Numerical prediction of the Q-switching threshold

In a general sense, stable cw modelocking can be characterized as a state where small deviations around the steady-state gain g_s and pulse energy E_s will be damped by the system, instead of being amplified as it is the case for Q-switching instabilities. Mathematically, these linearized equations for pulse energy E and gain g can be expressed in vector form⁴⁶,

where T_R denotes the cavity round-trip time, E_sat,L the saturation energy of the laser gain medium, τ_L is the upper-state lifetime of the gain medium (420 µs for Yb:CALGO³⁶) and G corresponds to the total gain (sum of all cavity gain and loss terms), which is zero for the steady state. For stability, the trace of this matrix needs to be negative⁴⁶. The Q-switching threshold can thus be defined as the pulse energy E_th for which trace(A) = 0. In cases where the only saturable losses arise from the SESAM, gain filtering is neglected and the laser is operating many times above threshold, i.e. (Eτ_L) ∕ (E_L,satT_R) >  > 1, the Q-switching threshold condition above leads to the following (often cited) simplified equation³⁹:

(E_sat,A: saturation energy of the SESAM, A_A: beam area on SESAM, F₂: inverse saturable absorption coefficient, ΔR: modulation depth). Evaluating this expression for the parameters of this 10-GHz laser yields a predicted intracavity pulse energy threshold of 9.9 nJ, corresponding to an average output power of 2.9 W (red dashed line in Fig. 4), which is about 4 times higher than we experimentally observe. Although Eq. ⁴ may be convenient to use, it does not provide the full picture.

We have developed a complete calculation of trace(A) in the steady-state for quasi-three-level modelocked lasers. We include soliton shaping, gain filtering using directly measured cross section data for Yb:CALGO⁴⁷, as well as the exact SESAM and PPLN responses, accounting for the transverse beam profile of the laser and pump beams in each intracavity component (a flat-top pump beam is assumed for simplicity).

Neglecting the PPLN crystal losses, this precise calculation already reduces the predicted threshold to an average power of ~900 mW. To further improve the accuracy, the total gain G needs to contain the residual nonlinear SHG losses in the PPLN crystal, which are analogous to inverse saturable absorption effects. Although the nonlinear losses are minimized by QPM design as described above, we need to assume random duty cycle (RDC) errors due to imperfect fabrication⁴⁸. Assuming a realistic QPM period jitter of 0.2 µm increases the nonlinear losses from ideally 0.01 to 0.05%. Including these RDC errors, the predicted Q-switching threshold output power drops to 724 mW, which is in very good agreement with the experimentally observed threshold.