures (1b,c) show the pulses for the m=3.2 and the m=1 solutions, respectively. The approximate m=1 pulse (curve c) turns on slightly sooner and is slightly higher than the exact solution (curve b). This small shift is inconsequential since the period is roughly six orders-of-magnitude greater than the pulse width. Simulations of eqs. (5) further show that at the initiation of the pulse the saturable absorber is bleached, that is D̄_b=-1, and that the gain is extracted, i. e. D_a<<D_b where D_a is the gain after the pulse.

 

Figure 1. Simulated pulse shapes as a function of time. (a), the three differential equations, eq. (5); (b), the gain differential equation for the actual microchip pa- rameters m=3:2, eq.(10); (c), gain differential equation for m=1, eq. (10); (d), the implicit solution, eq. (19).

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With these results in mind we set m=1 and proceed to derive various approximations to the exact pulse shape including the peak power and the FWHM. For m=1 the normalized gain differential equation, eq. (10), can be written as

We solve the above equation using integration by parts. The integral generated in this process can be approximated with little error by noting that most of the contribution comes from the region near η=1 where ln η≈2(η-1)=(η+1). Thus, continuing with the algebra, we obtain the implicit solution

accompanied by the intensity equation

where C=AD_bβ. The pulse shape embodied in eq. (12,13) is very close to the full simulations of eqs. (5) and in a moment we will make this comparison. But before doing so we derive equations for the peak power I_p and the pulse width (FWHM). At the pulse peak the intensity differential equation gives -1+AD_p+ĀD̄_p=0. Also, at the peak our numerical simulations show that the saturable absorber is transparent, i. e. D̄_p≈0. Thus, D_p≈1/A. Inserting this back into eqs. (12,13) gives the peak intensity I_p as

Consequently, the values for η at the halfwidth are obtained from

This equation has two roots which determine the half-width inversions during the rise of the pulse, η_r, and in the pulse tail, η_t. In the tail region η_t is small and in the rise region η_r is near one and thus eq. (15) yields the approximate equations

The half-width is then determined from eq. (12) by inserting η_r,t and forming the FWHM Δ_sFW=Δ_st-Δ_sr. Completing this task gives our final result

As we will show the FWHM given by eq. (17) is within 3% of the microchip laser experimental values[8].

Even though the implicit equations, eqs. (12,13), are approximate they are never-the-less a good representation of the exact pulse shape. However, our development would be more informative if we derive an approximate explicit form revealing the growth and decay time scales. To this end we find a piecewise solution. In the tail region where η<<1 eqs. (12,13) show that the intensity I_t is given by

This decay is relatively slow and depends only on the cavity decay rate. Its typical time scale is s-s*=O(1); s* is the integration constant and is unimportant in this development. In the pulse rise region substitution of the expansion ln η≈2(1-η)/(1+η) into eqs. (9,10), for m=1, followed by integration gives

Thus, the initiation of the pulse is fast and is clearly controlled by the pump in a complex manner. Its typical time scale is s-s*=O(1/(AD_bβ))<<1.

Next we briefly discuss the period T and initial inversion D_b in PQS lasers; for a more detailed discussion see reference [5]. Between pulses I is much less than one, consequently the gain equations in eq. (5) integrate into exponentials. Furthermore our simulations assure that after the pulse D_a;D̄_a<<1. Also, the fact that the period is much longer than the saturable absorber decay time implies that before the pulse D̄_b≈-1, as stated before. These assumptions lead to the inversion equation D=1-exp(-γs). After this equation is inserted into the flux equation in eq. (5) and integrated the resultant exponential function for the intensity I is small until s reached S where S satisfies

where X=_γS. When eq. (20) is satisfied a rapid pulse growth is initiated. This defines the period S. As we will see the value of S agrees with the experiment[8] to within a few percent.

In summary, modeling a particular experiment is straight forward. After all the physical constants and the pump flux have been identified, one forms parameters A, Ā, , and γ̄ using eqs. (6, 7, 8). Next, eq. (20) is solved for X from which the period T, and initial population inverison D_b can be calculated through T=X_τc/α_L and D_b=1- exp(-X), respectively. After D_b is determined the peak intensity is obtained using eq. (14) and the half-width using eq. (17). Finally, the pulse energy ε can be found through[7]

where t is the outcoupling intensity transmission, t+r=1, and hv is the laser photon energy. Of course, all this can be done as a function of the pump power. Finally, the pulse profile can be generated by solving one of the differential equations eq.(10) or eq.(11), or by plotting the implicit solution as a simple loop over η, see eq. (12). Alternatively, one could also use the explicit solution given by eq. (19).

3. Example

Now we model the microchip laser experiment[8] which uses Nd:YAG as a gain medium of length L_g=.05cm butt coupled to a Cr⁺⁴:YAG saturable absorber of length :025cm. The outcoupling reflectivity is given as r=94%. The Nd doping is at 1.6at.%. The absorption coefficient[8] of Cr is estimated at 6cm^-1 for a wavelength of 1.064µm. The experimental report is brief but lists a laser repetition rate of 6kHz, pulse width of 330ps, pulse energy of 11µJ, peak pulse power of 27kW at a peak intensity of 180MW/cm². This behavior is observed for a pump power of 1.2W with a threshold at .8W. The laser emits at a wavelength of λ_l=1.06µm and has an estimated beam waist of 70µm. The pump with a wavelength of λ_p=808nm is double passed since it reflects from the Nd-Cr interface.

Prior to discussing the predictions of this paper we summarize our previous modeling[7]. The full simulation used a patch technique[7] applied during the inter-pulse period when the laser energy is negligible and, consequently, the two inversion equations are easily integrable. In this work we estimated the values of the dimensionless parameters [7] as

Moreover, for a pump power of 1.2W, A=10.48 and the threshold (A_th≡1+Ā)=4.96. Also, we found D̄_b=-1, and D̄_a=10^-6, D_a=10^-4, and D_b=.759. Furthermore these simulation of eqs. (5) gave a period of 5.9kHz, a peak power of 29.8kW, the pulse width as 343psec, and a pulse energy of 11.9µJ. These values are very close to the above mentioned experimental values. Figure 2 shows the simulated pulse shape, the solid curve, compared to the experimental results shown as the dashed curve. Note that since our simulation does not include noise, the intensity drops to zero much more rapidly than in the experiment[8]. However, the simulation does capture the pulse rise and decay.

 

Figure 2. Pulse shape as a function of time in 200psec/division. Experimental, dashed curve, simulation, solid curve.

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Now we return to fig. (1) which summarizes the major results of this paper. For the sake of reference we repeat the full simulation[7] shown in fig. (2) as curve (a). As mentioned earlier this simulation gives a value of D_b=.759. At this point we employ our approximate equations. First, inserting the above values for A, Ā, γ, and γ̄ into eq. (20) gives a period of 5.64kHz, which is within 7% of the experiment. This value of the period leads to a value of D_b equal to .7584. We use this value as the initial condition for solving the inversion differential equation, eqs. (9,10) for the two values of m=αγ̄/γ=3.2 and m=1 which yields curves (b) and (c), respectively. The final curve (d) is our implicit solution [see eqs. (12,13)]. These four simulated pulse shapes are very close to one another. The two rigorous solutions, curve (a) and curve (b), are almost identical. The m=1 differential equation, eqs. (9,10), over estimates the peak and the width by about 7%. Our implicit solution, eqs. (12, 13), closely replicates the simulated solutions in curves (a) and (b). In fact, further simulations show that the pulse shape remains within 10% of the experimental values for 1<m<4.

The merit of our analysis lies in the application of our approximate equations. Now that we have determined the period and D_b, eq. (14) gives a peak intensity of I_p=206MW/cm², eq. (17) gives the half width of ΔsFW=321psec, and finally eq. (21) gives the energy ε=12.1µJ. The peak intensity matches the experiment to within less that 15% while the others agree to less than 10%.

Further information about the pulse shape is shown in fig. (3). This figure shows that the pulse rise time (1=(C-2)), obtained from eqs. (18,19), is a strong function of pump power. However, as we have mentioned the pulse decay depends only on the cavity decay time which is 216psec. Thus, as the pumping is changed the only pulse shape changes occurs in the early part of the pulse. Note that for large pumping eq. (17) shows that the minimum pulse width is governed by the cavity and is given by (τ_c/α_L) ln(2). This value, of course, is not a practical limit since it corresponds to a delta function like pulse growth.

 

Figure 3. The pulse rise time as a function of pump power.

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We conclude this comparison by mentioning that further numerical simulations show the above approximations are good for as much as 10 times above threshold before the error between the simulations and the approximations exceeds 20%. Also. these simulations show that the pulse shape, and its associated parameters such as the pulse energy, are mild function of m=σ ₊/γ. Specifically for.5<m<5 the pulse energy agrees with the m=3.2 solution to within about 15%. However, the pulse is strongly dependent on τ_g, τ_s, the pump small signal gain g_g=2_γN_TL_g, and the laser small signal gain g_s=2σ_aN₀L_s in the saturable absorber. Specifically eq. (21) shows that ε∝ (g_gP_γeτ_g-g_sτ_s/τ_g)/α_L.

Finally, we turn to a discussion of a recent experiment[14] which uses either Nd:YVO₄ or Nd:LSB as a gain medium and a SESAM (seminconductor saturable absorber mirror). When we compare their pulse differential equations with our normalized pulse form, eqs.(5), we find that: γ̄=9.3×10^-2, γ=3.7×10^-7, Ā=.36, A=4.08, α=4.×10^-3, and that m is large with a value of m=1000. After these values are inserted in the exact gain differental equation eq.(10), for D̄_b=-1, and in the m=1 differential equation eq. (11) we find that the two pulse shapes are virtually identical. The reason that the m=1 solution works so well is because the [1-η^m] term is multipled by 1/m, or β≈1. An absolute comparison with their experiment[14] is difficult since we must know cross sections, doping levels, losses, etc., and these are not readly available. Finally, as we mentioned in the introduction, we derived their formulas[15] published subsequent to their report[14].

4. Summary

We have developed accessible approximate analytic equations which characterize the pulse shape in microchip lasers. We have shown that it is straight forward to calculate the pulse period, peak power, pulse width, and pulse energy along with the pulse profile.

We would like to acknowledge Thomas Erneux at the Université Libre de Bruxelles, Optique Nonlinéaire Théorique, Campus Plaine, C. P. 231, 1050 Bruxelles, Belgium for his suggestions.