---
title: Unidirectional Guided Resonances (UGRs)
url: https://www.emergentmind.com/topics/unidirectional-guided-resonances-ugrs
type: topic
---

# Unidirectional Guided Resonances (UGRs)

Searching arXiv for recent and foundational papers on unidirectional guided resonances.
Unidirectional guided resonances (UGRs) are leaky eigenmodes of dielectric photonic-crystal slabs or other open periodic photonic structures that radiate entirely toward one side without the need for a reflective mirror on the opposite side [2203.02223]. In one rigorous formulation for a 1D-periodic structure with a single radiation channel, a UGR is a complex-frequency solution satisfying \(c_0^+=0\) and \(c_0^-\neq0\), so that all power leaks only downward; the opposite choice gives the upward-radiating case [2510.01965]. Closely related work in nanophotonics demonstrated unidirectional excitation of dielectric waveguide and surface plasmon-polariton modes from an all-dielectric nanoantenna, where asymmetrical excitation of higher multipoles creates a chiral near field that mimics a rotating emitter [1508.02731]. Across these settings, the unifying theme is one-sided coupling to guided or radiative channels by interference, symmetry breaking, and, in many platforms, topological polarization singularities.

## 1. Definition and distinguishing characteristics

In the photonic-crystal-slab literature, UGRs are usually defined as optical modes that radiate towards one side without the need for mirrors on the other, and are represented from a topological perspective by the merged points of paired, single-sided, half-integer topological charges [2203.02223]. The relevant observable is the asymmetry between the two radiation channels. One commonly used ratio is
\[
\eta \equiv \gamma_{\mathrm{top}}/\gamma_{\mathrm{bottom}} \to \infty,
\]
which expresses vanishing decay into one side at the UGR point [2203.02223]. Other papers use
\[
A = 10\log_{10}(P_{\mathrm{up}}/P_{\mathrm{down}})
\]
for radiation asymmetry [2306.09027], or
\[
\tau = |c_0^+/c_0^-|^2
\]
with UGR corresponding to \(\tau=0\) in the one-channel formulation [2510.01965].

A persistent misconception is that strict one-sided radiation in planar photonics requires a bottom mirror. The core UGR papers explicitly state the opposite: one-sided leakage is achieved with no mirror placed on the opposite side [2203.02223; 2306.09027]. Another misconception is that UGRs are identical to generic one-way waveguiding. In the gyromagnetic periodic dielectric chain, the reported effect is a unidirectional waveguide mode with \(\omega(k)\neq\omega(-k)\) produced by broken time-reversal and inversion-related symmetries, not a one-sided leaky resonance of the mirror-free type emphasized in UGR studies [1902.08054]. This suggests a useful distinction between nonreciprocal one-way propagation and reciprocal or quasi-reciprocal one-sided radiation.

## 2. Topological origin in photonic-crystal slabs

The modern topological picture starts from polarization singularities in momentum space. In the far-field polarization field, an integer topological charge is defined by
\[
q=\frac{1}{2\pi}\oint_C d\phi(k),
\]
where \(\phi(k)\) is the polarization angle around a closed loop \(C\) in \(k\)-space [2203.02223]. A conventional vortex center or V point has integer charge \(q=\pm1\), while a circular-polarization singularity or C point carries half-integer charge \(q=\pm \tfrac12\) [2203.02223].

In broken-symmetry photonic-crystal slabs, a symmetry-protected bound state in the continuum at \(\Gamma\) can split into half-integer topological charges. Yin and co-workers showed that when a pair of half-integer charges in the polarization field bounce into each other in momentum space, the downward channel is forced to vanish while the upward channel remains finite, producing a unidirectional bound state in the continuum, i.e. a UGR [1904.11464]. In that framework, the asymmetry
\[
A(k)=|r_t/r_b|^2=\gamma_t/\gamma_b
\]
diverges because \(d_b(k_{\mathrm{BIC}})=0\) while \(d_t(k_{\mathrm{BIC}})\neq0\) [1904.11464].

A particularly explicit realization was given by Wang et al. in a broken-\(C_2\) grating on 340 nm silicon-on-insulator. There, a symmetry-protected BIC at \(\Gamma\) carries an integer topological charge \(q=\pm1\). Breaking \(C_2\) symmetry in an L-shaped unit cell splits that integer charge into two half-charges \(q=\pm \tfrac12\), and by tuning the small-block etch depth \(h_1\) the two half-charges re-merge on one side of the downward radiation plane [2306.09027]. Since any downward radiation would have to carry both opposite circular polarizations simultaneously, it vanishes by destructive interference, whereas the upward path remains open [2306.09027]. The reported numerical asymmetry ratio \(P_{\mathrm{up}}/P_{\mathrm{down}}\) exceeds \(10^{65.8/10}\approx 3.8\times 10^6\), or about \(65\) dB [2306.09027].

## 3. Mechanisms beyond the canonical broken-\(C_2\) picture

Interband coupling provides a second major route to UGRs. Yin et al. analyzed two slab modes of opposite up-down mirror parity that cross in \(k\)-space and do not couple in the perfectly symmetric slab. Breaking that symmetry introduces a coupling \(\kappa(k)\tan\theta\), hybridizes the modes, and creates circular-polarization singularities from initially trivial polarization fields [2203.02223]. As the tilt angle increases, two half-charges can merge on a high-symmetry line and create a V point with topological charge \(+1\), producing a UGR with \(\eta\approx 70\) dB [2203.02223].

Lee et al. extended this mechanism to one-dimensional zero-contrast gratings. In that system, interband coupling between even-like and odd-like waveguide modes produces quasi-UGRs at small grating thickness and true UGRs when destructive interference in the unwanted port becomes nearly perfect [2312.00283]. With increasing \(h\), the directionality evolves from \(\eta\sim10\) dB at \(h=0.15\Lambda\) to \(\eta\approx81\) dB at \(h_u\approx0.2708\Lambda\), and then flips to a downward UGR at \(h_d\approx0.375\Lambda\) [2312.00283]. The same study also identifies exceptional points and quasi-BICs in the same folded-band setting [2312.00283].

A third route relies on anisotropy. In anisotropic planar anti-guiding waveguides, UGRs appear only when the so-called polar anisotropy-symmetry is broken; at the UGR point one transmission coefficient vanishes exactly while the other remains finite, and the canceled channel carries a phase singularity with \(2\pi\) winding [2105.08409]. Because the UGR condition depends on wavelength, in-plane propagation angle, and polar tilt, the radiation direction can be selected and switched by anisotropy orientation [2105.08409].

Rigorous perturbation theory places these observations on a broader footing. Yuan and Lu showed that, in the presence of a single radiation channel, a UGR has codimension \(1\): under a generic perturbation, one must tune one parameter to maintain \(c_0^+=0\) [2510.01965]. Starting from a generic bound state in the continuum, they further showed that a continuous family of UGRs can always be obtained by tuning one parameter, with
\[
Q \sim \mathrm{const}/\delta^2 \to \infty
\]
as the symmetry-breaking parameter \(\delta\to 0\) [2510.01965].

A more recent mechanism relaxes two assumptions that had become common in the field. Near the fourth stop band, competing first- and second-order Fourier harmonics can mediate out-of-plane radiation through two coherent channels; UGRs then arise when the channel cancellation occurs only in one direction [2606.27920]. The authors state explicitly that this mechanism enables UGRs without relying on in-plane symmetry breaking or interband coupling [2606.27920]. This suggests that UGRs are better regarded as a broader radiation-cancellation phenomenon than as the consequence of a single symmetry-breaking recipe.

## 4. Near-field and nanoantenna realizations

A near-field formulation predates the topological slab literature. In the comment on unidirectional excitation of guided modes under oblique circular illumination, the mechanism is described as the near-field analogue of a Huygens source engineered so that its evanescent spectrum couples asymmetrically into the two counter-propagating branches of a guided or surface mode [1306.5068]. In that picture, interference between electric and magnetic dipoles is central. For the slit geometry, the amplitude ratio can be written as
\[
A_+/A_-=(p_z+m_x/c)/(p_z-m_x/c),
\]
so equal magnitudes and a \(\pm 90^\circ\) phase shift can suppress one direction exactly [1306.5068]. The same comment argues that the magnetic dipole \(m_x\), omitted in the original interpretation, is essential, and that the dominant phase-shifted Huygens pair is \(p_z \leftrightarrow m_x\), not \(p_x \leftrightarrow p_z\) [1306.5068].

The all-dielectric nanoantenna realization of this idea uses a crystalline silicon sphere with \(\varepsilon'\approx 20\), sphere radius \(R_s=100\) nm, hemispherical notch radius \(R_n=60\) nm, and a point dipole located in the notch at distance \(a=135\) nm from the center and oriented along the \(z\)-axis [1508.02731]. Because the excitation is off-center, the nanoparticle lacks rotational symmetry and supports higher-order magnetic multipoles in addition to the dipole contribution [1508.02731]. At \(\omega/2\pi\approx470\) THz, the magnetic quadrupole dominates; at \(\omega/2\pi\approx600\) THz, the magnetic octupole prevails [1508.02731]. The resulting near field is chiral and similar to that of a circularly polarized dipole or quadrupole, which underpins unidirectional coupling to guided modes via spin-orbit interaction [1508.02731].

The relevant observables in this nanoantenna setting are the local density of states,
\[
\mathrm{LDOS}(\omega)=\frac{\Im\,G_{zz}(0,0,\omega)}{\Im\,G_{zz}^{(0)}(0,0,\omega)},
\]
and the front-to-back ratio,
\[
\mathrm{FBR}=\frac{|E_x(x=+X)|}{|E_x(x=-X)|},\qquad X=2000\ \mathrm{nm}
\]
[1508.02731]. In simulations, the front-to-back ratio reaches about \(5\) for a dielectric waveguide at \(470\) THz, \(5.5\) at \(595\) THz, and \(7.5\) for a plasmonic gold film at \(440\) THz [1508.02731].

## 5. Mathematical formalisms and observables

Several complementary mathematical descriptions recur across the UGR literature. In periodic slabs and gratings, temporal coupled-mode theory is the most common reduced model. For the topological grating coupler, the resonance amplitude \(a(t)\) obeys
\[
\frac{da}{dt}=\bigl(j(\omega-\omega_0)-(\gamma_u+\gamma_d+\gamma_i)\bigr)a+\sqrt{2\gamma_u}s_+,
\]
with outgoing fields
\[
s_{u-}=s_+-\sqrt{2\gamma_u}a,\qquad s_{d-}=-\sqrt{2\gamma_d}a,
\]
so the UGR limit is \(\gamma_d\to0\), hence \(s_{d-}=0\) [2306.09027]. On resonance and neglecting internal loss, the upward coupling efficiency approaches unity, and the insertion loss is \(L=-10\log_{10}(\eta)\) [2306.09027].

In interband-coupled ZCGs, the core object is a non-Hermitian \(2\times2\) Hamiltonian with Hermitian near-field coupling \(\alpha\) and anti-Hermitian far-field coupling \(\beta\), whose eigenvalues determine the complex frequencies and decay rates of the hybridized modes [2312.00283]. Directionality is then measured by
\[
\eta = 10\log_{10}(\Gamma_+/\Gamma_-),
\]
with \(|\eta|\gtrsim80\) dB taken as a true UGR and \(|\eta|\sim10\)–\(20\) dB as a quasi-UGR [2312.00283].

In the perturbative one-channel theory, the outgoing condition is enforced through the amplitudes \(c_0^\pm\), and the UGR condition is \(c_0^+=0\) with \(c_0^-\neq0\) [2510.01965]. The perturbation expansion introduces a tunable parameter \(\eta\) that must generally be slaved to a fixed perturbation \(\delta\) to maintain one-sided radiation to all orders [2510.01965].

These formalisms emphasize different aspects of the same object. TCMT is most convenient for device-level loss and coupling calculations, the polarization-singularity picture captures the topological structure of one-sided radiation, and perturbation theory clarifies codimension, tuning requirements, and the \(Q\)-factor scaling near a BIC.

## 6. Implementations, performance, and applications

The most mature UGR implementation in integrated photonics is the topological grating coupler demonstrated by Wang et al. on a 340 nm silicon-on-insulator platform. By engineering the dispersion and apodizing the grating, they reported a record-low-loss of \(0.34\) dB and bandwidth exceeding \(30\) nm at \(1550\) nm, with a \(20\times20\ \mu\mathrm{m}^2\) footprint, standard SOI 340 nm, planar CMOS-compatible, single-etch fabrication, and no mirror placed on the bottom [2306.09027]. The same work reports that a pair of grating couplers can function as an optic via interconnecting two stacked photonic chips with a loss of only \(0.94\) dB [2306.09027].

Topological photonic-crystal slabs support much larger single-sided quality factors. In the telecommunication regime, the experimentally demonstrated unidirectional bound states in the continuum achieved single-sided quality factor as high as \(1.6\times10^5\), equivalent to a radiation asymmetry ratio of \(27.7\) dB, with \(99.8\%\) emission upward [1904.11464]. In 1D ZCGs, true UGRs with \(\eta>80\) dB arise in a geometry that can be realized in a single lithography-etch step and is compatible with silicon-on-insulator or silicon-nitride platforms [2312.00283]. In the nanoantenna implementation, front-to-back ratios reached \(5.5\) for a dielectric waveguide and \(7.5\) for a plasmonic waveguide, demonstrating one-sided launching of guided modes from a subwavelength all-dielectric resonator [1508.02731].

| Platform | Representative result | Source |
|---|---|---|
| All-dielectric nanoantenna + dielectric waveguide | FBR\(_\text{field}\approx 5.5\) at \(595\) THz | [1508.02731] |
| All-dielectric nanoantenna + Au film | FBR\(_\text{field}\approx 7.5\) at \(440\) THz | [1508.02731] |
| Broken-symmetry photonic-crystal slab | Single-sided \(Q\) up to \(1.6\times10^5\); \(27.7\) dB directionality | [1904.11464] |
| Topological SOI grating coupler | \(0.34\) dB peak loss; \(>30\) nm 1 dB bandwidth | [2306.09027] |
| 1D zero-contrast grating | \(\eta\approx81\) dB at \(h_u\approx0.2708\Lambda\) | [2312.00283] |

The application space is correspondingly broad. The literature lists energy-efficient grating couplers and optical interconnects [2306.09027], photonic-crystal surface-emitting lasers and on-chip directional lasers with high efficiency and low threshold [1904.11464; 2203.02223], enhanced light trapping in photovoltaics [2203.02223], spin- and orbital-angular-momentum beam generation [1902.08054; 2203.02223], and quantum spin-photon interfaces, optical isolators, and nonreciprocal elements in integrated photonic and quantum networks [1508.02731]. Because several platforms are all-dielectric, low loss, and CMOS compatible [1508.02731; 2306.09027], UGRs are now treated not merely as a singular topological curiosity, but as a design principle for mirror-free, one-sided radiation control in nanophotonics and integrated optics.

Source: https://www.emergentmind.com/topics/unidirectional-guided-resonances-ugrs