Integrated photonics is an emerging technology that integrates multiple photonic components, such as waveguides, lasers, modulators, and detectors, onto a single chip to control and manipulate light [1–4]. By leveraging high-index materials like silicon, large-scale photonic integrated circuits (PICs) can be cost-effectively realized using current nanofabrication technology [5–9]. The grating coupler is a crucial component in integrated photonics, facilitating efficient coupling of light into and out of on-chip PICs [10–12]. Studies have shown that the coupling efficiency of grating couplers can be significantly enhanced by optimizing structural parameters using various geometric designs, including over-layered [13–15], slanted [16,17], interleaved [18,19], and L-shaped [20,21] configurations. In addition to optimizing structural parameters, eliminating unwanted downward radiation into the substrate remains a significant challenge for further improving coupling efficiency. Traditionally, this unwanted radiation can be suppressed by incorporating perfect reflectors such as metallic mirrors or multilayered photonic bandgap materials [22–24]. However, integrating these reflectors beneath the grating structures increases the overall bulkiness and introduces additional complexity to the fabrication process.

Unidirectional guided resonances (UGRs) [25–28] and bound states in the continuum (BICs) [29–33] are fascinating topological phenomena observed in planar photonic structures, such as one-dimensional (1D) gratings and two-dimensional (2D) photonic crystal slabs. BICs act as vortex centers carrying quantized integer topological charges and exhibit infinite radiative $Q$ factors because they do not radiate in either the upward or downward direction [34–36]. In contrast, UGRs also function as vortex centers and carry topological charges, but they exhibit finite $Q$ factors due to their ability to radiate exclusively in one direction [37–40]. Recently, it has been demonstrated that ultralow-loss grating couplers can be implemented by utilizing UGRs that originate from symmetry-protected BICs [41]. However, despite its enhanced coupling efficiency, the practical application of the proposed UGR-based grating coupler is limited by the negative-angle emission at the grating interface. Figure 1(a) compares the negative-angle ($\theta <0$) and positive-angle ($\theta >0$) emission from UGRs. For negative-angle emission, ${\mathbf{S}}_1·{\mathbf{S}}_2<0$, while for positive-angle emission, ${\mathbf{S}}_1·{\mathbf{S}}_2>0$, where ${\mathbf{S}}_1$ and ${\mathbf{S}}_2$ are the Poynting vectors for the UGR and emitted light, respectively. As illustrated in Fig. 1(b), light from the PIC enters the grating region through a waveguide and is subsequently emitted via UGRs. Conventionally, the emitted light from a grating coupler is directed into an optical fiber or fiber array block. In many practical applications, however, it is preferable that the optical fiber or fiber array block does not cross above the optical components constituting the PIC, making positive-angle emission more desirable.

In this paper, we investigate the formation and radiation of UGRs in two distinct L-shaped gratings, type I and type II, that lack both up-down mirror symmetry and in-plane $C_2$ symmetry. Our analysis employs rigorous finite element method (FEM) and finite-difference time-domain (FDTD) simulations. In type I gratings, quasi-UGRs with a notably high upward decay rate are readily observed in the lower band, whereas in type II gratings, such quasi-UGRs appear in the upper band. We show that by gradually varying the appropriate grating parameters, quasi-UGRs evolve into fully developed UGRs in the lower band for type I gratings and in the upper band for type II gratings. In type I gratings, UGRs exhibit negative-angle emission because their Poynting vectors are oriented antiparallel to their wavevectors, whereas in type II gratings, UGRs yield positive-angle emission due to the parallel alignment of their Poynting vectors and wavevectors. Moreover, systematic variation of the lattice parameters enables control over both the position and emission angle of UGRs. Our results provide valuable insights for the design of high-efficiency grating couplers that leverage UGRs.

We begin our investigation by examining a conventional 1D grating that exhibits in-plane $C_2$ symmetry, as depicted in Fig. 2(a). This partially etched geometry, commonly referred to as a zero-contrast grating (ZCG), has been widely utilized in various optical applications in recent years [42–44]. The grating consists of silicon with a refractive index of $n_{\mathrm{Si}}=3.48$, embedded in a surrounding medium with a refractive index of $n_{\mathrm{s}}=1.46$. Due to periodic variations in the dielectric constant, ZCGs exhibit photonic band gaps that emerge under the Bragg condition $k_x=k_{\mathrm{Bragg}}=jK/2$, where $k_x$ is the Bloch wavevector, $K=2\pi /\mathrm{\Lambda }$ is the magnitude of the grating vector, and $j$ is an integer representing the Bragg order. In principle, periodic structures support multiple leaky guided modes, including transverse electric (${\mathrm{TE}}_n$) and transverse magnetic (${\mathrm{TM}}_n$) modes, each characterized by its own dispersion curves and photonic band gaps. In this study, we focus on the BICs and UGRs associated with the second stop band created by the fundamental ${\mathrm{TE}}_0$ mode. Notably, recent investigations into BICs and UGRs have predominantly centered on the second band gap generated by the ${\mathrm{TE}}_0$ mode [36,45]. We note that our analysis is limited to the subwavelength region where only zeroth-order diffraction takes place.

Figure 2(b) presents the FEM-simulated dispersion curves and radiative $Q$ factors of Bloch modes near the second stop band. The structural parameters used in the simulations are $t=0.60\mathrm{\Lambda }$, $h=0.35\mathrm{\Lambda }$, and $w=0.3\mathrm{\Lambda }$. As shown in Fig. 2(b), the second band gap ($j=2$) opens at $k_x=0$. The simulated spatial electric field ($E_y$) distributions, displayed in the inset, reveal that the upper band edge mode, which exhibits a symmetric field profile, radiates out of the grating. In contrast, the lower band edge mode, with an asymmetric field profile, remains tightly confined within the grating. The presence of the symmetry-protected BIC is clearly confirmed by analyzing the radiative $Q$ factor. At the $\mathrm{\Gamma }$ point, the lower band exhibits a radiative $Q$ factor exceeding ${10}^{15}$, but the $Q$ values drop rapidly as $k_x$ deviates from the $\mathrm{\Gamma }$ point. Symmetry-protected BICs can occur at either the lower or upper band edge, depending on the lattice parameters. In principle, the position of symmetry-protected BICs can be influenced by all lattice parameters. However, previous studies suggest that the parameter $\alpha =w/\mathrm{\Lambda }$ plays a dominant role in determining which band edge mode becomes the BIC [46,47]. We therefore calculated the eigenfrequencies of the leaky edge mode and the BIC mode as a function of the parameter $\alpha =w/\mathrm{\Lambda }$, while keeping all other parameters fixed. Figure 2(c) shows that within the range $0<\alpha <1$, the second band gap closes at $\alpha ={\alpha }_{\mathrm{c}}=0.6219$. Before and after this band gap closure, the relative positions of the symmetry-protected BICs and leaky modes are reversed.

In gratings with in-plane $C_2$ symmetry, symmetry-protected BICs carry a topological charge of $q=+1$ or $-1$ [34]. However, by introducing in-plane asymmetry into the grating structure, as conceptually illustrated in Fig. 2(d), the symmetry-protected BIC splits into two paired circularly polarized $C$ points at $(k_x,\pm k_y)$, each carrying half-integer topological charges of $q=\pm 1/2$ [25,41]. One of these $C$ points exhibits left-handed circular polarization (LCP), while the other exhibits right-handed circular polarization (RCP). As the geometric in-plane asymmetry increases, the paired $C$ points move through momentum space. UGRs are formed when these two half-integer topological charges recombine, resulting in integer topological charges of $q=\pm 1$. Therefore, generating UGRs requires tracking the trajectories of the paired $C$ points in 2D momentum space. Alternatively, it has also been demonstrated that UGRs can be achieved by tracing the evolution of quasi-UGRs in 1D momentum space [39]. Figure 2(e) conceptually illustrates the transition from quasi-UGRs to genuine UGRs as geometric asymmetry increases. In Fig. 2(e), the radiation ratio is defined as $\eta ={\gamma }_{\mathrm{up}}/{\gamma }_{\mathrm{down}}$, where ${\gamma }_{\mathrm{up}}$ and ${\gamma }_{\mathrm{down}}$ represent the decay rates in the upward ($+z$) and downward ($-z$) directions, respectively. Eigenstates exhibiting $\eta \ge 70\mathrm{dB}$ are classified as UGRs, whereas those with $\eta <70\mathrm{dB}$, which are significantly greater than 0 and locally maximal, are identified as quasi-UGRs. In a previous study, Wang et al. experimentally realized an ultralow-loss grating coupler utilizing a UGR with $\eta \approx 65\mathrm{dB}$ [41]. It is worth noting that tracking the transition from quasi-UGR to UGR in 1D momentum space significantly reduces computational time and resources compared to tracing the trajectories of $C$ points in 2D momentum space. To identify UGRs, we perform detailed FEM simulations using the commercial software COMSOL Multiphysics. By varying the lattice parameters, we analyze the dispersion relations, radiative $Q$ factors, spatial electric field distributions, and radiation ratios $\eta$.

In recent years, polarization singularities in momentum space, such as $C$ points, BICs, and UGRs, have attracted significant attention due to their intriguing topological properties. The dynamics of these topological polarization singularities can be visualized by investigating the polarization vectors of the far-field radiation. When plotting the polarization vectors, each polarization singularity carries a topological charge defined as $$q=\frac{1}{2\pi }{\oint }_L\mathrm{d}\mathbf{k}·{\nabla }_{\mathbf{k}}\varphi (\mathbf{k}),$$(1)which describes how many times the polarization vector winds around the singularity [34]. Here, $\varphi (\mathbf{k})=\arg [c_x(\mathbf{k})+i{c}_y(\mathbf{k})]$ is the angle of the polarization vector, and $L$ is a closed path encircling the singularity in the counterclockwise direction. BICs cannot radiate because they are located at the vortex center where the polarization vector is undefined. In general, BICs possess integer-valued topological charges determined by the symmetry of the photonic lattices. In 1D gratings, for instance, accidental or symmetry-protected BICs near the second stop band can carry topological charges of $q=\pm 1$. Figure 3 schematically illustrates the polarization vector fields around a symmetry-protected BIC at the center of the Brillouin zone in a conventional 1D ZCG. As shown in Fig. 3(a), a topological charge of $q=+1$ leads to a $2\pi$ rotation of the polarization vector along a closed loop, whereas a charge of $q=-1$ induces a $-2\pi$ rotation as illustrated in Fig. 3(b). Because topological charge is a conserved quantity, the sum of individual charges remains unchanged, and the $C$ points are not destroyed under continuous variations of the lattice parameters. However, they are no longer vortex centers and therefore radiate in both the upward and downward directions. By continuously varying suitable lattice parameters, the two half-integer topological charges eventually recombine, giving rise to a UGR with a topological charge of $\pm 1$ at an off-$\mathrm{\Gamma }$$k$ point. The UGR is unique in that it behaves as a vortex center in one direction while remaining a radiating mode in the other direction.

We now examine the formation and radiation properties of UGRs originating from the symmetry-protected BIC at the upper band edge of the second stop band by employing a type I grating, which has a period $\mathrm{\Lambda }$ and a thickness $t=0.6\mathrm{\Lambda }$, as illustrated in Fig. 4(a). This type I grating is created by etching an additional small rectangular section (with depth $h_{\mathrm{a}}$ and width $w_{\mathrm{a}}$) into a symmetric grating that has a larger rectangular cross-section (with depth $h$ and width $w$). Figures 4(b)–4(d) depict the dispersion curves, radiative $Q$ factors, and radiation ratios $\eta$, respectively, for the symmetric grating with $h=0.48\mathrm{\Lambda }$ and $w=0.35\mathrm{\Lambda }$. We set $\alpha =0.35<{\alpha }_{\mathrm{c}}$ to ensure that a symmetry-protected BIC emerges at the lower band edge. In this study, we use the normalized wavevector $k_x/K$. The dispersion curves and spatial electric field distributions shown in Fig. 4(b) confirm that the lower band edge mode is a non-leaky, symmetry-protected BIC, while the upper band edge mode is a low-$Q$ leaky mode. The simulated radiative $Q$ factors further demonstrate that the lower band edge BIC exhibits an extremely high radiative $Q$ factor, exceeding ${10}^{14}$, whereas the upper band edge mode has a significantly lower $Q$ factor of approximately 20. Due to the broken up-down mirror symmetry, one might expect that the eigenstates of a grating, even though it preserves in-plane $C_2$ symmetry, would emit more electromagnetic energy in either the upward or downward direction. However, as shown in Fig. 4(d), simulated $\eta$ curves reveal that over the wavevector range $-0.15\le k_x/K\le 0.15$, neither quasi-UGRs nor UGRs are observed in either the lower band or the upper band.

Figure 4(e) illustrates the evolution of the radiation ratio $\eta$ curves as the in-plane geometric asymmetry increases. The geometric asymmetry is influenced by the parameters $w_{\mathrm{a}}$ and $h_{\mathrm{a}}$. To simplify the analysis, both parameters are varied simultaneously under the constraint $w_{\mathrm{a}}=h_{\mathrm{a}}$, thereby reducing the complexity of the study. With $w_{\mathrm{a}}=h_{\mathrm{a}}\ne 0$, type I gratings do not support symmetry-protected BICs at the lattice $\mathrm{\Gamma }$ point. Instead, as shown in Fig. 4(e), quasi-UGRs and a UGR emerge at off-$\mathrm{\Gamma }$$k$-points in the radiation ratio $\eta$ curves. For $w_{\mathrm{a}}=h_{\mathrm{a}}=0.1\mathrm{\Lambda }$, two quasi-BICs are observed in the blue-colored $\eta$ curve corresponding to the lower band. One of these quasi-BICs, with $\eta =16.5\mathrm{dB}$ at $k_x/K=-0.0263$, emits roughly 45 times more energy in the upward ($+z$) direction, whereas the other, with $\eta =-20.8\mathrm{dB}$ at $k_x/K=0.0747$, emits approximately 120 times more energy in the downward ($-z$) direction. Although type I gratings can support quasi-UGRs with both $\eta >0$ and $\eta <0$, this study focuses on the transition from quasi-UGRs to UGRs with $\eta >0$, since conventional grating couplers typically employ upward emission, away from the substrate. As $w_{\mathrm{a}}=h_{\mathrm{a}}$ increases beyond $0.1\mathrm{\Lambda }$, the $\eta$ values for quasi-UGRs increase, reaching 40.3 dB at $k_x/K=-0.0787$ when $w_{\mathrm{a}}=h_{\mathrm{a}}=0.175\mathrm{\Lambda }$. With a further increase of $w_{\mathrm{a}}=h_{\mathrm{a}}$ to $0.1805\mathrm{\Lambda }$, as shown in Fig. 4(e), a UGR is observed with $\eta =77.5\mathrm{dB}$ at $k_x/K=k_{\mathrm{u}}=-0.0831$. The simulated $E_y$ field distributions confirm that this UGR predominantly radiates in the upward direction. Figure 4(e) further reveals that the UGR reverts to a quasi-UGR with $\eta =40.4\mathrm{dB}$ at $k_x/K=-0.0859$ when $w_{\mathrm{a}}=h_{\mathrm{a}}$ is increased to $0.184\mathrm{\Lambda }$. We note that quasi-BICs and a UGR appear only in the lower band, as they originate from the symmetry-protected BIC at the lower band edge. Regardless of the values of $w_{\mathrm{a}}=h_{\mathrm{a}}$, neither quasi-UGRs nor UGRs are observed in the red-colored $\eta$ curves corresponding to the upper band. Additionally, the insets in Fig. 4(e) reveal no significant changes in the dispersion curves with variations in $w_{\mathrm{a}}=h_{\mathrm{a}}$.

As illustrated schematically in Fig. 1, light from the PIC is guided through a waveguide into the grating coupler and subsequently emitted outward via UGRs. The emission angle from the UGR at $k_x/K=k_{\mathrm{u}}=-0.0831$ is determined by the phase-matching condition, as depicted in Fig. 5(a), and is given by $$\theta =-\arcsin \left(\frac{|k_{\mathrm{u}}|}{k_{\mathrm{s}}}\right),$$(2)where $k_{\mathrm{s}}=n_{\mathrm{s}}{k}_0$ is the magnitude of the wavevector for the emitted light in the surrounding medium, with $n_{\mathrm{s}}=1.46$. In Eq. (2), the “$-$” sign indicates that the emission angle is negative, as defined in Fig. 1(a). In type I gratings, the Poynting vector is parallel to the group velocity, given by $${\mathbf{v}}_{\mathrm{g}}=\frac{\mathrm{d}\omega }{\mathrm{d}\mathbf{k}}.$$(3)

Consequently, the concave-down-shaped dispersion curves observed for the lower band near the $\mathrm{\Gamma }$ point indicate that the group velocity, and therefore the direction of electromagnetic energy flow, are opposite to the wavevector $\mathbf{k}$. Thus, in the lower band [Fig. 4(e) at $w_{\mathrm{a}}=h_{\mathrm{a}}=0.1805\mathrm{\Lambda }$], the electromagnetic energy of the UGR flows in the $+x$ direction, even though its wavevector points toward $-x$. To excite the UGR at $k_x/K=k_{\mathrm{u}}=-0.0831$, therefore, an incident wave should have a Poynting vector, and therefore a group velocity, that points in the $+x$ direction. We note that the phase-matching condition by itself does not lead to the emergence of UGRs and can be applied to determine emission angles for both UGR and non-UGR cases.

Using the normalized frequency $\omega =0.4189$ and the corresponding wavevector component $k_x/K=k_{\mathrm{u}}=-0.0831$ obtained from FEM simulations, Eq. (2) yields an emission angle of $\theta =-9.38°$. To confirm the light emission characteristics of the UGR, we conducted FDTD simulations using the open-source software package MEEP. Figure 5(b) displays the simulated spatial $E_y$ field distributions for the UGR at $k_x/K=k_{\mathrm{u}}=-0.0831$. A waveguide mode with a frequency of $\omega =0.4189$ was excited by an eigenmode source positioned at the left end of the waveguide region and propagating toward the grating region. The results in Fig. 5(b) demonstrate that the propagating waveguide mode effectively couples to the UGR within the grating, emitting light predominantly upward at a negative angle of $\theta =-9.38°$. An additional FDTD simulation was performed for the non-UGR mode, which shares the same eigenfrequency as the UGR but exhibits a wavevector in the opposite direction, i.e., $k_x/K=-k_{\mathrm{u}}=0.0831$. As shown in Fig. 5(c), a waveguide mode propagating in the $-x$ direction is launched at the right end of the waveguide region and couples to the non-UGR mode. However, this non-UGR mode emits light simultaneously in both the upward and downward directions, with an emission angle of $\theta =-9.38°$.

As conceptually illustrated in Fig. 1(b), many practical applications favor positive-angle emission from UGRs. We next demonstrate that UGRs in the upper band, where the Poynting vectors and wavevectors point in the same direction, exhibit positive-angle emission, using a type II L-shaped grating, as illustrated in Fig. 6(a). The type II grating is created by etching a smaller rectangular section (with depth $h_{\mathrm{a}}$ and width $w_{\mathrm{a}}$) into a symmetric grating that has a larger rectangular cross-section (with depth $h$ and width $w$). The difference between type I and type II gratings is as follows. In type I gratings, an additional small cross-sectional region is etched in the unetched Si area at the upper right of a symmetric grating, whereas in type II gratings, an extra small cross-sectional region is etched in the already etched area at the lower left. Figures 6(b)–6(d) present the dispersion curves, radiative $Q$ factors, and radiation ratio $\eta$ curves, respectively, for the symmetric grating with $h=0.27\mathrm{\Lambda }$ and $w=0.75\mathrm{\Lambda }$. We set $\alpha =0.75>{\alpha }_{\mathrm{c}}$ to ensure that a symmetry-protected BIC emerges at the upper band edge. The dispersion curves and spatial electric field distributions in Fig. 6(b) confirm that the mode at the upper band edge is a non-leaky, symmetry-protected BIC, whereas the mode at the lower band edge is a low-$Q$ leaky mode. The simulated radiative $Q$ factors further reveal that the upper band edge BIC exhibits an exceptionally high radiative $Q$ factor exceeding ${10}^{14}$, while the upper band edge mode has a significantly lower $Q$ factor of approximately 30. Moreover, as shown in Fig. 6(d), with the in-plane $C_2$ symmetry, neither quasi-UGRs nor UGRs are observed in the simulated $\eta$ curves over the wavevector range $-0.15\le k_x/K\le 0.15$.

Figure 6(e) depicts the evolution of the radiation ratio $\eta$ curves as the in-plane geometric asymmetry increases. In our FEM simulations, the parameters $w_{\mathrm{a}}$ and $h_{\mathrm{a}}$ are varied simultaneously under the constraint $w_{\mathrm{a}}=2h_{\mathrm{a}}$. When $w_{\mathrm{a}}=2h_{\mathrm{a}}\ne 0$, the gratings do not support symmetry-protected BICs at the lattice $\mathrm{\Gamma }$ point. Instead, quasi-UGRs and a UGR emerge at off-$\mathrm{\Gamma }$$k$-points in the $\eta$ curves. For example, when $w_{\mathrm{a}}=2h_{\mathrm{a}}=0.2\mathrm{\Lambda }$, two quasi-BICs are observed in the red-colored $\eta$ curve corresponding to the upper band: one exhibits $\eta =20.4\mathrm{dB}$ at $k_x/K=0.0296$, and the other shows $\eta =-7.5\mathrm{dB}$ at $k_x/K=-0.0128$. Here, we focus on the transition from quasi-UGRs to UGRs with $\eta >0$. As $w_{\mathrm{a}}=2h_{\mathrm{a}}$ increases beyond $0.2\mathrm{\Lambda }$, the $\eta$ values for quasi-UGRs rise, reaching 35.8 dB at $k_x/K=0.0686$ when $w_{\mathrm{a}}=2h_{\mathrm{a}}=0.245\mathrm{\Lambda }$. A further increase to $w_{\mathrm{a}}=2h_{\mathrm{a}}=0.2513\mathrm{\Lambda }$, produces a UGR with an $\eta$ value of 79.6 dB at $k_x/K=k_{\mathrm{u}}=0.0753$, with simulated electric field distributions confirming that this UGR predominantly radiates upward. Moreover, Fig. 6(e) reveals that when $w_{\mathrm{a}}=2h_{\mathrm{a}}$ is increased to $0.256\mathrm{\Lambda }$, the UGR reverts to a quasi-UGR with $\eta =37.8\mathrm{dB}$ at $k_x/K=0.082$. In Fig. 6(e), quasi-UGRs and the UGR appear exclusively in the upper band, consistent with their origin from the symmetry-protected BIC at the upper band edge, while no quasi-UGRs or UGRs are observed in the red-colored $\eta$ curves corresponding to the lower band. Additionally, the insets in Fig. 6(e) indicate that the dispersion curves remain largely unchanged with variations in $w_{\mathrm{a}}=2h_{\mathrm{a}}$.

Figure 7(a) illustrates the phase-matching condition used to determine the emission angle of the UGR at $k_x/K=k_{\mathrm{u}}=0.0753$ in the upper band. Note that the phase-matching condition for UGRs in the upper band is essentially identical to that for UGRs in the lower band. However, the emission angle for UGRs in the upper band is given by $$\theta =+\arcsin \left(\frac{|k_{\mathrm{u}}|}{k_{\mathrm{s}}}\right),$$(4)where the “+” sign indicates that the emission angle is positive, as defined in Fig. 1(a). In type II gratings, the concave-up-shaped dispersion curves observed in the upper band indicate that the group velocity, and hence the direction of electromagnetic energy flow, are parallel to the wavevector $\mathbf{k}$. Therefore, to excite the UGR at $k_x/K=k_{\mathrm{u}}=0.0753$ in the upper band [Fig. 6(e) at $w_{\mathrm{a}}=2h_{\mathrm{a}}=0.2513\mathrm{\Lambda }$], the incident wave should have a Poynting vector pointing in the $+x$ direction.

Applying the eigenfrequency $\omega =0.3967$ at $k_x/K=k_{\mathrm{u}}=0.0753$ to Eq. (4) yields a positive emission angle of $\theta =7.47°$. Figure 7(b) presents the FDTD-simulated spatial $E_y$ field distributions, demonstrating that at $\omega =0.3967$ and $k_x/K=k_{\mathrm{u}}=0.0753$, the UGR exhibits positive-angle emission. In this simulation, a waveguide mode propagating in the $+x$ direction is launched at the left end of the waveguide region and couples to the UGR in the type II grating. The simulated emission angle in Fig. 7(b) agrees well with the calculated value of $\theta =7.47°$. An additional FDTD simulation was conducted for the non-UGR mode, which shares the same eigenfrequency as the UGR but features a reversed wavevector, i.e., $k_x/K=-k_{\mathrm{u}}=-0.0753$. As shown in Fig. 7(c), a waveguide mode propagating in the $-x$ direction is launched at the right end of the waveguide region and couples to the non-UGR mode. Figure 7(c) confirms that this non-UGR mode emits light in both the upward and downward directions, with an emission angle of $\theta =7.47°$.

In Fig. 6, the UGR is generated by tracking the evolution of quasi-UGRs in 1D momentum space, with the parameters $w_{\mathrm{a}}$ and $h_{\mathrm{a}}$ varied simultaneously under the constraint $w_{\mathrm{a}}=2h_{\mathrm{a}}$, ensuring a fixed width-to-height ratio of $w_{\mathrm{a}}/h_{\mathrm{a}}=2$. We also investigated the evolution of quasi-UGRs in type II L-shaped gratings by varying $w_{\mathrm{a}}$ and $h_{\mathrm{a}}$ under different constraints, which yielded a range of width-to-height ratios. Our FEM simulation results demonstrate that UGRs with $\eta \ge 70\mathrm{dB}$ can be generated over various width-to-height ratios. Figure 8(a) presents the simulated $\eta$ curves for six selected constraints with distinct width-to-height ratios, varying from 1.6 to 3.6 in discrete increments of 0.4, thereby confirming the presence of UGRs in type II gratings. As the width-to-height ratio increases from 1.6 to 3.2, the position of the UGRs gradually shifts from $k_x/K=0.0352$ to 0.162. Moreover, our FEM simulations reveal that an increase in the width-to-height ratio leads to higher eigenfrequencies and larger emission angles for the UGRs. The calculated eigenfrequencies $\omega$ and corresponding emission angles $\theta$ are presented in Fig. 8(b). As the width-to-height ratio increases from 1.6 to 3.2, the emission angle increases from $\theta =3.52°$ (corresponding to $\omega =0.382$ and $k_x/K=0.0352$) to $\theta =15.06°$ (corresponding to $\omega =0.4272$ and $k_x/K=0.162$).

The inset in Fig. 8(a) shows that as the width-to-height ratio increases, UGRs emerge at larger values of $w_{\mathrm{a}}$. This shift reduces the effective index of the UGR mode because the fraction of high-index silicon within the type II grating decreases with increasing $w_{\mathrm{a}}$. Consequently, the lower effective index causes the UGRs to appear at higher frequencies (i.e., longer wavelengths) and larger wavevectors. Moreover, because the wavevector increases more significantly than the eigenfrequency, the UGR emission angles, as determined by Eq. (4), also increase with increasing $w_{\mathrm{a}}$. FDTD simulations further confirm the positive-angle emission of UGRs in type II gratings with varying width-to-height ratios. Figures 8(c)–8(e) present three representative cases with width-to-height ratios of 1.6, 2.4, and 3.6, respectively. These figures demonstrate that the propagating waveguide modes, excited at the left end of the waveguide region, couple to the UGRs at $k_x/K=0.0352$, 0.1056, and 0.162 and yield positive emission angles of 3.52°, 10.23°, and 15.06°, respectively, as determined by Eq. (4).

In this study, we focus on the emission of UGRs with $\eta \ge 70\mathrm{dB}$ because conventional grating couplers utilize upward emission away from the substrate. However, we believe that UGRs with $\eta \le 70\mathrm{dB}$ can be realized in type I and type II L-shaped gratings with appropriate lattice parameters, and this may be the focus of our future work. The coupling efficiency of grating couplers can be enhanced by employing specialized designs such as slanted or interleaved geometries. However, these conventional grating couplers typically incur an intrinsic loss of about 3 dB due to radiation in both upward and downward directions. By contrast, UGR-based grating couplers can achieve higher coupling efficiencies by suppressing unwanted downward radiation. Notably, Wang et al. experimentally demonstrated a UGR-based grating coupler that achieved a record-low insertion loss of −0.34 dB, with a sufficient bandwidth of approximately 30 nm at the 1550 nm telecom wavelength [41].

In conclusion, we have investigated the formation and radiation of UGRs in type I and type II L-shaped gratings where both up-down mirror symmetry and in-plane $C_2$ symmetry are broken. In type I gratings, quasi-UGRs are easily observed in the lower band, while in type II gratings they appear in the upper band. As the appropriate grating parameters are varied, these quasi-UGRs transform into fully developed UGRs; this evolution occurs in the lower band for type I gratings and in the upper band for type II gratings. In type I gratings, UGRs in the lower band, characterized by concave-down dispersion curves, exhibit negative-angle emission because their Poynting vectors are oriented antiparallel to their wavevectors. In contrast, in type II gratings, UGRs in the upper band, which display concave-up dispersion curves, exhibit positive-angle emission owing to the parallel alignment of their Poynting vectors and wavevectors. Since many integrated photonic applications favor positive-angle emission from grating couplers, we further demonstrated that by systematically varying the relevant lattice parameters, the position and emission angle of UGRs in type II gratings can be controlled. Our findings provide valuable insights for the design of high-efficiency optical interconnects that leverage UGRs.