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A method to simulate induced stresses for a laser crystal packaging technique and the consequent study of birefringent effects inside the laser cavities has been developed. The method has been implemented by thermo-mechanical simulations implemented with ANSYS 17.0. ANSYS results were later imported in VirtualLab Fusion software where input/output beams in terms of wavelengths and polarization were analysed. The study has been built in the context of a low-stress soldering technique implemented for glass or crystal optics packaging’s called the solderjet bumping technique. The outcome of the analysis showed almost no difference between the input and output laser beams for the laser cavity constructed with an yttrium aluminum garnet active laser crystal, a second harmonic generator beta-barium borate, and the output laser mirror made of fused silica assembled by the low-stress solderjet bumping technique.
© 2017 Optical Society of America
1. Introduction
Laser devices are nowadays widely spread over a whole range of different market sectors. The different existing laser market applications have been pushing laser devices requirements towards highly stringent demands of compact size, high efficiency and high reliability to be able to sufficiently perform in different device conditions. Moreover, the use of laser devices in automotive markets or for space applications has been challenging laser manufacturers to obtain more reliable and compact laser devices, but which should be able to perform under extreme conditions [1]. In terms of obtaining miniaturized devices with high reliability and good efficiency, the best candidates are still diode-pumped solid-state lasers (DPSSL) commonly assembled by adhesive means. However, devices that require high operation and storage temperature ranges, outgassing-free or vacuum compatibility, higher thermal and electrical conductivity or even radiation resistance assemblies have led to a need to look for new joining techniques. Several low stress soldering techniques can be utilized and implemented nowadays for such devices [2]. However, the packaging induced stress and consequent laser component birefringence have to be studied in order not to compromise the device miniaturization whilst also offering stress-free laser beam resonators. In this publication, we studied the stress packaging effects on laser crystals produced by the low-stress packaging laser-based solderjet bumping technique; furthermore, the method can be applicable to other packaging techniques utilized for laser devices.
The so called solderjet bumping technique (Fig. 1) uses spherical solder preforms of diameter range of 40 to 760 μm made of various soft solder alloys (e.g. tin-based lead-free solders, low melting point indium alloys or high melting point eutectic gold-tin, gold-silicon or gold-germanium solders). In favour of being able to join glass or crystals by a soldering technique to metallic or ceramic baseplates, this requires wettable metallization layers to be applied onto the optical components; which could be applied by physical vapour deposition (PVD) [3].
Although this technique guarantees a localized and minimized input of thermal energy making it suitable for joining glasses or in our case the study of laser crystals, the induced stresses have to be analysed to prevent possible laser resonator misbehavior, compromising of the laser beam quality or causing laser final power drop off.
Fig. 1 Soft solder alloys in a spherical form are transferred from the solder sphere reservoir to the jet capillary until they are melted and jetted to the desired components to be joined. The soldering device is mounted in a robotic arm able to solder components with 6 degrees of freedom [2].
2. Simulations methodology
For our case of study, we have chosen a plano-plano laser cavity represented by the most well-known and used laser materials in DPSSL devices (Fig. 2) ; an yttrium aluminium garnet or YAG (Y3Al5O12) active crystal, a second harmonic generator (SHG) beta-barium borate (β–BaB2O4 or BBO), and finally an output dichroic laser mirror made of fused quartz (SiO2). The selected soft solder alloy used to join the laser components to an aluminium nitride (AIN) baseplate was SnAgCu (SAC).
Fig. 2 Schematic of the studied DPSSL cavity. A pumping diode at 808 nm, and the plano-plano laser cavity represented by the three components; the YAG crystal, the SHG BBO and the output mirror.
The simulations were performed first with a Finite Element Method (FEM) using ANSYS 17.0 software to replicate the crystal packaging procedures and calculate the induced stresses. Then, the calculated stress induced birefringence was converted into the dielectric matrix thanks to each components piezo-optic tensor, to be finally imported to VirtualLab Fusion software to study the packaged components lasing capabilities.
2.1. FEM simulations by ANSYS
For the sake of simplicity, the optical components were created as independent 2 mm3 cubes soldered by two 760 μm diameter SAC alloy spheres. They were melted onto a 5 × 5×0.25 mm AIN baseplate with the ANSYS Design Modeler [Fig. 3]. Next, the components material properties were defined for each component as seen in Tables 1 and 2. In the case of the soldering alloy, instead of doing a complete phase change transition from liquid to solid that would increase the complexity of the simulations, some temperature dependent mechanical characteristics have been included in the analysis as seen in Table 2 and Fig. 4.
Fig. 3 An example of the designed geometry used for each laser component. In the case of the SHG BBO crystal, it was created by using two different coordinate systems (crystallographic and laboratory coordinates system). The two different coordinate systems were important to be able to define the material orthotropic characteristics (as seen in Table 1), but also to define the required crystal phase matching angle of 22.8° for SHG needs [4].
Fig. 4 Thermally dependent mechanical material properties. In (a) isotropic elasticity, Young’s Modulus. In (b), enthalpy for the alloy phase change. The alloy thermal dependent characteristics have been extracted by experimental data from the company Setaram Instrumentation (France).
Table 1. Main physical properties of laser materials used
Table 2. Main physical properties of soldering alloy and base plate used
Later a FEM transient thermal analysis was coupled to a static structural analysis in AN-SYS to study the cooling-down process from the SAC alloy (approximate melting temperature 217 °C) from 230 °C to 22 °C and the consequently induced stress on the components assembly. With a post processing analysis, we extracted the vector principal stresses along the optical beam path inside the laser components in order to study the components birefringence and possible lasing misbehaviour.
2.2. Stress induced birefringence approach
A mechanical stress induced onto a laser crystal simultaneously generates an anisotropic density distribution on the component, that creates differences in the material refractive indices defined mathematically by the indicatrix (represented as an ellipsoid that describes the different velocities of light passing through the material) [6]. The effect of different light velocities travelling inside a component that creates an optical anisotropy is also called birefringence. This effect can be described by changes on the material indicatrix Bij with [7]
with i, j = 1, 2, 3 respectively. The second-rank tensor B0,ij represents the free-of-stress indicatrix tensor, and ΔBij represents the indicatrix changes produced due to induced stress, which can be expressed as where k, l = 1, 2, 3, and the Einstein’s summation rule is applied here. The second-rank tensor σkl represents the induced vector principal stress, and πijkl is the fourth-rank piezo-optic constants tensor described for each material. With both Eqs. (1) and (2), we can calculate the indicatrix tensor Bij when certain stress σkl is present. Then, the dielectric constant tensor ∊ij can be calculated using the following relation and the resulting ∊ij is to be used for the subsequent optical simulation on the crystals. The relations in Eqs. (1)–(3) holds in any coordinate system. However, it should be emphasized that the tensors in each equation must be expressed in the same coordinate when being applied. For crystal materials, due to their symmetry properties, it is often easier to describe their properties in the crystalline coordinate system, for example, the piezo-optic tensor πijkl is usually only given in the bibliography in such systems [6]. On the other hand, it is convenient to describe the stress σij with respect to the actual crystal geometry in the lab coordinates system; and for the sake of subsequent optical simulation, the dielectric constant ∊ij needs to be given in the lab coordinate system. To treat the coordinate systems carefully, we firstly define two Cartesian coordinate systems x − y − z and x′ − y′ − z′ representing the lab and the crystalline coordinate systems respectively, and [aij] as the transformation matrix from lab to the crystalline system. Because stress is usually described in the lab system with respect to x, y, z, while the piezo-optic tensor is often given in the crystalline system with respect to x′, y′, z′. To apply Eq. (2), these two quantities must be expressed in the same coordinate system. Instead of transforming the fourth-rank piezo-optic tensor, we choose to transform the second-rank stress tensor into the crystalline system for simplicity. Thanks to the symmetry property, the stress is often expressed in the abbreviated manner, according to Nye’s conventions [6], as σn, with n = 1, . . . , 6. To apply the 3 × 3 coordinate transformation matrix, we first rewrite the abbreviated σn explicitly as σij, and then use the equation below to calculate the stress tensor with respect to x′, y′, z′, in the crystalline system. The coordinate transformation does not change the symmetry property, and the stress tensor σ′ij can also be abbreviated as σ′ according to Nye’s convention. Also, thanks to the symmetry property of crystals and using Nye’s convention [6], the tensors in Eq. (2) can be abbreviated and we can rewrite Eq. (2), in the crystalline coordinate system with respect to x′, y′, z′, as with m, n = 1, . . . , 6. In practice, the piezo-optic tensor is almost always given in the abbreviated manner as a 6 × 6 matrix in the crystalline system. After performing the calculation in Eq. (5), ΔB′m can be rewritten in the explicit form as ΔB′ij.Next, by using Eq. (1) the indicatrix with the influence of stress can be calculated. Due to the facts that 1) the tensor ΔB′ij obtained from Eq. (5) is given in crystalline system; and 2) the free-of-stress indicatrix tensor has a simple diagonal form in the crystalline system; we perform the calculation of Eq. (1) in crystalline system, and that gives
with where n′i is the principal refractive indices of the crystal. Having obtained B′ij in the crystalline coordinate system, we could obtain the dielectric constant tensor ∊′ij in the crystalline system, by directly inverting the matrix [B′ij] according to Eq. (3). However, since the values of ΔB′ij are much smaller than those of B′ij by magnitudes, a direct matrix inversion on [B′ij] would cause numerical errors. To take the influence of ΔB′ij, which is induced by the stress, into consideration correctly, we perform the matrix inversion according to [8], as below with defined as the average value of the matrix elements in [ΔB′ij]. Finally, after applying Eq. (8), we perform a coordinate transformation back to the lab system and obtain as the dielectric constant tensor in the lab coordinate system. Until this point, we have shown the practically complete method to relate the stress from the soldering process and the dielectric constant tensor that determines the optical properties. This enables further analysis of the optical effects that take place in such crystals materials.2.3. Electromagnetic field propagation in layered birefringent materials with Virtual-Lab Fusion
Strictly speaking, the stress distribution inside the crystal is in general inhomogeneous. Thus, we should use ∊ij (x, y, z) to fully characterize the optical properties of the material. However, in our situation, the beam radius is much smaller than the dimension of the crystal cube, and the beam path is centred along the z-axis, as shown in Fig. 5. This fact allows us to simplify the analysis by considering the inhomogeneity of ∊ij (z) only along the z-axis. Furthermore, the continuous inhomogeneity can be numerically approximated as a set of homogeneous layers [9] with different dielectric constant tensors , with the superscript (p) as the index of p-th layer.
Fig. 5 In (a), simulated 2 mm3 crystal cube with and internal laser beam represented as a red cylinder. The beam has a much smaller size than the dimension of the crystal cube. The cross-section marked within the yellow frame is shown in (b), where the inhomogeneity along z-direction is approximated as layered structures.
In the isotropic medium on the left side of the crystal where z < zin, we only need the transverse field vector components Ex and Ey to characterise the electromagnetic field [10]. To perform spectrum-of-plane-wave analysis for the general field, we first calculate the angular spectrum of the input field
with κ = (kx, ky), ρ = (x, y), and ℱ denotes a two-dimensional (2D) Fourier transform. Then, the input field can be treated as a superposition of plane waves with different weights that are determined by the Fourier transform.Next, each plane wave is to be propagated through the layered birefringent material. This process can be expressed as
with as the transmission coefficients matrix. To calculate T̃(κ), we use the numerically stable S-matrix method, and for that purpose, the knowledge on the plane waves in each layer is required. Based on the 4 × 4-matrix formulation from Berreman [11], Landry et al. developed it to a form that is preferable for numerical calculations [12]. We adopted their method and calculated the plane waves by solving the corresponding eigenvalue problem that is described by Eq. (28) in [12].But unlike the authors of [11,12], who build up the transfer matrix directly based on the eigen solutions, we sort out the plane waves according to their energy flowed directions, so to prepare them for the S-matrix calculation. For this sorting, we follow the criteria proposed by Li in Sec. 4.3 of [13]. Then the recursive S-matrix formulas can be applied and we will not repeat the well-developed S-matrix method here. Interested readers are suggested to refer to Eqs. (5)–(8) in [14] for more information.
Once the transmission coefficients matrix is obtained, the output angular spectrum can be obtained by using Eq. (12). Performing inverse Fourier transform on the output angular spectrum, we obtain the output field as
By combining Eqs. (11), (12) and (14), we implemented a numerical algorithm in the physical optics design software VirtualLab Fusion [15], by using its programming interface, and an automatic sampling technique [16] is applied to determined the sampling distance in angular spectrum domain.3. Results
3.1. ANSYS thermo-mechanical results
The transient thermal analysis simulated with ANSYS 17.0 showed an almost instant cooling-down process in all the three studied materials thanks to the low-stress solderjet bumping technique as seen in Fig. 6.
Fig. 6 Maximum soldering alloy phase change temperature during cooling-down process (green), and whole assembly minimum temperature (red). Here the almost instant cooling-down process on the BBO simulation is shown. FEM simulations carried out for YAG and fused quartz showed similar cooling-down ramps.
Fig. 7 In (a) Von-Mises stress calculated in MPa for the BBO FEM analysis. (b) vector principal stresses calculated along the laser beam propagation direction in MPa (maximum, middle and minimum principal stresses in red, green and blue, respectively). Similar results were obtained for the YAG crystal and the fused quartz laser output mirror.
Later, the transient thermal analysis results were coupled sub-step by sub-step to a static structural analysis where the internal stresses were calculated:
Then, the vector principal stresses in MPa were extracted for each of the simulations following a laser beam propagation path for further steps (Fig. 7.).
3.2. Piezo-optic tensor and crystal orientation
Following the mathematical steps defined before, in order to obtain ∊ij following the method in Sec. 2.2, we had to transform the induced stresses σn calculated along the laser beam by ANSYS from the laboratory coordinate systems to the crystal orientation system, σ′n. In the case of the YAG crystal, usually grown by the Czochralski method [Fig. 8(a)], an also cut along the [111] direction, we used the same transformations as in [7], where [aij] from equation Eq. (4) can be expressed as
where in our case, we assumed that x′ and y′ are perpendicular to the slab faces. In the case of the BBO with crystal grown and cut along the [001] direction, the resulted stress matrix had to be just rotated 22.8° along the y-axis [Fig. 8(b)] in order to accomplish the SHG phase matching. Doing so, [aij] from Eq. (4) is represented by where θ =22.8°. In the case of fused quartz, being an isotropic and amorphous material no transformation was required.Fig. 8 Coordinates transformations needed to move the coordinates from the laboratory to the crystal coordinate system. (a) crystal structure of cubic YAG, x′, y′, z′, coordinates in the crystallographic structure [7]. (b) crystal structure of BBO, x′, y′, z′, coordinates in the crystallographic structure and x, y, z, in the laboratory system [17].
Once the stresses have been transformed into the crystalline coordinate system and abbreviated as σ′n, we just needed to know the piezo-optical constant tensors π′mn (Table 3) to be able to apply Eq. (5). In our case, being the YAG a cubic m3m crystal, the BBO a trigonal 3m̄ crystal, and the fused quartz an isotropic material, we can express (thanks to the crystal symmetry and Nye’s convention the fourth-rank tensors expressed with 81 independent values can be reduced into a 36 independent values matrix [6]) the piezo-optic tensors as
Afterwards, we calculated the full indicatrix tensor with consideration of stress, according to Eq. (6). That requires the knowledge the free-of-stress refractive indices, as defined in Eq. (7). For YAG crystal, its refractive index is defined as [20]
for the BBO crystal, its ordinary and extra-ordinary refractive indices are defined as [17] and for the fused quartz [19] where λ is the wavelength given in micrometers. With all the information above, we are able to obtain ∊ij following the method described in Sec. 2.2, which will be used to analyse the stress induced effects on the laser beam in the next section.3.3. VirtualLab Fusion results
In contemplation of laser crystal lasing investigation, several cases per input wavelength and crystal conditions were evaluated as described in Table 4.
Table 4. Studied crystal types, laser resonator cavity produced beams, and stress conditions. The diode-pumping emission wavelength of 808 nm is avoided for being granted between both extreme 532 nm and 1064 nm laser cavity wavelengths.
Starting with the YAG crystal, and Ey-polarized input Gaussian @1064 nm in front of the crystal, the output field behind the crystal under the three different stress conditions can be seen in Fig. 9.
Fig. 9 Amplitude of the transmitted field behind the YAG crystal, with Ey-polarized Gaussian @1064 nm as the input. Column (a) ideal case without stress; column (b) with actual solderjet bumping packaging induced stress; column (c) with 10× increased stress. Upper row corresponds to the Ex-component and lower row the Ey-component. Note that column (c) differs only slightly from (b) in the central part of Ex-component.
Although the input field in front of the YAG crystal is linearly polarized along the y-direction, due to the possible polarization crosstalk that happens when light is refracted at the crystal surface, we obtain a non-zero |Ex| in the output field, even in the case without stress-induced birefringence, as shown in the column of Fig. 9(a). Besides the ideal case, we are interested in the case when stress is present. Comparing Fig. 9(b) with Fig. 9(a), the actual solderjet bumping packaging induced stress shows almost no influence on the output field; while if the stress values are increased to 10 times as the actual ones by design, |Ex| in Fig. 9(c) shows a slight difference at the central part. Obviously, the change in the central part of |Ex| is due to the stress-induced birefringence, while its major profile remains as in Fig. 9(a), which corresponds to the free-of-stress case.
Switching to the case with Ex-polarized input Gaussian @532 nm as input, we obtained the output field as in Fig. 10.
Fig. 10 Amplitude of the transmitted field behind the YAG crystal, with Ex-polarized Gaussian @532 nm as the input. Column (a) ideal case without stress; column (b) with actual solderjet bumping induced stress; column (c) with 10× increased stress. Upper row corresponds to the Ex-component and lower row the Ey-component.
In Fig. 10(a) in which no stress is present, we also see a non-zero |Ey|, although the input field is linearly polarized along x-direction. Nevertheless, in comparison to Fig. 9(a), the strength of polarization crosstalk for the beam @532 nm in the free-of-stress case is smaller. Therefore, the effects of stress-induced birefringence can be clearly seen in Fig. 10(b) and Fig. 10(c). Especially in Fig. 10(c), when the stress values are increased to 10 times of the actual ones, the induced birefringence is so strong that the distribution of |Ey| is very different from that in column Fig. 10(a), which corresponds to the free-of-stress case.
Next, in a similar manner, we investigated the BBO crystal. Starting with the Ey-polarized input Gaussian @1064 nm as the input. In our experiment, the BBO crystal is cut at the angle θ =22.8° and used in the o+o ⇒ e configuration for SHG. According to the geometry sketched in Fig. 8(b), an ordinary wave in the BBO crystal should be linearly polarized along y-direction, while an extra-ordinary wave polarized along x-direction. Therefore, we define the polarizations of both beams @1064 nm and 532 nm as in Table 4.
Unlike the case of YAG crystal, which is naturally isotropic, the BBO crystal is uniaxial anisotropic. Thus, the distribution of |Ex| in Fig. 11 shows a lateral shift, because polarization along x-direction corresponds to the extra-ordinary wave according to the geometry described in Fig. 8(b).
Fig. 11 Amplitude of the transmitted field behind the BBO crystal, with Ey-polarized Gaussian @1064 nm as the input. Column (a) ideal case without stress; column (b) with actual stress; column (c) with 10× increased stress. The upper row corresponds to the Ex-component and lower row the Ey-component.
When the Ex-polarized Gaussian @532 nm is used as the input, we obtained the output field as seen in Fig. 12.
Fig. 12 Amplitude of the transmitted field behind the BBO crystal, with Ex-polarized Gaussian @532 nm as the input. Column (a) ideal case without stress; column (b) with actual studied stress; column (c) with 10× increased stress. Upper row corresponds to the Ex-component and lower row the Ey-component.
We measured the centre of |Ex| distribution in Fig. 12 and it gives the values of 111 μm in our simulation. This lateral shift corresponds to a walk-off angle of 3.18°, which is in good agreement with the literature [21].
If we compare the case of BBO [Fig. 11 and 12] with that of YAG [Fig. 9 and 10], it is not hard to see that the strength of polarization crosstalk in BBO is much stronger than that in YAG, because BBO is naturally anisotropic. As a result, even when the stress values are increased by design, only slight changes are visible in Fig. 11(c) and 12(c).
For the fused quartz analysis, the results were similar to the above seen results for Ey and Ex, with respect to 532 nm and 1064 nm wavelengths for the different stress analysed cases. The above method used to investigate stress-induced birefringence produced in this case by the solderjet bumping technique for laser crystals packaging showed an almost negligible effect on the laser crystal capabilities for the materials and geometries used. The results showed a small-induced stress effect along the laser beam direction that did not compromise the output laser beams.
4. Conclusions and outlook
The above-mentioned method has been used in order to investigate the induced stress on laser resonator cavities packaged by the solderjet bumping technique; the results of a combined mechanical and optical study showed a small-induced stress in a commonly used DPSSL crystal geometries and laser materials. Subsequent studies can be carried out to reduce the laser crystal geometries, using different laser materials, or to use similar procedures to investigate alternative laser crystal packaging techniques.
From a theoretical point of view, the approximate description of the inhomogeneous birefringence within the layered medium is applicable only when the beam size is much smaller than the laser crystal. To also tackle the case with a reduced laser crystal size, we are also developing a technique that could handle the general inhomogeneous birefringence efficiently.
We have demonstrated in this paper that the stress-induced birefringence caused by our solderjet bumping technique has almost no influence to the laser beam that propagated through the crystal, but this is only a proof for the case of single-pass process. For a laser cavity in which the light travels back and forth for infinite rounds, what would be the possible influence of the stress-induced birefringence? This topic is still under investigation and we are going to apply the vectorial Fox-Li algorithm [22–24] for this study.
Furthermore, in previous authors’ publications the solderjet bumping technology was shown to be able to build similar miniaturized solid-state laser devices able to withstand extreme conditions thanks to the crystal soldering approach [25].
Acknowledgments
The authors want to acknowledge other members of the Fraunhofer IOF for their support, especially to Dr. Johannes Hartung and Henrik von Lukowicz; to the Professor Ulf Peschel from the Institut für Festkörpertheorie und Theoretische Optik (Jena) for his time and answered questions.
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