5.1: The Master Equation of Mode Locking
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Lets consider for the moment the cold cavity (i.e. there is only a simple linear medium in the cavity no lasing). The most general solution for the intracavity field is a superpositon of left- and rightward running waves
\[E^{(left)} (z, t) = Re \left \{ \sum_{n = 0}^{\infty} \hat{E}_n e^{j(\Omega_n t + K_n z)} \right \},\label{eq5.1.1} \]
and
\[E^{(right)} (z, t) = Re \left \{ \sum_{n = 0}^{\infty} \hat{E}_n e^{j(\Omega_n t - K_n z)} \right \},\label{eq5.1.2} \]
The possible values for the wavenumbers are \(K_n = n\pi /L\), resulting from the boundary conditions on metallic mirrors or periodicity after one roundtrip in the cavity. If the mirrors are perfectly reflecting, the leftward and rightward moving waves Eqs.(\(\ref{eq5.1.1}\)) and (\(\ref{eq5.1.2}\)) contain the same information and it is sufficient to treat only one of them. Usually one of the cavity mirrors is not perfectly reflecting in order to couple out light, however, this can be considered a perturbation to the ideal mode structure.
We consider the modes in Eq.(\(\ref{eq5.1.2}\)) as a continuum and replace the sum by an integral
\[E^{(right)} (z, t) = \dfrac{1}{2\pi} Re \{\int_{K= 0}^{\infty} \hat{E} (K) e^{j(\Omega (K) t - Kz)} dK \} \label{eq5.1.3} \]
with
\[\hat{E} (K_m) = \hat{E}_m 2L. \nonumber \]
Eq.(\(\ref{eq5.1.3}\)) is similar to the pulse propagation discussed in chapter 2 and describes the pulse propagation in the resonator. However, here it is rather an initial value problem, rather than a boundary value problem. Note, the wavenumbers of the modes are fixed, not the frequencies. To emphasize this even more, we introduce a new time variable \(T = t\) and a local time frame \(t' = t − z/v_{g,0}\), instead of the propagation distance \(z\), where \(v_{g,0}\) is the group velocity at the central wave number \(K_{n_0}\) of the pulse
\[v_{g, 0} = \dfrac{\partial \omega}{\partial k} |_{k = 0} = \left (\dfrac{\partial k}{\partial \omega} \right )^{-1} |_{\omega = 0} \nonumber \]
For introduction of a slowly varying envelope, we shift the frequency and wavenumber by the center frequency \(\omega_0 = \Omega_{n_0}\) and center wave number \(k_0 = K_{n_0}\)
\[k = K - K_{n_0}, \nonumber \]
\[\omega (k) = \Omega (K_{n_0} + k) - \Omega_{n_0}, \nonumber \]
\[\hat{E} (k) = \hat{E} (K_{n_0} + k), \nonumber \]
The temporal evolution of the pulse is than determined by
\[E^{(right)} (z, t) = \dfrac{1}{2\pi} Re \left \{ \int_{-K_{n_0} \to -\infty}^{\infty} \hat{E} (k) e^{j(\omega (k) t - kz)} dk\right \} e^{(\omega_0 t - k_0 z)}. \nonumber \]
Analogous to chapter 2, we define a slowly varying field envelope, that is already normalized to the total power flow in the beam
\[A(z, t) = \sqrt{\dfrac{A_{eff}}{2Z_0}} \dfrac{1}{2\pi} \int_{-\infty}^{\infty} \hat{E} (k) e^{j(\omega (k) t - kz)} dk. \nonumber \]
With the retarded time \(t'\) and time \(T\), we obtain analogous to Equation (2.6.12).
\[A(T, t') = \sqrt{\dfrac{A_{eff}}{2Z_0}} \dfrac{1}{2\pi} \int_{-\infty}^{\infty} \hat{E} (k) e^{j(\omega (k) t - v_{g, 0} k)T + kv_{g, 0} t'} dk. \nonumber \]
which can be written as
\[T_R \dfrac{\partial A(T, t')}{\partial T}|_{(GDD)} = j \sum_{n = 2}^{\infty} D_n \left (-j \dfrac{\partial ^n}{\partial t'} \right )^n A(T, t'), \nonumber \]
with the dispersion coefficients per resonator round-trip \(T_R = \dfrac{2L}{v_{g, 0}}\)
\[D_n = \dfrac{2L}{n!v_{g,0}^{n + 1}} \dfrac{\partial^{n -1} v_g (k)}{\partial k^{n - 1}} |_{k =0}.\label{5.1.13} \]
The dispersion coefficients (\(\ref{5.1.13}\)) look somewhat suspicious, however, it is not difficult to show, that they are equivalent to derivatives of the roundtrip phase \(\phi_R (\Omega) = \dfrac{\Omega}{c} n(\Omega) 2L\) in the resonator at the center frequency
\[D_n = -\dfrac{1}{n!} \dfrac{\partial^n \phi_{R}^{(n)} (\Omega)}{\partial \Omega^n} |_{\Omega = \omega_0}, \nonumber \]
Sofar, only the lossless resonator is treated. The gain and loss can be modelled by adding a term like
\[T_R \dfrac{\partial A(T, t')}{\partial T} |_{(loss)} = -l A(T, t') \nonumber \]
where \(l\) is the amplitude loss per round-trip. In an analogous manner we can write for the gain
\[T_R \dfrac{\partial A(T, t')}{\partial T}|_{(gain)} = \left (g(T) + D_g \dfrac{\partial ^2}{\partial t'^2} \right ) A(T, t'), \nonumber \]
where \(g(T)\) is the gain and and \(D_g\) is the curvature of the gain at the maximum of the Lorentzian lineshape.
\[D_g = \dfrac{g(T)}{\Omega_g^2} \nonumber \]
\(D_g\) is the gain dispersion. \(g(T)\) is an average gain, which can be computed from the rate equation valid for each unit cell in the resonator. The distributed gain obeys the equation
\[\dfrac{\partial g(z, t)}{\partial t} = -\dfrac{g - g_0}{\tau_L} - g\dfrac{|A(z, t)|^2}{E_L},\label{eq5.1.18} \]
where E is the saturation energy \(E_L = \dfrac{h \nu_L}{2^* \sigma_L} A_{eff}\) the upper state lifetime and \(\sigma_L\) the gain cross section. For typical solid-state lasers, the intracavity pulse energy is much smaller than the saturation energy. Therefore, the gain changes within one roundtrip are small. Furthermore, we assume that the gain saturates spatially homogeneous, \(g(z, t') = g(t')\). Then, the equation for the average gain \(g(T)\) can be found by averageing (\(\ref{eq5.1.18}\)) over one round-trip and we obtain
\[\dfrac{\partial g(T)}{\partial T} = -\dfrac{g - g_0}{\tau_L} - g \dfrac{W(T)}{E_L T_R}, \nonumber \]
where \(W(T)\) is the intracavity pulse energy at time \(t = T\)
\[W(T) = \int_{t' = -T_R/2}^{T_R/2} |A(T, t')|^2 dt' \approx \int_{-\infty}^{\infty} |A(T, t')|^2 dt'. \nonumber \]
Taking all effects into account, the linear ones: loss, dispersion, gain and gain dispersion, as well as the nonlinear ones like saturable absorption and self-phase modulation, we end up with the master equation of modelocking
\[\begin{array} {rcl} {T_R \dfrac{\partial A(T, t')}{\partial T} } & = & {-l A(T, t') + j \sum_{n = 2}^{\infty} D_n \left (j \dfrac{\partial ^n}{\partial t} \right )^n A(T, t')} \\ {} & + & {g(T) \left ( 1 + \dfrac{1}{\Omega_g^2} \dfrac{\partial ^2}{\partial t'^2} \right ) A(T, t')} \\ {} & - & {q(T, t') A(T, t') - j \delta |A(T, t')|^2 A (T, t').} \end{array} \nonumber \]
To keep notation simple, we replace \(t'\) by \(t\) again. This equation was first derived by Haus [4] under the assumption of small changes in pulse shape per round-trip and per element passed within one round-trip.