Generation of vector beams in planar photonic crystal cavities with multiple missing-hole defects

We propose a novel method to generate vector beams in planar photonic crystal cavities with multiple missing-hole defects. Simulating the resonant modes in the cavities, we observe that the optical fields in each defect have different phase and polarization state distributions, which promise the compositions of vector beams by the scattered light from the defects. The far-field radiation patterns of the cavity modes calculated via the Sommerfeld diffraction theory present vector beams possessing hollow intensity profiles and polarization singularities. In addition, the extraction efficiencies of the vector beams from the cavities could be improved by modifying the air-holes surrounding the defects. This planar photonic crystal cavity-based vector beam generator may provide useful insights for the on-chip controlling of vector beams in their propagations and interactions with matter.


Introduction
Vector beams, with spatially inhomogeneous polarization states [1], have attracted growing attentions due to their distinctive properties, including tight-focusing [2][3][4][5][6], orbital angular momentum (OAM) [7][8][9] and controllable nonlinear dynamics [10]. Such researches provide great application potentials of vector beams in surface plasmon excitation [11], superresolution imaging [12], optical manipulation [13], laser micromachining [14], and so on. Approaches based on diffraction optical elements [15][16][17] and interferometric techniques [18,19] have been demonstrated to convert fundamental laser modes to vector beams in free-space. On the other hand, generation of vector beams in photonic chips also attract significant interest, as it expands the applications of vector beams into photonic integrated circuits with improved stability and compact footprint. Already, with the fabrications of in-ring angular gratings [8] or surrounded nano-rods onto micro-ring resonators [20], vector beam emitters have been proposed and fabricated on silicon-on-insulator chips. Moreover, the ring-based emitters enable the manipulations of polarization states and OAM of the vector beams by changing the grating structures. More recently, Schulz et al. proposed an on-chip quantum computation with the OAM qudit states and sorters generated by ring emitters [9].
With their controllable photonic band structures, PPCs allow more reliable manipulations on light phases and polarization states via its Bloch modes, band-edge modes, and defect modes [26][27][28].
For instance, Noda et al. reported on-chip lasers of vector beams relying on the slow-light effect at the band-edge of PPCs [29][30][31]. Generation of vector beams in PPCs is expected to develop vector beam-based on-chip photonic circuits, classical and non-classical light sources, and coupled quantum dot cavity systems. In this paper, we demonstrate the generation of vector beams in PPC cavities by designing multiple missing-hole defects. Different from the vector beams generated from the superposition of Bloch modes of PPCs [29,30], the cavity-based vector beams are bound states of the PPC defects and have much smaller mode volumes in the wavelength-scale. These attributes indicate more reliable applications in micro-laser and on-chip integrations of linear and nonlinear devices of vector beams.

PPC cavity designs and theoretical analysis
We study the generation of vector beams in a triangular-lattice PPC structure with lattice constant of a, air-hole radius of r, slab thickness of d, and slab refractive index of n slab . The defect of the PPC cavity is introduced by symmetrically missing N separated holes with respect to the central air-hole, as shown in Fig. 1(a) for the case of N=6. In PPC cavities, as a result of the in-plane distributed Bragg reflection and the vertical total internal reflection, resonant modes are confined in the missing-hole defects. Moreover, relying on different geometry orientations of the defects, field distributions in each defect are expected to have different polarization directions with respect to the central air-hole. On the other hand, due to the existence of light components with k-vectors in the light cone of the PPC slab, the resonant modes would scatter out via the defects, which is responsible for the main energy-loss. The far-field light E j (x, y, z) scattered from the jth defect can be expressed as ( ) ( ) ( ) (1) where k=2π/λ is the wave number, R=[(x-x 0 ) 2 +(y-y 0 ) 2 +z 2 ] 1/2 and j=1, 2, …, N. A j (x 0 ,y 0 ,0)∝ E j (x 0 , y 0 , 0)µ j , where E j (x 0 , y 0 , 0) is the electric field of the near-field resonant mode in the jth defect and µ j is the unit vector describing the polarization direction. Based on the superposition principle of optical fields, the coherent combination of E j (x, y, z) scattered from the N defects will present a complex field pattern, which depends on near fields in each defect and their polarization states, as illustrated in Fig. 1(b).
To verify the formation of vector beams from the overall scattered light of resonant modes in the PPC cavity, we assign six dipole oscillations with different polarization states in the center of the cavity defects, as indicated by the red dots in Fig. 1

Numerical simulations and discussions
We then demonstrate the feasibility of above analysis by simulating resonant modes of the multi-defect PPC cavities with a three-dimensional finite element mode technique (COMSOL Multiphysics). To provide the reliability of simulation results in realistic applications, the PPC cavities are designed for a gallium arsenide (GaAs) slab with embedded indium gallium arsenide (InGaAs) quantum dots (QDs), which can provide an efficient internal light source [22] and a platform for studying the cavity quantum electrodynamics (QED) [25]. The thickness and refractive index of the GaAs slab are chosen as d=220 nm and n slab =3.46, respectively. The PPC structure has a lattice constant of a=280 nm and an air-hole radius of r = 0.3a.

Near-field resonant modes in a PPC cavity with N=6
Simulating a PPC cavity with six point-defects, we obtain two resonant modes at  respectively. In the two modes, the phase distributions of LH and RH components satisfy δ 2 =-δ 1 , and δ 1 = mϕ, where m and ϕ are the topological charge and the azimuthal angle to describe a helical phase structure, respectively. Therefore, the superposed field of the two components is derived as E=A 0 (cosδ ê x +sinδ ê y ), which implies their superposition has vector polarization states [35].

Generation of vector beams in a PPC cavity with N=6
With above analysis and the results shown in Figs where E 0 (x 0 , y 0 , 0) is the near-field in the PPC plane, ∇=∂/∂xê x +∂/∂yê y +∂/∂zê z , and n is the unit vector in z direction. Figure 3 shows the calculated far-fields in a transverse plane above the PPC slab, where the top and bottom images correspond to those of Mode1 and Mode2, respectively.
Modulated by the symmetric polarization states and step-phase structures of the two resonant modes, the far-field emission patterns present ring-and petal-like intensity distributions, as shown in Figs. 3(a1) and 3(a2). The hollow-core intensity indicates the existence of a polarization singularity at the center [19]. respectively. The topological charge of 2 for the second-order vector beam results in a larger hollow-core in the intensity profile than that of the first-order vector beam.
We then study the polarization states of the generated vector beams by calculating their Stokes parameters [35]. The four Stokes parameters are defined as where, E x and E y are the two polarization components along x-and y-axis, and the symbol "*" denotes the complex conjugation operation. With the Stokes parameters, an arbitrary polarization state can be described using the polarization ellipse. The orientation of the major axis ψ and ellipticity tanχ of the ellipse are governed by The ellipticity tanχ=0, +1 and -1 correspond to linear polarization, RH and LH circular polarization components, respectively.  Fig. 3(c2), the ellipticity distributions are discontinuous in the center region. The values of these discontinuous points have no physical meaning because the ellipticity of the polarization ellipse for the singularity is undefined. The polarization direction in Fig. 3(c1) is azimuthally varying, and that in Fig. 3(c2) has azimuthal and radial dependences, which shows first-and second-order vector character [3]. These results are consistent with results shown in Figs. 3(b1) and 3(b2).

Improvement of extraction efficiency in a PPC cavity with N=6
To implement the applications of the generated vector beams, such as in nano-lasers, single-photon sources, and cavity QED, it is desired to design the PPC cavity to have a moderately high extraction efficiency of the vector beams. Inspired by the improvement of off-chip coupling efficiency of cavity modes with perturbed air-holes [36,37], we enlarge the nearest neighbor air-holes around the defects. Here, the extraction efficiency η is defined as the ratio of the collected optical power in the far-field to the total power of the cavity near-field. The perturbed PPC cavity is schematically shown in Fig. 4(a) with the radius (r′) of the perturbed air-holes increased from 0.3a to 0.32a, which are denoted in red color. Figure 4(b) shows the near-field distribution of the resonant Mode1 for the perturbed cavity. Comparing with the result shown in

Vector beams generated in PPC cavities with N=3 and 4
The generation of vector beams are also studied in multi-defect PPC cavities with N=3 and 4, as shown in Figs. 5(a) and 5(b), respectively. Where, the top and bottom images depict the near-field resonant modes and the corresponding far-field radiation patterns. Similar to the case of N=6, each PPC cavity presents two resonant modes with spatially dependent polarization distributions. In the defects, the optical fields have azimuthal or radial polarizations for the two resonant modes. Modulated by the polarizations of the near-field modes, hollow-core intensity distributions are observed in the far-fields. However, because of the lower symmetry for the cases of N=3 and 4, the far-field radiation have non-circular mode patterns. The vector characters of the far-fields are also confirmed by analyzing the phase distributions of LH and RH circular polarization components and the Stokes polarization parameters.

Conclusions
In summary, we have demonstrated the generation of vector beams in PPCs by symmetrically missing multiple air-holes. With geometric symmetry of the defects relative to the cavity center, the resonant modes of PPC cavities show symmetric polarization states with respect to the central air-hole, and the vector characters are confirmed through the analysis of phase distributions of the decomposed LH and RH circular polarization components. Modulated with the polarization states and phase structures of the cavity modes, the far-field radiation patterns of the scattered light present vector beams with hollow-core intensity profiles. The vector characters of the far-field radiation patterns are analyzed carefully using the Stokes polarization parameters. In addition, the extraction efficiency of the vector beams from the PPC cavities could be improved by perturbing the air-holes around the defects. Comparing with other chip-integrated vector beam generators [8,9,20], the higher Q/V mode ratios of the PPC cavities promise the applications in on-chip