Frequency-doubled vortex beam emitter based on nonlinear Cherenkov radiation Download: 739次
Light beams with helical phase fronts are called vortex beams, and they carry an orbital angular momentum (OAM) of per photon[1]. Due to the unique properties of such beams, increasing attention has been paid in the area of modern optics and photonics in recent years. For example, the hollow intensity distribution of vortex beams provides a gradient force that can be used for optical manipulation[2], trapping[3], tweezers[4], as well as super-resolution microscopy[5]. Moreover, the OAM has infinite orthogonal states, and this property makes OAM beams suitable for high-capacity optical communications[68" target="_self" style="display: inline;">–
There are a variety of methods for generating vortex beams[10], and most of them are based on bulk optical elements such as mode converters using cylindrical lenses[11], spiral phase plates (SPPs)[12], spatial light modulators based on computer-generated holograms[13], q-plates[14], nano-antenna arrays[15], and OAM states from laser cavities[1618" target="_self" style="display: inline;">–
Nonlinear Cherenkov radiation (NCR) is the optical analog to the conventional Cherenkov radiation in particle physics. When the phase velocity of the nonlinear polarization wave (NPW) in the nonlinear medium is larger than that of the harmonic wave , then a coherent harmonic wave would be emitted along the Cherenkov angle . In nonlinear waveguides with ferroelectric domain structures, such as periodically poled waveguides, nonlinear Cherenkov radiation can be modulated by the reciprocal vectors provided by the structure. Let us take Cherenkov-type second harmonic generation (SHG) as an example, and the phase-matching condition shown in Fig.
Fig. 1. (a) Schematic of the phase-matching diagram for emitting vortex beams with doubled frequency; the orange arrow represents the polarization direction of the FW and the SH wave. (b) Cross section of the micro-ring resonator. (c) Illustration of the polar coordinate system and the emitted SH vortex beam with a helical phase front.
We choose a micro-ring resonator as the material platform to investigate the proposed scheme, as shown in Fig.
To analyze the electric field of the radiated wave, we establish the polar coordinate system based on the center of the micro-ring, with being the distance from the observation plane to the plane of the micro-ring as shown in Fig.
The micro-ring resonator supports both quasi-TE and quasi-TM polarization modes. During the study, we found that distribution of the NPW as well as the effective second-order nonlinear coefficient are quite complicated when the FW is in the TE mode. In this work, we only consider the situation when the TM mode is excited in the micro-ring resonator. In this case, the induced second-order nonlinear polarization only contains the component or component in the direction, which can be written as[29]where is the largest second-order nonlinear coefficient of . As the first-order backward reciprocal vector is utilized, we only consider the Fourier component in Eq. (
Based on Eq. (
Fig. 2. Simulated (a) intensity distribution and (b) phase of the emitted SH wave excited by TM polarized FW. The TC of the SH vortex beam is .
The normalized intensity of the radial, azimuthal, and vertical components of the SH wave are shown in Figs.
Fig. 3. Simulated intensity distributions of different components (a) , (b) , and (c) , with .
In addition to the field distribution of the generated SH vortex beam, we investigate the conversion efficiency for the frequency-doubled vortex beam emitter based on NCR. The energy stored in the micro-ring cavity can be evaluated from the definition of the -factor, [30], in which is the resonant frequency and is the lost power. Considering the low conversion efficiency of Cherenkov-type SHG, the depletion of the FW is neglected, and thus is approximately equal to the power coupled into the ring resonator . The amplitude of the FW can be written as
Substituting the field amplitude of the FW into Eq. (
The -factor of the micro-ring resonator is set to be , and the power coupled into the resonator is assumed to be 10 mW. The conversion efficiency varying with the poling periods is listed in Table
Table 1. Cherenkov Radiation Angle, Topological Charge, and Conversion Efficiency Varying with the Poling Period
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When the working temperature changes, the propagation constant of the FW will change accordingly. To ensure the azimuthal order of the resonant FW, which is determined by to be an integer, the temperature can only take discrete values. The azimuthal order of the FW will be increased by 1 when the temperature reaches 89°C, and the corresponding topological charge of the SH is increased from 976 to 978, the poling period being 500 nm. In addition, the radiation angle and the conversion efficiency will be changed to 44.9° and , respectively. Besides the platform used in the simulation, MgO-doped (MgO:LN) can be also used. The second-order nonlinear coefficients of and MgO:LN are almost the same[32]. However, the dispersion relation of the two crystals is different[33], and there is a slight refractive index difference between them. When the materials platform is changed to MgO:LN, SH vortex beams will still be emitted out because the Cherenkov phase-matching condition can be satisfied by automatically changing the radiation angle. For a fixed poling period of 500 nm, the radiation angle is changed from 45.0° to 45.6°, the topological charge of the SH vortex beam is changed from 976 to 964, and the conversion efficiency is slightly changed from to .
To conclude, we have proposed, in theory, a novel scheme to generate frequency-doubled vortex beams from micro-ring resonators. The material used in the theoretical model is a radially polled micro-ring resonator and the phase-matching diagram of the SHG process is of the nonlinear Cherenkov-type configuration. By analyzing the nonlinear polarization wave in the micro-ring resonator, we can determine the TC of the emitted SH vortex beam, which is the difference between twice the TC carried by the FW and the number of periodically inverted domain structures. The intensity and phase distributions of the emitted vortex beam are simulated by considering the dipoles around the micron-ring, and the emitted SH wave from the dipoles is a coherent superposition determined by the Cherenkov-type phase-matching diagram. In addition, we estimate the conversion efficiencies with different poling periods, which are mainly determined by the radiation angle related phase mismatch in the longitudinal direction and the effective nonlinear coefficient. The work presented in this Letter is a theoretical work. Regarding the recent developments in the fabrication of micro-ring resonators with a high quality factor up to [34], as well as the realization of a ferroelectric domain structure with a period being hundreds of nanometers in MgO:LN[35], it is possible to verify the proposed scheme in experiments. The vortex emitter based on nonlinear frequency conversion can not only work at the new wavelength, but also manipulate the topological charge of the emitted vortex beam by design of the ferroelectric domain structure. Further work can be extended to other nonlinear integrated photonic devices, such as terahertz vortex emitters based on difference frequency generation, and wavelength tunable vortex beam emitters exploiting frequency down conversions.
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Chen Lin, Yan Chen, Xiaoyang Li, Lei Yang, Rui Ni, Gang Zhao, Yong Zhang, Xiaopeng Hu, Shining Zhu. Frequency-doubled vortex beam emitter based on nonlinear Cherenkov radiation[J]. Chinese Optics Letters, 2020, 18(7): 071902.