---
title: Immersed Metagrating Advances
url: https://www.emergentmind.com/topics/immersed-metagrating
type: topic
---

# Immersed Metagrating Advances

Searching arXiv for the cited papers and closely related metagrating work.
An **immersed metagrating** is a metagrating realized within, rather than merely at the boundary of, a structured wave environment. In the clearest electromagnetic example, a **very small number of polarizable scatterers** are embedded directly inside a guided-wave discontinuity and synthesized with a metagrating-style semianalytical procedure so that their induced secondary fields redistribute power among the available channels of the host structure [2012.07150]. In a second, more recent usage, the term denotes a **reflective metagrating immersed in silicon** for high-order diffraction in spectroscopy, where subwavelength structure replaces a conventional blazed immersed grating and is optimized for efficiency and reduced polarization sensitivity over the SWIR-3 band [2509.21089]. Across these realizations, the common idea is sparse, analytically or numerically designed scattering embedded in a nontrivial electromagnetic environment—waveguide junction, high-index immersion medium, or other bounded host—so that the surrounding structure is an active part of the diffraction or mode-conversion mechanism rather than a passive backdrop.

## 1. Definition and scope

The term is exemplified by a waveguide-bend device in which a single passive polarizable scatterer is placed **inside the bend junction** of an abrupt H-plane rectangular-waveguide bend and designed by a metagrating-inspired semianalytical methodology to eliminate reflection of the dominant \(\mathrm{TE}_{10}\) mode [2012.07150]. In that setting, the available “channels” are not free-space diffraction orders but the finite set of propagating guided modes, so the metagrating concept is transplanted from plane-wave diffraction to embedded guided-wave scattering control.

A distinct but related usage appears in reflective spectroscopy, where an **immersed reflection grating** is implemented inside a high-index medium and redesigned as a subwavelength **immersed metagrating** rather than a conventional sawtooth blaze [2509.21089]. There, immersion refers to diffraction **inside a high refractive index medium**, specifically silicon, with the wavelength reduced to \(\lambda_\text{eff}=\lambda/n\), allowing high diffraction order and compact resolving-power scaling.

These two cases delimit the concept. One is an **embedded sparse scatterer in a guided discontinuity**; the other is a **subwavelength reflective grating embedded in a high-index immersion medium**. A plausible implication is that “immersed metagrating” is best understood as a design philosophy rather than a single geometry: sparse or subwavelength diffractive elements are placed within a structured host so that environment-mediated coupling becomes part of the synthesis.

## 2. Guided-wave realization in abrupt H-plane bends

In the guided-wave realization, the physical configuration is a rectangular waveguide with perfectly conducting walls, bent by an arbitrary angle \(\Phi\), with input region \((1)\), output region \((2)\), and a corner junction region \((j)\) [2012.07150]. The waveguide height is \(b\), the guide widths are \(a_1\) and \(a_2\), and operation is restricted to the single-mode regime under
\[
b<\lambda/2,\qquad \lambda/2<a_1,a_2<\lambda .
\]
Under these conditions, only the \(\mathrm{TE}_{10}\) mode propagates in each straight section, while higher-order \(\mathrm{TE}_{n0}\) modes are evanescent.

The passive element is a vertical inclusion aligned along the height direction \(y\), placed at cylindrical coordinates \((r_0,\phi_0)\) referenced to the corner point \(O\). For the modal derivation, the admissible region is restricted by
\[
r_0 \le \min(h_1,h_2),
\]
with
\[
\begin{dcases}
h_1 = AO = \frac{a_2+a_1\cos(\Phi)}{\sin(\Phi)} \\
h_2 = BO = \frac{a_1+a_2\cos(\Phi)}{\sin(\Phi)} .
\end{dcases}
\]
This ensures that the matching contours remain outside the source radius.

The metagrating interpretation is explicit. Conventional free-space metagratings use sparse polarizable meta-atoms whose induced currents are tailored so that their secondary fields redistribute energy among a small set of propagating Floquet channels. In the bend problem, the analogue of the metagrating meta-atom is the **single polarizable scatterer embedded in the junction**, first modeled as an infinitesimal current line and later realized as either a capacitively loaded wire or a metallic post. Near-field interactions are the strong coupling of the scatterer with the bend corner and PEC walls, including image effects and multiple scattering; far-field interactions correspond to how the induced current launches guided modal content toward the two ports. The device is sparse because it does not require a continuous matching section or a dense metasurface.

This realization is subject to several explicit limits. The walls are PEC, the inclusion is passive, linear, and ideally lossless during synthesis, and the theory is restricted to **H-plane bends** because the relevant port fields remain \(\mathrm{TE}_{n0}\) and the problem reduces to a 2D scalar form. Single-mode operation is essential; if additional propagating modes were present, one scatterer would generally not provide enough degrees of freedom to control all channels [2012.07150].

## 3. Semianalytical synthesis and perfect-transmission conditions

The analytical core is a semianalytical modal formalism combining rectangular-waveguide mode expansions in the ports with a radial-mode description in the bend junction [2012.07150]. In the input guide, the electric field is the incident \(\mathrm{TE}_{10}\) mode plus reflected \(\mathrm{TE}_{n0}\) modes; in the output guide, the transmitted field is expanded in \(\mathrm{TE}_{n0}\) modes. The scatterer contributes additive modal amplitudes \(A_n^{\mathrm{(sec)}}\) and \(B_n^{\mathrm{(sec)}}\), so the junction response is decomposed into bare-junction scattering and source-induced secondary scattering.

Inside the junction, the field is modeled in cylindrical coordinates as a 2D radial-waveguide problem with \(\partial/\partial y=0\). The inhomogeneous Helmholtz equation is
\[
\left(\mathbf{\nabla}^2+k^2\right)\mathbf{E}_\mathbf{j}=\mathbf{\hat{y}\, jk\eta \frac{I}{r_0} \delta(r-r_0)\delta(\phi-\phi_0),
\]
and the source-induced junction field is expanded in Bessel and Hankel functions over the wedge eigenvalue set
\[
M=\left\{\frac{m\pi}{\Phi}\mid m\in\mathbb{N}\right\}.
\]
A homogeneous junction field with coefficients \(C_\mu\) is added, and mode matching on the two interfaces yields closed-form matrix expressions for \(A_n\), \(A_n^{\mathrm{(sec)}}\), \(B_n\), \(B_n^{\mathrm{(sec)}}\), and \(C_\mu\).

The dominant-mode scattering parameters are then defined by
\[
{\left|S_{11}\right|}^2 = \left|\frac{A_1+A_1^{\mathrm{(sec)}}}{E_{\mathrm{in}}}\right|^2,
\]
\[
{\left|S_{21}\right|}^2 = \gamma^2\left|\frac{B_1+B_1^{\mathrm{(sec)}}}{E_{\mathrm{in}}}\right|^2,
\]
with
\[
\gamma = \left(\frac{\left(ka_2\right)^2 - \pi^2}{\left(ka_1\right)^2 - \pi^2}\right)^{\frac{1}{4}}.
\]
Perfect transmission is defined by
\[
|S_{11}|^2=0, \qquad |S_{21}|^2=1.
\]
The first condition enforces destructive interference in the input port,
\[
A_1 + A_1^{\mathrm{(sec)}} = 0,
\]
while the second expresses passivity and losslessness.

The design workflow mirrors metagrating synthesis. Candidate locations \((r_0,\phi_0)\) are scanned; for each location the zero-reflection current is solved as
\[
I_\mathrm{NR} = - \frac{A_1}{\left([R_{1\mu,2}^{(1)}] - [R_{1\mu,1}^{(1)}][\Lambda_1]^{-1}[\Lambda_2]\right)[\zeta_\mu]}.
\]
Passivity is then checked through
\[
\sigma(r_0,\phi_0) = \left|1-|S_{21}|^2\right|\bigg|_{|S_{11}|^2=0}.
\]
Locations with \(\sigma=0\) are the **perfect transmission locations** (PTLs). The paper also derives a semianalytical sufficient condition for PTLs,
\[
A_1^{\mathrm{(sec)}} = \pm je^{j\xi} \gamma B_1^{\mathrm{(sec)}},
\qquad
\xi = \arg(B_1^*A_1).
\]
In symmetric bends with \(a_1=a_2\), the axis \(\phi_0=\Phi/2\) is always a solution branch.

The branch structure is central. PTLs form continuous solution branches in the junction, but there are also “blind spots,” especially branch intersections, where
\[
A_1^{\mathrm{(sec)}} = B_1^{\mathrm{(sec)}} = 0,
\]
so no finite current can cancel reflection. Similar current divergence occurs near conducting walls and the corner due to image-theory cancellation. This suggests that immersed metagrating synthesis is strongly constrained by environment-dependent radiative coupling, not just by the scatterer polarizability itself.

## 4. Physical realizations and validated performance

Two practical scatterer geometries are proposed for the waveguide implementation: a **capacitively loaded wire** and a **cylindrical metallic post** [2012.07150]. The loaded wire is a vertical metallic wire with a cylindrical symmetric capacitive loading section. In the reported examples, the wire is copper, with fixed wire radius and plate thickness
\[
r_C = w_P = \lambda/50,
\]
while the adjustable dimensions are the plate radius \(r_P\) and plate separation \(d\). After identifying a PTL semianalytically, the geometry is retrieved by a short full-wave parametric sweep over \(r_P\) and \(d\).

For example, in an asymmetric \(105^\circ\) bend with \(a_1=0.95\lambda\), \(a_2=0.9\lambda\), a PTL at
\[
(r_0,\phi_0)=(0.398\lambda,51.13^\circ)
\]
is implemented with \(r_P=0.2\lambda\) and \(d=0.4b\), yielding over \(99\%\) transmission. In a symmetric \(75^\circ\) bend with \(a_1=a_2=0.85\lambda\), two different PTLs on different solution branches are implemented with loaded wires, both reaching over \(99\%\) transmission.

For the cylindrical metallic post, the paper proposes a semianalytical radius-retrieval rule, especially for symmetric bends. If the post is centered at \((r_0,\phi_0)\), an equivalent radius \(\tilde r_C\) is chosen from the one-point condition
\[
\mathbf{E_j}(r_0 + \tilde{r}_C,\phi_0) = 0,\qquad \tilde r_C>0.
\]
Over a family of right-angle bends, the optimized full-wave radius obeys approximately
\[
r_C \approx 1.0852\tilde{r}_C.
\]
This empirical calibration is most reliable for symmetric bends with \(\Phi\gtrsim 75^\circ\).

Validation is numerical and extensive. In a \(90^\circ\), \(a_1=a_2=0.9\lambda\) bend whose bare reflection exceeds \(80\%\), the selected PTL
\[
r_0=\frac{0.9\lambda}{\sqrt{2}},\qquad \phi_0=45^\circ
\]
with current
\[
I_\mathrm{NR} = 1.94e^{2.69j}\frac{E_{\mathrm{in}}b}{\eta}
\]
eliminates the reflection completely in HFSS, with excellent agreement to the semianalytical theory. Across loaded-wire and metallic-post designs, the full-wave examples routinely achieve **more than \(99\%\) transmission**, even when the bare junction reflects \(40\%\), \(70\%\), \(80\%\), \(90\%\), or in some cases more than \(99\%\) of the incident power. Demonstrations cover bend angles from \(45^\circ\) to \(120^\circ\), and both symmetric and asymmetric widths.

Bandwidth and tolerance are also reported. Exact matching is at the design frequency, but many examples maintain \(\gtrsim 90\%\) transmission over most of the single-mode band, especially on the symmetry branch; one opposing-branch example achieves \(|S_{21}|^2\ge 88\%\) over about \(2\,\mathrm{GHz}\) around \(10.6\,\mathrm{GHz}\). For fabrication offsets up to about \(1\,\mathrm{mm}\) (\(\sim \lambda/30\) at \(10\,\mathrm{GHz}\)), transmission remains very high, and in a symmetry-axis design, even shifts up to \(2\,\mathrm{mm}\) (\(\sim \lambda/15\)) keep \(|S_{21}|^2\ge 90\%\). Deviations orthogonal to the solution branch are more harmful than deviations along it.

## 5. High-order reflective immersed metagratings in silicon

A later development applies the term to **reflective immersed gratings** for compact SWIR spectroscopy, replacing the conventional immersed blazed reflection grating with a **sub-wavelength structured metasurface grating period** optimized by a modified covariance-matrix-adaptation strategy [2509.21089]. The grating equation is written as
\[
\sin\left(\theta_i\right) + \sin\left(\theta_d\right) = m \left(\lambda_\text{eff} / P\right),
\qquad
\lambda_\text{eff} = \lambda / n,
\]
and the resolving power scales as
\[
\mathcal{R} = mN.
\]
Because immersion reduces the wavelength inside the medium, the same physical period can support a higher diffraction order, or equivalently the same resolving power can be obtained with smaller \(N\).

The proposed device is a **reflective silicon-immersed grating** with **aluminum coating**, designed for the Sentinel-5-like SWIR-3 diffraction geometry. Its fixed parameters are
\[
P = 2.07\,\mu\mathrm{m},\qquad \Lambda=P/5=0.414\,\mu\mathrm{m},
\]
\[
\theta_i = 62.6^\circ,\qquad m=-5,\qquad \theta_d = -49.8^\circ
\]
at the design point. The SWIR-3 target band is
\[
2.304\,\mu\mathrm{m}\; \text{to} \;2.405\,\mu\mathrm{m},
\]
with mean wavelength
\[
\lambda_{\text{mean}} = 2.345\,\mu\mathrm{m}.
\]

Each period contains five rectangular nanopillars with common height \(H\) and widths \(b_{x_i}\), lengths \(b_{z_i}\), for \(i=1,\dots,5\), giving 11 design degrees of freedom. The reported optimized design is
\[
b_{X_1} = 0.163\,\mu\mathrm{m},\;
b_{Z_1} = 0.209\,\mu\mathrm{m},\;
b_{X_2} = 0.179\,\mu\mathrm{m},\;
b_{Z_2} = 0.171\,\mu\mathrm{m},
\]
\[
b_{X_3} = 0.152\,\mu\mathrm{m},\;
b_{Z_3} = 0.211\,\mu\mathrm{m},\;
b_{X_4} = 0.208\,\mu\mathrm{m},\;
b_{Z_4} = 0.174\,\mu\mathrm{m},
\]
\[
b_{X_5} = 0.177\,\mu\mathrm{m},\;
b_{Z_5} = 0.185\,\mu\mathrm{m},\;
H = 0.200\,\mu\mathrm{m}.
\]
All features remain above \(150\,\mathrm{nm}\), and the maximum aspect ratio is about \(1.3\).

The optimization target is the average target-order efficiency over three wavelengths across the SWIR-3 band,
\[
f(\mathbf{x}) = \frac{1}{3}\sum_{i=1}^3 \eta_m(\mathbf{x}, \lambda_i),
\]
with optimization performed at diagonal polarization so that S and P are balanced simultaneously. The polarization-sensitivity metric is
\[
\eta_{\text{pol}} = \frac{\eta_s - \eta_p}{\eta_s + \eta_p}.
\]
Simulations use **Lumerical FDTD:3D v.8.31.3683** with Bloch-periodic boundary conditions and 14 points per wavelength.

The performance improvement over the conventional immersed sawtooth/blazed grating is explicit. Average efficiencies over the SWIR-3 band are:

| Structure | S | P | D |
|---|---:|---:|---:|
| Immersed metagrating | 80.2% | 75.8% | 78.7% |
| Conventional immersed sawtooth grating | 65.9% | 58.1% | 62.0% |

Across the full SWIR-3 band, the metagrating maintains efficiency \(\gtrsim 74\%\) for both linear polarizations in the target \(-5\)th order. Polarization sensitivity remains within about **5%**, whereas the conventional sawtooth reaches values approaching **15%**. The target-order diffraction-angle difference between the metagrating and the sawtooth is reported to be less than
\[
0.08^\circ.
\]

A manufacturing-tolerance analysis applies systematic deviations \(\mathbf{x}\pm\Delta\mathbf{x}\) of \(\pm10\,\mathrm{nm}\), \(\pm20\,\mathrm{nm}\), and \(\pm25\,\mathrm{nm}\). The abstract reports a degradation of \(\sim 10\%\) at \(\pm25\,\mathrm{nm}\), almost negligible effect at \(-10\,\mathrm{nm}\), and \(\sim 5\%\) at \(+10\,\mathrm{nm}\). This suggests that immersed metagrating performance can be substantially improved relative to classical blaze while preserving practical manufacturability.

## 6. Related directions, misconceptions, and generalization

Several adjacent developments clarify what immersed metagratings are and are not. A **metagrating-assisted planar antenna** with a substrate-loaded sparse array of meta-wires and an embedded line source is highly relevant methodologically because it uses a **2D volume-surface integral equation** framework, purely reactive loads \(Z_n=jX_n\), and global optimization without a local-periodic approximation, but it is **not** an immersed metagrating in the sense of a device fully embedded in a bulk host or treated with layered-medium Green’s functions [2110.13000]. Its transferable value lies in explicit treatment of nonuniform embedded-source illumination, mutual coupling, substrate polarization currents, and finite-aperture diffraction.

Likewise, **“Scalable Metagrating for Efficient Ultrasonic Focusing”** develops a local-diffraction-order design framework for an ultrasonic reflective metalens in **air**, not in water or another liquid [2104.12937]. It focuses by locally controlling a single diffraction order with **adiabatically varying metagratings** and achieves \(\mathrm{FWHM}=0.364\lambda\) in simulation, but it is not an immersed-acoustic implementation in the strict sense.

A true fluid-immersed analogue is provided by **“Control water waves by metagratings”**, where a periodic array of rigid bridge-pillar-like obstacles embedded in shallow water is modeled as a non-resonant water-wave metagrating supporting a surface mode with \(|k_x|>k\) [2311.15795]. The equivalent anisotropic model uses
\[
u_x=0,\qquad u_y=\frac{a}{d}u_0,\qquad g'=\frac{d}{a}g_0,
\]
and the work reports the first water-wave metagrating experiment and the first observation of unidirectional propagation in such a system. This broadens the concept beyond electromagnetics: immersion can mean embedding the grating as rigid obstacles in the wave-bearing medium itself.

A common misconception is that any metagrating attached to a substrate or operating in a fluid is automatically “immersed.” The cited literature distinguishes more carefully among **substrate-loaded**, **integrated**, **airborne reflective**, **guided-wave embedded**, and **high-index immersed** implementations. Another misconception is that immersion merely rescales wavelength while leaving the design logic unchanged. The silicon SWIR work instead shows that polarization sensitivity, chromaticity, and manufacturability become central design variables, while the waveguide-bend work shows that local environment and blind spots can determine whether a passive solution exists at all.

Taken together, these studies indicate that immersed metagratings are best characterized by three recurring features: sparse or subwavelength scatterers, a structured host that strongly mediates scattering, and synthesis aimed at controlling a finite set of channels—guided modes, high diffraction orders, or bound surface-wave branches—through interference engineered within the immersion environment rather than only at an external interface.

Source: https://www.emergentmind.com/topics/immersed-metagrating