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In this paper, we report the generation of near-infrared/mid-infrared off-axis vortex fields with continuously tunable OAM states, formed through the coherent superposition of vortex and Gaussian modes within an idler-resonant KTiOAsO4 (KTA) OPO with a half-spherical cavity. This system produces signal and idler outputs with OAM, which is tunable in the range of 0–1 through precise adjustment in the resonant cavity length. The maximum energy outputs at wavelengths of 1.5 and 3.5 μm were 1.95 and 1.1 mJ, respectively, and were obtained at a maximum pump energy of 20 mJ.
A schematic of the experimental setup is shown in Fig. 1(a). A conventional Q-switched Nd:YAG laser (LS-2036, a pulse duration of 25 ns, a PRF of 50 Hz) producing an output with a Gaussian spatial profile at a wavelength of 1.064 μm was used as the pump source. A half-wave plate (HWP) was used to control the pump polarization to satisfy the phase matching condition into the KTA crystal. The laser output was converted to a first-order optical vortex beam with a diameter of 500 μm and ℓ = 1 by using a spiral phase plate azimuthally divided into 16 segments and a plano–convex lens (f = 750 mm). It was then focused into a type II non-critically phase matching KTA crystal with dimensions of 5 × 5 × 30 mm3, which was oriented with θ = 90° and φ = 0°. Both end faces of the crystal were antireflection-coated (R μm (idler field). As shown in Figs. 1(b) and 1(c), the pump beam exhibited an annular spatial profile with central phase singularity, as evidenced by the generation of a pair of Y-shaped fringes in a self-referenced interferogram. The pump beam was observed by a conventional CCD camera. The spatial forms and wavefronts of the signal and idler outputs were measured by using a pyroelectric camera (Spiricon Pyrocam III; spatial resolution: 75 μm). The output coupler (OC) was mounted on a one-dimensional translation stage. The cavity length was adjusted within a range of 35–85 mm by moving the OC along the optical axis. The input mirror (IM) and KTA crystal were then fixed. It is noteworthy that the cavity was stable at any cavity lengths, as evidenced by the cavity stability factor of 0.66–0.84.As detailed in our prior publications, an OPO, which is singly resonant for the idler field and has a plane-parallel (near-unstable) cavity configuration, prohibits the idler beam from lasing with a vortex mode (higher-order transverse mode). In these cases, unidirectional transfer of OAM from the pump field to the signal field occurs. It was found that the idler field is oscillated with a Gaussian mode, and the signal field, with a vortex mode for all tested cavity lengths, d, (from 35 to 85 mm); plots of the spatial profile of these modes are shown in Figs. 2(a1)–2(a3) and Figs. 2(b1)–2(b3), respectively. Note that the optical parametric generation without the cavity mirrors was never observed even at a maximum pump level of 21 mJ.The OPO in this work was singly resonant for the idler field and had a nearly half-symmetric cavity configuration. It was formed using a plane-concave IM(R = 500 mm), which was antireflection coated for the pump field and high-reflection coated for both the signal and idler field outputs, and a plane OC which was coated partially reflecting (R ≈ 80%) for the idler field, and high-transmitting for the pump and signal fields. The overall system was very compact with the resonant cavity having a length of 35 mm. This enabled vortex mode operation of the idler field. Its spatial profile is shown in Fig. 3(a1). The beam quality factor, M2, is generally defined as four times standard deviation obtained by fitting the spatial form of laser output to the Gaussian function. The ideal ℓth order LG mode exhibits M2 of ℓ + 1, as previously mentioned in Ref. 4343. S. Ramee and R. Simon, “ Effect of holes and vortices on beam quality,” J. Opt. Soc. Am. A 17(1), 84–94 (2000). https://doi.org/10.1364/JOSAA.17.000084. The beam quality factor M2 of the idler vortex output was measured to be ∼2.2 (as determined using the knife-edge method), which is very close to the ideal value (for the first order LG mode) of 2. With the idler field lasing as a vortex mode, the signal field is oscillated as a Gaussian mode as shown in Fig. 3(b1).Interestingly, extension of the idler cavity resulted in interaction between the idler and signal fields, and the generation of off-axis vortex modes, as evidenced by the asymmetric spatial forms of the idler and signal outputs with these having radial openings [Figs. 3(a3,4) and 3(b3,4)]. Further extension of the cavity out to 85 mm resulted in unidirectional transfer of OAM from the pump field to the signal field [Fig. 3(b6)], and the idler field is operated on a Gaussian mode [Fig. 3(a6)]. Such continuously tunable OAM transfer between the signal and idler fields has never been reported in prior vortex-pumped OPO research. This work highlights how the OAM of the off-axis vortex output could be continuously tuned within the range 0–1 by simply extending the cavity, as shown in Fig. 3.The output energies of the signal and idler fields as a function of the pump energy in both compact and extended cavity configurations are plotted in Fig. 4. In the compact cavity, the maximum Gaussian signal and vortex idler energies were measured to be 2.9 and 1.1 mJ at the maximum pump energy of 20 mJ, and these values correspond to slope efficiencies of 21% and 8.1%, respectively. When the cavity was extended, the maximum vortex signal and Gaussian idler energies were measured to be 1.95 and 0.65 mJ at the maximum pump energy of 20 mJ, corresponding to slope efficiencies of 23% and 6%, respectively. Extending the cavity reduced the output energies of signal and idler outputs owing to the severe diffraction loss. The output powers of signal and idler outputs were ranged within 97.5–145 and 32.5–55 mW, respectively. In general, the cavity extension increases the diffraction loss for the idler output, thus yielding high oscillation threshold. In fact, the measured oscillation threshold was ranged within 6.3–12.1 mJ. The idler output exhibited a temporal energy stability of less than 2% rms during a long observation time of over 2 h in both compact and extended cavities. It is also noteworthy that the degradation of spatial forms of the signal and idler outputs has never seen at any pump levels. The spectral bandwidths of the signal and idler outputs were measured to be Δλs = 0.12 nm (0.51 cm−1) and Δλi = 0.61 nm (0.55 cm−1), respectively, as measured using a scanning monochromator (SpectraPro HRS-500, 300 lines/mm, an aperture size of 30 μm, a spectrum resolution of 0.1–0.2 nm). The idler vortex output had typically a pulse width of 16 ns.The spatial coupling factor η(d) with respect to the pump, signal, and idler fields within the optical parametric cavity is given by4444. S. E. Harris, “ Tunable optical parametric oscillators,” Proc. IEEE 57(12), 2096–2113 (1969). https://doi.org/10.1109/PROC.1969.7495 η(d)=[WsdWi(d)WpWs2dWi2(d)+Ws2(d)Wp2+Wi2(d)Wp2]2,(1)where Wp, Ws(d), and Wi(d) are the mode radii of the pump, signal, and idler modes in the crystal, respectively. Also, the signal output radius is given by 1Ws2(d)=1Wp2−1Wi2(d).(2)To understand the observed OAM sharing effects between the signal and idler fields, which results from extending the cavity, the beam radius [Wi(d)] of the idler output as a function of the cavity length was calculated by LASCAD software under the assumption that the vortex mode is a Gaussian beam with M2 = 2. Also, the beam radius [Ws(d)] of the signal output was estimated by using Eq. (2). The resulting spatial coupling factors ηv(d) and ηG(d) among the vortex pump, Gaussian (vortex) signal, and vortex (Gaussian) idler fields were then estimated at various cavity lengths. Also, the pump beam radius was fixed at 250 μm. We observed that the spatial coupling factor ηv(d) degraded monotonically as the cavity length increased. Conversely, the spatial coupling factor ηG(d) increased with cavity length, exceeded ηv(d) at cavity lengths of > 58 mm [Fig. 5(a)]. These results support the experimental observations, the cavity extension enables the idler field to operate with an off-axis vortex mode as opposed to an integer vortex mode, and long cavities (cavity length > 85 mm) force the idler to operate with a Gaussian mode.The spatial form Fv(r, ϕ) and OAM of an off-axis vortex beam can be expressed as Fvr,ϕ=sin θ·CGe−r2+cos θ·Cvre−r2+iℓϕ+iξ2,(3)where Cv and CG are normalized coefficients, θ is the mixture parameter, ξ is the Gouy phase, and r and ϕ are the radial and azimuthal indices, respectively. Here, it is assumed that the beam radius is normalized to 1. It should be noted that the radial opening of an off-axis vortex beam rotates along the longitudinal propagation direction due to different Gouy phases. For the purpose of our examination, the Gouy phase ξ was fixed at π/4. Assuming that the amplitude mixture ratio (=tanθ) between the vortex and Gaussian modes of the generated off-axis vortex beams is directly determined by the spatial coupling factors ηv(d) and ηG(d), the spatial form and OAM of the generated off-axis vortex beams were numerically simulated. As shown in Figs. 5(b) and 5(c), the cavity extension (55–70 mm) forced the idler output to oscillate as an off-axis vortex with the radial opening. Further cavity extension (>80 mm) is resulted in near-Gaussian mode operation of the idler output, and the production of a signal field operates with an off-axis vortex. These simulated spatial profiles qualitatively support the observed experimental results.We have demonstrated the process of OAM sharing between idler and signal fields within an idler-resonant optical parametric oscillator. By simply extending or shortening the cavity, the off-axis vortex signal and idler outputs with non-integer OAM in the range 0–1 could be generated. The maximum energy outputs at wavelengths of 1.5 and 3.5 μm were 1.95 and 1.1 mJ, respectively, obtained for a maximum pump energy of 20 mJ. The continuous OAM sharing effect between the signal and idler outputs was theoretically modeled through a consideration of the spatial coupling factor among the pump, signal, and idler fields within the cavity. We believe that such continuous OAM sharing effects will be useful for a variety of applications, including OAM multiplication/division functional transformation integrated devices,4545. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “ Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–98 (2015). https://doi.org/10.1364/AOP.7.000066 tools for finely trapping and rotating microparticles,46,4746. S. H. Tao, X. C. Yuan, J. Lin, X. Peng, and H. B. Niu, “ Fractional optical vortex beam induced rotation of particles,” Opt. Express 13(20), 7726–7731 (2005). https://doi.org/10.1364/OPEX.13.00772647. J. Zeng, C. H. Liang, H. Y. Wang, F. Wang, C. L. Zhao, G. Gbur, and Y. J. Cai, “ Partially coherent radially polarized fractional vortex beam,” Opt. Express 28(8), 11493–11513 (2020). https://doi.org/10.1364/OE.390922 and living cells.2222. B. Gao, J. Wen, G. Zhu, L. Ye, and L. G. Wang, “ Precise measurement of trapping and manipulation properties of focused fractional vortex beams,” Nanoscale 14(8), 3123–3130 (2022). https://doi.org/10.1039/D1NR06163AAccording to the OAM conservation among the pump, signal, and idler outputs, the signal and its corresponding idler outputs have the possibility to exhibit off-axis vortex modes beyond the range of 0–1. In fact, in our previous publication,4848. R. Mamuti, S. Goto, K. Miyamoto, and T. Omatsu, “ Generation of coupled orbital angular momentum modes from an optical vortex parametric laser source,” Opt. Express 27(25), 37025–37033 (2019). https://doi.org/10.1364/OE.27.037025 the idler output exhibits unique wheel-shaped modes formed of the coherent superposition between Gaussian and higher-order (4th) vortex modes. However, such higher-order off-axis vortex modes exhibit typically a rather small spatial overlap with the pump beam in comparison with a lower-order off-axis vortex mode. Also, the generation of such higher-order off-axis vortex modes will require a pump source with short temporal coherence time. Thus, the present system prevents the generation of higher-order off-axis vortex with OAM > 1 or OAMThis work was supported by the National Natural Science Foundation of China (Grant Nos. 11664041 and 12264049), the Natural Science Foundation of the Xinjiang Uygur Autonomous Region (Grant No. 2021D01A114), and the Foundation of Xinjiang Normal University Key Laboratory of Mineral Luminescent Material and Microstructure of Xinjiang, China (Grant No. KWFG202202). This study was also funded by a Japan Society for the Promotion of Science (JSPS) KAKENHI (Nos. JP16H06507, JP18H03884, JP22H05138, and 22K18981), and JST Core Research for Evolutional Science and Technology (CREST) (No. JPMJCR1903).
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Taximaiti Yusufu: supervised the experimental process, corrected academic errors in the manuscript, and improved the academic quality of the manuscript; Yuxia Zhou: preliminary research, done all experiments and data analysis, and wrote the main manuscript text; Yuanyuan Ma: conducted the theoretical calculations; Takashige Omatsu: revised the manuscript and added the fruitful discussions. All authors contributed to the preparation of the manuscript.
Yuxia Zhou: Data curation (equal); Writing – original draft (equal). Taximaiti Yusufu: Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Yuanyuan Ma: Formal analysis (equal); Writing – review & editing (equal). Takashige Omatsu: Formal analysis (equal); Writing – review & editing (equal).
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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