Ultrashort Laser Pulse Phenomena, 2nd Edition

Mode-Locking Based on Nonresonant Nonlinearity 351

fundamental through SHG reduced the intensity to 30% of its initial value. Only

10% of that fundamental is reflected back through the second harmonic generating

crystal. However, because the full SH signal that was generated in the first

passage is reflected back, and because it has reversed phase with respect to the

fundamental, 30% of the initial fundamental is recovered. At the first passage, the

conversion to second harmonic should be sufficient to have a sizeable depletion

of the fundamental. Therefore, this method works best for high-power lasers. The

theoretical framework for the SHG has been set in Chapter 3 (Section 3.4.1) and

can be applied for a theoretical analysis of this type of mode-locking. A frequency

domain analysis of the mode-locking process using a nonlinear mirror can be

found in Stankov [42]. Available software packages, such as—for example—

SNLO software can be used to compute the transmission of fundamental and

second harmonic at each passage [43].

The electronic nonlinearity for harmonic generation responds in less than a

few femtoseconds. However, because of the need to use long crystals to obtain

sufficient conversion, the shortest pulse durations that can be obtained by this

method are limited to the picosecond range by the phase matching bandwidth.

The method has been applied successfully to flashlamp pumped lasers [44] and

diode pumped lasers [45–48]. A review can be found in Kubecek [49].

The same principle has also been applied in a technique of parametric mode-

locking, which can be viewed as a laser hybridly mode-locked by a nonlinear

process [50]. The third-order nonlinearity of a crystal applied to sum and differ-

ence frequency generation is used in the mode-locking process. The nonlinear

mirror can also be used to provide negative feedback instead of positive feedback

by adjusting the phase shift between fundamental and second harmonic by the

dispersive element [51].

6.4.2. Polarization Rotation

Nonlinear polarization rotation because of the nonlinear index associated with

elliptical polarization has been described in Section 5.4.2 as an example of a

third-order nonlinear process. Again, a second-order nonlinearity can also be

used for polarization rotation. As is the case when phase matched SHG is used,

the minimum pulse duration is determined by the inverse of the phase matching

bandwidth.

Under type II phase matching, the orientation of the fundamental field polar-

ization (assumed to be linear) at the output of the nonlinear crystal is directly

dependent on the relative intensity of the two orthogonal polarization compo-

nents. The crystal cut and orientation is assumed to perfectly fulfill the phase

matching conditions for SHG. If the linearly polarized incoming field is split into

two orthogonal components with strongly unbalanced intensity, then the wave of