1. Introduction

Nanosecond laser pulses are widely used in areas such as materials processing [1], spectroscopy [2], plasma research [3] and medical applications [4], where high pulse energies are required, and also in lidar systems [5]. The most established technology for achieving such pulses is active Q-switching. Since high pulse repetition rates are desirable for many applications [6], current research efforts are focussed on building actively Q-switched lasers which can operate at up to 1 MHz. At such frequencies and usual pump powers, large variations in output pulse energies which, in most cases, are detrimental to the intended application can ultimately damage optical elements inside the cavity. These energy variations are primarily the result of two distinct effects. Firstly, spontaneous emission is a stochastic process with known properties [7] which influences the shape of a pulse as well as its total energy ([8], Figs. 12 and 13). Secondly, it is already known from regenerative amplifiers [9] that the pulse-to-pulse dynamics of pumping and inversion depletion exhibits instability regions where pulse energies are elements of nontrivial limit cycles: strong pulses which entail substantial population depletion and therefore strongly weaken the subsequent pulse’s gain are followed by weak pulses which allow the populations to recover etc. In the case of actively Q-switched lasers, simulations ([10], Fig. 13) and experiments [11] have hinted at similar instabilities which ultimately yield (deterministically) chaotic behavior. For Yb-doped fiber lasers, the pulse-to-pulse instability regions at higher switching frequencies were studied comprehensively in [12], along with other phenomena such as stimulated Brillouin scattering and premature pulse build-ups powered by lower or stronger levels of stochastic seed from amplified spontaneous emission.

Hardware-based techniques have been applied to mitigate stochastic [13] and dynamic effects [14] but limit the flexibility of the system. These limitations could be avoided by employing active feedback control strategies as in [9], which in turn requires accurate modeling of the system’s behavior. A review of existing mathematical models of actively Q-switched lasers can be found in [15]. These models always involve some form of rate equation(s) for the lasing population densities and hence the optical gain of the active medium. A second set of equations then describes the evolution of optical energy within the cavity depending on the current gain and losses. On a more detailed level, we will differentiate mathematical models along three mutually independent key characteristics:

2. General model

2.1 Rate and transport equations

Contrary to regenerative amplifiers, the cavities of Q-switched lasers are constantly filled with continuous-wave optical energy, while the pulse is created via the Q-switching process and is usually of greater duration than the cavity round-trip time. Throughout this work, we will make the usual assumption that no two laser modes of different wavelengths are mutually coherent and thus that no interference phenomena such as mode-locking occur. Therefore, radiation will be represented by powers instead of electric fields. Our starting point will be the (multi-mode) traveling-wave equations, where we use the example of a forward pumped Nd:YAG crystal within a linear (Fabry-Perot) cavity. A sketch of the system is drawn in Fig. 1. The spatial coordinate inside the cavity along which light propagates is defined as $z$-direction, and the active medium with population densities $N_i(z,t)$ resides within the interval $[0,L]$. Population densities and optical powers will always be assumed to be transversally uniform, which is why there will be no spatial dependences other than on $z$. At $z = z_{PC}$, a Pockels cell with a reflection coefficient $R(t)$ performs the Q-switching. All optical components apart from the active medium and the Q-switch determine the boundary conditions at either end of the cavity, which are specified in Section 2.2. For Nd:YAG active media, the most interesting lasing transitions occur at wavelengths $\lambda _A$ = 946 nm and $\lambda _B$ = 1064 nm. Their technological relevance is due to a small quantum defect and hence high theoretical efficiency in the former and a large emission cross-section in the latter [24], which renders the 1064 nm transition the dominant one. Pumping can be achieved at $\lambda _p$ = 809 nm from the ground state to an elevated state, which quickly decays to the upper state for both lasing transitions considered. Thus, our full model of the active medium effectively consists of five population densities as visualised in Fig. 2. The rate equations for the population densities $N_i(z,t)$ with the mentioned underlying energy level scheme and a total of $M$ longitudinal cavity modes read [15]

 

Fig. 1. A sketch of the cavity, active medium (red area), Q-switch (Pockels cell, yellow area), forward pump power (blue) and forward- and backwards-running signal powers (green). The coupling of the two beam propagation directions as well as all other optical components are accounted for via the boundary conditions B.C. (green dots).

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Fig. 2. The five energy occupation levels that constitute our model and their transitions: pumping (blue) and two lasing transitions (red, green). The black arrows stand for nonradiative thermal transitions.

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Because active media usually have very similar refractive indices at comparable wavelengths $\lambda _A$, $\lambda _B$, and $\lambda _p$, the group velocities of radiation in this spectral range are all approximately equal to a common value $v$. The power transport equations can then be written as ([25], Cha. 8)

Note that this paper does not focus on modeling all possible effects occurring in Nd:YAG lasers, but rather on demonstrating, based on an accurate traveling-wave model, how to derive a simple set of equations (see Section 3) which describes the stochastically and dynamically influenced build-up of energy within the cavity. For a more detailed description, effects such as thermalization of state manifolds or excited state absorption can be additionally accounted for. In Section 3.2, we perform a thermalization analysis for the quasi-stationary case, while more elaborate approaches can be used similar to [27]. To model excited state absorption, as was done in [28], one can either introduce additional populations which are then pumped from the metastable state(s) or simply adapt the absorption loss coefficient $\alpha _p$ in Eq. (2a) to simulate the increase in threshold pump power and decrease in slope efficiency caused by excited state absorption.

2.2 Cavity boundary conditions

As can be seen in Fig. 1, optical elements (whose effective round-trip cavity efficiency is $\eta (\lambda _j)$) bring along that the pump and beam powers are subject to boundary conditions (B.C.)

2.3 Stochastic seeding via spontaneous emission

First, we define the total scattering cross-section

The dominant mode $\lambda _j$ of a pulse arises from a competition among modes around the maximum of $\sigma _{A}(\lambda _j)$ or $\sigma _{B}(\lambda _j)$ due to the greatest amount of spontaneously emitted photons in the early phase of the energy build-up. Bragg gratings and other wavelength-selective elements inside the cavity can shape $\eta (\lambda _j)$ in the boundary conditions Eqs. (3a–3c) such that the setup exhibits a few or many significant modes. As a consequence, the experimental output pulses will be approximately Bose-Einstein or Poisson distributed, as was shown in [30].

3. Model simplification

3.1 Transport equation solution and total rate equations

For control-related tasks, numerically solving the coupled rate Eqs.  (1a–1d) and power transport Eqs.  (2a,2b) equations with the boundary conditions Eqs.  (3a–4a) is not feasible as systematically investigating the pulse-to-pulse dynamics based solely on simulations is difficult. To eliminate spatial dependences, we introduce the total populations

We start the model simplification by choosing frames $t \rightarrow t \mp z/v$ in Eqs.  (2a,2b) which move with a point of the pump or signal beam. Additionally, we make the following two assumptions:

With initial conditions given by the incident powers $P_p(0,t_k) = P_p(t_k)$ and $P^{\pm }(\frac {L\mp L}{2},t_k,\lambda _j)$ for the pump and signal beams before transmission, respectively, and separating each two channels’ frequencies by one inverse cavity round-trip time, i.e. $\Delta \nu = 1/t_{RT}$, we show in the supplemental document that the power transport Eq. (2) can be solved analytically provided that the assumptions (A1.) and (A2.) hold. The solutions, i.e. the outgoing powers corresponding to any incoming ones, read

Assumption (A4.) may be implemented using lineshape separator functions $g_A(\lambda )$ and $g_B(\lambda )$ which are either 1 or 0 and switch support at some separation wavelength $\lambda _{AB}$. The effect of lineshape separators can be seen in Fig. 3. With assumptions (A1.)–(A4.), we show in the supplemental document that the total rate equations read as

 

Fig. 3. The overlap between lineshapes of scattering cross-sections $\sigma _A(\lambda )$ and $\sigma _B(\lambda )$ is eliminated by lineshape separators $g_A(\lambda )$ and $g_B(\lambda )$.

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3.2 State reduction via slow-fast dynamics

The model so far considers 5 populations of which only 4 are independent due to the algebraic condition involving $N_{dop}$. If the relaxation rate $\gamma _{ik}$ of an excitation state population $N_i$ is very large, the state is depopulated nearly instantly to $N_k$. In that case, which is usually a good approximation for low energy gaps $\Delta E_{ik}$ in the order of phonon energies, we can apply singular perturbation theory to reduce the order of the model. Assuming that the active medium is at thermal equilibrium (thus introducing a thermalization factor $B_{45} := \exp \left (\frac {-\Delta E_{45}}{k_BT}\right )$) [31] and employing a linear state transform, we show in the supplemental document that the total rate Eqs. (13a–13d) consist of a slow and a fast dynamic subsystem. The limit of infinitely fast relaxation yields a quasi-stationary model for the parameters

3.3 Continuous-time nonlinear model (CTNM)

Since we are mainly concerned with the evolution of the intracavity power rather than the transport effects inside the cavity, we will average the transmission through the active medium, the cavity losses and the Q-switch influence over half a cavity round-trip for a linear cavity, as was done in [23]. For this, the power transport solution Eq. (12b) and the boundary conditions Eqs. (3b,3c,4a) are utilized and for simplicity, $\sqrt {R(t)}$ is applied to both forward- and backward-running beams instead of $R(t)$ to the backward beam. We also assume that amplification and losses also apply to spontaneously emitted or backscattered photons. Then, half a cavity round-trip, i.e. a time increment of $\Delta t := t_{RT}/2 = L_{cav}/v$, affects the powers according to

Equation (20) and the reduced total rate Eq. (17) provide a very compact description of the system dynamics during a switching cycle and constitute what we will call the continuous-time nonlinear model (CTNM) of actively Q-switched lasers. In order to obtain the population and power after a switching cycle (and hence the pulse-to-pulse behavior of the system) from the CTNM, one would have to solve the coupled, nonlinear differential Eqs. (17) and (20), which is not possible without further approximations, as we will discuss in Section 3.5.

3.4 Comparison to existing models

It is an interesting observation that our modeling effort so far bridges the gap between spatially distributed traveling-wave modeling (where we started our investigation) and the more traditional point models which are spatially lumped. From the spatially lumped total rate Eq. (17), the rate equation used in traditional point models can be derived by isolating the pump and relaxation terms, additionally assuming losslessness and non-pumping-saturation, i.e. $\alpha _p =0$ and $LN_{dop} \gg N(t)$, whose solution yields Eq. (5) in [32], and assuming low single-transmission gains, i.e. $\exp (\sigma (\lambda _j) N(t) -\alpha L) \approx 1+(\sigma (\lambda _j) N(t) -\alpha L)$. This indicates that the extra terms in Eq. (17) are effects of the spatially distributed nature of the amplification process of the intracavity power as well as of the absorption of the pump power neglected by traditional point models. These modifications have a significant effect on the model’s behavior during a switching cycle and affect the pulse-to-pulse dynamics, especially for low to medium switching frequencies. Furthermore, the power equation used along with a point model rate equation in Degnan’s model [23], which has been used extensively for experimental laser optimisation [32], can be retrieved from Eq. (20) by neglecting the last terms accounting for spontaneous emission and backscattering and disregarding multi-mode behavior. To the best of the authors’ knowledge, the model proposed in this paper is the first one on Q-switched lasers where the stochastic nature of spontaneous emission is incorporated.

3.5 Pulse-to-pulse dynamics

As pointed out in Section 1, the output pulse energies of an actively Q-switched laser are subject to fluctuations due to stochastic and dynamic effects. The former mainly come from amplified spontaneous emission, while the latter are a result of unstable pulse-to-pulse dynamics due to a coupling of subsequent pulses. Since Q-switching (e.g. a Pockels cell represented by $R(t)$) operates in a cyclic manner, the pulse-to-pulse dynamics can be described by a discrete-time dynamic system where each switching cycle is a discrete event. If $t_m$ is the time immediately after the $(m-1)$-th pulse has been coupled out, one switching cycle evolves according to $t_{m+1} = t_m + 1/f_{switch}$. We use the $m$-th pulse energy $E_{m}$ given by Eqs. (5) and (21) and the abbreviation $N(t_m):=N_m$. While most generally, $N_m$ and $P_{j}(t_m)$ would both be dynamic variables, we can argue that $P_{j}(t_m) \approx 0$ for all $m$ and $j$ since all intracavity power has just been coupled out at those times, which reduces the dimension of the dynamical system from $1+M$ to $1$. With future control applications in mind, typical feedback quantities include the measured pulse energy, i.e. $E_m$ is the considered output quantity. As input quantity, we can choose the intracavity power $P_j(t_m+t_{pump}) =: u_{j,m}$ after the pumping phase and at the beginning of the $m$-th energy build-up. This input quantity is either given by spontaneous emission (uncontrolled case) or by a defined level from external seeding or controlled prelasing [13] (controlled case). Thus, defining the pulse-to-pulse and output maps by $f$ and $h$, respectively, the discrete dynamics are given by

4. Simulation results

In the following, we will show simulation results which demonstrate the interplay of stochastic and dynamic effects, agreement with first measurement results and the accuracy of the model simplifications proposed in the previous section.

The supplemental document contains the modeling parameters used in these simulations.

4.1 Pulse statistics and bifurcations

Apart from noise on the intracavity and output powers during a switching cycle, spontaneous emission also causes variations in the output energies even when the system is dynamically stable. To study the interplay of stochastic and dynamic effects in our general model discussed in Section 2. and given by (1–5), the duty cycle (DC) of the Pockels cell is varied at a fixed switching frequency ($f_{switch} =$ 1 MHz) and pump power ($P_p =$ 22 W). By plotting the output energy histograms and deterministic output pulses for comparison, a bifurcation diagram (Fig. 4) is obtained which displays how period-doubling bifurcations and increasing losses due to a longer build-up affect the energy distribution. Since a cavity design with large bandwidth was assumed, we have a large number of contributing modes and hence expect roughly Poissonian output energy statistics, as was pointed out in [30], which is indeed the case, cf. Fig. 5(b)). On top of the stochastic variations, period-doubling bifurcations at relative duty cycles of around 34% and 42% can be observed. From the third period-doubling onwards, we get deterministic chaos with an intermittent window of a 3-limit cycle between 53 and 56 %. The mean output energy diminishes at such build-up times since the amplification phase has no significant gain left and losses dominate instead.

 

Fig. 4. Distribution of output pulse energies of the full stochastic model (1–5) and its deterministic counterpart for varying relative duty cycles (DC). As the DC increases, losses lower the mean output energy and period-doubling bifurcations occur.

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Fig. 5. Comparison of the output energies $E_m$ of Eqs. (17),(20),(21),(5) to measured data and theoretical predictions [29]. a) shows the simulated steady-state energies $E_s$ and average energies from an experimental Yb:CaF2 laser. In b), the distribution of experimentally measured pulse energies is compared to simulations of an Nd:YAG laser. The bottom row compares the distribution of simulated normalized photon numbers Eq. (24) obtained from output energies of an Nd:YAG laser to c) a Poisson ($M=270$) and d) a Bose-Einstein distribution with $M=1$.

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4.2 Validity of the CTNM

The continuous-time nonlinear model (CTNM) given by Eqs. (17),(20) together with Eqs. (21),(5) simplifies the spatially distributed full model (1–5) by averaging all processes within the cavity over half a round-trip such that ordinary differential equations remain to describe the whole system dynamics. We will show in Section 4.3 (cf. Fig. 6) that this introduces only small model errors as the full model and the CTNM agree well both in stable and unstable regimes of operation. Figure 5 demonstrates that the CTNM can reproduce experimental data in terms of dependence on $P_p$ and the DC as well as theoretical predictions regarding the photon statistics of output pulses. The measured average pulse energies shown in a) stem from a double-pass diode-pumped Yb:CaF2 laser system at 200 kHz switching frequency and build-up times $t_{bu} = \textrm{DC}/f_{switch}$ of 1.4 $\mu$s and 1.5 $\mu$s, respectively. The corresponding simulation data represent steady-state output energies $E_s$ of (22) and were obtained using Eqs. (17),(20),(21),(5) without stochastic effects, with technical parameters adapted to the experiments and physical parameters similar to those from the literature (for reference, see the supplemental document for the Nd:YAG laser simulation parameters). One can see that the model fits the measured data very well both for different pump powers and for different build-up times.

 

Fig. 6. Collection of simulation results of full model ((1–5), blue), continuous-time nonlinear model (CTNM, Eqs. (17),(20),(21),(5), red), single-mode CTNM (yellow) and the discrete-time pulse-to-pulse model ((23), green). The two columns a) and b) correspond to a dynamically stable and unstable scenario. The rows display 1) total population inversions, 2) pulse energiese and 3),4) normalised pulse-to-pulse dynamic maps for populations and output energies.

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To compare the stochastic distributions of output pulse energies, we focused on a pulse-to-pulse-dynamically stable, non-saturating regime where dynamics effects do not interfere with the stochastics of amplified spontaneous emission. The bottom row of Fig. 5 shows the distribution of the normalized photon numbers

In b), a simulation of an Nd:YAG system at 1 MHz switching frequency and 6% DC is compared to experimental pulse energies. Here, our simulation exhibits slightly larger stochastic variations than the experimental observation. This is plausible since the theoretical prediction made significant approximations, such as assuming stochastic independence between the photon numbers of lasing modes to arrive at the $M$-fold degenerate Bose-Einstein distribution, which are expected to broaden the pulse spectrum.

4.3 Comparison of models

Since we have shown in Section 4.2 that the CTNM Eqs. (17),(20),(21),(5) is able to reproduce the slope efficiencies of experimental lasers and the photon statistics of output pulses at least qualitatively, we will now assess whether the CTNM differs significantly from the full model (1–5) and how further simplifications compromise the model’s accuracy by analysing a collection of simulation results shown in Fig. 6. One simplification lies in reducing the system to single-mode operation, neglecting all other cavity modes in Eqs. (17),(20) besides the dominating one (i.e. the mode subject to the highest gain). Another simplification of the CTNM is its approximate analytic solution, i.e. the discrete-time model (23) of the pulse-to-pulse dynamics. Figure 6 contains two exemplary cases for a system at $f_{switch}$ = 1MHz repetition rate and a pump power $P_p$ = 22.5 W exhibiting stable (a: 8% DC) and unstable (b: 35% DC) pulse-to-pulse dynamics. They are simulated using the full model (1–5), the CTNM Eqs. (17),(20),(21),(5), a single-mode version of the CTNM and the discrete-time pulse-to-pulse model (23) discussed in Section 3.5. Here, $N_s$, $u_s$ and $E_s$ stand for the steady-state population, uncontrolled intracavity power and pulse energy of the pulse-to-pulse dynamics (22). Row 3) of Fig. 6 demonstrates that increasing the DC value alters the shape of the pulse-to-pulse dynamics curve $f(N_m,u_s)/N_s$. Increasing one of the other two bifurcation parameters $f_{switch}$ or $P_p$ moves the steady state (position of the grey arrow) of the pulse-to-pulse dynamics further to the right of the dynamical map $f(N_m,u_s)$ (to higher $N_m$) such that eventually the absolute value of the slope becomes larger than one and the dynamics hence unstable. In Fig. 6, we can see that the single-mode approximation (yellow) introduces major modeling errors as all photons are emitted into the channel with maximum gain. This in turn leads to a premature energy build-up with enormous depletion and power losses. One can further see – most distinctly in the insets of Fig. 6(1a) and (1b) – that the population inversions of the full model and the CTNM differ only by about 1 percent. The discrete-time model matches these dynamics relatively well, but in the instability region of the pulse-to-pulse dynamics, the limit cycles for populations in Fig. 6(1b) as well as pulse energies in Fig. 6(2b) have smaller amplitudes. In terms of pulse energies $E_m$, row 2) again displays a good agreement between the CTNM and the full model. The normalised pulse-to-pulse dynamic maps in rows 3) and 4) of Fig. 6 for the full model, the CTNM and the discrete-time model stay in close proximity. However, it should be emphasized that the discrete-time model fits the dynamics well up to around 1.1 times the steady-state population $N_s$ in rows 3) and 4). At larger inversion, too much population is depleted in the discrete-time model, and the output map $h(N_m,u_s)$ overshoots significantly. This is due to the fact that losses can only be incorporated via operator splitting, as we showed in the supplemental document. In practice, the overshoot is not problematic at all as such high inversions are never reached if the cavity is being opened and closed periodically, but an analysis on a broad domain of inversion values is still useful for assessing the global dynamical behavior of the models.

5. Conclusion and outlook

In this work, a computationally efficient mathematical model for an actively Q-switched laser system including the effects of stochastic spontaneous emission is derived as a basis for the design of advanced control and estimation strategies. A spatially distributed multi-mode traveling-wave model, which is commonly used in the literature, serves as a starting point. Based on this model, the power transport equations were simplified by making physically feasible assumptions and the singular perturbation theory was applied to systematically reduce the number of relevant populations. Then all the processes within the cavity were averaged over half a cavity round-trip. As a result, a set of coupled, nonlinear ordinary differential equations were obtained, which describe the evolution of the inversion and each cavity laser mode at any point in time. Moreover, in order to separately study dynamic and stochastic effects on energy fluctuations, a discrete-time model of the pulse-to-pulse dynamics was derived by solving the differential equations in a semi-analytic way. Simulation results show that the model simplification steps only introduce small errors in the range of sub-percent, except for the discrete-time model at large populations, which does not constitute an issue for the practical application. In particular, the systematic incorporation of multi-mode behavior and spatially distributed amplification, which distinguishes the spatially lumped continuous-time nonlinear model presented in this work from traditional point models known from the literature, provides superior modeling accuracy with respect to the full model at moderate computational costs. Thus, the continuous-time nonlinear model accurately describes the behavior of actively Q-switched lasers during a switching cycle, while the presented discrete-time model provides a simple relation between the initial populations and the corresponding output energies of two different cycles. The simulation results show that stochastic seeding leads to amplified spontaneous emission noise whose pulse energy statistics correspond to theoretical results from the literature which the stochastic modeling was based on. A pulse energy distribution from simulation is compared to an experimentally obtained one. The simulated slope efficiencies of output energies for various build-up times agree with first measurement results presented in the paper.

Moreover, as it was expected from experimental observations and other papers from the literature, under certain conditions the nonlinear pulse-to-pulse dynamics exhibits period-doubling bifurcations and deterministic chaos. The model presented in this paper aims at laying the foundation to systematically design control and estimation strategies to achieve robustly stable Q-switched lasers at high repetition rates. In particular, the pulse-to-pulse dynamics can be stabilised or otherwise modified via active feedback methods and prelasing strategies can be employed to suppress stochastic seeding due to spontaneous emission.