Correlated nonreciprocity around conjugate exceptional points

The occurrence of exceptional points (EPs) is a fascinating non-Hermitian feature of open systems. A level-repulsion phenomenon between two complex states of an open system can be realized by positioning an EP and its time-reversal ($\mathcal{T}$) conjugate pair in the underlying parameter space. Here, we report interesting nonreciprocal responses of such two conjugate EPs by using a dual-mode planar waveguide system having two $\mathcal{T}$-symmetric active variants concerning the transverse gain-loss profiles. We specifically reveal an all-optical scheme to achieve correlative nonreciprocal light dynamics by using the reverse chirality of two dynamically encircled conjugate EPs in the presence of local nonlinearity. A specific nonreciprocal correlation between two designed $\mathcal{T}$-symmetric waveguide variants is established in terms of their unidirectional transfer of light with a precise selection of modes. Here, the unconventional reverse chiral properties of two conjugate EPs allow the nonreciprocal transmission of two selective modes in the opposite directions of the underlying waveguide variants. An explicit dependence of the nonlinearity level on a significant enhancement of the nonreciprocity in terms of an isolation ratio is explored by investigating the effects of both local Kerr-type and saturable nonlinearities (considered separately). The physical insights and implications of harnessing the features of conjugate EPs in nonlinear optical systems can enable the growth and development of a versatile platform for building nonreciprocal components and devices.

Figure 1

(a) Schematic design of ${\mathrm{WG}}_{\text{A}}$ and ${\mathrm{WG}}_{\text{T}}$ ($\mathcal{T}$ symmetric) based on the framework of a gain-loss assisted planar waveguide. Two arrows indicate their opposite propagation directions. Circular plus and minus signs in different segments are associated with the positive (loss) and negative (gain) imaginary indices as in Eq. (2). (b) Transverse background refractive index profile, i.e., $\mathrm{Re}\left[n\right(x\left)\right]$, (dotted black line; corresponding to the left vertical axis) along with normalized intensity profiles of two supported modes ${\mathrm{\Psi }}_{\text{F}}$ and ${\mathrm{\Psi }}_{\text{H}}$ (corresponding to the right vertical axis). (c) Transverse gain-loss distributions, i.e., $\mathrm{Im}\left[n\right(x\left)\right]$, for two $\mathcal{T}$-symmetric variants ${\mathrm{WG}}_{\text{A}}$ (solid green line) and ${\mathrm{WG}}_{\text{T}}$ (solid black line) for $\gamma =0.01$ and $\tau =2$. $w$ and $l(=l_m\times {10}^3)$ are considered in units of $\lambda$.

Figure 2

(a) Connections between the Riemann surfaces associated with ${\beta }_{\text{F}}$ and ${\beta }_{\text{H}}$, while varying the control parameters $\gamma$ and $\tau$, simultaneously. (a.1) and (a.2) show the distributions of $\mathrm{Re}\left[\beta \right]$ and $\mathrm{Im}\left[\beta \right]$, respectively. Dotted red and blue curves represent the trajectories of ${\beta }_{\text{F}}$ and ${\beta }_{\text{H}}$ for a chosen $\tau =3.1607$, which reveal the encounter of two conjugate EPs based on the coalescence and bifurcations in $\mathrm{Re}\left[\beta \right]$ and $\mathrm{Im}\left[\beta \right]$ at $\gamma =\pm 8.1\times {10}^{-3}$. Dotted blue squares separate the regions for ${\mathrm{WG}}_{\text{A}}$ and ${\mathrm{WG}}_{\text{T}}$. (b) Variation of $\langle {\mathrm{\Psi }}_{\text{F}}|{\mathrm{\Psi }}_{\text{H}}\rangle$ with respect to $\gamma$ (when $\tau =3.1607$), which shows the coalescence of ${\mathrm{\Psi }}_{\text{F}}$ and ${\mathrm{\Psi }}_{\text{H}}$ via $\langle {\mathrm{\Psi }}_{\text{F}}|{\mathrm{\Psi }}_{\text{H}}\rangle =1$ at both EP and EP*. (c) Parametric encirclement of two conjugate EPs in the ($\gamma ,\tau$)-plane following Eq. (3) (shown in the ground surfaces) and associated transfer process of ${\beta }_{\text{F}}$ and ${\beta }_{\text{H}}$ from their respective surfaces.

Figure 3

(a) Parametric loops to encircle an EP and its conjugate EP* in the ($\gamma ,\tau$) plane [following Eq. (3)]. (b) Associated dynamical variation of gain-loss profiles, i.e., two complex conjugate active potentials (separated via a transparent plane), experienced by two $\mathcal{T}$-symmetric waveguide variants ${\mathrm{WG}}_{\text{A}}$ and ${\mathrm{WG}}_{\text{T}}$. $w$ and $l(=l_m\times {10}^3)$ are considered in units of $\lambda$.

Figure 4

(a) Propagation dynamics of ${\mathrm{\Psi }}_{\text{F}}$ and ${\mathrm{\Psi }}_{\text{H}}$ through ${\mathrm{WG}}_{\text{A}}$ (upper panel) from $z=0$ to $z=l$ (associated with the CW dynamical EP encirclement) followed by the asymmetric conversions $\{{\mathrm{\Psi }}_{\text{F}},{\mathrm{\Psi }}_{\text{H}}\}\to {\mathrm{\Psi }}_{\text{H}}$; (lower panel) from $z=l$ to $z=0$ (associated with the CCW dynamical EP encirclement) followed by the asymmetric conversions $\{{\mathrm{\Psi }}_{\text{F}},{\mathrm{\Psi }}_{\text{H}}\}\to {\mathrm{\Psi }}_{\text{F}}$. (b) Similar modal dynamics through ${\mathrm{WG}}_{\text{T}}$ (upper panel) for the CW dynamical encirclement around the EP* with $z:l\to 0$, exhibiting the asymmetric conversions $\{{\mathrm{\Psi }}_{\text{F}},{\mathrm{\Psi }}_{\text{H}}\}\to {\mathrm{\Psi }}_{\text{F}}$; (lower panel) for the CCW dynamical encirclement around the EP* with $z:0\to l$, exhibiting the asymmetric conversions $\{{\mathrm{\Psi }}_{\text{F}},{\mathrm{\Psi }}_{\text{H}}\}\to {\mathrm{\Psi }}_{\text{H}}$. Intensities are renormalized at each step of evolution along $z$ to show the inputs and outputs clearly. $w$ and $l(=l_m\times {10}^3)$ are considered in units of $\lambda$.

Figure 5

A schematic analogy between a four-port optical device and our designed dual-mode waveguide operating with a dynamically encircled EP or EP* in the presence of nonlinearity. This analogy is essentially drawn to construct a $4\times 4S$-matrix [given by Eq. (7)] considering all the possible transmissions.

Figure 6

(a) Nonreciprocal transition of modes with the asymmetric conversions $\{{\mathrm{\Psi }}_{\text{F}},{\mathrm{\Psi }}_{\text{H}}\}\to {\mathrm{\Psi }}_{\text{H}}$ through ${\mathrm{WG}}_{\text{A}}$ which is active in the forward direction ($z:0\to l$; associated with the CW dynamical EP encirclement process). (b) Schematic nonreciprocal response of ${\mathrm{WG}}_{\text{A}}$ (which allows light to pass in the forward direction, however, blocks in the backward directions) along with outputs (O/P) at $z=l$ (for the allowed path $z:0\to l$) and $z=0$ (for the blocked path $z:l\to 0$. (c) Nonreciprocal transition of modes with the asymmetric conversions $\{{\mathrm{\Psi }}_{\text{F}},{\mathrm{\Psi }}_{\text{H}}\}\to {\mathrm{\Psi }}_{\text{F}}$ through ${\mathrm{WG}}_{\text{T}}$ which is active in the backward direction ($z:l\to 0$; associated with the CW dynamical encirclement of the EP*). (d) Schematic nonreciprocal response of ${\mathrm{WG}}_{\text{T}}$ (which allows light to pass in the backward direction, however, blocks in the forward directions) along with outputs (O/P) at $z=0$ (for the allowed path $z:l\to 0$) and $z=l$ (for the blocked path $z:0\to l$). For both ${\mathrm{WG}}_{\text{A}}$ and ${\mathrm{WG}}_{\text{T}}$, the self-normalized outputs are shown for their active directions, whereas relative outputs are shown in their blocked directions. $w$ and $l(=l_m\times {10}^3)$ are considered in units of $\lambda$.

Figure 7

Dependence of the isolation ratio (IR) on the local nonlinearity level ($N_l$), while considering (a) Kerr-type nonlinearity and (b) saturable nonlinearity, separately. Red square and blue circular markers show such a variation of IR for ${\mathrm{WG}}_{\text{A}}$ and ${\mathrm{WG}}_{\text{T}}$, respectively. The green arrows in both (a) and (b) indicate the largest values of the IRs, as achieved at the same $N_l$.