Fig. 1. Principle of SBOs in electric-field-driven synthetic temporal lattices. (a) Two fiber loops with slightly different lengths are connected by an OC to construct the temporal lattice. The incorporated PMs in the short and long loops can introduce the step-dependent phase shifts of ${\varphi }_u(m)$ and ${\varphi }_v(m)$. (b) Schematic of the synthetic temporal lattice mapped from the pulse evolution in two coupled fiber loops in panel (a) and the sketch of SBO trajectory denoted by the red curve. The purple curves denote the required dc- and ac-driving electric fields to induce SBOs, which are created simultaneously by imposing opposite phase shifts of ${\varphi }_u(m)$ and ${\varphi }_v(m)$ in short and long loops. (c) Effective time-averaging band structure of the synthetic temporal lattice at $E_{\omega }=1.8$, 3.8, and 5.3. The dashed curve is the original band structure without electric-field driving, where $\delta$ is the frequency detuning between the dc- and ac-driving electric field.

Fig. 2. Experimental setup. Panels (a) and (b) denote the long and short loops, respectively. All optical and electric components are as follows: PC, polarization controller; MZM, Mach–Zehnder modulator; OS, optical switch; OC, optical coupler; SMF, single-mode fiber; EDFA, erbium-doped fiber amplifier; VOA, variable optical attenuator; BPF, bandpass filter; AOM, acoustic optical modulator; PD, photodiode; OSC, oscilloscope; PBS, polarization beam splitter; PM, phase modulator; AWG, arbitrary waveform generator.

Fig. 3. Simulated and measured results of SBOs. (a) SBO oscillation amplitude A_SBOs as a function of the ac-driving amplitude $E_{\omega }$ and the inverse frequency detuning $1/\mathit{\delta }$. The golden spheres represent the measured results. (b) SBO oscillation period M_SBOs as a function of the inverse frequency detuning $1/\delta$. The curve and squares denote the calculated and measured results, respectively. The inset figure shows $M_{\text{SBOs}}$ as a function of $\mathit{\delta }$. (c) Initial oscillation phase of SBOs versus the initial Bloch momentum $k$ for $E_{\omega }=1.8$, $N=1$, $\phi =\pi /2$, and $\delta =\pi /150$. The solid curves and spheres denote the theoretical and experimental results, respectively. (d)–(g) Measured pulse intensity evolutions for $E_{\omega }=1.8$, 3, 3.8, and 5.3 under $\delta =\pi /150$. The white solid curves denote the averaging SBO oscillation trajectories $⟨\mathrm{\Delta }{n}_{\pm }(m)⟩$ obtained by fitting from the experimental results using the cosine function. The blue and green lines denote the SBO oscillation amplitude $A_{\text{SBOs}}$ and period $M_{\text{SBOs}}$, respectively. (h) Experimental pulse intensity evolution for $E_{\omega }=5.3$ and $\delta =\pi /90$.

Fig. 4. Fourier spectrum of SBOs. (a) The power ratio of SBOs with respect to all Fourier spectrum components as a function of the ac-driving amplitude $E_{\omega }$. The solid curve and spheres denote the theoretical and experimental results, respectively. (b) The standard deviation of the Fourier spectrum for $\omega >\alpha$ varying with $E_{\omega }$. (c)–(e) Fourier spectra of measured SBO trajectories at $E_{\omega }=1.8$, 3.8, and 5.3.

Fig. 5. Generalized SBOs under arbitrary-wave ac-driving fields. (a) Schematic of the sinusoidal-, rectangular-, and triangular-wave ac-driving electric fields. (b) SBO oscillation amplitude $A_{\text{SBOs}}$ versus the ac-driving amplitude $E_{\omega }$ under the sinusoidal-, rectangular-, and triangular-wave driving. The solid curves and spheres denote the theoretical and experimental results, respectively. (c)–(e) Measured pulse intensity evolutions under sinusoidal-, rectangular-, and triangular-wave driving, respectively. The ac-driving amplitude is taken as the collapse point for the rectangular-wave driving of $E_{\omega }=3$.

Fig. 6. Application of beam routing and splitting based on SBO collapse. (a) Packet oscillation displacements $⟨\mathrm{\Delta }{n}_{\pm }(m)⟩$ as a function of the driving amplitude $E_{\omega }$ for upper and lower band excitations. The solid curves and spheres represent the theoretical and experimental results, respectively. (b), (c) Measured pulse intensity evolutions for the upper band excitation at $E_{\omega }=1.8$ and 5.3, respectively. (d), (e) Measured pulse intensity evolutions for the simultaneous excitation of upper and lower bands with the power ratio of 65/35 under the ac-driving amplitudes of $E_{\omega }=1.8$ and 5.3, respectively.