9.3: Group- and Phase-Velocity of Solitons

The Kerr-effect leads to a change of phase velocity of the pulse, resulting in the self-phase shift of the soliton, \(\phi_o\), per round-trip. A change in group velocity does not appear explicitly in the solution of the NLSE. Self-steepening which becomes important for ultrashort pulses leads to an additional term in the NLSE and therefore to an additional term in the master equation (9.1.5)

\[L_{\text{pert}} = -\dfrac{\delta}{\omega_c} \dfrac{\partial}{\partial t} (|a(T,t)|^2 a(T, t)). \nonumber \]

The impact of this term is expected to be small of the order of \(1/(\omega_o \tau)\) and therefore only important for few-cycle pulses. However, it turns out that this term alters the phase and group velocity of the soliton like pulse as much as the nonlinear phase shift itself. We take his term into account in form of a perturbation. This perturbation term is odd and real and therefore only leads to a timing shift, when substituted into Eq.(9.1.5).

\[T_R \dfrac{\partial \Delta t (T)}{\partial T}|_{sst} = -\dfrac{\delta}{\omega_c} A_0^3 \text{Re} \left \{ \int_{-\infty}^{+\infty} \bar{f}_t^* (t) \dfrac{\partial}{\partial t} \left ( \text{sech}^3 \left ( \dfrac{t}{\tau} \right ) \right ) dt \right \} \nonumber \]

\[= \dfrac{\delta}{\omega_c} A_0^2 = \dfrac{2\phi_0}{\omega_c}. \nonumber \]

This timing shift or group delay per round-trip, together with the nonlinear phase shift leads to a phase change between carrier and envelope per roundtrip given by

\[\Delta \phi_{CE} = -\phi_0 + \omega_o T_R \dfrac{\partial}{\partial T} \Delta t (T)|_{selfsteep} = -\dfrac{1}{2} \delta A_0^2 + \delta A_0^2 = \dfrac{1}{2} \delta A_0^2. \nonumber \]

The compound effect of this phase delay per round-trip in the carrier versus envelope leads to a carrier-envelope frequency

\[f_{CE} = \dfrac{\Delta \phi_{CE}}{2\pi} f_R = \dfrac{\phi_0}{2\pi} f_R.\label{eq9.3.5} \]

The group delay also changes the optical cavity length of the laser and there- fore alters the repetition rate according to

\[\Delta f_R = -f_R^2 \Delta t (T)|_{selfsteep} = - 2 \phi_0 \dfrac{f_R}{\omega_o} f_R = -\dfrac{2}{m_0} f_{CE},\label{eq9.3.6} \]

where \(m_0\) is the mode number of the carrier wave. Eqs.(\(\ref{eq9.3.5}\)) and (\(\ref{eq9.3.6}\)) together determine the shift of the m-th line of the optical comb \(f_m = f_{CE} + m f_R\) due to an intracavity pulse energy modulation and a change in cavity length by

\[\Delta f_m = \Delta f_{CE} + m \Delta f_R = f_{CE} (1 - \dfrac{2m}{m_0}) \dfrac{\Delta w}{w_0} - m f_R \dfrac{\Delta L}{L_0}.\label{eq9.3.7} \]

Specifically, Equation (\(\ref{eq9.3.7}\)) predicts, that the mode with number \(m = m_0/2\), i.e. the mode at half the center frequency, does not change its frequency as a function of intracavity pulse energy. Of course, one has to remember, that this model is so far based on self-phase modulation and self-steepening as the cause of a power dependent carrier-envelope offset frequency. There may be other mechanisms that cause a power dependent carrier envelope offset frequency. One such effect is the group delay caused by the laser gain medium another one is the carrier-envelope change due to a change in carrier frequency, which gives most likely a very strong additional dependence on pump power. Nevertheless, the formula 9.82 can be used for the control of the optical frequency comb of a femtosecond laser by controlling the cavity length and the intracavity pulse energy, via the pump power.