1. Introduction

Optical vortex modes have an annular intensity profile and a helical phase structure, and have been shown to carry orbital angular momentum (OAM) [1]. The helical phase, and thereby the OAM, can act either clockwise or anticlockwise, which is referred to as the handedness of the vortex. Examples of stable vortex modes are the Laguerre Gaussian modes LG_pl (with l ≠ 0) where the sign of l indicates the handedness. The unique spatial and phase structural properties of optical vortex beams mean they have an array of applications such as in optical tweezers [2], increasing information density in free-space communication [3], and nanoneedle manufacture [4].

There are a range of specialist optics available to convert the Gaussian output of a laser into a vortex. Some commonly used examples include spiral phase plates (SPPs) [5], q-plates [6], cylindrical lenses [7], spatial light modulators (SLMs) [8] and Fresnel cone [9]. The commonly used spatial light modulator can provide highly configurable mode control, but suffers from a low damage threshold, often low conversion efficiency, and polarisation dependency [10]. Most commonly used for higher power operation are SPPs due to their high damage thresholds [11–13]. However, they have inherent restrictions, since they do not allow real-time handedness switching and only operate at a specified wavelength.

Direct output of vortex modes from a laser cavity, a so-called vortex laser, can have many benefits. Vortex lasers offer added simplicity for users as they have direct access to a vortex beam without the complication of additional optical systems, which may require precise alignment and with power losses due to the imperfect conversion efficiencies. Vortex lasers are an attractive prospect for efficient generation of high-quality beams in compact, environmentally robust, and turnkey systems. This would enhance the capability to translate vortex applications out of research laboratories and into commercial settings.

Vortex lasers have been demonstrated using a variety of techniques. Using specialised optics inside the laser cavity is one approach, for example by forcing vortex mode oscillation with a spot defect mirror [14] or internal mode conversion with a q-plate [15]. Alternatively using a pump with an annular intensity profile or off-cavity-axis pumping can favour the vortex as the lowest threshold cavity mode [16]. Similarly, by using coupled laser cavities sharing a gain medium, one cavity can deplete the centre of the gain region to favour the other to operate in a high order vortex mode [17]. Thermally induced lensing with spherical aberration in the gain medium has been used to favour vortex modes [18,19]. In most cases determining the handedness of the output vortex is unreliable without using additional controlling optics, such as the nanowires used by Lin et al. [16], making switching the handedness during operation difficult. Pulse vortex lasers using Q-switched cavities [20–23] and mode-locking [24] have been demonstrated but with limited performance as the high intracavity intensity and further cavity elements for pulsed operation adds to the design complexity.

In this work we demonstrate operation of a multi-Watt Q-switched vortex laser using an interferometric vortex output coupler (VOC). The VOC laser interferometrically extracts a vortex output from a laser operating on a fundamental Gaussian internal spatial mode. In our configuration, shown in Fig. 1, the VOC is a Sagnac type common-path interferometer, acting as the end mirror, comprised a 50% beamsplitter (BS) and three high reflectivity turning mirrors (M1, M2 and M3). The input Gaussian laser mode is split and then via the three mirrors recombines at the BS; however, in contrast to an aligned Sagnac interferometer, precise imbalancing adjustments are made to introduce relative shear and angular deviations between the recombining beams. As a result of the imbalance, not all of the power recycles back to the laser and some is extracted in a vortex mode in the transmission direction [25]. The first practical implementation of an interferometric VOC laser was demonstrated in our previous work [26].

 

Fig. 1. The concept of the VOC with a fundamental Gaussian input, which is partially transmitted as a vortex mode, with the remainder reflected unchanged.

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In this work, the VOC laser is operated in a Q-switched cavity configuration at a higher average power and efficiency. We further show that the VOC is capable of providing a self-mode-filtering to enforce intracavity fundamental Gaussian mode operation even when the equivalent conventional laser operates multimode. The laser is demonstrated to produce high quality LG_0±1-type output beams, which were verified through analysis of both the intensity and phase profile of the beam compared to the theoretical LG_0±1 form. We show that the VOC enables handedness switching of the output, during operation, without loss of power or quality. We also experimentally validate that the interferometric function of the VOC leads to suppression of higher order modes with a self-aperturing effect favouring TEM₀₀ mode operation inside the cavity with only a small reduction in output power.

The VOC uses standard, high-power handling optics, which makes it suitable for high pulse energy Q-switching. Additionally, its spatial filtering qualities will allow it to correct for intracavity beam degradation at high powers. This makes it an attractive approach for generating high energy and high-power vortex sources for applications such as high speed chiral nanoneedle production [4].

2. Experimental standard laser and vortex laser systems

The two laser cavities that were operated in this study are shown in Fig. 2: a standard linear laser cavity with plane mirror output coupler (OC), Fig. 2(a), and a variant of this cavity with a vortex output coupler (VOC), Fig. 2(b).

 

Fig. 2. The laser cavity and VOC operation. (a) Standard linear cavity, composed of: high reflectance (HR) back mirror (BM), Nd:YVO₄ crystal, acousto-optic modulator (AOM), intracavity lens and output coupler (OC). (b) Converted linear cavity with the VOC. The lens to OC and lens to M2 distances are matched in both cases. The inset intensity profiles show the beam at different points. (c) Schematic of a top view of the VOC showing how an angular misalignment is imparted between the two counter-propagating beams. (d) Side view of the AR plate, which can be tilted to introduce a spatial separation between the two counter-propagating beams.

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In the linear cavity, used as a starting point for the vortex laser, the gain medium was a 3 × 3x5 mm 0.5 at. % doped Nd:YVO₄ crystal, operating on the c-axis at 1064 nm. The cavity was formed between the back mirror (BM) and a 20% transmission plane output coupler (OC). An intracavity lens (f = 70 mm) provided mode size and stability control. An acousto-optic modulator (AOM) was used to Q-switch the cavity for pulsed operation. The gain medium was 1 mm from BM, the BM to lens distance was 225 mm, and the lens to OC distance was 80 mm. The Nd:YVO₄ crystal was end-pumped by a fibre-delivered laser diode module, with a nominal wavelength of 808 nm, through BM, which transmitted 808 nm and reflected 1064 nm. The pump beam was focused into the crystal to a minimum waist radius ${w_p}$ = 300 µm and with a super-Gaussian intensity profile. The pump beam propagation parameter was M² = 35. A maximum pump power of 12.8 W was used, or 12.2 W absorbed power accounting for 95% absorption in the crystal. The maximum pump power was set as a safety limit as there was crystal fracture observed with our Nd:YVO₄ crystal in its crystal holder when operating at much higher pump powers due to thermal stress in the gain medium but this is not expected to be a fundamental limit.

The linear cavity was converted into a vortex source by removing the OC and replacing it with a VOC, as shown in Fig. 2(b). The VOC cavity was made geometrically identical by placing the mid-mirror M2 of the Sagnac interferometer at 80 mm from the lens, identically matched to the lens to OC distance in the initial linear cavity. This meant that the mode size and cavity stability conditions were unchanged, so the linear cavity optimisations also applied to the VOC laser.

To configure the VOC for vortex output the Sagnac ring counter propagating beams must have a specific relative angular alignment of $\theta $ and a spatial displacement d between them in orthogonal axes, with the additional π phase difference needed ensured by the BS reflection properties. When these conditions are met it results in an LG₀₁ type output [27]. The controls used to achieve this configuration are shown in Fig. 2(c) and 2(d), where in the horizontal plane a relative angle ($\theta $) is introduced with rotation of M2 by $\theta /2$, and vertical displacement (d) is from vertical rotation of an anti-reflection (AR) coated plate with thickness t of 3 mm. Given a small rotation angle $\psi $ of the AR plate, each counter propagating beam is displaced by equal and opposite amount ${\pm} d \approx t\psi ({1 - 1/n} )$. The further condition for LG₀₁ type generation is for the relative magnitudes of $\theta $ and d to provide the canonical condition [26] $\frac{d}{{{w_0}}} ={\pm} \frac{{\pi {w_0}\theta }}{\lambda }$, where ${w_0}$ is the beam waist radius on M2, $\lambda $ is the wavelength, and the sign determines the vortex handedness. The magnitudes of normalised displacement $\frac{d}{{{w_0}}}$ (and correspondingly angle $\frac{{\pi {w_0}\theta }}{\lambda }$) should be small for ideal LG₀₁ output; in practice it has been shown that ratio values up to 0.5 will accurately reproduce the LG₀₁ mode [26]. The transmission T of the VOC is given by [26],

To operate the VOC as part of the laser cavity, so that it outputs a vortex, the canonical condition and ratio magnitudes must be maintained. In a laser, the varying thermal lens in the crystal can change the intracavity mode size at M2. Therefore, to maintain the canonical condition and transmission of the VOC output, small adjustments to $\theta $ and d were necessary when changing the pump power. In this investigation, the canonical condition was obtained by monitoring the output mode symmetry, where maximum azimuthal symmetry occurred when the canonical condition was satisfied. The transmission of the VOC was determined experimentally by measuring the intracavity power from the small known transmission fraction of M3.

3. Q-switched linear and VOC laser results

The standard linear cavity, see Fig. 2(a), was operated with a plane mirror output coupler with 20% transmission. At 150 kHz Q-switching rate, the average output power against absorbed pump power is shown in Fig. 3 (black circles). At 12.2 W of absorbed pump power the average output power was 5.4 W in a Gaussian mode, M² = 1.20, and it had a slope efficiency of 49%.

 

Fig. 3. Average output power at 150 kHz repetition rate and 20% output coupling against absorbed pump power. The VOC cavity results are in red and the linear cavity results are in black.

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To directly compare the performance of the VOC laser, the VOC was also operated with 20% transmission. The vortex laser results for 150 kHz Q-switching rate are also shown in Fig. 3 (red squares). The average vortex output power at 12.2 W pumping was 5.1 W, and it had a slope efficiency of 46%.

The VOC cavity and the linear cavities had similar lasing power, threshold and slope efficiency, showing that the implementation of the VOC negligibly impacted the cavity efficiency. This result shows the potential for the VOC as a general methodology for converting a standard TEM₀₀ laser cavity into a vortex laser. The VOC cavity efficiency was limited to that of the linear cavity, which is expected as the VOC cavity geometry was equivalent to the linear cavity and therefore had similar mode to pump size conditions. The equivalence of the plane output coupler and VOC, and the required fundamental intracavity mode, means that the VOC cavity can be designed and modelled using standard ABCD matrix analysis.

To operate the VOC the amount of shear and angular displacement introduced in the recombining beams must be matched to the beam waist size and the required transmission, as determined by Eq. (1). When operating with a measured transmission of 20% at 12.2 W of absorbed pump power the beam waist radius at M2 was measured to be 100 µm, and the AR plate had been rotated by $\psi = 2.0^\circ $ giving $d = 36.0\;$µ\textrm{m}. The normalised displacement was therefore $d/{w_0} = 0.36$. Using Eq. (1) this gives a theoretical transmission of $T = 20\%$, assuming the canonical condition is exactly satisfied, which precisely matches the measured transmission. From the values of d and ${w_0}$, it can be inferred that the relative horizontal beam angle was θ = 1.2 mrads, although this was not measured directly.

To further investigate the pulsed vortex laser the Q-switching repetition rate was varied from 25 to 175 kHz. The average vortex power against Q-switching rate is shown in Fig. 4(a). The average power of the vortex asymptotically increased with increasing repetition rate, from 4.5 W at 25 kHz to 5.1 W at 175 kHz.

 

Fig. 4. (a) Average vortex power of the VOC cavity against repetition rate of Q-switching at 12.2 W of absorbed pump power. (b) Pulse energy (black) and pulse full-width half-maximum (FWHM) duration (red) of the VOC cavity against pulse repetition rate at 12.2 W of absorbed pump power.

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The pulse energy and pulse duration against Q-switching rate is shown in Fig. 4(b), in black and red respectively, at 12.2 W of absorbed pump power. At 25 kHz, the pulse energy was 180 µJ with a pulse duration of 26.6 ns, and at 150 kHz the vortex pulses had an energy of 34 µJ and a duration of 108 ns. The spatial profile of the vortex output at 12.2 W of absorbed pump power and 150 kHz is shown in Fig. 5(a). The vortex had a beam propagation factor M² = 2.26. The intracavity mode was analysed by accessing the leakage from mirror M3 in the VOC, which revealed the intracavity mode had an M² = 1.12, Fig. 5(b). These values are close to the theoretical M² for the LG₀₁ mode (M² = 2) and a fundamental Gaussian (M² = 1). For comparison, with the same pumping and Q-switching conditions the standard linear cavity had an M² = 1.20.

 

Fig. 5. Spatial intensity profiles of the a) the vortex output and b) the intracavity mode of the VOC cavity. The M² of each beam is inset.

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To further verify the vortex mode quality and confirm the pure handedness the phase structure was measured using the method laid out by Takeda et al. [29]. This uses the interference of the vortex beam with a tilted planar reference beam to retrieve the field phase information, which was performed with a Mach-Zehnder interferometer. The retrieved phase images of the vortex outputs in the two handedness states is shown in the insets of Fig. 6, with (a) and (b) showing the two handedness results. The graphs show the corresponding circular cross section of the phase profile that was centred on the beam axis and at a fixed radius, matching the peak intensity of the annular ring. The retrieved phase images show the characteristic phase singularity in the centre, with an azimuthal phase variation totalling 2π around the beam centre. The circular cross sections verify that these match well to the theoretical linear phase change of the LG₀₁ mode, overlaid in red. The combination of close intensity and phase characteristics with the LG₀₁ mode verify the high purity of vortex mode generated.

 

Fig. 6. (left) Measured phase profile of a circular cross section around the phase singularity of the vortex with right handedness and (right) measured phase profile of the vortex with left handedness. The retrieved phase images for each handedness is inset. The measured phase data is shown in black, with the theoretical phase profile in red.

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The handedness of the vortex output was switched during operation by inverting the spatial displacement d through reversing the tilted AR plate angle $\psi $. By comparison of Figs. 6(a) and 6(b), it is clear that this reverses the handedness of the vortex, without any loss in quality. Additionally, there was no observed power loss incurred by switching. The beam propagation factor of the vortex after the switch was maintained with M² = 2.31 (compared to M² = 2.26), and the intracavity mode had M² = 1.11 (compared to M²= 1.12), values within the measurement accuracy.

4. Higher order mode suppression by VOC

It was observed experimentally that the beam propagation factor of the intracavity beam of the VOC laser was lower than that of the linear cavity. The theory behind the VOC is based on a perfect fundamental Gaussian input, despite this the VOC still produced high quality vortices from cavity geometries not optimised for fundamental Gaussian oscillation. This suggested that the VOC improved the intracavity beam quality, allowing it to function as intended.

To better understand the intracavity mode conditions needed for the VOC laser, the fundamental mode size at the gain medium was systematically reduced to mismatch the mode and pump diameters. This was achieved experimentally by moving the intracavity lens towards the BM. The effect of the lens movement on the power and beam propagation factor of both the standard linear cavity and VOC laser cavity were investigated.

The power and beam propagation factor results for the standard linear laser are shown in Fig. 7(a), including inset beam profiles at certain lens positions. The beam propagation factor increased with lens position from M² = 1.38 to M² = 7.32. This was expected because the fundamental mode size was becoming increasingly smaller than the pumped region allowing oscillation of higher-order modes, whilst the overall intracavity mode size at the pumped region was measured to remain constant. The transition from a Gaussian to an increasingly top-hat beam profile resulted in better matching to the top-hat pump profile, which is reflected in the output power increasing from 5 W to 5.4 W from 0 mm to 17 mm lens position. With further increasing lens position the output power decreased, which may have been the result of a higher proportion of the intracavity mode overlapping with the aberrated wings of the thermal lens.

The equivalent effects of intracavity lens movement in the VOC cavity are shown in Fig. 7(b), including inset beam profiles at certain points. In this figure the beam propagation factor is shown for both the output vortex and the intracavity beam. The results show two clear regions of operation with a transition point at a lens position of 25 mm, highlighted by a vertical dashed line in Fig. 7. The initial trend, below a lens displacement of up to 25 mm, was that the intracavity beam stayed approximately fundamental mode HG₀₀ and the vortex maintained its ideal value near M² ≈ 2. This high spatial purity was at the expense of some output power that reduced from 5 W to 3.8 W over the range. The measured intracavity fundamental mode radius decreased from 270 µm to 155 µm in this range, resulting in poorer gain region extraction (pump mode size 300 µm) and so a lower efficiency.

After 25 mm lens displacement, the VOC cavity could no longer maintain the fundamental Gaussian mode seen by its increasing M². The intracavity beam became increasingly multimode to fill the gain region, which resulted in a reduced rate of power loss. However, this interfered with the VOC operation, and the output vortex was also no longer LG₀₁ shaped and the output beam M² increased.

The lens position at the transition point of 25 mm, with intracavity M² = 1.1, corresponded to the equivalent standard linear cavity having an M² = 3.60. This shows that the addition of the VOC to the cavity caused it to maintain a high-quality fundamental Gaussian mode despite poor pump to mode size matching. In fact, it has been previously predicted that an unbalanced Sagnac interferometer has a higher loss for higher order modes [30]. Our results show a similar behaviour, so we attribute our results to the VOC having a higher transmission for higher order modes. This means the VOC will supresses these modes in the cavity due to their higher round-trip higher loss. In this way, the VOC spatially optimises the cavity with a self-mode-filtering due to the interferometric behaviour of the VOC. This provide a benefit that it does not necessarily require a fully optimised starting cavity and, additionally, this means that it could also reduce minimise spatial degradation of the intra-cavity mode during power scaling efforts.

 

Fig. 7. (top) The beam propagation factor (M²) and average power output of the linear cavity against detuning (lens position). (bottom) M² of the vortex and intracavity beam of the VOC and the average power of the vortex against detuning of the cavity. Dashed line highlights the point at which the VOC is no longer able to suppress the higher order modes in the cavity. Beam profiles for the VOC and linear cavities are inset, corresponding to the lens position at 7 mm, 25 mm and 33 mm.

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5. Conclusion

In summary, we have demonstrated a vortex output coupler that interferometrically converts the fundamental LG₀₀ intra-cavity mode into an LG₀₁ output, in a pulsed Q-switched cavity. An average power of 5.1 W was achieved at 150 kHz pulse frequency, with a slope efficiency of 46%. The vortex also had an average beam quality close to the theoretical ideal of M² = 2 and phase retrieval revealed that the phase structure of the vortex closely matched the phase structure of an LG₀₁. The handedness was fully determined and configurable during operation with a single control (tilt-plate) in the VOC. The design utilised the common-path Sagnac interferometer making it robust against environmental perturbations, which makes it ideal for commercial implementation. The simplicity of the design, employing only a beam-splitter and mirrors, means that the VOC can be used with any gain medium over any spectral range where these elements are available. Additionally, this makes it suitable for use in wavelength tuneable lasers. It also offers variable output coupling for cavity optimisation. The VOC is shown to suppress higher order modes in the cavity with a self-mode-filtering action, which means it can operate as part of a non-ideal cavity. The VOC components used are non-absorbing, non-diffracting and use dielectric coated interfaces, making it suitable for high power implementation. The VOC can replace the output coupler of any fundamental Gaussian mode cavity, and we have also shown that it effectively operates in high quality mode even in a standard cavity operating in multimode M² = 3.6. This makes it a highly effective solution for high power, wavelength tunable vortex lasers. Therefore, the VOC could be used to convert any high-power solid-state laser operating in fundamental mode or close to fundamental mode into a vortex laser, for example in thin-disk technologies [31,32]. Furthermore, as an alternative use, the VOC could be implemented solely for its property of acting as a self-filtering-mode selector in a laser to improve the fundamental Gaussian output.