1. Introduction

Manipulating the polarization state of light is of great interest in many applications, such as polarization controllers, life science microscopy, and displays [1]. Conventional polarization controllers are designed using anisotropic or chiral media; however, there are rigid thickness limitations and the approach necessitates quite bulky configurations [2]. In contrast, surface plasmonic resonance in noble metals has been utilized to develop miniaturized devices for polarization manipulation, such as planar chiral structures [3–5] and asymmetric microstructures [6–9]. For example, ultrathin chiral structures (total thickness Δ < λ/300) manipulating polarization conversion can be obtained by a combination of dielectric spacers and a metallic split-ring [10]. However, the substantial intrinsic loss and radiative loss of metallic structures reduces the transmitted power to such an extent that the potential applications of this technology are significantly limited. One possible means of overcoming the problems associated with these dissipative losses is to rely on low-loss dielectric structures, which utilize the localized resonance modes of high-index dielectric nanostructures to manipulate polarization [11]. Considerable efforts have been made recently to expand the applications of high-index all-dielectric sub-wavelength nanoparticles based on localized electric and magnetic resonance modes [12–16]. In particular, as a powerful concept for implementing mathematical operators based on co/cross-polarized light, optical all-dielectric metasurfaces have gained significant attention in recent years [17–20]. However, the development of metadevices based on all-dielectric building blocks is difficult using the conventional milling method. Fortunately, continuous high-index dielectric films (i.e., slab waveguides) with periodic perforated structures in one or two dimensions exhibit interesting optical properties due to the guided-mode resonance originating from the complex interaction between the in-plane guided waves and external radiations. Moreover, they can be fabricated simply with the conventional milling method [21]. Numerous applications of these structures have been demonstrated, such as optical filters [22, 23], perfect absorbers [24], and grating couplers [25]. However, optical polarization conversion based on guided-mode resonance has been scarcely reported thus far. Here, we investigate the polarization conversion of an all-dielectric silicon film with a square array of perforated L-shaped holes both theoretically and numerically. The broken symmetry of the L-shaped holes has the effect of dividing the degenerate guided-mode resonances of symmetrical square arrays of circular holes into four nondegenerate resonances under normally incident light due to coupling between the silicon slab waveguide modes and the diffracted waves at the Brillouin-zone boundaries via the localized resonances excited in the dielectric L-shaped holes. The transmitted spectra exhibit four resonance dips at each coupling point; thus, linear-to-linear and linear-to-circular polarization converters are obtained.

2. Material design and calculation

A schematic diagram of the all-dielectric structure composed of a square array of L-shaped holes introduced into a silicon film deposited on a silica substrate is shown in Fig. 1. The thickness t of the silicon film is 80 nm, the armlength l and width w of the L-shaped holes are 400 nm and 100 nm, respectively, and the area of a unit cell is p². The permittivities of the silicon film and silica substrate were set to Palik’s experimental dates and 1.5, respectively. The structure investigated here has a period of p = 600 nm. The ambient environment was assumed to be air with a permittivity of unity. The optical properties of the structure were calculated using the frequency-domain finite element method (FEM). This solver solves the problem for a single frequency at a time. Bloch boundary conditions were applied to mimic the two-dimensional nature of the geometry, and adaptive tetrahedral meshing with a minimum feature resolution of 1 nm was used in the simulations. All the simulations reached proper convergence.

 

Fig. 1 Three-dimensional schematic view of the all-dielectric L-shaped perforated structure. Geometrical parameters of the unitcell are shown in the inset.

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3. Results and discussion

For a plane wave with in-plane wavevectors ${\widehat{\beta }}_{in}$ incident on the proposed perforated slab waveguide, the diffracted surface wave from the periodic hole array propagates at the air–dielectric (silicon) interface. This surface wave has in-plane wavevectors ${\widehat{\beta }}_{in}+\widehat{G}(m,n)$, where $\widehat{G}$denotes the reciprocal lattice vectors of the periodic structures, which can be written as

Figures 2(a) and 2(b) show dispersion diagrams of the L-shaped perforation array for TE- and TM-polarized incident light, respectively. For comparison, we also calculated the dispersion of a circular perforation array with an equivalent value of f for TE- and TM-polarized incident light, as shown in Figs. 2(d) and 2(e), respectively. The transmitted spectra of the L-shaped and circular hole structures at normal incidence (${\widehat{\beta }}_{in}$ = 0) were extracted, and the results are shown in Figs. 2(c) and 2(f), respectively. It is evident that the transmittance dips for the L-shaped hole structure can be divided into the two groups labeled A and B, which correspond to single discrete transmittance dips for the circular hole structure. For the circular hole structure, the transmittance dip at a wavelength of about 1160 nm results from coupling between the TE mode of the waveguide and the diffracted surface waves with the reciprocal lattice vectors $\widehat{G}(1,0)$, which, theoretically, would occur at a wavelength of 1165nm. This dip is four-fold degenerate due to the superimposition of the reciprocal lattice vectors $\widehat{G}(\pm 1,0)$ and $\widehat{G}(0,\pm 1)$, as indicated by the red arrows shown in the inset of Fig. 2(c). However, due to the broken symmetry of the L-shaped holes, the diffracted waves with reciprocal lattice vectors of various directions can excite various localized resonance modes. Thus, the degenerate dip observed for the circular-hole array at a wavelength of 1160 nm splits into four nondegenerate resonance dips at wavelengths of 1115, 1127, 1162 and 1214 nm, which are grouped as A in Fig. 2(c). The localized resonance modes of the L-shaped hole array can be inferred from the electric field distributions shown in Fig. 3 [28, 29], which indicate that these dips are localized symmetric electric dipolar (Fig. 3c), symmetric electric quadrupolar (Fig. 3a), antisymmetric electric dipolar (Fig. 3d) and antisymmetric electric quadrupolar (Fig. 3b) resonances.

 

Fig. 2 Dispersion of an L-shaped perforation array [(a) and (b)] and a circular perforation array with an equivalent value of f [(d) and (e)] for TE- [(a) and (d)] and TM- [(b) and (e)] polarized incident light. The abscissa axes represent in-plane wavevectors k_x = k₀sinθ, where k₀ and θ are wavevectors in vacuum and the angle of incidence, respectively. The x-polarized transmitted spectra of the L-shaped and circular holes at normal incidence are given in (c) and (f), respectively. The red arrows indicate Rayleigh anomalies (RA) in the Si–SiO₂ interface. The inset in (c) represents wavevectors (arrows) in reciprocal space, where lowest-order wavevectors ( ± 1, 0) and (0, ± 1) (red arrows) and high-order wavevectors ( ± 1, ± 1) (blue arrows) are considered.

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Fig. 3 Electric field E_z distributions (a–d) for the four transmitted minima in group A (Fig. 2(c)) at wavelengths of 1115, 1127, 1162, and 1214 nm on the air–silicon surface, respectively. A white dashed line is plotted in (a) to easily distinguish symmetric and antisymmetric electric field distributions.

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For group B, the transmitted spectra are more complex due to the presence of multiple resonance dips. However, comparing the transmitted spectra of group B with those of the circular-hole array shown in Fig. 2(f), it can be seen that the group B transmittance minima can be attributed to hybridized modes between the TE-G(1,1) and TM-G(1,0) guided modes, and Rayleigh anomalies (RAs) in the Si–SiO₂ interface (for the circular-hole structure, these have theoretical values of 910, 895, and 900 nm, respectively).

Deviations from normal incidence (${\widehat{\beta }}_{in}$≠0) alters the in-plane wavevectors ${\widehat{\beta }}_{in}+\widehat{G}(1,0)$, ${\widehat{\beta }}_{in}+\widehat{G}(-1,0)$, ${\widehat{\beta }}_{in}+\widehat{G}(0,1)$, and ${\widehat{\beta }}_{in}+\widehat{G}(0,-1)$ [21, 30-31]. As a result, the degenerate resonance of the circular-hole array converts into four dispersive bands. The two dispersive bands labeled as 1 and 4 in Fig. 2(d) and one of the less dispersive bands labeled as 2 can be excited only by TE-polarized incident light, whereas the other weakly dispersive band labeled as 3 in Fig. 2(e) can be excited only by TM-polarized incident light (despite being a TE mode within the waveguide) [29]. Note that the TE-and TM-polarized incident light should not be confused with the TE and TM waveguide modes. Thus, the highly symmetric circular-hole structure includes three dispersive bands for TE-polarized incident light and one weakly dispersive band for TM-polarized incident light. However, the broken symmetry of the L-shaped holes facilitates coupling between the TE-and TM-polarized waveguide modes (i.e., the TE-polarized waveguide modes can be excited by the TM-polarized incident light and vice versa). Thus, optical polarization rotation can be induced in structures with broken symmetry [6, 32]. This leads to the superposition of TE- and TM-polarized transmission modes in the four dispersive bands of the L-shaped holes array (see Figs. 2(a) and 2(b)).

In general, the cross-coupling between the electric and magnetic resonances excited in the conventional metallic plasmonic structures enhances the cross-polarization transmission and suppresses the co-polarization transmission. As a result, optical polarization conversions can be achieved [33–35]. However, for metallic plasmonic structures, there are some disadvantages. First, the substantial intrinsic loss and radiative loss of metallic structures reduces the transmitted power of polarization conversions. Second, although some broadband and highly efficient optical rotations are achieved in multilayer metallic structures, the conversion efficiency of zero-order transmission in single-layer structures is a challenge. Third, it is difficult to integrate the metallic optical polarization converter into silicon-based photonic devices. Thus, to achieve a silicon-integrated photonic device, an all-dielectric single-layer optical polarization converter is preferred.

Here, the proposed all-dielectric L-shaped perforated structure can facilitate multiple optical polarization conversions through the coupling between TE- and TM-polarized modes in guided-mode resonances. That is, when TM-polarized light is incident on the structure, both the TE- and TM-polarized waveguide modes can be excited due to the scattering of the L-shaped holes with broken symmetry. Further, co-polarization (TM-polarized light) and cross-polarization (TE-polarized light) transmitted light are radiated via the L-shaped holes. The polarized states of the transmitted light can be obtained through superposition of the co-polarization and cross-polarization modes. The optical rotation angle (i.e., the angle between the long axis of the elliptically transmitted polarized state and the incident linearly polarized direction) and the ellipticity (i.e., the ratio of the short to long semiaxes of the polarization ellipse) can be calculated. The co-polarization E_xx and cross-polarization E_yx of transmitted normalized electric fields around the set of resonances labeled as A in Fig. 2(c) are shown on an enlarged scale in Fig. 4(a). Here, four transmitted peaks appear in the cross-polarization E_yx spectrum that correspond to the four transmittance minima in the co-polarization E_xx spectrum. Total transmittance ($T={\left|E_{xx}\right|}^2+{\left|E_{yx}\right|}^2$) is presented in Fig. 4(b), which shows four peaks at wavelengths of 1105.2, 1124.7, 1147.4 and 1209.9 nm with transmittance values of 75.2%, 67.8%, 69.7%, and 64.1%, respectively. The relationship between the E_xx and E_yx spectra leads to optical rotations and changes in ellipticities, as shown in Figs. 5(a) and 5(b), respectively. The optical rotation angles and ellipticities associated with the transmittance peaks at wavelengths of 1105.2, 1124.7, 1147.4, and 1209.9 nm are 11.7, −19.7, 16.86, and −23.9 degrees and 0.385, 0.435, 0.409, and 0.421, respectively.

 

Fig. 4 Co-polarization E_xx and cross-polarization E_yx transmission coefficients (a) and total transmittance (b) around the set of resonances labeled A in Fig. 2(c) on an enlarged scale.

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Fig. 5 Rotation angles (a) and ellipticities (b) of the guided-mode resonances labeled as group A in Fig. 2(c). (c) The polarization states associated with the wavelengths of the four ellipticity peaks (top) and ellipticity minima (bottom) in (b).

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Furthermore, linear-to-circular and linear-to-linear polarization converters [36] can be developed based on the proposed structure. This is illustrated by the polarization states of transmitted light shown at the top of Fig. 5(c) associated with the ellipticity peaks shown in Fig. 5(b) wavelengths of 1111.9, 1126.3, 1155.8, and 1212.6 nm, which have ellipticities of 0.893, 0.902, 0.94, and 0.758 and transmittances of 52.7%, 43.7%, 47.2%, and 38.3%, respectively. As such, these represent linear-to-circular polarization converters. This is further demonstrated by the polarization states of transmitted light shown at the bottom of Fig. 5(c) associated with the ellipticity minima shown in Fig. 5(b) at wavelengths of 1116.5, 1127.8, 1166.4, and 1215.3 nm, which have ellipticities of 0.0016, 0.008, 0.003 and 0.007 and transmittances of 31.1%, 24.7%, 23.9%, and 16.7%, respectively. The corresponding rotation angles of the four polarization states shown at the bottom of Fig. 5(c) are −45, 45.3, −45.1, and 45.5 degrees, respectively. As such, these represent linear-to-linear 45 degree polarization converters. Although a high polarization conversion efficiency could be achieved by employing the plasmonic structures, such structures are limited by fabrication complexity and the fundamental relations of ultrathin film at the near-infrared and visible wavelength range [37]. Our proposed polarization converters composed of continuous dielectric film with perforated L-shaped holes arrays can be simply and easily fabricated. Further, the proposed all-dielectric structures with CMOS-compatibility and easy integrability have great potential for the realization of highly efficient ultra-thin planar wave-based integrated photonics systems.

4. Conclusion

In conclusion, an all-dielectric silicon waveguide perforated by an array of L-shaped holes was proposed and investigated. Due to the broken symmetry of the L-shaped-hole structure, the low-order degenerate guided-mode resonances of symmetric unit cell structures are split into four nondegenerate guided-mode resonances under normally incident light. These resonances can radiate differently polarized light, resulting in the conversion of optical polarization. Multiple linear-to-linear and linear-to-circular polarization converters can therefore be achieved. The development of modern nanofabrication technology makes it possible to fabricate such elaborate structures using various techniques such as film-deposition, photoresist-based electron beam lithography, and focused ion beam (FIB) milling [8, 14, 15]. Therefore our results can be straightforwardly applied to the realization of miniaturized all-silicon optical systems based on optical polarization converters.