1. Introduction

Since the cubic nonlinear susceptibility χ⁽³⁾ is small compared to the quadratic nonlinear susceptibility χ⁽²⁾, direct third harmonic generation (THG) using a single crystal usually lacks efficiency. Hence, efficient THG is commonly understood as a cascaded process with two separate crystals generating the second harmonic and subsequently the sum frequency of the second harmonic with the fundamental. With this two-step scheme, above 80 % efficiency from the fundamental to the third harmonic is possible [1].

However, with the emergence of ultra-short pulse lasers featuring high peak power, direct THG in a single crystal has become feasible [2–5]. An efficiency of up to 11 % was obtained by Miyata et al. in a combination of the direct cubic nonlinear process with cascaded phase-mismatched quadratic processes [5].

Nevertheless, the demand for high conversion efficiencies still leads to the preference of a two crystal setup cascading second harmonic generation (SHG) and sum frequency generation. Craxton presented different high efficiency tripling schemes implementing Type I or Type II SHG and THG crystals, respectively. The optimum power ratio of 67 % of the second harmonic to the fundamental can be adjusted by a polarization-bypass using a Type I SHG crystal as well as a polarization-mismatch using a Type II SHG crystal [1]. Furthermore, the power ratio for optimum THG can also be optimized by using a phase-mismatch in the SHG crystal (Type I or Type II) reducing the conversion efficiency from the fundamental to the second harmonic. For birefringent phase-matching, the refractive indices and consequently the phase-mismatch can be adjusted by changing the angle (angle-detuning, see also [1]) or the temperature (temperature-detuning) of the nonlinear crystal, with the latter exploiting the dependence of the refractive indices from the temperature. Temperature-detuning can also be applied to quasi-phase-matching.

However, a phase-mismatched SHG process influences not only the conversion efficiency from the fundamental to the second harmonic but also has an impact on the phase of the residual fundamental wave [6–8]. This nonlinear phase shift is caused by the superposition of the non-converted fundamental with the fundamental generated by back conversion of the second harmonic [8]. This so-called cascaded χ⁽²⁾ : χ⁽²⁾ process is the reason for χ⁽³⁾-like self-action of the fundamental beam which had already been investigated theoretically by Ostrovskii in 1967 [6]. Experimental results on self-focusing and self-defocusing in potassium titanyl phosphate (KTP) were reported by DeSalvo et al. in 1992 using the Z-scan technique [7].

Since then, cascaded second-order effects have been thoroughly investigated leading to applications such as all-optical signal processing, pulse compression, solitons and mode-locking [9]. The latter was demonstrated by Cerullo et al. using self-defocusing in lithium triborate (LBO) in a Kerr-lens mode-locking scheme [10]. Beckwitt et al. reported on the compensation for self-focusing using the phase shifts induced by cascaded second-order processes to cancel the χ⁽³⁾-Kerr phase shifts [11]. However, to the best of our knowledge, the role of self-focusing and self-defocusing of the fundamental beam in a phase-mismatched SHG crystal on THG, in a two crystal frequency tripling scheme, has not been reported so far.

Recently, we reported on a deep-UV laser system at 191.7 nm by generating the seventh harmonic of a high-power Q-switched 1342 nm Nd:YVO₄ laser with four conversion stages [12]. The first two stages generated the second and third harmonic. The THG efficiency was optimized by using a phase-mismatch in the SHG crystal via temperature detuning. The resulting curve, efficiency versus temperature, was asymmetric with respect to the sign of the detuning from the point of optimum phase-matching.

In this paper, we present a detailed investigation of the THG optimization with respect to the phase-mismatch in the SHG crystal for different fluences of the fundamental beam in the SHG crystal. The asymmetry of these detuning curves increased with the fluence and was caused by self-focusing and self-defocusing of the fundamental beam in the phase-mismatched SHG crystal, which was verified by measuring the beam diameter in the THG crystal. By assuming a thin lens inside the SHG crystal, the dioptric power of the effective Kerr lens could be determined with the ABCD-matrix formalism. Additionally, the impact of this self-action on the beam quality of the third harmonic was investigated. Finally, the self-focusing and self-defocusing behavior was verified theoretically by numerically solving the coupled amplitude equations for SHG.

2. Experimental setup

The experimental setup, depicted in Fig. 1, consisted of a Q-switched 1342 nm laser and subsequent frequency doubling and tripling stages.

 

Fig. 1 Experimental setup. For details see text.

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The 1342 nm Nd:YVO₄ laser was actively Q-switched by an acousto-optic modulator and delivered 16.3 ns long pulses with 10 kHz pulse repetition frequency and low pulse energy fluctuations of σ < 1 %. The system provided 15.2 W average power with a Gaussian-shaped beam profile and a beam propagation factor of M² < 1.1. Details on the power characteristic and beam quality were published recently [12].

After two mirrors (M1 and M2, HR 1342) and a collimating lens L1, a half-wave plate and a thin-film polarizer (TFP) served as a variable attenuator. A telescope consisting of lenses L2 and L3 allowed the adjustment of the focus and therefore the fluence of the 1342 nm radiation in the SHG crystal. We used four different radii for our investigations: 265 μm, 231 μm, 201 μm and 179 μm (geometric mean of x-direction and y-direction). After the telescope and three mirrors (M3–M5, HR 1342), an average power of up to 13.3 W was available for SHG. The SHG crystal was a non-critically phase-matched (NCPM) bismuth triborate (BiBO) crystal (θ = 0°, ϕ = 0°, Type I) with a length of 15 mm and an aperture of 3 × 3 mm² (671/1342 nm AR coating). The temperature for phase-matching was stabilized around 244 °C. The THG crystal was a NCPM LBO crystal (θ = 0°, ϕ = 0°, Type II) with a length of 20 mm and an aperture of 3 × 3 mm², the entrance facet being AR coated at 671/1342 nm and the exit facet at 447/671 nm. The temperature of the THG crystal was stabilized around 174 °C for optimum phase-matching. Two different focusing schemes were used for THG. Without lens L4, the 1342 nm beam had a radius of 257 μm and the 671 nm beam a radius of 228 μm inside the THG crystal (measured at low pump power and perfect phase-matching), the crystal being positioned directly behind the SHG crystal in which the 1342 nm beam had a radius of 265 μm. In the other setup, the 1342 nm and the 671 nm beam were focused into the THG crystal with lens L4 whose position was adjusted for each focus in the SHG crystal, keeping the fluence inside the THG crystal approximately constant. At low pump power and perfect phase-matching, the 1342 nm beam had a radius of 169 μm and the 671 nm beam a radius of 126 μm (geometric mean of both directions) inside the THG crystal, the focus in the SHG crystal being 179 μm. Finally, two dichroic mirrors (M6 and M7) were used to separate the 1342 nm, 671 nm and 447 nm beams.

3. Results and discussion

To determine the influence of self-action of the fundamental wave in the SHG crystal on THG, the THG efficiency for SHG detuning, the beam radii in the THG crystal and the beam quality of the third harmonic were investigated.

In section 3.1 the impact of detuning of the SHG phase-matching on the THG efficiency is determined. Several foci of the fundamental beam in the SHG crystal were examined in order to identify the influence of the fluence on the shape of the detuning curve. The curve showed a distinctive asymmetry which increased with the fluence. In section 3.2 this can be attributed to self-action of the fundamental wave in the phase-mismatched SHG crystal affecting its beam diameter in the THG crystal. The dioptric power of the self-focusing and self-defocusing effect could be determined with the ABCD-matrix formalism by using the measured beam radii in the THG crystal. In section 3.3 the influence of the self-action on the beam quality of the third harmonic is presented. Finally, the observed self-action is verified theoretically in section 3.4 by solving the coupled amplitude equations numerically.

3.1. THG efficiency curve depending on temperature detuning of the SHG phase-matching

Recently, we published the phase-matching curves for SHG in BiBO and THG in LBO (see part 3.2 and 3.3 of [12]). Maximum SHG was obtained for a temperature of 244.2 °C of the SHG crystal with an acceptance bandwidth of 1.4 °C full width at half maximum (FWHM). Optimum phase-matching of the THG crystal occurred for a temperature of 174.44 °C, with the temperature acceptance bandwidth being 0.8 °C. At full pump power, both the SHG and THG phase-matching curves were asymmetric and deviated from the sinc²-like characteristic. This was caused by thermal effects at high pump powers which has been theoretically investigated by Okada et al. [13].

For optimum THG, it was not sufficient to only ensure perfect phase-matching in the THG crystal. Additionally, the temperature of the SHG crystal had to be optimized with respect to the tripling efficiency. The detuning of the phase-matching temperature of the SHG crystal had a huge impact on the efficiency of THG which is illustrated in Fig. 2. The red curves represent the obtained SHG power in the absence of THG whereas the blue curves display the resulting THG power, with the THG crystal being tuned to optimum phase-matching. Those curves were measured by decreasing the SHG temperature with a slope of 1 °C/min and simultaneously logging the SHG or the THG power, respectively.

 

Fig. 2 Dependence of the sum frequency power at 447 nm (blue) and the SHG power without THG (red) from the phase-matching temperature of the SHG crystal for different setups. The total pump power of the conversion units is kept constant at 13.3 W. Decreasing beam radii of the 1342 nm beam inside the SHG crystal were used for measurements (b)–(e) resulting in an increasing peak fluence. The summarized fluence of the 671 nm beam and the 1342 nm beam inside the THG crystal is approximately constant for measurements (b)–(e). Measurement (a) is conducted without lens L4 resulting in a lower fluence in the THG crystal. Beam radii and fluences F in the SHG crystal for the different setups (geometric mean of both axis): (a) 265 μm (F = 1.21 J/cm²), (b) 265 μm, (c) 231 μm (F = 1.59 J/cm²), (d) 201 μm (F = 2.1 J/cm²) and (e) 179 μm (F = 2.64 J/cm²). See text for further details. The THG data of part (e) has already been published in our prior work [12].

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The results in Fig. 2(a) were obtained by positioning the THG crystal directly behind the SHG crystal without lens L4. The focus in the SHG crystal was 265 μm resulting in a peak fluence of 1.21 J/cm² for a Gaussian beam. The THG efficiency showed a local minimum for perfect phase-matching in the SHG crystal. By detuning the SHG temperature the tripling efficiency increased resulting in a local maximum on both sides of the phase-matching peak. The increase in efficiency by detuning could be attributed to an improvement of the beam quality of the residual fundamental beam and the adjustment of the power ratio of the fundamental and the second harmonic. However, the asymmetry of the curve in terms of a different maximum THG power for positive and negative detuning could not be explained by these effects. By using lens L4 to generate a focus in the THG crystal, with the fluence in the SHG crystal being the same, the asymmetry of the curve was mirrored which is illustrated in Fig. 2(b). With lens L4, maximum THG was achieved for positive detuning, whereas without lens L4, optimum THG occurred for negative detuning. Hence, the asymmetry had to be caused by spatial effects. Additionally, the focusing in the THG crystal led to a higher conversion efficiency.

The asymmetry became stronger for decreasing beam radii and therefore increasing fluences in the SHG crystal (see Figs. 2(c)–2(e)). For the highest investigated fluence of 2.64 J/cm² (beam radius of 179 μm) in the SHG crystal, this asymmetry was rather distinctive (Fig. 2(e)). For positive detuning there was a narrow peak resulting in 7.25 W generated power at 447 nm. A much broader and lower local maximum was obtained for negative detuning providing 5.33 W THG power.

This behavior could be attributed to an effective Kerr lens in the SHG crystal caused by a cascaded χ⁽²⁾ process [7–9]. This led to self-focusing or self-defocusing of the fundamental beam, depending on the sign of the detuning, influencing the beam radius of the fundamental in the THG crystal, which is explained in the following section.

3.2. Dioptric power of the effective Kerr lens

The dioptric power of the effective Kerr lens and the resulting fundamental beam radius in the THG crystal were studied for the highest investigated fluence of 2.64 J/cm² in the SHG crystal. The beam radius of the fundamental and the second harmonic beam in dependence of the detuning from the phase-matching peak is shown in Fig. 3(a), with the focus in the SHG crystal being 179 μm. The beam radii were measured with moving slit beam profilers at the position of the THG crystal by separating and attenuating both beams directly behind lens L4. The radius of the second harmonic beam increased for positive and negative detuning similarly whereas the radius of the fundamental beam decreased to a distinguished minimum for positive detuning and increased to a maximum for negative detuning. The minimum radius was approximately coincident with the narrow THG peak for positive detuning. At this point, the beam radius of the fundamental and the second harmonic beam were almost the same, which led to a strong rise of the conversion efficiency. Furthermore, this equality can be expected to have a beneficial impact on the beam quality of the third harmonic (see section 3.3). The maximum of the beam radius was in coincidence with the broader and lower THG peak for negative detuning, with the large difference of the radii of the fundamental and the second harmonic leading to a much lesser rise of the efficiency. The THG efficiency was still increased due to a more beneficial power ratio and beam quality of the involved beams.

 

Fig. 3 (a) Dependence of the beam radii of the 1342 nm beam and the 671 nm beam inside the THG crystal from the phase-matching temperature for a beam radius of 179 μm in the SHG crystal. (b) Dioptric power of an effective Kerr lens inside the SHG crystal calculated from the beam radius at 1342 nm inside the THG crystal with the ABCD-matrix formalism using a thin lens inside the SHG crystal. Shifting the measured data by ΔT = 0.132 °C would lead to a vanishing lens for zero detuning.

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The dioptric power of the effective Kerr lens in the SHG crystal could be determined by assuming a thin lens in the center of the SHG crystal and simulating its impact on the beam propagation with the ABCD-matrix formalism. From the measured beam radius of the fundamental beam in the THG crystal the dioptric power of the lens in dependence of the phase-mismatch could be calculated which is presented in Fig. 3(b). The curve of the dioptric power was slightly asymmetric or shifted with respect to the phase-matching peak (ΔT = 0 °C), similar to the measured beam radii. This led to a non-vanishing Kerr lens for perfect phase-matching and to zero-crossing of the dioptric power for negative detuning. We think this result might be attributed to the asymmetry of the SHG phase-matching curve which is illustrated in Fig. 4(a). The deviation of the phase-matching peak from the mean value of the FWHM positions was ΔT = 0.147 °C. This difference had almost the same size as the shift of the zero-crossing point of the dioptric power curve with respect to the SHG phase-matching point. By shifting the measured curves of the radii and the dioptric power by ΔT = 0.132 °C zero detuning would lead to a vanishing Kerr lens. The effective Kerr lens was defocusing for positive detuning with a minimum dioptric power of −8.93 m⁻¹ and focusing for negative detuning with a maximum dioptric power of 9.09 m⁻¹.

 

Fig. 4 (a) Asymmetry of the measured phase-matching curve of the SHG stage. The deviation of the phase-matching peak from the mean value of the FWHM positions is ΔT = 0.147 °C. (b) Influence of the effective Kerr lens inside the SHG crystal on the propagation of the 1342 nm beam and its beam radius inside the THG crystal. The beam radii are calculated with the ABCD-matrix formalism.

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The impact of this self-focusing and self-defocusing on the subsequent propagation of the fundamental beam is depicted in Fig. 4(b). The beam radii of the fundamental beam were calculated with the ABCD-matrix formalism assuming a thin lens in the SHG crystal with a focal length of f = +110 mm (green, maximum for negative detuning) or f = −112 mm (magenta, minimum for positive detuning), respectively. Additionally, the propagation without any lens (black) is also illustrated. The effective Kerr lens affected the size and the divergence of the fundamental beam at lens L4 which thereupon influenced the beam radius of the fundamental beam in the THG crystal. In combination with lens L4, a defocusing Kerr lens led to a smaller beam radius in the THG crystal and a focusing Kerr lens to a larger one. Hence, self-defocusing had a beneficial impact on THG leading to a higher conversion efficiency.

3.3. Impact of the self-focusing and self-defocusing on the beam quality of the THG beam

The self-focusing and self-defocusing of the effective Kerr lens not only influenced the conversion efficiency of the sum frequency mixing process to the third harmonic, but also the beam quality of the generated beam at 447 nm which is shown in Fig. 5 for a focus of 179 μm in the SHG crystal and 13.3 W fundamental power.

 

Fig. 5 (Top) Beam profiles of the 447 nm beam for different phase-mismatches of the SHG process. (Bottom) Profiles in x-direction (black) with the corresponding Gaussian fit (green) and percentage of match. (a) ΔT ≈ −0.612 °C (Match of Gaussian fit: 79.9 %). (b) ΔT = 0 °C (SHG phase-matching peak, match of Gaussian fit: 88.3 %). (c) ΔT ≈ 0.186 °C (Match of Gaussian fit: 96.2 %).

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For a detuning of ΔT ≈ −0.612 °C from the SHG peak (broader and lower THG maximum in Fig. 2(e)), the resulting beam profile is presented in Fig. 5(a). The beam profile consisted of a peak on a broader background which was caused by the huge difference of the beam radii of the fundamental (approx. 225 μm) and the second harmonic beam (approx. 130 μm) in the THG crystal. The match of a Gaussian fit (green) with the profile in x-direction (black) was 79.9 %. At the phase-matching peak for SHG (ΔT = 0 °C), the wings of the beam profile were slightly broader compared to a perfect Gaussian beam which is illustrated in Fig. 5(b). The radius of the fundamental beam (approx. 140 μm) was slightly larger than the radius of the second harmonic beam (approx. 125 μm) resulting in the wider wings of the THG beam. The match of a Gaussian fit with the profile in x-direction was 88.3 %. An improved beam quality was obtained for a detuning of ΔT ≈ 0.186 °C (narrow maximum of THG in Fig. 2(e)) which is shown in Fig. 5(c). The beam radius of the fundamental (approx. 128 μm) and the second harmonic (approx. 123 μm) were nearly the same leading to a Gaussian-shaped beam profile of the generated beam at 447 nm, with the match of the Gaussian fit being 96.2 %. Hence, the self-defocusing of the fundamental beam by cascaded second-order effects in the phase-mismatched SHG crystal was beneficial for both the conversion efficiency in the subsequent THG stage and the resulting beam quality of the third harmonic beam.

3.4. Numerical treatment of the effective Kerr lens by phase-mismatched SHG

The induced nonlinear phase-shift of a phase-mismatched second harmonic (2ω) generation process on the fundamental wave (ω) can be determined by solving the coupled amplitude equations for SHG numerically. Additionally, analytical solutions in the limit of non-depleted pump approximation are presented for comparison. For Type I SHG, the coupled amplitude equations in the slowly varying envelope approximation for infinite plane waves in SI units are [14]:

The coupled amplitude equations were solved with a Runge-Kutta algorithm using the effective nonlinear coefficient d_eff = 2.2 pm/V [16] and the spatial and temporal peak intensity of a Gaussian beam I = 2P_Peak/(πw²) with the peak power P_Peak = 76.6 kW and the beam radius w = 179 μm.

The resulting normalized SHG power for 13.3 W average pump power (green curve) is shown in Fig. 6(a). The analytically obtained phase-matching curve (∝ sinc²(ΔkL/2), crystal length L) in the non-depleted pump approximation dA_ω/dz ≈ 0 is also plotted for comparison (black curve), the well known effect of acceptance bandwidth narrowing for high intensities being clearly observable in the numerical results. The bandwidth at full power was 1.3 °C which agreed well with the experimental value of 1.4 °C. The deviation was caused by the asymmetry of the experimental curve (see red curve in Fig. 2(e) and [12]) via thermal effects which were not considered in the numerical calculations. The bandwidth in the non-depleted pump approximation was significantly larger (approx. 2.27 °C). Hence, pump depletion has to be considered and it is to be expected that the numerical results describe the self-action effect more accurately than the analytical treatment which will be presented below.

 

Fig. 6 (a) Numerically calculated SHG phase-matching curve for a radius of 179 μm and 13.3 W fundamental power (green) and the phase-matching curve for the non-depleted pump approximation (black). (b) Numerically calculated nonlinear phase-shift per optical intensity η in dependence from the phase-mismatch of the SHG process at 13.3 W pump power (green). η is equivalent to the effective nonlinear refractive index $n_2^{\mathit{eff}}$ in the limiting case of non-depleted pump approximation (black). The experimental data on the dioptric power of the Kerr lens is overlaid for comparison (blue). The experimental data is shifted by ΔT = 0.132 °C for better comparability resulting in zero-crossing for vanishing dioptric power.

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The nonlinear phase-shift of the fundamental wave at the end of the SHG crystal is determined by the real and the imaginary part of A_ω:

The nonlinear refractive index $n_2^{\mathit{eff}}$ and η in dependence of the phase-mismatch are shown in Fig. 6(b). The maximum and minimum of η (green curve) occurred at a considerably smaller detuning ΔT = ±0.52 °C compared to $n_2^{\mathit{eff}}$ (black curve, ΔT = ±1.28 °C). Furthermore, the maximum and minimum value of η (±8.86 × 10⁻¹⁸ m²/W) was substantially lower than the extremal values of $n_2^{\mathit{eff}}$ (±14.66 × 10⁻¹⁸ m²/W). These characteristics of η are the result of pump depletion. The slope of the nonlinear phase-shift ΔΦ_NL(L) with respect to the input irradiance on the crystal decreases for growing fundamental pump power and shows a saturation behavior (see [8,9] for further details). Hence η, which is in principle the normalized nonlinear phase-shift, decreases for increasing fundamental pump power. Therefore, the maximum of η (green curve) in Fig. 6(b) is reduced compared to $n_2^{\mathit{eff}}$ (black curve).

For negative detuning from the phase-matching peak, η was positive resulting in self-focusing of the fundamental beam. Self-defocusing was achieved for positive detuning, η being negative. The experimental data on the dioptric power (blue curve) is overlaid in Fig. 6(b) for comparison with η. The data was shifted by ΔT = 0.132 °C for better comparability leading to zero dioptric power at the phase-matching peak. The characteristics of the η curve and the experimental data on the dioptric power of the Kerr lens were qualitatively in good agreement. The positions of the respective maxima for negative detuning and the minima for positive de-tuning coincided well.

4. Conclusion

In conclusion we reported for the first time, to the best of our knowledge, on the beneficial impact of self-action in a phase-mismatched SHG crystal on the THG efficiency and beam quality in a two crystal frequency tripling scheme. It was found that the temperature detuning curve was strongly asymmetric with respect to the SHG peak. For positive detuning, there was a narrow peak in THG efficiency whereas for negative detuning a shallow and broader maximum occurred (see Fig. 2(e)). This behavior could be attributed to an effective Kerr lens in the frequency doubling crystal caused by the phase-mismatched SHG process. This self-focusing and self-defocusing of the fundamental wave was induced by a cascaded χ⁽²⁾ : χ⁽²⁾ process and had a huge influence on the beam radius of the fundamental beam in the frequency tripling crystal (see Fig. 3(a)). The effective Kerr lens was defocusing for positive detuning, causing a smaller beam radius of the fundamental wave in the THG crystal in combination with the focusing lens in front of the THG stage. For negative detuning, the effective Kerr lens was focusing which led to a larger beam radius of the fundamental wave in the tripling crystal. The type of self-action (self-focusing or self-defocusing) was also in agreement with numerical calculations obtained by solving the coupled amplitude equations for SHG. The saturation behavior of the nonlinear phase-shift for high pump powers due to pump depletion could be verified. The position of the maximum and minimum phase-shift with respect to the detuning temperature agreed well with the experimental data of the dioptric power of the Kerr lens (see Fig. 6(b)). Additionally, the impact of this self-focusing and self-defocusing effect on the beam quality of the THG beam was investigated. It was found that the self-defocusing of the fundamental beam for positive detuning improved the beam quality essentially due to beam radii matching, leading to a Gaussian-shaped beam profile (see Fig. 5(c)). Finally, it is important to emphasize that self-action effects due to cascaded χ⁽²⁾ : χ⁽²⁾ processes have to be taken into account in designing successive frequency conversion stages.