1. Introduction

High-contrast gratings (HCGs), which are near-wavelength binary grating structures consisting of a high-index grating material surrounded by low-index materials [1], have attracted significant interest since the numerical [2] and experimental [3] demonstration of their diffraction-free, broadband, and high-reflectivity regimes. Thereafter, the extraordinary and intriguing properties of HCGs have been exploited to design and build various optical elements such as high-Q Fabry-Perot resonators [4], top mirrors in vertical-cavity surface-emitting lasers [5, 6], one- and two-dimensional hollow-core low-loss optical waveguides [7], slow-light waveguides [8], planar focusing reflectors and lenses [9, 10], monolithic cavity mirrors [11], saturable absorbers [12], resonant cavity-enhanced absorbers [13], polarizing beam-splitters [14], polarization-independent reflectors [15], and unidirectional transmission devices [16]. Properties such as geometrical simpleness and flexibility, large fabrication tolerance [17], small footprint, and well-known theoretical description makes HCGs promising candidates for various practical applications.

In a recent study, we have shown that a two-dimensional HCG structure with optimized geometrical parameters can be utilized for implementing a broadband circular polarizer [18]. The optimized geometrical parameters have been suggested by benefiting from the periodic dielectric slab waveguide interpretation, which is studied rigorously in [19]. The spectral transmission results obtained from the rigorous coupled-wave analysis [20] and finite-difference time-domain (FDTD) simulations (FDTD Solutions, Lumerical Inc.) have suggested that an operation percent bandwidth of 54% and 51% can be achieved, respectively, under the assumption that the proposed structure is said to be within the operation regime if the conversion efficiency exceeds 0.9. For the sake of maximizing the operation bandwidth in this theoretical study, the material that is employed outside the grating region as well as in the grooves between the silicon gratings has been assumed to be silicon dioxide.

2. Proposed Geometry

In this paper, we study the potential of the experimental realization and characterization of an HCG based broadband quarter-wave plate that can be designed by following the methodology given in [18]. By considering the potential challenges in the fabrication stage and evaluating the limitations of the devices used in the nanofabrication processes, without sacrificing significantly from the bandwidth, we decided to design an HCG structure with sapphire substrate and silicon gratings, where the grooves between the gratings are free-space as well as the region that the wave is incident onto the structure from. The proposed geometry is depicted in Fig. 1. The polarization states are defined such that a plane wave is transverse electric (TE) and transverse magnetic (TM) polarized if the magnetic field is in the x and y directions, respectively, where the incident wave propagates in the +z direction.

 

Fig. 1 Illustration of the proposed quarter-wave plate geometry. The dashed square box on the left denotes one period. The geometrical parameters are given by r = 220 nm, g = 350 nm, h_g = 320 nm, and Λ = 570 nm. In the theoretical and numerical consideration, for the sake of simplicity, regions I and III are assumed to be infinite in the −z and +z directions, respectively. The materials constituting region III and the ridges in region II are sapphire and silicon, respectively, whereas region I is free-space. n₀, n_si, and n_s represent the refractive indices of free-space, silicon, and sapphire, respectively. Grating direction is defined such that it corresponds to the y direction.

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In the present study, a theoretical solution of the specific HCG design problem is not provided due to the fact that the solution is fundamentally the same as the ones given in [18, 19]. However, we make a remark on the difference of this problem compared to the previously mentioned studies. As a result of the selection of different materials for filling regions I and III, i.e., free-space and sapphire, respectively, the wavenumbers of the zeroth reflection and transmission orders are expected to be different. The wavenumber of the zeroth reflection order is equal to k₀ and given by ${\gamma }_{0,\text{I}}=2\pi {\lambda }_0^{-1}$, whereas the wavenumber of the zeroth transmission order is equal to n_sk₀ and, therefore, given by ${\gamma }_{0,\text{III}}=2\pi n_s{\lambda }_0^{-1}$. This discrepancy results in unequal lateral magnetic field distributions in regions I and III [denoted by H^I(x, z ⩽ −t_g) and H^III(x, z ⩾ 0) in [19], respectively]. Therefore, ensuring the utilization of the correct lateral distributions while matching the boundary conditions at z = −h_g and z = 0 is of crucial importance and requires special attention.

The optimized HCG design is achieved with the aid of the theoretical calculations and the rigorous coupled-wave analysis for vertical binary gratings [20]. The utilized geometrical parameters are provided in the caption of Fig. 1. Subsequently, FDTD simulations (FDTD Solutions, Lumerical Inc.) are run in order to numerically characterize the transmission of the proposed HCG structure for TM and TE waves, while taking the effects arising because of the material dispersion into account, i.e., dependence of n_si and n_s on the wavelength of the incident light. In the simulations, the refractive index data given in [21] is employed for n_si and n_s. The two-dimensional simulations are constructed such that the periodic boundary condition is employed along the x direction, whereas the perfectly matched layer (PML) boundary condition is adopted along the z direction. The sapphire substrate is assumed to be extending to infinity in the +z direction for the purpose of avoiding the oscillations in the transmission spectrum that originate because of the Fabry-Perot modes created within a substrate of a finite thickness. Moreover, as a result of the typical short coherence length of broadband light sources, such an effect is not expected to be visible in the experimental results, unless a substrate with a thickness that is comparable to the operation wavelength is employed.

3. Numerical Results

The numerically obtained normalized transmitted intensities for the HCG for normally incident TM and TE waves (|T_TM|² and |T_TE|², respectively) are shown in Fig. 2(a). The proposed HCG geometry is optimized such that, in the close neighborhood of λ₀ = 1.55 μm, the conditions |T_TM| ≃ |T_TE| and ∠ (T_TM) − ∠ (T_TE) ≃ π/2 are satisfied simultaneously, which result in the transmission of a right-hand circularly polarized (RCP, +) and left-hand circularly polarized (LCP, −) wave assuming that the structure is illuminated by a normally incident plane wave that is linearly polarized with a polarization plane angle of 45° and −45° with respect to the x-axis on the xy-plane, respectively. The circular conversion coefficients for 45° polarization angle are subsequently calculated by C_± = 0.5(T_TM ∓ iT_TE) [22] and shown in Fig. 2(b). Obviously, T_TM and T_TE are complex numbers, which carry the phase information besides the magnitude. However, in the present experimental study, the phases of the numerical transmission coefficients are not explicitly given, since the phase information is not required for the obtaining of the experimental conversion coefficients. Instead, as it will be evident subsequently, these coefficients can directly be measured by the utilization of a linear polarizer and a quarter-wave plate. Finally, the conversion efficiency, which is shown in Fig. 2(c), is calculated as follows [18]:

 

Fig. 2 Numerical (a) TM and TE transmitted intensities, (b) circular conversion coefficients, and (c) conversion efficiency spectrum obtained via FDTD simulations. The wavelength interval of operation is denoted by Δλ. The numerical conversion efficiency spectrum yields a percent bandwidth of 42%.

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In addition to the conversion efficiency interpretation, one can define the conversion characteristics of the proposed design by calculating the ellipticity spectrum for transmitted waves. Under the assumption that the HCG is illuminated by a normally incident plane wave with a polarization plane angle of ±45°, the ellipticity, η, can be calculated as follows [24]:

4. Fabrication

The grating structure is fabricated onto a silicon-on-sapphire (SOS) wafer with a 100 mm diameter, which is purchased from Valley Design Corporation. The wafer consists of a 600 ± 60 nm thick < 100 > silicon layer that is grown epitaxially on a 0.53 ± 0.05 mm thick R-plane sapphire. Firstly, the SOS wafer is diced into square shaped chips with the dimension of 6 mm by 6 mm. The fabrication process starts with the reduction of the thickness of the silicon layer to 320 nm by means of the reactive ion etching (RIE) technique, while using sulfur hexafluoride (SF₆) as the etchant gas. Afterwards, polymethyl methacrylate (PMMA), which is an electron beam sensitive photoresist, is spun onto the sample. Subsequently, the desired grating structure is patterned onto the SOS sample using e-beam lithography. Following the development operation of the PMMA, the final step is accomplished by etching the unexposed silicon regions using the RIE system. The scanning electron microscope (SEM) micrographs of the fabricated HCG sample are shown in Fig. 3.

 

Fig. 3 (a) Zoomed out and (b) zoomed in top view SEM micrographs of the fabricated HCG structure. In (b), the legends V1, V2, and V3 denote the geometrical parameters g, r, and Λ, respectively.

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5. Experimental Setup

The outlines of the two experimental setups, which are constructed for the measurement of the linear transmission coefficients and the circular conversion coefficients of the HCG sample, are illustrated in Fig. 4 in detail. In the experiments, the sample is illuminated by a broadband near-infrared (NIR) light source (Spectral Products, ASBN-W100-F-L) and, with the aid of standard commercial NIR objectives and an adjustable aperture (not shown in Fig. 4), the spot size of the incident light is adjusted to be 50 μm. The normalization of the transmitted fields is performed with respect to a solely sapphire region that is obtained by etching the silicon layer on the SOS chip.

 

Fig. 4 Visual illustrations of the experimental setups that are utilized for the measurement of (a) linear transmission coefficients and (b) circular conversion coefficients. The arrows, which lie inside the HCG samples and point upwards, denote the grating direction that is defined in the caption of Fig. 1.

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In the first measurement, which is illustrated in Fig. 4(a), the linear intensity transmission coefficients (|T_TM|² and |T_TE|²) are obtained by the utilization of an adjustable linear polarizer (Thorlabs, DGL10) that is used for enabling the illumination of the grating region with TM and TE polarized light. In the second measurement, which is illustrated in Fig. 4(b) and conducted with the purpose of measuring the circular conversion coefficients, the grating region is illuminated by an RCP wave as a result of the employment of an adjustable linear polarizer (with its transmission axis making a 45° angle with the x – axis) followed by a quarter-wave plate (Thorlabs, WPQ05M-1550) that is oriented such that its fast axis is along the x direction. At the farfield, the transmitted wave through the grating region is passed through an adjustable linear polarizer that is oriented such that its transmission axis makes angles of +45° and −45° with respect to the x–axis. Afterwards, the intensities of the waves that are transmitted through the linear polarizer are measured using the spectrometer (Ocean Optics, NIR256-2.5) in order to find the square of the magnitudes of the RCP and LCP conversion coefficients (|C₊|² and |C₋|²), respectively. By the utilization of the Jones matrix formalism, it can easily be shown that by adjusting the linear polarizer to +45° and −45°, |C₊|² and |C₋|² can be measured directly, respectively. As a result of the period of the proposed HCG structure being equal to 570 nm, FDTD simulations and experiments are conducted between the free-space wavelengths of 1.2 μm and 2 μm with the purpose of avoiding the wavelengths for which the HCG is diffractive, i.e., λ₀ < n_sΛ.

The proposed HCG based quarter-wave plate can be combined with a HCG based [14], a wire grid [25], a thin-film grating [26], or a plasmonic [27] linear polarizer for the purpose of implementing a compact circular polarizer.

6. Experimental Results

The linear intensity transmission and circular conversion coefficients (for an incident wave with a polarization plane angle of 45°), and the conversion efficiency spectrum obtained from the experiments are shown in Fig. 5. Conversion efficiency data given in Fig. 5(c) suggests that the experimental lower and higher corner wavelengths are obtained as λ_L = 1.25 μm and λ_H = 1.75 μm. The usage of Eq. (2) results in an experimental percent bandwidth of 33%. In spite of the achievement of a smaller percent bandwidth compared to the numerical predictions, the experimental results exhibit a good agreement with the numerical results shown in Fig. 2.

 

Fig. 5 Experimentally obtained (a) TM and TE transmitted intensities, (b) circular conversion coefficients, and (c) conversion efficiency spectrum. The wavelength interval of operation is denoted by Δλ. The experimental conversion efficiency spectrum yields a percent bandwidth of 33%.

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We conjecture that the most significant factors responsible from the discrepancies between the numerical and experimental results are finite-size effect and finite substrate thickness. Finite-size effect originates from the usage of finite number of periods and finite grating length in the experiments (in contradiction with the two-dimensional periodic FDTD simulations, where the number of periods and the grating length are infinite) and possibly modifies the lateral and longitudinal wavenumbers of the excited modes inside region II and, additionally, this effect can result in diffraction from the edges of the sample. The utilization of a substrate with a finite thickness can result in multiple back-reflections inside the substrate, modifying the overall behavior of the corrugated surface and, in addition, Fresnel reflection from the back-side of the substrate can also modify the transmission characteristics of the sample.

In addition to the aforementioned effects, the trapezoidal grating profile achieved after the final etch step and possible deviations of the geometric parameters of the fabricated sample from the design parameters can conduce to the discrepancies between the numerical and experimental results. Furthermore, less significant effects such as the employment of non-ideal optical elements, i.e., linear polarizer and quarter-wave plate, probable small deviation of the actual refractive indices from the ones adopted in the simulations, and human related errors while setting the polarizer transmission axis angle and the angle of incidence might as well contribute to the deviations in the experimental results from the numerical ones.

7. Conclusions

In conclusion, we have shown the fabrication and the experimental characterization of a broadband quarter-wave plate that is based on two-dimensional silicon HCGs, where sapphire is employed as the substrate. It has been shown that a conversion efficiency that is higher than 0.9 can be achieved in a percent bandwidth of 42% and 33% numerically and experimentally, respectively. RCP and LCP waves can be obtained by adjusting the transmission axis of the linear polarizer to 45° and −45° with respect to the x-axis, respectively. In terms of practical applicability, the proposed structure can be employed in laser and nanoantenna applications, liquid crystal displays, and remote sensors.