1. Introduction

Flat optics [1–5] paves a new way to achieve high-efficiency wave manipulation within subwavelength thicknesses that are distinctly different from their conventional bulky counterparts [6–8]. Ease of fabrication, scalability of design, and convenience of integration make them a promising candidate, especially for long-wave applications. Among them, the subwavelength gratings (SWGs) have been extensively studied for a wide range of optoelectronic device applications, including lasers [9,10], filters [11,12], sensors [13–15], and beam splitters [16,17], etc. SWGs are composed of subwavelength periodic structures that enable highly efficient wave manipulation. Through engineering the SWGs’ parameters, such as the duty cycle, grating depth, and tapered sidewalls, extraordinary characteristics including high-efficiency wideband reflection or transmission [18], narrowband filtering [19], and ultrathin photodetector [20] can be obtained.

In order to achieve high-efficiency wave manipulation, several working mechanisms utilizing guided-mode resonance (GMR), Fabry-Pérot (F-P) resonance, and metagratings have been adopted in SWGs. SWG featuring GMR is composed of a dielectric grating deposited on a thin film waveguide and a substrate [21,22]. By specifically designing the grating so that the diffraction orders of the grating can be coupled with the propagating waveguide modes. Some of these guided modes are diffracted out of the waveguide and coupled back to radiation, and subsequently interfere with the uncoupled waves, leading to high-efficiency reflection or transmission. SWG featuring F-P resonance is composed of a single grating or bi-layered gratings to form an F-P cavity [23–26]. Due to the F-P interference between the waveguide array modes (WAMs) within the gratings or between the bi-layered gratings, high-efficiency reflection or transmission can occur. Metagrating [27,28] is a recently proposed concept to achieve high-efficiency wave manipulation. The electromagnetic response of each unit cell is specifically tailored to suppress undesired diffraction orders and reroute the incidence toward the desired diffraction order with high efficiency. Based on the concepts of GMR, F-P resonance, and metagratings, many dielectric optical components including wavelength-selective mirrors [29,30], bandpass filters [31–33], and polarizers [34,35] have been developed.

SWG featuring near-field Moiré effect by superimposing two one-dimensional SWGs with different periods has also been reported [36]. By inserting a silver slab between the two SWGs, the evanescent field can be enhanced via surface plasmon polariton (SPP) excitation. However, this wave manipulation scheme suffers large Ohmic losses within near-infrared and visible wavelength ranges due to the metallic material used for near-field enhancement. This could limit the device’s efficiency and hinder its high optical power applications due to thermal heating. Therefore, all-dielectric bi-layered SWGs incorporating near-field coupling that can support distinct optical resonances and high-efficiency wave manipulation are much more desired.

In this paper, a new strategy is proposed for designing the bi-layered composite grating with high first-order diffraction efficiency (∼90%). The bi-layered composite grating is formed by superimposing two rectangular dielectric SWGs with different grating periods, which are placed close to their near-field region to incorporate the near-field coupling effect. Oblique incidence is coupled with WAM of the bottom grating via wavevector-matching enabled by the top grating. Then, the resonant WAM is out-coupled to the first-order far-field diffraction of the composite grating, resulting in high-efficiency diffraction in the reflection region. The coupling might be similar to that of Wood’s anomalies on gratings [37]. Based on this concept, an effective design method is developed to obtain bi-layered composite gratings with high-efficiency first-order diffraction that fits over a broad spectral range. Simulation results of the bi-layered composite grating designs operated in both TE and TM modes are presented to verify the proposed design method. Moreover, parametric dependencies of these designs are also investigated. The bi-layered composite gratings could provide tunable bandwidth and adjustable wavelength functionalities by controlling the gap spacing of the SWGs and the incident angle of the input light, thus showing great application potential in sensing and communication. We envision that the design method proposed here could advance the tunable diffractive optical elements (DOEs) devices and promote their applications.

2. Configuration of the bi-layered composite grating

A cross-section of a unit cell of the proposed bi-layered composite grating is schematically shown in Fig. 1(a). The bi-layered composite grating locates in the x-y plane and the grating bars lie along the y axis. Each unit cell consists of two sets of free-standing parallel silicon SWGs separated by an air gap with gap spacing g. The gap spacing is set smaller than the incident wavelength to ensure the near-field coupling effect. The SWGs in the upper and lower layer are denoted by SWG1 (gray) and SWG2 (blue), respectively. Each SWG has the same crystalline-silicon (c-Si) film thicknesses t_i but different grating periods Λ_i, where i = 1, 2. The duty cycles of the gratings f_i are expressed as a_i /Λ_i with a_i as their silicon beam widths. SWG1 and SWG2 form an equivalent composite grating with grating period Λ. The composite grating is polarization-sensitive by its nature of 1D periodicity. Incident beams with E-field polarization parallel and perpendicular to the grating bars are referred to transverse electric (TE) and transverse magnetic (TM) polarizations, respectively. With a proper oblique incidence of a free-space linearly polarized (TE or TM) input beam onto the composite grating, diffraction beams can be found in the reflection region. By specifically designing the grating structure, high-efficiency diffraction (∼90%) of the first-order can be achieved. The periods of the SWGs are carefully designed so that at the resonant wavelength λ_R, the WAM in SWG2 would be fully excited by the incident light via SWG1 through wavevector matching.

 

Fig. 1. (a) Cross-section of the unit cell of the proposed bi-layered composite grating. (b) Calculated diffraction efficiency spectrum of the 0^th order and −1^st order diffraction in the reflection region. Insets: y component electric field profile at $\bar{\lambda } = 1.775$ (left) and at $\bar{\lambda } = 1.785$ (right). The color scales are normalized by the modulus of the maximum electric field amplitude. The normalized structural parameters for this plot are: ${\bar{\Lambda }_1} = 1/2$, ${\bar{\Lambda }_2} = 1/3$, ${\bar{t}_1} = {\bar{t}_2} = 0.9$, $\bar{g} = 0.115$; f₁ = f₂ = 0.5, n_si = 3.47, θ = 63°, TE polarization of incidence.

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We employ the finite element method (FEM) method to numerically investigate the spectral response of the composite grating. In our simulations, the unit cell shown in Fig. 1(a) is built. To achieve high simulation accuracy, we use one unit cell of the composite grating with an equivalent grating period of Λ in the simulation. In the x- and y-directions of the simulation region, periodic boundary conditions are applied, while in the z-direction, perfectly matched layers (PMLs) are adopted. For PML regions a mapped mesh is defined, while for other regions a free triangular mesh is used. The numerical calculations are performed with a good convergence condition by setting the maximum mesh size to be λ/20. The unit cell is illuminated with a broadband linearly polarized (TE or TM) plane wave with an incident angle of θ. The dispersion effect of the c-Si film is modeled with its refractive index n_si measured by ellipsometry [38]. Without loss of generality, the structural parameters are normalized in the following manner: ${\bar{\Lambda }_i} = {\Lambda _i}/\Lambda $, ${\bar{t}_i} = {t_i}/\Lambda $, ${\bar{a}_i} = {a_i}/\Lambda $, $\bar{g} = g/\Lambda $ and the working wavelength is normalized to be $\bar{\lambda } = \lambda /\Lambda $. In this way, all parameters become dimensionless and can be applied to the whole spectrum of electromagnetic waves.

In Fig. 1(b), we present our numerical results on the far-field diffraction efficiency in the reflection region of the composite grating to illustrate the working mechanism. The composite grating structure is obtained by the design method we developed (see Section 4). The normalized device structural parameters are given in Fig. 1. A pronounced high-efficiency –1^st order diffraction, manifesting a peak value of 89% with full width at half-maximum (FWHM) line width $\varDelta \bar{\lambda } = 0.01$ at $\bar{\lambda } = 1.785$(resonant wavelength) can be observed in Fig. 1(b). It can be seen from the right inset of Fig. 1(b) that at $\bar{\lambda } = 1.785$, TE₀ WAM in SWG2 is strongly excited. On the contrary, the left inset shows weak field confinement in the grating layers at $\bar{\lambda } = 1.775$. Thus, the formation of the significant high-efficiency –1^st order diffraction can be attributed to the coupling between the incident TE plane wave and TE₀ WAM in SWG2.

3. Principle of high-efficiency diffraction by the bi-layered composite gratings

In this section, we explore the physics behind the high-efficiency first-order diffraction of the proposed bi-layered composite gratings. First, the oblique incidence is coupled to a WAM of SWG2 via wavevector-matching enabled by SWG1. Then, the resonant WAM of SWG2 is out-coupled to the first-order far-field diffraction via wavevector-matching by SWG1 for a second time. Meanwhile, all the other diffraction orders (0^th order not included) are cut-off, resulting in high-efficiency first-order diffraction in the reflection region.

In the proposed bi-layered composite grating, the resonant SWG2 can be treated as a 1D photonic crystal operating at the bands below the light line [39]. It can be seen in the insets of Figs. 2(a) and 2(b) that SWG and 1D photonic crystal share similar TE₀ and TM₀ mode profiles, indicating that they can be unified with the same theoretical structure. By calculating the band diagram of SWG with a given grating profile, the normalized angular frequency ω of each WAM as functions of the normalized wavevector k_x can be obtained [shown in Figs. 2(a) and 2(b)]. These bands are symmetric about k_x = 0.5 due to Brillouin zone folding. We denote the WAMs as TE₀, TE₁, TE₂, etc. for TE incidence; and TM₀, TM₁, TM₂, etc. for TM incidence [40].

 

Fig. 2. Principle of high-efficiency diffraction of the bi-layered composite grating. (a) Calculated band diagram for the TE WAMs. Insets: normalized electric field profile exhibiting TE₀ mode profile of (i) SWG and (ii) 1D photonic crystal. (b) Calculated band diagram for the TM WAMs. Insets: normalized magnetic field profile exhibiting TM₀ mode profile of (i) SWG and (ii) 1D photonic crystal. The shaded regions represent radiated modes. Parameters for (a) and (b) are ${\bar{\Lambda }_2} = 1/3$, ${\bar{t}_2} = 0.9$, f₂ = 0.5 and n_si = 3.47. (c) Schematic of the wavevector-matching enabled by the near-field coupling between SWG1 and SWG2. (d) Wavevector diagram sketching the wavevector-matching conditions. Parameters for (c) and (d) are ${\bar{\Lambda }_1} = 1/2$, ${\bar{\Lambda }_2} = 1/3$, $\bar{\lambda } = 1.782$, θ = 63°, θ_d = 63°. The color scales are normalized by the modulus of the maximum electric field amplitude.

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For a basic understanding of the physics behind high diffraction efficiency, the schematic shown in Fig. 2(c) provides an intuitive insight into the principle. The grating parameters are carefully designed such that TE₀ WAM at $|{{k_x}} |= 0.5$ is coupled with the incident wave. When a linearly polarized (e.g., TE) plane wave obliquely incidents onto the bi-layered composite grating with an incident angle of $\theta$, the x-wavevector of the incoming light can be expressed as $k_{in}^\parallel{=} {k_0}\sin \theta$, where k₀ = 2π/λ₀ is the wavevector of the incident wave in free space with wavelength λ₀. After being diffracted through SWG1, an additional wavevector component of 2πm₁/Λ₁ will be added onto the incident x-wavevector, where m₁ is the diffraction order of SWG1 and we choose m₁ = −1 here. If the joint x-wavevector matches the wavevector of the WAM in SWG2 at k_x = −0.5, which we call the in-couple condition:

Light can be fully coupled to the WAM of SWG2 when the in-couple condition is satisfied. Then, the excited WAM in SWG2 will be further diffracted upwards or downwards by SWG2 and a wavevector component of 2πm₂/Λ₂ will be added onto the joint x-wavevector, where m₂ is the diffraction order of SWG2. The downward diffracted light cannot be further out-coupled to the far-field transmission region since the joint x-wavevector cannot match the wavevector in free space for any number of m₂. On the contrary, as for the upward diffraction, the out-couple condition can be fulfilled when the joint x-wavevector picks up another x-wavevector component 2πm′₁/Λ₁ provided by SWG1 where m′₁ is the order of diffraction by SWG1 for a second time.

In order to develop an effective way to describe the wavevector-matching conditions, a conceptual bi-layered composite grating model based on a wavevector diagram is shown in Fig. 2(d). The circle in the diagram represents the allowed propagation wavevector in free space. The horizontal arrows represent the x-wavevectors of the incident and diffracted waves as well as the grating vectors. The wavevector-matching conditions can be described by consecutively adding the grating vectors to the x-wavevector of the incidence by the head-to-tail method. Firstly, the x-wavevector of the incident wave $k_{in}^\parallel$ (red dashed arrow) is diffracted by the –1^st order of SWG1 (blue arrow), which fulfills the in-couple condition and leads to the excitation of a selected WAM in SWG2. Then, the wave is diffracted by the +1^st order of SWG2 (green arrow). At this point, the joint x-wavevector is greater than the allowed propagation wavevector in free space thus leading to the cut-off of all diffractions in the transmission region except the 0^th order. In the reflection region, on the other hand, the joint x-wavevector can pick up one additional –1^st order grating vector from SWG1 thus leading to the final joint x-wavevector (yellow arrow) that can be coupled to free space thus leading to the high-efficiency –1^st diffraction of the composite grating.

According to the above-analyzed wavevector-matching conditions, except for the –1^st order diffraction in the reflection region, all the other diffraction orders (0^th order not included) are cut-off. Hence, such high-efficiency –1^st order diffraction is mainly attributed to the wavevector-matching enabled by near-field coupling between SWG1 and SWG2. From the wavevector diagram in Fig. 2(d), it is clear that the out-couple condition is fulfilled in the reflection region when m₁ = −1, m₂ = +1, and m′₁ = −1, thus leading to:

The second half of Eq. (2) is the well-known grating equation representing the –1^st order diffraction of the equivalent bi-layered composite grating with diffraction angle θ_d. Here, from the above equation, the equivalent period Λ of the composite grating is defined as:

In the following design, the ratio of Λ₁/Λ₂ is chosen as 3/2 to simplify the introduction of the design method. For other ratios of the SWG1 and SWG2 periods, high-efficiency –1^st order diffraction can also be obtained by carefully selecting their grating periods following the wavevector-matching condition and optimizing their grating profiles.

4. Design method of the bi-layered composite gratings with high diffraction efficiency

To create a bi-layered composite grating with high-efficiency first-order diffraction, major structural parameters including the normalized grating periods, ${\bar{\Lambda }_i}$, the grating profile of SWG2 including t₂/Λ₂ and a₂/Λ₂, and the normalized gap spacing between the gratings $\bar{g}$ should be taken into consideration. Fig. 3 shows a flowchart of the proposed design approach.

 

Fig. 3. Flowchart of the proposed design procedure to obtain the bi-layered composite gratings with high-efficiency first-order diffraction.

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The design procedure is described as follows:

5. Results and discussions

To verify the proposed design method, two examples that employ respectively the TE₀ and TM₁ WAMs in the SWG2 are numerically investigated to achieve high-efficiency –1^st order diffraction of the composite grating. We design the bi-layered composite gratings operate at wavelengths around 1550 nm. The refractive index of the c-Si measured by ellipsometry is 3.47 within this wavelength range, and the extinction coefficient of the c-Si is around 10⁻⁶, which is negligible in our numerical simulations. We then calculate the –1^st order diffraction efficiency of the composite grating and run a parametric sweep to investigate the parametric dependence of both designs.

5.1 Design of the TE₀ mode based bi-layered composite grating

We first design a TE₀ mode based composite grating with its –1^st order diffraction efficiency as high as 90% in the reflection region. In this design, the grating profile of SWG2 is set to be t₂/Λ₂ = 0.897 and a₂/Λ₂ = 0.5. For the simplicity of the design, the grating period ratio Λ₁/Λ₂ is chosen at 3/2. As a result, the structural parameters of the bi-layered composite grating are determined to be Λ₁ = 435 nm, Λ₂ = 290 nm, Λ = 870 nm, t₁ = t₂ = 260 nm, and f₁ = f₂ = 0.5 according to the desired working wavelength around 1550 nm. Thus, the incident TE wave will be fully coupled with the TE₀ mode of SWG2 leading to high-efficiency –1^st order diffraction in the reflection region of the composite grating.

To make a systematic discussion of the TE₀ mode based bi-layered composite grating, the dependences of the –1^st order diffraction on various structural parameters are studied. Firstly, we investigate the influence of the variation of gap spacing on the –1^st order diffraction spectrum of the composite grating while all the other parameters are fixed. As shown in Fig. 4(a), the TE₀ mode supports a high-efficiency and narrow-bandpass performance. As the gap spacing increases from 50 nm to 300 nm when the incidence angle θ is 63° (corresponds to k_x = 0.5), the diffraction efficiency decreases from 89% to 69% with a decreased FWHM (Δλ) from 6.4 nm to 0.1 nm.

 

Fig. 4. (a) Simulated –1^st order diffraction efficiency spectrum of the composite grating in relation to the variation in gap spacing g when θ = 63° Insets: diffraction efficiency spectrum when g = 100 nm (top) and normalized electric field profile when g = 100 nm and λ = 1552 nm (bottom). The color scale is normalized by the modulus of the maximum electric field amplitude. (b) Simulated –1^st order diffraction spectrum of the composite grating in relation to the variation in incident angle θ when g = 100 nm. Inset: a plot of diffraction efficiency as a function of θ at λ = 1552 nm (marked by dotted gray line). (c) Calculated resonant wavelength of TE₀ mode when θ = 63° as a function of n_eff. (d) Calculated band diagram of the TE₀ mode of SWG2. The range of k_x when the incidence angle θ varying from 50° to 85° is marked by the black dotted lines.

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This phenomenon can be attributed to the near-field coupling of evanescent waves. Since the TE₀ WAM is predominantly confined within SWG2 [see the bottom inset of Fig. 4(a)], an increase of gap spacing could weaken the coupling between SWG1 and SWG2, thus resulting in a decrease in diffraction efficiency and narrowing in bandwidth. Moreover, as g increases, a monotonous blueshift of peak wavelength can also be found. This characteristic can be roughly estimated by modeling the upper region of SWG2 as an isotropic medium with a refractive index of n_eff according to the effective medium theory (EMT) [41]. With an increase of g, the n_eff will be decreased, thus resulting in a blueshift of the resonant wavelength of the TE₀ mode as shown in Fig. 4(c). In summary, by controlling the gap spacing, the diffraction efficiency and bandwidth of the composite grating can be adjusted.

Next, a calculated incident-angle-dependent –1^st order diffraction spectrum is provided in Fig. 4(b). Setting at a gap spacing of 100 nm between two SWGs, when θ varies from 50° to 85°, the spectrum shows significant stability in the diffraction efficiency and bandwidth. It is interesting to observe from the inset of Fig. 4(b) that the spectrum exhibits certain tolerance against the incident angle variation. The calculated FWHM angular range (Δθ) is 17.6° at λ = 1552 nm with a maximum diffraction efficiency of 89%. Such an angular characteristic can be attributed to the flat angular dependence response of the TE₀ mode at its band edge [seen from Fig. 4(d)]. This characteristic might be useful in applications requiring high angular stability.

Lastly, by scanning the duty cycles and thicknesses of the constituent SWGs while fixing all the other parameters, the influences of the SWG grating profiles on the –1^st order diffraction efficiency of the bi-layered composite grating are studied. The incident angle θ is fixed at 63° and the gap spacing g between the two SWGs is kept at 100 nm. As shown in Figs. 5(a) and 5(c), the performance of the composite grating shows remarkable robustness against minor variations in the grating profile of SWG1. On the contrary, as shown in Figs. 5(b) and 5(d), the performance of the composite grating is quite sensitive to the variations in the grating profile of SWG2. The explanation to this phenomenon is as follows: since SWG1 in the bi-layered composite grating system serves as a wavevector matcher as can be seen from Eqs. (1) and (2), it is the grating period rather than the grating profile of SWG1 that affects the wavevector-matching conditions. In comparison, changes in the grating profile of SWG2, where the resonant TE₀ WAM primarily resides, can affect the band structure and shift the resonant frequency of the WAM. As a result, the –1^st order diffraction efficiency spectrum of the bi-layered composite grating is more sensitive to the variations of the SWG2 grating profile.

 

Fig. 5. Parametric dependences of the designed TE₀ mode based bi-layered composite grating. Simulated –1^st order diffraction efficiency spectra of the composite grating in relation to the variations in (a) and (b) grating thicknesses and (c) and (d) duty cycles of SWG1 and SWG2 at θ = 63° and g = 100 nm. Dotted white lines in (a)–(d) represent the designed composite grating parameters.

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5.2 Design of the TM₁ mode based bi-layered composite grating

In the following design example, a high-efficiency TM₁ mode based bi-layered composite grating with its –1^st order diffraction efficiency as high as 90% is presented. In this design, the grating profile of SWG2 is set to t₂/Λ₂ = 0.795 and a₂/Λ₂ = 0.66. The grating period ratio Λ₁/Λ₂ is also chosen to be 3/2. As a result, structural parameters of the bi-layered SWGs are determined to be Λ₁ = 746 nm, Λ₂ = 497 nm, Λ = 1492 nm, t₁ = t₂ = 395 nm, f₁ = f₂ = 0.66 according to our proposed design procedure with its working wavelength setting at 1550 nm.

We also investigate the dependence of the composite grating performance on the gap spacing and incident angle in a similar way as presented in Section 5.1. As shown in Fig. 6(a), when the gap spacing increases from 250 nm to 600 nm with the incident angle remained at 32° (corresponds to k_x = 0.5), broadband diffraction will occur from 1.52 μm to 1.58 μm when g = 300 nm [see the bottom inset of Fig. 6(a)]. The bandwidth of the spectrum is gradually narrowed with the increase of g. This phenomenon can also be attributed to the near-field coupling of evanescent waves. As shown in the top inset of Fig. 6(a), the TM₁ WAM decays relatively slowly into the near-field of SWG2. Therefore, the coupling between SWG1 and SWG2 in a TM₁ mode composite grating is more tolerant of a larger gap spacing g when compared with a TE₀ mode composite grating.

 

Fig. 6. (a) Simulated –1^st order diffraction efficiency spectrum of the TM₁ mode based composite grating in relation to the variation in gap spacing when θ = 32° Insets: normalized electric field profile when g = 300 nm and λ = 1544 nm (top) and diffraction efficiency spectrum when g = 300 nm (bottom). The color scale is normalized by the modulus of the maximum electric field amplitude. (b) Simulated –1^st order diffraction spectrum of the composite grating in relation to the variation in incident angle when g = 300 nm. Inset: a plot of diffraction efficiency dependency on θ at λ = 1529 nm (marked by dotted gray line) showing high angular tolerance. (c) Calculated band diagram of the TM₁ mode of SWG2. The range of k_x when θ varying from 10° to 60° is marked by the black dotted lines.

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Next, a calculated incident-angle-dependent –1^st order diffraction spectrum is provided in Fig. 6(b). When θ varies from 10° to 60° with the gap spacing g fixed at 300 nm, the high-efficiency region forms a curved strip as shown in the figure. The diffraction efficiency reaches its maximum (89.5%) at the center of the strip, where θ = 32° and λ = 1548 nm. This corresponds to the k_x = 0.5 and ω = 0.33 point in the TM₁ mode band diagram shown in Fig. 6(c). The band edge of the TM₁ mode exhibits a concave curve when θ varying from 10° to 60° [marked by the black dotted lines in Fig. 6(c)], which represents the variation trajectory of the resonant frequency of the TM₁ mode in the frequency domain. Therefore, the concave curve of TM₁ mode in the frequency domain will translate into a convex strip region of high-efficiency –1^st order diffraction in the wavelength domain. Moreover, the spectrum exhibits a high tolerance against the incident angle variation, and the calculated FWHM angular range (Δθ) is 42.5° at λ = 1529 nm with its maximum diffraction efficiency of 85% [see the inset of Fig. 6(b)].

Lastly, by scanning the duty cycles and thicknesses of the constituent SWGs while keeping all the other parameters unchanged, the influences of the SWG grating profiles on the –1^st order diffraction efficiency of the bi-layered composite grating are studied when θ and g are kept at 32° and 300 nm, respectively. As shown in Figs. 7(a) and 7(c), the performance of the composite grating shows good robustness against minor variations in the grating profile of SWG1. On the contrary, as shown in Figs. 7(b) and 7(d), the –1^st order diffraction efficiency of the composite grating is more sensitive to the variations in the grating profile of SWG2 as expected.

 

Fig. 7. Parametric dependence of the designed TM₁ mode based composite grating. Simulated –1^st order diffraction efficiency spectra of the composite grating in relation to the variations in (a) and (b) grating thicknesses and (c) and (d) duty cycles of SWG1 and SWG2 respectively when θ = 32° and g = 300 nm. Dotted black lines in (a)–(d) represent the designed composite grating parameters.

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5.3 Fabrication discussion

The proposed designs can be fabricated using CMOS-compatible micro/nano fabrication processes. First, a layer of the dielectric SWG pattern (SWG2) can be achieved on a silicon-on-insulator (SOI) wafer by lithography and dry etch. Next, a silicon dioxide (SiO₂) layer can be deposited on the patterned SOI wafer, planarized using chemical mechanical polishing (CMP), and thinned down to the desired thickness. The SiO₂ layer acts as a sacrificial layer to achieve the required gap spacing between the two SWGs. Next, a thin silicon layer is deposited on the SiO₂ layer. With another lithography and dry etch process, another layer of SWG pattern (SWG1) can be obtained. Finally, the structure can be released through HF etching and CO₂ critical drying to remove the sacrificial SiO₂ layer and the buried oxide layer. Thus, the designed bi-layered composite grating structure can be achieved.

For the proposed bi-layered structure, the effects of alignment errors between the upper and lower layer are also investigated. The relative position of the bi-layered structure will be determined by the alignment of the two lithography steps during the device fabrication. Considering the matured micro/nano fabrication technology, alignment accuracy down to sub-micron scale can be easily achieved nowadays [42]. Misalignment occurring in the y direction [see Fig. 1(a)] makes no difference to the –1^st order diffraction because the composite grating structure can be treated as sufficiently long in the y direction. On the other hand, for the misalignment occurring in the x-direction Δx, the influence of Δx on the –1^st order diffraction efficiency of the designed TE₀ and TM₁ mode based composite gratings is investigated and the simulation results are provided in Figs. 8(a) and 8(b), respectively. It is obvious that both designs show significant robustness against the misalignment occurring in the x-direction.

 

Fig. 8. Simulated –1^st order diffraction efficiency spectra of the designed (a) TE₀ and (b) TM₁ mode based composite gratings in relation to the variations in misalignment of the two grating layers in the x direction Δx. Parameters for plot (a) are Λ₁ = 435 nm, Λ₂ = 290 nm, Λ = 870 nm, t₁ = t₂ = 260 nm, f₁ = f₂ = 0.5, θ = 63° and g = 100 nm; Parameters for plot (b) are Λ₁ = 746 nm, Λ₂ = 497 nm, Λ = 1492 nm, t₁ = t₂ = 395 nm, f₁ = f₂ = 0.66, θ = 32° and g = 300 nm.

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6. Conclusion

In this letter, we propose a new strategy to design the bi-layered composite gratings with high-efficiency first-order diffraction in the reflection region. Although metasurface could also diffract the incident light to a desired order diffraction with high efficiency [43–45], the proposed bi-layered composite gratings are different from metasurface in both design method and working mechanism. The proposed composite grating structure is composed of two different SWGs with thicknesses smaller than the operation wavelength, making the composite grating compact and integration-friendly. We build a simple and intuitive theoretical model to describe the bi-layered composite grating that yields high-efficiency first-order diffraction. Due to near-field coupling, oblique incident light is coupled with the WAM of the bottom grating by satisfying wavevector-matching conditions enabled by SWG1 and out-coupled to the first-order diffraction in the far-field reflection region. An effective design procedure based on the theoretical model is developed to obtain the desired bi-layered composite grating with high efficiency. In order to verify the proposed design method, two composite grating designs working on the TE₀ or TM₁ mode are demonstrated. Through controlling the gap spacing and the incident angle, both of the designed composite gratings could provide adjustable wavelength responses and show potentials in tunable bandwidth and diffraction efficiency. These results demonstrate the feasibility of the proposed design strategy for high-efficiency wave modulations. Such a bi-layered dielectric thin film structure can be effectively fabricated and characterized by several well-established techniques [46]. We believe that the rich properties of the bi-layered composite gratings could pave the way for the novel design of flat optical devices and aid in their future applications.