Generation of vortex beams directly from the laser source can be limited in power and efficiency, or to specific pump sources and gain media. Here, we propose a new high power and high efficiency vortex laser methodology with interferometric mode transformation as output coupling, which uses high power handling and low loss optics that have wavelength versatility. Experimental demonstration is made in a diode-pumped Nd:YVO4 laser using an imbalanced Sagnac interferometer as output coupler producing high quality vortex output beams (M2 = 2.07) with fully selectable control of handedness whilst the intracavity mode is maintained as a fundamental Gaussian. Vortex output power >3W is produced with only small reduction in efficiency compared to the equivalent TEM00 laser. Continuous variation of vortex output coupling transmission and the quality of the vortex is investigated experimentally showing good agreement with theory. This work reveals a new approach to high power structured laser radiation direct from the source through interferometric spatial mode transformations.
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An optical vortex has a phase singularity point with zero intensity in an optical field that is surrounded by a spiralling phase front. Since the identification of the orbital angular momentum in optical vortex modes, for example the Laguerre-Gaussian (LGpl, l ≠ 0) , they have found a wide array of applications [2–4]. However, there remains the challenge of efficiently generating high quality vortex modes, particularly at high power and energy levels.
A typical route to generating an optical vortex is to use the Gaussian output beam of a laser and convert it using specialist optics [5–7]. These are usually wavelength specific and can have limited optical damage thresholds, particularly in the widely used computer controlled spatial light modulators , and are sensitive to input beam dimensions and alignment.
An advantageous approach is to directly output the required vortices from the laser source. This can yield efficient, high power, low cost and robust optical vortex sources, which would make them more practical devices. Sources with these properties will be required to take optical vortex applications into industrial environments and out of specialist optical laboratories .
To construct an optical vortex laser the Laguerre-Gaussian (LG0l) vortex modes can be selected directly in a laser resonator [10–15]. A drawback of these methods is that they are typically limited in the range of cavity designs or gain media that can be used, are degraded by thermally induced lensing effects, and often imprecise handedness control.
In this work we use an alternative approach, which is to use an optical device to convert the internal fundamental Gaussian mode into a vortex output, whilst feeding back and preserving the internal Gaussian mode in the laser. We term this type of device a vortex output coupler (VOC) and the concept is shown in Fig. 1(a). This approach has the advantage of being compatible with existing lasers operating with standard pumping and cavity designs that target the fundamental Gaussian mode. Previous implementations of the VOC concept have relied on bespoke optical components [16,17], whereas here we present a method to achieve this interferometrically that requires only mirrors and a beamsplitting element.
The key advantages of our VOC are that it uses standard high-power capable optics, the vortex handedness is fully determined and easily changeable, the additional optics introduce negligible loss making it high efficiency, and it could replace the output coupler of any standard laser cavity. This technique enables the conversion of standard lasers into vortex sources and is suitable for continuous-wave or pulsed operation, and can be used with wavelength tunable gain media with broad gain linewidths.
2. Concept of a laser with a Vortex Output Coupler
In this investigation a VOC was implemented using a misaligned Sagnac interferometer, as shown conceptually in Fig. 1(a). This is a common path interferometer, which makes it robust to environmental perturbations and the interference condition works at any input wavelength. In a Sagnac interferometer, consisting of a 50/50 beamsplitter (BS) and turning mirrors (M1–M3), the input beam is split into equal parts and propagate in opposite directions around the ring. For the aligned case, upon recombination at the BS the beams destructively interfere in the transmission direction due to a relative π phase difference, so the input beam is totally reflected. Instead, the conditions to transmit a vortex beam can be obtained by unbalancing the Sagnac interferometer .
If the two returning Gaussian beams are oppositely vertically displaced by a distance ±d and deviated by a horizontal angle ±θ, relative to their common aligned path, they interfere in the transmission direction to produce a ring shaped intensity pattern with a vortex azimuthal phase profile . This interference pattern is equivalent to an LG01 type vortex with an exp(±iϕ) azimuthal phase term when , where w0 is the beam waist diameter and λ is the wavelength. These conditions can be obtained through unbalancing the Sagnac interferometer by rotating mirror M2 horizontally by an angle θ/2, see Fig. 1(b) top view, and opposite vertical tilts to mirrors M1 and M3 can give the displacement d . In our implementation we take a different approach by rotating a glass plate of thickness t vertically by angle ψ producing a displacement d ≈ tψ(1 − 1/n), see Fig. 1(b) side view, where n is the refractive index of the glass. The advantage of our configuration is that it achieves an identical effect but has the simplicity of providing single controls for both d and θ.
The key feature of the imbalanced Sagnac interferometer VOC is that a controllable fraction of the incident Gaussian power is transmitted into the vortex output. The remaining power is reflected in beam that has approximately the same intensity and phase distribution as the incident Gaussian, and can be aligned to be collinear with the input beam. This allows it to function as an output coupler in a laser cavity, where the internal Gaussian mode is preserved as a stable cavity mode whilst extracting the vortex output. If the VOC was used externally to a laser the reflected Gaussian beam power would be wasted as loss and the conversion efficiency would be equal to the transmission; however, by returning this reflected power to the oscillator to be amplified as a part of the cavity the conversion efficiency is approximately 100 %.
3. Experimental vortex generation
In this work we experimentally demonstrate the VOC to output an LG01 type mode from a linear laser resonator using the cavity design shown in Fig. 2. The cavity was formed with a high reflectance back mirror (BM), Nd:YVO4 crystal, dichroic turning mirror (TM1), intracavity lens of focal length f = 150 mm for mode size control, and the VOC completed the linear cavity by acting as the end reflector. The VOC had a perimeter of 200 mm and an internal AR plate with a thickness of 3 mm. The gain medium was a 2 mm square by 4 mm long 0.5 at.% doped Nd:YVO4 crystal operating on the c-axis at 1064 nm, with the internal laser mode and outputs being linearly parallel to the c-axis throughout this work. It was end-pumped through the dichroic TM1 with a nominal 808 nm fibre delivered laser diode module with up to 13.6 W optical power. The crystal absorbed 78 % of the incident pump power due to an actual central pump wavelength of 806 nm that did not accurately match the 808 nm absorption line in Nd:YVO4. The laser cavity could be redirected away from the VOC to use a standard plane output coupler (OC) by inserting a turning mirror (TM2), where the distance from TM2 to the OC was equal to the path length between the insertion point of TM2 to M2.
The transmission T of the VOC is defined as the fraction of incident power from the laser cavity that was converted into vortex output. The intra-cavity power was deduced from the known 0.7 % transmission of the BM. The VOC was first operated with a transmission of T = 16 % at 8.1 W of absorbed pump power, which gave a vortex output power of 2 W. The intra-cavity mode and vortex output intensity profiles are shown in Fig. 3. The vortex output coupler had negligible effect on the intra-cavity mode, which remained approximately TEM00 with an M2 < 1.3.
The M2 beam quality parameter can be used to determine the likely dominant transverse mode component in the vortex output, where a pure LG01 mode has M2 = 2. The average beam qualities of both the left and right handed vortex outputs were M2 = 2.07 in each axis, which indicates a close match to the intended LG01 mode. To further assess the quality of the vortex output the intensity profile was compared to that of the theoretical LG01 mode. Circular cross sections were taken to calculate the average and standard deviation of the beam intensities at different radial distances from the beam centre. The measured and fitted theoretical intensity profiles are shown in Fig. 4. The experimental vortex outputs were excellent matches to the LG01 mode for both left and right handed outputs, which verifies their high purity. The intensity minima were both below 2 % of the peak intensities, limited by the accuracy of the CMOS camera measurement equipment. The beams had excellent symmetry, with both having a standard deviation of 6 % in intensity at the peak.
To reveal the vortex phase structure of the output, a Mach-Zehnder interferometer was constructed to self-interfere the output with a curved wavefront reference beam. The single spiral interferograms in Fig. 3 reveal a high quality first order vortex phase structure. The handedness of the vortex output produced by the VOC is determined by the relative signs of d and θ, which makes the handedness controllable by reversing either the AR plate or M2 rotation direction. Reversing the vortex handedness did not affect the vortex output power or intensity profile, see Fig. 3, with the handedness reversal being confirmed with the opposing spiral directions in the interferogram. These results confirm that the vortex handedness control is fully independent from the internal Gaussian mode, it can be performed in place without replacing intra-cavity optics or requiring realignment of the laser cavity.
To fully determine the mode components of the vortex outputs modal decomposition methods can be used . However, our analysis shows that the measured vortex outputs had excellent intensity matching to the LG01 mode with the correct spiral phase. This is sufficient to confidently conclude that the outputs have a high LG01 mode component.
The performance of the VOC laser was compared to an equivalent laser with a standard partially reflecting mirror output coupler (OC) by inserting TM2 to redirect the cavity, see Fig. 2. The laser output power versus input pump power is shown in Fig. 5 for the VOC and standard OC lasers, where the VOC was matched to the standard OC transmission of 16 %.
The VOC and standard OC lasers had almost identical laser thresholds at 1 W, which highlights the low insertion loss of the VOC. The slope efficiencies of the VOC and standard OC lasers were 34 % and 40 %, respectively. This resulted in similar maximum output powers at 10.7 W pump power absorbed of 3.2 W and 3.7 W, respectively. At the highest pump power the standard OC laser had an M2 = 1.5, and this increasing higher order mode content may have improved the mode matching efficiency to the pumped region giving it a higher efficiency than the VOC laser. It has been shown that a Sagnac ring with an imaging asymmetry creates a larger loss for higher order transverse modes , so it is likely that the VOC also suppresses higher order modes.
The efficiency of the VOC laser was limited to that of the standard OC cavity in this work. These first results have demonstrated that the VOC laser will operate at the same powers and efficiencies of standard linear cavities, therefore it will be capable of operating at the same elevated powers and efficiencies of more optimised cavity designs. These results highlight the suitability of the VOC as a simple replacement for a standard OC and its inherently low loss design from non-absorbing and non-diffracting components.
A key benefit of incorporating the VOC device inside the laser cavity is the increase in vortex generation efficiency when compared external use. As a simple example, if the VOC was used externally on the output of the standard OC laser in Fig. 5 with T = 16 %, a vortex beam of power 0.59 W would have been generated. This is over a factor of 5 times less than the power from the VOC laser of 3.2 W. However, incorporating the VOC into the cavity provides benefits beyond just efficiency increases. If used externally, the VOC requires precise alignment of the input beam to the device, in particular ensuring a beam waist is formed on M2. When used internally, these alignments are automatically satified as a result of the conditions for a stable cavity mode.
4. Interferometric mode transformation
A key property of an output coupler is its transmission, which in this instance is the fraction of input Gaussian power that is converted into the vortex output. The transmitted vortex electric field given in  can be integrated and related to the input power to give the transmission T of the VOC asFig. 6 with the calculated output far field intensity profiles shown at various points. The beam is exactly equal to the LG01 mode for (d/w0) → 0; however, the vortex output maintains a close similarity to the ideal LG01 mode up to approximately T = 30 %, which is suitable for high-power implementations. Above this transmission it becomes increasingly two-lobed due to decreasing accuracy of the small angle approximation that relates the exact output form to the LG01 mode. Despite this, the spiral phase profile and central phase singularity are maintained even at these high transmission values.
The continuously variable transmission is unusual in an output coupler design and is advantageous for optimising the output power from a laser . To demonstrate this feature the output coupler transmission was varied from 3 % to 41 % at a fixed input pump power of 4.1 W, with the resulting vortex powers shown in Fig. 7. It should be noted that changing the transmission was a simple procedure using only two controls - the AR plate and mirror M2 rotation angles to control and maintain the condition. The transmission was continuously tunable, with an optimum power output found at T = 22 %.
The vortex intensity profiles are shown at selected transmission values in Fig. 7. From T = 3 % to 30 % the output vortex maintained good symmetry as expected from the theory. Above T = 30 % the vortex intensity profile became stretched in the horizontal plane as predicted by the theoretical calculations in Fig. 6. The central intensity minimum of the vortex arises from the destructive interference condition of the Sagnac interferometer, so it maintained its high contrast to the peak intensity. For example, at the maximum transmission of T = 41 % the central intensity minimum was below 2 % of the peak. It was experimentally verified with spiral interferograms that the output maintained its vorticity for all transmission values, which is consistent with the theoretical beam profiles shown in Fig. 6 that have spiral phase profiles.
Both the rotation angles of M2 and the AR plate are functions of the beam waist size on M2 through the VOC transformation condition . The Gaussian beam waist radius was approximately w0 = 150 μm throughout this work, which for a VOC transmission of T = 20 % gives θ/2 = 0.4 mrad for M2 and ψ = 3° for the AR plate. These are both easily and accurately achievable with common optical mounts, in particular the M2 rotation is within typical alignment tolerances for a laser cavity mirror and did not need frequent realignment.
The concept of interferometric mode transformation presented in this work is not limited to the production of vortex modes as in the VOC. Simply by altering the input mode or the interference properties of the beams diverse and unique outputs can be generated directly from lasers. For example, the VOC in this work can be configured to output a HG10 or HG01 (Hermite-Gaussian) mode if either of d or θ are zero, respectively, see Fig. 8.
In summary, we have demonstrated a vortex output coupler that interferometrically converts the typical LG00 internal mode of laser cavities into an LG01 type vortex output. The vortex mode generated had an excellent average beam quality of M2 = 2.07, and the handedness was fully determined and controllable through a single control on the Sagnac interferometer. The intra-cavity mode remained approximately Gaussian. The transmission of the output coupler was continuously adjustable and was used to optimize the laser output power. The design utilised a Sagnac interferometer, making it robust against environmental perturbations and suitable for use broad bandwidth or wavelength tunable lasers. This technique only requires a beamsplitter and mirrors so it can be used with any laser gain medium in wavelength ranges that may not be accessible with other methods, for example because the required optics or pump geometries are not available. It could replace the output coupler of any fundamental mode linear laser cavity to convert it into a vortex source.
The interferometric vortex output coupler is suitable for higher average output powers beyond the 3 W demonstrated in this work in addition to high peak power pulsed applications. The design is inherently low loss through introducing non-absorbing and non-diffracting components, and in this configuration uses dielectric coated interfaces that are a proven technology to withstand both high peak and average incident power. The output coupler can be implemented in high power laser designs provided that they operate with the fundamental Gaussian mode. Additionally, we have observed that this device can aid in suppressing higher-order cavity modes, which will further increase the power scaling potential for these cavities.
The interferometric output coupling method presented here is not limited to a Sagnac design, and could be implemented with other suitable interferometers, for example Mach-Zehnder or Michelson. Each interferometer will have differing advantages, and allows this VOC approach to be suitable for both linear and ring resonators. Additionally, this interferometric approach is not limited to LG01 type vortex output and can be adapted to generate other beam structures depending on the interferometer configuration and the intra-cavity mode input. For example, the presented configuration can increase the orbital angular momentum density of the input by outputting a higher order vortex mode. A detailed investigation of the mode conversion technique is planned for a future work. The interferometric output coupling method presented in this work enables high power and efficient generation of vortex beams and other mode structures at diverse wavelengths that may not be currently possible through other methods.
Engineering and Physical Sciences Research Council (EPSRC); EPSRC Quantum Systems Engineering Skills and Training Hub studentship, Imperial College London (EP/P510257/1).
Portions of this work were presented at the Advanced Solid-State Lasers conference in 2018, AW2A.2 .
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