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The key importance of the sign of the stress-optic anisotropy ratio for reducing thermally induced depolarization in cubic crystals with 432, and m3m symmetry is addressed. A simple method for measuring the stress-optic anisotropy ratio (including its sign) was proposed and verified in CaF2 and TGG crystals by experiment. The ratio at room temperature for the wavelength 1076 nm was measured to be −0.47 and + 2.25, respectively. In crystals with a negative value of this parameter thermally induced depolarization may be reduced significantly by choosing crystal orientation. In a CaF2 crystal with the [111] orientation a 20-fold reduction of thermally induced depolarization as compared to the [001] orientation was obtained in experiment, which is very promising for using CaF2 as an active element in high-average-power lasers.
©2012 Optical Society of America
1. Introduction
Тhermally induced depolarization in active еlements (AEs) is one of the key factors restraining enhancement of average power of solid-state lasers. Hereinafter we will speak about the depolarization in AEs implying that all the results may be extended to any optical elements made of cubic crystals. The thermally induced birefringence caused by the photoelastic effect in AEs makes the initially optically isotropic medium – glass, cubic crystal or ceramics – anisotropic. Thermally induced eigenpolarizations are linear and orthogonal to each other, but different at different points of the cross-section. They are oriented along and across the temperature gradient in glass, and are more complicated in crystals and in ceramics (see the text below). The phase difference (the magnitude of birefringence) is also a function of transverse coordinates. As a result, the radiation is depolarized on transmission through the sample.
Under depolarized radiation we understand the radiation whose polarization is constant in time, but varies from point to point of the cross-section. The degree of local depolarization of radiation Г is understood as the ratio of the intensity in the depolarized component to the total intensity in two polarizations in each point of beam cross-section. The ratio of the corresponding powers gives the integral depolarization degree γ.
Negative consequences of thermally induced birefringence are quite apparent. There are power losses from depolarization in the polarized radiation. Besides, on passing the polarizer the now polarized radiation is amplitude (e.g., Maltese cross) and phase (e.g., astigmatism) modulated as a result of the significantly inhomogeneous depolarization degree over the cross-section. Thus, the power losses caused by birefringence in the initial spatially polarized mode, for example, in the linearly polarized Gaussian beam, are much larger than the depolarization degree.
Investigations into thermally induced depolarization in AEs were initiated back in the 1960s and are still continued to date. The depolarization in glass was studied in detail [1, 2]. In cubic, cylindrically shaped crystals it was first investigated in [3–7]. Only the [111] orientation was considered in all those works. The influence of the orientation of cubic crystals with 432, and m3m symmetry (i.e., garnets, fluorides, and the like) on thermally induced depolarization was first considered in [8], where a technique for computing dielectric impermeability tensor ΔB in Cartesian coordinates was described for a cylindrical AE of arbitrary orientation. Based on that technique the authors of [8] performed numerical computations of homogeneous distribution of heat generation (pump) in a YAG crystal. However, the results presented in [8] are true only for the [111] orientation due to the error made by the authors. They supposed that directions of eigenpolarizations coincide with radial and tangential directions, i.e., with directions of the principal stresses. The same erroneous assertion can be found in [6, 9] and in the classical book that ran through five editions [10]. Actually, the directions of eigenpolarizations coincide with the axes of the reference frame in which the dielectric impermeability tensor ΔB has a diagonal form. This error was pointed out in [11].
In a number of subsequent works thermally induced depolarization was studied for the crystal orientations other than [111]. In Section 2 we present a brief analysis of the results of the cited works and discuss the key significance of the stress-optic anisotropy ratio ξ. We show that the value (and especially the sign) of this parameter is of principal importance for the crystals used in high-average-power lasers. Section 3 concerns measurements of ξ, including determining its sign. Specifically, the values of ξ for TGG and CaF2 crystals were measured by an original and simple method, with the ξ for CaF2 proved to be negative. The unique properties of crystals with ξ < 0 to suppress thermally induced depolarization are discussed in Section 4. In Section 5 these properties are confirmed by experimental results for a CaF2 crystal.
2. Stress-optic anisotropy ratio
The stress-optic anisotropy ratio ξ was first introduced in the paper [12] in the form
where πij (i,j = 1,2…6) are piezooptic coefficients, i.e., the elements of the piezooptic tensor of the fourth rank in the two-index Nye notation [13]. It was shown in [12] that, given ξ < 0, at plane stress (e.g. in a thin disk) there exists a crystal orientation at which thermally induced depolarization vanishes under definite conditions. This effect was demonstrated experimentally in [12, 14] on an example of BaF2 and CaF2 windows for powerful middle and far IR lasers. Those remarkable works were evidently left unnoticed by the scientific community concerned with high-power lasers with the wavelength of about 1 micron, as at that time fluorides were not regarded to be a promising active medium in this range. Note that no expressions for the depolarization degree at arbitrary orientation were derived in those works, and the problem of choosing crystal orientation for ξ > 0 (typical, e.g., for YAG) was not even discussed.Analytical expressions for depolarization in long cylindrical crystals with [001] orientation were derived in [11, 15, 16], where it was shown that it may be less than in crystals with [111] orientation, if optimal laser radiation polarization is chosen. The authors of [15] introduced the parameter
that was later called in the book [16] optical anisotropy parameter of a crystal. Here pij (i,j = 1,2..6) are photoelastic coefficients (elements of elastooptic tensor in Nye notation [13]). The anisotropy of elasticity tensor is neglected in nearly all the works concerned with the thermal effects in cubic crystals. In this case, parameters ξp and ξ are identical. This approximation is fulfilled well, for example, for YAG and rougher for CaF2. Rigorous allowance for elasticity tensor anisotropy is a sophisticated problem that will be considered elsewhere. But for a single crystal with [001] orientation and axial symmetric heat source, if we use the parameter ξ introduced according to Eq. (1), the elasticity tensor anisotropy has no influence on the analytical expression for thermally induced depolarization. Therefore, in the current paper we will speak about stress-optic anisotropy ratio ξ. Note that ξ = 1 for all glasses.The depolarization in the [011] orientation was investigated theoretically and analytical expressions for direction of eigenpolarizations and phase difference between them were derived in [17]. However, the calculations of depolarization presented in [17] had been made with an error due to incorrect reduction of the integration interval over polar angle. As a result, a wrong optimal angle of inclination of radiation polarization was given in [17]; hence, the depolarization values obtained were greatly underestimated.
The problem of thermally induced depolarization for arbitrary orientation of a cubic crystal with 432, and m3m symmetry was first solved analytically in a general form in [18]. A series of theorems on physical specificity of the [001], [111] and [110] orientations were proved, the problem on the best and worst orientations was solved, and the experimental results confirming the theoretical conclusions were presented in [19].
We will omit the cumbersome expressions from [18] for the phase difference δ of thermally induced birefringence and angle Ψ between eigenpolarization and radiation polarization directions at arbitrary orientation for any axially symmetric density distribution of heat generation power. We will only point out that δ and Ψ depend on crystal orientation (Euler angles α, β and θ), on the profile of heat generation source, on the ratio of the crystal and heat source radii, on the polar coordinates r and φ, on the stress-optic anisotropy ratio ξ, and on the normalized power of heat generation source p:
where ν is Poisson ratio, E is Young modulus, αT is thermal expansion coefficient, κ is thermal conductivity, n0 is the “cold” index of refraction for wavelength λ, and Ph is the power of heat generation.It is worthy of notice that all the material constants of the medium (κ, αT, n0, E, ν and πij) except ξ enter only the normalized heat generation power p (Eq. (3)) which is a multiple of δ and does not affect Ψ. This means, for example, that an increase of κ or a decrease of αT allows heat generation power Ph to be increased proportionally for arbitrary crystal orientation without changing p. At the same time, both δ and Ψ have a complex dependence on the stress-optic anisotropy ratio ξ. Specifically, the choice of optimal orientation greatly depends on ξ.
It was shown in [18] that the parameter ξ is universal for any orientation and a unique characteristic of the medium affecting the dependence of thermally induced birefringence on crystal orientation. In particular, expressions for δ and Ψ for the [111] orientation may be obtained from the expressions for the [001] orientation by formal substitution [20, 21]:
The theory of thermal effects in laser ceramics was constructed in [22–24]. The theory takes into consideration that the orientation of crystallographic axes (hence, of the axes of thermally induced birefringence) is random in each grain. The theoretically predicted effect of small-scale spatial modulation of thermally induced depolarization and wave front of the beam transmitted through ceramics was observed experimentally in [25, 26]. This effect is an exceptional feature of ceramics; it has no analogs either in glasses or in single crystals, and is determined by the size of grains and parameter ξ.
The study of the thermally induced birefringence in Faraday isolators revealed that parameter ξ retains its universality and uniqueness in magnetooptical media [18], ceramics included [27].
The parameter ξ is a decisive parameter not only for rod geometry, but for disks [16, 21, 28, 29] and slabs [16, 30] too. Note also that, besides depolarization, thermal lens astigmatism is ξ-dependent as well [16].
Thus, ξ is not a mere combination of piezooptic coefficients, but rather a key characteristic of a crystal, determining thermally induced distortions in arbitrarily oriented crystals and in ceramics also.
Consequently, information about the value of ξ for crystals used in high-average-power lasers is of principal significance. In Section 3 we will describe a simple method for measuring ξ, including determining its sign.
3. Measuring stress-optic anisotropy ratio ξ
With known piezooptic coefficients πij at required wavelength and temperature, the value of ξ is calculated by Eq. (1). For πij measurements, special laboratory equipment and proven measurement and calibration technique are required to provide good measurement accuracy. For lack of these, πij values are unknown for many crystals, for other crystals diverse data are cited in different papers. As a result, they may give values of ξ greatly differing not only in value but in sign too for the same material. For example, for a CaF2 crystal one can find in the literature ξ = −0,49 [31] and ξ = −0,64 [32], and “-” [31–33] and “+” [15]). Below we will show that the sign of stress-optic anisotropy ratio is of principal importance.
3.1. Determining the absolute value of stress-optic anisotropy ratio ξ
A simple method of assessing stress-optic anisotropy ratio based on measurements of thermally induced depolarization degree γ integral over the beam cross-section in a rod with [001] orientation was described in [34] and is widely used in different works [18, 20, 35, 36]. The dependence of the integral thermally induced depolarization degree on the angle between the incident polarization and one of the crystallographic axes γ(θ) was measured for γ<<1; after that the ratio of the maximum to minimum depolarization degree γmax/γmin was calculated. As was shown in [16, 34], γmax/γmin = ξ2, if |ξ|>1 and γmax/γmin = 1/ξ2, if |ξ|<1. Hence, for estimating |ξ| one needs only to know whether it is smaller or greater than unity. If γ(θ = π/4) = γmax, then |ξ|<1, and if γ(θ = 0) = γmax, then |ξ|>1. Note that for the unknown position of the crystallographic axes in the plane of the rod end (i.e., for unknown θ origin), it is possible to determine whether |ξ|>1 or |ξ|<1 by rotating the crystal about the axis perpendicular to the rod axis [18].
Thus, one can readily measure |ξ|, but the sign of ξ remains unknown, as it was not determined experimentally in the earlier works. A method of measuring ξ sign and its experimental implementation will be described below.
3.2. The concept of the method of experimental finding of ξ sign
The proposed method of determining the sign of ξ is also based on measuring thermally induced depolarization in a rod with [001] orientation as a function of the angle between the incident polarization and one of the crystallographic axes. However, for determining ξ sign it is necessary to measure transverse distribution of the local depolarization Γ(r,φ), rather than the depolarization degree γ integral over the beam cross-section. For explanation of the concept of the proposed method let us write for the [001] orientation an expression for Γ(r,φ) [16, 18]:
whereand h(r) is the form-factor determined by the radial dependence of crystal heating. For weak birefringence (δ<<1), the substitution of Eq. (6) into Eq. (5) yieldsAnalysis of Eq. (7) readily shows that the distribution Г(r,φ) is the so-called Maltese cross for arbitrary θ and ξ. At the same time, direction of the symmetry axes of the Maltese cross strongly depends on θ and ξ. From Eq. (7) it is clear that Г = 0 at φ = φ0, and φ0 is defined by the following expressionInvestigation of the extremes of function φ0(θ) shows that they occur at the angles θ satisfying the conditionFrom Eq. (9) it follows that for ξ > 0 a change in the angle θ leads to variation of the angle φ0 in the interval between minimum and maximum values. In other words, when a crystal is rotating around the z-axis (θ is changing) the Maltese cross “oscillates” between two positions corresponding to two extremums [Fig. 1 (а) and Fig. 1(b) (Media 1)]. If ξ<0, then according to Eq. (8), function φ0(θ) has no extremums and, with a continuous variation of the angle θ in the interval [0,2π], the angle φ0 will continuously vary from 0 to 4π. The Maltese cross will not oscillate in this case; instead, it will rotate nonuniformly with double frequency [Fig. 1(c) and Fig. 1(d) (Media 2)].
Fig. 1 Plots of φ0(θ) for ξ = 2.25 (a) and for ξ = −0.47 (c). And single-frame excerpts from video recordings of distribution Г(r,φ) as a function of angle θ for ξ = 2.25 (b) (Media 1) and for ξ = −0.47 (d) (Media 2).
Thus, by rotating a crystal with [001] orientation about the z-axis and watching the Maltese cross one can unambiguously determine ξ sign: if the cross is oscillating, then ξ>0, and if it is rotating, then ξ<0. For glasses (ξ = 1) and ceramics, the Maltese cross will be stationary (the oscillation amplitude vanishes to zero).
By measuring φ0(θ) and matching it to (8), then treating ξ as a fitting parameter one can also measure |ξ|, but accuracy of such measurements is lower than of the technique described in Section 2.1.
3.3. Measuring stress-optic anisotropy ratio
For our experiment we took two samples: a terbium gallium garnet (TGG) crystal and a calcium fluoride (CaF2) crystal. The schematic of the experiment is shown in Fig. 2 . A 300 W cw Yb-fiber laser (IRE-Polus) at the wavelength of 1076 nm was used as a source of linearly polarized laser radiation. The intensity distribution over the beam cross-section had a Gaussian profile. Laser radiation was used for heating and as a probe signal. As absorption in CaF2 at the wavelength of 1076 nm is very weak, we used a samarium doped crystal (0.04% Sm) for observing well pronounced thermal depolarization. Such a small concentrations of samarium substantial change only the value of absorption coefficient and almost don’t change refractive index, thermal conductivity, elastic parameters and thermal shock parameter of the CaF2.
Fig. 2 Schematic of the experiment: 1 – calcite wedge, 2 – absorbers, 3 – crystal under study, 4 – quartz wedges; 5 – Glan prism, 6 – CCD camera. On the right – crystal geometry.
Calcite wedge 1 ensured ideal linear polarization. Wedges 4 of fused quartz attenuated the radiation. Glan prism 5 cross-polarized with the calcite wedge was adjusted to a minimum transmitted signal whose intensity distribution was measured by CCD camera 6.
To begin with, we measured the directions of the crystallographic axes in the studied crystals by means of an X-ray diffractometer. Then, we measured values of |ξ| for both crystals using the technique described in Section 2.1. Plots for γmax(p) and γmin(p) are given in Fig. 3 . The value for TGG (|ξ| = 2.25) obtained from the γmax/γmin ratio coincided with those measured earlier [18, 20]. The value for CaF2 was measured to be |ξ| = 0.47 ± 0.02.
Fig. 3 Experimental dependences (circle) and theoretical curves for γmax and γmin as a function of laser radiation power for TGG (a) and for CaF2 (b) with [001] orientation.
After that, we determined the sign of ξ using the technique described in Section 2.2. The results are illustrated in Fig. 4 . In the course of TGG crystal rotation the Maltese cross was oscillating Fig. 4(b) (Media 3), i.e., ξ>0; when the CaF2 crystal was rotating, the cross was rotating too Fig. 4(d) (Media 4), i.e., ξ<0.
Fig. 4 Experimental (red circles) and theoretical (blue curve) dependences φ0(θ) for TGG (a) and CaF2 (c), and single-frame excerpts from video recordings of experimental distribution Г(r,φ) as a function of angle θ for TGG (b) (Media 3) and for CaF2 (d) (Media 4)
To the best of our knowledge, values of photoelastic coefficients have not been estimated for the TGG crystal so far. Thus, we were the first to determine the sign of ξ for this crystal. For the CaF2 crystal, no measurements by means of thermally induced depolarization have been made before, and, as shown above, data on piezooptic coefficients are diverse. Note that the value of ξ calculated by Eq. (1) is greater than zero for many garnets [37], like for TGG, and smaller than zero for fluorides, like for CaF2 (ξ = −0.192 for ВaF2; ξ = −0.196 for SrF2 [33]).
Thus, we have rigorously proved that the CaF2 crystal has negative stress-optic anisotropy ratio ξ. Such crystals possess specific features that will be considered in Section 4.
4. Unique properties of crystals with negative stress-optic anisotropy ratio
It was shown in [12, 38] that for ξ<0 a thin disk or a long rod has one specific orientation at which the eigenpolarization direction Ψ depends neither on r nor on φ. Using the notation adopted in [38], we will designate this orientation [[C]], and when speaking about the specific orientation will imply a set of physically equivalent orientations. The fact that Ψ does not depend on transverse coordinates means that the thermally induced eigenpolarizations have the same orientations over the crystal aperture. Consequently, if the active element has ξ<0, then by choosing the specific orientation [[C]] and radiation polarization parallel to any of the eigenpolarizations of the medium, it is possible to eliminate thermally induced depolarization.
In the ξ<0 range, that is attractive for practical applications, there exists a unique value of ξ, namely, ξ = −0.5 at which a widely used [111] orientation is the specific orientation [[C]]. Moreover, when ξ = −0.5, the phase difference δ vanishes to zero in the [111] orientation, i.e., there is absolutely no birefringence, which means that the depolarization is zero at any (!) radiation polarization. This is readily seen in Eqs. (5),(6) with allowance for the substitution of Eq. (4), as p tends to zero. The same conclusion may be derived from the formulas for the [111] orientation presented in [3, 4, 6, 7].
The form of the distribution Ψ(x,y) depends on crystal orientation in a rather sophisticated manner. The profiles of Ψ reduced to the ± 45° range are presented in Fig. 5 for α = 45°, β = 0…90°, ξ = −0.192 (BaF2). For this value of stress-optic anisotropy ratio markedly different from −0.5, the [111] and [[C]] orientations differ significantly. In the [[C]] orientation (β = 66.34°), Ψ(x,y) = 0.
Fig. 5 Ψ profiles for α = 45°, β = 0…90°, ξ = −0.192 in a long rod heated by a flat-top beam with a diameter equal to 0.7 diameter of the crystal.
According to the results of measurements presented in Section 3, CaF2 is a crystal with negative ξ, with ξ = −0.47 being very close to −0.5. This means that the [[C]] orientation is very close to [111]. The calculation by the method proposed in [38] demonstrated that the difference is less than 1 degree: the Euler angles must be not α = 45°, β = 54.7°, like for [111], but, for example, α = 45°, β = 55.6° (the [131 131 127] orientation or equivalent ones). This property of the CaF2 crystal was first mentioned in [12, 14].
The function γmin(β) in CaF2 (ξ = −0.47) and in BaF2 (ξ = −0.192) is plotted in Fig. 6 for α = 45° and values of p varying from 2 to 16 and, for reference, for an infinitely large p. The calculations were done for the long-rod geometry and flat-top heating and probe beams with diameters equal to 0.7 diameter of the crystal. It is clear from the figure that the degree of beam polarization in CaF2 is much less for the [111] orientation than for the [001] orientation, even at large thermal losses, which is especially important in terms of practical applications. The depolarization degree for the [[C]] orientation is zero in this approximation.
Fig. 6 Curves for γmin(β) in (a) CaF2 (ξ = −0.47) and (b) BaF2 (ξ = −0.192) for α = 45° and different p. The characteristic orientations are shown by arrows.
Thus, thermally induced depolarization may be suppressed substantially in the CaF2 crystal. Results of experimental verification of the effect will be presented in Section 5.
It is worth noting that for CaF2 (i.e., for ξ = −0.47) the depolarization is much larger in ceramics than in a single crystal with [111] orientation. As was shown in [23], that a ceramics sample with stress-optic anisotropy ratio ξ has the similar polarization properties as a crystal with [111] orientation with ξeff = 152(ξ−1)/162 + 1. For CaF2, we have ξ = −0.47, i.е., ξeff = −0.29. It is clear from Eq. (7), with the substitution of Eq. (4) taken into account, that the depolarization degree at p << 1 for [111] orientation is proportional to (1 + 2ξ)2. Hence, at small thermal loads, the depolarization degree in CaF2 ceramics is higher than in a crystal with [111] orientation by (1 + 2ξeff)2/(1 + 2ξ)2 = 48 times, which is definitely an important advantage of the CaF2 single crystal over the CaF2 ceramics.
5. Experimental verification of thermally induced depolarization suppression in CaF2
For comparison of values of thermally induced depolarization at different CaF2 crystal orientations two samples of the same length were cut from closely spaced sites of one boule of samarium doped (0.04%) fluorite, which has homogeneous distribution of the absorption coefficient in the aperture. This was done to ensure equality of power absorbed in these samples at the same power of the incident laser radiation. The absorbed power was less than 1% of incident power. One sample had [001] orientation, the other [111] orientation. The accuracy of crystal cutting was ± 4°.
For these two samples γ(p) were measured. The scheme of the experiment in shown in Fig. 2 and results of the measurements are presented in Fig. 7 . Minimum depolarization γmin(p) was measured for the CaF2 crystal with [001] orientation, and γ(p) for the crystal with [111] orientation.
Fig. 7 Integral depolarization for two orientations of crystallographic axes of a CaF2 crystal. The [001] orientation in minimum depolarization: experiment – blue circles, theory – blue dashed curves. The [111] orientation: experiment – red circles, theory – red dashed curves, taking into account that one of the Euler angles differs from its value in the [111] orientation by 2 degrees. Theoretical calculation for the [111] orientation – black dashed curve.
One can see from Fig. 7 that the integral depolarization degree in the CaF2 crystal with [111] orientation is 20 times less than in the same crystal with [001] orientation, which is a clear demonstration of the advantage of the [111] orientation predicted above for CaF2 material.
Comparison of (7) for θ = 0 and θ = π/4 corresponding to the minimum and maximum thermally induced depolarization in the [001] orientation, and with the substitution (4) corresponding to the transition to the [111] orientation, shows that for p << 1 the [111] orientation is advantageous to [001] in terms of thermally induced depolarization in a rather broad range ξ = (−2…−0.2). Specifically, for CaF2 (ξ = −0.47), the [111] orientation is two orders of magnitude more advantageous to [001] in terms of integral depolarization degree, independent of the ratio of the pump beam and signal beam diameter and diameter of the crystal. Outside this range, in particular for ξ > 0, the optimal orientation for p << 1 is [001] [19].
The discrepancy between the magnitude of depolarization suppression obtained in the experiment for the [111] orientation and the theoretical value may be attributed to inaccurate cutting of the [111] crystal. In particular, the red dashed line in Fig. 7 is the theoretical plot for the case when one of the Euler angles differs from its value in the [111] orientation only by 2 degrees.
Thus, the unique properties of the crystals with negative stress-optic anisotropy ratio ξ have been confirmed experimentally on an example of CaF2 that is one of the most widely used laser crystals.
6. Conclusion
A technique for experimental determination of the sign of stress-optic anisotropy ratio ξ of cubic crystals with 432, and m3m symmetry (garnets, fluorides, and others) was proposed and implemented experimentally.
Stress-optic anisotropy ratio ξ in CaF2 and TGG crystals at the wavelength of 1076 nm were measured at room temperature to be −0.47 and + 2.25, respectively.
It was confirmed experimentally on an example of CaF2 that crystals with negative ξ possess a unique property that allows reducing depolarization substantially by choosing crystal orientation: the thermally induced depolarization in a CaF2 crystal with [111] orientation is 20 times less than with [001] orientation.
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