Q-switching with single crystal photo-elastic modulators

An overview is given about experiments with a new method for Q-switching lasers at a constant pulse repetition frequency. It uses inside the laser resonator a Single Crystal Photo-Elastic Modulator (SCPEM). This consists of one piezo-electric crystal electrically excited on a mechanical resonance frequency. In resonance mechanical stresses are induced that lead via the photo-elastic effect to a strongly modulated birefringence. Polarized light going through such an oscillating crystal will experience a significant modulation of its polarization and of transmission through a polarizer. Suitable materials should not be optically active, as it is for example the case for SiO2, and should allow the excitation of a longitudinal oscillation with an electric field perpendicular to the travelling direction of the light. Crystals of the group 3m, like LiTaO3 and LiNbO3, proved to be ideally suited for SCPEMS for the NIR- and VIS-region. For the infrared GaAs can be used. We demonstrated SCPEM-Q-switching for a Nd:YAG-fiber, a Nd:YVO4-slab- and a Nd:YAG-rod-laser with typical pulse repetition rates of 100-200kHz, pulse enhancement factors of ~100 and pulse durations ~1/100 of the period time. Typically the average power during pulsed operation is nearly the same as the cw-power, when the modulator is switched off. The most stable results were achieved up to now with the Nd:YVO4-slab-laser at 10W average power, 1.1 kW peak power, 127 kHz pulse repetition rate, and 70ns pulse durations.


Introduction
Q-switching is a common technology to produce laser pulses. It is based on a fast optical switch in the laser resonator blocking laser light while pumping energy is stored in the gain medium. Usually electro-or acousto-optic Q-switches are used for this task. We introduce here Single Crystal Photo-Elastic Modulators (SCPEM) that combine features of both technologies, namely first the change of polarization in case of electro-optics, and second the use of the photo-elastic effect as in acousto-optics. These resonant devices allow running lasers on a constant pulse repetition frequency determined by the crystal size and excited eigenmode.
GaAs as the optical material. Its special shape allows together with the two actuators a superposition of a vertical and horizontal longitudinal mode. Further Canit & Badoz (1983) proposed an advanced design for a conventional PEM, where small actuators are glued on the sides of the glass piece ( Fig. 1, right) to excite the 3 rd harmonics of the longitudinal horizontal mode. Disadvantages of conventional PEMs are: -High precision is needed to adjust the system such that it oscillates with high merit. -Furthermore the device is very large compared to its useful aperture. -No superposition of frequency adjusted higher modes is possible the gluing process of the actuator(s) can lead to stresses in the glass and hence to an unwanted stray birefringence

Definition and description of a SCPEM
A Single Crystal Photo-Elastic Modulator (SCPEM, Fig. 1) is a piezoelectric optical transparent crystal that is electrically excited on one of its resonance frequencies. Many possible configurations are described by Bammer (2007). A necessary feature is that the polarization of light of any wavelength passing this crystal must not be changed when the crystal is at rest. This for example cannot be fulfilled with the quartz crystal of a conventional PEM, due to its optical activity (Nye, 1985), which would need to be compensated by a second reversely oriented quartz crystal as proposed in an early patent on PEMs (n.n., 1925).
SCPEMs for the MIR are based on 43m crystals and are experimentally and theoretically described by Weil & Halido (1974). Fig. 2 shows now one favourable configuration based on a crystal with symmetry 3m. The light travels along the optical axis (= z-axis), the exciting electrical field points into the y-direction, and in most cases the longitudinal x-eigenmode is used. Two further important eigenmodes are the longitudinal y-eigenmode and the shear yz-eigenmode. Of course there exist infinitely many higher eigen-modes and frequencies.
In most cases a polarizer (analyzer) oriented at 45° is placed behind the modulator. Fig. 2 further shows a photo diode to get a transmission signal T and a resistor (here with 100 Ω) to generate a measure for the piezo electric current I generated by the crystal.

Use as a Q-switch
We define now for monochromatic light with the wavelength λ the retardation as where L is the z-dimension of the crystal and n x , n y are the refractive indices for x-and ypolarized light. They are calculated in chapter 2.2 for the photo-elastic effect and in chapter 2.6 for the electro-optic effect. The transmission T(δ) for the situation of Fig. 2 is The parameters T min and T max take into account that in reality no perfect optical "on" or "off" is possible. For the discussion in this chapter we set T min = 0 and T max = 1 and the Eq. 2 will reduce to the theoretical formulas found in elementary books like Bass, (1995) or Maldonado, (1995). Based on Eq.
(2) and with a retardation course ω t T δ 1 = 0.5π transmission courses as depicted in Fig. 3 can be realized (ω… angular frequency of one resonance). The choice δ 1 = π leads to quite sharp transmission peaks, while during the off-time a second peak with ~10% transmission evolve. When this is used for Q-switching it can be expected that the small off-transmission will not lead to an emission while the sharp transmission peaks will cause defined pulsing of the laser. This was first demonstrated for a low power fibre laser (Bammer & Petkovsek, 2007). For high gain lasers however this will not longer work, typical problems like pre-/post-lasing or spiking are encountered, and a second eigenmode must be added to get even sharper transmission peaks as discussed later. Fig. 4 shows typical resonance curves of crystal-current and crystal-deformation (e.g. the movement of one x-facet or the angle of the z-facet in case of a shear-mode) at any (well working) resonance frequency.

Fig. 4. Typical resonance curves of a piezo crystal for current and deformation
One important fact here is that the relation between deformation and current is approximately constant within the FWHM-resonance bandwidth (usually in the range 1/1000-1/10000 of the resonance frequency). In fact it changes linearly with the frequency, but since the frequency change is so small within the bandwidth and since the modulator is not useful outside of this bandwidth, a constant value can be assumed. For Q-switching this is sufficiently precise, for other applications, e.g. ellipsometry more accuracy is needed. The reason why we point out this relation between current and deformation, which is directly connected to the retardation 1 , is the following: The crystal frequency and the damping can change during operation, e.g. when the crystal heats up, expands and has then increased contact and damping forces with the contacts. Since this directly influences the relation between voltage amplitude and retardation amplitude, it is better to track and control the current amplitude during in order to keep the retardation amplitude on a desired value. (Bammer, 2007; give formulas for the calculation of these relations with ~10% accuracy, such that practically one has to measure the real values anyway. Typically the retardation per current of the most used material for SCPEM, namely LiTaO 3 , is around 50nm/mA corresponding to ~0.5rad/mA for λ = 632nm. Regard that not al 3m-materials will show the x-oscillation discussed now. For BBO it seems that this x-oscillation cannot be excited , where only a yz-sheareigenmode seems to work (Fig. 6), due to its degenerated elasticity matrix.

Dual mode operation
Besides the x-eigenmode sketched in Fig. 2 a y-excitation can also excite longitudinal y-modes and shear yz-modes. In case of the y-modes different shapes are found corresponding to different acoustic waveguide modes (Dieulesaint & Royer, 2000). E.g. for a LiTaO 3 -crystal with the dimensions 28x9.5x4mm Fig. 5 shows two different y-eigenmodes at the frequencies 258.8 kHz and 273.2 kHz. The higher mode shows additionally a strong xyshear component. Eigenmodes as shown in Fig. 5 can now be superposed to the x-oscillation. This makes sense if the higher frequency is exactly three times higher than the basic frequency. Here "exact" means that the 3x-multiple of the basic frequency must be within the FWHM-bandwidth of the higher frequency. For the crystal simulated in Fig. 5 it was possible to tune the first harmonic from initially 90.8 kHz to 91.4 kHz by grinding the x-length from initially 28.8mm to 28.6mm, such that the frequency was one third of the eigen-frequency 274.2 kHz ). Fig. 6 shows as a further example the yz-shear mode of a y-excited BBO-crystal with dimensions 8.6 x 4.05 x 4.5mm at 131.9 kHz . A similar mode is also possible for LiTaO 3 and in (Bammer & Petkovsek, 2008) a dual mode operation of a LiTaO3-crystal (21x7x5mm) with the yz-shear mode (381kHz) and the x-mode (127kHz) was demonstrated.

Simulating SCPEM-Q-switching
In this chapter a simple numerical model for the description of the laser dynamics will be introduced and applied to a Nd:YAG-rod laser.

The laser photon life time t p
The life-time t p of a photon in a laser cavity with intrinsic transmission T c , outcoupler reflection R out , modulated internal transmission T(t) and roundtrip frequency f c is given by If the internal transmission is due to a SCPEM with transmission course as in Fig. 7, then a strong modulation of the photon life time t p is produced. This parameter is of utmost importance in the laser rate equations describing the laser dynamics.

The laser rate equations
The simplest model to describe laser-activity is based on two coupled rate equations for the average population density n (of the upper laser level) and the average photon density P in the laser gain medium (see for an introduction e.g. Siegmann, 1986).
with c…velocity of light in the gain medium, σ…cross-section of induced emission, M…fraction of light emitted by spontaneous emission which travels in a direction that contributes to the laser mode by ASE (amplified spontaneous emission) and is also responsible for the start of laser operation, t n …life time of the upper laser level, p = P abs /E pp /V g (P abs …absorbed pumping power, E pp …pump photon energy, V g …volume of gain medium) … pumping rate when the upper laser level is empty (number of excited states per m³ and s), p (1 -n(t)/n max )… reduced pumping rate when the upper laser level becomes filled using n max the number of laser active atoms per m³, n th …thermal excitation of the lower laser level (important for quasi-three level systems like Yb:YAG, where the lower laser level has little energetic distance to the ground level and is therefore filled in thermal equilibrium according to the Boltzmann-statistics), Γ … laser mode overlap factor (He et al., 2006). Neglecting all losses between laser gain medium and out-coupler the laser power P l of the laser is given by where A is the effective emitting area, T out is the out-coupler transmission, and E pl is the laser photon energy. Fig. 8 shows a very simple setup based on a side-pumped Nd:YAG-rod with diameter D = 3mm, length L g = 75mm and dotation 1.1 at. %. Between the laser back mirror and the rod a SCPEM together with a PBSC (polarizing cube beam splitter) are placed. Pumping is done at 808nm, the laser emits at 1064nm. The following parameters are used for the simulation: resonator length L = 0.3m, maximum number of places for Nd-atomes in the YAG-host: 1.36 10 28 m -3  with dotation 1.1% the number of active atoms becomes n max = 1.496 10 26 m -3 , thermal population of lower laser level n th = 0, out coupling surface A = D²π/4, mode factor Γ = L g /( n g L g + L -L g ) = 0.2075 2 , intrinsic cavity transmission T c = 0.9 (taking into account the depolarizing effect of the laser rod, leading to high loss with the PBSC, further the PBSCtransmission is rated only >95%), out coupler transmission T out = 0.1, life time of upper laser level t n = 230µs, cross-section for stimulated emission σ = 28 10 -24 m -3 (Koechner, 1999). ASEfactor M: to cause some amplification one spontaneously emitted photon must go after on laser-mirror-reflection through the whole laser rod. Hence M must be smaller than two 2 This formula for the mode factor is obtained by assuming a constant laser mode cross-section A along the resonator, equal to the cross-section A of the gain. Further it must be considered that all generated photons are generated on the laser mode volume. A more detailed calculation needs a numerical integration over the photon density in the laser mode. times the projection of a quarter of the effective out-coupling surface on the unit sphere in the gain medium 2(A/4)/(L/2)² divided by the surface of the unit sphere 4π:

Simulating a Nd:YAG-rod-laser
M < n g 2 A/π/L²/2 ~ 0.75 10 -4 (8) This holds for loss free "perfect fitting" spontaneous emission from the centre of the laser cavity, which is never the case. We choose therefore a much smaller M = 10 -5 and remark that changing this figure has little influence on the results presented latter. For the SCPEM the chosen parameters are T min = 0.01, T max = 0.99, first resonance frequency f R = 91kHz, δ 1 =1.2 π, δ 3 = 0. The absorbed pumping power is first assumed to be P abs = 100W. By setting zero the left sides of the rate equations in Eq. 6 and using T max for the SCPEM-transmission the stationary values of the laser power P l0 and the inversion n 0 can be calculated. The left graph in Fig. 9 shows this stationary laser output power P l0 versus the absorbed pumping power P abs , the right graph in Fig. 9 shows the laser output power P l versus the out-coupler transmission T out .  Fig. 9. cw-laser power versus P l0 pumping power P l0 (left); cw-laser power P l0 versus outcoupler transmission (right) The vertical lines in Fig. 9 indicate the actual values of P abs and T out chosen in the simulation.
Obviously the performance of this laser is poor due to the high loss in the cavity (T c = 0.9). Fig. 10 shows now the simulation result of Eq. 6 with the parameters given above. Fig. 10. Simulation results of Nd:YAG-rod laser based on Fig. 8  The simulation starts with the steady state values for SCPEM-off. First the laser power drops to zero and remains zero during several SCPEM-cycles while the inversion n increases. Then pulsing starts and rapidly a quasi-stationary situation sets with constant pulsing at every transmission window with the following pulse parameters: peak power P peak = 275W, FWHM-pulse duration t FWHM = 150ns, pulse energy 46µJ, average power P av = 8.44W. The poor average power is due to the fact that the pulses occur too late, shortly after the transmission peak, such that the laser-light is not polarized linearly and experience strong loss in the PBSC.

Chaotic behavior
Not every modulator frequency can be imposed to the laser. In many case chaotic behaviour is found, i.e. randomly varying pulse emission. Chaotic behaviour in connection with a SCPEM has interesting attractors in Poincare-maps (indicating deterministic chaos) and bifurcation diagrams showing that low modulator frequencies do not allow stable operation (Fig. 11).  In Fig. 11 a side-pumped Yb:YAG-slab laser (slab size 10 x 5 x 0.5mm) with 50mm resonator length and 400W pumping power was simulated based on the rate equations 6. Fig. 12 shows one setup that was used for a realization of SCPEM-Q-switching. The gain unit uses a Nd:YAG rod with diameter: 5 mm, length: 110 mm, doping concentration: 1.1% and polished ends (plano-plano ended) with an AR-coating for 1064 nm with reflection < 0.25%. It is side pumped from two sides with linear diode stacks, each one using 8 diode laser bars emitting at 808nm.  13 shows stable pulsed operation with pulse frequency 190.2 kHz, average power 2.1 W, peak power 70 W, and pulse width 333ns. When the modulator is switched off the laser emits continuous wave with 2.8 W. This configuration, however do not allow stable operation at higher power. Fig. 14 shows the laser performance at higher power. The pulses are emitted irregularly with strongly varying pulse parameters.

Experimental results
Higher stable pulsed power can be achieved with a higher modulator frequency since low SCPEM-frequencies tend to produce chaotic output as indicated by Fig. 11.

Conclusion
The possible use of SCPEMs for Q-switching was discussed. With a proper choice of parameters stable pulsed operation is possible with high efficiencies, little loss in average power, peak power 100 times higher then the average power, pulse repetition rates 100-300 kHz, and pulse durations down to 25ns. The important advantage is the simplicity of the solution making it interesting for quasi-cw applications, where no change of pulse repetition rate is needed. Especially frequency doubling and tripling should be possible with high efficiency with this type of laser operation.
crystal current I(t) laser power P l (t) laser power P L (t) for SCPEM-off www.intechopen.com