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Abstract
In response to growing demand from optical vortex (OV) beam applications, numerous generation techniques have been developed competing in power scalability, purity, and wavelength flexibility. Direct vortex emission from lasers typically grants access to efficient, high power, and pure mode generation. In this work we demonstrate a compact, unidirectional Nd:YVO4 ring laser with an intracavity wedge-plate shearing interferometer (WPSI) as an output coupler, which converted the internal Gaussian mode to LG01 (Laguerre-Gaussian OV) output. It directly generated a watt-level LG01 OV with high mode purity (98%) in a single longitudinal mode. The monolithic WPSI has advantages in stability and simplicity compared to other designs. The system is compact and cheap, using off-the-shelf components, and can be readily adapted to any gain media, widening the scope for OV generation at wavelengths currently unobtainable using competing methods.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. Introduction
An optical field exhibits vorticity when the wave-fronts possess a cork-screw like helical phase structure surrounding an intensity null referred to as the phase singularity. The azimuthal rotation of the light field allows the optical vortex (OV) to possess a well defined orbital angular momentum (OAM) [1]. The discovery of OAM in OVs has since garnered considerable interest. In recent decades a plethora of applications have emerged; OVs have been used in cryptography and quantum information [2], measurement of single photon OAM [3] and free-space optical communications [4,5]. Observational astronomy has used OVs in phase coronagraphs [6,7], and OAM diagnostics can be used to detect rotating black holes [8]. High resolution imaging beyond the diffraction limit in has been achieved in STED microscopy [9–12]. Material processing has made use of OVs in producing chiral needles [13,14], and OVs have been used in optical manipulation in the form of optical tweezers, vortex traps and imparting angular momentum onto target particles [15–17].
The most commonly used form of OVs are the propagation invariant Laguerre-Gaussian (LG) modes, particularly those with radial index $p=0$ and azimuthal index $l=1$. LG modes can be generated by converting an input fundamental Gaussian beam from a laser source using a mode conversion device, examples include spiral phase plates [18], computer generated holograms [19], diffractive optics [20], q-plates [21,22], spatial light modulators [23]. These techniques have advantages in OV handedness selection and simplicity but are commonly limited by wavelength tunability, sensitivity to misalignments and are usually limited in their power scaling capabilities due to low damage thresholds.
Alternatively, OV beams can be directly emitted from laser cavities. Intracavity methods can offer improved power scaling, higher mode purities and efficiency. They can also exploit the self-adaption of a laser cavity to low-loss regimes to reduce alignment sensitivity. Examples include intracavity q-plates [24] and spatial light modulators [25], spot defect mirrors [26], coupled-cavities [27], annular pumping of the gain medium [28] and off-axis pumping of the gain medium [29–31]. Similar techniques can be employed in optical parameteric oscillators [32,33]. Intracavity methods may suffer from limitations in the use of high loss components, require custom pumping geometries, difficulties in consistent handedness selection, and distortions due to thermal lensing effects.
There is a demand for high purity, high power, narrow frequency vortex sources. Interferometric sensing applications exist exploiting the helicity of vortex beams, where enhanced coherence length sources provide advantages for remote probing, for example rotational doppler interferometery [34,35] or surface polarimetry [36]. Free space optical communications would benefit from narrow emission spectra [5]. Narrow bandwidth vortex sources can be generated external to laser cavities with the usual methods, however they retain the same drawbacks previously discussed. There have been some demonstrations of direct single frequency vortex generation. An annular pumped gain medium has been demonstrated in linear [37] and ring [38] cavity configurations. A non-planar ring oscillator has been demonstrated [31] using off-axis pumping. These methods were successful in their implementations, but are not necessarily adaptable to other gain media due to bespoke pumping arrangements or crystal geometries. In this work, we present a cavity design that uses standard techniques and cavity geometries to construct a single longitudinal mode source, but is converted to a high quality vortex source by the simple inclusion of the mode-transforming WPSI output coupler. In addition to being a high quality vortex source in itself, it is also highly adaptable to other gain media and cavity designs to meet application requirements.
In this work we use an interferometric technique to convert the typical fundamental Gaussian internal cavity mode into a OV output, a device termed a vortex output coupler (VOC). These devices are highly compatible with standard laser cavity designs. Interferometric methods use standard low-loss, high damage threshold mirrors so are an appealing alternative to the use of lossy specialist devices. Previous VOCs have been implemented with a Sagnac interferometer [39–41]. We recently demonstrated the non-standard use of a WPSI for interferometric OV generation [42]. In this paper we present, for the first time, the WPSI implemented as a VOC in a unidirectional Nd:YVO$_4$ ring laser. We obtain watt-level, high-purity (98$\%$) LG$_{01}$ output in a single longitudinal mode. The WPSI presents advantages as a VOC over multiple element interferometers in its simplicity of operation and alignment, reduced insertion losses through fewer optical interfaces, and stability as a monolithic element.
2. WPSI configuration
In this section we summarise the mode conversion technique and configuration of the WPSI, and introduce a new method to correct for beam waist location. A more detailed examination of the WPSI can be found in our previous paper [42]. The WPSI is a wedged optic that has a nominal thickness $t_0$ and is wedged along the vertical direction with a small wedge angle $\alpha$ as shown in Fig. 1. An incident TEM$_{00}$ fundamental Gaussian mode is reflected off the front and rear surfaces, these interferometrically combine to form a propagation stable LG$_{01}$ mode when setting the correct incidence angle.
Fig. 1. Incident TEM$_{00}$ is converted into an LG$_{01}$ by introducing a shear displacement and angular misalignment in the reflected beam. (a) Top view: the tilted faces of the WPSI are parallel separated by $t_0$. An incident beam at $\theta _i$ is reflected with a shear separation of 2d$_x$. (b) Side view: Wedge angle $\alpha$ reflects the incident beam in opposing directions resulting in a total angular separation of 2$\theta _i$.
The two reflected beams are each deviated by a half angle $\theta _y=n\alpha$ in the vertical plane by the non-parallel faces, where $n$ is the refractive index of the WPSI medium. The faces are parallel in the horizontal plane and a shear separation between the reflected beams, $d_x=t_0 \theta _i / n$, is introduced when the incident beam is at an angle $\theta _i$ to the surface normal. The shear and angular deviations are set according to the canonical condition
where $w_0$ is the waist radius of the incident mode, $\theta _0$ is its divergence, and $\epsilon$ is used as a normalised measure of the beam offsets [43,44]. For a given a WPSI and input beam the required incidence angle ($\theta _i$) in the horizontal plane isThe phase difference between the front and rear reflected beams must result in destructive interference. The WPSI can be translated ($\Delta y$) in the vertical direction (y-axis) to adjust the optical path through the WPSI, allowing for fine tuning of the phase. The total phase difference ($\Phi$) is
where the additional $\pi$ term is due to the internal reflection from the rear surface and $k=2\pi /\lambda$ is the wavenumber of the incident beam with wavelength $\lambda$. The $\alpha \Delta y$ term is due to thickness changes from the wedge angle. Typically $t_0 \gg \Delta y$, so this term can be neglected in other WPSI thickness calculations but must be included for the phase. This allows independent control of the interference condition through $\Phi$ without affecting the other vortex generation configurations.The prior shear and angular offsets are required at the beam waist, so if the WPSI is positioned at the beam waist then the vortex generation conditions are satisfied. However, if the WPSI is positioned a distance $z$ from the beam waist an inherent shear in the vertical plane, $d_y$, is introduced at the beam waist
which is due to the vertical wedge angle and determined by projecting the reflected beams to the beam waist location. The addition of $d_y$ at the beam waist can result in imperfect vortex generation, so should be eliminated. This is simple to achieve through introducing a vertical angle of incidence $\theta _{i,y}$ to counteract the shear which returns the system to ideal vortex generation alignment. The vertical correction angle is typically much smaller than the horizontal angle of incidence.The power reflectance of the WPSI in the LG$_{01}$ mode is $R_{01} = 2 R (2 - R)^2 \epsilon ^2$, where $R$ is the surface power reflectance [42]. For an uncoated WPSI using Fresnel reflection $R_{01}$ is typically of the order of a few percent. This allows the WPSI to act as an output coupler, because the transmitted power remains in the fundamental Gaussian beam passing unchanged through the WPSI.
3. Cavity design
The ring vortex laser design is shown in Fig. 2. The gain medium was a 3 mm thick 0.5 % doped Nd:YVO$_4$ crystal end pumped by a 808 nm fibre coupled diode laser. The ring cavity was 80 mm $\times$ 50 mm with four turning mirrors; a pump-through dichoric mirror (DM), two high-reflectivity mirrors (HR) and a partially reflective (PR) mirror with transmission, T=1.3 %. Mode size control was achieved with a $\mathrm {f} = {500}\;\textrm{mm}$ intracavity lens placed at an equal distance of 13 cm to the crystal on each side. The cavity internally oscillated on the fundamental Gaussian mode.
Fig. 2. Ring shaped vortex cavity with a pump-through dichoric mirror (DM), gain medium (GM) in copper housing, WPSI, partially-reflective mirror (PR), retro-reflector (RR), high-reflectivity mirrors 1&2 (HR1&2) and a 50 cm intracavity lens (f). The internal mode is Gaussian, with LG$_{01}$ vortex mode output from the WPSI.
The ring resonator forms two travelling wave modes due to the clockwise (CW) and anti-clockwise (ACW) emission from the gain medium. Typically, the CW and ACW modes are optically uncoupled and compete for gain. The counter propagating modes have similar thresholds, so due to transient perturbations and gain clamping the oscillator will rapidly switch from CW to ACW emission. This also effectively halves the emission power available in one emission direction. A retro-reflecting mirror returned the ACW mode through the PR mirror back in the CW direction through the gain medium, in this way CW operation was baised, making the cavity unidirectional [45]. The unidirectionality ratio of CW to ACW cavity power was 20:1, measured by comparing transmitted power through the HR-mirror.
The vortex output was generated with the WPSI, which was the commercially available Thorlabs SI100P shearing interferometer. It had a thickness of 2.6 mm, measured wedge angle of 0.22 mrad, and constructed of uncoated UVFS, with Fresnel reflection providing the reflectance properties. It was mounted with a vertical translation stage for phase control and tilt controls for the horizontal and vertical angle of incidence. The thickness and wedge angle of this WPSI were in the correct range for vortex beam conversion for the intracavity beam radius and required angle of incidence.
The beam radius near the WPSI was configured to be 0.35 mm. This was selected for combined optimal mode conversion properties for the WPSI and laser performance with the pump size in the crystal. The resulting mode conversion parameters for the WPSI were a horizontal angle of incidence of $\theta _{i}={3.74}^{\circ}$ and vertical correction angle $\theta _{i,y}={0.2}^{\circ}$. The canonical factor was $\epsilon =0.33$, which is sufficient for good quality vortex generation.
The alignment procedure of the WPSI is to first measure the cavity mode diameter, with or without the WPSI present. Using the equations in Sec. 2 the required angle of incidence is calculated and the WPSI aligned to this angle. Fine adjustment can then be performed to account for experimental measurement uncertainty in the alignment and to fine-tune the output quality.
4. Results
The LG$_{01}$ output power against absorbed pump power for the laser is shown in Fig. 3. The laser generated a 1.15 W vortex beam for 4.73 W of absorbed power. At maximum power the intracavity flux was 50 W, calculated using the known transmission of the PR-mirror of $T = {1.3}\%$. Despite the high flux we observed no measurable degradation in WPSI performance due to its high damage threshold, low loss and monolithic design. The reflectance, and therefore the output coupling fraction, of the WPSI was measured to be $R_{01} = {2.3}\%$. The intracavity and output vortex modes were both vertically linearly polarised.
Fig. 3. Vortex laser power curve shows laser threshold at 0.6 W of pump power and a slope efficiency of 21 %.
The vortex laser had a slope efficiency of 22.6 %. Findley-Clay analysis was used to investigate cavity losses present within the system, with a measured total cavity round-trip loss of 4.0 %. The combined measured losses through the imperfect mirror coatings was 1.5 %. We attribute the remaining losses to imperfect AR coating on the laser crystal and unavoidable hair-line crystal fractures from prior work. Eliminating these losses would further increase performance.
The intensity profile of the output vortex mode is shown in Fig. 4(a), it had a measured beam propagation parameter of M$^2_{x,y}=1.94,2.09$ in the horizontal and vertical axes, respectively, which matches excellently with the theoretical value of M$^2_{x,y}=2$ for a pure LG$_{01}$ mode. To isolate the phase structure of the beam a spiral interferogram, shown in Fig. 4(b), confirms the annular phase profile of the vortex. The interferogram was generated by interfering the vortex beam with a spherical wavevfront reference beam in a Mach-Zehnder interferometer. The number of spiral arms reveals the azimuthal LG index, $l$ = 1, as expected for an LG$_{01}$ mode.
Fig. 4. Results for watt-level vortex generation. (a) OV at 1 W output power. (b) Spiral interferogram confirming OV annular phase profile. (c) Intracavity Gaussian mode sampled from HR leakage. (d) FP signal of the laser emission spectrum shows the emission to be SLM. (e) The LG$_{p,l}$ modal decomposition of the vortex output in (a)
The beam propagation parameter and spiral interferogram are key markers that can be used to identify the primary modal composition of the vortex beam. To quantify the mode purity of the beam, a forked interferogram was generated by interfering the vortex beam and a planar reference beam. The interferogram was analysed using a phase retrieval algorithm based on [46] to reconstruct the vortex beam’s phase profile, intensity profile and perform modal decomposition [42]. This technique conclusively determined the vortex mode to have 97.8 % LG$_{01}$ mode power content at maximum power. The modal decompositional chart can be seen in Fig. 4(e).
Also required for high purity vortex generation is a pure internal fundamental Gaussian mode, where distortions or higher order modes will reduce the LG$_{01}$ purity. This was confirmed through beam propagation parameter measurements, the internal mode had M$^ 2=1.03$ across both axes and is shown in Fig. 4(c).
Spectral analysis on the laser emission was conducted using an external Fabry-Perot interferometer (FPI) diagnostics setup. For a single frequency input, the FPI generates a repeating pattern of concentric circular fringes that diminish in interference order with increasing radial distance and are separated by the free spectral range (FSR) of the FPI. Multiple fringes inside one FSR will occur for multi-frequency input. Our FPI transmission spectrum can be seen in Fig. 4(c). The ring separation was verified to be consistent with the 3.4 GHz FSR of the FPI, so this result confirms the single longitudinal mode emission of the laser. The precise measurement of the emission bandwidth was limited by the resolution of the FPI (finesse of 50), but was confirmed to be at least less than 50 MHz, and is expected to be comparable to similar neodymium doped vanadate ring lasers [45].
5. WPSI phase control
The reflectance properties of the WPSI are determined by the interference of the front and rear surface beams. If the interference is constructive the reflectance is maximised, if it is destructive it is minimised. With the WPSI configuration implemented, constructive interference results in TEM$_{00}$ reflection and destructive is LG$_{01}$ reflection - the desired mode.
It can be seen from Eq. (3) that the laser wavelength affects the phase relationship $\Phi$, which results in the relationship between wavelength and WPSI reflectance computed in Fig. 5(a). The reflectance maxima and minima correspond to constructive and destructive interference, respectively, and resemble a low-finesse Fabry-Perot etalon. The laser must operate at the minima to result in LG$_{01}$ reflection, as indicated on the insets. Due to the loss avoiding nature of the laser oscillator, the laser adapts its wavelength to operate in the destructive interference condition. This makes WPSI operation in the laser simple as the phase condition is met by default by the laser cavity, as much as is possible.
Fig. 5. (a) WPSI reflectance corresponding to shifts in incident wavelength relative to 1064nm. LG$_{01}$ and TEM$_{00}$ reflections found when destructive and constructive interference conditions are satisfied respectively. (b) Reflectance plot zoomed in to near OV condition. The red vertical lines indicate the position of cavity longitudinal modes. The WPSI spectrum (blue) can be adjusted in position via $\Delta y$.
A further consideration for the laser wavelength is that the cavity is restricted to certain longitudinal modes. With the inclusion of the WPSI, these are given by
where $L$ is the total cavity optical path length excluding the WPSI, $t$ is the WPSI thickness, and $M$ is the resonant mode order which must be integer.An example longitudinal mode pattern for the laser in this work is shown in Fig. 5(b), where the red lines show cavity longitudinal mode positions. Due to the relative thickness of the WPSI and cavity perimeter, it is possible that there is not a valid longitudinal mode at the correct destructive interference condition. However, this is easy to correct through vertical translation of the WPSI to change its thickness and phase condition, through Eq. (3), to fine-tune the alignment. Once the alignment is achieved the laser will naturally operate at the ideal phase condition and vortex generation.
Translating the WPSI will also change the total optical path length in the cavity, which will in turn also cause the longitudinal mode spectrum to spectrally shift. For the cavity in this work, the longitudinal mode pattern moved a factor of 1/50 compared to the WPSI, so it was a negligible effect. A significant difference will always occur when the WPSI thickness and cavity lengths are significantly different, so can usually be neglected.
In the presented cavity a WPSI vertical translation of 0.5 mm was necessary for a resulting 2$\pi$ phase change. So in practice the necessary adjustments were easy to achieve. It should also be noted that if the cavity perimeter were larger, or the WPSI thinner, the spectra would be much more likely to have a valid longitudinal mode at the destructive interference condition, making these considerations and adjustments unnecessary.
For the laser presented in this work, after an initial warm up time of 30 minutes, the laser was recorded to operated stably in a single longitudinal vortex mode for over 2 hours in a controlled environment with no adjustments necessary. Longer stable operation times are possible. Beam pointing stability measurements saw positional fluctuations of <1 % over the course of the 2 hour run. Longitudinal mode frequency drift was the most significant factor over long time periods, but could be removed with additional frequency stabilising elements, for example peizo-electric cavity mirrors to fine tune cavity length, to further stabilise the design if necessary.
6. Discussion
The single pass conversion of the intracavity power with the WPSI was 2.3 %; however, in the laser cavity the unconverted power was recycled in the next cavity round trip, which effectively increases the conversion efficiency to 100 % (minus small round trip losses). This presents a significant advantage over external use of the WPSI. The single pass conversion efficiency, and therefore the output coupling fraction, can be increased through using a higher $\epsilon$ parameter or adding reflective coatings to the WPSI surfaces. Up to approximately 10 % is possible, depending on purity requirements [42]. This can also benefit overall laser output power and efficiency.
The implemented WPSI interferometric mode converter can generate first order vortex modes (LG$_{01}$). With a Gaussian input other output modes can be configured, for example vortex dipoles [47] or Hermite-Gaussian modes [41]. Superposition states of higher order vortex modes can be made with different intracavity input modes [44]. These examples demonstrate the adaptability of the interferometric mode conversion method.
The presented laser was designed as a demonstration of a compact device with a 80 mm $\times$ 50 mm optical path. The WPSI can be added to any laser cavity oscillating on the fundamental Gaussian mode, so the underlying laser design can be tailored flexibly to each use case. Larger cavities can be used, particularly if space for additional intracavity components is required, for example wavelength or temporal modulators. The vortex output will adopt the same properties as the base laser with the same output coupling, allowing power scalability of the presented concept.
7. Conclusion
In this paper we present, for the first time, a WPSI used as a vortex output coupler in a ring cavity for watt-level OV generation with 98 % $LG_{01}$ mode purity. Our setup boasts a compact and simple design using readily available and inexpensive optics, making it an appealing alternative to systems requiring specialist optical components. The cavity operated in single longitudinal mode which was verified using a FPI diagnostics setup and was made unidirectional through the use of a retro-reflective mirror. The WPSI was used as both the mode conversion device and output coupler, with 2.3 % output coupling fraction. The WPSI was shown to operate at high intracavity powers (50 W) with no detected degradation in performance or surface damage due to its high optical and thermal damage thresholds. This system can be adapted to any gain media granting access to direct generation of previously unattainable optical vortex wavelengths. We found the beam to be stable and maintained its transverse geometry over a span of over 2 hours run time after warm up, making the system ideal for high stability applications such as particle trapping and levitation.
Funding
Imperial College London (Imperial College Research Fellowship); Engineering and Physical Sciences Research Council (EP/T51780X/1).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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