Panels (e) and (f) show analogous plots when the pump is on the lower bistability branch. The dashed lines are calculated for noninteger values of | ℓ S |. Squares and circles are the values corresponding to the modes of integer angular momenta | ℓ S |. The eigenvalues are labeled according to the Bogoliubov norm ∥ v S, I ± ∥ of the associated eigenvectors. (b), (c) Real and imaginary parts of the Bogoliubov eigenvalues ω S for a pump in the upper bistability branch as a function of the signal angular momenta ± | ℓ S |. The big, red arrow represents the incident pump, while the small arrows describe the probing with the signal in the forward (dark blue) and in the reverse (cyan) directions. A sketch of the device is found in the upper-right part of panel (a). The vertical dashed-dotted and dashed lines signal respectively the real parts of the Bogoliubov eigenvalues corresponding to the reverse- and forward-propagating signals. 3a1, this allows us to harness the device as an optical isolator. The different role of FWM for signals propagating in opposite directions leads to different transmittance spectra in the upper bistability branch. Panels (a) and (d) display the results when we pump in the upper and lower bistability branches, respectively. The pump is close to resonance with the mode of angular momentum ℓ P = 133. (a), (d) Transmittance T of a weak signal beam in the forward (dark blue) and reverse (cyan) directions across a coherently illuminated passive ring resonator in add-drop configuration as a function of the signal frequency ω S in the vicinity of the resonator modes with absolute angular momenta | ℓ S | = 131. In the forward case, the different frequencies of pump, idler, and signal allow us to employ a band-pass filter to isolate the transmitted signal beam at the system's output. This can lead to a zero transmitted field a S − ( out ) = 0 while a S + ( out ) ≠ 0 at the same frequency. The absence of FWM coupling pump and signal in reverse operation produces a different phase shift with respect to the forward case, where FWM is possible and an idler field a I + (dashed blue arrow) is generated. The field amplitudes of these modes inside the resonator are given by a S ±. In the forward and reverse operation direction shown in panels (b) and (c), the system is probed by the external fields a S ± ( in ) (solid blue and cyan arrows). The field amplitude a P + corresponds to the large intensity pump mode inside the resonator. In both forward and reverse configurations, the resonator can be pumped in the CCW direction by an external field a P + ( in ) (thick red arrow). The ring resonator case is recovered by setting t S = 1 and k S = 0. ![]() The dashed green (orange) rectangles indicate directional couplers coupling the ring resonator to the bus waveguides (S-shaped element), with transmission and coupling amplitudes t w ( t S) and i k w ( i k S), respectively. General scheme of a ring or TJR operated in (b) forward and (c) reverse directions. This allows us to set the CCW (CW) modes as the forward (reverse) operation direction of our optical isolator. ![]() While FWM (symbolized by the gray arrows) can effectively couple CCW-propagating pump, signal, and idler modes, it cannot couple the pump P + with the signal S − because the resulting idler would completely fall out of resonance. The green dashed lines are the dispersion relations for CCW ( ℓ > 0) and CW ( ℓ < 0) modes including the curvature given by Eq. ( 3). (a) Dispersion relation of the resonant frequencies ω ℓ ( 0 ) of a ring or Taiji (TJR) resonator as a function of the angular momentum ℓ of the mode. A few most relevant setups realizing our proposal are specifically investigated, such as a coherently illuminated passive ring resonator and unidirectionally lasing ring or Taiji resonators. Taking advantage of a close analogy with fluids of light, our proposed isolation mechanism is physically understood in terms of the Bogoliubov dispersion of collective excitations on top of the strong pump beam. The mechanism underlying optical isolation is based on the breaking of optical reciprocity induced by the asymmetric action of four-wave mixing processes coupling a strong propagating pump field with copropagating signal and idler modes but not with reverse-propagating ones. Such devices can be straightforwardly realized in state-of-the-art integrated photonics platforms. In this work we propose and theoretically characterize optical isolators consisting of an all-dielectric and nonmagnetic resonator featuring an intensity-dependent refractive index and a strong coherent field propagating in a single direction.
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