5.3: Active Mode-Locking by Phase Modulation

Side bands can also be generated by a phase modulator instead of an amplitude modulator. However, the generated sidebands are out of phase with the carrier, which leads to a chirp on the steady state pulse. We can again use the master equation to study this type of modelocking. All that changes is that the modulation becomes imaginary, i.e. we have to replace \(M\) by \(jM\) in Eq.(5.2.1)

\[T_R \dfrac{\partial A}{\partial T} = \left [g(T) + D_g \dfrac{\partial ^2}{\partial t^2} - l - j M (1 - \cos (\omega_M t)) \right ] A. \nonumber \]

The imaginary unit can be pulled through much of the calculation and we arrive at the same Hermite Gaussian eigen solutions (5.2.5,5.2.6), however, the parameter \(\tau_a\) becomes \(\tau_a'\) and is now complex and not quite the pulse width

\[\tau_a' = \sqrt[4]{-j} \sqrt[4]{D_g/M_s}. \nonumber \]

The ground mode or stationary solution is given by

\[A_0 (t) = \sqrt{\dfrac{W_s}{2^n \sqrt{\pi} n! \tau_a'}} e^{-\tfrac{t^2}{2\tau_a^2} \tfrac{1}{\sqrt{2}}(1+j)}, \nonumber \]

with as before. We end up with chirped pulses. How does the pulse shortening actually work, because the modulator just puts a chirp on the pulse, it does actually not shorten it? One can easily show, that if a Gaussian pulse with chirp parameter \(\beta\)

\[A_0 (t) \sim e^{-\tfrac{t^2}{2\tau_a^2} \tfrac{1}{\sqrt{2}}(1+j\beta)}, \nonumber \]

has a chirp \(\beta > 1\), subsequent filtering is actually shortening the pulse.