1. Introduction

Diffractive gratings have been crucial components in the broad area of optics for centuries [1]. In the recent decade, researchers have shown extensive interest in an operation regime where the grating period Λ is comparable or slightly smaller than the wavelength λ [2–6], where strong field interaction among the near-wavelength structures can provide extraordinary optical behaviors. One interesting type of grating, known as the high-contrast grating (HCGs), is made of periodic high-index materials surrounded by materials with much smaller indices [6]. HCGs have been used as ultrabroadband reflectors [7] and high-Q resonators [8]. They can be integrated as compact, high-performance tunable mirrors in optoelectronic devices, such as vertical-cavity surface-emitting lasers (VCSELs) [3, 9].

HCGs are referred as one-dimensional (1D) if the periodicity is only in one direction (e.g. periodic bars, stripes, or grooves). In this case the fields can be decoupled into the transverse-electric (TE) and transverse-magnetic (TM) polarizations. An incident wave with one polarization will not excite an HCG mode with an orthogonal polarization. For 2D HCGs, structures are periodic in two directions and the modes supported by the grating have hybrid polarizations, that is, they are not separable into TE and TM. The modes in 2D HCGs are also more closely-packed spectrally. In other words, with a given incident wave, many more modes can be excited in a 2D HCG than in a 1D HCG. The diffracted modes, also known as the Floquet modes, have one extra degree of freedom for the propagation direction, which makes it very difficult to predict the optical properties of a 2D HCG. Moreover, there are many more structural parameters in 2D HCGs, which largely complicate the design and optimization procedure. Up to now, majority of the gratings in use are still one-dimensional.

However, there are applications which require beam-steering or focusing in both directions, or generating optical vortices [10, 11]. In these cases, 2D HCGs are more advantageous. Extensive experimental work has been done in demonstrating the functionalities of 2D gratings [2, 10]. In this work, we examine the physics of 2D HCGs and the mechanisms which allow high-performance operation. The dual-mode analysis we propose can largely simplify the searching of the initial design and the subsequent optimization. A top-down design procedure is provided. To model 2D HCGs, the analytical mode-matching method [6,12], which was previously used in 1D HCGs, is rather inefficient and requires complex root-searching for coupled nonlinear equations. An in-house developed rigorous coupled-wave analysis (RCWA) [13, 14] program is shown to be very convenient for design purposes, and both the convergence and accuracy are verified.

One important application for 2D HCGs is the 2D phase plate, which normally requires non-periodic designs. However, the high-index material can largely confine the field around the local position, where effective-medium approximation is applicable [11, 15, 16]. Based on our optimized design for periodic 2D HCGs, we can apply the grating parameters as the local parameters for the non-periodic grating, according to the design requirement at the local position. Our designed non-periodic phase plate is then verified through the finite-difference time-domain (FDTD) simulation. The results show excellent agreement with our designs.

There is much recent progress on beam shaping using plasmonic metastructures [2]. The localized resonance behavior resulting from the plasmonic effect also enables a wide range of beam engineering. However, the metallic structures supporting the plasmonic resonance also introduce much optical loss. HCGs, on the other hand, have intrinsically much lower loss. Using our designs we demonstrate excellent power efficiency (above 90%).

2. Optical properties of 2D high-contrast gratings

We first study the underlying physics of 2D high-contrast gratings. Figure 1(a) shows a general 2D grating on a substrate (optional for suspended grating membranes). The incident plane wave is characterized by the polar and azimuthal angles of its wave vector k_i = (k_ix,k_iy,k_iz) = (k₀ sinθ cosϕ, k₀ sinθ sinϕ, k₀ cosθ). The polarization of the incident electric field should also be characterized by two angles, that is, E_i = (E₀ sinα₁ cosα₂, E₀ sinα₁ sinα₂, E₀ cosα₁). Yet the electric field is transverse to the wave vector for propagating waves, thus one angle α is sufficient to describe the polarization. For oblique incidence, it is convenient to decompose the fields into s- and p-polarizations, which will be discussed in later sections.

 

Fig. 1 (a) Schematic of a 2D high-contrast grating on a rectangular lattice with a substrate. The plane-wave incident polar and azimuthal angles are θ and ϕ, respectively. Parameters: Λ_x and Λ_y (periods in $\widehat{x}$ and ŷ), D_x and D_y (grating widths in $\widehat{x}$ and ŷ), n_g and n_s (grating and substrate indices), and t_g and t_s (grating and substrate thicknesses). (b) Schematic of a 2D high-contrast grating on a hexagonal lattice with a substrate. Parameters: period Λ, rod diameter d, grating and substrate indices n_g and n_s, grating and substrate thicknesses t_g and t_s.

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Design parameters include grating periods and widths in both $\widehat{x}$ and ŷ directions (Λ_x, Λ_y, D_x, and D_y), as well as thicknesses and refractive indices of both the high-index material and the substrate (n_g, n_s, t_g, and t_s), as shown in Fig. 1(a). We define duty cycles for the high-index material as η_x = D_x/Λ_x and η_y = D_y/Λ_y.

For applications where optical fields express rotational symmetry, 2D circular gratings on a hexagonal lattice, as shown in Fig. 1(b), are more advantageous. Design parameters are simplified to grating period (Λ), rod diameter (d), thicknesses and refractive indices of both the grating and substrate (t_g, t_s, n_g and n_s). The duty cycle is defined as η = d/Λ. Gratings on a hexagonal lattice, being more closely packed, require $\mathrm{\Lambda }<2{\lambda }_0/\sqrt{3}$ in order to have only the main harmonic being propagating, as compared to Λ_x,y < λ₀ for gratings on a rectangular lattice.

We first look at the incident and transmitted regions, which are typically homogeneous. According to the Floquet theorem, the scattered field consists of Floquet modes with different orders of tangential wave vectors, as a result of the phase matching between the homogeneous region and the 2D grating with the translational symmetry. In rectangular lattice, the (m,n)-th order Floquet mode is a plane wave with a wave vector being

Then the total transverse field in the grating layer can be expressed in spectral domain as

Similar to Eq. (5) we can expand the transverse magnetic field in terms of the magnetic components ${\tilde{\mathbf{H}}}_t^{(i)}$ of the eigenmodes ( ${\tilde{\mathbf{H}}}_t^{(i)}$obtained from ${\tilde{\mathbf{E}}}_t^{(i)}$). We can then express z-dependent transverse electric and magnetic fields including forward and backward propagation,

Figure 2(a) shows the (0,0)-th order reflection spectra for a 2D HCG under normal incidence. Results are calculated with the finite-element method (FEM) using COMSOL Multiphysics, finite-difference time-domain method (FDTD) using Lumerical Solutions, as well as an in-house developed 2D rigorous coupled-wave analysis (RCWA) program. Good agreement is shown among three methods and good convergence of our RCWA program upon total number of 2D spectral orders (N) is observed. Moreover, calculating each frequency point using RCWA with N = 625 is only 90 seconds on a personal computer with an Intel Core i7-3520M processor, whereas the computation cost is much heavier for FDTD and FEM due to the full discretization of the 3D solution domain. When λ < max{Λ_x,Λ_y}, higher spectral orders become propagating, and extracting each spectral order accurately from the total field solution in FDTD or FEM becomes more challenging.

 

Fig. 2 (a) Comparison among the (0,0)-th order reflectivity spectra of a 2D high-contrast grating (HCG) under normal incidence calculated using finite-element method (blue star), finite-difference time-domain (red dots), and rigorous coupled-wave analysis (RCWA) using N = 289 (magenta), N = 441 (green), and N = 625 (blue). HCG parameters are Λ_x = 1μm, Λ_y = 0.5μm, D_x = 0.6μm, D_y = 0.3μm, and t_g = 0.5μm. (b) Spectra of normalized scattered power in the $\widehat{z}$-direction for (0,0), (+1,0), and (−1,0) spectral orders (blue, red, and black, respectively), and the sum of the three spectra (green).

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We also verify that our RCWA simulation is energy-conserving. We calculate the scattered power flux (reflection and transmission) normalized by the incident power flux for the (0,0), (+1,0), (−1,0) modes, as shown in Fig. 2(b). The power scattered into the (0,0) mode drops from unity when λ < 1μm, indicating power transfering into higher diffraction orders. Nonetheless, the total scattered power remains unity, satisfying the conservation of energy.

We can also study the 2D HCG properties under oblique incidence. The field polarization is defined relative to the incidence plane, which is formed by the surface normal and the incidence wave vector. We use s- and p-polarization for electric field perpendicular and parallel to the incidence plane, respectively. Figure 3(a) shows the (0,0)-th order reflectivity and transmissivity as functions of the incidence polar angle θ with wavelength λ = 2μm and incidence azimuthal angle ϕ = 0. In this case, all higher spectral orders remain evanescent for the whole 90-degree range of θ. Thus the sum of reflectivity and transmissivity remains unity. Results from our 2D RCWA program and FEM software agree very well.

 

Fig. 3 (a) Incident polar angle dependent reflectivity (black), transmissivity (magenta), and their sum (green) for the (0,0) fundamental order under p-polarized incidence with λ = 2μm calculated using rigorous coupled-wave analysis, and comparison with finite-element method (red and blue crosses). The same grating structure is solved under (b) s-polarized and (c) p-polarized incidence with wavelength being λ = 0.6μm. Normalized scattered power fluxes in the $\widehat{z}$-direction for the (0,0)-th, (+1,0)-th, (−1,0)-th, (−2,0)-th, and (−3,0)-th spectral orders are shown as the blue, red, black, green, and cyan lines, respectively. The magenta line indicates the total normalized scattered power flux in the $\widehat{z}$-direction. Arrows indicate the cutoff angles for spectral orders to appear or disappear.

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As the wavelength decreases, the higher spectral orders may be propagating. The critical condition for the (m,n)-th spectral order to transition between propagating and evanescent is

Figures 3(b) and 3(c) show the normalized scattered power in the $\widehat{z}$-direction for different spectral orders under s- and p-polarized oblique plane wave incidence, respectively. The cutoff angles for the (−2,0), (+1,0), and (−3,0) modes are calculated from Eq. (9) to be 11.54°, 23.58°, and 53.13°, which agree with the results indicated by the arrows in Figs. 3(b) and 3(c).

We also observe resonance behavior around the cutoff angles for p-polarized but not the s-polarized incidence. Such resonance is not a numerical artifact since it remains as the numerical accuracy is varied (N in RCWA). It is indeed the Wood’s anomaly which has been studied extensively since 1902 [17–22]. Hessel and Oliner [21] classified the anomaly into two types: a Rayleigh type where a diffraction order appears or disappears, and a resonance type which is related to the coupling between the diffracted wave and the leaky wave inside the grating. The two types may occur very closely or separately. The second type is dependent on the incident field polarization, which requires the surface reactance of the grating to be capacitive or inductive for resonance to occur. The surface reactance is affected by grating thickness, which explains why for a range of thicknesses, the anomaly only happens to one polarization. Previous investigation on Wood’s anomaly in 1D gratings is well applicable to 2D gratings, though rigorous theoretical analysis could be rather involved. In this case, our 2D RCWA program can be a convenient tool to study this effect. One thing to note is that, many studies [21, 22] on 1D grating defined s- and p-polarizations relative to the grooves of the grating, while in this work on 2D grating, polarizations are defined relative to the incidence plane. Thus the analysis on “s-anomalies” in [21] is applicable to the p-polarized case in Fig. 3(c).

Regardless of the anomalies, the energy conservation is still satisfied, as confirmed by our RCWA results. The total scattered power flux in the $\widehat{z}$-direction remains a smooth function and has a cosine dependence on the incident polar angle when normalized by the total incident power flux, as shown by the magenta lines in Figs. 3(b) and 3(b).

Even using only the fundamental order of reflection or transmission with normal plane wave incidence, the 2D grating can be designed as various optical components, such as polarizers and waveplates. Figure 4(a) shows the magnitude of the reflected, transmitted, and total scattered wave under normal plane wave incidence in the subwavelength regime (λ > max {Λ_x, Λ_y}). The scattered wave is decomposed into two polarizations, one being parallel and the other being perpendicular to the incidence. As the incident polarization rotates relative to the grating about the z-axis, the magnitude of the parallel component varies relative to the perpendicular component. This provides the possibility of using 2D gratings as polarizers. Moreover, the phase difference between the two components is also dependent on the azimuthal angle of the incident electric field, as shown in Fig. 4(b). This enables us to design quarter-wave plates. In this example, the parallel and perpendicular reflected waves have a 90° phase difference when the incident polarization angle is 27.7°. If such angle is 52.7°, the two transmitted components have a −90° phase difference.

 

Fig. 4 (a) Reflectivity (blue), transmissivity (red), and the normalized scattered power (green) of a normal incident plane wave with λ = 1.55μm as functions of the azimuthal polarization angle. The arrows indicate the components parallel and perpendicular to the incident wave. (b) Phase differences between the parallel and perpendicular components for reflection (red) and transmission (blue) as functions of the azimuthal polarization angle. The 90° phase differences correspond to polarization angles of 27.7° and 52.7° for reflection and transmission types, respectively.

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3. Design principles of 2D high-contrast gratings

We have verified our 2D RCWA program as a design tool and studied the optical properties of 2D HCGs, yet the design of 2D HCGs remains a challenge. The structural parameters include n_g, t_g, Λ_x, Λ_y, η_x, and η_y. The incidence condition includes wavelength (λ), propagation direction (ϕ, θ), and polarization (α). Here we present a design procedure which largely simplifies the searching of initial parameters and the optimization process. As shown in Fig. 5, we start from the eigenmodes in the 2D HCG, which is considered as an infinitely long waveguide along the $\widehat{z}$-direction and periodic in the xy-plane. Based on the dispersion relations of the eigenmodes, we can obtain initial designs for the grating structure in the xy-plane. Then using the dual-mode analysis we can find the resonance condition, and thus the values of grating thickness to provide high and possibly perfect reflection or transmission. With the initial structural parameters, we can then optimize the design for various applications. For reflectors, filters and resonators, we are concerned with locations of the passband, stopband, and transition band. By checking the frequency spectra and adjusting the grating thickness, we can shift these bands to our desired frequency range. For polarizers and waveplates, we want to obtain desired amplitude selectivity and relatively phase shift between two orthogonal polarizations. Both can be achieved by rotating the incident polarization relative to the 2D HCG, as shown in Fig. 4. For 2D phase plate, we need to obtain a full 2π phase tuning range by varying the HCG transverse structure (e.g. Λ or η). At the same time, we need to maintain high reflection or transmission for designs to be practical. More details about the phase plate are in the next section.

 

Fig. 5 Design procedures of 2D high-contrast gratings.

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3.1. Eigenmodes in 2D high-contrast gratings

Eigenmodes in 2D dielectric gratings are hybrid modes, which cannot be separated into transverse electric (TE) and transverse magnetic (TM), as in the case of 1D gratings. Furthermore, eigenmodes are more closely spaced in the frequency domain due to one extra degree of freedom (one more mode number), which largely complicates our analysis and prediction. However, the majority of the modes with strong confinement have dominant transverse field components, namely E_x/H_y-dominant (EH-like) and E_y/H_x-dominant (HE-like). These modes in periodic gratings resemble the EH and HE modes in the rectangle dielectric waveguides, using Marcatili’s approximation [23]. We can analyze the two separately under the approximation that grating modes with one dominant polarization only couple strongly with the free-space Floquet modes with the same polarization. Thus we can reduce the number of modes to analyze by almost half. Moreover, the field distribution of the incident plane wave possesses even symmetry in the xy-plane across the unit cell under normal incidence. Therefore eigenmodes with odd symmetry will not be excited and we can further narrow down our analysis. Figure 6(a) shows the dispersion curves of the eigenmodes with even symmetry and H_x-component dominant over H_y-component. In fact, the four modes shown in Fig. 6(a) are very much like the HE modes in rectangular dielectric waveguides when they are well-confined between the light lines (k_z = ω/c and k_z = n_gω/c). Similarly, we can separate out E_x/H_y-dominant eigenmodes with even symmetry, as shown in Fig. 6(b). The H_y field profiles for EH₀₀-like and EH₂₀-like modes at $\omega =0.8\frac{2\pi c}{{\mathrm{\Lambda }}_x}$ indeed are very close to those in dielectric waveguides, as shown in Fig. 6(c).

 

Fig. 6 (a) HE-like even eigenmodes in a 2D rectangular grating on a rectangular lattice. The two dashed lines indicate the k_z = ω/c and k_z = n_gω/c light lines. (b) EH-like even eigenmodes in a 2D rectangular grating on a rectangular lattice. (c) ℜe[H_y] for the EH₀₀-like eigenmode and EH₂₀-like eigenmode at $\omega =0.8\frac{2\pi c}{{\mathrm{\Lambda }}_x}$, as indicated by the green dots in (b).

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Now we are able to focus on the modes which have major contribution to the HCG behavior. For instance, between the cutoff frequencies ω_c20 and ω_c02 (for the EH₂₀-like and EH₀₂-like modes, respectively), we have a spectral region where only two key modes exist. This means within this region we can engineer the interference between the two modes to produce desired functionalities.

In other cases, 2D gratings on hexagonal lattice are preferred. Our RCWA program can be easily modified for this case. As shown in Fig. 7(a), the unit cell can be chosen as the yellow rectangle with a fixed aspect ratio of $\sqrt{3}:1$. We can still separate eigenmodes with H_x-dominant and H_y-dominant polarizations. The dispersion curves of the first few H_x- and H_y-dominant eigenmodes are shown in Figs. 7(b) and 7(c), respectively. We further identify the eigenmodes which are symmetric with the translation along $\frac{1}{2}\widehat{x}+\frac{\sqrt{3}}{2}\widehat{y}$ from those which are anti-symmetric, as indicated by the red and blue lines in the dispersion curves. The H_y components of the first symmetric, first anti-symmetric, and second symmetric H_y-dominant modes are shown in Figs. 7(d)–7(f), respectively.

 

Fig. 7 (a) Circular high-contrast grating on hexagonal lattice. The yellow rectangle indicates a choice of the unit cell. Parameters: Λ = 1μm, η = 0.6. Dispersion curves of (b) H_x-dominant and (c) H_y-dominant eigenmodes possessing symmetry (red) and anti-symmetry (blue). ℜe[H_y] for (d) the first symmetric, (e) the first anti-symmetric, and (f) the second symmetric modes at frequency $\omega =0.5\frac{2\pi c}{\mathrm{\Lambda }}$, as indicated by the green dots in (c).

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3.2. Dual-mode analysis for perfect reflection and transmission

In many cases, the incident wave only strongly couples to a few eigenmodes due to mismatch of the field polarization or symmetry. We are particularly interested in the case when only two eigenmodes are strongly excited. Then we can engineer the structure to produce nearly perfect interference between them. In such case we can simplify our study to a dual-mode analysis. The transverse field in Eq. (5) is rewritten as

Using our dual-mode analysis, the dimensions of matrices and vectors in Eq. (11) are reduced to contain only two modes,

In Fig. 8(a) we see the reflection and transmission spectra of a 2D circular HCG on a hexagonal lattice under E_x/H_y-polarized normal incidence at λ = 1.55μm. We have perfect transmission (zero reflection) at two wavelengths and ultra-high transmission in between. Using our dual-mode analysis, we can easily extract the contribution to the reflected (0,0) mode from the back-coupling of each of the two strongly excited eigenmodes, as well as the contribution from all other modes, as shown in Fig. 8(b). Furthermore, we can calculate the phase difference of the contribution from the two modes. We can see that at zero reflection, indicated by the dashed lines, the coupling magnitudes of the two modes are almost the same, and the phase difference is almost ±π. This indicates the interference of the two modes is very close to but not yet perfectly destructive. We still see some coupling from other eigenmodes. This is because the eigenmodes in 2D HCG are hybrid modes and we have no perfect selectivity of polarizations. The result is a slight difference between the wavelengths for zero-reflection and the dual-mode perfect interference. Nonetheless, this analysis provides a very good estimation of the zero-reflection condition. The zero-transmission case can be analyzed similarly when we look at the forward-coupling to the transmitted region.

 

Fig. 8 (a) Reflectivity (red) and transmissivity (blue) of a 2D circular high-contrast grating on a hexagonal lattice with Λ = 750nm, λ = 1.55μm, and t_g = 713nm. Blue dashed lines indicate the wavelengths for perfect transmission. (b) Back-coupling magnitude from the first (red), second (black) symmetric eigenmodes and all other eigenmodes (green) in the grating to the (0,0) mode in the incident region. The dashed lines in (b) are at the same wavelengths as in (a). Blue solid line is the phase difference between the reflected waves back-coupled from the two eigenmodes.

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3.3. Resonance conditions in 2D high-contrast gratings

For a given desired wavelength, we can also adjust the HCG thickness t_g to study the resonance condition. We first look at a rectangular HCG with the same structure as in Fig. 2 under E_x/H_y-polarized normal incidence. The dual-mode eigen-equation in Eq. (12) yields two eigen-solutions which we will name as supermode 1 and supermode 2. The phases of the eigenvalues (φ_1,DM and φ_2,DM), which are the round-trip phases for the two supermodes, are functions of the grating thickness. Figure 9(a) shows the half-trip phases ( ${\phi }_{1,\mathrm{DM}}^{\text{half}}$ and ${\phi }_{2,\mathrm{DM}}^{\text{half}}$), and we record the HCG thicknesses at which the half-trip phases are even and odd multiples of π, indicated by the empty boxes and solid circles, respectively. At each given wavelength, we can solve this dual-mode eigen-problem and obtain the thicknesses for resonance to occur. Figure 9(b) shows the contour plot for the resonance conditions. Red and black lines indicate the half-trip phases ${\phi }_{1,\mathrm{DM}}^{\text{half}}$ and ${\phi }_{2,\mathrm{DM}}^{\text{half}}$ being odd multiples of π, respectively. Blue and green lines indicate the half-trip phases ${\phi }_{1,\mathrm{DM}}^{\text{half}}$ and ${\phi }_{2,\mathrm{DM}}^{\text{half}}$ being even multiples of π, respectively.

Figure 10(a) shows the reflectivity contour plot as a function of the grating thickness and wavelength. From Fig. 6(b) we can find the cutoff frequencies for the EH₂₀ and EH₀₂ to be ${\omega }_c^{{\mathrm{EH}}_{20}}=0.392\frac{2\pi c}{{\mathrm{\Lambda }}_x}({\lambda }_c^{{\mathrm{EH}}_{20}}=2.55{\mathrm{\Lambda }}_x)$ and ${\omega }_c^{{\mathrm{EH}}_{02}}=0.827\frac{2\pi c}{{\mathrm{\Lambda }}_x}({\lambda }_c^{{\mathrm{EH}}_{02}}=1.21{\mathrm{\Lambda }}_x)$. Between these two frequencies we see a dual-mode spectral window from the dispersion curves. These two frequencies are also indicated by the black lines in Fig. 10. We can see the ultra-high-reflection regions are mostly within the dual-mode window. Figure 10(b) shows the overlap between the resonance lines in Fig. 9(b) and the reflectivity contour in Fig. 10(a). Excellent agreement is shown which indicates that our dual-mode analysis can successfully predict the resonance and high-reflection conditions in a 2D HCG.

 

Fig. 9 (a) Half-trip phases of the supermodes 1 (blue) and 2 (red) in the same grating as in Fig. 2 under λ = 2.1μm normal incidence. Empty boxes and solid circles indicate half-trip phases being odd and even multiples of π, respectively. (b) Resonance conditions for the grating thicknesses at given wavelengths. Red and blue lines indicate the half-trip phase of supermode 1 being odd and even multiples of π, respectively. Black and green lines indicate the half-trip phase of supermode 1 being odd and even multiples of π, respectively.

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Fig. 10 (a) Reflectivity contour plot of a rectangular high-contrast grating (same as in Fig. 2) as a function of the thickness t_g and wavelength λ. The solid lines indicate λ = 2.55Λ_x and λ = 1.21Λ_x, corresponding to the cutoff frequencies ${\omega }_c^{{\mathrm{EH}}_{20}}=0.392\frac{2\pi c}{{\mathrm{\Lambda }}_x}$ and ${\omega }_c^{{\mathrm{EH}}_{02}}=0.827\frac{2\pi c}{{\mathrm{\Lambda }}_x}$ for the EH₂₀- and EH₀₂-like modes, as shown in Fig. 6(b). (b) Overlap between the resonance contour plot (white) and the reflectivity contour plot.

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For a circular HCG on a hexagonal lattice, we can also identify a dual-mode spectral window where only two symmetric modes with the same dominant polarization are strongly excited, as shown in Fig. 11(a). From the transmissivity contour plot of our designed transmission-type HCG in Fig. 11(b), we see wide high-transmission regions inside the dual-mode window. These high-transmission regions are mostly bounded by or along the resonance lines, as shown in Fig. 11(c). We again see an excellent overlap between the dual-mode resonance lines and the contour plot. From the resonance lines, we can easily identify the design regions for broadband or sharp-transition applications, such as reflectors, filters, and resonators.

 

Fig. 11 (a) Dual-mode dispersion curves in a hexagonal-lattice grating and the dualmode window (same as in Fig. 7) is indicated by the blue dashed lines at $\omega =0.44\frac{2\pi c}{\mathrm{\Lambda }}$ and $\omega =0.6\frac{2\pi c}{\mathrm{\Lambda }}$. (b) Transmission contour plot as a function of the grating thickness t_g and wavelength λ. The dual-mode window indicated by the two black lines. (c) Overlap between the resonance lines (white) and the transmission contour plot.

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4. Engineering of 2D phased arrays using high-contrast gratings

With the knowledge of the resonance conditions and the mechanisms for ultra-high reflection or transmission, we can also apply the initial design to 2D phase plates using HCGs. Now we fix the wavelength and grating thickness, but vary the transverse structure to find a full 2π reflection (or transmission) phase tuning range while maintaining the power efficiency. Figure 12 shows an example of a transmission-type hexagonal-lattice HCG operating at λ = 1.55μm with a thickness of 0.8μm. We are able to find an optimum design range by fixing Λ = 0.75μm, and tuning η from 10% to 63%, as indicated by the black lines. Thus we have a mapping between the transmission phase and the transverse geometry. The design of the phase plate is essentially aimed at generating a transverse position-dependent phase alternation profile, that is, ΔΦ(x,y). This can be translated into the design of the η(x,y) profile. In many applications, we would need substrates for the HCGs. Figure 13(a) shows the magnitude and phase tuning by the HCG duty cycle with different substrate thicknesses (t_s = 0, λ/4, and λ/2). We can see the full 2π phase tuning range is still obtainable except the phase discontinuity occurs at different locations. The magnitude remains almost unchanged for thickness being multiples of half-wavelength, but is strongly perturbed at quarter-wavelength. In the situation when the grating transmission is not close enough to unity, we can use the substrate to further improve the transmission, and find the optimum value of t_s, as shown in Fig. 13(b).

 

Fig. 12 Contour plots as functions of the grating period Λ and duty cycle η for the (a) magnitude and (b) phase of the transmission through a hexagonal-lattice grating under λ = 1.55μm normal incidence. The black lines indicates a 2π phase range at Λ = 750nm. The white lines indicate the contour for 90% transmission.

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Fig. 13 (a) Tuning of the transmission magnitude (solid) and phase (dots) by grating duty cycle with thickness being 0 (blue), 258nm (black), and 517nm (red). Same structure as in Fig. 7 and λ = 1.55μm. (b) Transmission magnitude (blue) and phase (green) as functions of the substrate thickness with 48% duty cycle.

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By introducing a phase alternation ΔΦ based on the transverse spatial location (x,y), we can have 2D HCGs behaving like many conventional optical components. For example, having $\mathrm{\Delta }\mathrm{\Phi }(x,y)=\frac{2\pi }{\lambda }(x\cos \varphi \sin \theta +y\sin \varphi \sin \theta )$, we can steer the normal-incident plane wave into an oblique direction of $\widehat{x}\cos \varphi \sin \theta +\widehat{y}\sin \varphi \sin \theta +\widehat{z}\cos \theta$, where θ and ϕ are the polar and azimuthal angles of the transmitted wave vector. For a lens with a focusing length f, we need $\mathrm{\Delta }\mathrm{\Phi }(x,y)=\frac{2\pi }{\lambda }\left(f-\sqrt{f^2+x^2+y^2}\right)$. For an axicon [24] with an opening angle α_axi, an index n_r and a radius R, we have $\mathrm{\Delta }\mathrm{\Phi }(x,y)=\left(R-\sqrt{x^2+y^2}\right)(n_r-1)\tan {\alpha }_{\text{axi}}$. For a focusing lens that generates orbital angular momentum (OAM) of $m\overline{h}(m=0,\pm 1,\pm 2\dots )$, we have $\mathrm{\Delta }\mathrm{\Phi }(x,y)=\frac{2\pi }{\lambda }\left(f-\sqrt{f^2+x^2+y^2}+m\varphi \right)$. Using Fig. 13 we can design the local duty cycle η(x, y) for the 2D HCG based on the desired phase alternation ΔΦ(x, y).

Once we have the design for 2D HCGs, we use the 3D FDTD method to verify the performance. Figure 14(a) shows a normal-incident x-polarized plane wave toward the $+\widehat{z}$-direction at λ = 1.55μm passing through our designed HCG beam steering plate with a 15° steering angle. The two dashed lines indicate the normal and 15° from normal directions. The transmitted wave is indeed traveling along the desired direction with high power efficiency. Figure 14(b) shows a normal-incident x-polarized Gaussian beam with a beam waist w₀ = 7μm transmitting through a 10-μm-radius HCG phase plate, which is designed to have a focal length f = 15μm. The magenta dashed lines indicate the focusing behavior of an equivalent lens with a numerical aperture of 0.5547. Figure 14(c) shows that the same Gaussian beam passes through a HCG phase plate, which acts like an axicon [25], and becomes a Bessel beam. The magenta dashed lines indicate an Bessel beam opening angle of 2θ_BS = 0.4rad, which is the result of an equivalent axicon with a radius R = 10μm, an index n_r = 1.5, and an axicon opening angle α_axi = θ_BS/(n_r − 1) = 0.4rad. Such a Bessel beam has a maximum propagation distance z_max ≈ w₀/θ_BS = 35μm, within which the beam will have no diffraction. This maximum distance can be increased by reducing the axicon opening angle α_axi.

 

Fig. 14 ℜe[E_x] in the xz-plane for an x-polarized normal-incident wave toward $+\widehat{z}$ at λ = 1.55μm transmitting through HCG phase plates designed for Gaussian beams to: (a) be deflected by 15°; (b) focus at a 15-μm distance; (c) be converted to Bessel beams. Black solid lines indicate the grating layer. Blue dashed lines indicate the source locations. Other dashed lines are for visual aids. All field intensities are normalized by the incident field intensity.

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We can allow our 2D HCG phase plate to focus and simultaneously generate orbital angular momentum as a beam passes through. Such modification can be easily done by adding a spatial-dependent, or more specifically, an azimuthal angle-dependent phase delay. Using the phase information from Fig. 13(a) we arrive at a 2D HCG design in Fig. 15(a) on top of a glass substrate, with an operation wavelength at λ = 1.55μm. The transmitted field intensity profiles at z = f − 4λ, z = f, and z = f + 4λ are shown in Figs. 15(b)–15(d), respectively, where the focal length is designed to be 20μm. We can see a clear intensity null at the center, which is necessary for nonzero OAM. The incident beam has a Gaussian distribution with a beam waist of 7μm and zero OAM, as shown in Fig. 15(e). The focusing behavior is observed by comparing the beam waists and the peak intensities 4λ below, 4λ above, and exactly at the focal plane, as well as those of the source. Our design provides high transmissivity with a power efficiency above 90%. Figures 15(f)–15(h) show the phase distribution profiles at z = f −λ/3, z = f, and z = f + λ/3. The spatial-dependent phase distribution already indicates nonzero OAM. The phase of the transmitted wave increases from −π to π as the azimuthal angle increases in a 2π range, indicating the OAM is $+1\overline{h}$ per photon. Alternatively, we see the phase profile rotates clockwise by 2π/3 as the beam propagates by Δz = λ/3 in the $+\widehat{z}$-direction, showing a left-hand helical phase front. Therefore the OAM is indeed $+1\overline{h}$ per photon.

 

Fig. 15 (a) Structure of the 2D high-contrast grating phase plate for focusing at f = 20μm and generating $+1\overline{h}$ orbital angular momentum. Field intensities at z = f − 4λ, z = f, and z = f +4λ are shown in (b), (c), and (d), respectively. (e) Gaussian source intensity with a beam waist of 12μm. Phase distributions (in π) of the transmitted wave at z = f − λ/3, z = f, and z = f +λ/3. are shown in (f), (g), and (h), respectively.

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5. Conclusion

We have investigated the physics of 2D high-contrast gratings (HCGs). Our in-house developed 2D rigorous coupled-wave analysis (RCWA) program is shown to be an efficient and accurate tool for understanding the optical behavior of 2D HCGs with various structural and incidence parameters. We further demonstrate the design rules for various optical applications using 2D HCGs, based on the HCG mode properties we obtained from RCWA. Using our dual-mode analysis, the design process can be largely simplified. Once our high-performance initial design is obtained, it can be optimized according to the desired functionalities, such as broadband reflection, high-Q resonance, filtering, polarizing, etc. At last, we discuss the design of phase plates using 2D HCGs. Our designed HCG phase plates can function as conventional optical components, such as lenses, deflectors, axicons, spiral phase shifters, with excellent agreement and ultra-high performance.