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Abstract
We report experimental observations of thermal lens effects in a diamond Raman laser operating up to 1.1 kW output power in a quasi- steady-state regime. Measured changes in the output beam parameters as a function of output power, including beam quality factor and beam divergence after a fixed focusing lens, are compared to modelling enabling us to track the development of a thermal lens up to 16 diopters at maximum output power. Analysis shows that good agreement between model and experiment is obtained by considering the power deposition profile and the spatial overlap with the laser mode. The results clarify previous work that raised questions about thermal lens effects in the diamond gain medium and provides increased confidence in thermal models for determining the power limits for the current design.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Lasers generating average powers above the kilowatt level with high beam quality are of broad interest in applications such as materials processing, power beaming, defence and high energy physics [1–4]. Diamond Raman lasers (DRLs) are an emerging high-power laser technology offering kilowatt beam powers [5–7], large brightness enhancement [8] and efficient wavelength conversion across a wide range [9–15], all in a compact solid-state platform. High beam powers and brightness are enabled by diamond’s extremely-high thermal conductivity and low thermal expansion coefficient [16]; while efficiency and wavelength versatility are aided by its high Raman gain coefficient, pure single-line Raman spectrum and wide transparency range [5]. Output powers have steadily increased since the availability of high-purity and low-birefringence CVD diamond samples [17], from the 1 W level in 2011 [18] through 10 W [19] and 100 W [20] levels in 2012 and 2014, respectively, and recently surpassing the kW level in a quasi-steady-state regime [6].
Thermal lensing is a primary factor impacting upon DRL beam power and brightness, and for designing systems for operating at multi-kW power levels. Thus accurate knowledge of thermal lens development is required. In diamond plates subjected to a high power laser beam, the thermal lens strength formed as a result of absorption at a wavelength 10.6 µm was analysed in [21]. Recent work in pulsed and quasi-continuous wave DRLs have also shown evidence of possible thermal lensing [6,22]. However, detailed analysis of thermal lens effects in DRLs over a wide power range is needed to increase confidence in models for predicting performance scalability and developing thermal mitigation strategies.
Techniques for measuring the thermal lens in the active medium of end-pumped solid-state lasers include laser probing of media [23–25] and through analysis of the output beam properties [26,27], some of which have been applied to Raman lasers [28,29]. Probe techniques have the important advantage that the lens information is obtained directly and uninfluenced by the effects of gain guiding or higher-order spatial mode excitation. In the case of continuously pumped external cavity diamond Raman lasers, however, it was found that the first signs of thermal lensing at high power were associated with an increase in beam quality and without any notable decrease in slope-efficiency [6]. With this, and given the low gain and low-order spatial mode properties of the isotropic laser medium, measurements of the output beam characteristics enables thermal lens properties to be deduced more directly compared to most other laser media.
Here we report a beam analysis investigation into the thermal lens strength in a DRL operating at up to 1.1 kW output power in a quasi-steady-state regime. Measured output beam parameters as a function of output power (quality factor and divergence angle) are compared to beam propagation calculations, enabling us to track the development of the diamond thermal lens strength up to a value of approximately 16 diopters at 1.1 kW output power. This result clarifies a previous anomaly for thermal lensing in the DRL of reference [30] and enables improved modelling of diamond Raman lasers as powers increase beyond the kilowatt level.
2. Experimental setup
The DRL pump configuration and cavity design was identical to that of our recent report for a 1.23 kW DRL [6]. The DRL was pumped by a free-running, linearly-polarized, flashlamp-pumped Nd:YAG laser operating at 1064 nm with a 4 Hz pulse repetition frequency. The pump pulses were amplitude-modulated to the pulse width of 100 µs using an acousto-optic modulator (QS 27-20-S, Gooch & Housego). The pump beam quality was maintained at $M^{2} = 15$ using a fixed voltage amplitude on the flashlamp. The pump power incident on the DRL was controlled using an external attenuator consisting of a half-wave plate and linear polarizer. The pump beam was focused into the crystal to a spot radius of 130 µm using a beam-expanding telescope and a focusing lens. The diamond was an 8.6 mm-long type IIa CVD-grown crystal cut for propagation along the $\langle 110\rangle$ crystallographic axis with $4 \times 1.2$ mm input and output surfaces. The diamond was positioned in a 242 mm-long concentric cavity formed by a pair of concave mirrors with radii of curvature of 150 mm and 92 mm (see Fig. 1). The cold-cavity fundamental Stokes mode waist radius was calculated to be 80 µm using resonator design software (LasCAD). Any residual pump transmitted through the output coupler was filtered from the Raman laser output beam using a long-pass filter. The residual pump power after double-passing the cavity was detected on a calibrated photodiode after rejection from a pump beam isolator port.
Fig. 1. Experimental setup of the DRL and output beam characterisation. Inset shows the pump beam profile at focus.
The output Stokes beam was focused by an $f$ = 150 mm N-BK7 plano-convex lens and analyzed using a CCD camera. A beam-sampling optic with 5% reflectivity at 1240 nm was used to direct a low-power portion of the Stokes beam to the CCD camera, and absorptive ND filters were used between the beam sampler and the camera for further attenuation. The camera was fixed on a rail allowing the acquisition of beam profiles through the Stokes beam caustic. The majority of the Stokes beam power (transmitted through the beam sampler) was incident on an energy meter for measurement of Stokes pulse energy while an InGaAs photodiode simultaneously captured time traces of the Stokes output (detection bandwidth of 35 MHz). The CCD camera was configured to the second-moment beam profile measurements.
3. Model
We used a model and experimental approach to determine the thermal lens in the DRL as a function of output power. For the model, we used measured optical power values (depleted pump and output Stokes) to determine the heat load in the diamond, and then used an expression for the thermal lens strength for an end-pumped laser rod. Details of the thermal lens model are given below in this Section. These model lens values are then compared with experimental thermal lens values found by measuring the Stokes output $M^{2}$ and beam divergence.
The thermal lens was modelled as a gradient refractive index lens with the fixed refractive index along the optical axis and parabolic with distance $r$ from beam center in the transverse plane, as
where $n_0=2.39$ is the refractive index of diamond at 1240 nm [16], and $n_2$ is the parabolic parameter for the refractive index. The assumption of a constant refractive index along the optical axis is reasonable given that the confocal parameter for the pump beam (waist radius 130 µm, $M^{2} = 15$, $n = 2.39$) is 15.9 mm, approximately twice the crystal length, and the confocal parameter of the Stokes beam is much longer. The value for $n_2$ is calculated from the focal length of an equivalent lens (much greater than the crystal length) which is given by $f^{-1}=n_2 L$, where $L=8.6$ mm is the diamond length. The focal length of the thermal lens was calculated using a modification of the expression for continuous-wave end-pumped rod lasers derived for a top-hat pump beam profile [31]:The modification we introduce to this model is a correction factor that accounts for the effect of mismatched pump and laser modes for either top-hat or Gaussian pump beam profiles [32]:
where $x=w_s/w_0$ is the ratio of the Stokes and pump waist radii (valid for $2>x\geq 0.4$), and $r$ is the ratio of the corrected thermal lens focal length to the focal length calculated using the standard equation above. Given the measured profile shape of the pump in the diamond (Fig. 1), we focus here initially on a correction factor for top-hat pump profiles. Thus, for example, for the case of equal pump and Stokes waist radii ($x=1$), the corrected thermal lens focal length will be 1.58 times longer than that calculated with the standard equation.The power deposited as heat in the medium is the sum of power converted to Raman phonons and the absorption of the Stokes field. This is given by the difference in power depleted from the pump and the Stokes output power, and is determined experimentally as a function of pump power in [6].
Beam propagation calculations to determine the effect of the lens on the external beam properties (and vice versa) were performed using the ABCD method (LASCAD). For these we assumed that the lens profile is constant along the optical axis, which was deemed reasonable due to the larger confocal parameter of the Stokes and pump beams compared to the crystal length and the weak dependence of the pump waist and divergence on the thermal lens strength. The fixed input parameters for the beam propagation calculations were the spatial dimensions of the setup, refractive indices and curvatures of the optical elements, and the operational wavelength (1240 nm). This enabled us to deduce the relationship between $M^{2}$ beam quality parameter for the Stokes beam, the thermal lens (characterized by $n_2$) and the output beam divergence.
4. Results
Lensing in the diamond was investigated as a function of the Stokes output power by monitoring the beam divergence and the $M^{2}$ beam quality parameter of the output Stokes beam. The $M^{2}$ was measured after the external lens by measuring the waist size and the far-field beam divergence half-angle. The beam quality improved from $M^{2}~=~2.20$ at the output power of 280 W to $M^{2} ~=~1.25$ at the highest characterized power of 1.1 kW as shown in Fig. 2(a). Measurement uncertainty was reduced by averaging between at least 10 measurements at each point.
Fig. 2. (a) Measured Stokes beam quality parameter as a function of Stokes output power. (b) Comparison of the pump waist radius and calculated Stokes mode radii (for $M^{2} = 1$, 2 and 3 cavity modes), as a function of thermal lens strength.
The steady increase in Stokes beam quality as a function of output power indicates that the thermal effects are present in the laser cavity and is attributed to an increasing positive thermal lens in the diamond. Figure 2(b) shows the calculated Stokes mode radii for $M^{2} = 1$, 2 and 3 cavity modes, as a function of diamond thermal lens strength, along with the pump waist radius. The cavity design (mirror curvatures and separation) provides a cold-cavity fundamental Stokes mode radius of 80 µm, which is smaller than the pump waist radius of 130 µm, thereby providing substantial amplification to higher-order Stokes transverse modes. The effect of an increasing positive thermal lens is to increase the waist size of the Stokes modes (Fig. 2(b)). A range is shown up to 16 diopters, the point at which the mode-sizes increase rapidly with increasing strength and just prior to the cavity going unstable. For increasing laser power, the fundamental Stokes transverse mode depletes an increasing portion of the pump beam until the Stokes fundamental mode size reaches or exceeds the pump mode size, by which point the laser output beam is predominantly the fundamental transverse mode ($M^{2} < 1.3$). Note that the pump beam size varies by less than 1% through the thermal lens for strengths up to 16 m$^{-1}$.
The increase in beam quality with power is also accompanied by a measured decrease in the output divergence as shown in Fig. 3 (the half-angle after the focusing lens). These values are in good agreement with the calculated values for cavity beams of $M^{2}$ in the range 1 to 3.
Fig. 3. Stokes beam divergence half-angle after the focussing lens. Values in brackets are the measured $M^{2}$ beam quality factors. Curves represent cavity mode calculations for $M^{2}$ between 1 and 3 for a thermal lens obtained using Eqs. 3 and 4.
Experimental values for the thermal lens strength as a function of Stokes power obtained from the beam propagation calculation are plotted in Fig. 4 alongside model values obtained using Eqs. (2–4) for the range $M^{2}$ = 1 - 3. At low Stokes powers, the agreement between model and experiment is best for higher $M^{2}$ values as expected from Fig. 3. For higher powers, the model supports the trend towards increasing lens strength and higher beam quality, and best agreement is obtained for the top-hat power deposition profile.
Fig. 4. Thermal lens strength as a function of Stokes output power. The experimental values (red circles) use the measured $M^{2}$ and divergence values as input to the Stokes beam propagation calculations to determine the thermal lens strength. Lines show $f^{-1}_{\textrm {corr}}$ (Eq. 4), with two families of curves corresponding to top-hat (dashed) and Gaussian (dotted) pump beam profiles. Each family includes several $M^{2}$ values in the range of interest.
5. Discussion
The results presented above show good agreement between trends in the experimentally measured Stokes beam parameters and thermal modelling of the laser cavity as a function of Stokes output power, considering the assumptions in the model and the large thermally-affected power range explored. The large increase in lens strength observed above 400 W is deduced to result from the combined increases in the thermal load and Stokes beam quality (and hence reduced correction factor $r$ in Eq. 4). Discrepancies between the model and experimental values can largely be attributed to the assumptions regarding the pump profile used to determine a value for $r(x)$. We have made the assumption in the model that the pump beam profile $I_{\textrm {p}}(r,z)$ is a top-hat and matches the power deposition profile $\mathcal {P}_{\textrm {dep}}(r,z)$. Though the pump beam profile is near top-hat (see profile shown in Fig. 1), the power deposition is more strictly given by the pump depletion profile [16]:
The results above highlight some subtle differences in the way that thermal effects manifest in high-power DRLs. They explain the improvement in beam quality at higher power, the absence of rollover in efficiency, and also help explain a previous anomaly observed for a high power fiber-laser-pumped DRL in which the lens strength was over-estimated [5,30]. In this latter case, the effect of mismatched pump and Stokes beams on the thermal lens strength was greatly exacerbated due to the fact that the pump beam was significantly smaller than the Stokes beam (20 µm pump radius and 55 µm cold-cavity Stokes radius) and because the pump beam profile was Gaussian. Using the Gaussian expression for the correction factor (Eq. 4) with $x=55/20=2.75$ gives $r=16$. Applying this to the quoted thermal lens strength ($f^{-1}=170$ diopters) reduces it to levels that are much more moderate and more consistent with the observed efficiency characteristics observed for that laser.
6. Conclusion
Beam divergence and $M^{2}$ has been measured for a DRL operating at steady-state output powers up to 1.1 kW. A correction factor analysis of the thermal lens has been used to account for mismatched pump and Stokes beam sizes resulting in good agreement between measurements and calculations. Complexities affecting the heat deposition profile due to pump and Stokes transverse profiles, relative sizes and pump diffraction effects, have been identified as important considerations affecting the accuracy and application of the model. These results provide a method for better predicting DRL performance at powers in the thermal-affected regime.
Funding
Australian Research Council (LP160101039); Air Force Office of Scientific Research (FA2386-18-1-4117).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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