Ultrashort Laser Pulse Phenomena, 2nd Edition

Mode-Locking Based on Nonresonant Nonlinearity 349

6.4. MODE-LOCKING BASED ON NONRESONANT

NONLINEARITY

Various techniques of mode-locking using second-order nonlinearities have

been developed. A first method is a direct extension of Kerr lens mode-locking,

which has been analyzed in the previous chapter. A giant third-order susceptibility

can be found near phase matching conditions in SHG, not unlike the situation

encountered with a third-order susceptibility, which is seen to be enhanced near

a two photon resonance [36,37]. In this method, called cascaded second-order

nonlinearity mode-locking, the nonlinear crystal is used in mismatched conditions

with a mirror that reflects totally both the fundamental and SH waves. The cascade

of sum and difference frequency generation induces a transverse focusing of the

fundamental beam in a way similar to Kerr self-focusing. This method has been

applied to solid-state lasers by Cerullo et al. [38] and Danailov et al. [39]. The

resonance condition (the phase matching bandwidth) implied in this method does

not make it applicable to the fs range.

Another technique was introduced by Stankov, [40,41] who demonstrated

passive mode-locking in a Q-switched laser by means of a nonlinear mirror con-

sisting of a second harmonic generating crystal and a dichroic mirror. Dispersion

between the crystal and the dichroic mirror is adjusted so that the reflected SH

is converted back to the fundamental.

A third method, based on polarization rotation occurring with type II second

harmonic generation, is the equivalent of Kerr lens mode locking in fiber lasers.

It has also been applied to some solid-state lasers. The last two methods will be

discussed in more detail in the following subsections.

6.4.1. Nonlinear Mirror

The principle of operation of the nonlinear mirror can be understood with the

sketch of Figure 6.4, showing the end cavity elements that provide the function of

nonlinear reflection. A frequency doubling crystal in phase matched orientation

Laser

Cavity

Nonlinear

crystal

B/2

2

ab





Figure 6.4 End cavity assembly constituting a nonlinear mirror. The end mirror is a total reflector

for the SH and a partial reflector for the fundamental.

350 Ultrashort Sources II: Examples

is combined with a dichroic mirror output coupler that totally reflects the SH

beam and only partially reflects the fundamental. These two elements form a

reflector, whose reflectivity at the fundamental wavelength can either increase or

decrease, depending on the phases of the fundamental and SH radiation. These

phase relations between the first and second harmonics can be adjusted inserting

a dispersive element between the nonlinear crystal and the dichroic mirror. The

dispersive element can be either air (the phase adjustable parameter is the distance

between the end mirror and the crystal) or a phase plate (of which the angle can

be adjusted).

At low intensity, the cavity loss is roughly equal to the transmission coefficient

of the output coupler at the fundamental wavelength. At high intensities, more

second harmonic is generated, reflected back and reconverted to the intracavity

fundamental, resulting in an increase in the effective reflection coefficient of the

crystal output coupler combination. The losses are thus decreasing with intensity,

just as is the case with a saturable absorber. Figure 6.5 shows the variation of

intensities of the fundamental and second harmonic in the first (left) and second

(right) passage through the second harmonic generating crystal. Depletion of the

abbaA

F

SH

M:

R

W

0.1

R

2W

0.99

0.4 0.8

1.0

0.8

0.6

0.4

0.2

0

1.0

0.8

0.6

0.4

0.2

1.2

R

NL

R

W

2.2 1.8 1.4 

0



0

BA

F, SH Relative intensities

Figure 6.5 Variation of intensity of the fundamental (F, solid line) and the second harmonic

(SH, dashed line) for two successive passages, A and B, through the nonlinear crystal. The entrance

and exit surfaces a and b are labeled in Fig. 6.4. A fraction R = 10% of the fundamental intensity

is reflected back into the crystal, together with the entire second harmonic. After propagation for a

distance B in air, the phase of the second harmonic with respect to the fundamental has undergone a

shift of π, resulting in a reconversion of second harmonic into fundamental at the second passage.