## Abstract

We present the measurement results for coefficients of linear thermal expansion (CTE) and thermo-optic coefficients of uniaxial yttrium orthovanadate YVO_{4} crystals in the temperature range 80 - 320 K. The CTE data were obtained for the directions parallel (*α***_{||}**) and perpendicular (

*α*) to the optical

_{⊥}**-axis of YVO**

*c*_{4}. The corresponding polynomial expressions for the observed temperature dependencies were derived. At 80 K the CTE values

*α*and

_{⊥}~0.11∙10^{−6}K^{−1}*α*

_{||}*~2.43∙10*, are approximately 5.3% and 28% of those at room temperature, respectively. Thermo-optic coefficients corresponding to ordinary and extra-ordinary beams were measured for two wavelengths, 633 and 1570 nm. To the best of our knowledge, the CTE and

^{−6}K^{-}*dn/dT*data for this important laser host in the temperature range 80 - 320 K are reported for the first time.

©2012 Optical Society of America

## 1. Introduction

Recent developments in cryogenically cooled, eye-safe lasers based on Er^{3+}-doped YVO_{4} and GdVO_{4} crystals, demonstrated record high optical-to-optical laser efficiencies and a great potential for further power scaling [1,2]. It is well understood that the thermal conductivity, the coefficient of linear thermal expansion (CTE) and the thermo-optic coefficient (*dn/dT*) are the key thermo-optic properties of the laser material which ultimately define its power scaling potential with preserved beam quality [3]. In the case of cryogenic solid-state laser development, the knowledge of these three parameters in the *80 – 320 K* temperature range is critically important for laser design. Thermal conductivity data for orthovanadate crystals in this range are available from a few sources (e.g., [4–6]). Meanwhile, to the best of our knowledge, the data on CTE and *dn/dT* below room temperature has not been reported, though it is readily available in the temperature range from *+ 20* to *+ 1000 °C* [4,7–10]. Based on growing interest in rare-earth doped orthovanadates, viable competitors to conventional YAG as cryogenic laser hosts, systematic measurements of their thermal and thermo-optic properties from the temperature of liquid nitrogen to a room temperature are useful in order to remedy the dearth of data.

The purpose of our work was to eliminate this data vacuum for the YVO_{4} single crystal. Reported here are the measurement results for the coefficients of linear thermal expansion and thermo-optic coefficients for YVO_{4} in the *80 – 320 K* temperature range. The measurements were carried out using Twyman-Green interferometry and the obtained results were compared with the published room temperature data on CTE and *dn/dT*.

## 2. Experiments and results

The temperature-dependent amplitude of lattice vibrations affects the mean inter-ion bond length. As a result, linear thermal expansion and refraction indices, which define optical path length inside a laser crystal, become temperature-dependent as well. Any thermally induced change in the optical path length can be measured by interferometric methods with high precision. If two parallel faces of the crystal under investigation form an interferometer, then it can be probed with a polarized laser beam to produce an interference fringe pattern. The interval between the maxima and minima of this pattern corresponds to a round-trip optical path length change equal to half the wavelength (*λ*) of the probe beam and can be determined as:

*L*is the crystal length,

*T*is the crystal temperature

*n(λ,T)*is its refractive index and

*dT*is the temperature increment.

The temperature-induced shift of interference fringes causes modulation of the output of the photodiode, positioned behind the slit in an opaque screen, see Fig. 1
. The fractional change in the optical path length, *γ(λ,Τ),* can be calculated from Eq. (1):

*dn/dT*is the thermo-optic coefficient, i.e., the temperature change of the refractive index (at the wavelength of a probe beam

*λ*) and

*α*is the linear thermal expansion coefficient, CTE. Thus, as follows from Eq. (2), the measurements of the

*γ(λ,Τ)*value will simultaneously reflect the changes in both CTE and

*dn/dT*.

The YVO_{4} is a uniaxial crystal with tetragonal symmetry **(**space group D_{4h}), and is known to have strong anisotropy in thermal and optical properties along and perpendicular to its optical *c**-*axis. Therefore, thermal expansion and thermo-optic coefficients have to be characterized by four parameters: *α _{⊥}*,

*α*,

_{||}*dn*and

_{o}/dT,*dn*, where subscripts || and

_{e}/dT*⊥*denote directions of thermal expansion parallel and perpendicular to the optical

**-axis, respectively. For refraction indices we use the conventional subscripts -**

*c***(ordinary) and**

*o***(extraordinary) which are defined by the vector of the electric field, normal and parallel relative to the**

*e***-axis. Thus, for a uniaxial crystal, four independent measurements are required. Because the refractive indices**

*c**n*and

_{o}*n*are functions of temperature and wavelength, their derivatives

_{e}*dn*and

_{o}/dT*dn*are also temperature and wavelength dependent.

_{e}/dTFigure 1 shows an experimental set-up for the measurements of *γ(λ,T)*. Either a helium-neon (He-Ne) laser (*633 nm*) or a *TEM _{00}*, fiber coupled diode laser (

*1570 nm*) was used as a probe source. Every tested crystalline sample was mounted inside the liquid nitrogen cryostat and was initially cooled down to

*77 K*. The temperature of the sample was then raised very slowly (~

*0.5° K /min*) in order to maintain a uniform temperature distribution in the sample. The crystal was single-point fixed to a cold-plate in order to provide full “freedom” of its thermal expansion/contraction with changes in temperature. The temperature of the sample was monitored with a calibrated silicon diode sensor (model DT-470-SD, Lake Shore Cryotronics) attached to the top of the crystalline slab. The sensor was connected to a temperature controller (model 335, Lake Shore Cryotronics). The sinusoidal photodiode output and the temperature of the sample were recorded simultaneously.

Shown in Figs. 1a–1c are three different orientations of the uniaxial YVO_{4} crystal with respect to the light polarization and the propagation direction of the probe beam. For every chosen wavelength of the probe beam, these orientations allow us to obtain three independent functions *γ* = *f(T)* containing different combinations of two CTEs and two thermo-optic coefficients:

In order to find all four aforementioned components${\alpha}_{\left|\right|}$,${\alpha}_{\perp}$,$\frac{1}{{n}_{o}}\cdot \frac{d{n}_{o}}{dT}$ and $\frac{1}{{n}_{e}}\cdot \frac{d{n}_{e}}{dT}$, either *α _{||}* or

*α*has to be known in advance or measured independently. With the first probe beam, it is necessary to measure all three coefficients of

_{⊥}*γ*

_{1},

*γ*

_{2}and

*γ*

_{3}. While using the second beam (with a different wavelength), it is sufficient to measure only two fractional coefficients because CTEs

*α*and

_{||}*α*are wavelength-independent.

_{⊥}As was already mentioned, there is no available data on the linear thermal expansion of YVO_{4} below room temperature. Therefore, in order to determine CTE *α _{⊥}(T)* in this range, we built a free space interferometer by carefully mounting two small fused silica wedge-prisms on top of the YVO

_{4}crystal, see Fig. 1, inset

**. This interferometer was probed with the He-Ne beam at**

*d**633 nm*in a manner similar to that described above.

Figure 2a
shows the measured temperature behavior of *α _{⊥}(T)*. Figures 2b–2d show fractional changes (

*γ*,

_{1}(T)*γ*and

_{2}(T)*γ*) in the optical path length, for the three different crystal orientations with respect to the probe beam propagation direction and the probe beam polarization (shown in Figs. 1a, 1b, 1c). The measurements were performed on a

_{3}(T)*20.1 mm*long,

*3.07 mm*thick, sample provided by Photop Technologies, Inc.

In all our measurements, crystalline samples were initially cooled down to approximately 77.7 – 77.8 K and then their temperatures were slowly brought up to approximately 320 −325K. For each recorded temperature point in the above range, there was a corresponding point of the intensity of the fringe pattern. The resulting “sinusoidal” function had a variable period, which shortens with the rising temperature. However, it is an extremely rare event, when the minima (or maxima) of the fringe pattern coincide with the lowest temperature in the available temperature range. In addition, it was difficult to identify a peak (or a valley) of the fringe pattern for temperatures below ~100K, especially for very “slow” changes in thermal expansion. Therefore, in Fig. 2, for fractional changes *γ*, we presented our experimental data from the temperature of 80K and up, see Figs. 2b–2d; but for the CTE *α _{⊥}* - only from 100K.

The fractional changes *γ _{1}(T)*,

*γ*and

_{2}(T)*γ*, measured with the same wavelength of the probe beam, and the CTE

_{3}(T)*α*can be fit to a second order polynomial respectively:

_{⊥}(T)*α*, shown in Fig. 2a, were reliably measured only above 100K, a linear approximation of this data with Eq. (6) can easily be extended down to the temperature of 80K.

_{⊥}(T)Using the expressions (3)–(6), the temperature dependence of the CTE *α*_{||} can be defined as:

Figure 3
depicts temperature dependencies of CTEs *α _{⊥}* and

*α*plotted according to expressions (6) and (7). We can compare the measured values of

_{||}*α*and

_{⊥}*α*with published data [4,7–9]. As it was mentioned before, a comparison can only be done for room temperature values. There is a noticeable spread in the CTE data obtained by different authors, especially for the

_{||}*α*. Indeed,

_{⊥}*α*values were reported in the range of (

_{⊥}*1.69 – 4.43*)∙

*10*while α

^{−6}K^{−1}*values were reported in the range of*

_{||}*(8.19 – 11.4)∙10*. Our measurements of

^{−6}K^{−1}*α*=

_{⊥}*2.2·10*(expansion normal to the

^{−6}K^{−1}**-axis) and**

*c**α*=

_{║}*8.4·10*(expansion parallel to the

^{−6}K^{−1}**-axis) are falling well within the published range and are in the best agreement with the data for YVO**

*c*_{4}at

*300 K*published in [8].

It should be noted that at cryogenic temperatures the measured values of *α _{⊥}* and

*α*are only

_{||}*5.3%*and

*28%*of the corresponding CTE values measured around

*300 K*. Their ratio, which reflects the degree of optical anisotropy, becomes even larger with temperature reduction.

A similar fitting procedure for *γ*_{1}*(T)* and *γ*_{2}*(T)* dependencies measured with the *1570 nm* probe beam (see Figs. 2b, 2c) in the *80 - 320 K* range yields:

After subtracting Eq. (6) from Eqs. (3), (4), (8) and (9) and subtracting the expression (7) from expression (5), the refractive indices *n _{o}* and

*n*were derived by integrating simple first order differential equations. The initial values of these indices at room temperature were taken from [10]: ${n}_{o}^{}(633nm)=1.99279$,${n}_{e}^{}(633nm)=2.21533$,${n}_{o}^{}(1570nm)=1.94426$and ${n}_{e}^{}(1570nm)=2.14809$. Then, using the expressions (1)–(7) and the corresponding refractive indices of the YVO

_{e}_{4}, the thermo-optic coefficients $\frac{dn}{dT}$ can be calculated.

Figures 4
and 5
show the behavior of refractive indices (*n _{o}, n_{e}*) and thermo-optic coefficients ($\frac{d{n}_{o}^{}}{dT}$, $\frac{d{n}_{e}^{}}{dT}$) for the

*633 nm*and

*1570 nm*wavelengths We believe that these data are reported for the first time. However, for the room temperature and above, the thermo-optic coefficients were measured at the

*633 nm*wavelength in [10]: $\frac{d{n}_{o}^{}}{dT}~17.4\cdot {10}^{-6}{K}^{-1}$, $\frac{d{n}_{e}^{}}{dT}~12.5\cdot {10}^{-6}{K}^{-1}$ and can be compared with our room temperature data. It appears that our measurements yielded slightly lower values: $\frac{d{n}_{o}^{}}{dT}(633nm)=15.6\cdot {10}^{-6}{K}^{-1}$ and $\frac{d{n}_{e}^{}}{dT}(633nm)=9.5\cdot {10}^{-6}{K}^{-1}$.

The obtained temperature dependencies can be approximated by the expressions:

It can be seen from Fig. 5 that the thermo-optic coefficients of YVO_{4} are considerably lower at cryogenic temperature compared to their room temperature values (3.5 – 5 times). There is also a noticeable anisotropy in the thermo-optic coefficients at room temperature: $\frac{d{n}_{o}^{}}{dT}$ >> $\frac{d{n}_{e}^{}}{dT}$. At cryogenic temperatures, anisotropy remains strong at 1570 nm, but becomes significantly smaller at 633 nm. A comparison of thermo-optic coefficients of the YVO_{4} crystal at different wavelengths (*633 nm* and *1570 nm*), but at the same temperature, shows that these coefficients are always higher at *1570 nm*. At the same time, at cryogenic temperatures, thermo-optic coefficients of YVO_{4} are very small for both wavelengths.

## 3. Conclusions

We report the measurement results for coefficients of linear thermal expansion and thermo-optic coefficients of a uniaxial YVO_{4} crystal over a wide range of temperatures from room- down to the liquid nitrogen temperature. To the best of our knowledge, CTE and *dn/dT* values for this important laser host, in the *80 - 320 K* temperature range, are reported for the first time. The CTE data were obtained for two directions - parallel (*α _{║}*) and perpendicular (

*α*) to the optical axis of the crystal. The corresponding polynomial expressions for the observed temperature dependencies of CTE were derived. Thermo-optic coefficients

_{⊥}*dn/dT*were measured for two wavelengths,

*633 nm*and

*1570 nm*, for ordinary and extraordinary beams. The CTE values at cryo temperatures (~

*80K*)

*α*and

_{⊥}~0.11∙10^{−6}K^{−1}*α*are approximately 5.3% and 28% of those at room temperature. CTE of YVO

_{║}~2.43∙10^{−6}K^{−1}_{4}in the direction perpendicular to the optical axis is one of the lowest among known laser hosts, offering significant advantage in power scaling with preserved beam quality compared to conventional YAG.

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