4.6: Q-Switched Mode Locking

To understand the regime of Q-switched mode locking, we reconsider the rate equations (4.4.18) to (4.4.20). Figure 4.28 shows, that we can describe the laser power on two time scales. One is on the order of the Q-switching envelope and occurs on multiple round-trips in the laser cavity, \(T = mT_R\). Therefore, it is on the order of microseconds. The other time scale \(t\) is a short time scale on the order of the pulse width, i.e. picoseconds. Assuming a normalized pulse shape \(f_n (t)\) for the n-th pulse such that

\[\int_{-T_{R}/2}^{T_{R}/2} f_n (t - nT_R) dt = 1, \nonumber \]

we can make the following ansatz for the laser power

\[P(T, t) = E_P (T) \sum_{n = -\infty}^{\infty} f_n (t - nT_R). \nonumber \]

Here, \(E_P (T = mT_R)\) is the pulse energy of the m-th pulse, which only changes appreciably over many round-trips in the cavity. The shape of the m-th pulse, \(f_m (t)\), is not yet of further interest. For simplicity, we assume that the mode-locked pulses are much shorter than the recovery time of the absorber. In this case, the relaxation term of the absorber in Eq.(4.4.21) can be neglected during the duration of the mode-locked pulses. Since the absorber recovery time is assumed to be much shorter than the cavity round-trip time, the absorber is unsaturated before the arrival of a pulse. Thus, for the saturation of the absorber during one pulse, we obtain

\[q(T = mT_R, t) = q_0 \exp \left [-\dfrac{E_P (T)}{E_A} \int_{-T_R/2}^t f_m (t') dt' \right ].\label{eq4.6.3} \]

Then, the loss in pulse energy per roundtrip can be written as

\[q_P(T) = \int_{-T_R/2}^{-T_R/2} f_m (t) q (T = m T_R, t) dt = q_0 \dfrac{1 - exp [-\tfrac{E_P(T)}{E_A}]}{\tfrac{E_P (T)}{E_A}}.\label{eq4.6.4} \]

Equation (\(\ref{eq4.6.4}\)) shows that the saturable absorber saturates with the pulse energy and not with the average intensity of the laser, as in the case of cw-Q-switching (4.4.21)). Therefore, the absorber is much more strongly bleached at the same average power. After averaging Eqs.(4.4.18) and (4.4.19) over one round-trip, we obtain the following two equations for the dynamics of the pulse energy and the gain on a coarse grained time scale \(T\):

\[T_R \dfrac{dE_P}{dT} = 2 (g - l - q_P (E_P)) E_P,\label{eq4.6.5} \]

\[T_R \dfrac{dg}{dT} = -\dfrac{g - g_0}{T_L} - \dfrac{gE_P}{E_L}.\label{eq4.6.6} \]

This averaging is allowed, because the saturation of the gain medium within one pulse is negligible, due to the small interaction cross section of the solid-state laser material. Comparing Eqs.(4.4.18), (4.4.19) and (4.4.21) with (\(\ref{eq4.6.3}\)), (\(\ref{eq4.6.5}\)) and (\(\ref{eq4.6.6}\)), it becomes obvious that the stability criterion (4.4.22) also applies to Q-switched mode locking if we replace the formula for cw-saturation of the absorber (4.4.21) by the formula for pulsed saturation (\(\ref{eq4.6.4}\)). Then, stability against Q-switched mode locking requires

\[-2E_P \dfrac{dq_P}{dE_P}|_{cw-mod} < \dfrac{r}{T_L}|_{cw-mod},\label{eq4.6.7} \]

with

\[-2E_P \dfrac{dq_P}{dE_P}|_{cw-mod} = 2q_0 \dfrac{1 - \exp[-\tfrac{E_P}{E_A}] (1 + \dfrac{E_P}{E_A})}{\tfrac{E_P}{E_A}}. \nonumber \]

When expressed in terms of the average power \(P = E_P/T_R\), similar to Eq.(4.4.29), we obtain

\[-2T_L E_P \dfrac{dq_P}{dE_P}|_{cw-mod} = 2T_L q_0 \dfrac{1 - \exp[-\tfrac{P}{\chi_P P_L}] (1 + \dfrac{P}{\chi_P P_L})}{\tfrac{P}{\chi_P P_L}}, \nonumber \]

where \(\chi_P = \chi T_A\) describes an effective stiffness of the absorber compared with the gain when the laser is cw-mode-locked at the same average power as the cw laser. Thus, similar to the case of cw-Q-switching and mode locking it is useful to introduce the driving force for Q-switched mode locking

\[QMDF = \dfrac{2q_0 T_L}{\chi_P}.\label{eq4.6.10} \]

Figure 4.29 shows the relation (\(\ref{eq4.6.7}\)) for different absorber strength. In going from Figure 4.18 to Figure 4.29, we used \(T_A = 0.1\). We see, that the short normalized recovery time essentially leads to a scaling of the abscissa, when going from Figure 4.18 to Figure 4.29 while keeping all other parameters constant. Comparing Eqs.(4.4.30) with (\(\ref{eq4.6.10}\)), it follows that, in the case of cw-mode locking, the absorber is more strongly saturated by a factor of \(1/T_A\), which can easily be as large as 1000. Therefore, the Q-switched mode locking driving force is much larger than the mode locking driving force, MDF, Accordingly, the tendency for Q-switched mode locking is significantly higher than for cw Q-switching. However, now, it is much easier to saturate the absorber with an average power well below the damage threshold of the absorber (Figure 4.29). Therefore, one is able to leave the regime of Q-switched mode locking at a large enough intracavity power.

Image removed due to copyright restrictions.

Please see:
Kaertner, Franz, et al. "Control of solid state laser dynamics by semiconductor devices." Optical Engineering 34, no. 7 (July 1995): 2024-2036.

Figure 4.29: Visualization of the stability relations for Q-switched mode locking for different products \(2q_0T_L\). The assumed stiffness for pulsed operation is \(\chi_P = 10\), which corresponds to \(T_A = 0.1\). The functional form of the relations for cw Q-switching and Q-switched mode locking is very similar. The change in the stiffness, when going from cw to pulsed saturation, thus essentially rescales the x-axis. For low-temperature grown absorbers, \(T_A\) can be as small as \(10^{-6}\)

Image removed due to copyright restrictions.

Please see:
Kaertner, Franz, et al. "Control of solid state laser dynamics by semiconductor devices." Optical Engineering 34, no. 7 (July 1995): 2024-2036.

Figure 4.30: Self-Starting of mode locking and stability against Q-switched mode locking

We summarize our results for Q-switched mode locking in Figure 4.30. It shows the stability boundary for Q-switched mode locking according to eq.(\(\ref{eq4.6.7}\)), for different strengths of the saturable absorber, i.e. different values \(2q_0T_L\). One may also derive minimum critical mode locking driving force for self-starting modelocking of the laser MDFc due to various processes in the laser [24][25][27][28]. Or, with the definition of the pulsed stiffness, we obtain

\[\chi_{p, c} \le \dfrac{2q_0 T_L}{MDF_c} T_A. \nonumber \]

Thus, for a self-starting laser which shows pure cw-mode locking, we have to design the absorber such that its MDF is greater than this critical value. Or expressed differently, the pulsed stiffness has to be smaller than the critical value \(\chi_{p,c}\), at a fixed value for the absorber strength \(q_0\). There is always a trade-off: On one hand, the mode locking driving force has to be large enough for self-starting. On the other hand the saturable absorption has to be small enough, so that the laser can be operated in a parameter regime where it is stable against Q-switching mode locking, see Figure (4.30).