## Abstract

Thermally induced depolarization and thermal lens of three Konoshima Chemical Co. laser-ceramics samples Yb^{3+}:Lu_{2}O_{3}(C_{Yb}≈1.8 at.%), Yb^{3+}:Y_{2}O_{3}(C_{Yb}≈1.8 at.%), and Yb^{3+}:Sc_{2}O_{3} (C_{Yb}≈2.5 at.%) were measured in experiment at different pump power. The results allowed us to estimate the thermal conductivity of the investigated ceramic samples and compare their thermo-optical properties. The thermo-optical constants P and Q and its sign measured for these materials at the first time.

© 2013 OSA

## 1. Introduction

High-average-power laser systems have rigid restrictions on active-gain materials. Heat generation in optical elements leads not only to increased mean bulk temperature, but also to temperature gradients, which, in turn, give rise to elastic stresses, that may destroy the elements and give rise to parasitic thermal effects – the thermal lens and thermally induced birefringence. These effects, in turn, degrade the quality of the generated radiation and are one of the limiting factors of further laser power enhancement [1].

One of the ways to control parasitic thermal effects is a search for alternative optical materials with optimal laser, spectral, and thermo-optical properties. In recent years cubic sesquioxide crystals have been used for fabrication of high-average-power CW and subpicosecond lasers. As was shown in the works [2–6], rare-earth metal sesquioxides, such as Y_{2}O_{3}, Lu_{2}O_{3}, and Sc_{2}O_{3} may be promising candidates by virtue of their high, as compared to Y_{3}Al_{5}O_{12}, thermal conductivity and large absorption and stimulated emission peak cross-sections. In such media 141 W of average output power in mode-locked operation with 738 fs pulses and up to 300 W in CW operation were demonstrated [5, 7].

However, all thermo-optical and mechanical constants of these media responsible for the thermal lens and thermally induced birefringence have been insufficiently studied in the literature or have significant variations in the values. One reason why sesquioxides are poorly known and underutilized in the manufacture of lasers, is the difficulty in manufacturing and the unavailability of such materials. It is next to impossible and very expensive to grow large-size high-quality single crystals from these materials because of very high melting temperatures and high-temperature phase transition (Y_{2}O_{3}). Nowadays it is possible to grow high-quality sesquioxides single-crystal samples of volume up to 5 cm^{3} [8]. At the same time, the present-day technology of producing optical ceramics enables manufacturing large aperture elements with size of up to 100 cm in diameter and 2 cm in thickness of high optical quality [9, 10] from nearly all materials, including sesquioxides [11–13].

In the present work we studied samples of Yb^{3+}-doped Y_{2}O_{3} (10x10x1 mm), Lu_{2}O_{3} (10x10x0.5 mm), and Sc_{2}O_{3} (6.7x5.2x1.2 mm) Konoshima Сhemical Co. ceramics with different concentrations of lasant ions ≈1.8 at.%, ≈1.8 at.%, and ≈2.5 at.%, respectively. Thermo-optical properties responsible for the thermal lens and the thermally induced birefringence considering the influence of the photoelastic effect of these samples were investigated and compared.

## 2. Estimation of thermo-optical constant Q

One of the key parameters in studying of thermo-optical properties is the heat release power caused by linear absorption of transmitted radiation in the medium, quantum defect, reabsorption of luminescence radiation in active elements, and so on. Knowing which part of the laser power in the studied elements is converted into heat, one can measure thermal effects.

For assessing the power of heat generation the following experiment was conducted. The studied sample was thermally connected to a buffer element that was an Y_{3}Al_{5}O_{12} single crystal 15 mm in diameter and 6 mm in thickness with a dielectric coating on one side reflecting at wavelengths of 940 nm and 1030 nm. According to our estimates the heat transfer coefficient of connection studied sample - buffer element was two order more than convective heat transfer between studied sample and atmosphere. Therefore, we assumed that all of the absorbed heat is distributed in the system: the buffer element – the studied sample. The volume of the buffer element was several times more that the studied sample. The temperature on the surface of the buffer element was measured with and without pump radiation. The buffer element temperature did not change at switched on heating radiation in the absence of the studied sample, which indicates that absorption in dielectric mirrors is weak and absorption in the studied sample is the only source of heat. Assuming that all the absorbed heat is distributed uniformly over the buffer element – the studied sample volume during the times less than the measurement time, supposing that the heat stored in the studied sample is negligible compared to that stored in the buffer element, and knowing the heat capacity, volume and density of the buffer element material and the dependence of temperature on time at switched on heating radiation and at its passive cooling, one can readily determine the heat release power in the studied sample. The pump-power-into-heat conversion coefficient at wavelength 940 nm for our samples were measured and are presented in Table 1

In the case of heat generation in an optical element there arise temperature gradients which, in turn, lead to thermal stresses. Due to the photoelastic effect these stresses result in changes in the index of refraction that depends on the magnitude and direction of the stresses and, for the optical element made of crystal material, on the orientation of its crystallographic axes. Thus, as linearly polarized radiation is passing through a thermally loaded optical element to each point of its cross section, its polarization state changes, which results in appearance at the output of the optical element of a portion of orthogonally polarized radiation with the intensity distributed over the cross section [14, 15].

A ceramics consist of a large number of monocrystalline granules with average size 1-100 μm (depending on process of fabrication) separated by thin (~1 nm) boundaries. A significant distinction between such ceramics and a single crystal is that the crystallographic axes in each granule are oriented randomly. The orientation of crystallographic axes in a given ceramic grain is determined by three Euler angles α, β, and γ, which are random variables within their respective intervals. In the absence of thermal load, the ceramics from cubic crystal is almost free from depolarization.

The expression for thermally induced polarization distortions introduced by a ceramic element to linearly polarized radiation consists of two components, one of which does not depend on the number of grains in the beam path (it is polarization distortions averaged over cross-section and all possible directions of the crystallographic axes in each grain), and the other is inversely proportional to it [16, 17]. In the approximation of a large number of grains in the beam path, the second term can be neglected and the thermally induced depolarization is defined by

where*λ*is the wavelength of probe radiation;

*κ*is the thermal conductivity;

*P*is the power of heat generation;

_{h}*A*is the constant, depended on profiles of pump and signal beams and equal 0.017 when both beam have Gaussian intensity distribution and equal radius,

*α*is the thermal expansion coefficient;

_{T}*n*is the index of refraction at the wavelength

_{0}*λ*;

*E*is Young’s modulus;

*ν*is Poisson’s ratio;

*π*are the elements of the piezo-optic tensor in the two-index Nye notation [18]; and

_{ij}*S*is a factor which for ceramics element depends on assumptions about the statistical properties of directions of crystallographic axes in each grain:

_{1}*S*= (75 + 53

_{1}*ξ*)/128 for the Euler angles

*α*,

*β*, and

*γ*uniformly distributed, respectively, in the ranges [-

*π*,

*π*], [0,

*π*], and [-

*π*,

*π*] [16]; and

*S*= (2 + 3

_{1}*ξ*)/5 for the Euler angles

*α*and

*γ*distributed uniformly and

*β*with the probability density sin(

*β*)/2 [17], where

*ξ*=

*π*/(

_{44}*π*-

_{11}*π*) is the stress-optic anisotropy ratio. For all glasses

_{12}*S*= 1, for single crystals with [111] orientation

_{1}*S*= (1 + 2

_{1}*ξ*)/3. The typical ceramics crystallite size that are made by this technology of the order of 1 micron, so the approximation that we have done, performed for all our ceramics samples. The values of the most of material constants in the Eq. (2) for the sesquioxides are unknown. So we have made estimates of the thermo-optical parameter

*Q*based on experimental measurements of thermally induced depolarization by using Eq. (1).

_{eff}The Eq. (1) and (2) holds true for the case of a side-cooling optical element, when the heat flow to the end surface is negligible compared to the heat flow to the side of the sample (rod geometry). In our experiments we have met this requirement by providing efficient heat transfer from the side surface of the samples. Equation (1) was obtained in [16, 17] for materials with symmetry m3m, 432, $\overline{4}3m$(garnets, fluorides and others). The crystal lattice of the studied sesquioxides has m3-symmetry point group. For single crystals of this symmetry, the expressions for the thermally induced depolarization are different, but Eq. (1) for *γ* in the case of ceramic optical elements remains valid without any changes (theoretically, it will be shown in our nearest publications).

The radiation wavelength and the heat release power *P _{h}* are known. Then, by measuring the dependence of integral depolarization on the power of heat generation in a ceramic sample and using the Eq. (1) we can readily find the value of

*Q*. The stress-optic anisotropy ratio

_{eff}/κ*ξ*may be found by measuring thermally induced polarization distortions [19] or by measuring values of the elements of piezo-optical tensor. Both these measurements can be done on a single crystal sample that is hard to obtain.

In ceramics, *Q _{eff}/κ* is a quantitative characteristic of the material in terms of thermally induced polarization distortions. The less the

*Q*value, the less (at a fixed power of heat generation) the level of polarization distortions induced by the thermally loaded optical element, and therefore is less losses in the fundamental mode and higher quality of the transmitted radiation.

_{eff}/κThe thermally induced depolarization was measured using the scheme presented in Fig. 1(а).

The sample was heated by continuous radiation of a diode laser at 940 nm wavelength. The maximum power of the heating radiation was 75 W. A continuous ytterbium fiber laser at 1076 nm was used as a source of linearly polarized probe radiation. The intensity distributions of heating and probe radiation over beam cross section in the area of the studied sample had a profile close to a Gaussian one [see Fig. 1(b)]. A calcite wedge ensured linearity of the polarization and good contrast of the scheme (less than 2∙10^{−6} for the entire power range of the heating radiation). Fused quartz wedges attenuated radiation. Glan prism was adjusted to a minimum of the transmitted signal whose intensity distribution was measured by a CCD camera. By turning the Glan prism by 90° we could record using the CCD camera intensity distribution of the main component of the field. The value of integral depolarization was calculated as a ratio of the radiation power incident on the CCD camera at crossed calcite wedge and the Glan prism to the total power of probe radiation incident on the Glan prism.

The dependence of integral depolarization on heat release power measured in experiment for three sesquioxide ceramic samples are shown by the squares in Fig. 2. The straight lines are for the theoretical curves plotted according to Eq. (1), and the values of *Q _{eff}/κ* at which the experimental and theoretical data agree best are presented in Table 2. The integral depolarization at small heat generation power arising due to the inhomogeneity and imperfection of these polycrystalline ceramics small (<10

^{−5}) and close to the value of “cold” depolarization in good optical quality single crystal. Local distribution of depolarization at small heat generation power was uniformly distributed across the whole aperture of the each samples.

The sign of *Q _{eff}* may be determined by the behavior of the local depolarization distribution as a function of the character of ellipticity of incident radiation polarization. When small ellipticity is induced in the incident radiation the four-leaf symmetry of spatial distribution of the depolarization will change in different ways, depending on the sign of

*Q*. Knowing the sign of this coefficient for an Y

_{eff}_{3}Al

_{5}O

_{12}single crystal and comparing the behavior of local depolarization distribution in Y

_{3}Al

_{5}O

_{12}and in sesquioxide samples we found that they have the same sign of thermo-optical constant

*Q*. Now we know

*Q*/

_{eff}*κ*and its sign for all three sesquioxide materials at room temperature and at wavelength 1076 nm.

## 3. Estimation of thermo-optical constant P

Another thermal effect studied in this research is thermal lens. As was said above, heat generation results in appearance in the crystal bulk of temperature gradients and related stresses. There are three principal effects influencing the thermal lens strength in an optical element with temperature gradients: thermal expansion of a sample, the variation of the index of refraction as a function of temperature, and the variation of the index of refraction as a function of thermal stress. The characteristic features of the thermal lens in ceramic elements were studied theoretically in [20] and experimentally in [21]. It was shown in those works that, like in the case of polarization distortions, the behavior of a ceramic optical element in terms of phase distortions is similar to that of [111] single crystal and the isotropic part of phase distortions related to changes of the index of refraction is defined by

*k*= 2

*π*/

*λ*is the wave number,

*r*and

*r*are the polar radius and the radius of heating radiation beam,

_{h}*κ*is the thermal conductivity,

*P*is the heat release power,

_{h}*ξ*is the stress-optic anisotropy ratio,

*S*is the factor dependent on the statistical characteristics of crystallographic axes orientation in each grain:

_{2}*S*= 11/64 for the Euler angles

_{2}*α*,

*β*, and

*γ*uniformly distributed in their ranges [20];

*S*= 1/5 for the Euler angles

_{2}*α*and

*γ*distributed uniformly and

*β*with the probability density sin(

*β*)/2. For glass

*S*= 0, for a single crystal with [111] orientation

_{2}*S*= 1/3.

_{2}*P*and

*Q*are the thermo-optical constants [22]:

The parameter *P* is proportional to the wave aberration averaged for two orthogonal polarizations and includes changes of the refractive index as a function of temperature stresses. Not all material constants for the studied sesquioxides materials in the Eq. (4) are known. So we have made estimates of the thermo-optical parameter *P* based on experimental measurements of thermal lens.

For the parabolic temperature distribution in the sample *P* is proportional to the strength of the thermal lens. The Eq. (4) was derived [20] for axisymmetric heating radiation with flat-top intensity distribution over the cross section and holds true for polar radii smaller than the radius of heating radiation *r*<*r _{h}*. The phase incursion constant over the cross-section is neglected in this expression, as it does not contribute to the thermal lens strength. For the majority of optical materials the magnitude of thermo-optical constant

*P*is an order of magnitude larger than the magnitude of thermo-optical constant

*Q*, hence the second term in the square brackets in (3) may be omitted and the expression will be written in the form

It will be clear from results of the experiments that this simplification is justified. The Eq. (5) does not take into account distortions caused by the change of the geometrical dimensions of the sample. Under the same conditions and approximations, the contribution to the thermal lens strength related to the sample expansion may be described by the following expression

where*α*= 1/

_{T}*L*∙d

*L*/d

*T*is the thermal expansion coefficient of the material. The phase multiplier is defined as

*F*is the focal distance of the thermal lens.

For thick samples the optical force of the thermal lens is determined by the thermo-optical constant *P* and the contribution of thermal expansion can be neglected. For thin samples, when the contributions of sample expansion and of refractive index changes to the optical force of thermal lens 1/*F* are comparable, necessary to measure its contributions separately. We measured phase distortions using the method of phase-shift interferometry described, for example, in [21, 23]. The schematic of the experiment is shown in Fig. 3(а). The studied sample was heated by continuous radiation of a diode laser at the wavelength of 940 nm. The beam radius of the heating radiation was 1.08 mm and had flat-top profile. A diode laser with central wavelength of 1060 nm was used as a probe signal source. Michelson interferometer was assembled for measurements. The measured sample was placed in one of its arms, and in the second arm the radiation was reflected from a reference mirror whose position was varied within half wavelength by applying voltage to piezo-electric stacks. A series of interferograms were recorded that were then processed to find phase distortion distributions [24]. Interference patterns were measured with and without pump radiation, at beam interference from the reference mirror and from three surfaces: the front surface of the studied sample, its rear surface and the mirror behind the sample by turn [Fig. 3(b)].

By subtracting the phase distributions obtained without pump radiation from the distortion with pump radiation we obtained difference phase distributions Δ. Taking into consideration that Δ_{1}, Δ_{2} and Δ_{3} were obtained at the interference from the front and rear surfaces of the studied sample and from the mirror behind the sample, respectively, we can obtain the following expressions

Taking into account that

for the change of sample’s length and refractive index we obtainThus, on finding phase distortions in the three above measurements we can separate the heat induced contributions associated with the change of sample’s geometrical dimensions and with the volume changes of the index of refraction. For further estimates we used the area of phase distortion distribution that did not exceed the transverse size of heating radiation. The experimental data were approximated by the paraboloid of revolution in the sense of minimizing root mean square deviation. The characteristic distributions of phase distortions and of the approximating paraboloid are depicted in Fig. 4.

The strength of the thermal lens was calculated from parameters of the approximating paraboloid. The approximation was done for phase distortion distributions Δ*L* and Δ*n* separately for all values of power in each studied sample. The dependence of thermal lens strength on heat release power is plotted in Fig. 5 for all the studied samples.

Using Eq. (5)-(7) and values of thermal lens strength for the studied samples of sesquioxide ceramics we determined values of the *P*/*κ* and *α _{T}*/

*κ*that are listed in Table 2. The values of

*κ*,

*α*and d

_{T}*n*/d

*T*found in the literature for all studied materials are also given in the Table 2.

The Eq. (5) was derived neglecting the term containing *Q*. Comparison of the data from Table 2 shows that the thermo-optical constant *P* is about an order of magnitude larger than *Q* (exact values of the stress-optic anisotropy ratio *ξ* for sesquioxides are unknown, as was mentioned above, but generally they were of order 1).

It is well known that impurity concentration and the method of manufacturing material greatly influence thermal conductivity [5]. The data obtained from the experiment measurements depend on the value of thermal conductivity, unknown for our samples. As was shown in [25, 26], the coefficients of linear expansion of the single-crystal and ceramic samples are the same and are practically independent of the concentration of the Yb^{3+} (at reasonable of its values 0-10 at.%). Using the values of the coefficients of linear expansion from [6, 25] we can estimate the thermal conductivity of our ceramic samples (see line 4 in Table 2).

The presence of ceramic grains and the boundaries between them in ceramics by the theory could leads only to a reduction of the thermal conductivity value. But the small fraction of volume boundaries in the ceramic material and the comparable number of inclusions and inhomogeneity of structure may result in the ceramics to the same value of thermal conductivity, as in the single crystal. Therefore, at all other equal conditions the value of thermal conductivity allow to judge the quality of the ceramic material. As can be seen from Table 2, the estimates of the thermal conductivity obtained for Y_{2}O_{3} and Lu_{2}O_{3} ceramic samples are very close to presented in [5] for single crystals with the same concentration of Yb^{3+}. This indicates comparable quality of investigated ceramic samples. The Sc_{2}O_{3} ceramic sample demonstrated quite smaller value of thermal conductivity coefficient than single-crystal.

Using the obtained values of the thermal conductivity from experimental data can be estimated value of thermo-optical constants of *P* and *Q _{eff}* (see line 5 and 6 in Table 2).

## 4. Discussion

Parasitic thermal effects (thermal depolarization and thermal lens) of three samples of Konoshima Chemical Co. laser ceramics Yb^{3+}:Lu_{2}O_{3} (C_{Yb}≈1.8 at.%), Yb^{3+}:Y_{2}O_{3} (C_{Yb}≈1.8 at.%), and Yb^{3+}:Sc_{2}O_{3} (C_{Yb}≈2.5 at.%), were measured in experiments as a function of laser radiation power. The results of measurements were used to calculate thermo-optical constants *κ*, *P* and *Q _{eff}*.

We’d like to note, that all three samples of sesquioxide ceramic samples had very good optical quality (did not distort profile of the transmitted radiation, had a weak “cold” scattering). Analyzing the data from Table 2, we’d like to mention, that our estimations of the thermal conductivity for studied sesquioxide ceramic samples are close to thermal conductivity of a single crystal with same concentration of Yb ion. It may be an additional proof of the high (comparable to the single crystals) quality of the investigated ceramic samples.

It could be seen, that the Lu_{2}O_{3} and Sc_{2}O_{3} for the same heat release power have very similar to each other *Q _{eff}/κ* value and for Y

_{2}O

_{3}sample

*Q*value is 30% less. Since the thermally induced depolarization is proportional to the squared

_{eff}/κ*Q*, this difference leads to more than twofold decrease of the integral depolarization value with the same heat dissipation.

_{eff}/κComparing the *Q _{eff}* values, and taking into account the dependence of the thermal conductivity of these materials on the concentration of Yb

^{3+}[5], we can say that the active elements with a high concentration of Yb

^{3+}in terms of thermally induced polarization distortions preferable Lu

_{2}O

_{3}, since at concentrations greater than 4 at.% thermally induced depolarization in this material will be less than in Y

_{2}O

_{3}and in Sc

_{2}O

_{3}. On the contrary, in the active elements with a low concentration value Yb

^{3+}is preferable to use a material Sc

_{2}O

_{3}, since its

*Q*less on 15% and the thermal conductivity with decreasing concentration of Yb

_{eff}^{3+}grows significantly faster than in Y

_{2}O

_{3}. At concentrations of less than 1 at.% of Yb

^{3+}thermally induced depolarization arising in Sc

_{2}O

_{3}is less than in the material of Y

_{2}O

_{3}or Lu

_{2}O

_{3}. In comparison for Y

_{3}Al

_{5}O

_{12}

*Q*= −12,8∙10

_{eff}^{−7}1/K at room temperature and wavelength of 1030 nm [27]. It should be noted that the above arguments are valid only on the assumption that the thermal conductivity in the sesqui-oxide ceramic samples has the same dependence on the concentration of Yb

^{3+}, as in the single crystals, which in itself requires experimental confirmation. I’d only note that good quality Y

_{3}Al

_{5}O

_{12}ceramics shows the same amount of thermal conductivity as a single crystal with the same dopant concentration [28].

Comparing the value of thermo-optical constant P, we note material Sc_{2}O_{3}, where its value is much smaller than in the materials Y_{2}O_{3}, and Lu_{2}O_{3}. As with the polarization distortion, power of the thermal lens is inversely proportional to thermal conductivity, which is a function of the concentration of Yb^{3+}. For Sc_{2}O_{3} value for *P*/*κ* less than in Y_{2}O_{3} and in Lu_{2}O_{3}, for the whole range of Yb^{3+} concentrations, represented in [5]. The lower the value thermo-optical constant *P*, the smaller the power of induced thermal lens in the volume of optical element. For comparison we give value the of P for Y_{3}Al_{5}O_{12}, which is 9,4∙10^{−6} [1/K] at room temperature for radiation at 1030 nm wavelength [27].

## 5. Conclusion

Summarizing the comparison of studied sesquioxide ceramics it can be said that Lu_{2}O_{3}, from the viewpoint of thermally induced effects is promising for use at high concentration of Yb^{3+} ions, which is the usual condition for the thin disc geometry of active element. Sc_{2}O_{3} is a promising material for the manufacture of active elements with a low concentration of Yb^{3+} ions, which is a common requirement for the elements with rod or slab geometry. At these geometries the small thermal lens strength, caused by the change of the refractive index due to the gradient of temperature and thermal stresses can be one of the decisive factors in the choice of material. Y_{2}O_{3} material took an intermediate position between the two previous media from the viewpoint of thermally induced effects and the value of thermo-optical constants and has nothing special to stand out. At the same time all three investigated sesquioxides demonstrated comparable or superior thermal properties compared with Y_{3}Al_{5}O_{12}.

## Acknowledgments

Authors acknowledge Program of Presidium of Russian Academy of Science “Extreme light and its application” and Mega Grant of Government of the Russian Federation “Diagnosis of new optical media for the advanced lasers” Nº 14.B25.31.0024.

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