Floquet $\pi$ mode engineering in non-Hermitian waveguide lattices

Floquet topological systems exhibit rich physics associated with quasienergy band structures and new topological states; nevertheless, they are usually explored in Hermitian systems. Recent studies have shown the capability of non-Hermiticity in engineering topological states, while the interplay of Floquet topological phases and non-Hermiticity remains unclear. Here, we reveal that the non-Hermitian modulation can induce the phase transitions between trivial and nontrivial topological Floquet states. Our study theoretically predicts that the non-Hermitian modulation can create a Floquet $\pi$ mode in an originally topological trivial system according to the reopening of quasienergy band gap (i.e., the $\pi$ gap), which is well confirmed experimentally in the silicon waveguide platform. Our approach shows the powerful capability of non-Hermitian modulation in engineering topological modes in Floquet photonics systems and would inspire different possibilities in optical field manipulation in open systems.

Figure 1

(a) Schematic illustration of non-Hermitian periodically curved waveguide array, where the red and blue colors indicate the gain and loss respectively. The inset shows the scheme of the gap $D\left(z\right)$ between adjacent waveguides. (b) $D\left(z\right)$ as function of $z$ in a full period under different waveguide profiles, where “Designed” corresponds to our inverse designed waveguide profile to meet the requirement of coupling strength $c_0+\delta \left(z\right)$ and “Cosine curved” corresponds to widely used cosine function modulated waveguide. (c) Quasienergy spectrum under OBCs with 60 waveguides as a function of $\omega /c_0$. (d) Coupling strength of $[c_0+\delta \left(z\right)]/k_0$ of different waveguide profile, where “Cosine coupling” corresponds to the exact theoretical model with cosine function modulated coupling term. Our designed waveguide profile can exactly realize the cosine function modulated coupling as required by the Floquet model.

Figure 2

(a) Real part of quasienergy spectrum under OBCs with 60 waveguides as a function of $\gamma /c_0$. ${\gamma }_{\mathrm{c}}=0.142c_0$ (marked by the black dashed line) is the threshold of gain/loss strength of recovering $\pi$ mode. The $\pi$ modes are highlighted in red. The bottom panel depicts the absolute value of numerically calculated $G_{\pi }$. (b) Calculated $G_{\pi }$ and phase diagram of $\pi$ mode as a function of $\gamma /c_0$ and $\omega /c_0$. $\pi$ mode exists in region II, and disappears in I and III. Orange lines indicate the boundary of different topological phases and the black dotted line indicates the trajectory of topological phase transition of increasing $\gamma$, corrsponding to (a). Quasienergy band under PBCs for a Hermitian $\gamma =0$ (c) and non-Hermitian $\gamma =0.005k_0$ (d) systems, respectively. $k$ is the Bloch wave vector and $a$ is the transverse period. Inset depicts the band structure near the $\pi$ gap for better visualization. Here we set $c_0=0.02k_0, {\delta }_0=0.015k_0$, and $\omega =1.33c_0$.

Figure 3

(a), (b) Quasienergies of 60 waveguides in the Hermitian $\gamma =0$ (a) and non-Hermitian $\gamma =0.005$ (b) cases. Only ten quasienergies around the first Floquet Brillouin zone edge are shown. Inset shows the imaginary part of quasienergy. In the non-Hermitian system, $\pi$ gap opens and a pair of $\pi$ modes appear, including one lossy mode and one gain mode, corresponding to the blue and red modes in the inset in (b), respectively. (c) Normalized intensity at $z=0$ of modes in the non-Hermitian cases [circled by red dotted line in (b)]. (d) The dynamic evolution of recovered $\pi$ modes for 60 waveguides.

Figure 4

(a) SEM image and enlarged regions of the non-Hermitian sample. Left: Full view of the sample, consisting of a grating coupler, a taper, and the waveguide array. Right top: Input waveguide. Right bottom: Cr on the waveguide. (b) CMT calculated optical field propagation in Hermitian (left) and non-Hermitian (right) waveguide arrays and corresponding normalized output intensities (bottom). (c) Experimental detected output intensities of Hermitian (left) and non-Hermitian (right) sample. Top: CCD recorded output signal. Bottom: Output intensities, which are normalized to maximum. Scale bar = 10 μm.

Figure 5

(a) Coupling coefficient under different waveguide gaps (edge to edge). The coupling strength does not depend on the gap linearly. Inset shows the inverse designed gap from the relation between gap and coupling strength. (b) $D\left(z\right)$ of adjacent waveguides for cosine coupling modulation. $D1\left(z\right)$ and $D2\left(z\right)$ correspond to $c_0+\delta \left(z\right)$ and $c_0-\delta \left(z\right)$ respectively. (c) Scheme of designed waveguide array. Only ten waveguides are plotted.

Figure 6

Fabrication flow of the photonic silicon waveguide array deposited with Cr.

Figure 7

Imaginary part of quasienergy spectrum with increasing gain/loss strength γ. The Floquet frequency is set to $\omega =1.33c_0$.

Figure 8

Experimentally captured output signals with different wavelengths. (a)–(h) Correspond to $\lambda =1480$, 1490, 1500, 1510, 1520, 1570, 1580, and 1600 nm, respectively. Scale bar = 30 μm.