1. Introduction

Vortex laser beams may be described as having an annular beam profile and orbital angular momentum [1–3]. These unique characteristics have seen vortex laser beams applied in a range of cutting edge applications such as optical trapping and manipulation [3], fabrication of nano-needle structures [4, 5], quantum information [2], and ultra-resolution microscopy techniques such as stimulated emission depletion (STED) microscopy [6].

Vortex laser beams may be produced by converting a Gaussian beam to a Laguerre Gaussian (LG) beam using extra cavity methods such as computer-generated holograms [7], spiral phase plates [8] and spatial light modulators [9]. Each of these methods can however, be prone to misalignment and can suffer poor beam quality and/or losses through the conversion process [10, 11]. In recent years there has been considerable effort devoted to the direct generation of vortex laser beams in solid-state lasers; progress in this area has been recently reviewed by Senatsky et.al [12]. A variety of methods have been employed to force the laser to oscillate on an LG mode instead of a Gaussian mode, including pumping with an annular shaped beam [13, 14], using thermal lensing [15], using an intracavity spiral phase plate [16], or by using a defect spot on one of the resonator mirrors [10, 11]. Until now, the laser wavelengths for which vortex beams have been directly generated are fairly limited, and are based on He-Ne [11], Nd [10] and Er [17] gain media.

Methods of frequency conversion, such as optical parametric oscillation (OPO) and sum-frequency generation (SFG) are widely used for frequency extension of conventional lasers, yet frequency conversion of optical vortex lasers is not well advanced and is quite a complex topic. As described in [18], conservation of orbital angular momentum in the case of second harmonic generation (SHG) requires two photons, each with angular momentum of lћ, to yield a second harmonic photon with orbital angular momentum 2lћ, and such behaviour has been observed experimentally [18, 19]. Similar topological charge considerations have been shown to apply for four-wave mixing [20] and OPOs [21–23], and the generation of LG₀₁ modes by frequency conversion is not typical.

In this paper we report intracavity Raman frequency conversion of a continuous wave Nd:GdVO₄ laser oscillating on an LG₀₁ mode at 1063 nm, to deliver output at the Stokes wavelength of 1173 nm on the same LG₀₁ mode. The LG₀₁ mode for the fundamental field was preferentially excited by the use of a defect spot on the resonator output coupler, in common with the approach in [10, 11]. The LG₀₁ mode for the Stokes field result was favoured by a combination of an annular gain profile and the defect spot. We show using interferometry that the helical wavefront of the resonating fundamental field is preserved, and is also present for the resonating Stokes field. Hence we demonstrate for the first time (to the best of our knowledge) the application of SRS for intracavity wavelength conversion of a vortex laser beam, and that using SRS as a means of frequency conversion can enable the frequency-converted (Stokes) beam to have the same orbital angular momentum as the fundamental beam.

2. Experiment

The self-Raman laser cavity was formed using a 20 mm long 0.3 at. % doped Nd:GdVO_4, a-cut crystal which had a high-reflecting (HR) coating (M1) applied to its input facet (R> 99.99% from 1063 to 1173 nm) and an anti-reflection coating R<0.05% from 1063 to 1173 nm applied to its output surface, and a 250 mm radius of curvature output coupler (M2) which was HR coated (R = 99.91% at 1063 nm and R = 99.40% at 1173 nm). In this self-Raman configuration, the Nd:GdVO₄ crystal performs the dual functions of laser medium (to generate the fundamental field) and Raman medium (to generate the first-Stokes field). The cavity length was typically 22-32 mm. The output coupler was laser micro-machined so as to remove areas of the HR coating to produce regions of low reflectivity (defect spots) which would prevent oscillation of the lowest order Gaussian mode, and tend to force the laser to oscillate on LG modes. The use of similar defect spots to generate LG modes in He-Ne and Nd:YAG lasers has been reported elsewhere [10, 11]. In order to optimise the diameter of the defect spot for robust generation of the LG₀₁ modes, a 3 × 3 array of spots was produced (with inter-hole spacing of 400 µm), one row consisted of spots of 40 µm diameter, and the other rows consisted of spots which increased in diameter by 20 µm steps from 60 µm to 160 µm. The output coupler was mounted on a 2-axis translation stage to enable alignment of the system with different defect spots. The Nd:GdVO₄ crystal was end pumped by a 30 W, 879 nm fibre-coupled (0.22 NA, 100 µm core diameter) laser diode which was focussed to a diameter of 300 µm. The laser resonator layout is shown in Fig. 1.

 

Fig. 1 System layout.

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The behaviour of the vortex Raman laser was investigated in terms of the fundamental and Stokes field output powers, beam quality and the wavefront characteristics. The order and direction of the vortex beam was determined using the interferometer shown in Fig. 1. A forked interference pattern was produced by expanding the beam (using a × 5 beam expander arrangement with f = 25 mm and f = 125 mm converging lenses) in one arm of the interferometer and combining it with the beam in the other arm. Also, a spiral interference pattern was produced by creating a spherical wavefront in one of the arms of the interferometer [24] by removing the f = 125 mm lens, and then combining this with the beam in the other arm. The interference patterns generated by the interferometer were recorded using a CCD camera (Cohu 4800). The spatial profile of the laser beam was obtained by simply blocking the arm of the interferometer containing the pair of lenses.

3. Laser performance

Laser action was first investigated without a defect spot, in order to provide a baseline for performance. Fundamental field output at 1063 nm was produced for absorbed pump powers above 80 mW, in a TEM₀₀ mode. For absorbed pump powers above 1 W, Stokes field output at 1173 nm was generated, also in a TEM₀₀ mode.

Lasing of the fundamental field, in the absence of SRS, was then examined with the cavity aligned to lase centred on individual defect spots (through careful positioning of the output coupler). The cavity mode was only able to oscillate on one defect spot at a time as the diameter of the resonator mode was smaller than the spacing between defect spots. Generation of an annular output could only be achieved with the system aligned on a 40 µm diameter defect spot, and with a total cavity length of 32 mm (spacing of 12 mm between the end of the Nd:GdVO₄ crystal and the output coupler). Here a threshold absorbed pump power of only ~100 mW was required to oscillate the LG₀₁ fundamental (1063 nm), only slightly above the 80 mW threshold observed without the defect spot.

Next we investigated intracavity SRS of the fundamental field. As the absorbed pump power was increased, the fundamental field output power increased linearly until an absorbed diode pump power of ~2 W was reached, at which point oscillation of the Stokes field was observed. The Stokes field was also observed to have an annular output profile. This threshold was twice the SRS threshold observed without the spot defect, and this is not unexpected given that the resultant fundamental field power density for the SRS process is lower when the system is lasing on an LG₀₁ mode compared to TEM₀₀.

The power scaling characteristics of the fundamental and Stokes fields are shown in Fig. 2. The spectral content of the output was investigated using a fibre coupled spectrometer, and it consisted of only two wavelengths: the fundamental at 1063 nm and the first Stokes at 1173 nm. Both fields are linearly polarised along the c-axis of the Nd:GdVO₄ crystal and there are no contributions from other Raman-shifts or orthogonal polarisations within the system. Both the fundamental and Stokes fields maintained their annular profiles for absorbed pump powers up to ~7 W, beyond which the output became multi-lobed, no longer resembling an LG₀₁ mode. We believe that this is primarily due to significant thermal lensing manifesting within the Nd:GdVO₄ self-Raman crystal; strong thermal lensing is well known in self-Raman systems [25, 26]. As this thermal lens increases in strength with higher pump powers, it causes a contraction of the cavity mode at the output coupler and hence a change in the cavity mode to defect spot diameter ratio, preventing oscillation of the LG₀₁ mode. A maximum fundamental field output power of 400 mW was achieved for 6.8 W absorbed diode pump power. The maximum Stokes power generated from the system was 380 mW for 6.8 W absorbed diode pump power, representing a diode to Stokes generation efficiency of 5.6%. This generation efficiency is lower than what has been reported previously for continuous-wave intra-cavity Stokes lasers [26, 27]. Compared to the output powers from the laser without the spot defect, which generated 454 mW of fundamental and 420 mW of Stokes output for an absorbed diode pump power of 6.8 W, the output power at the fundamental and Stokes wavelengths were approximately 10% lower when the system lased with an LG₀₁ mode. Again, these results are not unexpected given that the fundamental field power density for the SRS process is lower when the system lases on an LG₀₁ mode.

 

Fig. 2 Power scaling characteristics of the fundamental and Stokes fields as a function of absorbed pump power.

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4. Beam characteristics

The spatial beam profiles and interferograms of the fundamental and Stokes fields are shown in Fig. 3 for the case when each field is operating near-threshold. Figures 3(a) and 3(d) show the annular profiles produced at both the fundamental and Stokes fields. The interferograms 3(b) and 3(e) show a single forklet indicating the presence of a first order (LG₀₁) phase singularity in the centre of the beams [24], and this is supported by the spiral interferograms of Figs. 3(c) and 3(f) which exhibit a single spiral pattern propagating clockwise in both cases. Note that the contrast of the fork and spiral interferograms in the case of the Stokes field is not as good as that obtained for the fundamental field, and this is due to the reflectivity of the mirrors used in the Mach-Zehnder interferometer being lower at the Stokes wavelength.

 

Fig. 3 Images (a), (b), (c) showing the spatial profile, fork-interferogram, and spiral interferogram respectively for the fundamental field; and (d), (e), (f) showing the spatial profile, fork, and spiral interferograms respectively for the Stokes field. Note that the images show some distortion and diffraction rings, caused by debris on the optics used to capture each image and slight clipping of the beam within the interferometer.

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Once lasing, the laser would operate with a randomly chosen spiral direction, with both the fundamental and Stokes spiral directions always being the same as one another. The fundamental and Stokes field output powers were independent of the spiral direction. This direction did not change with pump power, and could only be changed (albeit randomly) by restarting the laser or making major changes to the alignment or resonator length. As was reported in [10], the rotation direction of the spiral (indicating the sign of the first order vortex), could not be changed in a controlled way; for example there was no apparent regularity with which the spiral direction would change as the resonator length was increased. Figure 4 shows clockwise and anti-clockwise spiral patterns that were generated by the laser cavity at the fundamental wavelength, as the cavity length was changed.

 

Fig. 4 Images showing (a) clockwise; and (b) anti-clockwise spiral waveform generated at the fundamental wavelength when the cavity length was changed.

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Measurements of the beam quality factor (M2) were undertaken to provide an indicator of the quality of the generated vortex beam, as described in reference [10]. We note an alternative (and possibly more quantitative) method could be to evaluate the fidelity (quality of the amplitude and phase) of the generated vortex using an element such as a spatial light modulator [28], however the process is complex and difficult to implement due to spatial uncertainty in the location of the singularity (dark spot) and was not attempted in this work. The beam quality of the laser output was measured for incident diode pump powers just above the threshold for oscillation of the fundamental (~100 mW incident diode pump power) and the Stokes (~2 W incident diode pump power) fields. In the case of the fundamental field, a beam quality factor (M²) of 2.18 was measured. When pumping just above 2 W, the beam quality of the fundamental field increased to M² ~3.12, while the beam quality of the Stokes field was M² ~2.21. As the absorbed diode pump power was further increased to 6.8 W, the beam quality of the Stokes field changed only slightly, increasing to M² ~2.51. The beam quality factors of the fundamental and Stokes fields being M²~2 is consistent with a first order LG mode being generated within the cavity [10]. It should be noted that while the M² of both the fundamental and Stokes fields increased with incident diode pump power, when examining the interferogram produced by the beam across the whole power scaling range, a single fork dislocation was observed for all cases, which indicates that the vortex mode remained single-order [24]. The fact that the beam quality factor of the fundamental mode increased with pump power suggests that a mixed mode (comprising a first-order LG mode with some higher-order modes) is being produced in this case; it is extremely difficult to de-convolve each of these modes and examine each in isolation.

5. Discussion

It was found that an annular output beam profile (LG mode) could only be achieved when the resonator was aligned to lase on the smallest defect spots, ie those with a diameter of 40 µm, and only for absorbed diode pump powers below ~7 W. At higher absorbed pump powers and/or larger defect spot sizes, the output beam profile was multi-lobed and resembled higher order Hermite Gaussian (HG) modes. We simulated the change in resonator mode diameter using ABCD resonator modelling [29] and found that for pump powers up to 7 W the resonator mode diameter at the output mirror varied from 280 µm to 240 µm, for estimated thermal lens focal lengths between 500 and 140 mm in the Nd:GdVO₄ crystal. For 7 W pump power, the ratio of the defect spot diameter to the diameter of the resonator mode at the output coupler is therefore ~0.17 for an intracavity mode diameter of 240 µm. This value is slightly lower than the 0.21 observed in [10] which used an Nd:YAG laser crystal. For higher pump powers, the stronger thermal lens induced within the Nd:GdVO₄ crystal causes significant contraction of the resonator mode at the output coupler and the defect spot is no longer effective at forcing oscillation of an LG mode. Here we note the thermal lensing in a self-Raman laser is substantially stronger than for the corresponding fundamental laser [25, 26], this being due at least in part to the inelastic nature of SRS which contributes additional thermal load.

In the vortex Raman laser, there are at least two contributing mechanisms for generating the Stokes field in a LG₀₁ mode. The first is the spatially-varying Raman gain profile, dictated by the annular profile of the fundamental field. The second is the defect spot on the output coupler, which acts as a spatially-dependent, on-axis loss for the Stokes field.

It is interesting to consider the manner in which orbital momentum is conserved during each SRS event, and this is depicted in Fig. 5. Using a similar approach to that outlined in [18], in which photons are considered to have orbital angular momentum of lћ, the orbital angular momentum states for the fundamental, Stokes and phonon before the scattering event are denoted l_F, l_S and l_R respectively, and those after the event are denoted l_F^’ l_S^’ and l_R^’ . The value of l is frequently called the topological charge. By the nature of the stimulated process, the scattered Stokes photon must have the same l number as the incident Stokes photon, ie. l_S^’ = l_S. Conservation of orbital angular momentum requires:

= l'_R)

 

Fig. 5 Concept diagram showing orbital angular momentum states of participating photons and vibration states before and after a SRS event.

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The fact that SRS enables the frequency-converted (Stokes) light field to have the same topological charge as the fundamental field is highly significant, as is the fact that the frequency converted beam has an LG₀₁ mode. To the best of our knowledge there has been only one previous report [23] of a frequency-converted beam having a topological charge of l = ± 1, that being an OPO in which the topological charge of l = 1 was transferred from the pump to the signal within an OPO, and the idler had a topological charge of zero. More usually [21, 22] an OPO excited by a LG₀₁ mode would generate a signal and idler each with l = 1/2. Similarly, for the nonlinear processes of SHG and SFG the topological charge of the output beam is also different to that of the fundamental/excitation beam. Accordingly Raman lasers offer an exciting approach for substantially increasing the wavelength versatility of directly-generated LG₀₁ vortex beams. Given the wide variety of laser gain materials and Raman-active media, coupled with intracavity and extracavity Raman laser configurations, and the ability to operate in CW and pulsed regimes, there is excellent potential for obtaining vortex laser beams across a wide range of wavelengths.

4. Conclusion

We have demonstrated a continuous-wave Nd:GdVO₄ self-Raman laser operating with first-order vortex output at both the fundamental and Stokes wavelengths. 400 mW and 380 mW of output power were achieved at the fundamental and Stokes wavelengths respectively for 6.8 W incident diode pump power. The LG₀₁ mode for the fundamental field was excited by using a defect spot on the output coupler, and this resulted in an annular distribution of Raman gain in the Nd:GdVO₄ crystal. This, in combination with the defect spot on the output coupler, resulted in the Stokes field also occurring in a LG₀₁ mode. We considered the conservation of orbital angular momentum in SRS, and found it to be consistent with our observation of LG₀₁ modes for the fundamental and Stokes fields. This result is significant as it demonstrates the application of intracavity SRS to generate high power, high beam-quality vortex beam emission operating at a wide range of wavelengths from a compact all solid-state laser resonator.

Acknowledgments

This work was in-part performed (supported by) at the OptoFab node of the Australian National Fabrication Facility.

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