4.3: Stability and Relaxation Oscillations

How does the laser reach steady state, once a perturbation has occured?

\[g = g_s + \Delta g \nonumber \]

\[P = P_s + \Delta P \nonumber \]

Substitution into Eqs.(4.1.9-4.1.10) and linearization leads to

\[\dfrac{d \Delta P}{dt} = \pm 2 \dfrac{P_s}{T_R} \Delta g \nonumber \]

\[\dfrac{d \Delta g}{dt} = -\dfrac{g_s}{E_{sat}} \Delta P - \dfrac{1}{\tau_{stim}} \Delta g \nonumber \]

where \(\tfrac{1}{\tau_{stim}} = \tfrac{1}{\tau_L} (1 + \tfrac{P_s}{P_{sat}})\) is the stimulated lifetime. The perturbations decay or grow like

\[\left ( \begin{matrix} \Delta P \\ \Delta g \end{matrix} \right ) = \left ( \begin{matrix} \Delta P_0 \\ \Delta g_0 \end{matrix} \right ) e^{st}. \nonumber \]

which leads to the system of equations (using \(g_s = l\))

\[A \left ( \begin{matrix} \Delta P_0 \\ \Delta g_0 \end{matrix} \right ) = \left ( \begin{matrix} -s & 2 \dfrac{P_s}{T_R} \\ -\tfrac{T_R}{E_{sat} 2 \tau_p} & -\tfrac{1}{\tau_{stim}} - s \end{matrix} \right ) \left ( \begin{matrix} \Delta P_0 \\ \Delta g_0 \end{matrix} \right ) = 0 \nonumber \]

There is only a solution, if the determinante of the coefficient matrix vanishes, i.e.

\[s \left ( \dfrac{1}{\tau_{stim}} + s\right ) + \dfrac{P_s}{E_{sat} \tau_p} = 0, \nonumber \]

which determines the relaxation rates or eigen frequencies of the linearized system

\[s_{1/2} = -\dfrac{1}{2\tau_{stim}} \pm \sqrt{\left ( \dfrac{1}{2\tau_{stim}} \right)^2 - \dfrac{P_s}{E_{sat} \tau_p}}. \nonumber \]

Introducing the pump parameter \(r = 1 + \tfrac{P_s}{P_{sat}}\), which tells us how often we pump the laser over threshold, the eigen frequencies can be rewritten as

\[s_{1/2} = -\dfrac{1}{2\tau_{stim}} \left ( 1 \pm j \sqrt{\dfrac{4(r - 1)}{r} \dfrac{\tau_{stim}}{\tau_p} - 1} \right ), \nonumber \]

\[= -\dfrac{r}{2\tau_L} \pm j \sqrt{\dfrac{(r - 1)}{\tau_L \tau_p} - \left ( \dfrac{r}{2\tau_L} \right )^2} \nonumber \]

There are several conclusions to draw:

• (i): The stationary state \((0, g_0)\) for \(g_0 < l\) and \((P_s, g_s)\) for \(g_0 > l\) are always stable, i.e. \(\text{Re} \{s_i\} < 0\).
• (ii): For lasers pumped above threshold, \(r > 1\), the relaxation rate becomes complex, i.e. there are relaxation oscillations
  \[s_{1/2} = - \dfrac{1}{2\tau_{stim}} \pm j \sqrt{\dfrac{1}{\tau_{stim} \tau_p}} \nonumber \]
  with frequency \(\omega_R\) equal to the geometric mean of inverse stimulated lifetime and photon life time
  \[\omega_R = \sqrt{\dfrac{1}{\tau_{stim} \tau_p}} \nonumber \]
  There is definitely a parameter range of pump powers for laser with long upper state lifetimes, i.e. \(\tfrac{r}{4\tau_L} < \tfrac{1}{\tau_p}\)
• If the laser can be pumped strong enough, i.e. \(r\) can be made large enough so that the stimulated lifetime becomes as short as the cavity decay time, relaxation oscillations vanish.

The physical reason for relaxation oscillations and later instabilities is, that the gain reacts to slow on the light field, i.e. the stimulated lifetime is long in comparison with the cavity decay time.

Example \(\PageIndex{1}\) diode-pumped Nd: YAG-Laser

\[\lambda_0 = 1064\ nm, \sigma = 4 \cdot 10^{-20} cm^2, A_{eff} = \pi (100 \mu m \times 150 \mu m), r = 50\nonumber \]

\[\tau_L = 1.2 ms, l = 1%, T_R = 10ns\nonumber \]

From Eq.(4.1.4) we obtain:

\[I_{sat} = \dfrac{hf_L}{\sigma \tau_L} = 3.9 \dfrac{kW}{cm^2}, P_{sat} = I_{sat} A_{eff} =1.8 W, P_s = 91.5W\nonumber \]

\[\tau_{stim} = \dfrac{\tau_L}{r} = 24 \mu s, \tau_p = 1 \mu s, \omega_R = \sqrt{\dfrac{1}{\tau_{stim} \tau_p}} = 2\cdot 10^5 s^{-1}. \nonumber \]

Figure 4.4 shows the typically observed fluctuations of the output of a solid- state laser with long upperstate life time of several 100 μs in the time and frequency domain.

One can also define a quality factor for the relaxation oscillations by the ratio of imaginary to real part of the complex eigen frequencies 4.29

\[Q = \sqrt{\dfrac{4\tau_L}{\tau_p} \dfrac{(r - 1)}{r^2}},\nonumber \]

which can be as large a several thousand for solid-state lasers with long upper-state lifetimes in the millisecond range.

Image removed due to copyright restrictions.

Please see:

Keller, U., Ultrafast Laser Physics, Institute of Quantum Electronics, Swiss Federal Institute of Technology, ETH Hönggerberg—HPT, CH-8093 Zurich, Switzerland.

Figure 4.4: Typically observed relaxation oscillations in time and frequency domain.