Incident Direction Independent Wave Propagation and Unidirectional Lasing

We propose an incident direction independent wave propagation generated by properly assembling different unidirectional destructive interferences (UDIs), which is a consequence of the appropriate match between synthetic magnetic fluxes and the incident wave vector. Single-direction lasing at spectral singularity is feasible without introducing nonlinearity. UDI allows unidirectional lasing and unidirectional perfect absorption; when they are combined in a parity-time-symmetric manner, the spectral singularities vanish with bounded reflections and transmissions. Furthermore, the simultaneous unidirectional lasing and perfect absorption for incidences from opposite directions is created. Our findings provide insights into light control and may shed light on the explorations of desirable functionality in fundamental research and practical applications.

Figure 1

(a) Schematic of uniform resonator chain with one side-coupled resonator, ${\omega }_{\alpha }={\omega }_c+V_{\alpha }$. The blue and green arrows indicate the optical path lengths for counterclockwise mode (black arrow) photons tunneling between resonators in opposite directions. (b) Right UPA: right perfect absorption and left resonant transmission. (c) Left UPA: left perfect absorption and right resonant transmission. The upper panels illustrate UPA configurations with opposite synthetic magnetic fluxes, $k$ is the incident wave vector, and $V_{\alpha }=-e^{ik}$. The central panels illustrate destructive interference at the side-coupled resonator $\alpha$ from one incident direction, and the lower panels illustrate perfect absorption from the opposite incident direction. The dotted line indicates the equivalently decoupled part. (d) Schematic of uniform resonator chain with two side-coupled resonators. $j$ is the resonator index, ${\omega }_{\alpha ,0,\beta }={\omega }_c+V_{\alpha ,0,\beta }$.

Figure 2

(a), (b) Schematic of the incident direction independent wave propagation. (c), (d) Simulations of single-direction lasing at $V_{\alpha }=-e^{-ik}$, $V_{\beta }=-e^{ik}$ in Fig. 1. The Gaussian wave packet is $|\mathrm{\Psi }(0,j)⟩=(\sqrt{\pi }/\sigma {)}^{-1/2}\sum _j{e}^{-({\sigma }^2/2){(j-N_c)}^2}{e}^{i{k}_{c}j}|j⟩$, centered at $N_c$, where $k_c=\pi /3$ is the wave vector, $\sigma =0.1$, and $j$ is the resonator index. The resonator chain is cut at $j=\pm 100$. The blue curves in (c), (d) depict the wave intensities $P(j)=|\mathrm{\Psi }(t,j){|}^2$ at time $t=60/J$. Other parameters are ${\varphi }_{\alpha }={\varphi }_{\beta }=\pi +k$, $V_0=-2i\sin k$, and $k=\pi /3$.

Figure 3

Density plots of (a) $|r_L|$, (b) $|t_L|$, and (c) $|t_R|$ as functions of ${\varphi }_{\alpha }$ and ${\varphi }_{\beta }$. $r_R=0$ for $|{\varphi }_{\alpha }|\ne 2\pi /3$. (d) $|t_L|$ ($|t_R|$) for ${\varphi }_{\alpha }=-2\pi /3$ $(2\pi /3)$ and $|r_R|$ at when $r_L$ diverges. Solid white (dashed yellow) lines indicate the divergence (zero) at the SS. SSs coincide with unity values at the marked points in (b), (c). Color bars are all in (c), $|r_L|$ and $|t_{L,R}|$ are cut to five. The parameters are $V_{\alpha }=-e^{-i\pi /3}$, $V_{\beta }=-e^{i\pi /3}$, $V_0=0$, and the incident wave vector is $k=\pi /3$.

Figure 4

Snapshots of the wave intensities and schematics of the equivalent systems for (a)–(f) transmissionless and (g)–(i) reflectionless unidirectional lasing and perfect absorption for a Gaussian wave packet of $\sigma =0.1$, $k_c=\pi /3$. The arrows indicate the phase directions with values inside, $k=\pi /3$.