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Abstract
In this paper, we present a model to predict thermal stress-induced birefringence in high energy, high repetition rate diode-pumped Yb:YAG lasers. The model calculates thermal depolarisation as a function of gain medium geometry, pump power, cooling parameters, and input polarisation state. We show that model predictions are in good agreement with experimental observations carried out on a DiPOLE 100 J, 10 Hz laser amplifier. We show that single-pass depolarisation strongly depends on input polarisation state and pumping parameters. In the absence of any depolarisation compensation scheme, depolarisation varies over a range between 5% and 40%. The strong dependence of thermal stress-induced depolarisation on input polarisation indicates that, in the case of multipass amplifiers, the use of waveplates after every pass can reduce depolarisation losses significantly. We expect that this study will assist in the design and optimisation of Yb:YAG lasers.
Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. Introduction
Laser systems capable of amplifying picosecond to nanosecond pulses to high energies (ranging from a few J to kJ) are required for a wide range of scientific and industrial applications. Direct applications include processing of industrial materials, such as laser shock peening [1], inertial confinement fusion [2,3] and research into the extreme states of matter found inside stars and at the core of planets [4]. High energy lasers are also used as pump sources for petawatt-class femtosecond amplifiers, required for the generation of high brightness and compact laser-driven radiation (x-ray, $\gamma$-ray) [5] and particle (electron, proton, ion, muon) [6–8] sources. Unique features such as small source size and short pulse duration make laser-driven secondary sources more attractive compared to traditional accelerators for remote, high resolution imaging of fast-moving objects [9] and medical diagnostics and treatment [10]. Recently, it has been demonstrated that diode-pumped solid state lasers (DPSSLs) are capable of delivering multi-J pulse energy at multi-Hz repetition rate. Yb$^{3+}$-doped Yttrium Aluminium Garnet (Yb:YAG) has been identified as one of the most suitable active media for high-energy, high repetition rate DPSSLs [11]. Cryogenically-cooled ceramic Yb:YAG is used, for example, for the production of 105 J pulses at 10 Hz (> 1 kW average power) in the DiPOLE100 laser [12] and of 1.5 J pulses at 500 Hz (750 W average power) [13].
The substantial thermal load experienced by laser materials in high average power DPSSLs is the ultimate limitation for scaling average power. In particular, thermal gradients in optical components lead to detrimental effects, including aberrations and stress-induced birefringence [14]. For the latter, the material behaves as a non-uniform retardation element, where the effect on the polarisation state of the incoming light varies across the aperture of the optic. This causes beam depolarisation, i.e. a degradation of the polarisation purity of a beam propagating through the optic. Low polarisation purity reduces the efficiency of processes involving polarisation-sensitive elements such as frequency conversion in nonlinear crystals.
In this paper, we describe theoretical and experimental studies carried out on the DiPOLE100 laser, designed and developed at the STFC Rutherford Appleton Laboratory for the European XFEL in Hamburg, described in a previous publication [15]. The system is based on a diode-pumped, multi-slab, cryogenically-cooled Yb:YAG amplifier architecture [16,17]. Thermal gradients result from the interplay between heating of the material due to the pumping process and the cooling action by the flow of cryogenic helium gas. This paper describes a model to calculate the magnitude of thermal stress-induced depolarisation as a function of gain medium geometry, pump power, cooling parameters and input polarisation state. Model predictions are compared with experimental data. We show that model predictions are in good agreement with experimental observations. To the best of our knowledge, this is the first time that thermal stress-induced depolarisation has been studied – both theoretically and experimentally – in a high energy, high repetition rate cryogenically-cooled Yb:YAG amplifier. Previous publications [18–22] provide only theoretical descriptions of depolarisation in Yb:YAG lasers. Our model is a useful tool to minimise thermal stress-induced depolarisation through system design optimisation and to evaluate the effectiveness of depolarisation compensation schemes.
2. Thermal stress-induced depolarisation model
The model described in this section is divided into three parts, namely:
- 1. Heat distribution model, to calculate deposition of heat in the slabs as a function of gain medium geometry and pumping parameters;
- 2. Finite element thermal and stress model, to calculate temperature and stress in the slabs as a function of heat distribution, cooling parameters and material properties;
- 3. Thermal stress-induced birefringence model, to evaluate the impact of temperature, stress and input polarisation state on the polarisation purity of a beam propagating through the optic.
The following sections provide a more detailed description of each modelling step.
2.1 Heat distribution model
As a result of the pumping process, part of the power delivered by the diode laser pumps is deposited as heat in the gain medium. In the DiPOLE amplifiers, slabs are face-pumped on both sides by two 940 nm wavelength pumps [12,15,16]. Slabs have a central Yb:YAG region, surrounded by a Cr:YAG absorber cladding to suppress parasitic oscillations (Fig. 1(a)). Heating arises from two processes: the quantum defect, which causes heat deposition in the pumped volume, and the absorption of 1030 nm fluorescence radiation in the Cr:YAG absorber cladding. In DiPOLE amplifiers, the time scales of the heat transport are much longer than the pump repetition period. Therefore, the system can be regarded as steady-state and only the average power needs to be considered. The amount of heat deposited in each slab and its distribution depend on both gain medium characteristics and pumping parameters.
Fig. 1. (a) Photo of a ceramic Yb:YAG/Cr:YAG slab used in the 100 J amplifier. The red dotted square marks the pumped region. (b) Heat deposition map.
The average pump power delivered to the gain medium is the total pulse energy delivered by the two sources multiplied by the repetition rate. Assuming that the Yb-doping concentration of the slabs has been chosen so that all pump power is absorbed by the slab set and that each slab absorbs the same amount of pump energy [16], the power absorbed by each slab $P_{slab}$ is the total power divided by the number of slabs. The fraction of power converted into heat in the pumped volume as a result of the quantum defect is:
where $\lambda _{l}$ and $\lambda _{p}$ are the emission and pump wavelength, respectively. The amount of pump power converted into fluorescence radiation is: where $P_{extr}$ is the power extracted by the seed beam undergoing amplification through stimulated emission. In the following sections, the value of $P_{extr}$ is set to zero to consider a worst case scenario where heat load is maximum. Fluorescence occurs in all directions; the majority of it does not leave the gain medium slabs and is eventually absorbed and converted into heat in the Cr:YAG cladding. Ray tracing modelling in Zemax OpticStudio [23] is carried out to calculate the distribution of power deposited as heat in the pumped region and in the cladding. In the model, pump power is absorbed in the pumped volume at the centre of each slab (Fig. 1(a)). The model assumes uniform absorption along the pump propagation direction. It also considers a perfectly uniform, square pump beam with a top-hat intensity profile. In the pumped volume, heating due to power $P_{QD}$ occurs uniformly. The pumped volume acts as a source of fluorescence radiating a power $P_{fluo}$ in a spatially uniform and angularly isotropic manner. This way, the thermal power loading per unit of volume is computed. Figure 1(b) shows the heat distribution in one of the six slabs used in a DiPOLE 100 J amplifier [15]. Each slab is 8.5 mm thick and has a 100 mm square Yb:YAG region surrounded by a 10 mm wide Cr:YAG absorptive cladding with an absorption coefficient of $3 \pm 1$ cm$^{-1}$ at 1030 nm. Diode laser pumps irradiate a 79 x 79 mm$^{2}$ region at the centre of the slabs. The slab set is pumped by two diode laser sources, each emitting 0.65 ms pulses with a pulse energy of 188.5 J at 10 Hz repetition rate, corresponding to 3.77 kW of total average pump power. Each slab absorbs 628.3 W. As a consequence of the quantum defect, 54.9 W are deposited at in the pumped region. The remainder is converted in fluorescence emission, mostly absorbed by the Cr:YAG cladding and the rest escaping the slab and not being converted into heat. According to the ray-tracing model, the heat deposited in the cladding is 470.4 W.2.2 Finite element thermal and stress model
The result from the heat distribution model is used as the input for a finite element analysis (FEA) simulation to calculate temperature and stress distribution within the gain medium slabs [24]. The calculation is carried out numerically using ANSYS FEA software. The model uses heat flow equations widely used and described in the literature [24,25]. The edges of the slabs are treated as thermally adiabatic, meaning no heat transfer occurs at those boundaries. This approximation is justified by the fact that the edge of the slab only has a small area and little thermal contact with its mount, as the slab is held only by a narrow beryllium-copper spring strip which allows for thermal contraction and expansion [26]. The mounting configuration, shown in Fig. 2, was designed to minimise the holding force and to minimise static stress in the slabs. Heat transfer is therefore assumed to occur by convection only through the faces of the slab. In this case, the heat transfer coefficient is governed by the properties of the turbulent flow of the cooling gas and is calculated through a full CFD simulation taking into account the spatial variation of both the heat transfer coefficient and of the temperature of the cooling fluid [26]. In addition to the heat transfer coefficient information, the FEA model is fed YAG material properties. Based on data published by Aggarwal [27], the density of YAG is $4.56 \cdot 10^{3}$ kg/m$^{3}$ and the specific heat capacity of YAG, in the units of J/(kg $\cdot$ K), is fitted over a temperature range between 80 K and 300 K by the following third-order polynomial:
Fig. 2. (a) 3D rendering of a gain medium slab held in a mounting vane. (b) and (c) Details of the beryllium-copper spring strip arrangement, which is the same on all sides of the slab.
Based on [27–29], the thermal conductivity of Yb-doped single crystal YAG can be calculated, in W/(m $\cdot$ K) and over a 80 K to 300 K temperature range, as:
As no information about Young’s modulus and Poisson ratio at cryogenic temperatures is available, the room temperature values of, respectively, 308 GPa and 0.233 were used [32]. The tensile strength of crystalline YAG is 175 MPa [33]. At the time of writing, the value of tensile strength of ceramic YAG is not available in the literature. Therefore, the value for crystalline YAG is used for the simulations.
Based on this input data and on the thermal power loading distribution computed as described in Section 2.1, the numerical FEA simulation provides temperature and stress distribution within the slab. FEA simulations were performed considering two heat loading scenarios. The first considers a total average pump power of 3.77 kW and gives rise to the heat map shown in Fig. 1(b). The second scenario considers halved heat load (1.89 kW). Two values of helium mass flow rate (i.e., different cooling powers) were also used, namely 135 g/s and 180 g/s. Temperature and stress were calculated for the resulting four scenarios, further referred to as “135 g/s, 1.89 kW”, “135 g/s, 3.77 kW”, “180 g/s, 1.89 kW” and “180 g/s, 3.77 kW”. For all scenarios, slabs were cooled with a stream of helium gas at a temperature of 150 K and at a pressure of 10 bar. Figure 3 and Fig. 4 show the resulting temperature and shear stress distributions for these scenarios, obtained by averaging temperature and stress values along the z-direction (beam propagation direction). The shear xy stress component is shown here because it is the main contributor to depolarisation [34]. The vertical asymmetry observable in the temperature and stress distributions is caused by the direction of the cooling flow (top to bottom), causing spatial variation of both the heat transfer coefficient and of the temperature of the cooling fluid. As expected, the highest temperatures and stresses are observed in the case of high heat load and low cooling power (“135 g/s, 3.77 kW” scenario).
Fig. 3. Temperature distributions; flow rate and pump powers are, respectively: (a) 135 g/s, 1.89 kW, (b) 135 g/s, 3.77 kW, (c) 180 g/s, 1.89 kW and (d) 180 g/s, 3.77 kW. The color scales are the same for all images shown.
Fig. 4. XY shear stress distributions; flow rate and pump powers are, respectively: (a) 135 g/s, 1.89 kW, (b) 135 g/s, 3.77 kW, (c) 180 g/s, 1.89 kW and (d) 180 g/s, 3.77 kW. The color scales are the same for all images shown.
2.3 Thermal stress-induced birefringence model
In the absence of stress, YAG has a body-centred cubic Bravais lattice [35]. This structural symmetry makes YAG isotropic (i.e., its properties are the same in all directions). However, stress causes deformation (strain). As a result, if stress is not hydrostatic, the lattice is no longer symmetrical and its optical properties are no longer isotropic. This is the well known elasto-optic effect [36]. In the presence of stress, the dielectric impermeability tensor of the material changes as follows [37]:
As no information on the values of the piezo-optic tensor components of ceramic YAG at cryogenic temperatures is available, the room temperature values were used are [40,41]:
Calculations of thermal stress-induced birefringence were performed following the procedure detailed in [18]. The simulation provides, for each finite element, the values of the principal indices of refraction $n_{\pm }(x,y,z)$ and of the relative angle between the crystal axes $\theta (x,y,z)$ and the laboratory coordinate system of the birefringent material. Effectively, every voxel behaves as a retardation element, which, according to the Jones matrices calculus [43], is described by the following matrix:
Fig. 5. Schematic showing the representation of a gain medium slab as a collection of retardation elements, each described by Jones matrix $\textbf {M}$.
3. Measurement of stress-induced depolarisation
3.1 Measurement setup
Experimental characterisation of thermal stress-induced depolarisation was performed on a DiPOLE 100 J amplifier, described in [12,15]. Depolarisation was measured over a single pass through the amplifier head using the setup shown in Fig. 6(a).
Fig. 6. (a) Setup used for the measurement of stress-induced depolarisation in the 100 J cryogenic main amplifier (M1-M15 = mirrors, QW1, QW2 = quarter-waveplates, HW1, HW2 = half-waveplates, L1-L8 = lenses, CAM1, CAM2 = cameras). (b) Cross-sectional rendering of the amplifier head.
The radiation source used for the measurement is a 1030 nm CW diode laser whose beam is shaped into a super-Gaussian 24 mm square flat-top profile. The beam propagates through a polariser followed by a quarter- and a half-waveplate (QWPO-1030-20-4-R3 and QWPO-1030-20-2-R3, respectively, Manx Precision Optics), further referred to as QW1 and HW1, allowing the selection of the desired input polarisation state. The leakage through mirror M1 is down-collimated by lenses L1 and L2 and collected by a camera (AV MANTA G-145B POE W270 RCG, Allied Vision Technologies), further referred to as CAM1. Correct orientations of the waveplates were established to provide $0^{\circ }$, $30^{\circ }$, $45^{\circ }$, $60^{\circ }$ and $90^{\circ }$ linear polarisation states. Using an additional polariser after HW1, a polarisation extinction ratio >1000:1 was measured for each of the linear states. Circular polarisation was generated by changing the orientation of QW1 by $45^{\circ }$. The beam then propagates through a periscope (M4, M5), which raises the height of the beam with respect to the plane of the optical table. The beam then propagates through a Keplerian telescope (lenses L3 and L4) for magnification to a size of 75 x 75 mm$^{2}$. The beam then propagates through the amplifier head, followed by a Keplerian 1:1 telescope for spatial filtering and image-relaying. Figure 6(b) shows the amplifier head optics (six gain medium slabs, two sapphire windows, two fused silica windows). The periscope and out-of plane propagation between M9 and M10 each introduce rotation of the beam and therefore of the polarisation. However, the reflection angles are chosen such that the beam rotation at the point of the amplifier head is zero. Finally, the beam is steered out of the 100 J amplifier multipass setup using mirrors M14 and M15 and is down-collimated using lenses L7 and L8. The depolarisation diagnostic consists of a quarter- and a half-waveplate (WPMQ10M-1064 and WPMH10M-1064, respectively, Thorlabs) - QW2 and HW2 - both mounted on rotatory mounts to control their orientation, a polarising beam splitter cube (CM1-PBS253, Thorlabs), further referred to as analyser, and camera CAM2, identical to CAM1, collecting the beam transmitted through the analyser. The beam is relay imaged from the centre of the amplifier head to CAM2.
3.2 Considerations on the input polarisation state
Several components in the system uniformly (i.e., in the same way across the entire beam cross-section) modify the polarisation state of the beam even before light reaches the gain medium slabs. The setup contains three main sources of uniform polarisation change, namely high-reflectivity mirrors, a periscope and sapphire windows. The high-reflectivity coating applied to mirrors introduces a small phase delay between s- and p-polarisation components, which depends on the particular coating design. At the time of writing, the overall effect of mirrors on the polarisation state of the beam is unknown because the 100 J amplifier contains high reflectivity mirrors with a range of different coating designs with phase delays not yet characterised. Figure 7 shows the theoretically calculated phase delay ($\sim 3^{\circ }$ and $\sim 9^{\circ }$ for angles of incidence of $45^{\circ }$ and $42^{\circ }$, respectively) for one type of mirror used in the 100 J main amplifier.
Another cause of polarisation change is the periscope composed by mirrors M4 and M5. If it is assumed that periscope mirrors have 100% reflectivity and do not introduce any phase delay between s- and p-polarisation components, the effect of the periscope is a $90^{\circ }$ rotation of the polarisation plane.
Fig. 7. Calculated phase change upon reflection for s- and p- polarisation components as a function of incidence angle for a high reflectivity mirror (courtesy of Manx Precision Optics).
A third cause of polarisation change is off-axis propagation of the beam through the z-cut, 20 mm thick sapphire pressure windows. In the setup of Fig. 6(a), the beam propagates through each window with an azimuthal angle = $4.5^{\circ }$ and a polar angle = $5^{\circ }$. The effect of off-axis propagation through the sapphire material on the polarisation state can be calculated by using the procedure detailed in [44]. Figure 8 shows the effect of one window on the input polarisation states considered in the measurements.
Fig. 8. Polarisation state after propagation at $4.5^{\circ }$ and $5^{\circ }$ azimuthal and polar angles through one z-cut, 20 mm thick sapphire window. The polarisation state of the incident beam is given in the title of each graph.
The effect of the mirrors on the polarisation state of the beam has not been characterised because of access issues. As a result, the polarisation state of the beam just before the gain medium slabs is not known. For this reason, in the following sections “input polarisation" refers to the polarisation state of the beam after HW1.
3.3 Measurement procedure
The measurement is carried out by first of all adjusting the orientation of the extraordinary axes of QW1 and HW1 to obtain the desired input polarisation state. Images of beam profiles are acquired and analysed using RayCi beam profiling software (Cinogy) and their power is integrated over the beam area. A Python script is used to calculate, in real time, the normalised power $P_{2 \: norm}$ as:
where $P_{CAM1}$ and $P_{CAM2}$ are the integrated signals from CAM1 and CAM2, respectively. This allows for compensation of power fluctuations of the CW source. At first, the orientation of QW2 and HW2 is adjusted for maximum extinction (i.e., minimum $P_{2 \: norm}$, further referred to as $P_{2 \: norm,min}$). Then the analyser is rotated by $90^{\circ }$ to achieve maximum transmission (i.e. maximum $P_{2 \: norm}$, further referred to as $P_{2 \: norm, max}$). Instead of rotating the analyser by $90^{\circ }$, it is also possible to measure the transmitted and rejected powers at the same time. However, this approach was not adopted because of space constraints. The degree of depolarisation of the output beam is calculated as:Value $DEP$ is the average depolarisation, in percentage, across the beam area and gives a direct indication of the losses incurred by a polarisation sensitive optic or process. If the beam is perfectly uniformly polarised then $DEP = 0\%$ independently on the actual polarisation state of the beam. Indeed, the aim of the depolarisation measurement is to measure the uniformity of polarisation across the beam aperture, not to determine the mean or predominant polarisation state. Any "pure" polarisation state can be transformed into any other desired state with the use of a quarter- and of a half-waveplate. The uniformity is measured by using QW2 and HW2 to bring the average polarisation of the beam as close as possible to, for example, a $90^{\circ }$ linear state and then by measuring the transmission through the analyser transmitting $0^{\circ }$ linear state. In the worst case $DEP = 50\%$ (i.e. the polarisation state varies across the beam area, with equal portions of two orthogonal polarisation states). The same procedure is then repeated for the other input polarisation states detailed above to measure the dependence of $DEP$ on the input polarisation state.
3.4 Measurement of thermal stress-induced depolarisation
Without any amplifier head optics in the beam path and after optimising QW2 and HW2 to minimise $R_{1}$, the signal on CAM2 was below the detection limit. The transmission minimum (the waveplate setting where a certain polarisation component is maximised) can be reliably found. This means that none of the remaining optics introduces significant depolarisation. Initially, depolarisation was measured, in the absence of heat load, with all amplifier head optics in the beam path. Under typical operating conditions, the amplifier is pressurised to 10 bar absolute. The inner surfaces of the sapphire windows are exposed to this pressure, whereas the outer surfaces are exposed to vacuum. Figure 9 shows the measured average depolarisation over the beam area as a function of input polarisation. A measurement in the absence of heat load (black bars in Fig. 9) shows that the depolarisation averaged across the beam aperture depends on the input polarisation, varying between 3.2%, for $90^{\circ }$ linear polarisation, and 7.7%, for $60^{\circ }$ linear polarisation. This measurement quantifies the amount of static stress-induced depolarisation affecting the 100 J amplifier at operating conditions. Further investigations, outside of the scope of this paper, showed that this static stress is caused by the method of bolting the sapphire windows onto the pressure vessel, the process of bonding a Cr:YAG cladding to the Yb:YAG region of the gain medium slabs and the presence of a pressure difference of 10 bar across the sapphire windows. It is worth bearing in mind that, when thermal stress-induced depolarisation measurements were performed, sources of static stress were not (and could not be) eliminated. As a result, it was not possible to isolate the contribution of thermal stress from that of static stress. Measurements were collected at 150 K cooling temperature and using 10 Hz pump pulse repetition rate and a helium gas pressure of 10 bar. The timing of image acquisition was chosen such that the camera signal was not affected by pump light or fluorescence coming from the gain medium. Depolarisation was measured for the four scenarios discussed in section 2.2 and measurement results are shown in Fig. 9. Data shown in this article is available at [45].
Fig. 9. Average depolarisation across the beam area as a function of input polarisation, pump power and helium gas flow rate.
This data shows that depolarisation increases with the pump power, as more heat is deposited in the slabs, and decreases when the helium gas mass flow rate is increased, as the cooling action becomes more effective. Comparison between Fig. 4 and Fig. 9 confirms that higher shear xy stress causes higher depolarisation. In the case of the scenario used for operation in DiPOLE100 (180 g/s, 3.77 kW), minimum and maximum depolarisation values of 9.5% and 34.7% are observed for $90^{\circ }$ linear and circular input polarisation, respectively. Figure 9 also reports the data for static stress-induced depolarisation when there is no heat load on the amplifier. At 1.89 kW pump power, the contribution of thermal stress-induced depolarisation to the total depolarisation is negligible for $0^{\circ }$ and $90^{\circ }$ linear polarisation. For the other polarisation states, thermal stress contribution to the overall depolarisation is 2 to 3 times bigger than the static stress contribution. In the two scenarios with 3.77 kW pump power, thermal stress-induced depolarisation is higher than static stress-induced depolarisation by a factor between 3 and 8, depending on the input polarisation state. The data shows that the choice of input polarisation can make a large difference (using $0^{\circ }$ or $90^{\circ }$ linear input polarisation state generally results in significantly lower depolarisation). It is worth noting that there is a potential for even further reduction when a larger set of polarisation states is tested.
The strong dependence of thermal stress-induced depolarisation on input polarisation state is also easily observed from the images recorded with CAM2 at maximum extinction. Figures 10(a)–10(d) show depolarisation for $0^{\circ }$ linear and Figs. 10(e)–10(h) show depolarisation for $45^{\circ }$ linear input polarisation for the 4 different scenarios. Images show that depolarisation patterns recorded for $45^{\circ }$ linear input polarisation present brighter features.
Fig. 10. Depolarisation images for $0^{\circ }$ linear input polarisation for “135 g/s, 1.89 kW” (a), “135 g/s, 3.77 kW” (b), “180 g/s, 1.89 kW” (c) and “180 g/s, 3.77 kW” (d) and with $45^{\circ }$ linear input polarisation “135 g/s, 1.89 kW” (e), “135 g/s, 3.77 kW” (f), “180 g/s, 1.89 kW” (g) and “180 g/s, 3.77 kW” (h). The colour scales are the same for all images.
One main conclusion from the measurements is that input polarisation has a large influence on overall depolarisation. Optimising input polarisation is thus an easy-to-implement measure to minimise depolarisation. However, this measure is not sufficient to achieve full depolarisation compensation which, at least in theory, can instead be achieved by more complicated schemes [46,47].
4. Comparison with theoretical model
The experimental results of thermal stress-induced depolarisation were compared with predictions provided by the theoretical model described in section 2. The model was used to calculate depolarisation for all four different cooling and pumping combinations considered in section 3.4. The model considers normal propagation of the beam through the gain medium slabs, which was however not possible to implement experimentally. For comparison with experimental results, the model uses the same input polarisation states used for the measurements (as measured after HW1). The effects of sapphire windows and mirrors were not included in the model.
Measurements in the absence of heat load provided static stress-induced depolarisation values, but not the full set of birefringence parameters (expressed in terms of space-variant retardation and orientation of the principal axes). For this reason, inclusion of static stress in the numerical model, although preferable, was not possible. Full characterisation of static stress-induced birefringence through techniques such as the six-step method [48] will be implemented in the near future to enable the inclusion of static stress effects in the model. These measurements will include the assessment of the effects of the mount on static stress in the gain medium slabs. Figure 11 shows the comparison between model predictions (yellow bars) and experimental data (green bars) for the four scenarios.
Fig. 11. Experimental and theoretical depolarisation values for the “135 g/s, 1.89 kW” (a), “135 g/s, 3.77 kW” (b), “180 g/s, 1.89 kW” (c) and “180 g/s, 3.77 kW” (d) scenarios. All graphs also show the level of depolarisation due to solely static stress (black stars).
The experimental and theoretical results show different trends of depolarisation versus input polarisation state. For the sub-set of input polarisation states considered in the study, experimental results show that minimum depolarisation values are obtained by setting the input as either $0^{\circ }$ or $90^{\circ }$ linear and that maximum depolarisation values are instead observed for either $45^{\circ }$ linear or circular polarisation. In the case of theoretical results, the trend appears inverted, with minimum depolarisation predicted for linear $45^{\circ }$ input polarisation and high depolarisation values obtained with $0^{\circ }$ and $90^{\circ }$ linearly and with circularly polarised light. A lower depolarisation value for linear $45^{\circ }$ input polarisation was also theoretically predicted and experimentally demonstrated in the case of square-shaped Yb:YAG gain medium slabs by Starobor [21].
It is worth noting that the magnitude of maximum and minimum depolarisation values in the experimental and theoretical data are fairly similar, as shown in Fig. 12. Theoretically predicted depolarisation values are consistently lower than experimental measurement results. An explanation for this difference is probably the effect of static stresses, not considered in the model.
Fig. 12. Maximum and minimum depolarisation values in the case of experimental (blue squares) and theoretical (red dots) results. Labels next to each marker indicate the input polarisation state for which each data point was recorded.
Figure 13 shows a comparison between experimentally measured depolarisation patterns (Figs. 13(a)–13(f)) and the depolarisation patterns calculated in the model (Figs. 13(g)–13(l)) for the whole set of input polarisation states considered for the “180 g/s, 3.77 kW” scenario. The following conclusions, drawn from this case, are also valid for the other three scenarios. It is possible to notice that experimental and theoretical depolarisation patterns obtained for circular input polarisation state are very similar (Figs. 13(f) and 13(l)). This is in agreement with the observation that, both experimentally and theoretically, maximum depolarisation values are observed for circular input polarisation. It is also possible to notice a similarity between the experimental depolarisation patterns for $0^{\circ }$/$90^{\circ }$ linear input polarisation states (Figs. 13(a) and 13(e)) and the theoretically calculated depolarisation pattern for $45^{\circ }$ linear input polarisation state (Fig. 13(i)). This is in line with the observation that, in the case of experiments, minimum depolarisation values are observed for either $0^{\circ }$ or $90^{\circ }$ linear input polarisation state, whereas, in the case of theoretical predictions, depolarisation is minimised for linear $45^{\circ }$ input polarisation state. Similar depolarisation patterns are also observed for $30^{\circ }$ and $60^{\circ }$ linear input polarisation states (Figs. 13(b), 13(d), 13(h), and 13(j)), which both experimentally and theoretically show intermediate depolarisation values. The similarities between depolarisation patterns highlighted in Fig. 13 allow an approximate estimate of the polarisation state "seen" by the gain medium slabs in the experiment (unknown due to the reasons described in Section 3.2) to be obtained.
Fig. 13. Experimental (upper row) and theoretical (lower row) depolarisation patterns for the “180 g/s, 3.77 kW” scenario and for a number of input polarisation states. Red dashed lines link images with similar depolarisation patterns. The colour scales for experimental and theoretical patterns are the same for all experimental and theoretical images, respectively.
Comparison between theoretical predictions and experimental observations shows that the model is able to estimate fairly well the range of depolarisation values expected for a given combination of cooling and pumping parameters. Therefore it can be used for the optimisation of future DiPOLE systems and, more in general, for other YAG lasers.
5. Conclusion
This paper provided a description of a theoretical model for the calculation of thermal stress-induced birefringence and presented experimental results of stress-induced depolarisation in a DiPOLE 100 J, 10 Hz laser amplifier. Experiments showed that DiPOLE 100 J amplifiers are affected by thermal stress-induced depolarisation effects. Depolarisation values strongly depend on the input polarisation state, with some polarisation states leading to significantly reduced depolarisation levels. Tests reported in this paper were limited to six different input polarisation states. There is a potential for even further depolarisation reduction when a larger set of input polarisation states is tested (possibly via multi-channel depolarisation diagnostics to reduce data acquisition time). Further modelling and experiments will verify whether depolarisation levels can be minimised beyond the values observed with the six input polarisation states considered so far. In the case of multipass amplifiers such as DiPOLE100, polarisation control measures relying on waveplates at every pass could significantly reduce depolarisation. This approach will be experimentally verified in the near future. As the beam propagates through the same amplifier four times, the approach of rotating the plane of polarisation by $90^{\circ }$ at every pass using a polarisation rotator is possible in principle [46]. However, in reality, this method would not be very effective on DiPOLE100 because the polarisation change caused by mirrors and sapphire pressure windows is different on every pass. Comparison between predictions from the model and experiments showed that the model predicts fairly well the range of depolarisation values for a given combination of cooling and pumping parameters. This shows that the model represents a useful tool for the optimisation of future laser systems.
Funding
Royal Commission for the Exhibition of 1851 (Industrial Fellowship); Laserlab-Europe (654148); Engineering and Physical Sciences Research Council (1979259); Horizon 2020 Framework Programme (739573).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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