Integrated vortex beam emitter in the THz frequency range: design and simulation

Compact vortex beam emitters have emerged as new light sources for novel applications in areas including spectroscopy, particle manipulation and communications. Reported devices depend on linear optical phenomena and emit light in the near-infrared regime. Here, we propose and numerically evaluate a nonlinear vortex beam emitter that functions in the THz regime. The design utilises a LiNbO3 microring, a Si microdisk, and a Au second-order top grating to convert waveguide-coupled infrared light into a freely propagating THz beam via difference-frequency generation. The output beam carries a topological charge that is tuneable with input wavelengths. Three devices are evaluated in a test frequency range from 9 THz to 13.5 THz, and the topological charge can change from -2 to 4. A frequency shift accompanies the change in the topological charge, and its magnitude depends on the planar dimensions of the emitter. Th is is the au tho r’s pe er re vie we d, ac ce pte d m an us cri pt. H ow ev er , th e o nli ne ve rsi on of re co rd w ill be di ffe re nt fro m thi s v er sio n o nc e i t h as be en co py ed ite d a nd ty pe se t. PL EA SE C IT E TH IS A RT IC LE A S DO I: 1 0.1 06 3/5 .00 10 54 6

advances in THz research, e.g. THz wireless communications and THz-driven electron acceleration, have made the development of a compact, surface-emitting THz vortex beam source highly desirable. 18,19 This work introduces a nonlinear, tuneable vortex beam emitter that could be useful in many of these emerging applications. By utilising resonance-enhanced difference-frequency generation, the device retains the compact size of previously reported vortex beam emitters whilst also extending the functional regime into the THz. The integration of photonics and THz research has witnessed the use of nano-and microstructures at every stage of THz generation, propagation, modulation and detection. [20][21][22] This work demonstrates a new approach of the integration that allows for creating a tuneable topological charge in the generation of THz light. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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Page 4 of 18 cross section outside the microdisk, with the edges of the Si microdisk and the Au grating shown in dashed lines. Figure 1 illustrates the microstructures of the THz vortex beam emitter. The emitter is a waveguidecoupled, multi-layered microdisk that converts two infrared waveguide modes into free-space THz light through the nonlinear process of difference-frequency generation. LiNbO 3 (lithium niobate) is chosen as the waveguide material because of its high nonlinear coefficient [23][24][25][26] and increasing importance in integrated photonic circuits. 27,28 The waveguides have two segments, a straight bus waveguide and a microring resonator. They are in the same plane, and have the same rectangular cross section that is 1 µm in width and 800 nm in height. The microring has a bend radius, defined as being from the inner boundary of the waveguide, of 18.68 µm, and a gap with the straight waveguide of 200 nm. The LiNbO 3 waveguides are embedded in a SiO 2 cladding layer, a common configuration of lithium niobate on insulator (LNOI) platforms. 27,28 The thickness of the cladding layer is 4.6 µm in total and 2 µm beneath the LiNbO 3 waveguides. The bottom of the SiO 2 layer is fully covered by an unstructured Au base film, which has a thickness of 1 µm. A Si microdisk and a Au angular grating, which are concentric to the The THz emitter is pumped by two infrared light beams with angular frequencies of ω 1 and ω 2 . The infrared light is evanescently coupled from the straight waveguide into the ring waveguide, where it forms whispering gallery mode resonances. Due to the resonance enhancement, the microring has a levitated effective nonlinear efficiency 29,30 in the generation of the THz light ω THz , where ω THz = ω 1ω 2 . It is worth noting that, the microring configuration can be used to enhance many different nonlinear processes, among which difference-frequency generation is the only second-order nonlinear process that generates THz light. As shown below, similar to the pump light circulating along the LiNbO 3 microring, the THz light can form a whispering gallery mode that traces the circumference of the Si This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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microdisk. The THz light is further confined by the Au angular grating at the top and the unstructured Au base film at the bottom, and leaky radiation from this subwavelength-thick cavity forms the output beam. As the unstructured Au base film, a closed end of the leaky cavity, blocks any emission downwards, only the radiation above the emitter is studied.
A key aspect of the design in the generation of topological charge is the Au top grating. Top gratings have been used on a variety of light sources, such as light-emitting diodes (LEDs) 31 , vertical-cavity surface-emitting lasers (VCSELs) 32 and quantum cascade lasers (QCLs). 33,34 These gratings can control both the local optical density of states and the flow of light, and are adopted mainly for improving the efficiency and the directionality of light emission. More recently, second-order surface gratings with rotational symmetry have been used to create vortex beams in linear light emitters. [35][36][37][38][39][40] In contrast to these recent works, which are all in the near-infrared regime, this work adopts a different type of surface grating for use in the THz regime: it is metallic and covers a large area of the top surface of the light emitter. This approach benefits from the low ohmic loss in metals in the THz regime as compared to the near-infrared regime. It can also suppress direct light emission (i.e. emission that bypasses any interactions with a top grating) that can hinder the generation of pure vortex modes.
The performance of the device was numerically evaluated by using a commercial FDTD (finitedifference time-domain) solver (Lumerical FDTD Solutions). For linear properties, the permittivity of the materials, including LiNbO 3 , SiO 2 , Si and Au, was fitted based on experimental values. 41,42 The fitting covered the whole range from the infrared to the THz, and satisfied the Kramers-Kronig relations as required by the FDTD method (see Table S1 in the Supplementary Material for representative values). LiNbO 3 was the only nonlinear material in the device, and its nonlinear coefficient 23-26 was set as d 33 = 170 pm/V. The two input infrared waves had equal power and were launched 40 µm away from the 200 nm gap (i.e. the location where the microring and the straight waveguide were the closest). Both waves were polarised in the same direction (TM polarisation, magnetic field parallel to the substrate) in the z-cut LiNbO 3 waveguides, in order to access this coefficient. Although this polarisation configuration forbids birefringence phase matching, quasi-phase matching could be used in future This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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Page 6 of 18 works to boost nonlinear efficiency (e.g. via decorating the microring with nanostructures). To elucidate the mechanism of the creation of topological charge, the results in the main text were obtained by eliminating the nonlinear generation in the straight waveguide. This modification enables establishing a clear link between the near-field distribution and the far-field radiation, and results without this modification are presented in Fig. S2 in the Supplementary Material as a comparison.   43 with the material loss attributed as the This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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main limiting factor. The average transmission is -8.3 dB at λ 1 and λ 2 , and the average coupling efficiency is 85% under the assumption of zero scattering loss at the coupling section. 29 The wavelengths of the two input light beams are set at these resonances, in order to benefit from the resonance enhancement of nonlinear effects in the microring.

IV. Generation of topological charges
The input light is confined by the LiNbO 3 waveguide and the SiO 2 embedding layer, which dictate the input properties of the device. In contrast, the THz light generated inside the LiNbO 3 microring has a much longer wavelength (~25 μm in free space) and penetrates into the Si microdisk. As shown in where l is the topological charge of the radiated light and q is the number of the grating elements. These two equations lead to which provides the basis for controlling the topological charge l in this work. For a given device where q is fixed, l can be tuned by adjusting the value of m 1m 2 . As indicated in Fig. 2b, this work chooses to demonstrate the tuning of l by fixing m 1 (i.e. fixing λ 1 ) and changing m 2 (i.e. changing λ 2 ).
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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Page 8 of 18 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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Page 10 of 18 A profound contrast is observed between the non-vortex beam (Fig. 4a) and the vortex beams (Figs. 4b-4d): both the phase and the amplitude are twisted in the latter, while no such feature is discernible in the former. The small features very close to the centre of the phase maps of Fig. 4a are attributed to numerical errors, as the local field intensity is almost zero. It is worth noting that the doughnut-shaped amplitude profile seen in Fig. 4a does not imply the existence of a finite topological charge, and similar features have been reported previously in non-vortex beams produced by linear vortex beam emitters. 36 In contrast to Fig. 4a, the beams in Figs. 4b-4d all carry a finite value of topological charge l. The value of l is related to the twist of the fields and can be retrieved from the maps by using two different methods.
The first method is to trace the rotation of the field in a xy plane: along a circle centred on the beam axis, the absolute value |l| equals the cycle number of the phase or the amplitude, and the sign of l determines the direction (either clockwise or anti-clockwise) that the phase increases. The second method is to trace the rotation of the field along the z direction: the field rotates by 2π/|l| for a distance of λ, and the sign of l determines the rotation direction for both the phase and the amplitude.

V. Output spectra of three different designs
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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Page 11 of 18 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. Figure 5a shows the output spectra of the device in the frequency range from 9 THz to 13.5 THz. As l changes from 0 to 3, the peak frequency changes by 3.61 THz, from 9.58 THz at l = 0 to 13.19 THz at l = 3. This change originates from the energy conservation in the difference-frequency generation and follows the change of λ 2 . The output power also depends on l. With both input light set at a test value of 10 W, the output reaches the highest value of 2.0 × 10 -5 W at l = 2, and drops to 14% of this value at l = 0 and 8% at l = 3. This change in output power is directly related to the change in frequency, as the latter affects key characteristics such as light confinement of the microdisk and phase matching of the nonlinear process. All the peaks show a finite width, and some have small but discernible side bands.

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These features are artefacts associated with the FDTD method; both the peak width and the sidebands become smaller with increasing the simulation time (i.e. with waves better approximating perfect, monochromatic waves).
For this design architecture, a change in output frequency always accompanies a change in the topological charge due to the difference-frequency generation. Nevertheless, the magnitude of the frequency change can be controlled by modifying the planar size of the design. As long as the pump wavelengths (λ 1 and λ 2 , both ~1.5 μm in the current design) are significantly larger than the FSR (~10 nm in the current design), the frequency change Δf that accompanies a unit change in l (i.e. Δl = ±1) is approximately Δf ≈ c × FSR / (λ 1 × λ 2 ) ≈ c × FSR / λ 2 (4) where c is the speed of light, and λ is the average of λ 1 and λ 2 . For a microring resonator, the FSR is inversely proportional to the round trip length along the microring. 29 If the radius r is significantly larger than the width of the waveguide, the FSR can be approximated as FSR ≈ λ 2 / (n g × 2 × π × r) (5) where n g is the group velocity. This leads to the conclusion that by adjusting the radius of the microring, Δf can be controlled as Δf ≈ c / (n g × 2 × π × r) The value of Δf scales inversely with the radius of the microring r.
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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Page 13 of 18 To numerically verify this conclusion, the planar dimensions of the emitter are modified and the output spectra are shown in Figs. 5b and 5c. From the current design (Fig. 5a), the radii of the microring, defined from the middle of the waveguide, and the Si microdisk are increased by 50% in Fig. 5b, reaching 28.77 μm and 33 μm, respectively. For the third design in Fig. 5c, the increase is 100%, and the radii of the microring and the Si microdisk are 38.36 μm and 44 μm, respectively. In order to create similar values of l in the same frequency range for the three designs, the element number of the Au grating is increased from q =8 in Fig. 5a to 13 and 18 in Figs. 5b and 5c, respectively. All the other dimensions are the same for these three designs.
For the two new designs, the output spectra and the topological charges are calculated using the same methods discussed above (see Fig. S3 in the Supplementary Material for transmission spectra). Figure   5 compares the three designs in the same frequency range from 9 THz to 13.5 THz. The number of peaks is 4, 5 and 7 in Figs. 5a, 5b and 5c, respectively, increasing with the planar size of the emitter.
The frequency difference between adjacent peaks Δf shows the opposite trend of change, which is 1.20

VI. Conclusion
To conclude, we have demonstrated a THz vortex beam emitter via numerical simulation. The emitter consists of a LiNbO 3 microring for difference-frequency generation, a Si microdisk for near-field confinement of the generated THz, and a Au second-order top grating for creation of topological charge.
The device is pumped by two infrared light beams from a coupling waveguide, and emits THz light vertically into free space. The value of the topological charge can be tuned by changing the wavelengths of the incident light, and ranges from -2 to 4 in the simulated frequency range from 9 THz to 13.5 THz This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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Page 14 of 18 for one of the three designs. The output spectra, in particular the frequency change that is associated with the change in topological charge, can be adjusted by tuning the planar dimensions of the emitter.
As a proof-of-principle demonstration, this work concentrates on the topological charge and the frequency shift, while parameters such as conversion efficiency, including its balance among multiple output spectra peaks, could be further improved by exploring the vertical dimensions (e.g. via creating high-quality THz whispering-gallery modes). 46 The emitter has a planar dimension that is comparable to its functional wavelengths, and has the capability to impart a tuneable topological charge on a freely propagating THz light beam. These features can prove useful in a range of emerging THz applications such as THz wireless communications and electron acceleration.

Supplementary material
See the supplementary material for representative permittivity values used in the numerical simulation, planar dimensions of the smallest device, influence of the straight waveguide on THz generation, and the linear transmission spectra of the two larger devices.