Design and analysis of broadband guided-mode resonant reflectors with coated triangular and trapezoidal profiles in TE polarization

: A low-refractive-index grating layer with symmetrical triangular/trapezoidal grooves covered with a high-refractive-index Si layer is used to design a broadband guided-mode resonant reflector. Software Rsoft is used to simulate the reflection and transmission spectra as well as the internal electric field distribution at the resonant wavelength. It is discovered that the interaction between resonant modes promotes the formation of a wideband spectrum. The reflector has been proven to provide wideband ( ∆λ > 450 nm) and high reflectivity (R > 98.4%) spectra over a wide range of base angles from 44° to 72°, and the maximum high reflectivity (R > 99%) spectral range in transverse electric polarization is 458 nm, spanning 1422 to 1880nm. The results not only demonstrate excellent tolerance to the base angle and grating depth but also provide more possibilities for the design of broadband reflectors.


Introduction
Most optical mirrors with high reflectivity have been extensively investigated in the fields of imaging, telecommunications, and sensors.Traditional metal mirrors have low reflectivity owing to the limitations of material absorption loss.Dielectric stacking mirrors require 10-100 quarterand/or half-wave layers to achieve high reflectivity, and the layer thickness and refractive index are strictly controlled.To improve the reflectivity and reduce the number of layers, photonic crystals, subwavelength gratings, and metamaterial structures [1][2][3][4][5] have been developed to replace metal reflectors and dielectric stack reflectors.In particular, the periodic subwavelength grating reflector, also known as the guided-mode resonance (GMR) reflector, can induce the GMR effect and yield high reflectivity though a few layers with optimal parameters [2,[6][7][8][9].In addition, a reflectivity close to unity in a broadband range can be achieved by the excitation and interaction of multiple guided-mode resonances.These facts indicate that GMR reflectors offer the advantages of simple structure and ultra-high diffraction efficiency; hence, they are used extensively in lasers, wave plates, detectors, and couplers [10][11][12][13][14][15][16].
Transverse magnetic (TM) and transverse electric (TE) grating reflectors based on GMR effects have been extensively investigated previously.The polarization-dependent boundary condition of GMR grating determines the difference of the distribution of resonant modes in TE and TM, which leads to the differences in optimal parameters and spectrum performance [17][18].The modal curves in the TM case present sufficient curvature and spectral proximity [17], and are more capable of providing broadband reflection function than TE case, so the research of broadband GMR reflectors is mainly focused on TM polarization.For example, Chang-Hasnain et al. designed a high-contrast grating (HCG) structure [9] that yielded high reflectivity (R > 99%) over a wide spectral range of approximately 520 nm in TM case, and the large refractive index difference between materials contributed to the generation of a broadband spectrum.By contrast, Magnusson et al. proposed zero-contrast grating (ZCG) reflectors, in which the refractive index of the grating layer matched the homogeneous layer, and the bandwidth was reported to be ∼700 nm [8].Subsequently, based on the two structures above, researchers designed many ultra-wideband grating reflectors with different profiles, periods, and materials in TM polarization, such as multi-period [19], three-layer [20], hybrid [21], double-layer subwavelength [22], and symmetric triangle/trapezoid grating reflectors [23][24].For the TE polarization, a multi-subpart profile grating reflector with four subparts was proposed to demonstrate a bandwidth of 630 nm [25].However, the bandwidths of the simpler two-subpart periodic ZCG [26] and HCG [2] were narrower, i.e., 380 and 125 nm, respectively.Herein, we propose a new GMR grating structure for TE polarization to ameliorate this condition.
GMR reflectors with triangular/trapezoidal profiles comprising a low-refractive-index grating layer and a high-refractive-index film layer are proposed herein.We point out that although they are all broadband reflectors with triangular or trapezoidal groove, the waveguide layer structure and the polarization performance presented here are completely different from those in [24].We simulated the mode evolution corresponding to wideband reflection spectra formation and discovered that the number and location of resonance modes changed with the structural parameters, thereby realizing reflection bandwidth tuning.It has been proven that the structure shows not only a broadband high reflectivity spectrum in a wide range of base angles under TE polarization, but also a better tolerance to the device parameters.To the best of our knowledge, for the simpler two-subpart GMR reflectors reported for TE polarization, our design offers the best performance in terms of bandwidth and reflectivity.

Device structure
The proposed GMR reflectors are shown in Fig. 1.The low-refractive-index SiO 2 layer (n s = 1.48) serves as a substrate and a symmetrical triangular and trapezoidal groove grating with a period of Λ, base angles α and β, and depth of groove d g .The yellow high-refractive-index Si material (n = 3.48) covered above acts as a waveguide layer with a height of d L , with a gradient distribution of equivalent refractive index.
For the rectangular groove broadband reflectors with fixed index, the structural difference between TE [26] and TM [8] polarization is reflected in the parameters: it has smaller period and larger thickness for TM reflectors [18].However, the structural difference of triangular/trapezoidal groove broadband reflectors with gradient index is reflected not only in the parameters, but only in the distribution of equivalent refractive index along the z-direction.The effective refractive for each layer of broadband triangular grooves grating with same parameters in TE and TM polarization using effective medium theory are shown in Fig. 2. The TM reflectors in [24] is topped with a triangular groove grating with a gradient equivalent refractive index from n to n c , but it shows different concavity than the top grating of TE reflectors designed here due to the difference of the equivalent refractive index as expressed in Eq. ( 1) and (2).On this basis, TE reflectors additionally introduce a low refractive index triangular groove grating at the bottom of the structure, with an effective refractive index gradient opposite to the top grating, from n s to n.Therefore, the TE reflectors can be equivalent to an inverted double grating structure along the z-direction.Nano-imprinting fabrication method is considered to implementing the coated triangular and trapezoidal profiles grating which is technically easier than mechanical ruling considered to fabricate the grating in [24].

Evaluation of resonant modes
"DiffractMOD" in RSoft was used to obtain the optimal parameters and field distribution of the GMR grating.DiffractMOD is a tool for the design of diffractive optical structures, in which the rigorous coupled-wave analysis (RCWA) method [27] is used to simulate the electromagnetic wave diffraction of periodic structures.The incident plane wave can have an arbitrary direction and polarization, and all types of simulation results can be output, including diffraction efficiency, near field, and far field results.
We remark that the period Λ, base angle α, and groove thickness d g of the reflector with a triangular profile satisfy d g = Λ tan(α/2).Moreover, the grating groove will evolve from a triangle to a trapezoid with a decrease in the thickness of the grating layer d g , while maintaining a constant base angle.For simplification, we fixed the thickness of the high-refractive-index film to 400 nm, and the two base angles of the triangular and trapezoidal grooves were symmetrical.
Figures 3(a)-(h) present the transmission contour maps T 0 (λ, d g ) on a logarithmic scale with the wavelength and groove depth for the base angles of 20°, 40°, 44°, 45°, 46°, 60°, 72°, and 75°, respectively.In these contour maps, the red region represents high transmission, and the yellow region represents high reflectivity caused by the GMR.As shown in Fig. 3(a), at α = 20°, three vertically distributed modes were observed, i.e., at 1.2, 1.6, and 2.4 µm, and their positions were almost independent of the grating thickness.With an increase in the base angle, a new mode appeared, and the modes at both ends shifted to the middle.However, the modes distributed at different grating depths shifted at different speeds, resulting in the bending of the modes, as shown in Fig. 3(b).When the three modes shifted by an appropriate distance, they interacted with each other to produce a high reflection spectrum (R > 99%) from 1430 to 1882nm at the maximum grating layer thickness, which corresponds to a broadband reflector with a triangular groove, as shown in Fig. 3(c).When α = 45°, as shown in Fig. 3(d), the two modes on the right region combined to form a mixed mode, and when α = 46°, as shown in Fig. 3(e), the mixed mode bent and shifted downward, forming a broadband spectrum at d g = 0.326 µm.Subsequently, the mode shown on the left side of Fig. 3(f) bent in the long-wavelength direction under the attraction of the mixed mode, and the distance between the two modes decreased gradually as the base angle increased.At α = 72°in Fig. 3(g), the two modes shifted by the least distance, resulting in a strong interaction between them; hence, an ultra-high reflectivity (R > 99.9%) can be achieved in a broadband range (308 nm).As illustrated in Fig. 3(h), the resonant modes rearranged and combined to form new mode distributions, and the two new modes repel each other, thereby weakening the interaction between them; consequently, the characteristics of wideband and high reflection of the GMR reflectors will not be maintained.
The GMR reflectors with triangular or trapezoidal profiles can yield high reflectivity over a wide bandwidth, and the characteristics are jointly contributed by the three resonant modes.The strong interaction between the close resonant modes ensures high reflectivity but reduces the bandwidth, whereas the weaker interaction between the distant resonant modes yields a wider bandwidth but a lower reflectivity.The relative distance between the resonant modes can be controlled by the structural parameters, such as the base angle and grating thickness, to tune the bandwidth and reflectivity.In particular, the structure parameters corresponding to the broadband reflection are not unique, but can be realized for any base angles between 44°and 72°, as shown in the yellow area in Fig. 3.

Result
The GMR reflector can realize wideband reflection over a wide range of base angles.To highlight this point, we simulated the reflection and transmission spectra at different base angles, and only three representative values were selected for simplification.
Figure 4 shows the optimized reflection and transmission spectra on the linear and logarithmic scales of the GMR reflectors with the coated trapezoidal groove at α = 46°, 60°, and 72°.As shown in Fig. 4(a), the GMR reflectors with optimized parameters α = 46°, and d g = 326 nm exhibited high reflectivity (R > 99%) over a 458 nm bandwidth in the 1422-1880nm range.In the band range, three transmission dips were observed, each of which represents a GMR, at λ = 1435, 1663, and 1857nm.The spectral bandwidths demonstrated exceeded previously reported results for simpler two-part periodic reflectors operating in TE polarization.This mainly arise from the fact that the incident light with different wavelengths can be effectively coupled into the waveguide layer at the cross-sectional area of different duty cycle.For the GMR reflector with optimized parameters α = 60°, and d g = 359 nm, as shown in Fig. 4(b), a high reflectivity (R > 98.4%) from 1416 to 1870nm spanning a 454 nm range was obtained.Similarly, three guided-mode resonances that occur at λ=1430 nm, 1678 nm, and 1824nm, and the distance between two modes on the short wavelength direction induces a decrease in local reflectivity at λ=1491 nm.As shown in the dotted curve in Fig. 4(c), when the base angle increased to 72°and the grating thickness increased to 475 nm, a strong interaction occurred between the resonant modes, resulting in ultra-high reflectivity (R > 99.9%) in a wide band range (308 nm) from 1493 to 1747nm.Moreover, when the grating depth was 448 nm, a high reflectivity (R > 98.4%) bandwidth of 464 nm from 1415 to 1879nm was indicated, as represented by the solid curve in Fig. 4(c).In addition, we noted that the broadband GMR grating reflector designed here only work in TE polarization, with several independent high reflection peaks in other polarization instead.Conversely, the broadband reflectors proposed in [24] only work in TM polarization, and present a few independent narrow reflection peaks under TE polarization.
The electric field amplitude distribution inside and around the grating at the resonant wavelengths is illustrated in Fig. 5.At the positions of the short resonant wavelength corresponding to Figs. 5(a), (d), and (g), the field was distributed in both the high-and low-refractive-index grating layers and was gradually pushed to the coated layer as the base angle increased.In Figs.5(b), (e), and (h), for λ = 1.663, 1.678, and 1.588 µm, the field was pushed to the coated layer and the boundary between the high-and low-refractive-index layers, showing the characteristics of TE 1 .For the long wavelength shown in Figs.5(c), (f), and (i), the field was only distributed at the bottom of the high-refractive-index layer.

Effect of grating thickness
The high reflectivity bandwidth of the reflectors was affected by the grating thickness because the position of the GMRs varied with d g .Figure 6(a) presents a zero-order reflectance contour map R 0 (λ, d g ) depicting the wavelength and grating depth for α = 46°, and d L =400 nm.When the grating depth d g was less than 320 nm, the reflection spectrum was partitioned into three regions and did not exhibit the characteristics of broadband reflection.As the grating depth increased, the resonant modes at both ends shifted closer to the middle gradually, and they interacted with each other to form a widest high reflection spectrum at d g = 326 nm.Furthermore, when the grating thickness d g was larger than 326 nm, the three modes continued to approach each other under the attraction force; therefore, the reflection bandwidth decreased as the grating thickness increased.To clarify the tolerance to grating depth, Figs.6(b) and (c) present the linear reflection and logarithmic transmission spectra of the reflector under different grating depths.The reflection bandwidths are listed in Table 1 corresponding to reflectivity greater than 99% and 97%, separately.For d g = 316 nm, as represented by the blue curve, the two resonant modes on the right were extremely distant from each other, causing the reflectivity to decrease to approximately 97% at λ = 1817nm; in this case, R > 97% can be achieved in the 510 nm band range from 1408 to 1918nm.The widest reflection bandwidth corresponding to d g = 326 nm was 458 nm; additionally, as the grating thickness increased to 388 nm in the purple curve, the proximity of the three resonant modes resulted in a narrowing of the reflectivity bandwidth, yielding R 0 > 99% across a 321 nm bandwidth.Therefore, the reflectors exhibited high reflectivity (R > 99%) over a wide range of grating depths (∆d g = 62 nm), and its reflection bandwidth decreased gradually from 458 to 321 nm as the grating thickness increased.Compared with the case of α = 46°, the two resonant modes on the right were opposite each other when α = 44°, as presented by the blue curve, which resulted in a wider bandwidth but a decrease in the local reflectivity at λ = 1882nm.However, when α = 48°, corresponding to the yellow curve, the two resonant modes on the right shifted toward each other, thereby ensuring high reflectivity but a narrower bandwidth.The bandwidth corresponding to reflectivity greater than 99%, 98%, and 96% are presented in Table 2.When the grating structure was asymmetric, as shown by the purple curve, i.e., α ≠ β, the characteristics of broadband reflection was preserved.
In this case, the reflectivity was greater than 98% from 1418 to 1901nm (spanning a 483 nm band range), which effectively reduced the effect of the manufacturing error of the grating base angle on the reflection spectrum.Table 3 shows that when the thickness of the coated layer changed by 10 nm, the variation range of the high reflectivity bandwidth was approximately 156 or 182 nm; therefore, the reflectivity spectra were sensitive to the film depths compared with the base angle and grating thickness.

Conclusions
We treated broadband GMR reflectors operating in TE polarization for normally incident light in the telecommunication spectral region.The reflectors were composed of a grating layer with triangular and trapezoidal ridge shapes on a substrate, and the high-refractive-index film layer of Si formed a conformal coverage on it.By simulating the logarithmic transmission contour map of the devices, we discovered three resonant modes in the considered range, and the interaction between them contributed to the formation of the wideband spectrum.The reflector was designed to provide a 458 nm spectral width for 99% reflectance at α = 46°, which was the best result obtained from single-period GMR reflectors with a simpler structure in TE polarization reported to date.Furthermore, it exhibited the characteristic of a broadband high reflectivity spectrum (R > 98.4% and ∆λ > 450 nm) over a wide range of base angles from 44°to 72°.We simulated the variation in the reflection spectrum based on the equipment parameters and discovered that it exhibited different sensitivities to the grating depth, base angle, and coated layer thickness-it exhibited a higher sensitivity to the depth of the coated layer.In summary, the proposed structure provided more selectivity for the design of ultra-broadband GMR reflectors.Disclosures.The authors declare no conflicts of interest.
Data availability.Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Fig. 1 .
Fig. 1.Schematic view of broadband GMR reflectors with coated (a) triangular and (b) trapezoidal profile.In the structure, high refractive index film layer of Si (n = 3.48) forms conformal coverage on low refractive index grating layer of SiO 2 (n s = 1.48).Light of TE polarization is incident on structure normally from air (n c = 1).

Fig. 2 .
Fig. 2. The effective refractive index for each layer of broadband triangular grooves grating reflectors in TE and TM polarization using effective medium theory.The inset displays the structures of TE and TM reflectors with same parameters.Parameters: Λ = 0.75µm, d g = 0.375µm, d h = d L = 0.4µm.

Figure 7 (
Figure7(a) shows the reflectance contour map R 0 (λ, α) for the base angle and wavelength for d g = 326 nm, and d L = 400 nm; as shown, the reflection bandwidth is associated closely with the base angle.For better clarity, the reflection spectra on a linear scale and the transmission spectra on a logarithmic scale at four base angles are presented in Figs.7(b) and (c), respectively.Compared with the case of α = 46°, the two resonant modes on the right were opposite each other when α = 44°, as presented by the blue curve, which resulted in a wider bandwidth but a decrease in the local reflectivity at λ = 1882nm.However, when α = 48°, corresponding to the yellow curve, the two resonant modes on the right shifted toward each other, thereby ensuring high reflectivity but a narrower bandwidth.The bandwidth corresponding to reflectivity greater than 99%, 98%, and 96% are presented in Table2.When the grating structure was asymmetric, as shown by the purple curve, i.e., α ≠ β, the characteristics of broadband reflection was preserved.In this case, the reflectivity was greater than 98% from 1418 to 1901nm (spanning a 483 nm band range), which effectively reduced the effect of the manufacturing error of the grating base angle on the reflection spectrum.

Funding.
Fudan University-CIOMP Joint Fund (Y9S133H190); International Cooperation Project (E01142UZH0); Jilin Province Science and Technology Innovation Project of China (20190201021JC); Special fund project for high-tech industrialization of scientific and technological cooperation between Jilin Province and the Chinese Academy of Sciences (2020SYHZ0033); National Natural Science Foundation of China (NSFC) (62005273).