## Abstract

Faraday rotation is arguably the most widely studied nonreciprocal phenomenon, observable in the form of polarization rotation as waves travel in magneto-optical materials. It is at the basis of the realization of various forms of isolators and circulators. However, magneto-optical materials are bulky and difficult to integrate, and inherently associated with losses. Here we leverage nonlinear phenomena in conventional optical crystals to mimic Faraday rotation in a resonant cavity excited with two pumps with opposite circular polarizations and slightly detuned frequencies. The effect can be observed in basic resonant structures with no need for material resonances or tailored dispersion, arbitrary frequency of the pumps, and low loss, yielding an interesting path towards the realization of compact magnet-free nonreciprocal optical elements.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

The realization of magnetless nonreciprocal components has been a major theme in recent photonics research. Nonreciprocity is useful for source protection and necessary to improve optical chip packing density. Magnetic materials, while popular in the microwave regime, are known to be lossy and difficult to integrate with current photonic technologies, though there has been recent progress with silicon-compatible materials, such as graphene and Ce:YIG [1,2]. Spatiotemporal modulation has been explored as an effective way to create nonreciprocal microwave components, with interesting pathways towards the implementation of magnet-free nonreciprocity [3–6]. In optics, beyond electro-optical modulation which is fundamentally limited in terms of speed, this effect can be realized with a traveling pump wave in nonlinear media [7]. For example, a Kerr or optomechanical ring resonator can be used to break degeneracy between counter-propagating modes, therefore breaking time-reversal symmetry [8–10]. Nonlinear resonators [11–15] can provide isolation without external modulation, but have limited dynamic range, and are fundamentally limited by time-reversal symmetry and dynamic reciprocity [16,17]. Given the limitations of current technologies, research in improved nonreciprocal responses that do not rely on magneto-optics is essential for progress in nanophotonic and quantum technology.

The Faraday effect is a widely known non-reciprocal phenomenon, wherein a magnetized sample rotates the polarization angle of a propagating field. The rotation handedness depends on the applied bias and not on the direction of propagation, hence leading to a nonreciprocal response. Due to microscopic reversibility, a circularly polarized optical pump can also induce magnetization in magneto-optical materials, a phenomenon known as inverse Faraday effect [18,19]. Such a response can in principle be used to rotate the polarization of a probe signal with a single optical pump, but it still requires magnetic media, which are generally not ideal for integration and practical implementation. A circularly polarized pump can also induce Faraday rotation in non-magnetic media, but it requires either a coherent interaction between a pump at the same frequency as the probe [20] or breaking Kleinmann symmetry, which is only possible in lossy, resonant media, posing an important tradeoff between the amount of rotation and the transmitted power through the media [20–23]. Recently, this concept has been explicitly explored by coupling to phonons [24,25]. While the effect may be strong, it requires careful tuning of pump and probe frequencies and the mechanical resonance, and it is narrowband in nature, being limited by the mechanical resonance frequency.

Here, we introduce an alternative approach to magnetless Faraday rotation in media with instantaneous third-order nonlinearity, by biasing the system with two optical pumps with opposite circular polarizations and detuned frequencies. Our approach does not require material loss, nor phase-matching between pump and probe signals, making it insensitive to the actual frequency of the pump signal and material dispersion, and widely applicable to waves propagating in opposite directions. We discuss a possible implementation in a compact Fabry–Perot resonator supporting degenerate modes, but alternative configurations are also available, for instance, in microresonators supporting degenerate modes and coupled to a waveguide. In addition to its theoretical significance as an alternative way to induce Faraday rotation via optical pumping, the proposed approach may be used in the design of magnetless isolators and other nonreciprocal devices for nanophotonic and quantum applications.

## 2. THEORY

We start our analysis with a brief discussion of the different mechanisms that can lead to Faraday rotation via optical pumping. First, consider the fundamental conservation laws in a scattering event: energy, angular momentum, and linear momentum. Linear momentum is dependent on the dispersive material properties, but photon frequency and spin are well-defined and sufficient to capture the behavior we describe below. Polarization is an important consideration for nonlinear processes and results in selection rules for a given lattice geometry [23]. We further assume that no energy or momentum is transferred to the lattice, as in a lossless isotropic medium. We decompose the linear signal ${\omega}_{s}$ into right-handed (RH, ↻) and left-handed (LH, ↺) components, defined with respect to a fixed axis instead of the wave propagation direction, and assume a RH pump at ${\omega}_{p}$ [20,26]. Consider the first transition in Fig. 1(a): the relevant ${\chi}^{(3)}$ component is ${\chi}_{\mathrm{RRLR}}({\omega}_{s};{\omega}_{p},-{\omega}_{p},{\omega}_{s})$, where the L component arises because conjugating a circularly polarized wave reverses its handedness. The second transition in Fig. 1(b) is described by ${\chi}_{\mathrm{LRLL}}({\omega}_{s};{\omega}_{p},-{\omega}_{p},{\omega}_{s})$. Away from resonances, when Kleinmann symmetry holds, we can perform the symmetry permutations

Equation (1) is the general permutation relation for any nonlinear process, while Eq. (2) follows from Kleinmann symmetry [21]. The material is isotropic, ${\chi}_{\mathrm{RRLR}}={\chi}_{\mathrm{LLRL}}$, which follows after performing a 180° rotation, hence the transitions in Figs. 1(a) and 1(b) are identical. This is unsurprising when one thinks in terms of $x$- and $y$- polarization components, due to symmetry. It is known that the refractive-index increase from a strong pump in the presence of instantaneous third-order nonlinearity is independent of the relative phase of pump and probe. Thus, the phase difference between the $x$- and $y$- components of the pump should not matter, and there can be no first-order all-optical Faraday effect in such a medium.

Two other possible nonlinear processes are the conversion of a LH probe into a RH photon at $\omega ={\omega}_{s}+2{\omega}_{p}$, and the conversion of a RH probe to a LH photon at $\omega ={\omega}_{s}-2{\omega}_{p}$. These new photons will oscillate at their own frequencies and can then be converted back to photons at ${\omega}_{s}$, in principle interfering with the signal at ${\omega}_{s}$. In order for this effect to be significant, the up/down-converted signals need to be phase-matched with the signal at ${\omega}_{s}$ or, in the case of a resonator, they need to couple to one of the resonances. Such an approach requires carefully calibrated resonant frequency spacings and widths, which may be difficult to realize in practice. One idea to overcome this problem may be to use a small ${\omega}_{p}$, so that the converted signals fall within the same resonance as the original signal, but generally there are not powerful sources at low enough frequency to induce such an effect.

We propose instead a different approach to induce strong Faraday rotation based on nonlinear processes and optical pumping based on two detuned pumps. The idea is to induce a nonlinear polarization current in the resonator that rotates in time, analogous to non-reciprocity induced by synthetic angular momentum bias using spatio-temporal modulation [27]. Rather than creating the artificial rotation with a low frequency electro-optical bias needing its own integration into the structure as proposed in [25], we show that beating between two optical waves can create a similar angular-momentum bias in the nonlinear medium, inducing strong polarization conversion and nonreciprocity based on all-optical modulation. Consider the interaction with two circularly polarized pumps with arbitrary frequency, opposite handedness, and with a small frequency detuning $\mathrm{\Delta}$. The second pump induces the additional parametric interactions illustrated in Figs. 1(c) and 1(d), and with them photons at ${\omega}_{s}\pm \mathrm{\Delta}$. When $\mathrm{\Delta}$ is comparable with the resonance linewidth, the converted signals couple to the same resonance mode as the original signal. As shown in the following, this approach offers a number of benefits: small footprints and the absence of phase-matching requirements compared to traveling wave structures, as well as complete freedom in the choice of the absolute frequencies of the pumps, since only their difference matters.

In order to analyze the scenario described above, we use coupled-mode theory. The two pump waves are defined as

Using the standard assumption of slowly varying ${a}_{\pm}(t)$, the temporal evolution of the system is governed by

After taking the second derivative of (7) (again neglecting $\frac{{\partial}^{2}a(t)}{\partial {t}^{2}}$), plugging into Eq. (5), we find the closed-cavity coupled-mode equations

Equation (8) manifests the photon interactions of Figs. 1(c) and 1(d), and illuminates the underlying phenomena behind our approach, which we will now study in the time-domain. Since the system is linear in the small-signal approximation, we can decompose the cavity components as ${a}_{\pm}(t)={a}_{\pm ,0}(t)+{a}_{\pm ,s}(t)$, where ${a}_{\pm ,0}(t)$ are the primary cavity fields, excited from the incident fields and therefore sharing their handedness, while ${a}_{\pm ,s}(t)$ are the secondary fields excited only through coupling terms in Eq. (8) and thus of opposite handedness. The secondary fields are therefore zero before the moment at which the optical pump is turned on ($t=0$ in Fig. 2). For RH and LH excitations, the initial conditions are ${a}_{+,0}(t=0)=1,{a}_{-,0}(t)=0$ and ${a}_{-,0}(t=0)=1,{a}_{+,0}(t)=0$, respectively. This approach highlights the effect of the pumps on right- and left-handed signals (initial fields) independently. For simplicity, we assume $|\frac{\partial a}{\partial t}|<{\omega}_{0}|a|$, and so we can neglect the first term in the bracket on the right-hand side of Eq. (8). Hence, the impulse response of the envelopes for RH and LH excitations is given by

Figures 2(a) and 2(b) show the amplitude of the envelope functions when the initial field is entirely right-handed. It can be seen that, as the system evolves in time, there is a periodic exchange of power between the primary RH and the secondary LH signals with frequency $x/2$. When $\mathrm{\Delta}$ is small compared to $|D|{\omega}_{0}^{2}$, there is a large transfer of power into the sideband ${a}_{-,s}$, whereas this transfer is smaller as $\mathrm{\Delta}$ increases. This power exchange between the primary and secondary signals is accompanied by rapid changes in the phase of the primary signals, as seen in Figs. 2(c) and 2(d), showing the arguments of ${a}_{\pm ,0}(t)$ for the same $\mathrm{\Delta}$ as in Figs. 2(a) and 2(b). Importantly, the phase accumulation is opposite for RH and LH signals, leading to a nonreciprocal response, driven by the pumps breaking time-reversal symmetry. This property can be better understood by realizing that, for RH excitation, the RH photons in the cavity are coupled by the pump to lower frequency LH photons, and vice versa. The transition rates are essentially identical for the two processes. However, during their duration in the sidebands, the up-converted photons acquire more phase than the photons at the original frequency, while the down-converted photons acquire less phase. Both sidebands then couple back to the original frequency, bringing the accumulated phase with them and resulting in opposite phases for RH and LH signals, as in Figs. 2(c) and 2(d). The difference in phase, and therefore the strength of the nonreciprocal response, is both a function of the frequency difference $\mathrm{\Delta}$ and the proportion of photons that participate in the up- and down- conversion processes.

We can then use this intuition to explain the response of the system under a linearly polarized excitation. A linear input will have equal RH and LH components. When the cavity is driven close to ${\omega}_{0}$, the RH and LH resonances will be excited roughly equally, and the field at the fundamental frequency will rotate in time with an angle given by $\theta (t)=[\mathrm{arg}({a}_{+,0}(t))-\mathrm{arg}({a}_{-,0}(t))]/2$. After the fields spend a time in the cavity equal to the modal lifetime determined by the $Q$-factor, the mode will leak out. We can phenomenologically add the coupling into and out of the cavity using coupled-mode theory [28], assuming negligible material loss. For a two-sided cavity, with modal input power ${S}_{\text{in},i}$ from one side of the cavity, and leakage rate $\gamma $, we have

## 3. RESULTS

For a linear input at $\omega ={\omega}_{0}+\mathrm{\Omega}$ ($\mathrm{\Omega}$ is the detuning of the probe frequency from the resonance frequency of the resonator, accounting for the Kerr frequency shift), we can solve for steady-state Fourier amplitude ${\tilde{a}}_{+}(\mathrm{\Omega})$ that couples to ${\tilde{a}}_{-}(\mathrm{\Omega}-\mathrm{\Delta})$, and then separately for ${\tilde{a}}_{-}(\mathrm{\Omega})$ and ${\tilde{a}}_{+}(\mathrm{\Omega}+\mathrm{\Delta})$. For a system with no direct pathway and no material loss, the transmission coefficient ${S}_{21,\pm}=\sqrt{\gamma}{a}_{\pm}$, so

Exactly at the original resonance ($\mathrm{\Omega}=0$), both RH and LH components have identical transmission and reflection coefficient magnitudes [Figs. 3(a) and 3(b)] but in general different phases. A linearly polarized input wave at this center frequency will thus remain linearly polarized, but with a different polarization angle. This rotation is calculated in Figs. 3(c) and 3(d), which show the transmission coefficients for an incident $x$-polarized wave, highlighting the emergence of a cross-polarized component of the transmitted field. The corresponding polarization rotation is independent of the direction of propagation, hence nonreciprocal. In the figures, in order to define $D$, for the cavity modes ${A}_{m}(z)$ we have used the standing wave fields in a Fabry–Perot resonator (the longitudinal mode order does not matter) with refractive index $n=3.4$ and ${\chi}^{(3)}=3\times {10}^{-18}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{m}}^{2}/{\mathrm{V}}^{2}$, close to values for silicon, and we have assumed the pumps to be nonresonant, collinear traveling waves. For such a system, $D\approx \frac{{\chi}^{(3)}\sqrt{{I}_{1}{I}_{2}}}{c{n}^{3}{\epsilon}_{0}{\omega}_{0}}$, with ${I}_{i}$ being the optical intensity of the pump beams. More fundamentally, these powers represent the intensities of the pump circulating in the cavity. One benefit of the described approach, also stressed above, is that the absolute frequency of the pumps does not matter. Hence, it should be quite feasible to enable a different, wider-linewidth resonance for the pumps to enable strong field enhancement in the cavity for the pump fields, while maintaining a detuning larger than the resonance linewidth of the probe.

One drawback of this design is the fundamental limit in polarization conversion efficiency stemming from symmetries. An ideal Faraday rotator has scattering coefficients $|{S}_{xx}|=|{S}_{yy}|=1/\sqrt{2}$ and ${S}_{yx}={S}_{xx}{e}^{\pm j\pi /4}=-{S}_{xy}$. On the other hand, in the structure described above, for an $x$-polarized incident field, light will leak equally to both sides of the above symmetric cavity, so that $|{S}_{21,yx}|=|{S}_{11,yx}|\le 0.5$. To achieve better rotation efficiency, the longitudinal symmetry of the cavity must be broken. If we consider a one-sided cavity, in which transmission is eliminated, as in Fig. 1(f), the coupled-mode equations are only minorly modified, and the new reflection coefficients are

*reciprocal*component due to nonlinearity-induced birefringence [21], which we have ignored from Eq. (6) in our previous analysis. At extremely large pump powers, the rotation saturates at 0°, independently of the detuning.

To better map the efficiency as a function of pump detuning, we show in Fig. 5 the scattering coefficients for $\mathrm{\Omega}=0$ and $Q=5\times {10}^{4}$, with equal intensities in each pump beam, versus detuning and pump power. For a given pump detuning, the rotation angle monotonically increases with increasing pump power, while the reflected wave remains linearly polarized. This allows efficient, nonreciprocal polarization shift at any angle with sufficient detuning and pump power. We stress that in all these examples we have assumed a non-resonant pump; utilizing resonance enhancement for the pump can increase the coupling coefficient $D$ with the same input power.

To confirm the validity of our approach, we implemented a one-and-a-half-dimension (one spatial dimension and two polarizations) finite-difference time-domain (FDTD) solver utilizing auxiliary differential equations [30]. A dielectric resonator was chosen to mimic lossless silicon, and the resonator was backed by a perfect-electric conductor (PEC), while the front mirror was realized as a thin slab with Drude material dispersion, so that it is reflective at the probe frequency but transparent to the pump waves. While this particular example was chosen for numerical simulation simplicity, similar effects may be reasonably achieved experimentally with frequency-selective surfaces or Bragg mirrors. The probe frequency was chosen to be at the center of a Fabry–Perot resonance peak, with $Q$-factor $\sim 2\times {10}^{3}$. Figure 6 shows numerical calculations of this structure, which match well with the coupled-mode analysis. The probe frequency was chosen to track the resonance center with increasing pump power due to the Kerr effect, a frequency shift of order of a linewidth for the highest pump powers used. The spectrum (not shown) confirms that sidebands exist at only ${\omega}_{0}\pm \mathrm{\Delta}$, as required by conservation laws, minimizing insertion loss due to frequency conversion. Overall, this approach offers tunability over the entire range of rotation angles while maintaining low insertion loss and ellipticity, important properties of Faraday rotators.

## 4. CONCLUSION

In this paper, we have explored the use of instantaneous nonlinear effects to mimic Faraday rotation in resonators, with a number of benefits over other approaches based on inverse Faraday phenomena or nonresonant pumping. In steady-state, these structures are linear and non-reciprocal to small signals, unlike static nonlinear isolators. The beating of two pumps in the nonlinear resonators establishes an effective synthetic angular momentum bias in the structure, which lifts degeneracy and replaces a magnetic bias to break reciprocity. The interactions we describe can be used to preferentially transmit RH or LH light and create a circulator, or as a polarization rotator. The proposed platform is easy to adapt to a wide range of settings, as the cavity structure can be simple and there are no requirements for the dispersion of the materials or the relative frequencies of pump and probe beams. This may be particularly important for utilization with silicon, which has a strong ${\chi}^{(3)}$ nonlinearity but suffers from two-photon absorption for pumps at telecom wavelengths. Furthermore, unlike nonreciprocal devices based on traveling wave modulation in which the nonreciprocal effect essentially exists in only one direction due to phase-matching, in our system the rotation may be applied in both directions in the transmitting resonator. Because it does not rely on directional-dependent phase-matching, the proposed structure can fit in small footprints that lend themselves to modern nanophotonic systems. We believe that these findings open important pathways for integrated nonreciprocal devices in classical and quantum computing and processing systems.

## Funding

Air Force Office of Scientific Research; National Science Foundation; Simons Foundation.

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