Dispersion and anisotropy of thermo-optic coefficients in tetragonal GdVO<sub>4</sub> and YVO<sub>4</sub> laser host crystals: erratum

P. A. Loiko; K. V. Yumashev; V. N. Matrosov; N. V. Kuleshov

doi:10.1364/AO.54.004820

We found errors in the values of thermo-optic coefficients, $d n / d T$ , for tetragonal vanadate laser host crystals, ${YVO}_{4}$ and ${GdVO}_{4}$ , reported recently by us [1]. The values of $d n / d T$ were measured by a laser beam deviation method for a medium with a linear thermal gradient. It was proposed in [2]; a detailed description of this procedure can be found in [3]. The above mentioned errors occur in the measurements of the temperature gradient, $Δ T$ , in the studied sample. Vanadates have a relatively large thermal conductivity ( $κ \sim 10 W / mK$ [4]). In addition, we used a relatively small sample (height: 4–5 mm). This resulted in an unexpected and strong heat flow through the sample leading to the reduction of the $Δ T$ value. This effect was not significant in our previous experiments with double tungstates, $XRE {({WO}_{4})}_{2}$ [5], which possess much lower thermal conductivity ( $κ \sim 1 - 2 W / mK$ ).

To produce a linear thermal gradient in our rectangular samples, we used two massive copper blocks. They were attached to two opposite lateral faces of the samples (heat grease was used to provide thermal contact). The blocks were designed so that the size of their faces in contact with the sample was the same as the sample face itself. In this work, we drilled small holes ( $\sim 1 mm$ diameter) in the sample/block interface (one hole for the “cold” block and a second one for the “hot” block). Two sensitive calibrated thermocouples (type K, chromel–alumel) were inserted into these holes filled with heat grease. The precision of the determination of the $Δ T$ value was $\sim 1 K$ .

The measurements of $d n / d T$ coefficients were performed close to room-temperature. The temperature of the “cold” sample surface was $\sim 0 ° C$ and the temperature of the “hot” one was $\sim 30 ° C$ .

With the renewed setup, we carefully repeated the measurements described in [1]. The results on the principal thermo-optic coefficients (TOCs), $d n_{o} / d T$ and $d n_{e} / d T$ , for ${YVO}_{4}$ and ${GdVO}_{4}$ crystals are shown in Fig. 1 as solid circles. Here, the error bars are smaller than the symbols. The total error of the measurement of the $d n / d T$ value is $\sim 0.5 \times 10^{- 6} K^{- 1}$ . For both vanadates, thermo-optic coefficients obey the relation $d n_{o} / d T > d n_{e} / d T$ and decrease with the wavelength. This is in agreement with [1]. However, their values are substantially larger than reported in [1]. In particular, at 1.06 μm, $d n_{o} / d T = 13.8$ and $d n_{e} / d T = 8.0 \times 10^{- 6} K^{- 1}$ for the ${YVO}_{4}$ crystal, and $d n_{o} / d T = 12.9$ and $d n_{e} / d T = 7.9 \times 10^{- 6} K^{- 1}$ for the ${GdVO}_{4}$ one.

Fig. 1. Dispersion of principal thermo-optic coefficients for (a) ${YVO}_{4}$ and (b) ${GdVO}_{4}$ crystals: symbols are the experimental data (solid circles, this study; open circles and triangles, adopted from [6–8]) and curves are fittings of our data with Eq. (1).

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The dispersion of newly measured TOCs was fitted using the model described in detail in [5]. The following expression was used (it was not described in [1]):

d n / d T = - α_{vol} \frac{(n_{\infty}^{2} - 1)}{2 n (λ)} \frac{λ^{2}}{λ^{2} - λ_{g}^{2}} - \frac{1}{E_{g}} \frac{d E_{g}}{d T} \frac{(n_{\infty}^{2} - 1)}{2 n (λ)} {(\frac{λ^{2}}{λ^{2} - λ_{g}^{2}})}^{2} .

Here,

λ

denotes the light wavelength,

E_{g}

is the bandgap and

d E_{g} / d T

is its temperature derivative,

λ_{g} [μm] = 1.2398 / E_{g}

[eV],

α_{vol}

is the volumetric thermal expansion,

n (λ)

corresponds to the Sellmeier equation for the refractive index, and

n_{\infty}

is the refractive index in the long-wavelength limit. Experimental data in Fig. 1 were fitted with Eq. (1), and

E_{g}

and

d E_{g} / d T

were the variable parameters. The expressions for

n (λ)

were taken from [6];

α_{vol}

was determined as

2 α_{a} + α_{c}

. Here,

α_{a}

and

α_{c}

are the linear coefficients of thermal expansion along the

a

- and

c

-axes, respectively. The following values of

n_{\infty}

were used:

n_{o \infty} = 1.9206

and

n_{e \infty} = 2.1216

for

{YVO}_{4}

,

n_{o \infty} = 1.9457

and

n_{e \infty} = 2.1514

for

{GdVO}_{4}

.

Thermal expansion coefficients were remeasured with a horizontal dilatometer; see details in [1]. The precision was $\sim 0.05 \times 10^{- 6} K^{- 1}$ . The values are $α_{a} = 1.90$ and $α_{c} = 8.34 \times 10^{- 6} K^{- 1}$ for ${YVO}_{4}$ , $α_{a} = 1.19$ and $α_{c} = 8.10 \times 10^{- 6} K^{- 1}$ for ${GdVO}_{4}$ . These parameters are in good correspondence with the results from an independent study by Sato and Taira [6].

The best-fitting curves for TOCs are shown in Fig. 1. Typical values of the variable parameters are $E_{g} = 3.8 \dots 4.2 eV$ and $d E_{g} / d T = - 0.8 \dots 2 \times 10^{- 4} eV / K$ . The modeling allows us to derive simple analytical thermo-optic dispersion formulas [5]:

d n / d T = [A_{0} + \frac{A_{1}}{λ^{2}} + \frac{A_{2}}{λ^{4}} + \frac{A_{3}}{λ^{6}}] \times 10^{- 6} K^{- 1} .

Here,

λ

is expressed in micrometers and

A_{0 - 3}

are the expansion coefficients. Their values are listed in Table 1. The equivalence between Eqs. (1) and (2) is

\sim 0.2 \times 10^{- 6} K^{- 1}

. Thermo-optic dispersion formulas are valid for the spectral range of 0.4–2 μm.

Table 1. Expansion Coefficients in the Thermo-Optic Dispersion Formulas^a for ${YVO}_{4}$ and ${GdVO}_{4}$ Crystals

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Previously, $d n / d T$ coefficients in ${YVO}_{4}$ and ${GdVO}_{4}$ were studied by Zelmon et al. [7,8] with a conventional minimum deviation method and by Sato and Taira [6] using a highly accurate interferometric setup. These data are shown in Fig. 1 (as open circles and triangles), and they are in a good agreement with the proposed dispersion formulas. A more detailed comparison of our data with the results from [6–8] at a reference wavelength of 1.1 μm is performed in Table 2. The difference between three sets of $d n / d T$ values obtained by three independent methods does not exceed $2 \times 10^{- 6} K^{- 1}$ .

Table 2. Comparison of Thermo-Optic Coefficients^a for ${YVO}_{4}$ and ${GdVO}_{4}$ Crystals Reported So Far

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The data for the thermal coefficients of the optical path [TCOP, $W = d n / d T + (n - 1) α$ ] obtained during the evaluation of thermo-optic coefficients are shown in Fig. 2 (here, points are the experimental data and curves are their fitting; for details refer to [1]). The exact values at the wavelengths of 633 and 1064 nm are summarized in Table 3. Here, notations $a$ -cut and $c$ -cut represent the light propagation direction (along the $a$ axis and $c$ axis, respectively).

Table 3. Thermal Coefficients of the Optical Path (TCOP)^a for ${YVO}_{4}$ and ${GdVO}_{4}$ Crystals

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Fig. 2. Dispersion of thermal coefficients of the optical path (TCOP) for $a$ -cut and $c$ -cut (a) ${YVO}_{4}$ and (b) ${GdVO}_{4}$ crystals: symbols are the experimental data and curves are their fitting.

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As a final remark, we discuss the total error for the evaluation of the $d n / d T$ coefficient by a laser beam deviation method. This evaluation contains two steps. First, TCOP is determined on the basis of the measured beam deviation $Δ θ$ [9]:

TCOP = Δ θ \frac{H}{L} \frac{1}{Δ T} .

Here,

H

is the sample height along the thermal gradient,

L

is the sample length along the light propagation direction, and

Δ T

is the temperature gradient.

H

(

\sim 6 mm

) and

L

(

\sim 9 mm

) dimensions are determined with a precision of a few micrometers;

Δ θ

is determined with a precision of

\sim 0.03 mrad

(a CCD camera is used). This value is limited by temporal temperature instabilities in the sample leading to a jitter of the beam spot on a CCD camera. Second significant error arises from the above mentioned problem of correct determination of

Δ T

(with the precision of

\sim 1 K

). Finally, the error for the TCOP value is

\sim 0.5 \dots 1 \times 10^{- 6} K^{- 1}

. The second step is the evaluation of the

d n / d T

value itself as

d n / d T = TCOP - (n - 1) α

. Here, the sources of error are the values of the refractive index

n

and thermal expansion coefficient

α

that are typically determined independently. For refractive indices calculated from the Sellmeier formulas, a typical precision is

\sim 0.002

, and for the measurement of

α

performed in this work, the precision was

\sim 0.05 \times 10^{- 6} K^{- 1}

. Thus, total error in the determination of thermo-optic coefficient was

\sim 0.5 \dots 1 \times 10^{- 6} K^{- 1}

.

The above mentioned error in the determination of the temperature gradient results in an underestimation of TCOP values in our previous work [1] as compared with the present study (the TCOP coefficients reported in [1] are $\sim 2.5$ times lower). In addition, as $d n / d T$ coefficients are not measured directly but evaluated as $d n / d T = TCOP - (n - 1) α$ , the difference between their values reported in [1] and in this study is relatively high. In particular, for ${YVO}_{4}$ at a wavelength of 1.06 μm, we reported $d n_{o} / d T = 3.1$ and $d n_{e} / d T = 0.6 \times 10^{- 6} K^{- 1}$ [1]; compare these data with the corrected values from Table 2.

References

1. P. A. Loiko, K. V. Yumashev, V. N. Matrosov, and N. V. Kuleshov, “Dispersion and anisotropy of thermo-optic coefficients in tetragonal GdVO₄ and YVO₄ laser host crystals,” Appl. Opt. 52, 698–705 (2013). [CrossRef]

2. S. Vatnik, M. C. Pujol, J. J. Carvajal, X. Mateos, M. Aguiló, F. Díaz, and V. Petrov, “Thermo-optic coefficients of monoclinic KLu(WO₄)₂,” Appl. Phys. B 95, 653–656 (2009). [CrossRef]

3. P. Loiko, F. Druon, P. Georges, B. Viana, and K. Yumashev, “Thermo-optic characterization of Yb:CaGdAlO₄ laser crystal,” Opt. Mater. Express 4, 2241–2249 (2014). [CrossRef]

4. Y. Sato and T. Taira, “The studies of thermal conductivity in GdVO₄, YVO₄, and Y₃Al₅O₁₂ measured by quasi-one dimensional flash method,” Opt. Express 14, 10528–10536 (2006). [CrossRef]

5. P. A. Loiko, K. V. Yumashev, N. V. Kuleshov, G. E. Rachkovskaya, and A. A. Pavlyuk, “Thermo-optic dispersion formulas for monoclinic double tungstates KRe(WO₄)₂ where Re = Gd, Y, Lu, Yb,” Opt. Mater. 33, 1688–1694 (2011).

6. Y. Sato and T. Taira, “Highly accurate interferometric evaluation of thermal expansion and dn/dT of optical materials,” Opt. Mater. Express 4, 876–888 (2014). [CrossRef]

7. D. E. Zelmon, J. J. Lee, K. M. Currin, J. M. Northridge, and D. Perlov, “Revisiting the optical properties of Nd doped yttrium orthovanadate,” Appl. Opt. 49, 644–647 (2010). [CrossRef]

8. D. E. Zelmon, J. M. Northridge, J. J. Lee, K. M. Currin, and D. Perlov, “Optical properties of Nd-doped rare-earth vanadates,” Appl. Opt. 49, 4973–4978 (2010). [CrossRef]

9. P. A. Loiko, K. V. Yumashev, N. V. Kuleshov, and A. A. Pavlyuk, “Thermo-optic coefficients and thermal lensing in the Nd-doped KGd(WO₄)₂ laser crystals,” Appl. Opt. 49, 6651–6659 (2010). [CrossRef]

		Expansion Coefficients ^b
Crystal	TOC	$A_{0}$	$A_{1}$ , ${μm}^{2}$	$A_{2}$ , ${μm}^{4}$	$A_{3}$ , ${μm}^{6}$
${YVO}_{4}$	$d n_{o} / d T$	11.86	1.952	0.0780	0.0391
	$d n_{e} / d T$	6.34	1.872	−0.0418	0.0693
${GdVO}_{4}$	$d n_{o} / d T$	11.73	2.165	0.0345	0.0558
	$d n_{e} / d T$	6.78	2.213	−0.1429	0.1012

	${YVO}_{4}$		${GdVO}_{4}$
Ref.	$d n_{o} / d T$	$d n_{e} / d T$	$d n_{o} / d T$	$d n_{e} / d T$
This paper	13.8	8.0	12.9	7.9
Refs. [7,8]	14.0	9.0	11.8	9.0
Ref. [6]	15.5	8.4	15.2	9.9

		${YVO}_{4}$		${GdVO}_{4}$
$λ$ , nm	Polariz.	$a$ -Cut	$c$ -Cut	$a$ -Cut	$c$ -Cut
633	$E ∥ c$	15.1	—	15.3	—
	$E ⊥ c$	20.5	26.2	20.0	27.5
1064	$E ∥ c$	10.4	—	9.3	—
	$E ⊥ c$	15.9	22.0	14.1	21.4

		Expansion Coefficients ^b
Crystal	TOC	$A_{0}$	$A_{1}$ , ${μm}^{2}$	$A_{2}$ , ${μm}^{4}$	$A_{3}$ , ${μm}^{6}$
${YVO}_{4}$	$d n_{o} / d T$	11.86	1.952	0.0780	0.0391
	$d n_{e} / d T$	6.34	1.872	−0.0418	0.0693
${GdVO}_{4}$	$d n_{o} / d T$	11.73	2.165	0.0345	0.0558
	$d n_{e} / d T$	6.78	2.213	−0.1429	0.1012

	${YVO}_{4}$		${GdVO}_{4}$
Ref.	$d n_{o} / d T$	$d n_{e} / d T$	$d n_{o} / d T$	$d n_{e} / d T$
This paper	13.8	8.0	12.9	7.9
Refs. [7,8]	14.0	9.0	11.8	9.0
Ref. [6]	15.5	8.4	15.2	9.9

Dispersion and anisotropy of thermo-optic coefficients in tetragonal GdVO₄ and YVO₄ laser host crystals: erratum

Abstract

References

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Figures (2)

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Applied Optics