Standing-wave laser resonators define a set of longitudinal (axial) modes which have an integer number of half-wavelengths within the length of the cavity. The frequency difference between neighbouring axial modes, the axial mode spacing was given by equation (2.1) in Chapter 2. In homogeneously broad- ened gain media, the different axial modes compete for gain so typically only a few modes oscillate near the maximum gain frequency. By introducing an appropriate modulation of the signal within the cavity, it is possible to fix the phase relationship between axial modes so that energy is transferred from the stronger modes in the centre of the gain bandwidth to the weaker modes in the wings. Under ideal conditions all the modes within the gain bandwidth oscillate and are locked in phase to one another. In the time domain this trans- lates to a situation where all the energy circulating in the cavity builds up into a narrow pulse whichseesthe minimum loss in the modulator. There are two
main types of mode-locking: passive and active. In the former, a non-linear optical element, which acts as a saturable absorber, provides an intensity de- pendent loss which is lower for higher intensity pulses. In this technique, once the mode-locking is established, the pulse becomes shorter and more intense until it becomes balanced by the pulse broadening caused by the limited gain bandwidth of the gain medium and other broadening mechanisms such as etalon effects and, in the case of ultrashort pulses, dispersion and self-phase modulation [8].
Passive mode-locking can potentially produce gain bandwidth limited pulse durations for many solid-state laser materials (tens of fs to a few ps). The most common types of passive mode-lockers consist of either Semiconductor- Saturable-Absorber-Mirrors (SESAMs) [9] or Kerr-lens mode-lockers. SESAMs are typically based on Bragg mirrors which have a layer of semiconductor such as InGaAs on the surface which acts as a quantum-well absorber. Kerr-lens mode-locking is achieved by placing a Kerr medium (or in some cases the laser medium itself) in the cavity which acts as a lens with intensity dependent focal length. This modifies the mode profile so that, when combined with an aper- ture (hard or soft) placed in an appropriate location in the cavity, this can lead to a loss which is lower for higher intensity pulses [10].
Whereas with passive mode-locking, the laser pulse itself activates the modu- lator to reduce the loss, with active mode-locking the modulator is activated by an external rf signal, which causes either a phase or amplitude modula- tion of the laser field. In this case the modulation frequencyfm and the cavity
length must be matched so that the circulating pulse is synchronised with the modulator’s low loss condition. This implies that
fm=
c
2lc
, (3.2)
where lc is the cavity length. From equation (2.1) we can see that this is the
same as the axial mode spacing of the resonator which leads to coupling of ax- ial modes. There are two distinct types of active mode-locking, utilising either phase modulation, which alters the frequency (FM), or amplitude modulation (AM). In the FM case an electro-optic phase modulator (EOM) is normally used which introduces a time varying doppler shift in frequency caused by rapidly varying the phase of light passing through the modulator. If this frequency
shift is enough to push the radiation away from the peak of the gain band- width of the laser material then this causes a time varying loss in the cavity which gives rise to mode-locking when equation (3.2) is satisfied. Since the phase shift varies approximately quadratically about the pulse arrival time, a frequency ‘chirp’ over the duration of the output pulse is introduced. While this causes a small amount of broadening of the pulse duration it can be advan- tageous in systems where even shorter pulse durations are required because it allows pulse compression using dispersive elements such as prism or grating pairs. By this technique, the pulse duration can, in principle, be made band- width limited.
AM active mode-locking, which is used in this work, is normally achieved using an acousto-optic modulator (AOM) which is normally comprised of a piezoelectric transducer bonded to a transparent medium like glass or quartz. An acoustic standing wave is set up in the modulator which modifies the lo- cal refractive index and causes time varying diffraction of radiation out of the cavity. The resulting amplitude modulation results in side-bands developing at±fmabout the carrier frequency of the laser radiation. When the axial mode
spacing matches the modulation frequency these side-bands become amplified and in turn develop new side-bands until all the axial modes in the gain band- width are coupled. However, compared to passive mode-locking, in practice the maximum achievable bandwidth is normally less due to external condi- tions which cause minor changes in cavity length. This, combined with lower modulation depths achievable with AOMs compared to SESAMs for exam- ple, results in longer pulse durations for most actively mode-locked systems. The pulse durations achievable using active mode-locking, with acousto-optic or electro-optic modulators, is typically tens to several hundred ps [11]. One advantage of active mode-locking, over passive mode-locking, is that mode- locked operation is often easier to initiate. Some passively mode-locked sys- tems require additional active modulators to initiate mode-locking which adds considerable complexity. For (AM) mode-locking, Kuizenga [12] used the bound- ary condition that, for steady-state operation, the pulse shape is unchanged after one cavity round trip to derive a pulse duration (tp) of
tp =γ
(glg)1/4
(δmfm∆ν)1/2
where γ is a constant equal to 0.53 for Bragg diffraction, g is the saturated gain coefficient of the laser, lg is the length of the gain medium, δm the mod-
ulation depth and∆νthe gain bandwidth of the laser medium (∼250GHz for Nd:YVO4 ). The saturated gain coefficient is described in more detail in Chap-
ter 4. It depends on the material parameters of the laser medium, the pump power and also the circulating power in the resonator, which in turn depends on the output coupler transmission T. We know that under lasing conditions the gain equals the loss in the cavity, so the approximation that2glg ≈T+L[8]
can be used. The AOM used in this work was based on a quartz crystal with Brewster angled faces to give low loss transmission. By placing the AOM as close as possible to one of the end mirrors in the cavity, it was possible to dou- ble pass the circulating pulse through the AOM within the low loss region of the modulation cycle. The AOM drive frequency, required to set up a suitable diffractive acoustic standing-wave, was set at 50MHz. The condition for low loss in the modulator occurs twice every rf cycle which means the correspond- ing modulation frequency was 100MHz. From equation (3.2) this leads to a required cavity length of 1.5m. This rather long cavity length constraint de- mands careful design of the resonator to accomplish stable diffraction limited operation.