1. Introduction

The heat deposit in laser rods due to optical pumping causes stress, which leads to a refractive index change inside the rods via the photoelastic effect. This thermally induced birefringence results in depolarization of a beam passing the laser crystal [1]. Furthermore, the refractive power of the thermal lens depends on the polarization and causes bifocusing. Both effects limit the output power of linearly polarized Nd:YAG lasers [2]. In an oscillator consisting of two crystals and an imaging optic between the crystals, a 90° quartz rotator can be applied to compensate for the thermally induced birefringence. This approach reduces the depolarization losses and compensates for the bifocusing [3]. Koechner and Rice [4] and Soms et al. [5] have shown that the amount of depolarization depends on the Nd:YAG crystal orientation. Therefore, crystal orientations other than the conventional [111]-cut could be an option to reduce the depolarization intrinsically. Shoji and Taira [6] suggested the use of [110]-cut crystals in combination with small beam size in the high pumping regime to reduce depolarization. This effect was demonstrated in TGG crystals by Mukhin et al. [7]. However, it is only relevant at kilowatt pump power levels and for laser beam diameter no larger than 2/5th of the crystal diameter.

In our previous calculations and experiments [8], we compared the [100]-, [110]- and [111]-orientations in a single pass configuration. For pump power levels up to 200 W, the [100]-cut crystal yields the least depolarization - about 1/6 of the [111]-cut crystal. In this pump power regime the [110]-cut crystal causes more depolarization than the [100]- or [111]-cut crystal. Note that other than in the case of a [111]-cut crystal, the depolarization of [110]- and [100]-cut crystals depends on the orientation of the linearly polarized incident light with respect to the crystal orientation. Therefore, the input polarization has to be aligned for minimum depolarization when comparing the crystals. In this paper, we compare the performance of the different crystal orientations in an oscillator configuration containing one or two crystals.

2. Thermal lens

Besides the depolarization, the thermal lens and its shape is an important property for laser operation. There are two major contributions to the thermal lens: the direct temperature-induced and the stress-induced refractive index change [9]. In Nd:YAG crystals only the stress contribution depends on the crystal axes. We will refer to this contribution as stress lens.

For a [100]-cut crystal aligned for least depolarization, the stress lens adds 17% to the direct thermal lens in polarization direction and subtracts 13% perpendicular to it [8]. This is smaller and more asymmetric than the stress lens in the [111]-cut crystal, which adds 20% and subtracts 3% [9]. For the [110]-cut crystal, there is no analytic expression to compare the stress lens with the [111]-cut crystal. To visualize the shape of the stress lenses, we compared the optical path length added or subtracted (optical path difference) by the stress-induced refractive index change (Fig. 1 ). We calculated these graphs based on the program core we used to simulate the depolarization [8].

 

Fig. 1 Contribution of the stress lens to the optical path length at 120 W pump power and a pump beam radius of 1 mm. Top to bottom: [111]-, [100]-, [110]-cut crystal. Left: local orientation of principal axes; middle and right: optical path difference for axis 1 and axis 2 (radial and tangential in case of the [111]-cut crystal). Orientation of the x-y axes is the same as in Ref. [8].

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In all crystals, there are two different optical path lengths depending on the input polarization and the lateral position. For the [111]-cut crystal, one can treat it as two different lenses, one for radial and one for tangential polarization [9]. This effect is also known as bifocusing. It becomes more complex for the [100]- and [110]-cut crystals. These crystals cannot be treated like two lenses for radial and tangential polarization, because the principal axes are not oriented in radial and tangential directions (Fig. 1, left). Besides that, the optical path differences introduced by the stress term are also not rotationally invariant like in the [111]-case. Both cuts have different symmetries: the stress lens of the [100]-cut crystal is reproduced when the crystal is rotated by 90°. The stress lens of the [110]-cut crystal is reproduced after a rotation of 180°. We will see the consequences of this characteristic in section 3.

3. Oscillator with a single Nd:YAG crystal

We investigated the crystals in an asymmetric standing wave resonator (Fig. 2 ). It emits continuous wave radiation in the fundamental mode with beam propagation factor M²<1.2. To achieve fundamental mode operation, we exploited the fact that the aberrated thermal lens’ focal length increases with the radial position as described by Clarkson [10] and Winkelmann et al. [11]. All crystal rods were 3 mm in diameter and 54 mm long including 7 mm undoped endcaps with a coating for pump light double pass. The pump spot radius was 0.8 mm and the laser beam radius was 0.5 mm. The Nd-doping concentration was 0.1 at. %.

 

Fig. 2 Setup of the single crystal standing wave resonator (HR: high reflection coating, AR: anti reflection coating, T: transmission).

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First, we examined the influence of the non-circular stress lenses in an oscillator without an intracavity Brewster plate. The beam profile of the fundamental mode and the next higher order mode, which is combination of TEM₀₁ and TEM₁₀, is shown in Fig. 3 . The deformation of the donut-shaped mode in the oscillators employing the [100]- or [110]-cut crystals was clearly visible. Therefore, the non-circular stress lenses have major impact on the beam profile even without an intracavity Brewster plate.

 

Fig. 3 Comparison of the different beam profiles emitted by the laser without an intracavity Brewster plate. Left to right: Using [111]-, [100]- and [110]-cut crystals. Top: Fundamental mode, bottom: next higher order mode.

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We inserted a Brewster plate into the resonator to introduce polarization dependent losses. For single transverse mode operation at 140 W pump power, we had to take the different thermal lenses of the different crystal orientations into account. To keep the same working point for all crystal orientations, we adjusted the resonator’s shorter armlength [10]. We achieved an M² < 1.2 and circular beam profiles without recognizable distortions using this method. From the resonator lengths required for fundamental mode operation, we calculated the mode size inside the resonator. By this means we determined the thermal refractive power of the [100]-cut crystal to be approximately 5% smaller than in the [111]-cut crystal. This is in agreement with our theoretical result [8]. The refractive power of the [110]-cut crystal was 5% stronger than in the [111]-cut case.

To measure the losses caused by depolarization in dependence of input polarization, we rotated the crystal around the z-axis and measured the light reflected by the Brewster plate in one direction (Fig. 2).

In a [111]-crystal, the depolarization does not depend on the input polarization. Thus, using a [111]-crystal and rotating it did not change the output power nor the losses at the Brewster plate (Fig. 4(a) ). Using a [100]-cut crystal (Fig. 4(b)), the losses at the Brewster plate changed as we rotated the crystal. It was possible to reduce the losses to 20% of the [111]-cut case, while the laser output power was slightly increased.

 

Fig. 4 Single sided Brewster plate losses and laser power depending on crystal orientation. The angles are arbitrary.

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The depolarization losses caused by the [110]-cut crystal (Fig. 4(c)) were larger than in all other cases. In contrast to the [100]-cut crystal the output power was not reproduced until the crystal was rotated by 180° (note the different output power levels at 40° and 130° in Fig. 4(c)). This cannot be caused by depolarization, because depolarization loss reproduces after 90°, which can be seen in single pass experiments [8, Fig. 8]. As noted before, the [110]-cut crystal’s stress lens reproduces after 180°. A different stress lens also causes a different mode size inside the crystal. The output power changes as well, because the laser efficiency depends on the overlap with the pump light.

To simulate the depolarization losses, we applied the single pass method we already used to simulate our probe beam measurements [8] (Fig. 4, dashed lines). We also adapted it to a double pass by propagating through the crystal twice (Fig. 4, dotted lines). We assumed a linearly polarized input beam, derived its resonator internal power from the measured laser output power, propagated it through the birefringent crystal and then calculated the losses at the Brewster plate. The measured depolarization loss is higher than calculated in the single pass theory, since in reality the crystal is passed more than once. The measured values are lower than estimated in the double pass simulations, because the losses for the spatially resolved polarization eigenstate established in the resonator are less than the losses for the linearly polarized beam, which we assumed in the simulation. Thus, exact results cannot be expected using this approach. However, these simulations are a good approximation for the upper and lower limits of the depolarization in an oscillator configuration.

For a [110]-cut crystal, the stress lens’ 180°-symmetry causes one linearly polarized resonator mode to run more efficiently. Therefore, the beam is intrinsically polarized even without the intracavity Brewster plate. However, the polarization extinction ratio will be limited by the crystal’s depolarization properties. To analyze the output polarization in dependence of the crystal angle, we rotated the crystal and used a half wave plate to align the output polarization to the polarizer again (Fig. 5 ). The polarization ratio stayed the same and the half wave plate had to be rotated by half the crystal angle. This indicates that the orientation of the polarization is naturally oriented to the crystal axes.

 

Fig. 5 Power levels in S- and P- polarization using a [110]-cut crystal without intracavity Brewster plate (red/black). As the crystal was rotated the orientation of the output polarization was adapted to the axes of the extra cavity polarizer with a half wave plate (blue curve).

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4. Oscillators with two crystals

The output power of the described system can be scaled by use of a multi-rod resonator. We compared the performance of the different crystals in a two crystal setup (Fig. 6 ) with a birefringence compensation as described by Lü et al [3]: a 90° quartz rotator between the crystals swaps the radial and tangential polarization components and 4-f imaging optics ensures the same beam diameter in both crystals. We compared the output power levels of the system with and without the birefringence compensation, i.e. with and without the quartz rotator, using the same pump (0.8 mm) and seed (0.5 mm) beam radii as in the resonator with one crystal.

 

Fig. 6 Two crystal standing wave resonator with birefringence compensation. For some experiments we omitted the quartz rotator. Typical output curves are shown in the diagram. Fundamental mode operation can be found at 140 W pump power per crystal.

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At a pump power of 140 W per crystal, an output power of about 70 W has been obtained from all three crystal orientations in the compensated resonator with a loss of 1 to 4 W at the Brewster plate. The input-vs-output power curves were quite similar for all three crystals (Fig. 6, diagram). The low output power at 90 W pump power is caused by the asymmetric resonator design. At this pump power level the resonator is unstable for all modes [11]. The beam quality was identical for each case: M²-factor ≈1.1 and circular beam profile. A representative beam profile is shown in Fig. 7(a) . The [100]- and [110]-cut crystals needed to be rotated with respect to each other, in order to align the crystal axes to each other and generate identical depolarization.

 

Fig. 7 Beam profile and M²-factors in an asymmetric standing wave two-crystal resonator. (a) [111]-cut crystals with quartz rotator, (b) [111]-cut crystals without quartz rotator, (c) [100]-cut crystals without quartz rotator, (d) [110]-cut crystals without quartz rotator.

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Without the birefringence compensation, fundamental mode operation at 140 W pump power was not possible when using [111]-cut crystals in our resonator configuration (Fig. 7(b)). The reason for this is bifocusing as described by Murdough and Denman [2], which limits the obtainable fundamental mode output power from Nd:YAG lasers with [111]-cut crystals. The radially polarized higher order mode becomes stable before the actual fundamental mode, which needs to be stable for radial as well as tangential polarization. The donut shaped beam profile was attenuated at top and bottom (Fig. 7(b)) due to the orientation of the Brewster plate.

The use of [100]-cut crystals without birefringence compensation resulted in an output power of approximately 70 W. The loss at the Brewster plate (one direction) was less than 4 W. Due to the non-circular thermal lens, the beam profile was elliptical and the beam quality degraded to M²≈1.3 in one axis (Fig. 7(c)). This degradation is related to our resonator design, specifically its mechanism of mode selection. Due to the different thermal lenses, the mode sizes and stability in the vertical and horizontal axes are shifted with respect to each other. The TEM₀₀ needs to be stable in both axes. Therefore, the laser can only be operated in the overlapping region. In this region the axis with the stronger thermal lens can already be stable for higher order modes. In this case, adequate higher order mode suppression is not possible via resonator stability, which leads to a degradation of the beam quality. Therefore, the observed degradation in one axis can possibly be avoided with a different approach for mode selection (e.g. mode selective pumping).

The existence of an overlapping region depends on the pump power level. If such a region does not exist, fundamental mode operation is not possible. This was not the case in our resonator as indicated by the M²-factor of 1.05 and 1.3.

For the [110]-cut crystals the output power without birefringence compensation was 57 W and the loss at the Brewster plate was 12 W. The M²-factor was <1.2 for both axes (Fig. 7(d)). Since the thermal lens is less elliptical than in the [100]-cut case, the beam profile was nearly circular. Because the beam quality was not limited by bifocusing or the ellipticity of the stress lens, it was possible to scale the output power by increasing the pump power. In a resonator with adjusted arm lengths, we measured an output power of 65 W at a pump power of 180 W per crystal with unchanged beam quality. Further scaling of the output power was only limited by the available pump power.

5. Conclusion

We have experimentally investigated the performance of [111]-, [100]- and [110]-cut Nd:YAG crystals in single- and double-crystal lasers at a pump power levels of 140 W per crystal. [100]-cut crystals can be used to decrease the depolarization by approximately 80% in comparison to uncompensated [111]-cut crystals. However, the elliptical stress lens has to be taken into account. When using the [110]-cut crystal, the lowest depolarization loss was on par with the [111]-orientation.

In a birefringence compensated two crystal system pumped with 140 W per crystal, the use of [111]-cut crystals was the most simple way to reach output power levels of about 70 W and circular beam profiles. Using the other crystal orientations, we achieved the same output power and beam quality in this setup, but the crystal axes needed to be aligned to each other.

In the same system without birefringence compensation, fundamental mode operation was not possible with the [111]-cut crystals due to bifocusing. With the [100]-cut crystals, output power levels comparable to the compensated system, but with an elliptical beam profile have been reached. With the [110]-crystal we achieved decent beam quality, but depolarization loss was large.

Overall, the [100]-cut crystal can be used for an 80% reduction of the depolarization loss in an oscillator with about 50 W output power and even higher output power levels can be achieved, if elliptical beam profiles can be tolerated or compensated.