We report on direct generation of optical vortices from a continuous-wave (cw), Gaussian beam pumped doubly resonating optical parametric oscillator (DRO). Using a 30-mm long MgO doped periodically poled lithium tantalate (MgO:sPPLT) crystal based DRO, pumped in the green by a frequency-doubled Yb-fiber laser in Gaussian spatial profile we have generated signal and idler beams in vortex mode of order, l = 1, tunable across 970-1178 nm. Controlling the overlap between the Gaussian pump beam with the fundamental cavity mode of the resonant signal and idler beams of the DRO through the tilt of the pump beam and/or the cavity mirror in transverse plane, we have generated both signal and idler beams in vortex and vortex dipole spatial profiles. Using the theoretical formalism for the vortex beam generation through the superposition of two Gaussian beams we have numerically calculated the spatial profile of the generated beam in close agreement with our experiment results. The generic experimental scheme can be used to generate optical vortex across the electromagnetic spectrum and in all time scales (cw to ultrafast) using suitable OPO.
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Optical vortices, the doughnut shaped beams carrying orbital angular momentum (OAM), have drawn a great deal of interest in recent times with new applications in various fields of science and technology including quantum information and quantum entanglement , lithography , high resolution microscopy  and micro-particle manipulation . Typically, the optical vortices are generated through the mode conversion of Gaussian laser beam using the conventional mode converters such as cylindrical mode converter based on astigmatic lenses , spiral phase plates (SPPs) having transverse thickness variation corresponding to the spiral phase pattern of the vortices , q-plates , liquid crystal based spatial light modulators (SLMs) and computer generated holograms . Among all mode converters the SPPs, cylindrical mode converters and the etched-glass diffractive optical elements, are simple and stable devices to generate high power optical vortices required for various applications including nonlinear optics , gravitational wave detection  and micromachining . On the other hand, due to their wide wavelength coverage, the low damage threshold mode converters including q-plates and SLMs are conventionally used for applications requiring low power vortex beams. However, the future advancement of optical communication with a demand of high-capacity and multi-channel operation requires high power laser source with wavelength and OAM-tunable vortex beam output over a wide spectral range. Efforts have been made in recent times to generate optical vortices from the lasers . While the direct generation of high power optical vortices from lasers eliminates the need of phase elements for mode conversion, however, the limited wavelength tunability of the lasers is the major challenge towards the generation of vortex beams over a truly wide wavelength range.
On the other hand, optical parametric oscillators (OPO), based on nonlinear optical effects, have been established as the source of coherent radiation tunable across the electromagnetic spectrum in all time scales (continuous-wave (cw) to femtosecond) [13–15]. Therefore, the direct generation of optical vortex from the OPOs can be the most straightforward way of addressing the requirement of wavelength and OAM-tunable vortex beam directly from the source. Efforts have been made in recent times to generate optical vortex from the OPOs by transferring the OAM of the pump beam to one of the generated beams owing to the OAM conservation in nonlinear frequency conversion processes [16–18]. Similarly, the control of the cavity losses among the resonating beam has also enabled the selective transfer of the pump OAM modes to one of the OPO output beams . However, due to the need of pump beam in vortex spatial profile, all the OPO based vortex sources [16–18] till date demand external mode converter for the pump wavelength. Additionally, the vortex pumped OPO has higher operation threshold as compared to the Gaussian beam pumped OPO. Therefore, it is imperative to generate vortex beam directly from the Gaussian beam pumped OPOs.
Given the vortex beam generation through the superposition of orthogonal Hermite-Gaussian (HG) modes with suitable phase relation and the operation of OPOs in higher order TEM modes , one can in principle generate vortex beam directly from a Gaussian beam pumped OPO. Here we report, for the first time to the best of our knowledge, the direct generation of tunable optical vortex beam from a Gaussian beam pumped cw OPO. By controlling the overlap of the Gaussian pump beam and the fundamental OPO cavity modes through the external pump beam tilt, we have generated optical vortex of order, l = 1, tunable across 970-1178 nm. The experimental results also show a close agreement with theoretical treatment used for vortex beam generation from the overlap of two Gaussian beams .
2. Experimental setup
The schematic of the experimental setup is shown in Fig. 1. A continuous-wave (cw), single-frequency, Yb-fibre laser, providing output power up to 50 W at 1064 nm in Gaussian intensity distribution is used as the fundamental laser. Using a lens, L1, of focal length, f = 150 mm, we have focused the laser at the centre of a 50 mm long, MgO-doped periodically poled congruent grown lithium tantalate (MgO: PPCLT) crystal (C1) of single grating of period, Λc = 7.65 µm, to frequency-double into green at 532 nm with power of 4.5 W in a Gaussian intensity profile. The half-wave plate (λ/2) is used to adjust the polarization of the fundamental beam with respect to the poling period direction of the MgO:PPCLT crystal for optimum phase-matching. A lens, L2, of focal length, f = 150 mm is used to collimate the generated green and subsequently extracted from the undepleted pump using a wavelength separator, S1. The power attenuator comprised with a λ/2 plate and a polarizing beam splitter (PBS) cube is used to adjust the input green pump power to doubly-resonating OPO (DRO). The subsequent λ/2 plate controls the polarization of the green beam as required for OPO operation. The DRO is configured in a standing wave V-shaped cavity comprised of two plano-concave mirrors, M1-M2, of radius of curvature, r = 100 mm, and a plane mirror, M3. All the cavity mirrors are high reflective (R>99.5%) across 850-1200 nm ensuring doubly resonant condition for signal and idler beams and high transmission (T>95%) at pump wavelength, 532 nm. A 30 mm long and 2x1 mm2 in aperture, MgO-doped periodically poled lithium tantalate (MgO:sPPLT) crystal (C2) of single grating period of, Λc = 7.97 µm, is used as the nonlinear crystal for OPO operation. The crystal is housed in a temperature oven whose temperature can be varied up to 200°C with a stability of ± 0.1°C. The out-coupled signal and idler radiation of mirror, M3, are wavelength separated using the separator, S2, for further studies. Since the mirror reflection changes vortex sign, we have used even number of reflections to both signal and idler beams before characterization. The input Gaussian pump beam is focused at the centre of MgO:sPPLT crystal using a convex lens, L3, of focal length, f = 150 mm, to an estimated beam waist radius of wp = 55 μm. The separation of the curved mirrors, M1 and M2, placed on the translation stages, was adjusted to match the confocal parameter of the resonating signal to that of the pump beam for optimum performance of the DRO . The beam waist radius of the signal is estimated to be ws = 78 μm with the optical length of the DRO cavity of, L = 315 mm.
3. Results and discussions
To verify the generation of vortex, beam directly from a Gaussian beam pumped DRO, we adjusted the crystal temperature at 62°C producing signal and idler at 1002 nm and 1134 nm respectively, away from the degeneracy to study the beams individually. Pumping the DRO with Gaussian beam we have measured the spatial profile of the signal and idler for different cavity alignments with the results shown in Fig. 2. As expected, for exact alignment of the OPO cavity ensuring optimum overlap of the pump beam with the fundamental cavity mode, the Gaussian pump beam produces signal and idler beams, as shown in Figs. 2(a)-2(b), resembling Gaussian spatial profiles. To confirm the absence of any topological charge in the resonant beams, we have recorded the self-interference pattern of the signal and idler beams using a Mach-Zehnder interferometer (MZI). The straight line interference fringes, as shown in Figs. 2(c)-2(d), confirm both signal and idler oscillating in fundamental cavity modes resulting the Gaussian spatial intensity distribution.
Keeping the cavity parameters fixed, we have introduced a tilt between the Gaussian pump beam and the fundamental cavity mode by tilting the focusing lens, L3, in the transverse plane and recorded the intensity profile of the signal and idler beams with the result shown in Figs. 2(e)-2(f), respectively. As evident from Figs. 2(e)-2(f), both the signal and idler are transformed into beam with doughnut shaped intensity distribution. To confirm the presence of topological charge, we have recorded the self-interference of the signal and idler beams using the MZI with the results shown in Figs. 2(g)-2(h), respectively. The presence of characteristic fork in the interference pattern, as evident from Figs. 2(g)-2(h), confirm both the signal and idler carrying optical vortices of same order, ls = li = 1 and same sign. Given the OAM conservation of nonlinear processes [16,17], one can expect the Gaussian pump beam, lp = 0, to produce signal and idler vortices of same order but opposite in sign satisfying, lp = ls + li. However, the generation of signal and idler beams with same vortex order and sign from the Gaussian pump beam can be attributed to the generation of higher order cavity mode due to the pump beam tilt and not due to the OAM conversation of the nonlinear process of the OPO.
To understand the vortex beam generation mechanism of the Gaussian pumped DRO, we have investigated the generation of vortex and dipole using superposition of two transversely scaled Gaussian beams .21]. Therefore, we can assume both the beams to have same beam waist radius i.e., α = 1, and numerically simulated the intensity distribution of the superposed beams by varying the linear phase difference ϕ along x-axis, y-axis and 45◦ to the x-axis with the results shown in Fig. 3. As evident from Figs. 3(a)-3(d), the superposed beam maintains a Gaussian intensity distribution for linear phase difference, ϕx,y = 0, and transform into HG10 mode, HG01 mode and HG01 mode along 45° due to the linear phase difference ϕx,y along x-axis, y-axis and 45° to the x-axis, respectively. The HG01 mode along 45° can be transformed into optical vortex as shown by the inset of Fig. 3(d) using the astigmatic mode converter based on tilted curved mirrors .
To verify the theoretical proposal experimentally, we pumped the DRO at constant power of 1 W producing signal (idler) at 1002 nm (1134 nm). Controlling the spatial overlap of the pump beam and the cavity mode by tilting the pump focusing lens, L3, to introduce linear phase along x and y-axis, we have recorded the spatial intensity profile of both signal and idler beams with the results shown in Figs. 3(e)-3(h), and Figs. 3(i)-3(l), respectively. As evident from Figs. 3(e) and 3(i), both signal and idler beams have Gaussian spatial intensity profile in absence of any tilt in the lens, L3, similar to the theoretical result of Fig. 3(a). The measured operation threshold as low as ~60 mW, confirms the DRO operation in fundamental cavity mode and the signal and idler carry Gaussian spatial mode. However, the introduction of linear phase difference between the pump beam with the cavity mode through the tilt of the lens, L3, by ~3′ along x-axis and y-axis, transfer the pump energy to the resonating beams (signal and idler) in HG10 mode, and HG01, as shown in Figs. 3(f) and 3(j) and Figs. 3(g) and 3(k), respectively. In addition, the presence of higher order spatial mode for the signal and idler beams is also be confirmed from the increase of DRO threshold to ~280 mW in both cases. Since the HG01 mode along 45° can be represented as the superposition of HG10 and HG01 modes, and can be transformed into optical vortex, we have introduced linear phase along 45° by tilting the lens along both x-axis and y-axis equally. As evident from Figs. 3(h) and 3(l), both the signal and idler have doughnut shaped intensity distribution. While we expect the signal and idler beams to oscillate with HG01 mode along 45°, however, the tilted curved spherical mirrors of the DRO cavity act as an astigmatic mode converter  to transform the HG mode into optical vortex. On the other hand, one can generate signal and idler beams in HG01 mode along 45° by reducing the astigmatism through the use of low angle of incidence at the curve mirror and transform into vortex beams using external mode converter. In such case, we expect both the signal and idler beams to carry vortex spatial profile with same order and sign similar to the current experimental results. From Fig. 3, it is evident that experimental results are in close agreement with the theoretical prediction. Although we have created linear phase by tilting the lens, one can in principle observe the same effect by tilting the plane mirror, M3, and the curve mirrors, M1-2, of the DRO cavity. The tilt of the nonlinear crystal can modify the output modes of an OPO, however, it is not as sensitive as compared to the tilt of the lens and cavity mirrors.
On the other hand, the crystal tilt has more serious implications to the OPO performance in terms of wavelength change and crystal damage. We have measured the pump beam shift at the center of the nonlinear crystal due to the tilt of the focusing lens. Using a 4-f imaging system we have imaged the pump beam position at the center of the crystal in a CCD camera and measured the shift of the Gaussian beam by 12 µm along 45° to x axis for the lens tilt of 3′ as used to generate vortex beam. It is to be noted that the beam waist size of the pump beam at the center of the nonlinear crystal is 55 μm.
We have further recorded the spatial profile of the signal and idler beams across the tuning range of the DRO. The results are shown in Fig. 4. Changing the crystal temperature from 55°C to 80°C, we have generated signal and corresponding idler tunable across 1023 - 970 nm and 1108 - 1178 nm, respectively, with total output power varying from 68 mW to 48 mW with a maximum of 84 mW at 995 nm for a fixed pump power of 4.3 W. As evident from the Figs. 4(a)-4(f), the signal and corresponding idler beams carry doughnut shaped intensity profiles resembling optical vortex across the tuning range. To confirm the presence of phase singularity and determine the vortex order, we have recorded the self-interference of the beams using MZI. The presence of characteristic fork, as shown in Figs. 4(g)-4(l), confirm both the signal and idler beams to carry optical vortex of order, l = 1, and same sign. Relatively poor image quality of the idler beams can be attributed to the inferior sensitivity of the CCD camera beyond 1100 nm. Due to unavailability of suitable wavelength separator to extract the signal from the idler, we could not study the spatial form of the beams in the wavelength range of 1024-1107 nm. However, it is important to note that, we have recorded the spatial profile in doughnut shape in the wavelength range 1024-1107 nm but could not observe self-reference interference fringes due to the obvious reason.
As evident from , by controlling the beam overlap one can generate vortex dipole. To verify the possibility of controlled generation of vortex and vortex dipole we have designed the DRO in a standing wave X-cavity configuration (not shown here) comprised with two curved mirrors and two plane mirrors. Although the use of additional plane mirror increases the OPO threshold to 112 mW due to the extra loss, however, we have observed the X-cavity configuration enabling easy control over the cavity modes and beam overlapping. Keeping the curved mirror separation of 127 mm corresponding to α = 1 and optimizing the DRO to produce both signal and idler beams in Gaussian mode, further, we have tilted the plane mirrors and recorded the signal and idler beams with the results shown in Fig. 5. As evident from Figs. 5(a) and 5(b), the signal and idler beams have doughnut shaped intensity profile which further confirmed to carry vortex beams from the self-interference images recorded using MZI as shown in Figs. 5(c) and 5(d). Due to the mechanical constraint of the X-cavity setup, the signal has faced one extra reflection as compared to the idler beam. As a result, the fork pattern of signal and idler are in opposite directions. However, counting the number of reflections in each beam path, we confirm that both signal and idler carry vortex beam of same order, ls = li = 1, and sign. Interestingly, the change in curved mirror separation from 127 mm to 118 mm changing the transverse scaling factor from α = 1 to α = 1.2, keeping rest of the cavity parameters unchanged, we still observe the signal and idler beams to have doughnut shaped profile as shown in Figs. 5(e) and 5(f). However, the presence of oppositely oriented fork pattern in the self-interference images as recorded in Figs. 5(g) and 5(h), confirm the generation of vortex dipole of unit charge in both signal and idler wavelengths. Such transformation of vortex into vortex dipole can be attributed to the change in the transverse scaling factor, α, of the superposed beams and also to the increased beam astigmatism due to the presence of two curve mirrors of the DRO in X-cavity configuration. To measure the spatial stability of the resonating vortex beams, we have recorded 20 continuous frames in the CCD camera and estimated the shift in the dark core of the beam in each frame. We have observed a maximum variation of 60 µm and 62 µm along x and y directions respectively, from the mean position of vortex minima. Such small fluctuations of the beam position can be attributed to the instability of the DRO. Since the DROs have higher cavity instability as compared to singly-resonant OPOs configurations, using suitable cavity locking scheme one can improve the stability and mode quality of the output vortex beam. We have also studied the output power characteristics of the generated beam and recorded 84 mW of total output power at pump power of 4.3 W at a slope efficiency of 2.1%. While the increase of mirror transmission might produce higher output power, however, the increase of cavity loss can restrict the DRO operation in Gaussian mode.
In conclusion, we have demonstrated the direct generation of optical vortex and vortex dipole from a Gaussian beam pumped DRO by simply controlling the overlap of the pump beam with the cavity modes of the resonant beams. Pumping the DRO at green we have generated signal and idler beams in vortex spatial profile with same order and sign tunable across 970-1174 nm with a practical output power as high as 84 mW. Using the simple theoretical model explaining the vortex and vortex dipole generation from the superposition of two Gaussian beams, we have numerically simulated the beam overlap of the Gaussian pump beam with the cavity mode of the DRO in close agreement with the experimental results. The generic experimental scheme can be used to generate vortex and vortex dipole across the electromagnetic spectrum in all time scales (cw to ultrafast).
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