Theory of laser self-locking in the presence of host dispersion

A theory is developed of laser mode locking due to saturable absorbers when the dispersive property of the active material is taken into account. The electric field within the cavity is expanded in terms of cavity modes, the problem is treated under the rate-equation approximation, and only third-order nonlinear polarization terms of the dye are considered. The field is assumed to be made of
2N + 1
oscillating modes. It is shown that, owing to dispersion, the phase φ
l
of the modes (
l
running from
-N
to
+N
) has a term that is proportional to l
²
and a term (smaller than the previous one) that is proportional to l
⁴
. In agreement with the experimental results, the term proportional to l
²
increases the pulsewidth of the total electric field over the case of perfect mode-locking and it also gives a positive linear sweep of the field frequency. The pulsewidth increase is expressed by a factor γ that is proportional to the fourth power of the oscillating bandwidth, to the square of the relaxation time T
1
of the dye, and to the product of the length of the active material times a quantity ε that is related to dispersion of the material.