The output power of linearly polarized Nd:YAG lasers is typically limited by thermally induced birefringence, which causes depolarization. However, this effect can be reduced either by use of some kind of depolarization compensation or by use of crystals which are cut in - and -direction, instead of the common -direction. Investigations of the intrinsic reduction of the depolarization by use of these crystals are presented. To our knowledge, this is the first probe beam-experiment describing a comparison between -, - and -cut Nd:YAG crystals in a pump power regime between 100 and 200 W. It is demonstrated that the depolarization can be reduced by a factor of 6 in -cut crystals. The simulations reveal that a reduction of depolarization by use of a -cut crystal in comparison with a -cut crystal only becomes possible at pump powers in the kW region. Analysis also shows that the bifocusing for -cut is slightly smaller and more asymmetrical than for -cut.
© 2010 OSA
In optically pumped laser rods, a transverse thermal gradient causes stress, which leads to a refractive index change inside the crystals via the photoelastic effect. This thermally induced birefringence results in depolarization, which is defined as the ratio of depolarized power to the initially linearly polarized power, and bifocusing, since the refractive power of the thermal lens depends on the polarization. Both effects limit the output power of linearly polarized Nd:YAG lasers . To compensate for the thermally induced birefringence in a two head resonator, a 90° quartz rotator and an imaging optics between the crystals can be applied . Simpler schemes to partly compensate for moderate thermal depolarization have been presented by Clarkson et al.  and Morehead . Koechner and Rice  have shown that for Nd:YAG crystals the depolarization depends on the growth direction of the crystal, but because of a mistake they concluded that depolarization is the same for all crystal cuts in the high power limit. Soms et al.  derived the correct results for the  case, whereas Shoji and Taira  concentrate on high pump power levels and the -cut. In this paper, different crystal cuts are theoretically and experimentally investigated in the low pump power regime in order to intrinsically reduce the depolarization in this operating regime.
2. Theoretical background
YAG has a cubic crystal structure and is thus optically isotropic. Stress (from nonuniform thermal expansion, for example) breaks this isotropy and induces birefringence. The inverse of the dielectric tensor can be expressed as I/n02 + B, where I is the 3-dimensional identity matrix, n0 is the unstressed index of refraction, and B implies the change of index with stress. It is usually expressed in terms of the elastic strain ε, B = p·ε, where B and ε are 3 × 3 matrices (rank-2 tensors) and the elasto-optic tensor p is rank-4. For YAG’s symmetry class m3m there are only three independent elements: p11 = −0.029, p12 = 0.009, and p44 = −0.0615  (here, the Nye-notation  is used, see appendix). The principal indices of refraction are approximately 
The power depolarization in propagating a distance L is
For a uniform cylinder uniformly heated, following Shoji and Taira  one can write this phase shift as7] list and our appendix derives the principal directions and the birefringence for the three major orientations. If ϕ is the angular coordinate with respect to the x-axis, the principal directions are
For  the x-axis is taken along crystal axis and for  it is along a 45° diagonal (e.g. ) with y along a crystal axis. For  symmetry implies these results are independent of the direction of the x-axis.
The birefringence coefficient is
First, the case of low depolarization is considered, for which the phase shift ψ is much less than one cycle. In Fig. 2 we will see this is less than approximately 400 W absorbed pump for our crystals. The appendix shows that if we choose the x-direction along the input polarization (γ = 0), then the depolarization of a ray is approximatelyEqs. (24) and (25)], the constants of proportionality are
Hence, for a plane wave, the ratio of depolarizations for  and  is
Also, by studying the diagonal element B xx, the appendix shows that for low depolarization, -oriented YAG with polarization at 45° has a stress-induced lens which adds 17% to the thermal lens in the polarization direction and subtracts 13.6% in the perpendicular direction. This is somewhat smaller and more asymmetric than the familiar , which adds 20% to the polarization direction and subtracts 3% from the other .
In the general case, the depolarization of a given ray [Eq. (3)] can be integrated over the cross section to find a beam’s depolarization. We consider a Gaussian beam of radius r g. In that case three of the four orientations of interest can be integrated in closed form (see appendix).
These have the low-depolarization ratios of (12) and the high-depolarization asymptotes of
3. Numerical simulations
With the results of the previous section, we can carry out numerical simulations and introduce the parameters which will be used later in the experiments (section 4). The numerical calculations allow for complicated and more realistic pump light distributions and for arbitrary crystal-cuts. Figure 1 shows an example calculation for a single pass setup in an end-pumped configuration with 180 W of pump light (coating for pump light double-pass) in a 40 mm long YAG rod with 3 mm in diameter and 0.1 at.% Nd-dopant concentration. This crystal configuration will be used for the experimental investigations in the following section. The diameter of the diffraction limited Gaussian probe beam is set to 700 µm in this example and the pump spot radius to 1.2 mm. The shape of the pump spot is assumed to be Gaussian with 0.15% background, since these parameters correspond to the typical pump light distributions in the experiments shown in the following sections. Due to the quantum defect we estimated 26% of the pump light to be transferred to heat. The heat distribution in the laser rod is calculated according to Wilhelm et al. . The minimal depolarization not only for the special cases analyzed in the previous section, but for all crystal orientations is shown as a color-coded spherical surface. -, - and -crystal axes are marked in the figure. We assumed the probe beam orientation to be optimized for each case. In our experimental investigations we focus on crystals with these cuts.
In contrast to the case of a -cut crystal, the depolarization of -cut and -cut crystals depends on the orientation of the linearly polarized incident light with respect to the crystal orientation.
The comparison of Bxy for the different crystal-cuts is sufficient to find the differences in the depolarization [Eq. (10)]. Figure 2 shows the evolution of depolarization with increasing pump power for polarization states with minimal and maximal depolarization. Low power pumping behavior as given in Eq. (12) is demonstrated on Fig. 2(a). At pump powers below 2 kW, the -cut crystal with 45° between the incident polarization and the crystallographic axes yields the lowest depolarization. At higher pump powers [Fig. 2(b)] the depolarization converges to constant values for all crystals.
The depolarization also depends on the ratio of probe beam radius to crystal radius. For - and -cuts, Eq. (14) shows that the depolarization depends on the squared heating power Pheat, which is scaled by the squared ratio rg/r0. Therefore, the depolarization of a -cut crystal saturates at the same value of 25%, independently of the probe beam size rg, although the point of saturation will be shifted to higher pump powers for smaller probe beam radii [Fig. 3(a) ].
The saturation of the depolarization in a -cut crystal occurs at lower powers [Fig. 3(c)]. Here, the depolarization saturation value depends on the probe size.
For the simulations in Fig. 3, a uniform pumping case has been assumed. However, in order to maintain a good overlap between pump- and laser light, variations in the beam radius rg must be accompanied by variations in the pump spot size. For -cut and -cut crystals, the depolarization increases when decreasing the pump spot radius rpump and reaches a maximum at rpump = rg. In case of the -cut, there is no local maximum, but the depolarization increases further when decreasing the pump spot. Furthermore, in case of -cut and -cut crystals the depolarization remains at a constant value for rg = rpump. The actual depolarization value depends on the crystal configuration. The depolarization of a -cut crystal decreases with the size of the probe- and pump beam radius.
4. Single pass experiments
All three crystal-cuts were used in a single-pass setup with a linearly polarized Nd:YLF probe laser source with an output power of approximately 1 W at a wavelength of 1053 nm as shown in Fig. 4 . This wavelength was chosen, because it will not be amplified in the pumped Nd:YAG crystals. On the other hand it is close to the Nd:YAG laser wavelength and therefore only small errors compared to 1064 nm occur, whereas the functionality of the AR-coatings on the crystal end faces and the HR-coated mirrors is ensured. The Nd:YAG test crystals consist of a 40 mm 0.1 at.% doped section and 7 mm long undoped endcaps at both ends to reduce surface stresses due to thermal expansion. With a highly reflective coating for the pump light at the backside, a smoothed longitudinal temperature distribution can be achieved, while 96% of the pump light is absorbed. The crystals were optically pumped at 808 nm by a fiber coupled OEM module with a maximum output power of 210 W. The input polarization was varied using a half-wave plate (HWP 1) in front of the pumped crystal. A second half-wave plate (HWP 2) was installed behind the crystal to rotate the polarization back. Both waveplates were rotated by the same angle. Alternatively, one could rotate the crystal itself, but this would be mechanically much more imprecise. The depolarization ratio was measured with a photodiode behind a polarizing beam splitter (PBS) cube. The probe beam power of the unpumped crystal was compared with the probe beam power under pumped conditions. To investigate the spatial structure of the depolarization, a CCD camera was used. The beam profile, as it occurs at the end face of the crystal, was imaged via a telescope between the crystal and the polarizing beam splitter on the CCD.
The general shape of the depolarization pattern did not depend on the pump- and probe beam sizes for - and -cut crystals. However, we used a pump spot diameter of 2 mm and a probe beam diameter of 1300 µm, since these parameters are the optimized dimensions for laser operation.
The -cut crystal produces a cloverleaf-like depolarization pattern, as shown in the inset close to the CCD in Fig. 4. In these crystals, the axes of the local birefringence are orientated in radial and tangential direction. Therefore, the depolarization pattern is independent of the probe beam polarization angle, but the orientation of the pattern follows the orientation of the input polarization.
The -beam pattern [Fig. 5(a) ] also shows this cloverleaf-like form, but its shape changes with the probe beam polarization, since the principal axes of the refractive index ellipsoides are no longer oriented in radial and tangential direction. It can be seen that near a 45° angle between the incident polarization and the crystallographic axes the “leaves” are thinner than for example at 0° and, thus, the depolarization is reduced.
The -cut crystal [Fig. 5(b)] produces an elliptical polarization pattern for an input polarization of 45°. The principal axes for this crystal are given by Eq. (7). Note the dependence on beam radius r in this case, which causes a different depolarization pattern for various beam sizes. For r0>>rg, the last term of the denominator in Eq. (7) becomes large, which means that the principal axes are almost orthogonal with respect to the - and the -axis in the center of the crystal. Thus, the change of the polarization is large near the center in this case, while for the other crystal types no depolarization occurs in the center. The experimental observations can be well explained by our theoretical simulations as demonstrated in Fig. 5.
The depolarization ratio has been measured for different input polarizations at beam radii of rg = 150 µm, rg = 350 µm, rg = 650 µm, and rg = 900 µm. The graphs shown in Fig. 6 , 7 , and 8 for the -, -, and -cut crystals have been measured at a pump power of 140 W. The -cut crystal (Fig. 6) did not show any change in depolarization with variation of the input polarization angle. Small beam sizes led to less depolarization, as one can expect from the radial phase shift distribution with a gradient from low to higher values in radial direction.
In -cut crystals the depolarization depends on the input polarization (Fig. 7). The minimum has been found for all probe beam sizes at 45° between probe beam polarization and - and -axis, respectively. This is caused by the fact that the principal axes are rather oriented towards the diagonal directions than towards the radial and tangential directions (Fig. 7, right). Therefore, the geometric factor sin2 (2(θ-π/4)) = cos2 (2θ) in Eq. (3) becomes smaller. Where the geometric factor is not small (along the crystal axes), the phase shifts and thus the birefringence is small.
In -cut crystals (Fig. 8) the minimum depolarization can be found at an incident polarization of 0°, if rg < 0.5r0. At rg ≥ 0.5r0 this behavior is inverted and minimum depolarization occurs at 45°, which can be understood from the depolarization pattern (Fig. 5). 0° corresponds to the common clover leaf form with no depolarization in the center, and hence low depolarization for small beams. The depolarization figure at 45° corresponds to an ellipse with the depolarization maximum in the center of the rod, but less depolarization in the outer regions. Therefore, less depolarization requires large probe laser beam sizes.
Again, the experimental results are in good agreement with the theory for all crystal types. The numerical simulations in Fig. 6–8 include the fact that the crystals are end pumped, while the analytical results presented in section 2 are derived for homogeneous pumping. However, homogeneous pumping is already a good approximation.
Figure 9 shows the measured depolarization against the pump power for the polarization orientations leading to the most and the least depolarization for a probe beam with a diameter of 650 µm. As expected, the depolarization of the -cut crystal can be reduced to about 1/6 of the -cut crystal depolarization. In the worst case, the depolarization is about 1.6 times the depolarization for the -cut crystal. The best results we achieved with the -cut crystal are comparable with the worst case of the -cut crystals, and worse than all the other crystals with worst polarization orientation. The fraction of depolarization still increases almost linearly with the pump power, confirming the low power pumping case. The comparison of the experimental curves in Fig. 9 with the theoretical curves in Fig. 2 (left) shows a good agreement in terms of the qualitative differences between the crystals. Since the calculations had been done with a homogeneous pump light distribution assumed, the simulated numbers are lower than the measured ones. According to Fig. 2 (right) the lowest depolarization can be expected for the -cut at pump power levels above 2 kW. Due to limited pump power, this regime was not accessible in our experiments. However, an experimental verification of reduction of depolarization by use of a -cut crystal has been recently reported by Mukhin et al. .
In the theory section analytical results for the depolarization of a Gaussian beam for -crystals and for favorable and disadvantageous orientations of -crystals under different pumping conditions were derived. We deduced an equation that makes comparisons of the depolarization of the three orientations easy, once the stress-optic tensor is known, since it mainly depends on the off-diagonal elements Bxy of the impermeability matrix B. We also found the stress-induced lens for -oriented crystals.
The depolarization behavior in -, -, and -cut Nd:YAG rods was experimentally investigated by a probe beam. An excellent agreement between theory and experiment was found. For less than 2 kW pump power, the least depolarization has been achieved in a  cut crystal. Using this crystal, a decrease of depolarization by a factor of 6 has been demonstrated.
We expect also that the depolarization in linearly polarized laser operation can be reduced by use of non-conventionally cut crystals. While for -cut crystals the polarization direction does not matter, it will be necessary to take care of the orientation of -cut or -cut crystals with respect to the laser’s polarization direction to achieve minimal depolarization.
This appendix provides results presented in Section 2: the form of the strain, the principal directions and values of the birefringence matrix, the form of the elasto-optic tensor in the four orientations of interest, the value of the stress-induced thermal lens for 45°-oriented, -cut crystal, and the average of the depolarization over a Gaussian beam for this case.
For a rod of radius r0 and length L, which is uniformly heated by power Pheat, the principal directions of the elastic strain are radial, angular, and longitudinal with magnitudes :
with α the coefficient of thermal expansion, κ the thermal conductivity and ν Poisson’s ratio. Translating these to transverse Cartesian coordinates, with ϕ the angle to the x-axis,
Substituting in Eq. (16), the Cartesian components are
in units of S.
The same process can be used in reverse with the birefringence tensor. Call its principal values B 1 and B 2 and say principal direction 1 makes angle θ with the x-axis. Then assuming the form Eq. (17) and using double-angle formulas, we find
which imply principal directions and values
These relations allow for the simplification of the depolarization [Eq. (3)] for small birefringence. From Eq. (19) we have that (B 1-B 2)sin(2θ) = 2B xy. Using this and Eqs. (1) and (4) we can approximate for small ψ
which is reasonable since B xy is the coupling between polarizations.
The elasto-optic tensor p is rank-4. The two rank-2 tensors it relates are symmetric and three-dimensional, so they have six independent components each. Thus p can be written as a 6 × 6 matrix. Following Nye , to account for double-counting the symmetric strain, its off-diagonal elements get a factor 2. That is, we call the vectors
YAG is a cubic crystal of symmetry class m3m. Nye  shows that the form of the birefringence tensor is
For isotropic materials (e.g., glass), p 44=(p 11-p 12)/2. This provides a good check of all nontrivial formulas derived.
To find the form of the elasto-optic tensor in orientations not along the crystal axes we can rotate p as a rank-4 tensor. Alternately, knowing how vectors B and ε rotate, Eqs. (17) and (22), we can directly rotate the 6 × 6 matrix p. Write the transverse part of the birefringence tensor as
in terms of a matrix P and a vector p. The results for the four orientations of interest are
Diagonal elements B xx and B yy describe the focusing of x- and y-polarized light, respectively. Quadratic terms are the lens power. Substituting Eq. (18) into Eq. (24) we can express, for x-polarized light, B xx = (S/r 0 2)(c xx x 2 + c xy y 2). For  and for -45° we find
Compare this with the direct thermal lens. The two lenses are
where Q =P heat/(π r 0 2 L) is the heat density and dn/dT = 7 × 10−6 /°C. Thus in each direction the ratio of stress to thermal lens is
By a handy coincidence, the coefficient of c equals 1 for YAG. So the coefficients c [Eq. (26)] describe the strength of the stress-induced lens relative to the direct thermal lens. Diagonal terms (e.g., c xx) are lens in the polarization direction and off-diagonal are lens in the opposite spatial direction. The results for  are long known (see, for example ). The stress lens for -45° is smaller and more asymmetric than for .
Finally, we average the depolarization of -45° over a Gaussian beam to derive the result given in Eq. (14). The calculation for -0° is similar and for  is easier. In the depolarization product [Eq. (3)] the geometrical factor is
where the radial integral
This radial integral can be expressed as the sum of three Gaussian integrals and evaluated to yield
Using a special case of Eq. (3.615.1) of ,
we obtain the final result, Eq. (14).
The authors would like to thank Gerald Mitchell from Precise Light Surgical, who encouraged this work and also introduced us to start this collaboration. This work was supported by the German Volkswagen Stiftung.
References and links
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