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Abstract
We report on an analysis of the nonlinear absorption in lithium triborate (LBO) used for second and third harmonic generation of ultrashort laser pulses at average powers in the order of kW and with sub-picosecond pulse duration. Thermographic imaging of the LBO crystals together with a simple analytical model revealed the presence of nonlinear absorption in both harmonic generation processes. Subsequent processing with a numerical model considering the nonlinear mixing, the absorption, and the heat conduction was used to estimate the absorption coefficients. Average powers exceeding 100 W in the ultraviolet and 400 W in the visible spectral range were obtained while maintaining a good beam quality by avoiding excessive nonlinear absorption.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Ultrafast laser sources emitting in the visible and ultraviolet spectral region are promising for a wide range of applications, ranging from industrial material processing to scientific investigations of light-matter interaction. While material processing applications benefit from the possibility to fabricate high-quality microstructures with high efficiency and high productivity [1,2], scientific applications such as high harmonic generation into the extreme ultraviolet (XUV) spectrum benefit from higher conversion efficiencies due to an enhanced single-atom response at shorter wavelengths [3,4]. All of these applications benefit from a good beam quality, a high average power as well as high pulse energy and short pulse duration. The combination of these properties constitutes a highly demanding challenge on the design of the laser source. State of the art sub-ps near-infrared (NIR) lasers with average powers in the kilowatt range offer all of these features [5–8]. For this reason, subsequent frequency conversion utilizing nonlinear crystals is interesting to reach the visible and UV spectral region while maintaining these properties.
Despite the recent significant advances of the NIR ultrafast lasers, only a few sub-ps lasers in the visible spectral range were reported to exceed an average power of a few hundred watts and a pulse energy of a few tens of microjoule [9–11]. To the best of our knowledge only one laser with sub-ps pulses and one with a pulse duration of a few ps were reported to reach or exceed an average power of 100 W in the UV [12,13]. While strong thermal degradation of the beam profile was shown to occur already at a power of around 120 W [13] for the LBO-based picosecond UV laser, 100 W of average power in the UV with a nearly diffraction-limited beam quality and at a repetition rate of 3.5 MHz were achieved with sub-ps pulse durations using a BBO crystal [12]. This impressive result was enabled by reducing the thermal gradients in the thin BBO crystal by bonding it to transparent sapphire heat-spreaders, which is a well-known approach also used to minimize thermal lensing in rod lasers [12,14–16]. Severe threshold-like degradation of the beam quality was reported when standard crystals without heat spreaders were used instead of the described composite structure [12]. While a bonded sapphire-BBO structure was required in these experiments to maintain good beam quality at high average power, to our knowledge no comparable results were reported for commercially available nonlinear crystals. In [12] the excessive thermal load was assumed to be caused by an elevated absorption coefficient in the UV as compared to the ones in the visible and IR spectral ranges. The threshold-like behavior at the low average UV power of a few tens of Watts however suggests that further contributions must be present beyond the sole linear absorption commonly measured for the BBO and LBO crystals. At a wavelength of 355 nm the linear absorption coefficient of both crystals was reported to amount to less than $5 \times {10^{- 4}}\,1/\textrm{cm\; }$[17,18], which cannot explain the excessive heating reported in [12].
We therefore used thermographic measurements in combination with an analytical and numerical model to identify and further analyze the absorption processes in LBO crystals in a real-world experimental setup for second and third harmonic generation of a high-power ultrafast Yb:YAG laser.
Our investigation focused on LBO as it offers a range of excellent properties for high-power conversion into the visible and UV spectral range. Compared to BBO, which offers similarly low linear absorption in the spectral range of 343 nm to 1030 nm [18], the approx. 20% larger bandgap of 7.78 eV [19] and the about twice as high thermal conductivity of approx. $3.5\,\textrm{W}/({\textrm{m} \times \textrm{K}} )$ (multidirectional average) [20] are especially interesting for applications involving high intensity and high average power. Depending on the crystal cut and nonlinear process these advantages come at the cost of an approx. 2 to 5 times lower nonlinear coefficient compared to BBO. For THG we used type I rather than type II phase matching due to the 5 times higher temperature range, the 50% higher nonlinear coefficient, and the 30% increased spectral bandwidth [21].
For our experiments we used a thin-disk multipass amplifier which emitted sub-ps pulses at a wavelength of 1030 nm and delivered an average power of approx. 780 W. The laser systems’ repetition rate and pulse energy were adjustable, which allowed for the analysis of a large range of average power and pulse peak power. From these experiments we identified the occurrence of nonlinear absorption in both frequency conversion processes, being especially critical for THG.
After a short introduction into the theory and the methods in section 2 the experimental results are presented in section 3 which is followed by a discussion of the limitations imposed by nonlinear absorptions and possible strategies to alleviate them in Section 4.
2. Theory and methods
2.1 Nonlinear absorption
Nonlinear absorption describes the phenomenon that two or more photons are absorbed at the same time which leads to an intensity-dependent absorption coefficient. This effect can occur when the sum of the photon energies is sufficient to overcome the energy gap between the valence and the conduction band of a material which otherwise would be transparent at the wavelength of interest. For LBO with an energy gap of 7.78 eV [19] between the valence and the conduction band at least three-photon absorption at a wavelength of 343 nm (10.84 eV) or four-photon absorption at a wavelength of 515 nm (9.63 eV) is required to cross the energy gap, as can be seen from Table 1. A phenomenological description of nonlinear absorption is provided by the generalized Beer-Lambert law [22]
Table 1. Sum of the Photon Energies in eV (Bold When Larger Than the Band Gap of LBO) of Different Nonlinear Absorption Processes, Where “nPA” Refers to the n-Photon Absorption at the Given Wavelength.
To avoid confusion between different processes, in the following, we will consider absorption processes up to the four-photon absorption and use the notation $\mathrm{\alpha },{\;\ \mathrm{\beta} },{\;\ \gamma },$ and $\mathrm{\delta }$ instead of ${\alpha _1}$ (linear-), ${\alpha _2}$ (two-photon-), ${\alpha _3}$ (three-photon-), and ${\alpha _4}$ (four-photon-absorption coefficient), respectively. When required a subscript denoting the vacuum wavelength in nanometers will be added (e.g.: ${\mathrm{\gamma }_{343}}$ for the three-photon absorption coefficient at a wavelength of 343 nm). The total absorption coefficient $\mathrm{\psi }$ therefore reads
The linear absorption coefficients of the LBO bulk material were reported to be ${\mathrm{\alpha }_{1030}} < \,50 \times {10^{- 6}}\,1/\textrm{cm}$, ${\mathrm{\alpha }_{515}} < \,150 \times {10^{- 6}}\,1/\textrm{cm}$ and ${\mathrm{\alpha }_{343}} < 500 \times {10^{- 6}}\,1/\textrm{cm}$ for the wavelength of 1030 nm, 515 nm, and 343 nm, respectively [17,18]. By using multimode nanosecond laser pulses at a wavelength of 355 nm the three-photon absorption coefficient was found to range between $\,{\mathrm{\gamma }_{355}} = 0.8\,\textrm{x }{10^{- 3}}\; \textrm{c}{\textrm{m}^3}/\textrm{G}{\textrm{W}^2}$ and $\,{\mathrm{\gamma }_{355}} = 6\,\textrm{x }{10^{- 3}}\; \textrm{c}{\textrm{m}^3}/\textrm{G}{\textrm{W}^2}$ [18].
2.2 Analytical model and measurement method
Figure 1 schematically shows a nonlinear crystal in a temperature-controlled mount. In our experiments, the mount was made of copper and set to a constant temperature on its outer surface by means of a heater and a control loop. The inner surfaces are in thermal contact with the crystals’ lateral surfaces (highlighted in light blue). Due to this lateral heat transfer, the heat conduction in the nonlinear crystal exhibits a certain similarity to the one in rod lasers.
Fig. 1. Schematic illustration of a nonlinear crystal in the temperature-controlled mount. The surfaces of the nonlinear crystal that are in thermal contact with the temperature-controlled mount are highlighted in blue.
As a first-order approximation we therefore used an analytical solution of the heat conduction equation for a radially cooled, rod-shaped crystal, which is homogeneously heated in a cylindrical volume around the axis of symmetry [23]. The analytical model was used to qualitatively identify the type of absorption processes from the temperature increase as well as for an approximate, qualitative estimation of the absorption coefficients. The analysis was then further detailed using the numerical model presented in section 2.3, which additionally takes into account the temporal evolution of the pulses and the spatial intensity distributions. In a rod arrangement, the absolute temperature in the center of the cylinder is given by [23]
Introducing $\mathrm{\Delta T} = {\textrm{T}_0} - {\textrm{T}_\textrm{C}}$ and $\mathrm{\xi } = \left[ {1 - 2\textrm{ln}\left( {\frac{{{\textrm{R}_\textrm{p}}}}{{{\textrm{R}_\textrm{s}}}}} \right)} \right]$ in Eq. (3), one obtains the temperature difference
By identifying the fractional heat load per length $\mathrm{\eta }/{\textrm{L}_\textrm{p}}$ as the absorption coefficient $\mathrm{\psi }(\textrm{I} )$ allows reformulating Eq. (5) as
Inserting Eq. (2) this reads
The experimental procedure described in section 2.4 involves the comparison of two experiments: one in which the average power was changed by varying only the repetition rate and keeping the intensity constant and one in which the average power was changed by changing the pulse energy but with constant repetition rate, constant pulse duration τ, and unchanged beam diameter db. Under these specific conditions, and only then, the average power is proportional to the intensity and Eq. (7) can be changed to
When the average power P is varied by changing the repetition rate and the intensity is kept constant, Eq. (7) reduces to
By measuring the surface temperature of the crystal’s facet and using either Eq. (9) or Eq. (10), it is possible to determine which absorption mechanisms govern the absorption coefficient $\mathrm{\psi }(\textrm{I} )$. For this purpose, we used a thermographic camera (InfraTec Variocam HD 900) which operates in the wavelength range of 7.5 to 14 µm and has a measurement accuracy of $\, \pm 1.0\,\textrm{K}$ in the investigated temperature interval. As LBO exhibits high absorption of radiation at a wavelength $\mathrm{\lambda } > 2.6\mathrm{\mu m}$ [24,25], we expect that the measured thermal radiation originates from the surface of the crystal and a thin layer underneath it. The measured surface temperature was corrected by the experimentally determined emissivity $\,{\mathrm{\varepsilon }_{\textrm{LBO}}} = 0.75$, which was measured to be constant for temperatures ranging from 45°C to 70°C by using the reference temperature technique [26].
2.3 Numerical model
As the analytical model assumes a constant, cylindrically shaped distribution of the heat density and does not consider the temporal and spatial distribution of the intensities, a numerical model was used for a more precise quantitative investigation. The analytical model was used to qualitatively identify the dominant multiphoton absorption mechanism from experimental measurements of ΔT as a function of P and I. The numerical model additionally considers the temporal and transverse spatial variation of the intensity and the resulting 3D distribution of the heat density in the crystal and therefore provides a more accurate quantitative determination of the absorption coefficients when fitting the model to experimental data. For this numerical simulation we used the commercial software MATLAB to model the frequency conversion processes and the finite-element software COMSOL to solve the heat conduction equation. The frequency conversion processes were modeled by numerically solving the 3D coupled differential equations that describe the three-wave mixing in the slowly varying envelope approximation. The model described in [27] was used and extended with the frequency domain treatment of phase-matching from [28] for precise modeling of dispersion effects. This results in the coupled differential equations
The FFT split-step beam propagation method presented in Ref. [27] was implemented in MATLAB to solve the Eqs. (11)–(13). Dispersion effects such as phase mismatch, group delay and group delay dispersion etc. were accounted for by a frequency-dependent phase in the Fourier domain by using the Sellmeier equation from Ref. [29].
The infrared laser pulses incident on the nonlinear crystal were modeled by initializing the electric field in the frequency domain using the measured spectrum. The FWHM pulse duration of the modeled pulse was then adapted to fit the measured pulse duration by applying positive group delay dispersion. The transversal spatial intensity distribution was assumed to be Gaussian.
The second harmonic generation was modeled as a degenerated sum-frequency generation process. A subsequent sum-frequency generation step was added for the third harmonic generation process, in which the output of the frequency-doubling simulation was used as an input. As in the experiments, the distribution of the energy between the frequency-doubled pulse and the infrared pulse was optimized for the highest conversion efficiency.
The distribution of the heat density within the nonlinear crystal was calculated by applying the nonlinear absorption coefficients to the intensity distribution of the interacting beams. For a first simulation run guessed values for the nonlinear absorption coefficients were used which were subsequently refined in iterative steps as explained below.
The calculated distribution of the heat density was then transferred to the finite-element model shown in Fig. 1 to compute the distribution of the temperature inside the nonlinear crystal. The nonlinear absorption coefficients were manually optimized by minimizing the difference of the computed and measured dependence of the temperature increase $\mathrm{\Delta T}$ on the crystal surface on the applied intensity and average power. The thermal contact coefficient was manually adapted at the same time to match the computed and measured temperatures on the crystal’s surface. Since the contributions of convective and radiative heat transfer are negligible, only pure conductive heat transport was considered.
2.4 Experimental procedure
To identify and quantify nonlinear absorption in a real-world frequency conversion application, we used a typical frequency conversion setup and recorded the surface temperature of the nonlinear crystal for different pulse peak intensities and average powers. The identification of the contributions from the different nonlinear absorption coefficients was accomplished by varying the average power in two different ways while keeping the beam diameter and pulse duration constant. The first experiment serves as a benchmark (we will refer to it in the following as “benchmark”) where the pulse intensity is held constant and the average power is increased by increasing the repetition rate of the injected pulse train. As the pulse intensity is constant in this case, the absorption is constant too and according to Eq. (10) the surface temperature increases linearly with increasing optical power. The second experiment probes the intensity dependence of the absorption (we will refer to it in the following as “probe”) and is based on increasing the pulse intensity by increasing the pulse energy at a constant repetition rate, pulse duration, and beam diameter. In this configuration, the temperature increases according to Eq. (9). For purely linear absorption ($\mathrm{\mathrm{\beta} } = \mathrm{\gamma } = \mathrm{\delta } = 0$), the two experiments result in an identical and linear increase of the surface temperature with the optical power. In contrast, a nonlinear increase of the surface temperature is expected for the probe experiment when nonlinear absorption is present.
To quantify the nonlinear absorption coefficients with the numerical model, we additionally conducted probe experiments (varying intensity) with different repetition rates and crystal lengths to generate a larger and more general database.
The LBO samples were manufactured and coated by Cristal Laser and are of commercially available high quality for high-power industrial and scientific applications.
2.5 Experimental setup
The thin-disk multipass amplifier (TDMPA) that was used for the experiments emitted sub-ps laser pulses at a wavelength of 1030 nm and was a modified version of the systems which are described in detail in [8,11]. Its output power was fixed to approx. 780 W for all experiments, the other laser parameters are listed in Table 2. As seen from the setups in Fig. 2, the pulse energy and average power incident on the frequency-doubling crystals were reduced by means of a half-wave plate (HWP) and a thin-film polarizer (TFP). The TDMPA is capable of emitting pulses with constant pulse energy at an integer divider of the base repetition rate.
Fig. 2. Experimental SHG (a) and THG (b) setups. A thin-disk multipass amplifier (TDMPA) delivers infrared pulses at 1030 nm. HWP: Half-wave plate, TFP: Thin-film polarizer, DC: Dichroic Mirror
Table 2. Parameters of the Infrared Laser
Two curved mirrors were used to create a slightly divergent beam with a beam diameter of approx. 1.8 mm (D4σ) on the SHG LBO crystal. The HWP in front of the SHG crystal was used to adjust the direction of polarization of the fundamental beam to the axis of the crystal.
The frequency-doubled and the fundamental beam were separated by a dichroic mirror behind the SHG crystal. All LBO crystals had an aperture of 10 mm x 10 mm and were cut for type I (oo-e) phase matching. The details are listed in Table 3.
Table 3. Parameters of the Nonlinear LBO Crystals
For the THG the frequency-doubled and the fundamental beam were both led through the THG crystal, as depicted in Fig. 2 b). The size of the fundamental beam on the THG LBO crystal was increased to approx. 2.3 mm (D4σ) by the slight beam divergence. The frequency-tripled beam was separated from the frequency-doubled and the fundamental beams by dichroic mirrors behind the THG crystal.
3. Results
The advent of nonlinear absorption in the frequency conversion processes was first verified by fitting the simple analytical model to the measured temperatures. The numerical model was used in a second step for a more precise determination of the value of the different absorption coefficients.
3.1 Experimental verification of nonlinear absorption
3.1.1 Second-harmonic generation
Figure 3 shows the measured increase $\mathrm{\Delta T}$ of the peak temperature on the surface of a 1 mm long LBO crystal as a function of the generated second-harmonic power for the benchmark experiment (constant intensity) and the probe experiment (varying intensity). The experiments were conducted with a pulse energy of up to 1.52 mJ and a repetition rate of up to 500 kHz.
Fig. 3. Peak temperature increase on the exit facet of a 1 mm long SHG LBO crystal (sample S2, see Table 3) as a function of the generated SH power for the benchmark and probe experiments. The benchmark experiment (constant intensity) used a fixed IR pulse energy of 1.52 mJ and varying repetition rates from 50 kHz to 500 kHz. The probe experiment (varying intensity) used a varying pulse energy of up to 1.52 mJ and a fixed repetition rate of 500 kHz as well as constant pulse duration and beam diameter. The error bars mark the measurement accuracy of $\, \pm 1.0\,\textrm{K}$ of the thermographic camera.
At the maximum power, the intensity and power were identical for both experiments. The different curvatures clearly indicate a nonlinear characteristic of the absorption. The solid lines represent a fit of the equations (10) (benchmark with constant intensity) and (9) (probe experiment with varying intensity but constant frep, τ and db) to the measured data. For the latter, it was assumed that only linear- and four-photon absorption can occur ($\mathrm{\mathrm{\beta} } = \mathrm{\gamma } = 0)$, as follows from the energy gap of LBO and the photon energies listed in Table 1. The influence of possible absorption of the incident IR radiation was investigated by measuring the temperature of the crystal when no SHG process was taking place but the full IR power passed the crystal. This condition was implemented in two different ways. First, we used an IR beam with an orthogonal polarization to the one used for SHG to suppress the generation of the SH beam. Second, we used an IR beam with the correct polarization for SHG but significantly detuned the critical crystal axis. In both cases, no temperature increase was measured when injecting infrared average powers and pulse energy of up to 780 W and 780 µJ. Hence, we conclude that the absorption of the IR beam is negligibly small.
3.1.2 Third harmonic generation
To measure only the influence of the nonlinear absorption from the frequency-tripled radiation, effects of nonlinear absorption of the incident frequency-doubled laser pulses in the THG crystal were avoided as described at the end of this subsection. The pulse energy was reduced in this experiment by operating the laser at a repetition rate of 2 MHz. In order to maintain a high SH conversion efficiency, a 1.5 mm long LBO crystal was used in the SHG stage to compensate for the reduced IR pulse intensity. The THG experiment was conducted with a 1 mm long LBO crystal (sample C1, see Table 3). The laser was operated at an infrared pulse energy of up to 284 µJ and a repetition rate of up to 2 MHz. The results of the THG benchmark- (constant intensity) and probe experiment (varying intensity but constant frep, τ, and db) are shown in Fig. 4. The clearly nonlinear characteristic of the probe experiment confirms the presence of nonlinear absorption. The solid lines represent fits of eq.(9) (probe experiment with varying intensity) and (10) (benchmark experiment with constant intensity) to the measured data.
Fig. 4. Peak temperature increase on the exit surface of a 1 mm long THG crystal (sample C1, see Table 3) for the benchmark and probe experiment as a function of the generated TH Power. The benchmark experiment (constant intensity) used a fixed IR pulse energy of 284 µJ and varying repetition rates from 100 kHz to 2000 kHz. The probe experiment (varying intensity) used a varying IR pulse energy of up to 284 µJ and a fixed repetition rate of 2000 kHz as well as constant pulse duration and beam diameter. The error bars mark the measurement accuracy of $\, \pm 1.0\,\textrm{K}$ of the thermographic camera.
Only linear- and three-photon absorption were considered due to the band gap of LBO and the photon energies listed in Table 1. In order to avoid nonlinear absorption of the incident frequency-doubled laser pulses, the intensity of the frequency-doubled pulses was significantly reduced compared to the experiments for the determination of the absorption at a wavelength of 515 nm. This was accomplished by increasing the beam diameter on the THG crystal from 1.8 to 2.3 mm (decreasing the intensity by 40%) and by reducing the maximum IR pulse energy from 1.52 to 0.284 mJ (decreasing the maximum applied intensity by another factor of about 5). With $\mathrm{\psi } \propto {\textrm{I}^3}$ this leads to a significantly reduced nonlinear absorption of the frequency-doubled pulses. In fact, no temperature increase of the THG crystal was measured, when operating it out of phase-matching but injecting 421 W of infrared and 147 W of frequency-doubled power with a repetition rate of 2 MHz. As only negligible amount of UV light was generated (P<<1 W) in this case, instead of 90 W for the case of optimized phase matching, we conclude that the absorption of the fundamental and second-harmonic pulses was negligibly small using these parameters.
3.2 Determination of the nonlinear absorption coefficients
A larger number of probe experiments was conducted in order to generate a broader data set which is suitable to quantify the nonlinear absorption coefficients by fitting the numerical model to the experimental data. For SHG we used crystals with a length of 1.0 mm and 1.5 mm and repetition rates ranging from 500 kHz to 2 MHz. The THG probe experiments were carried out with crystal lengths of 1.0 mm and 2 mm at a repetition rate of 2 MHz. The beam size on the crystal was kept constant for all experiments.
Figure 5 and Fig. 6 show the measured temperature rise as a function of the average power (varying pulse energy but constant frep, τ, and db for each of the curves) together with the results of the numerical model for SHG and THG, respectively. Table 4 summarizes the determined values of the absorption coefficients. The given uncertainties are discussed in the following. The performance of the frequency conversion is summarized in Table 5 as a reference. The efficiency is given at the maximum of the generated harmonic power and refers to the incident infrared laser beam. Additional figures depicting the evolution of the conversion efficiency and output power as a function of the IR power as well as measurements of the M2 at maximum output power are located in the Supplement 1.
Fig. 5. Measured and computed temperature increase $\mathrm{\Delta T}$ on the exit surface of the SHG crystal as a function of the generated SH Power for crystals with a length of 1.0 and 1.5 mm. The repetition rate, pulse duration, and beam diameter remained constant, hence $P \propto I$ holds. The lines were added to guide the eye. The specifications of the used crystal samples S1 and S2 are given in Table 3.
Fig. 6. Measured and computed temperature increase $\mathrm{\Delta T}$ on the exit surface of the THG crystal as a function of the generated TH Power for a set of different crystals with a length of 1.0 and 2 mm. The repetition rate, pulse duration, and beam diameter remained constant, hence $P \propto I$ holds. The insets show the simulated (left) and measured (right) temperature distribution on the crystal’s surface at an average power of 116 W. The lines were added to guide the eye. The specifications of the used crystal samples C1, C2, C3 are given in Table 3.
Table 4. ${\boldsymbol \alpha },{\boldsymbol \gamma },$ and ${\boldsymbol \delta }$ as Determined by Fitting the Numerical Simulation to the Measured Temperature Increase ΔT.
Table 5. Parameters of the Frequency-Converted Laser Pulses.
The fitted numerical model is in excellent agreement with the experiments, as seen in Fig. 5 and Fig. 6. Nonlinear absorption was identified in both the SHG and the THG processes, being more pronounced in the latter. Due to the dominance of the nonlinear over the linear absorption and the comparatively low average power on the order of 100 W in the THG experiments, the simulations were very sensitive with respect to the nonlinear absorption coefficient but only little influenced by variation of the linear absorption coefficient over a large range. The shaded area illustrated in Fig. 7 allows to estimate the accuracy of the determined values of the absorption coefficients. Due to the smaller impact of the linear absorption, we can only state that ${\mathrm{\alpha }_{343}} < 1.5 \times {10^{- 3}}\; 1/\textrm{cm}$ as seen from the gray shaded area resulting from a corresponding variation of ${\mathrm{\alpha }_{343}}$. Varying ${\mathrm{\gamma }_{343}}$ around the value of $5.8 \times {10^{- 4}}\textrm{c}{\textrm{m}^3}/\textrm{G}{\textrm{W}^2}$ which corresponds to the best fit, one finds that it is most probably accurate to within about ±10% (blue shaded area).Compared to the nonlinear absorption coefficients in Ref. [18], which were found to range between $\,{\mathrm{\gamma }_{355}} = 0.8 \times {10^{- 3}}\; \textrm{c}{\textrm{m}^3}/\textrm{G}{\textrm{W}^2}$ and $\,{\mathrm{\gamma }_{355}} = 6 \times {10^{- 3}}\; \textrm{c}{\textrm{m}^3}/\textrm{G}{\textrm{W}^2}$, the value of the three-photon absorption coefficient resulting from our measurements is somewhat lower. This difference can be caused by the different experimental conditions, as the measurements in [18] were conducted with multimode nanosecond pulses, with a highly complex intensity profile of the beam, which complicates the evaluation. Additionally, the more than three orders of magnitude differing pulse duration might lead to different dynamic effects influencing the nonlinear absorption coefficients.
Fig. 7. Measured and computed temperature increase $\mathrm{\Delta T}$ on the exit surface of the THG crystal C3 as a function of the generated TH Power for a range of linear (gray) and nonlinear (blue) absorption coefficients.
Furthermore, the accuracy is directly determined by the investigated maximum intensity, which was approx. one order of magnitude larger in our experiments. We therefore expect our results to provide a more robust database for ultrashort pulse durations. Compared to the result of a theoretical study of LBO [30], in which $\gamma $ was calculated to range between $\mathrm{\gamma } = 0.4 \times {10^{- 5}}\; \textrm{c}{\textrm{m}^3}/\textrm{G}{\textrm{W}^2}$ and $\mathrm{\gamma } = 1 \times {10^{- 5}}\; \textrm{c}{\textrm{m}^3}/\textrm{G}{\textrm{W}^2}$, the value determined from our measurements is one to two orders of magnitude larger. The contribution of nonlinear absorption to the heat load during SHG was weak compared to the one observed with THG and mainly became important for the combination of very high pulse intensities and average powers. As the theoretical probability of an n-photon process scales with ${W_n} \propto {({{E_B}/{E_c}} )^{2n}}$ [22,31], where ${E_B}$ is the amplitude of the electric field of the laser beam, and ${E_c}$ is a characteristic electric field of the crystal, we attribute this significantly reduced contribution to the lower likeliness ${W_4}$ of four-photon absorption compared to three-photon absorption.
Due to this lower nonlinear contribution and a higher average power, the simulation was sensitive to both, the linear- and nonlinear absorption coefficients. The shaded area illustrated in Fig. 8 allows to estimate the accuracy of the determined values of the absorption coefficients. The linear absorption coefficient ${\alpha _{515}}$ was estimated to be accurate within about ${\pm} 200 \times {10^{- 6}}\,1/\textrm{cm\; }$, as seen from the gray shaded area resulting from a corresponding variation of ${\alpha _{515}}$ around the value of $1000 \times {10^{- 6}}\,1/\textrm{cm}$ (best fit). As indicated by the blue shaded area illustrating a variation of $\delta $, we estimated an accuracy of about ±10% around the value of ${\delta _{515}} = 6 \times {10^{- 9}}\,\textrm{c}{\textrm{m}^5}/\textrm{G}{\textrm{W}^3}$ (best fit). The four-photon absorption coefficient of ${\mathrm{\delta }_{515}} = 6 \times {10^{- 9}}\; \textrm{c}{\textrm{m}^5}/\textrm{G}{\textrm{W}^3}$ determined in our work is reasonably close to the theoretically predicted value of $\mathrm{\delta } \approx 1 \times {10^{- 10}}\,\textrm{to\; }5 \times {10^{- 10}}\textrm{c}{\textrm{m}^5}/\textrm{G}{\textrm{W}^3}$ [30]. In contrast, the linear absorption coefficients of $700 \times {10^{- 6}} < \,{\mathrm{\alpha }_{515}} < 1000 \times {10^{- 6}}\,1/\textrm{cm}$ are significantly higher than the values of $10 \times {10^{- 6}} < {\mathrm{\alpha }_{515}} < 100 \times {10^{- 6}}\,1/\textrm{cm}$ reported previously [10,18,32]. These comparatively high linear absorption coefficients require further investigations and are suspected to be related to the nonlinear absorption as the temperature increase due to absorption of the infrared and second-harmonic radiation was not measurable for inhibited harmonic generation. Such a relation could possibly be explained by absorption from an excited state which is populated by a multi-photon process from the valence band (referred to as multi-photon assisted excited-state absorption in the following). Depending on the allowed radiative transitions from the excited state, the single-photon absorption of one-, two-, or even all of the involved frequencies may be increased [33,34]. Due to the long lifetime on the order of milliseconds [35] and the high applied pulse repetition rates on the order of one 1 MHz multiple subsequent laser pulses may be affected by a multiphoton-absorption induced increase of the linear absorption coefficient. The determined linear absorption coefficients listed in Table 4 therefore explicitly only apply as long as nonlinear absorption is present due to a harmonic generation process. Furthermore, the excited state again may allow for multi-photon processes to energetically even higher lying states. However, as the probabilities of multi-photon processes are low compared to single-photon processes, the contributions of multi-photon processes from the excited state were considered negligible in the presence of single-photon processes and high average power. As the current experimental setup and model did not allow further investigation of multi-photon assisted excited state absorption, future investigations should include additional measurements (e.g. pump-probe absorption- and spectroscopic measurements) and rate-equation-based modeling of excited-state absorption. Furthermore, it is worth pointing out that absorption in crystals, in general, depends on the orientation of the crystal with respect to the direction of propagation and polarization of the considered radiation. The values specified in Table 4 are therefore only valid for the crystal orientations specified in Table 3 and the associated polarization direction employed for type I (oo-e) phase matching.
Fig. 8. Measured and computed temperature increase $\mathrm{\Delta T}$ on the exit surface of the SHG crystal S2 as a function of the generated SH Power for a range of linear (gray) and nonlinear (blue) absorption coefficients. The dash-dotted line was added to guide the eye.
While nonlinear absorption in the coating on the exit facet may also increase surface temperature, we expect that this was not a major contribution in our experiments. This expectation is based on the data of the two crystal lengths of 1 mm and 2 mm in Fig. 6, which both show high agreement to our model. As the length of the two crystals differed by a factor of 2, a deviation of one of the two datasets from the model would be expected if absorption in the coating was significant. Since we neither observed hotspots in any of our thermograms nor measurable temperature increase for inhibited harmonic generation, we expect that absorption related to the quality of the coatings was also not an issue in our experiments. Nonetheless, future investigations should include additional measurements of uncoated samples to enable direct separation of effects originating from the bulk and the coated surfaces.
4. Discussion of limitations and strategies for power and energy scaling
As nonlinear absorption occurs in all materials, the results of our study are not limited to LBO. Possible thermal limitations therefore need to be considered for all harmonic generation processes and in transmissive optics at high average powers and high intensities. The impact of the effect obviously depends on the properties of the material (thermal conductivity, bandgap energy) and the laser parameters (wavelength, intensity, and average power). As the probability of multiphoton absorption rapidly decays for higher-order processes, a larger energy gap is helpful to alleviate the heating by nonlinear absorption. E.g. comparing THG to a wavelength of 343 nm (3.61 eV) in BBO (6.43 eV [27]) and LBO (7.78 eV), LBO is expected to perform better than BBO, as three photons instead of two photons are needed to cross the energy gap.
For our specific case of harmonic generation, where the encountered thermal limitations were caused by the nonlinear absorption of the frequency-converted radiation itself, only a decrease of its intensity reduces the absorption and therefore detrimental thermal effects for a given material. For fixed parameters of the laser pulse, the reduction of the intensity can only be achieved by an increase of the area of the injected beam(s), which effectively lowers both, the nonlinear absorption and the conversion efficiency. A decreased conversion efficiency can partly be compensated for by the use of longer crystals but is limited by temporal walk-off and spectral phase mismatch. A non-collinear phase-matching geometry which allows for improved or even perfect group-velocity matching and enables the use of longer crystals [36] might therefore prove beneficial. The crystal length and beam size however need to be well adapted to avoid detrimental effects due to the spatial walk-off inherent to non-collinear phase matching.
Alternatively, the well-established coherent beam combination technique could be used to distribute the average power and hence the heat load among several identical nonlinear crystals while using a simple collinear phase-matching geometry as proposed by Tsubakimoto et al. [37]. Although increasing the complexity and cost of the overall system, this approach provides straightforward power scalability as shown in recent years for ultrafast fiber amplifiers [6,38]. Depending on the generated frequency, the problem might however only be shifted from the nonlinear crystal to the transmissive optical components of the beam combination stage.
The commonly used high-power capable nonlinear crystals exhibit a low thermal conductivity on the order of a few $\textrm{W}/\,({\textrm{m} \times \textrm{K}} )$, which causes a large temperature increase at already small heating powers. An improved thermal management concept as demonstrated in Ref. [12] can therefore be beneficial to reduce thermal gradients and the related degradation of the beam quality. Depending on the pulse parameters and the material of the transparent heat spreaders, the benefits and drawbacks of adding transmissive heat spreaders in the optical path need to be weighed carefully [39]. Alternatively, operation at cryogenic temperatures could be beneficial as an increase of the thermal conductivity by more than one order of magnitude was measured at temperatures below approx. 50 K for LBO [40], and a significant reduction of the absorption of deep-UV radiation was measured for BBO [41]. Cryogenic cooling could therefore push the boundaries of the currently encountered power and energy scaling problems to a new level but requires extensive research as the optical properties at these temperatures are unknown.
5. Conclusion
In summary, we studied the nonlinear absorption of lithium triborate crystals under real-world application conditions for frequency doubling and frequency tripling of a sub-picosecond Yb:YAG laser with high average and high peak power. Nonlinear absorption of the generated harmonic radiation was observed for both harmonic generation processes. While being clearly the dominant heating mechanism during third-harmonic generation, the severity during second-harmonic generation strongly depends on the pulse parameters and was only significant when simultaneously applying high pulse peak intensity and high average power.
Four-photon absorption was found to be the main nonlinear absorption mechanism during second-harmonic generation. Three-photon absorption was found to be the dominant absorption process during third-harmonic generation. Although the absorption per pulse was found to be low ($\le 0.5\; \%)$, significant thermal load was accumulated over many pulses due to the high average power. As a consequence, the simultaneous generation of high peak intensity and high average power is especially critical.
While nonlinear absorption is well-known in the field of deep-UV generation [42], to the best of our knowledge, this effect so far was not linked to the thermal distortions and limitations observed for frequency-doubling and –tripling of high-power ultrafast Ytterbium lasers [12,13,15]. Identifying this limitation enables the development of dedicated concepts to avoid excessive nonlinear absorption facilitating the scaling of average power and pulse peak power.
Funding
Horizon 2020 Framework Programme (687880).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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