Evolution and Nonreciprocity of Loss-Induced Topological Phase Singularity Pairs

Singular behaviors are increasingly being found in structures supporting bound states in the continuum (BICs), while the nonreciprocity with spectral phase singularity has not been reported. Here, we demonstrate the origin, evolution, and application of topological phase singularity pairs (TPSPs) resulting from BICs in nonreciprocal and non-Hermitian systems. The nonreciprocity contributes to creating accidental BICs asymmetrically, each of which can split into TPSP with topological charges $\pm 1$ in reflection phases by inserting loss. The formation, annihilation, and revival processes of these TPSPs can be selectively controlled via either material or radiative loss. The criteria to predict both number and angular positions of asymmetric BICs are established. Near-complete violation of Kirchhoff’s law of thermal radiation has been correspondingly manifested over a wide angular range. The unveiled physics synergizing nonreciprocity and topology will bring new opportunities in multidisciplinary areas like thermal science, magneto-optics, or topological metasurfaces.

Figure 1

Generation of TPSPs and nonreciprocal tuning mechanisms. (a) Sketch of the planar structure, illuminated by TM waves $(E_x,k_y,H_z)$. (b) Compared dispersion surfaces of ${\omega }_i-\beta$ for lossless (left) and lossy (right) cases without $\mathbf{B}$. (c) Shaping the angular and spectral distributions of TPSPs via loss ($\mathrm{\Gamma }$, middle) and thickness ($t_e$, right) of the ENZ layer in $\omega -\theta$ spaces. The BICs and TPSPs are denoted by the blue “$\times$” and colored circles (or triangles). The gray (red) arrows show the moving directions when increasing (decreasing) the value of $\mathrm{\Gamma }$ ($t_e$). The leftmost plot shows dispersion curves with or without $\mathbf{B}$. (d) $Q$ factors for the case at $\mathbf{B}=0.3\mathrm{T}$ and $t_e=0.5\mu \mathrm{m}$. The bottom inset shows charge conservation between BIC and TPSP. (e) Phase diagrams ${\phi }_r$ and vector flows of $r$ when $\mathrm{\Gamma }=0$ (left) and $\mathrm{\Gamma }=5\times {10}^{10}\mathrm{rad}/\mathrm{s}$ (right). Results in (b)–(e) are analyzed without the dielectric spacer.

Figure 2

Evolution of asymmetric TPSPs when changing the ENZ layer’s thickness and loss. (a) Trajectories of TPSPs from three BICs (blue “$\times$”). Three dashed lines are dispersion curves for different $t_e$: $\text{0.15\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$ (purple), $\text{0.5\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$ (orange), and $2\mu \mathrm{m}$ (yellow). The gray (red) arrows show the moving directions of TPSPs marked in colored triangles (circles) when increasing (decreasing) the value of $\mathrm{\Gamma }$ ($t_e$), at a fixed $t_e=0.5\mu \mathrm{m}$ ($\mathrm{\Gamma }=5\times {10}^{10}\mathrm{rad}/\mathrm{s}$). The inset gives $Q$ factors at $t_e=0.5\mu \mathrm{m}$. (b) A typical case supporting three coupled TPSPs at $t_e=0.5\mu \mathrm{m}$ and $\mathrm{\Gamma }=5\times {10}^9\mathrm{rad}/\mathrm{s}$. Two sequential annihilation processes when (c) increasing the $\mathrm{\Gamma }$ to $5\times {10}^{10}\mathrm{rad}/\mathrm{s}$ and (d) further reducing the thickness to $t_e=0.1\mu \mathrm{m}$. (e) Charges revival at a large thickness $t_e=2\mu \mathrm{m}$. All data are analyzed at $\mathbf{B}=0.3\mathrm{T}$ and $L^n=1.75$.

Figure 3

Number and position selection criteria for asymmetric BICs. (a) Blooming flower patterns for predicting the angles $\theta$ and number of BICs when ${ϵ}_d=2$. (b) Projection of $L^n-\theta$ plane for typical cases: ${ϵ}_d=2$, $\mathbf{B}=0.3\mathrm{T}$ (left) and ${ϵ}_d=1$, $\mathbf{B}=0.2\mathrm{T}$ (right). The number of BICs is marked in each shaded region. (c) Influence of ${ϵ}_d$ in angular distributions of BICs in $\mathbf{B}-\theta$ space. The red dashed lines in (b)–(c) denote a typical case at $L^n=1.75$, where the BICs are marked with “$\times$.”

Figure 4

Nonreciprocal thermal radiation based on asymmetric TPSPs. (a) Compared principles to realize perfect nonreciprocal thermal emitters. (b) Asymmetric emission and absorption spectra with two TPSPs (white circles) at $L^n=1.75$, $t_e=0.5\mu \mathrm{m}$, and $\mathbf{B}=0.3\mathrm{T}$, along with a radiation pattern at ${\omega }_A$ (right). The full lines and dots are results obtained by scattering matrix and full-wave simulation. (c) Evolution of emission and absorption spectra via continuously tuning of $L^n$ at $\mathbf{B}=0.1\mathrm{T}$ and $t_e=5\mu \mathrm{m}$. (d) The corresponding working angles for cases in (c). The inset shows a typical angular diagram at $L^n=0.25$.