8.3: Break-up into Multiple Pulses

In the treatment of mode locking with fast and slow saturable absorbers we only concentrated on stability against energy fluctuations (Q-switched mode locking) and against break through of cw-radiation or continuum. Another often observed instability is the break-up into multiple pulses. The existience of such a mechanism is obvious if soliton pulse shaping processes are present. If we assume that the pulse is completely shaped by the solitonlike pulse shaping processes, the FWHM pulse width is given by

\[\tau_{FWHM} = 1.76 \dfrac{4|D_2|}{\delta W}.\label{eq8.3.1} \]

where \(W\) denotes the pulse energy. \(D_2\) the negative disperison and \(δ\) the self-phase modulation coefficient. With increasing pulse energy, of course the absorber becomes more strongly saturated, which leads to shorter pulses ac- cording to the saturable absorber and the soliton formula. At a certain point, the absorber will saturate and can not provide any further pulse stabilization. However, the Kerr nonlinearity may not yet saturate and, therefore, the soliton formula dictates an ever decreasing pulse width for increasing pulse energy. Such a process continues, until either the continuum breaks through, because the soliton loss becomes larger than the continuum loss, or the pulse breaks up into two pulses. The pulses will have reduced energy per pulse and each one will be longer and experiences a reduced loss due to the finite gain bandwidth. Due to the reduced pulse energy, each of the pulses will suffer increased losses in the absorber, since it is not any longer as strongly saturated as before. However, once the absorber is already over saturated by the single pulse solution, it will also be strongly saturated for the double-pulse solution. The filter loss due to the finite gain bandwidth is heavily reduced for the double-pulse solution. As a result, the pulse will break up into double-pulses. To find the transition point where the break-up into multiple pulses occurs, we write down the round-trip loss due to the gain and filter losses and the saturable absorber according to 6.35

\[l_m = \dfrac{D_f}{3\tau_m^2} + q_s (W_m), \nonumber \]

where, \(q_s (W_m)\) is the saturation loss experienced by the pulse when it propagates through the saturable absorber. This saturation loss is given by

\[q_s (W) = \dfrac{1}{W} \int_{-\infty}^{+\infty} q(T, t)|A_s (t)|^2 dt.\label{eq8.3.3} \]

This expression can be easily evaluated for the case of a sech-shaped steady state pulse in the fast saturable absorber model with

\[q_{fast} (t) = \dfrac{q_0}{1 + \tfrac{|A(t)|^2}{P_A}}, \text{where } P_A = \dfrac{E_A}{\tau_A}. \nonumber \]

and the slow saturable absorber model, where the relaxation term can be neglected becauseof \(\tau_A \gg \tau\).

\[q_{slow} (t) = q_0 \exp \left [-\dfrac{1}{E_A} \int_{-\infty}^{t} |A_s (t')|^2 dt' \right ]. \nonumber \]

For the slow absorber 8.8 the absorber losses (\(\ref{eq8.3.3}\)) can be evaluated independent of pulse shape to be

\[q_{s.slow} (W) = q_0 \dfrac{1 - \exp [-\tfrac{W}{E_A}]}{\tfrac{W}{E_A}}.\label{eq8.3.6} \]

Thus for a slow absorber the losses depend only on pulse energy. In contrast, for a fast absorber, the pulse shape must be taken into account and, for a sech-shaped pulse, one obtaines [14]

\[q_{s, fast} (W) = q_0 \sqrt{\dfrac{1}{\alpha (1 + \alpha)}} \text{tanh}^{-1} [\sqrt{\dfrac{\alpha}{1 + \alpha}}], \text{ with} \alpha = \dfrac{W}{2P_A \tau},\label{eq8.3.7} \]

and the pulse energy of one pulse of the multiple pulse solution. The energy is determined from the total gain loss balance

\[\dfrac{g_0}{1 + \tfrac{m W_m}{P_L T_R}} = l + l_m. \nonumber \]

Most often, the saturable absorber losses are much smaller than the losses due to the output coupler. In that case the total losses are fixed independent of the absorber saturation and the filter losses. Then the average power does not depend on the number of pulses in the cavity. If this is the case, one pulse of the double pulse solution has about half of the energy of the single pulse solution, and, therefore, the width of the double pulse is twice as large as that of the single pulse according to (\(\ref{eq8.3.1}\)). Then the filter and absorber losses for the single and double pulse solution are given by

\[l_1 = \dfrac{D_f}{3\tau_1^2} +q_s (W_1), \nonumber \]

\[l_2 = \dfrac{D_f}{12\tau_1^2} + q_s (\dfrac{W_1}{2}). \nonumber \]

The single pulse solution is stable against break-up into double pulses as long as

\[l_1 \le l_2 \nonumber \]

is fulfilled. This is the case, if the difference in the filter losses between the single and double pulse solution is smaller than the difference in the saturable absorber losses

\[\dfrac{D_f}{4\tau_1^2} < \Delta q_s (W) = q_s (\dfrac{W}{2}) - q_s (W).\label{eq8.3.12} \]

Figure 8.12 shows the difference in the saturable absorption for a single pulse and a double pulse solution as a function of the ratio between the single pulse peak power and saturation power for a fast absorber and as a function of the ratio between the single pulse energy and saturation energy for a slow absorber. Thus, for both cases the optimum saturation ratio, at which the largest discrimination between single and double pulses occurs and, therefore, the shortest pulse before break-up into multiple pulses occurs, is about 3. Note, that to arrive at this absolute number, we assumed that the amount of saturable absoption is neglegible in comparison with the other intracavity losses, so that the saturated gain level and the gain and filter dispersion are fixed.

Image removed due to copyright restrictions.

Please see:
Kartner, F. X., J. A. d. Au, and U. Keller. "Mode-Locking with Slow and Fast Saturable Absorbers--

What's the Difference." Selected Topics in Quantum Electronics 4 (1998): 159.

Figure 8.12: Difference in loss experienced by a sech-shaped pulse in a slow (- - -) and a fast (____) saturable absorber for a given pulse energy or peak power , respectively.

At this optimum operation point, the discrimination against multiple break-up of a fast absorber is about 50% larger than the value of the slow absorber. Since the minimum pulsewidth scales with the square root of \(\Delta_{q_s} (W)\), see Equation (\(\ref{eq8.3.12}\)), the minimum pulsewidth of the slow absorber is only about 22% longer than with an equally strong fast saturable absorber. Figure 8.12 also predicts that a laser modelocked by a fast saturable absorber is much more stable against multiple pulse break-up than a slow saturable absorber if it is oversaturated . This is due to the fact that a fast saturable absorber saturates with the peak power of the pulse in comparison with a slow saturable absorber, which saturates with the pulse energy. When the pulse breaks up into a pulse twice as long with half energy in each, the peak power of the individual pulses changes by a factor of four. Therefore, the discrimination between long and short pulses is larger in the case of a fast saturable absorber, especially for strong saturation. Note that Figure 8.12 is based on the simple saturation formulas for fast and slow saturable absorbers Eqs. (\(\ref{eq8.3.6}\)) and (\(\ref{eq8.3.7}\)). We compare these predictions with numerical simulations and experimental observations made wiht a Nd:glass laser [15][16].

The Nd:glass laser described in ref. [15] was modelocked by a saturable absorber which showed a fast recovery time of \(\tau_A = 200\) fs, a modulation depth of \(q_0 = 0.005\) and a saturation energy of \(E_A = 17\ nJ\). The other laser parameters can be found in [16]. Without the solitonlike pulse formation (GDD and SPM is switched off), the laser is predicted to produce about 200 fs short pulses with a single pulse per round-trip, very similar to what was discussed in the fast saturable absorber mode locking in Chapter 6. The dynamics becomes very much different if the negative GDD and positive SPM are included in the simulation, (see Figure 8.13)

Image removed due to copyright restrictions.

Please see:
Kartner, F. X., J. A. d. Au, and U. Keller. "Mode-Locking with Slow and Fast Saturable Absorbers--

What's the Difference." Selected Topics in Quantum Electronics 4 (1998): 159.

Figure 8.13: Each trace shows the pulse intensity profile obtained after 20,000 cavity round-trips in a diodepumped Nd:glass laser according to [15]. When the laser reaches the double-pulse regime the multipel pulses are in constant motion with respect to each other. The resulting pulse train is not any longer stationary in any sense.

With increasing small signal gain, i. e. increasing pulse energy, the soliton shortens to 80 fs due to the solitonlike pulse shaping, (Figure 8.13).

Image removed due to copyright restrictions.

Please see:
Kartner, F. X., J. A. d. Au, and U. Keller. "Mode-Locking with Slow and Fast Saturable Absorbers--

What's the Difference." Selected Topics in Quantum Electronics 4 (1998): 159.

Figure 8.14: Steady state pulse width (\(\circledR\)) and time-bandwidth product (o) for a Nd:glass laser modelocked by a saturable absorber with a 200 fs recovery time with GDD and SPM included, shown as a function of the intracavity pulse energy. The time-bandwidth product is only meaningful in the single pulse regime, where it is shown. The pulses are almost transform limited sech-pulses. The pulse width in the multiple pulseing regime is only unique in the parameter region where multiple pulses of similar height and width are achieved. The pulses break up into multiple pulses when the absorber is about three times saturated.

The pulse width follows nicely the soliton relation (\(\ref{eq8.3.1}\)), (dash-dotted line). The pulses become shorter, by about a factor of 2.5, than without GDD and SPM before the pulse breaks up into longer double-pulses. The pulse break-up into double-pulses occurs when the absorber is about two times saturated, close to the point where the shortest pulse can be expected according to the discussion above. Figures 8.13 shows, that the break-up point for the double pulses is also very close to the instability for continuum break-through. Indeed the first pulse train after break-up at a small signal gain of \(g_0 = 0.09\) shows the coexistance of a longer and a shorter pulse, which indicates continuum break-through. But the following five traces are double pulses of equal height and energy. For even stronger saturation of the absorber the double-pulses break-up into triple pulses and so on. Then the dynamics becomes even more complex. This behavior has been observed in detail in a Nd:glass laser [15], (see Figure 8.15), as well as in Cr4+:YAG lasers [17]. The simulations just discussed match the parameters of the Nd:glass experiments.

Image removed due to copyright restrictions.

Please see:
Kartner, F. X., J. A. d. Au, and U. Keller. "Mode-Locking with Slow and Fast Saturable Absorbers--

What's the Difference." Selected Topics in Quantum Electronics 4 (1998): 159.

Figure 8.15: Pulsewidth in a Nd:glass laser [15] as a function of intracavity stored energy, i.e. pulse energy for a single pulse per round-trip. Dots mea- sured values and solid line fits for a single and double-pulse solitonlike pulse stream.

Figure 8.15 clearly shows the scaling of the observed pulse width according to the soliton formula until the pulses break up at a saturation ratio of about 2. Notice, that the absorber recovery time of 200 fs is not much shorter than the pulse width achieved. Nevertheless, the optimum saturation ratio is close to the expected one of about 3. The break-up into pure double and triple pulses can be observed more clearly if the absorber recovery time is chosen to be shorter, so that continuum break-through is avoided. Figure 8.16 shows the final simulation results obtained after 20,000 round-trips in the cavity, if we reduce the absorber recovery time from 200 fs to 100 fs, again for different small signal gain, e.g. intracavity power levels and pulse energies. Now, we observe a clean break-up of the single-pulse solution into double-pulses and at even higher intracavity power levels the break-up into triple pulses without continuum generation in between. Note that the spacing between the pulses is very much different from what has been observed for the 200 fs response time. This spacing will depend on the details of the absorber and may also be influenced by the dynamic gain saturation even if it is only a very small effect in this case [17].

Image removed due to copyright restrictions.

Please see:
Kartner, F. X., J. A. d. Au, and U. Keller. "Mode-Locking with Slow and Fast Saturable Absorbers--

What's the Difference." Selected Topics in Quantum Electronics 4 (1998): 159.

Figure 8.16: Each trace shows the pulse intensity profile obtained after 20,000 cavity round-trips for an absorber with a response time \(\tau_A = 100\) fs for different values of the small-signal gain. The simulations are always started with a 1 ps initial pulse shown as the first trace. Note that only the single pulse solutions are stationary.