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Abstract
Collinear phase matching of the Stokes ↔ anti-Stokes interaction for Raman-active crystals with different birefringence was studied theoretically as well as experimentally. It was shown that collinear phase matching of the Stokes ↔ anti-Stokes interaction in low-birefringent crystals can be insensitive to angular mismatch if a phase matching angle is higher than 60°. We have developed and experimentally realized an extracavity parametric Raman anti-Stokes laser based on a low-birefringent SrWO4. Cyan 507-nm anti-Stokes conversion from green (532 nm) pump radiation of a 5-ns, 1-mJ second harmonic Nd:YAG laser has been obtained. Laser setup with a single beam excitation made it possible to use an output face of the SrWO4 crystal as an output coupler because of wide (6°) angular tolerance of collinear phase matching that resulted in an increase of slope efficiency of anti-Stokes generation higher than 3% at the anti-Stokes energy output of a 10-µJ level.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Stimulated Raman scattering (SRS) in crystals presents a simple and non-expensive method for laser frequency conversion into Stokes-shifted wavelengths that are not readily available from solid-state lasers [1]. Parametric four-wave mixing (FWM) of frequency components in crystalline Raman lasers on the same resonant Raman nonlinearity allowed generation of not only Stokes, but also anti-Stokes SRS components as far as a phase matching condition was fulfilled [2–10]. For example, it can help to generate multicolor laser pulses in the red-green-blue spectral range having potential for data storage, RGB micro-sized projectors and color displays, biomedical techniques, etc. However, parametric Raman conversion in the visible range, where crystals have high dispersion, was low-efficient ($\mathrm {<}$ 1 %) because the interaction of the SRS components has to be non-collinear to compensate the dispersion wave mismatch that was critical to angular mismatch [5,8]. Using the birefringent nonlinear crystal can help to realize collinear phase matching for orthogonally polarized radiation components that was widely used for three-wave mixing in nonlinear optics of media with quadratic nonlinearity. A similar idea of collinear phase matching has been proposed in work [11] for parametric FWM of SRS components in a birefringent Raman-active crystal, but such parametric Raman conversion was low-efficient [10] because of narrow angular tolerance of phase matching and a walk-off effect for the extraordinary waves limiting the effective interaction length. In order to increase the effective interaction length, the non-collinear wave interaction compensating the walk-off effect has been also proposed and realized in [11]. In the work [8], to widen the angular tolerance of FWM of the SRS components in the highly birefringent crystal, the non-collinear phase matching condition for orthogonally polarized SRS components in the crystalline parametric Raman anti-Stokes laser was studied in detail. It was found that the phase matching of parametric Raman interaction is tangential and insensitive to the angular mismatch if the Poynting vectors of the pump and anti-Stokes waves are collinear. Using the highly birefringent CaCO${}_{3}$ crystal, such tangential phase matching condition compensating the anti-Stokes walk-off has been experimentally realized using an additional non-collinear input probe wave having the same frequency as the pump wave. It allowed to increase the conversion efficiency into the anti-Stokes wave up to 4 % in the single-pass and extracavity schemes of the parametric Raman anti-Stokes laser [8,9]. Disadvantage of such schemes is complexity of the system having two non-collinear orthogonally polarized (pump and probe) input beams.
In this study, we will present in detail the conditions of collinear phase matching of FWM of orthogonally polarized SRS components for crystals with different birefringence and demonstrate operation of a crystalline parametric Raman anti-Stokes laser with collinear phase matching. We used a SrWO${}_{4}$ crystal having low birefringence which was cut at a phase matching angle that allowed obtaining efficient anti-Stokes generation in the simplest collinear extracavity system with a single input beam and even without a separate output coupler.
2. Collinear phase matching of parametric Raman interaction
The idea is to use birefringent (uniaxial) Raman-active crystals for anti-Stokes parametric Raman generation at fulfillment of collinear phase matching condition for FWM of orthogonally polarized SRS radiation components [11]:
In expressions (1) we also took into account that the phase-matching type should be eeoo (ooee) [8–11]. It is caused by the fact that the most efficient parametric Raman interaction occurs at following conditions. Firstly, the $\nu$${}_{R}$ Raman mode of the crystal should be coherently driven by the SRS interaction of the pair of the equally polarized pump and Stokes waves. Secondly, the process of scattering of the probe wave into the anti-Stokes wave should take place at their equal polarization too. In order to fulfill the collinear phase matching condition, their polarization has to be orthogonal to the polarization of the pump and Stokes waves.
We can write expressions for the collinear propagation wave mismatch $\Delta$k of the eeoo or ooee type:
Expanding expressions (2) to Taylor series we obtain the approximate expressions:
Collinear phase matching can also be characterized by the angular tolerance ($\Delta \Theta_\textrm {pm}$) corresponding to the coherence condition |$\Delta$k | $\mathrm {\le }$ $\mathrm{\pi}$/L [8] where L is the crystal length:
As it follows from expressions (6), at |$\Delta {k}_{0}$| $\mathrm {\le }\pi /L$ (the crystal is short enough to fulfill the coherence condition) the angular tolerance is the widest, it is centered at the zero angle $\Theta$ = 0, and therefore all interacting SRS components become ordinary waves. This scheme was realized in work [12] in the CaCO${}_{3}$ at the pump/probe wavelength of $\lambda$${}_\textrm {pump}$ = $\lambda$${}_\textrm {probe}$ = 1.338 µm close to the zero dispersion wavelength of CaCO${}_{3}$ of $\lambda$${}_{d}$ = 1.35 µm (for an ordinary wave) corresponding to |$\Delta$k${}_{0}$| = 0.9 cm${}^{-1}$ at the crystal length of L = 1 cm. This setup resulted in efficient anti-Stokes generation at parametric Raman interaction of ordinary waves because using the Sellmeier equation for CaCO${}_{3}$ from the first formula in expressions (6) with the sign ’+’ we obtain $\Delta \Theta _\textrm {pm}~\mathrm {\approx }~6\mathrm {{}^\circ }$.
It also follows from expressions (6) that at |$\Delta {k}_{0}$| $\mathrm {>}\pi /L$ we obtain collinear phase matching centered at the phase matching angle $\Theta _\textrm {pm}$ (expressions (5)) with the narrow value of $\Delta \Theta _\textrm {pm}$ (expressions (6) with the sign ’-’) for interaction of the orthogonally polarized (eeoo or ooee) SRS components which can be also approximately expressed (using the derivative from (4) [10]) as
One more “angular” issue of collinear phase matching of orthogonally polarized SRS components is the walk-off effect for the SRS components playing the role of extraordinary waves limiting the effective interaction length [10]. The walk-off angle for extraordinary waves (it is the angle between the extraordinary wave vector and its Poynting vector in the crystal) is [13]
Let’s make calculations using expressions (2)-(8) for Raman-active crystals with different birefringence: CaCO${}_{3}$ and SrWO${}_{4}$. For comparison, we consider the crystals with an equal length of L = 1 cm. We select the pump/probe wavelength of $\lambda$${}_\textrm {pump}$ = $\lambda$${}_\textrm {probe}$ = 0.532 µm that is essentially lower than $\lambda$${}_{d}$ for both crystals. We use Sellmeier equations for CaCO${}_{3}$ from [14] and for SrWO${}_{4}$ from [15]. CaCO${}_{3}$ is a negative uniaxial crystal with high birefringence of $n_{\textrm {pump}}^{\textrm {e}} -n_{\textrm {pump}}^{\textrm {o}}$ = -0.175 and with a Raman frequency of $\nu$${}_{R}$ = 1086 cm$^{-1}$ [11]. SrWO${}_{4}$ is a positive uniaxial crystal with low birefringence of $n_{\textrm {pump}}^{\textrm {e}} -n_{\textrm {pump}}^{\textrm {o}}$ = 0.011 and with a Raman frequency of $\nu _{R}$ = 921 cm$^{-1}$ [16].
Figure 1 shows the dependences of the wave mismatches $\Delta$k${}_{eeoo}$ and $\Delta$k${}_{oo}{}_{ee}$ on the light propagation angle $\Theta$ calculated by expressions (2) for the high-birefringent CaCO${}_{3}$ (Fig. 1(a)) and low-birefringent SrWO${}_{4}$ (Fig. 1(b)) crystals at $\lambda$${}_\textrm {pump}$ = $\lambda$${}_\textrm {probe}$ = 0.532 µm.
Fig. 1. Calculated dependences of the wave mismatches $\Delta$k${}_\textrm {eeoo}$ and $\Delta$k${}_\textrm {ooee}$ on the light propagation angle $\Theta$ calculated by expressions (2) for (a) high-birefringent CaCO${}_{3}$ and (b) low-birefringent SrWO${}_{4}$ crystals at $\lambda _\textrm {pump}~=~\lambda _\textrm {probe}$ = 532 nm.
It can be seen from Fig. 1(a) that, in the high-birefringent CaCO${}_{3}$ crystal, the collinear phase matching condition ($\Delta$k = 0) is fulfilled at low phase matching angle of $\Theta _{\textrm {pm}}^{\textrm {eeoo}}$= 10.8$\mathrm {{}^\circ }$ that is close to the value of $\Theta _{\textrm {pm}}^{\textrm {eeoo}} \mathrm {\approx }$ 10.7$\mathrm {{}^\circ }$ from the approximate expression (5). The phase matching type is eeoo because the CaCO${}_{3}$ crystal is negative uniaxial. From the approximate expression (7) ($\Delta$k${}_{0}$ = 58 cm$^{-1}$ is higher than $\mathrm{\pi}$/L) the angular tolerance for CaCO${}_{3}$ has a narrow value of $\Delta \Theta _{\textrm {pm}}^{\textrm {eeoo}} \mathrm {\approx }$ 0.6$\mathrm {{}^\circ }$. The walk-off angles are $\beta _{\textrm {probe}}^{\textrm {e}}$= 45.1 mrad and $\beta _{\textrm {aS}}^{\textrm {e}}$= 45.5 mrad for the extraordinary probe and anti-Stokes waves, respectively, and so the walk-off at the crystal output face is large and amounts L $\cdot$ tan$\beta _{\textrm {probe}}^{\textrm {e}} \mathrm {\approx }$ 0.45 mm at L = 1 cm. That explains low-efficiency of parametric Raman conversion in CaCO${}_{3}$ with the focused pump beam [10].
Figure 1(b) shows that, in the low-birefringent SrWO${}_{4}$ crystal, the collinear phase matching angle is significantly higher and amounts $\Theta _{\textrm {pm}}^{\textrm {ooee}}$= 68.3$\mathrm {{}^\circ }$ (from the approximate expression (5) it is about $\Theta _{\textrm {pm}}^{\textrm {ooee}} \mathrm {\approx }$ 69.0$\mathrm {{}^\circ }$). The phase matching type is ooee because the SrWO${}_{4}$ crystal is positive uniaxial. From the approximate expression (7) ($\Delta$k${}_{0}$ = 80 cm$^{-1}$ is higher than $\mathrm{\pi}$/L) the angular tolerance for SrWO${}_{4}$ has a very high value of $\Delta \Theta _{\textrm {pm}}^{\textrm {ooee}} \mathrm {\approx }$ 6$\mathrm {{}^\circ }$. The walk-off angles for the extraordinary waves are low and amount $\beta _{\textrm {pump}}^{\textrm {e}}$= 3.8 mrad and $\beta _{\textrm {S}}^{\textrm {e}}$= 3.7 mrad therefore the walk-off at the crystal output face is low (L $\cdot$ tan$\beta _{\textrm {pump}}^{\textrm {e}}\mathrm {\approx }$ 0.04 mm at L = 1 cm) too.
Figure 2 demonstrates an overview diagram of phase matching characteristics ($\Theta _\textrm {pm}$ and $\Delta \Theta _\textrm {pm}$) of parametric Raman anti-Stokes generation at $\lambda$${}_\textrm {pump}$ = $\lambda$${}_\textrm {probe}$ = 532 nm calculated from expressions (5) and (7) for several uniaxial Raman-active crystals in dependence on their birefringence $\left |\, n_{\textrm {pump}}^{\textrm {e}} -n_{\textrm {pump}}^{\textrm {o}} \right |$. Data for LiIO${}_{3}$ ($\nu$${}_{R}$ = 770 cm$^{-1}$, $n_{\textrm {pump}}^{\textrm {e}} -n_{\textrm {pump}}^{\textrm {o}}$= - 0.15), CaWO${}_{4}$ ($\nu$${}_{R}$ = 911 cm$^{-1}$, $n_{\textrm {pump}}^{\textrm {e}} -n_{\textrm {pump}}^{\textrm {o}}$ = 0.017), and PbWO${}_{4}$ ($\nu$${}_{R}$ = 905 cm$^{-1}$, $n_{\textrm {pump}}^{\textrm {e}} -n_{\textrm {pump}}^{\textrm {o}}$ = - 0.083) were taken from [16–19].
Fig. 2. Overview diagram of phase matching characteristics ($\Theta _\textrm {pm}$ and $\Delta \Theta _\textrm {pm}$) of parametric Raman anti-Stokes generation at $\lambda _\textrm {pump}$ = $\lambda _\textrm {probe}$ = 532 nm for known uniaxial Raman-active crystals in dependence on their birefringence.
It can be seen from Fig. 2 that the crystals with high birefringence of $\left |\, n_{\textrm {pump}}^{\textrm {e}} -n_{\textrm {pump}}^{\textrm {o}} \right |\mathrm {>}$ 0.08 have a low phase matching angle $\Theta _\textrm {pm}$ $\mathrm {<}$ 30$\mathrm {{}^\circ }$ and narrow angular tolerance $\Delta \Theta _\textrm {pm}$ $\mathrm {<}$ 0.6$\mathrm {{}^\circ }$. Decreasing the crystal birefringence lower than $\left |\, n_{\textrm {pump}}^{\textrm {e}} -n_{\textrm {pump}}^{\textrm {o}} \right |\mathrm {<}$ 0.017 results in increasing the phase matching angle $\Theta _{pm}$ $\mathrm {>}$ 54$\mathrm {{}^\circ }$ and widening the angular tolerance $\Delta \Theta_{pm}$ $\mathrm {>}$ 2.8$\mathrm {{}^\circ }$. Among the described crystals, the SrWO${}_{4}$ crystal with the lowest birefringence of $\left |\, n_{\textrm {pump}}^{\textrm {e}} -n_{\textrm {pump}}^{\textrm {o}} \right |$= 0.011 has the widest angular tolerance ($\Delta \Theta_{pm}$ = 6$\mathrm {{}^\circ }$) and the lowest walk-off ($\mathrm {<}$ 4 mrad) because the phase-matching angle $\Theta {}_{pm}$ = 69$\mathrm {{}^\circ }$ is quite close to noncritical 90$\mathrm {{}^\circ }$-phase matching. We can conclude that collinear phase matching of Stokes $\mathrm {\leftrightarrow }$ anti-Stokes interaction in low-birefringent crystals can be insensitive to angular mismatch if a phase matching angle is higher than 60$\mathrm {{}^\circ }$. We can check also other known crystals with lower birefringence, for example, SrMoO${}_{4}$ ($\nu$${}_{R}$ = 888 cm$^{-1}$, $n_{\textrm {pump}}^{\textrm {e}} -n_{\textrm {pump}}^{\textrm {o}}$ = 0.005) [20] and BaWO${}_{4}$ ($\nu$${}_{R}$ = 926 cm$^{-1}$, $n_{\textrm {pump}}^{\textrm {e}} -n_{\textrm {pump}}^{\textrm {o}}$ = - 0.002) [15]. However, the phase matching condition cannot be fulfilled for these crystals according to expression (5) because of too low birefringence.
3. Experimental setup
We propose an optical scheme of the extracavity parametric Raman crystalline anti-Stokes laser with collinear phase matching for a Raman-active crystal with low birefringence presented in Fig. 3. It was realized experimentally for the Raman-active SrWO${}_{4}$ crystal cut at 69$\mathrm {{}^\circ }$.
Fig. 3. An optical scheme of the extracavity parametric Raman SrWO${}_{4}{}_{\ }$anti-Stokes laser and a photo of the system with cyan anti-Stokes output beam. HT: high transmission, HR: high reflectivity.
The parametric Raman laser was excited by a laboratory-designed oscillator-amplifier nanosecond Nd:YAG laser system frequency-doubled to 532 nm. As the master oscillator for the this laser system we used a quasi-continuous laser-diode-pumped oscillator based on a 2.4 at.%-doped Nd:YAG active crystal and passively Q-switched by a Cr:YAG saturable absorber. Linearly polarized $\mathrm {\sim }$1 mJ output pulses were further amplified using a flashlamp-pumped Nd:YAG amplifier to an energy level up to 10 mJ. The repetition rate was limited by a power supply to 10 Hz. A second harmonic generator (a 30-mm KDP crystal) was used to generate $\mathrm {\sim }$5 ns pulses at $\lambda$${}_\textrm {exc}$ = 532 nm in a TEM${}_{00}$ mode (M$^2$ $\mathrm {\sim }$1.1).
The collinearly phase matched parametric Raman laser based on a low-birefringent SrWO${}_{4}$ crystal was developed in the simplest configuration of single beam excitation in contrast to the tangentially phase matched parametric Raman lasers based on a high-birefringent CaCO${}_{3}$ crystal [8,9]. In the previous case of CaCO${}_{3}$, the pump laser radiation was split into orthogonally polarized pump and probe waves with different incidence angles. Here, the pump and probe input waves are collinear because of collinear phase matching. The energy ratio between horizontally (extraordinary pump wave) and vertically (ordinary probe wave) polarized parts of the excitation laser radiation was tuned by a half-wave ($\lambda$/2) plate rotating the linear polarization of the excitation radiation to the angle $\mathrm{\varphi}$.
The uncoated SrWO${}_{4}$ crystal ($\nu$${}_{R}$ = 921 cm$^{-1}$) having a length of 1.3 cm was cut at the collinear phase matching angle of $\Theta {}_\textrm {pm}$ $\mathrm {\approx }$ 69$\mathrm {{}^\circ }$. A lens with a focal length of 300 mm was used for focusing of the input excitation radiation into the active crystal of the parametric Raman laser. The pump laser beam diameter at the focal plane was about 250 µm.
The Raman laser cavity was formed by the input (pumping) flat mirror having high transmission (HT) at the pump and probe excitation ($\lambda$${}_\textrm {exc}$ = $\lambda$${}_\textrm {probe}$ = $\lambda$${}_\textrm {pump}$ = 532 nm) and anti-Stokes ($\lambda$${}_{aS}$ = 507 nm) wavelengths (for its single-pass interaction) and high reflectivity (HR) in a wavelength range of 550-700 nm for extracavity SRS oscillation of the Stokes Raman radiation. As for the output coupler, during the initial experiments we used a mirror with reflectivity of 70 % at 500-700 nm. The mirrors were placed as close as possible ($\mathrm {\sim }$ 2 mm) to the active SrWO${}_{4}$ crystal. After the optimization, it was possible to remove the output mirror and use the uncoated output face of the SrWO${}_{4}$ crystal as an output coupler. Behind the Raman laser output there was placed a birefringent wedge splitting the orthogonally polarized components of the output radiation.
4. Experimental results
Initially, we used a separate output coupler (R = 70 % at 500-700 nm) to study sensitivity to angular mismatch of phase matching. In this case, we observed efficient SRS generation of many (first, second, third, and fourth) Stokes SRS components at wavelengths of 559, 590, 624, and 662 nm. At the same time, only one anti-Stokes component at 507 nm was generated, but efficiency of conversion into the anti-Stokes wave was lower than 1 % in comparison with efficient SRS generation into many Stokes components. Rotating the SrWO${}_{4}$ crystal in the horizontal plane, we found that the 507-nm anti-Stokes generation was extremely insensitive to angular mismatch in agreement with theoretical prediction of very wide angular tolerance of $\Delta \Theta _\textrm {pm}~\mathrm {\approx }$ 6${}^{o}$. This made it possible to remove the output coupler (R = 70 %) and therefore the laser cavity was formed by the pumping mirror and the SrWO${}_{4}$ output face with the Fresnel reflection coefficient of $\mathrm {\sim }$9%. This setup was possible because inaccuracy of the crystal orientation at $\Theta _\textrm {pm}~\mathrm {\approx }$ 69${}^{o}$ was significantly lower than $\Delta \Theta _\textrm {pm}~\mathrm {\approx }$ 6${}^{o}$. This output coupling optimization was directed to weaken the SRS generation of the high-order Stokes components competing with the required anti-Stokes generation. As a result, we have obtained the most efficient anti-Stokes generation at 507 nm that is presented below. Analysis of the anti-Stokes wave polarization is demonstrated by a photo in Fig. 3: behind the birefringent wedge splitting the orthogonally polarized components of the output radiation, only vertically polarized spot of the cyan 507-nm light (transmitted through an appropriate bandpass filter Thorlabs FB508.5-10) was observed in spite of presence of both the horizontally (the pump) and vertically (the probe) polarized light spots for the residual green (532 nm) spectral components in the output radiation.
Figure 4 shows generation spectrum of the parametric Raman SrWO${}_{4}$ laser and photo of the optical components decomposed by dispersion prisms. The spectrum was measured by a fiber-coupled spectrometer (Ocean Optics HR2000 with resolution $\mathrm {\sim }$1 nm at FWHM). The actual intensity shown by the spectrometer depends strongly on the exact position of the fiber in the generated beam.
Fig. 4. Generation spectrum of the parametric Raman SrWO${}_{4}$ laser excited at 532 nm and photo of the optical components decomposed by the dispersion prisms.
The anti-Stokes part of the generated spectrum included only the first anti-Stokes component at $\lambda$${}_{aS}$ = 507 nm (cyan color) without generation of higher order anti-Stokes components (485 nm, 464 nm, etc.) because the phase matching condition of parametric Raman interaction was fulfilled only for this component. The Stokes part of the generation spectrum included two bright Stokes frequency components generated by SRS as $\lambda _{\textrm {S}} =(\lambda _{\textrm {exc}}^{\, -\, 1} -\nu _{\textrm {R}} )^{-\, 1}$= 559 nm (lime color) and $\lambda _{\textrm {S2}} =(\lambda _{\textrm {exc}}^{\, -\, 1} -2\, \nu _{\textrm {R}} )^{-\, 1}$= 590 nm (orange color), and also very weak near-threshold SRS generation at $\lambda _{\textrm {S}3} =(\lambda _{\textrm {exc}}^{\, -\, 1} -3\, \nu _{\textrm {R}} )^{-\, 1}$= 624 nm (red color).
Beam quality and beam divergence of the Stokes SRS radiation components were worse than for the excitation 532-nm beam because of SRS generation in the short external cavity. As shown in Fig. 4, the lime-color spot is larger than the green-color spot. The anti-Stokes beam quality was comparably high as for the excitation beam because the parametric Raman anti-Stokes generation was single pass. The anti-Stokes cyan-color spot has a very good beam profile and an equal size with the green-color spot of the excitation light presented in Fig. 4.
Figure 5 demonstrates experimental optimization of the excitation light polarization angle $\mathrm{\varphi}$ relative to the horizontal achieved by the half-wave plate rotation. We have found that an optimum range of this angle is $\mathrm{\varphi}_\textrm {opt}\mathrm {\approx }$ 20-30${}^{o}$ corresponding to an optimum range of the probe/pump energy ratio of ${(W_\textrm {probe}/W_\textrm {pump})}_\textrm {opt}$ = tan${}^{2}\mathrm{\varphi}_\textrm {opt}$ $\mathrm {\approx }$ 0.13-0.33. This result is similar to the tangentially phase matched parametric Raman anti-Stokes CaCO${}_{3}$ lasers presented in [8,9]. In [8] this optimum was explained the following way. Phase-matched parametric generation of the anti-Stokes wave from the probe wave takes place on the medium vibration forced by another pair of waves - the pump and Stokes waves. On the other hand, parametric coupling of the pump and Stokes waves occurs by means of the medium vibration forced by the anti-Stokes and probe waves. These vibrations have a phase shift of $\mathrm{\pi}$, and therefore they coherently subtract each other suppressing the anti-Stokes generation at an excessive increase of the probe wave intensity.
Fig. 5. Experimental optimization of the excitation light polarization angle $\mathrm{\varphi}$ relative to the horizontal polarization at different values of the excitation pulse energy of 0.28, 0.45, and 0.68 mJ.
The output/input energy characteristics at the optimum value of $\mathrm{\varphi}_\textrm {opt}$ = 25${}^{o}$ together with an analysis of the output radiation polarization state are presented in Fig. 6. It can be seen that anti-Stokes wave had clearly vertical polarization, but the Stokes radiation had both horizontally and vertically polarized components similar to the input excitation radiation. We can see that the generation threshold energy of $\mathrm {\sim }$200 µJ was equal for both the anti-Stokes and Stokes components because the collinear phase matching condition was fulfilled. Initial growth of the anti-Stokes output energy has a slope efficiency of $\mathrm{\eta}_\textrm {slope}$ = 3.2 % against the Stokes slope efficiency (31+7 %). Further input energy increase to the values higher than 400 µJ resulted in saturation of the anti-Stokes output energy growth presented in Fig. 6(a). Maximum optical-to-optical efficiency of the anti-Stokes generation was as high as 1.8 % at the output and input energies of 8 and 450 µJ respectively, but the highest anti-Stokes output energy of 10 µJ has been obtained at the input energy of 700 µJ corresponding to the lower optical-to-optical efficiency of 1.4 %. The anti-Stokes beam spot measured by a CCD beam profiler (DataRay WinCamD) is presented in Fig. 6(a). The profile was Gaussian with minor distortions on its margin.
Fig. 6. Dependences of output pulse energies of (a) the anti-Stokes component and (b) all the Stokes components (horizontally and vertically polarized parts) on the excitation pulse energy at the optimum value of $\mathrm{\varphi}_\textrm {opt}$ = 25${}^{o}$.
Output radiation temporal characteristics were measured using photodiodes (EOT ET-2000) connected to an oscilloscope (Tektronix DPO4104, analog bandwidth of 1 GHz). Typical oscillograms are presented in Fig. 7. The measured anti-Stokes pulse duration was about 1.8 ns (FWHM) which is approximately three times shorter than the pump pulse duration. The pulse shortening can be explained by generation of the anti-Stokes pulse in the temporal region of overlap of the Stokes pulse with the depleted pump pulse because of FWM mechanism of the anti-Stokes generation [12].
Fig. 7. Temporal profiles of excitation, 1${}^\textrm {st}$ Stokes, and anti-Stokes radiation components.
5. Mathematical modeling
Saturation effect of the growth of the anti-Stokes pulse energy should be explained using mathematical modeling because there are two possible reasons for competition between conversion into the anti-Stokes and Stokes waves. Firstly, the Stokes energy is converted into higher order Stokes components at increasing energy input. Secondly, Stokes SRS generation depletes all the input excitation radiation including the probe wave which should be scattered into the anti-Stokes wave. Both these issues were solved by us before for the tangential phase matching in [9]. Firstly, we have used separate probe wave having no conversion into the Stokes wave. Secondly, the output coupler reflectivity for the 2${}^\textrm {nd}$ Stokes component was decreased. It resulted not only in a linear growth of the anti-Stokes energy output, but also in higher slope efficiency of 5.4 %. Now we will explain how these effects can affect the process.
In [9] we have developed a model of the external cavity parametric Raman anti-Stokes laser with two non-collinear (pump and probe) beam excitation. Now we have only single beam excitation system. We assume that the input laser beam with the wavelength of $\lambda _\textrm {exc}$ is the excitation wave for Raman conversion into the Stokes waves with wavelengths of $\lambda _{\textrm {S}} =(\lambda _{\textrm {exc}}^{\, -\, 1} -\nu _{\textrm {R}} )^{-\, 1}$ and $\lambda _{\textrm {S}2} =(\lambda _{\textrm {exc}}^{\, -\, 1} -2\, \nu _{\textrm {R}} )^{-\, 1}$, but in the FWM process of parametric Raman generation of the anti-Stokes wave ($\lambda _{\textrm {aS}}^{\, -\, 1} =\lambda _{\textrm {probe}}^{\, -\, 1} +\lambda _{\textrm {pump}}^{\, -\, 1} -\lambda _{\textrm {S}}^{\, -\, 1}$), this input laser beam is divided between the pump extraordinary wave exciting the parametric FWM process and the probe ordinary wave ($\lambda$${}_\textrm {probe}$ = $\lambda$${}_\textrm {pump}$ = $\lambda$${}_\textrm {exc}$) converting into the anti-Stokes wave in this process. So, the model is
The boundary conditions can be written as
For the SrWO${}_{4}$ crystal, the Raman gain coefficient is g = 16 cm/GW at 532-nm excitation [21]. The crystal internal loss coefficient was measured to be b = 0.01 cm${}^{-1}$. We use $\Delta$k = 0 because the phase matching condition is fulfilled. We also select the optimum value of the excitation light polarization angle $\mathrm{\varphi}$ = 25${}^{o}$.
Figure 8 demonstrates the modeling results at two values of the input excitation pulse fluence $F_{\textrm {exc}}^{\textrm {in}} =\int _{-\infty }^{\infty }I_{\textrm {exc}}^{\textrm {in}} (t)dt$ equal to 1.4 and 1.6 J/cm${}^{2}$. Taking into account the experimental value of the excitation beam diameter of 250 µm these values of fluence correspond to the excitation pulse energies of 687 and 785 µJ that provide the saturated regime of the anti-Stokes conversion in the experiment (presented in Fig. 6(a)).
Fig. 8. The modeling results at two values of the input excitation pulse fluence: 1.4 and 1.6 J/cm$^{2}$.
It can be seen from Fig. 8 that the output probe pulse ($I_{\textrm {probe}}^{\textrm {out}}$) is depleted due to not only anti-Stokes, but also Stokes conversion at both values of the input fluence. Therefore the anti-Stokes energy growth should be lower than for the two-beam excitation system [9] where the probe pulse was depleted only by anti-Stokes conversion. So, this is not the reason for the anti-Stokes growth saturation effect, but it gives us lower initial slope of the growth. However, we can also see that the output fluence of the anti-Stokes pulse is the same ($F_{\textrm {aS}}^{\textrm {out}}=0.03~J/cm^{2}$) at both values of the input excitation pulse fluence, and therefore we have saturation of the anti-Stokes energy growth. We can conclude that the reason is the 2$^{nd}$ Stokes generation. Thus we also made modeling for the case of single Stokes SRS generation at $R_{\textrm {S2}}^{\textrm {out}}$= 0 to reveal a role of the 2$^{\textrm {nd}}$ Stokes wave in the process.
Figure 9 shows the calculated dependences of the output anti-Stokes, 1${}^{st}$ Stokes, and 2${}^\textrm {nd}$ Stokes pulse fluences on the input excitation pulse fluence. Dotted lines correspond to the case of the single Stokes SRS generation at $R_{\textrm {S2}}^{\textrm {out}}$= 0. Solid lines correspond to the case of the double Stokes SRS generation at $R_{\textrm {S2}}^{\textrm {out}}$= $R_{\textrm {S}}^{\textrm {out}}$ = 9 %.
Fig. 9. The calculated dependences of the output anti-Stokes, 1${}^\textrm {st}$ Stokes, and 2${}^\textrm {nd}$ Stokes pulse fluences on the input excitation pulse fluence (the dotted lines are for $R_{\textrm {S2}}^{\textrm {out}}$= 0).
It can be seen that in the case of single Stokes SRS generation (dotted lines) we have linear growth of the 1$^\textrm {st}$ Stokes and anti-Stokes fluences without saturation. But we can see an interesting effect that the initial anti-Stokes energy growth (the blue solid line) is faster than in the case of the single Stokes SRS generation (the blue dotted line). It can be explained by that the 2$^\textrm {nd}$ Stokes generation leads to back-conversion of a part of energy into the excitation and probe pulses giving more efficient anti-Stokes conversion. This process started at the threshold of the 2$^\textrm {nd}$ Stokes generation ($F_\textrm {exc}^\textrm {in}\mathrm {\sim }$ 1 J/cm$^{2}$ in Fig. 9) but it is stopped at $F_\textrm {exc}^\textrm {in}\mathrm {\sim }$ 1.5 J/cm$^{2}$ when the 2$^\textrm {nd}$ Stokes fluence has become comparable with the 1$^\textrm {st}$ Stokes fluence that resulted in full modulation of the SRS components with a period equal to a round-trip time of the Raman laser cavity (see Fig. 8(b) where this modulation is full in contrast to Fig. 8(a)). So, this modulation induced by the efficient 2$^\textrm {nd}$ Stokes generation is the reason for saturation of the anti-Stokes energy growth (the blue solid line in Fig. 9 at $F_\textrm {exc}^\textrm {in}\mathrm {>}$ 1.5 J/cm$^{2}$).
6. Conclusions
In conclusion, we have theoretically and experimentally studied characteristics of collinear phase matching of Stokes $\mathrm {\leftrightarrow }$ anti-Stokes interaction for Raman-active crystals with different birefringence. It was shown that collinear phase matching of Stokes $\mathrm {\leftrightarrow }$ anti-Stokes interaction in low-birefringent crystals can be insensitive to angular mismatch if a phase matching angle is close to 90$\mathrm {^\circ }$. We have developed and experimentally realized the extracavity parametric Raman anti-Stokes laser based on a low-birefringent SrWO$_{4}$ crystal which was cut at the phase matching angle of 69$\mathrm {^\circ }$. Cyan 507-nm anti-Stokes conversion from green (532 nm) pump radiation of the 5-ns 1-mJ second harmonic Nd:YAG laser has been obtained. Application of the laser setup with a single beam excitation made it possible to use output face of the low-birefringent SrWO$_{4}$ crystal as the output coupler because of wide (6$\mathrm {{}^\circ }$) angular tolerance of collinear phase matching that resulted in an increase of slope efficiency of anti-Stokes generation higher than 3 % at the anti-Stokes energy output of a 10-µJ level. Saturation of anti-Stokes output energy growth in the collinearly phase matched parametric Raman laser was observed in contrast to linear growth of anti-Stokes output energy in the tangentially phase matched parametric Raman laser [9]. Mathematical modeling has shown that this issue is caused by competition between generations of the anti-Stokes and 2$^\textrm {nd}$ Stokes waves. It can be eliminated by further improvement of the laser system.
Funding
European Regional Development Fund/European Social Fund “Center of Advanced Applied Sciences” (CZ.02.1.01 / 0.0 / 0.0 / 16_019 / 0000778).
Disclosures
The authors declare no conflict of interest.
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