Laser Method for Studying Temperature Distribution within Yb:YAG Active Elements

Abstract

Currently, laser systems based on active elements doped with Yb³⁺ with simultaneously high pulse repetition rates and high peak power are in demand for many applications. High thermal load of active elements is the primary limiting factor for average power scaling. Experimental investigation of temperature distribution in active elements is of particular importance for estimation of cooling efficiency and for thermal processes’ monitoring. In the present work, the method of dynamic laser thermometry is proposed for temperature distribution investigation within cryogenically cooled Yb³⁺-doped active elements. The method is based on the dependence of the Yb³⁺ ion absorption cross-section on temperature at a wavelength of 1030 nm. The method was tested to study the 2D temperature map of the Yb:YAG active element of the high-power, diode-pumped, cryogenically cooled laser amplifier. The best measurement accuracy ±3 K is achieved at the maximal temperature 176 K. The results of numerical simulation are in good agreement with the experimental data. On the basis of the investigation, the quality of the cooling system is evaluated. The advantages and other possible applications of the method are discussed.

1. Introduction

Currently, laser systems based on active elements doped with Yb³⁺ with simultaneously high pulse repetition rates and high peak power [1,2,3,4] are in demand. The radiation of such systems can be used for pumping Ti:Sa laser systems [5], optical parametric chirped pulse amplification [6,7], high harmonic generation [6,8,9,10], and other applications [11,12].

In high-power systems, high temperature of active elements is the primary limiting factor for further power scaling. Strong nonuniform heating of the active elements deteriorates spatial beam profile and decreases the power of radiation. In order to suppress thermal effects, aside from complex cooling systems, special geometries of active elements are proposed [13], and the active elements with non-uniform distribution of dopant ions are designed [14,15]. A combination of factors involved in the thermal load requires experimental investigation of temperature distribution in active elements. At room temperatures, infrared cameras are often used for this purpose [16,17,18,19]. However, in the case of cryogenic cooling, most of the materials used for windows of vacuum chambers are opaque for thermal radiation (at the temperature of 100 K, the wavelength of thermal radiation is about 30 μm). Temperature in active elements doped with Yb³⁺ can be obtained from the upconversion spectrum of Er³⁺ impurities [20,21]. In the case of cryogenically cooled active elements, the temperature dependence of fluorescence spectrum is often used. Both the ratio between the intensities at different wavelengths [22] and the entire spectrum in a certain range can be investigated [23]. To reach a high spatial resolution in analyzing the spectrum, it is necessary to use a confocal microscope [23]. Various temperature investigation methods based on the temperature dependence of fluorescence spectrum were compared in the cases of cryogenically-cooled Yb:YAG [24] and Yb:YLF [25] active elements. A technique for investigation of temperature distribution, based on temperature dependence of the absorption cross-section at the wavelength of the zero-phonon line, was reported [26].

In this contribution, we present a method for investigation of the temperature distribution in Yb:YAG active elements. The method was tested to study 2D temperature map of the active element of high-power, diode-pumped, cryogenically cooled multipass multidisk laser amplifier [27]. In the experiment, it is shown that, at the maximal temperature ~176 K, the value can be determined with as high precision as ±3 K. On the basis of the investigation, the quality of the cooling system is evaluated.

2. Laser Thermometry Method

The method of dynamic laser thermometry is based on the dependence of the Yb³⁺ ion absorption cross-section on temperature at the wavelength of 1030 nm. Previously, in the test experiment, the feasibility of the method was shown [28]. In this work, the method applicability for the measurement of 2D temperature distribution of active element in laser amplifier with high-power diode pumping is demonstrated. The underlying principle of the method is that laser radiation is absorbed in an active element. The absorption is dependent on temperature according to equation:

$$I\left(x,y\right)=I_0\left(x,y\right)\cdot \exp \left(-{\displaystyle \underset{0}{\overset{L}{\int }}\alpha \left(T\left(x,y,z\right)\right)dz}\right),$$

(1)

where z is the beam axis, the x–y plane is perpendicular to z, I₀(x,y) is the probing beam intensity distribution before passing the element, I(x,y) is the intensity distribution after passing the element, α(T) is the temperature-dependent absorption coefficient, T(x,y,z) is the volumetric temperature distribution, and L is the optical path length.

The optical power transmission coefficient γ is defined as

$$\gamma =\frac{{\displaystyle \underset{S}{\iint }{I}_0\left(x,y\right)\cdot \exp \left(-{\displaystyle \underset{0}{\overset{L}{\int }}\alpha \left(T\left(x,y,z\right)\right)dz}\right)dxdy}}{{\displaystyle \underset{S}{\iint }{I}_0\left(x,y\right)dxdy}},$$

(2)

where S is the interaction area of the radiation spot and the active element.

The temperature distribution in the element is inherently smooth and continuous, and, using the mean value theorem, the integral in the numerator can be written as

$${\displaystyle \underset{0}{\overset{L}{\int }}\alpha \left(T\left(x,y,z\right)\right)dz}=\alpha \left(T\left(x,y,z_c\right)\right)L=\alpha \left(T_c\left(x,y\right)\right)L,$$

(3)

where z_c is the coordinate that satisfies the equality in range [0, L], and T_c is the temperature value in this point. Considering that S is a simply connected domain and is limited, the expression (2) can be written as

$$\gamma =\frac{S}{P_0}I\left(x_c,y_c\right)\cdot \exp \left(-\alpha \left(T_c\left(x_c,y_c\right)L\right)\right)=\frac{P\left(T_v\right)}{P_0},$$

(4)

where (x_c,y_c) is the the point inside S, T_v is the temperature value of T_c in this point, and P₀ is the power of the incident radiation. From (4), it immediately follows that, in terms of total absorption, an element with an arbitrary temperature distribution always has an equivalent element uniformly heated to the temperature T_v. If the probing beam spot size is small enough in comparison to a characteristic size of temperature distribution features, the intensity distribution in (4) can be omitted:

$$\gamma \approx \exp \left(-\alpha \left(T_{v}L\right)\right).$$

(5)

Therefore, by forming a spot small enough compared to the area being examined and scanning in the x–y plane, it is possible to obtain a two-dimensional distribution of the equivalent temperature T_v over the element.

At the first stage of the experiment, the dependence of transmission coefficient γ on temperature is determined in the absence of pumping. The measurements are conducted under slow uniform cooling of the active element. There are optical losses independent from temperature but dependent on the beam spot position. To take them into account, the dependence of the transmittance on the scanning laser spot position is measured when the temperature of the element is steady.

Next, the measurements are carried out in the presence of pump radiation. To separate the processes of absorption and amplification of the probing signal, it is necessary to measure the transmittance after the end of the pump pulse. In the case of a high pulse repetition rate or continuous pumping, it is necessary to interrupt the pumping for a short period of time. The lifetime of the excited state of the Yb³⁺ ion is ~1 ms, and, therefore, a break between pulses of the order of 5 ms (~10⁻² of initial magnitude) is sufficient to eliminate population inversion influence on the transmittance. At the same time, in bulk Yb:YAG elements, heat transfer processes have characteristic times of the order of seconds. Thus, it follows that the absorption dynamics in the absence of pump radiation depends only on thermal processes.

The experimental results can be used to verify numerical simulation data. In turn, with the help of numerical simulation, it is possible to calculate the three-dimensional temperature distribution in the active element. In this work, a temperature field model is used [29]. In the model, the system of balance equations is solved jointly with the heat equation, which makes it possible to take into account the temperature dependence of the absorption cross section of Yb³⁺ ions at the pump wavelength (936 nm) along with the temperature dependencies of the laser level population. The three-dimensional non-stationary heat equation includes the calculation of temperature distributions in the active element and in the components of the cryogenic cooling system [14].

3. Results

In this work, the method of dynamic laser thermometry was used to study the temperature distribution in the pumped region of the active element of a multidisk amplifier [27]. The active elements of the amplifier are the diffusion-welded YAG-Yb:YAG active mirrors. The thickness of the doped part (9.8 at.%) is 3.75 mm, and the undoped part is as thick as 2 mm. The disk diameter is 25.4 mm. The elements are attached to a copper cubes (45 mm × 45 mm × 50 mm). The copper cube in turn is attached to the cryogenic cooler head.

To measure the transmittance map, an optical scheme with two movable mirrors was used (Figure 1). The first mirror displaces beam in the horizontal direction, and the second in the vertical direction. The scanning laser is a Yb:YAG continuous wave laser with the spectral width of 1.6 nm, centered at 1030 nm. The radiation was focused into the active element into a waist with a diameter of 260 ± 14 μm at the level 1/e², which corresponds to a Rayleigh length of about 10 cm.

The beam power of the scanning laser in the crystal was 6 mW. The beam was attenuated to minimize heating of the scanned area caused by the absorption of scanning radiation by the active element. The volumetric power density of the scanning laser was estimated to be at the level of 10 W/cm³. The average power of the diode pump laser in the experiments was 108 ± 3 W, and the beam diameter was 2 mm, which corresponds to a volumetric power density on the order of 10 kW/cm³. Thus, the temperature of the active element is governed by the parameters of the pump radiation. To measure the power of the probing beam, photodiodes with a time constant of no more than 10 µs were used.

The active element was gradually cooled from room temperature to a temperature of 40 K. The temperature of the active element was measured by a thermocouple placed inside the copper cube just behind the active mirror. Based on the cooler passport data and the known parameters of the cooling system, the cooling rate of the system was estimated to be 0.14 K/s. The main contribution to the heat capacity of the system is made by the massive copper cube and cryogenic cooler own heat capacity. Using a thermocouple, the cooling rate of the active element was measured; it is 0.12 K/s, which is close to the calculated value. The transmittance is determined by the power ratio $\gamma =P_2/P_1$, where P₁ and P₂ are the radiation power recorded by power meters 1 and 2, respectively. The dependence of the transmittance of the system on the temperature of the active element is shown in Figure 2.

Since absorption at the wavelength of 1030 nm at temperatures below 60 K is practically absent in Yb:YAG, the entire curve is normalized to the power ratio at a temperature of 40 K. The curve was fitted with the error function:

$$\gamma \left(T\right)=\frac{1}{2}\left[\left(A-1\right)\cdot erf\left(\frac{T-T_0}{\Delta T}\right)+A+1\right],$$

(6)

where erf(T)—the error function, parameters A = 0.29, T₀ = 209.24 K, ΔT = 65.70 K. For calculation, the equivalent temperature from transmittance (6) is inverted:

$$T\left(\gamma \right)=T_0+\Delta T\cdot erfinv\left(\frac{2\gamma -\left(A+1\right)}{A-1}\right),$$

(7)

where erfinv(γ)—the inverse error function.

It has been experimentally established that the shape of the transmittance curve on temperature does not depend on the position of the beam in the crystal, and the differences between the fitted curves do not exceed 1%. A map of the transmittance at 40 K versus the coordinates of the scanning beam in the crystal is shown in Figure 3. The measurements were averaged over 30 s with a sampling frequency of 10 kHz for each point of the map. The standard deviation of the transmittance values in any point does not exceed 1%. The region of the active element around the expected center of the pump radiation spot was measured with a smaller step. The decrease in transmittance along x~−1 mm is associated with defects in the optical system.

Next, the dependence of the transmittance on time with pumping was studied. The dependence of the average temperature on the position of the scanning laser beam in the pumped mode is shown in Figure 4 (left). The pump pulse duration was as short as 0.6 ms, the repetition rate was 1000 Hz, and the beam diameter was 2 mm at the 1/e² level with a supergaussian profile (Figure 4, right).

The measurement of transmittance took 30 seconds with a sampling frequency of 100 kHz for each point. To separate absorption and amplification processes, the pump radiation was interrupted at a frequency of 1 Hz for 10 ms. The transmittance was measured during the last 3 ms of each interruption. Since only 10 of 1000 pulse cycles are skipped, the average absorbed power is decreased by 1% and the temperature remains unchanged. During the interruption, the temperature of the components also changes slightly. The experimentally measured heatsink cooling rate is 0.12 K/s, so the temperature change over 10 ms is close to 1.2 mK. The simulation shows that the maximal temperature drop inside the active element during the interruption is not greater than 0.1 K.

In Figure 5, the equivalent temperature distributions in vertical cross-section are shown, for the scanning laser position at the points x = 0 mm (at the maximum), x = 0.6 mm, and x = 1.2 mm (pump spot edge) in comparison with the simulated results. The plateau is caused by the supergaussian pump beam profile.

The dependencies for horizontal sections are shown in Figure 6. The scanning laser radiation enters the active element at an angle of 6° in horizontal plane (Figure 1) with respect to the normal to the surface, and is perpendicular to the surface in vertical plane. Thus, the scanning beam walk-off after passing through the active element is 0.64 mm. The beam walk-off and its final size lead to a smoothing of the curve of the dependence of the equivalent temperature on the scanning laser spot position.

The temperature value at the maximum is 176 ± 3 K. The temperature of about 135 K outside the pumping region agrees with the temperature of the copper cooler measured using the thermocouple (130 K). The error in determining the temperature is calculated taking into account random errors in measuring the transmittances in the mode without pumping and in the mode with pumping. The measurement error is calculated using the variance formula. The minimal reached temperature measurement error was estimated to be ±3 K. At lower temperatures, the absorption is evanescent thus increasing measurement errors.

The results of numerical simulation of equivalent temperature distribution in the active element are in good agreement with the experimental data. Cooling occurs from the entire surface of the active element in contact with the coolant. The temperature outside the pumping region is close to the temperature of the copper cooler measured with a thermocouple. It follows from the fact that the thermal contact between the active element and the cooling system is uniform, and the temperature difference at the boundary does not exceed 5 K.

The sensitivity of the method depends on the thickness of the active element and the dopant concentration: the larger they are, the greater the absorption in the active element, which allows the method to be used at lower temperatures. For the temperatures in the range of 120–250 K, the dependence of the transmittance has a greater steepness, which reduces the measurement error. Increasing the power stability of scanning radiation decreases the measurement errors. The spectral properties of the scanning laser radiation are also important. The linewidth narrowing increases the dependence slope and hence improves the method sensitivity. Monochromatic radiation is ideal, with a wavelength coinciding with the local maximum of the Yb:YAG absorption peak at about 1030 nm. For media with large aspect ratios, such as rods, the spatial resolution decreases due to large averaging along beam propagation. In addition, the angle of incidence of scanning beam should be as small as possible to prevent the smoothing of measured curves.

Laser thermometry can be carried out to measure equivalent temperature distribution in any laser medium that exhibits measurable dependency of absorption cross-section near the lasing wavelength on temperature. Many Yb-doped and common rare-earth doped media have such dependencies, for example, Yb:YLF [30,31,32], Yb:KYW [30], Yb:KGW [30], Yb:YAP [30], Yb:LuAG [33], Yb:YLO ceramic [34], Nd:YAG [35].

4. Conclusions

The dynamic laser thermometry method is proposed for temperature distribution investigation within Yb³⁺-doped active elements. The method was tested to study the 2D temperature map of the Yb:YAG active element of the high-power, diode-pumped, cryogenically cooled laser amplifier. The best measurement accuracy ±3 K is achieved at the maximal measured equivalent temperature 176 K.

The method of dynamic laser thermometry is simple, reliable, and robust. Measurement of the temperature distribution of the active element is possible without interrupting the operation of the amplifier, and also in the pumped region. The very important feature of the method is the usage of the seed radiation for measurement, and, thus, there is no need to use an auxiliary laser or special equipment.

Aside the verification of modelling results, the method makes it possible to qualitatively analyze the cooling efficiency of both the element as a whole and of selected parts. This is especially important in laser systems, where thermal processes play a significant role.