An optical leaky wave antenna with silicon perturbations for electronic control

An optical leaky wave antenna (OLWA) is a device that radiates a light wave into the surrounding space from a leaky wave (LW) guided mode or receives optical power from the surrounding space into a guided optical mode. In this work, we propose and provide a 3D analysis of a novel CMOS compatible OLWA made of a silicon nitride (Si3N4) waveguide comprising periodic silicon perturbations which allow electronic tuning capability. The analysis presented here includes the effect of the number of semiconductor perturbations, the antenna radiation pattern and directivity. We show that the number of the silicon perturbations has to be large to provide a long radiating section required to achieve radiation with high directivity. In other words, the proposed structure allows for a very narrow-beam radiation. Preliminary results are confirmed by exploiting leaky wave and antenna array factor theory, as well as verified by means of two full-wave simulators (HFSS and COMSOL). Our purpose is to ultimately use PIN junctions as building blocks for each silicon implantation for the electronic control of the radiation. In particular, the electronic tunability of the optical parameters of silicon (such as refractive index and absorption coefficient) via current injection renders itself the ideal platform for optical antennas that can facilitate electronic beam control, and boost the efficiency of optoelectronic devices such as light-emitting diodes, lasers and solar cells, and bio-chemical sensors.


INTRODUCTION AND STATEMENT OF THE PROBLEM
An optical leaky wave antenna (OLWA) is a device that radiates a light wave into the surrounding space from a leaky wave (LW) guided mode or, vice-versa, it couples receiving optical power from the surrounding space into a guided optical mode. Optical antennas have the capabilities to enhance the interaction between light and matter, and thus have the potential to boost the efficiency of optoelectronic devices such as light-emitting diodes, lasers and solar cells, and bio-chemical sensors [1][2][3].
In [4], we have proposed the design of a dielectric (silicon nitride) OLWA with periodic semiconductor (silicon) perturbations, capable of producing narrow beam radiation, with electronic tuning capability. The optical antenna cases studied up to date mostly focus on the local field distribution control of the device. Those kinds of antennas are fabricated in a sub-wavelength dimension and some exceptional physical phenomena, such as super-resolution effect and near-field enhancement [5], are realizable due to the resonant modes. In the optoelectronics domain, silicon-on-insulator (SOI) devices have been widely utilized recently to deliver chip scale active and passive photonic devices such as amplifiers, switches and modulators with potential optoelectronic integration on the same platform [6][7]. The silicon on insulator (SOI) based planar waveguide provides tight mode confinement due to a large index contrast, which is essential for miniaturization, and simultaneous low loss operation [8]. In particular, the use of silicon offers the electronic tunability for optical parameters of silicon (such as refractive index and absorption coefficient) via current injection. Combination of optical antennas with silicon perturbation can facilitate electronic beam control for OLWAs [9][10][11]. Hence, the OLWA can function as a passive transmitter or receiver as well as beam steering device [4].
The OLWA proposed here radiates a narrow beam because a leaky wave (LW) with low attenuation constant is excited at one end of the dielectric waveguide with perturbations. Accordingly, the periodicity of the silicon perturbations of the waveguide produces a spatial harmonic of the guided mode in the visible region. Here we show that pointing angle, beam-width, and operational frequency are all related to the LW complex wavenumber, whose value depends on the amount of silicon perturbations in the waveguide [4]. Also, the propagation constant and the attenuation coefficient of the LW in the periodic structure are extracted from full-wave simulations. The number of silicon perturbations and their effect on the radiation pattern is investigated to obtain a directive compact emitter.
The structure of the paper is as follows. The structure of the proposed OLWA is presented in Sec. 2. Then, the two theoretical models used to predict the radiation pattern of the OLWA are summarized in Sec. 3. In Sec. 4 we describe the performance of the OLWA, and we observe the radiation pattern tunability due to the usage of electronic control of PIN junctions in Sec. 5.

PROPOSED STRUCTURE AND FULL-WAVE SIMULATION RESULTS
The first difficulty in the 3D implementation arises from the design of a single mode rectangular dielectric waveguide (it is well known that the fundamental mode cannot be avoided in this kind of waveguides); we adopt approximate formulation for optical fibers on the normalized frequency V to have a starting point for the design of the wave guiding structure. Also, the waveguide cross section cannot be too small otherwise it cannot confine most of the energy injected into the structure in the core. Taking care of all the previous conditions, the proposed structure is reported in Fig. 1. The waveguide is made of silicon nitride (Si 3 N 4 ) with refractive index equal to 1.67, is positioned along the x direction and is 49.285 μm long; it has a square cross section of w w× , with 1 μm w = in the yz plane. The silica glass computational domain has a refractive index equal to 1.45, is 49.285 μm long along the x direction and 6 μm in both the y and z directions. To provide direct radiation, we choose to utilize 35 N = elements, made of silicon, with refractive index equal to 3.48, positioned periodically along the x direction 14.03175 μm away from the antenna's input side (on the left in Fig. 1). The perturbation is periodic along x with period d (each element has a length equal to d/2), has a height of

THEORETICAL MODELS
Here we adapt two theoretical models (equivalent aperture model and array factor model) to predict the radiation pattern of the OLWA and to provide guidance for further design modifications.

Floquet theory
As previously mentioned in Sec. 2, the guided mode excited by the incoming field on the left side of the waveguide becomes leaky in correspondence of the periodic silicon perturbations. Such a mode decays exponentially along the structure, even for a lossless structure, and a leakage phenomenon (radiation) takes place.
The electric field is polarized along z and travels along the x direction. The z-component of the guided electric field satisfies the quasi-periodic property ( where d is the period of the periodic structure and ,0 x k i β α = + is the wavenumber along the propagation direction. Here, β and α are the phase and attenuation constants, respectively, of the LW. Note that, before introducing the perturbations, the original guided mode has a phase propagation constant very close to β , whereas the attenuation constant of the waveguide would be much smaller than α (without perturbation the attenuation is caused only by material losses and scattering by irregularities of the structure). Instead, with perturbations, the value of α can be significantly large because of radiation, though to have a directive OLWA this value has to be maintained low. Accordingly, in terms of a Fourier series expansion, the electric field at any place along the periodic structure can be represented as the superposition of Floquet spatial harmonics where , x n k is the Floquet wavenumber, n is the order of the Floquet spatial harmonic, and ( ) y n E is the weight of the nth harmonic [12][13][14]. Each Floquet wave number , has a value very close to the one of the bound mode in the unperturbed waveguide, and thus it is not radiating.

Equivalent aperture (EA) model
Based on the Floquet field expansion (1), the electric field along the periodic structure can be described as a superposition of field terms, and only the 1 n = − is relevant for radiation. Therefore, its expression evaluated on the "aperture" (i.e, a section from where we assume radiation is propagating away [15]) is where L is the total length of the silicon perturbation, and that in the case of large L α (either when the antenna is long, or when the attenuation constant is large) can be further simplified.

Array factor (AF) model
In the array factor method instead, the periodic perturbation can be considered as an array of identical scatterers. The farfield pattern is then obtained by multiplying the pattern of the single element (quasi isotropic) and the array factor (AF) of the array (very angle-selective) [16]. As usual for directive antennas the total far field radiation pattern is approximated as φ . For the OLWA under consideration, the magnitude of the array factor where we have assumed that L Nd = . An analogous procedure was applied in [17] to find the directive radiation pattern of a narrow slit in a corrugated silver film.

ANTENNA RADIATION CAPABILITIES
The accuracy of theoretical models is compared with full-wave numerical simulations. In particular we use 3D modeling by using HFSS and COMSOL as described below.

Guiding properties and scattering parameters
The maps of the electric field magnitude at the input side and along a xy plane section in the center of the waveguide are shown in Fig. 2; the former shows good power confinement within the core, whereas the latter shows that the leaky wave slowly decays along the silicon perturbations. We also computed the scattering parameters of the antenna, considering two waveports ( 4 μm 4 μm × ) located at the beginning and at the end of the waveguide. The reflection coefficient 11 15 dB S = − shows that a negligible amount of the power injected into the antenna is reflected back. The transmission coefficient 12 21 dB S = − shows that a radiation phenomenon is taking place, since the dielectric materials are considered to have negligible losses. In case power is still present at the end of the waveguide, a higher number of silicon perturbations could be required to radiate all the power injected into the antenna.

Extraction of the leaky wave parameters
As stated in Sec. 3.1, the excited leaky mode decays exponentially along the structure, even for a lossless structure, and a leakage phenomenon (radiation) takes place.
To better extract the leaky wave theory parameters 1 β − and α we simulate a 70 elements long perturbation by resorting to PEC/PMC symmetry planes at the right end side of the waveguide. Indeed, we know that the local field magnitude decays exponentially along the perturbed waveguide with attenuation constant .
α Moreover, its phase shift is linearly dependent to the propagation constant, as can be seen looking at the phase term 1 x β − . We perform a linear fitting of the extracted data reported in Fig. 3 (more precise estimation would however be based on finding the complex wavenumber in a fully periodic waveguide). The leaky wave theory parameters are ) (bottom waveguide) extracted by the decaying field along the perturbations shown in Fig. 3. This result proves that the field decays in the same way in every section of the waveguide.
We then compare in Fig. 4 the decaying field in the bottom side of the waveguide from HFSS with the one obtained by a COMSOL simulation. The leaky wave theory parameters from COMSOL are

Radiation pattern
The radiation patterns from the HFSS and COMSOL simulations from previous section are then compared to the theoretical fields with the extracted parameters from HFSS field decay (bottom waveguide), with leaky wave parameters A side-lobe is present in the full-wave simulations (not predicted by theory); its presence is due to a certain phase coherence at the beginning of the silicon perturbations. This coherence can be explained by the fact that the first few elements of the perturbation form a subarray that radiates at a certain maximum direction (i.e. the first side-lobe level, around 53 −°) different from the maximum radiation direction of the remaining part of the perturbation (main-lobe, around °−93 ).
To understand how directive the radiation is, we look at the 3 dB beam-width, defined as 3dB 3dB 3dB , where F is defined as in (4). By looking at either the simulated HFSS or COMSOL result,

Reducing the number of silicon perturbations to have a compact emitter
We decrease the number of silicon perturbations and we compute and compare the radiation patterns in Fig. 6. It can be observed that the beam widens by decreasing the number of silicon elements (to remark that a long radiating section is needed to obtain high directivity). As such, we computed the 3 dB beam width for the three shown cases, and we got about

ELECTRONIC CONTROL OF THE RADIATION
The use of silicon and SOI technology provides opportunities for opto-electronic integration. As an indirect bandgap material, silicon is highly transparent for optical communication wavelengths. The use of free carrier plasma dispersion effect [18] can induce changes in refractive index and absorption coefficient, which are two critical parameters in OLWA radiation pattern. Up to date plenty of work has been conducted to utilize this nonlinear effect to realize modulation in electro-optic device [7,19]. Here we propose to use silicon perturbations with PIN junction diode structure in OLWAs to control the radiation pattern. Figure 7 illustrates the figurative representation of a single silicon perturbation cell with a PIN diode structure. To ensure low loss operation the intrinsic region is placed under the silicon nitride waveguide region. In order to realize the electronic tuning of OLWAs, the carrier density in the intrinsic region is changed by current injection. . The ohmic contact is assumed at anode and cathode sides. The doping profile of each region is defined as an abrupt step junction for simplicity. The concentration dependent mobility of electrons and holes is calculated based on empirical equations given in [20]. The current and the ambipolar lifetime of the PIN diode are calculated to derive more accurate carrier distribution in the intrinsic region, furthermore, to achieve better estimation of the possible refractive index change and attenuation. The relationship of PIN current with respect to bias voltage is plotted in Fig. 8   We then consider the current flow due to electron-hole recombination in I region. The current density (n for electrons and p for holes) is given by [20] , , 0 , and p w is the width of intrinsic region in the z direction (as in Fig. 1). Here we assume that the carrier concentration throughout the intrinsic region is approximately constant and diffusion current can be neglected as well.

PIN junction modeling
The carrier distribution in intrinsic region can be estimated based on equation (7), from simulated current density and lifetime results. In Fig. 9, electrons and holes distribution can both reach 19 3 10 cm − at current of However, in this last case, Auger processes, which is neglected here, should be taken into account for accurate estimation [21].

Electronic control capabilities
From previous analyses, we know that the radiation pattern at the transverse direction highly depends on the attenuation coefficient of the antenna and the effective index. According to Fig. 9, if we set the doping concentration to 18 3 10 cm , respectively. This carrier density can alter both the real and imaginary parts of the Si refractive index (unitless) as described by the Drude's model [22] ( ) ( ) The case without carrier injection had the following parameters: ). It can be observed that the values of 1 β − in absence and in presence of carrier injection follow the approximate model of small perturbations mentioned above. Regarding α however, the linear fitting method is very sensible and does not allow the estimation of small variations: future studies will involve the estimation based on finding the complex wavenumber in a fully periodic waveguide.
Practically, as stated in the previous section, the density of electrons and holes in Si can be increased up to 19 3 10 cm − in optoelectronic devices via current injection or photo generation before Auger process starts to mitigate the device performance [23]. Future studies will take this process into account, trying to reach concentrations up to 20 3 10 cm − to facilitate more pronounced tunability. Also, future works will involve amplifying weak perturbations in the optical domain by placing perturbations into optical resonators so that these perturbations will be experienced by the optical field during multiple roundtrips [24]. Indeed, it has been already demonstrated theoretically and experimentally that weak perturbations created via injection of carriers into silicon optical resonators alter the resonance wavelength and the quality factor of the resonator and provide modulation of the optical signals [24]. Similarly, by integrating OLWAs into a racetrack or Fabry-Perot integrated cavities, it is expected to amplify the variation in the radiation pattern of an amount which is proportional to the quality factor of the cavity. Future research should be dedicated to quantify the perturbations in the radiation pattern in resonators with practical geometries.

CONCLUSION
We have shown the performance of an optical leaky wave antenna (OLWA) able to produce very directive radiation at 193 THz. With a periodic set of PIN junction perturbation little control is exerted on the radiated beam. Enhanced control will be studied in the future by inserting the OLWA into a ring resonator.