Trapezoidal Slab Gratings

• Trapezoidal slab gratings are periodic photonic structures with trapezoidal cross-section bars that control guided modes and diffraction phenomena.
• Researchers use the Tapered Simplified Modal Method (TSMM) to discretize the tapered profile and rigorously model modal coupling and reflection properties.
• These gratings are applied in planar cavities and sensors, showcasing giant reflection bands, anomalous transmission, and tunable resonance features.

A trapezoidal slab grating is a non-rectangular, periodic photonic structure characterized by trapezoidal cross-section bars or grooves embedded within a dielectric slab. Unlike purely rectangular gratings, these systems exhibit significant complexity in their modal structure, coupling behavior, and spectral response due to the graded geometry. Trapezoidal slab gratings play a critical role in advanced photonic devices, particularly for manipulating guided mode resonances, giant reflection bands, anomalous transmission, and bound states in the continuum (BIC). New analytical and computational methods (notably the Tapered Simplified Modal Method) have enabled rigorous modeling and efficient design, illuminating rich physical phenomena resulting from geometric asymmetry and mode interference.

1. Analytical Foundations and Maxwellian Formalism

The electromagnetic response of trapezoidal slab gratings is governed by the generalized Maxwell's equations. For a periodic grating system, the dielectric function $\epsilon(\omega,\mathbf{r})$ is expanded in a Fourier basis along the periodic axis. The field equations reduce, after elimination of magnetic components, to

$$\nabla \times \nabla \times \mathbf{E}(\mathbf{r}) = \frac{\omega^2}{c^2} \mathbf{D}(\mathbf{r}),$$

where $\mathbf{D}(\mathbf{r})$ incorporates the detailed permittivity profile. In slab or self-sustained gratings, the profile is parametrized by a filling factor $f_x = L_x/d$, and the dielectric contrast $\Delta\epsilon(\omega)$. The Fourier decomposition leads to a matrix formulation for eigenmodes:

$$\sum_{G'} M_{G,G'}(\omega) \phi_n(G') = k_n^2 \phi_n(G), \quad M_{G,G'}(\omega) = \frac{\omega^2}{c^2} \epsilon_{G,G'} - q_x^2(G) \delta_{G,G'},$$

where the summation encompasses all Fourier (Bragg) components, including fundamental, guided, evanescent, and divergent contributions (Pilozzi et al., 2012). The coupling strength between modes depends on both material dielectric contrast and the geometric filling factor, and the interplay among these yields the system’s unique reflective and transmissive properties.

2. Tapered Simplified Modal Method (TSMM) for Trapezoidal Gratings

Modeling the non-rectangular (trapezoidal) grating profile necessitates discretization along the propagation ($z$) axis into $N$ layers, each approximated as a rectangular grating with constant local duty cycle. Within each layer $i$ of thickness $h_i = H/N$ (total grating depth $H$), transverse fields are expanded as: $$E_t(x,z) = \sum_{q=1}^M [a_q e^{\Gamma_q z} e_q(x) + a_{-q} e^{\Gamma_{-q} z} e_{-q}(x)],$$ with modal amplitudes governed by mode-coupling differential equations: $$\frac{dA_q}{dz} - \Gamma_q A_q = \frac{1}{2} \sum_p A_p (K_{pq} + \widetilde{K}_{qp}),$$ where $K_{pq}$ and $\widetilde{K}_{pq}$ are overlap integrals encoding the coupling across the tapered geometry (Li et al., 2016). Boundary conditions at input/output surfaces enforce continuity for both fields and result in a transfer matrix formulation for modal amplitudes, enabling simultaneous computation of both transmission and reflection.

Numerical simulations demonstrate that the TSMM accurately predicts the diffraction efficiency and field distributions in trapezoidal gratings under both Littrow-mounting (Bragg-matched) and off-Bragg conditions, with convergence controlled by the discretization parameter $N$. Compared to full triangular profiles, truncated (trapezoidal) gratings exhibit enhanced reflection due to increased index mismatch at the boundaries.

3. Giant Reflection Bands and Anomalous Negative Transmission

In self-sustained rectangular or trapezoidal dielectric grating slabs, a “giant reflection band” occurs predominantly in P polarization near the Brewster angle $\theta_B$, given by

$\tan \theta_B = \sqrt{\bar{\epsilon}/\epsilon_0},$

where $\bar{\epsilon}$ is the effective permittivity (Pilozzi et al., 2012). Near this regime, Fabry–Perot oscillations are suppressed, and the interaction between the travelling fundamental and quasi-guided diffracted modes (those with strong Bragg components) dominates. The result is a destructive interference that cancels outgoing fundamental wave contributions over a broad spectral range; reflected intensity approaches unity for extended frequency regions. The phase-matching condition that governs this broad band is

$$\tan(\phi_{-1}) = \frac{q_x(-G)}{k_2} \approx m\frac{d}{L_z}, \quad m \in \mathbb{Z},$$

with $\phi_{-1}$ the phase accumulated by the quasi-guided mode.

Associated with these bands is the phenomenon of anomalous negative transmission: at certain photon energies, the -1st transmission order carries nearly all transmitted power at a negative (“deflected”) output angle, originating from Bragg diffraction rather than material negative-index behavior. These effects are robust in self-sustained gratings, especially compared to engraved (surface) gratings, due to strong volumetric coupling and the involvement of higher-order Fourier components.

4. Guided Mode Resonance, Wood's Anomalies, and Bound States in the Continuum

Trapezoidal slab gratings support guided mode resonance (GMR) and related phenomena including Wood’s anomalies and bound states in the continuum (BIC). The GMR condition is described by

$k_i = k_0 \sin\theta - i \frac{2\pi}{D} = \beta,$

where $D$ is the grating period, $\theta$ the incident angle, and $\beta$ the waveguide mode propagation constant (Ruan et al., 2022). Structural modifications—such as cutting off corners or tilting trapezoidal elements—break symmetry and induce leakage, manifesting as Fano resonances (quasi-BICs) in the reflectance spectrum.

Topologically, perfect symmetry supports a polarization vortex with charge $q=1$ at the BIC; asymmetry splits this into two half charges $q=1/2$, resulting in pronounced changes in the polarization distribution in momentum space. The resonance linewidth, quantifying the spectral sharpness, follows

$$\Delta\lambda_{FWHM} = \frac{\lambda D \gamma}{\pi},$$

where $\gamma$ is the leakage rate. These features underpin the device applications of trapezoidal gratings in narrowband filtering and sensing.

5. Ultra-Broadband Reflectivity and Angular Performance

Subwavelength trapezoidal slab gratings enable engineering of ultra-broadband high reflectivity through coherent interference among Fabry–Perot, guided slab, and waveguide array modes (Cui et al., 2015). The propagation constant for mode $m$ is

$$k_z^2 = \left( \frac{2\pi n}{\lambda} \right)^2 - \left( \frac{2\pi m}{\Lambda} \right)^2,$$

for refractive index $n$ and grating period $\Lambda$. Broadband reflectivity is achieved when guided and FP mode phase accumulation satisfy $$\Delta\phi = \phi_{guided} - \phi_{FP} \approx 2\pi p$$, $p \in \mathbb{Z}$.

Reflectivity $R > 0.99$ is sustained for fractional bandwidths exceeding $30\%$ and for incident angle ranges exceeding $40^\circ$. Integration with asymmetric (semiconductor substrate) waveguide architectures preserves high reflectivity, facilitating VCSEL and quantum cavity applications.

6. Applications: Planar Cavities, Sensing, and Photonic Devices

Trapezoidal slab gratings, especially in self-sustained configurations, are highly effective as planar photonic cavity mirrors, vertical micro–cavity reflectors, and wavelength-selective components. Their broad reflection bands and tunable resonances support cavity resonances with high $Q$-factors, suitable for lasers and quantum devices. The transition from BIC to quasi-BIC, engineered via asymmetry parameters $M_1$, $M_2$, allows ultra-narrow Fano resonances and is exploited for advanced sensing (Goos–Hänchen shift, $$S_{GH} = -\frac{\lambda}{2\pi} \frac{\partial \phi_r}{\partial \theta}$$), switches, and multiplexers (Ruan et al., 2022). Enhanced reflection and controllable transmission are enabled by design choices in duty cycle, geometry, and substrate integration, directly impacting device functionality and performance.

7. Comparative Analysis: Rectangular vs. Trapezoidal and Engraved Slab Gratings

Traditional rectangular or engraved (surface) gratings differ fundamentally from trapezoidal slab gratings in diffraction mechanism and spectral characteristics. In engraved gratings, weak coupling between guided and diffracted modes at the slab interface leads to second-kind Wood’s anomalies (SKWA) with relatively narrower reflection bands. Trapezoidal and self-sustained grating slabs, by contrast, support volumetric diffraction and strong mode mixing, generating broader and more robust reflection and transmission effects. The improved coupling, wider stop bands, and greater angular tolerance render trapezoidal slab gratings superior for many advanced photonic applications.

---

Trapezoidal slab gratings thus represent a versatile, rigorously analyzable photonic architecture for controlling electromagnetic field confinement, resonance phenomena, and broadband reflectivity. Their implementation leverages advanced modal and eigenvalue analysis and offers tunable device properties for next-generation photonic technologies.