Photonic Modal Circulator Using Temporal Refractive-Index Modulation with Spatial Inversion Symmetry

It has been demonstrated that dynamic refractive-index modulation, which breaks time-reversal symmetry, can be used to create on-chip nonreciprocal photonic devices. In order to achieve amplitude nonreciprocity, all such devices moreover require modulations that break spatial symmetries, which adds complexity in implementations. Here we introduce a modal circulator, which achieves amplitude nonreciprocity through a circulation motion among three modes. We show that such a circulator can be achieved in a dynamically modulated structure that preserves mirror symmetry, and as a result can be implemented using only a single standing-wave modulator, which significantly simplifies the implementation of dynamically modulated nonreciprocal devices. We also prove that in terms of the number of modes involved in the transport process, the modal circulator represents the minimum configuration in which complete amplitude nonreciprocity can be achieved while preserving spatial symmetry.

Figure 1

Dynamically modulated two-port system. The fields at each port consist of multiple frequency sidebands.

Figure 2

(a) A slab waveguide design. Blue region represents the static waveguide surrounded by air. The modulation is applied to $1/3$ of the waveguide shown as the yellow region. (b) Band structure of the slab waveguide. The waveguide supports three modes. The modulation frequencies ${\mathrm{\Omega }}_1$, ${\mathrm{\Omega }}_2$, and ${\mathrm{\Omega }}_3$ are chosen to couple $|1⟩\leftrightarrow |2⟩$, $|2⟩\leftrightarrow |3⟩$, and $|1⟩\leftrightarrow |3⟩$, respectively. Panels (c) and (d) show the phase factors associated with transitions between mode $|1⟩$ and mode $|3⟩$. By choosing modulation phases such that $({\varphi }_1+{\varphi }_2)-{\varphi }_3=\pi /2$, (c) for the up-conversion from mode $|1⟩\to |3⟩$, the yellow and the blue pathways constructively interfere, and (d) for the down-conversion from mode $|3⟩\to |1⟩$, the two pathways destructively interfere. The transition between mode $|1⟩$ and $|3⟩$ is nonreciprocal. (e) Unidirectional circulation among the three modes.

Figure 3

Waveguide simulation results. The top panels are the normalized $E_z$ field distributions with different input modes from the left port. The solid black lines represent the boundaries of the waveguide and the yellow shaded regions represent the modulation regions. The bottom panels are the normalized photon number flux ${|a_n(x)|}^2$ as a function of the propagation distance $x$. The blue, orange, and green lines represent the photon number flux of modes $|1⟩$, $|2⟩$, and $|3⟩$, respectively. Panels (a)–(c) show the complete conversions from $|1⟩$ to $|3⟩$, $|2⟩$ to $|1⟩$, and $|3⟩$ to $|2⟩$, respectively.

Figure 4

Coupled-ring system. (a) The coupled-ring system preserves mirror symmetry about the $xz$ plane. The system consists of three identical ring resonators coupled with coefficient $\mu$. Only the yellow ring is coupled to a straight waveguide and is modulated such that its resonant frequency varies as ${\omega }_0+{\omega }_m(t)$. (b) The frequency spectrum for the coupled-ring system. The coupled-ring system supports three supermodes with frequency separation $\sqrt{2}\mu$, which is far smaller than the free spectral range (FSR). (c) Transmission of different frequency components as a function of the input frequency $\omega$. The solid lines are the coupled mode theory (CMT) fitting results and the dots represent the simulation results. When the incident light has frequency $\omega \approx {\omega }_0-\sqrt{2}\mu \approx 2\pi \times 194.682\mathrm{THz}$, there is a strong output at $\omega +2\mathrm{\Omega }$, indicating the transition $|1⟩\to |3⟩$. Similarly, the $|2⟩\to |1⟩$ occurs at input frequency around 194.722 THz and the $|3⟩\to |1⟩$ occurs at input frequency around 194.762 THz.