Synthetic dimensions, typically formed by a set of atomic [1, 2] or optical modes [3–9], allow simulations of complex structures that are hard to do in real space, as well as high-dimensional systems beyond three-dimensional Euclidian space. Therefore, synthetic dimensions provide opportunities to investigate and predict, in a controlled manner, a wide range of physical phenomena occurring in e.g. ultracold atoms, solid state physics, chemistry, biology, and optics [3, 10–12]. Exploration of synthetic dimension using optics has been of particular interest in recent decades, leveraging wide range of degrees of freedom of light, including space [4–6, 13], frequency [14–25], time [9, 26], and orbital angular momentum [24, 27].

Integrated optics is an ideal platform for creating synthetic dimensions in the frequency domain, due to the high frequency and bandwidth of light, availability of strong nonlinear interactions, good stability and coherence of the modes, scalability, and excellent reconfigurability [11]. Furthermore, the ability to tailor the gain and loss within an optical system naturally allows the investigation of non-Hermitian physics which are typically hard to explore in other physical systems. Frequency synthetic dimensions in photonics has recently been experimentally investigated, including the measurement of band structure [14] and density of states of frequency crystals up to four dimensions [18], realization of two synthetic dimensions in one cavity [7], dynamical band structure measurement [15], topological windings [8] and braiding [28] in non-Hermitian bands, spectral long-range coupling [19], high-dimensional frequency conversion [21], frequency diffraction [16], and Bloch oscillations [17, 29, 30]. With a few exceptions [18, 31], investigations of synthetic frequency dimensions on photonic chip have not been extensively studied. In particular, one of the most fundamental phenomena– the reflection of light by synthetic mirrors -- has not been investigated yet in frequency synthetic dimensions.

Here we study, both theoretically and experimentally, reflection and interference of optical energy propagating in a discretized frequency space, i.e. one dimensional frequency crystal, caused by frequency-domain mirrors introduced in such a frequency crystal. The lattice points of the frequency crystal are formed by a set of frequency modes inside a thin-film lithium niobate (TFLN) micro-resonator, and the lattice constant is determined by the free spectral range (FSR) of the resonator (for a single spatial mode) [18]. Applying a continuous-wave (CW) electro-optic phase modulation to the optical resonator (Fig. 1a), at a frequency equal to the FSR (microwave-frequency range), results in coupling between adjacent frequency modes. Photons injected into such crystals can hop from one lattice site to another, leading to a tight-binding crystal [11, 18]. The coupling strength between nearest neighbor lattice points, \({\Omega }\), (Fig. 1b) is proportional to the voltage of the microwave driving signal and is related to the conventional modulation index \(\beta\) of a phase modulator as \(\beta =2\pi \frac{{\Omega }}{FSR}\) [32]. As a result, when injecting a CW optical signal into one of the crystal lattices sites (cavity resonances), optical energy spreads along the frequency synthetic dimension. A defect introduced in the frequency crystal can break the discrete translational symmetry of the lattice, resulting in reflection of light in the frequency domain (Fig. 1b). The defect can serve as a mirror in the frequency crystal, which is the frequency analog of the mirror in real space (Fig. 1c). The frequency mirror can be introduced by a mode splitting that is induced by coupling specific lattice points to additional frequency modes (Fig. 1d). These additional modes can be different spatial or polarization modes, clockwise and counterclockwise propagating modes of a cavity, or modes provided by additional cavities. In this work, we first use coupling between the traverse-magnetic (TM) and traverse-electrical (TE) modes to realize mirrors for the latter. Then, we show that the frequency mirrors can also be realized using coupled resonators, an approach that allows better control, reconfigurability, and is more tolerant to fabrication imperfections.

We first theoretically investigate the frequency crystal dynamics for a reflection. For a conventional electro-optic frequency crystal without mirrors [18], the Hamiltonian is described by \(\text{H}={\sum }_{\text{j}=-\text{N}}^{\text{N}}\left({\omega }_{j}{a}_{j}^{†}{a}_{j}+{\Omega }\text{cos}{\omega }_{m}t\left({a}_{j}^{†}{a}_{j+1}+h.c. \right)\right)\) in which \({a}_{j}\) represents each frequency mode and \({\omega }_{m}\) is the modulation frequency that equal to the \(FSR\). In the rotating frame of each mode \({a}_{j}\to {a}_{j}{e}^{-{i(\omega }_{L}+j{\omega }_{m})t}\), they are all frequency-degenerate with a tight-binding coupling, i.e. \(\text{H}={\sum }_{\text{j}=-\text{N}}^{\text{N}}\left(\frac{{\Omega }}{2}\left({a}_{j}^{†}{a}_{j+1}+h.c. \right)\right)\). As a result, Lorentzian resonances of the resonator are broadened and have a profile corresponding to the density of states (DOS) of the crystal (Fig. 2a) [18]. Therefore, varying the laser detuning \({\Delta }={\omega }_{L}-{\omega }_{0}\), where \({\omega }_{L}\) is the laser frequency and \({\omega }_{0}\) is the 0th resonance of the resonator that the laser is pumping, changes the excitation energy (\(E=\text{\hslash }{\Delta }\) in the rotating frame of the 0th resonance) of the pump signal (Fig. 2b, blue curve on the left side). This corresponds to the excitation of different modes of the band structure of the crystal (Fig. 2b, blue curve on the right side), leading to two synthetic Bloch waves with wave vectors \({k}_{\pm }\) given by

$${k}_{\pm }=\pm \frac{1}{a}{\text{cos}}^{-1}\frac{{\Delta }}{{\Omega }}$$

in which \(a=FSR\) is the lattice constant of the frequency crystal. For example, when \({\Delta }=0\), two Bloch waves with wave vectors \({k}_{\pm }=\pm 0.5\frac{\pi }{a}\) will be excited, representing waves that propagate along the positive and negative direction in frequency crystal (Fig. 3a) with a propagation phase of \({\varphi }_{p}={k}_{\pm }\times a=\pm 0.5\pi\) for a single hopping. To form a frequency mirror, additional mode \(b\) is used to break the periodic translation symmetry. We assume mode \(b\) (with a linewidth \({\kappa }_{b}\)) is placed at frequency \({\omega }_{mr}\) that is frequency-degenerate with the crystal mode \({a}_{mr}\) (with a linewidth \(\kappa\)) and the coupling strength between \(b\) and \({a}_{mr}\) is \(\mu\). This additional mode \(b\) plays the role of the mirror with a reflection coefficient

$$r=-\frac{1-{\xi }^{2}}{1+{\xi }^{2}}$$

in which \(\xi \approx -\frac{1}{1+\left(1+C\right)u}\). The parameter \(C=4{\mu }^{2}/{\kappa }_{b}\kappa\), analogous to the cooperativity in cavity quantum electrodynamics, is used to qualify the strength of the mirror (\(C\tilde200\) in our work. See supplementary materials for details), and we assumed \(u\equiv \frac{\kappa }{{\Omega }}\ll 1\) (see details in supplementary materials). This leads to interference between the forward propagating and the reflected waves (Fig. 1b) resulting in the final state

$$\psi \left(x\right)\tilde {e}^{ikx}+ r {e}^{ik{x}_{mr}}{e}^{-ik(x-{\text{x}}_{mr})}$$

where \(k={k}_{\pm }+\text{i}\frac{\alpha }{2}\), \({x}_{mr}\) represents the position of the mirror, and \(\alpha\) is related to the propagation loss of the Bloch wave in the frequency domain. The propagation loss \(L={e}^{-\alpha a}\) is defined as the power loss for a single hop and determined by the coupling strength \({\Omega }\) and linewidth of the resonator \(\kappa\):

$$L\approx {\left|\left(1-\frac{u}{2}-\frac{{u}^{2}}{8}\right)\right|}^{2}$$

In our TFLN platform we estimate the propagation loss \(L\) is\(0.2 \text{d}\text{B} \text{p}\text{e}\text{r} \text{l}\text{a}\text{t}\text{t}\text{i}\text{c}\text{e} \text{p}\text{o}\text{i}\text{n}\text{t}\) with \(u=0.048\)(Fig. 2c), which is low enough to observe the interference and trapped state effects. With the above expression for the final state \(\psi \left(x\right)\), we show such interference causes an oscillation of energy distribution \({\left|\psi \left(x\right)\right|}^{2}\) along the frequency dimension and the oscillating period is determined by the wave vector \(k\) (Fig. 2d). Using the Heisenberg-Langevin equation, we numerically show that constructive/destructive interference in the frequency domain (see Supplementary Materials) leads to trapped states using multiple mirrors (Fig. 2e). The mirror provides a sharp cut-off to the propagation and a \(20 \text{d}\text{B}\) power drop after passing the mirror.

The first approach we realize frequency domain mirrors based on polarization mode coupling inside a dispersion-engineered TFLN micro-resonator. This requires both refractive index and frequency degeneracy of TE-like and TM-like modes (from here on referred to as TE and TM modes, respectively) propagating inside the ring (Fig. 3a and 3b). The group index degeneracy provides large \(\mu\) while frequency degeneracy leads to mode splitting. Due to the birefringence of lithium niobate, the TE modes that propagate along the y-direction and z-direction (of the thin-film lithium niobate crystal axes) have different indices \({n}_{o,TE}\) and \({n}_{e,TE}\) while the indices of TM modes are \({n}_{o,TM}\) for both directions. Therefore, by optimizing the cross-section of a x-cut lithium niobate micro-resonator, the value of \({n}_{o,TM}\) can be designed to be between the values of \({n}_{o,TE}\) and \({n}_{e,TE}\) over a broad range of wavelengths (Fig. 3c). When the TE mode circulates inside the micro-resonator, it experiences different averaged TE indices ranging from \({n}_{e,TE}\) to \({n}_{o,TE}\) at different bending points of the resonator. As a result, the TM modes can have an index degeneracy with the TE modes over a broad wavelength range (Fig. 3c). Frequency degeneracy was accomplished using a Vernier effect caused by the difference in FSR of TE and TM modes: the TM modes come in resonance with TE modes periodically, leading to periodic mode splitting that gives rise to periodic frequency mirrors (Fig. 3b). This coupling can be observed in the transmission spectrum of the TE modes (Fig. 3d).

To experimentally verify the presence and reflection of Bloch waves, we excite the frequency crystal at different values of detuning \({\Delta }\). By pumping at \({\Delta }=0\) on the device without mirrors (no polarization induced splitting) the energy propagates along the frequency dimension (Fig. 3e) without a reflection. However, when the device with engineered polarization-splitting is used, propagating wave is reflected by polarization-splitting induced mirror, and interference between the two waves leads to a constructive/destructive pattern at every other lattice points due to the propagation phase of \({\varphi }_{p}=\pm 0.5\pi\). Note that the constructive interference results in a flat spectrum of generated comb signal which could be of interest for frequency comb applications. By varying the laser detuning \({\Delta }\), we show varying interference fringes, due to the change of wave vector \(k\) (Fig. 3e).

Even better control of defects in the synthetic frequency dimension can lead to realization of frequency mirrors with controllable reflection strength and position in the crystal, as well as more complex arbitrary multi-mirrors configuration. Such control can be achieved in TFLN using the coupled-resonator platform (Fig. 4a). In our design, a long racetrack cavity (cavity 1) with a FSR1 = 10.5 GHz is used to generate the frequency crystal through electro-optic modulation, while a small square-shaped cavity (cavity 2) with a FSR2 = 302.9 GHz is coupled to the racetrack cavity to provide frequency mirrors through the resultant mode splitting. Interestingly, in our system, the coupling strength between two cavities \(\mu\) can be larger than FSR1 and as a result, a single resonance mode of the cavity 2 couples to multiple resonances of cavity 1 (Fig. 4b). This does not lead to a conventional two-mode-splitting but instead results in dispersive interactions that gradually reduce FSR1 in the frequency range around the resonances of cavity 2 (Fig. 4b). Indeed, the transmission spectrum of the device shows that the FSR1 gradually varies from ~ 10.5 GHz to ~ 8.5 GHz and back to ~ 10.5 GHz in the wavelength around 1628.8 nm (Fig. 4c), corresponding to a 20% variation of the FSR1. To verify that this large change of FSR1 originates from the formation of multi-hybrid modes due to the presence of cavity 2, we measured the wavelength-dependence of FSR1, and found that it is periodic with a period equal to FSR2 (Fig. 4d). Finally, with the existence of multiple frequency mirrors, we verified the trapped state with constructive/destructive interference at every other lattice point in the coupled-resonator device (Fig. 4e). The strong mirror provides a cut-off of > 30 dB for the energy propagation in the frequency crystal. Despite the strong cut-off produced by the frequency mirrors, it is difficult to see multiple roundtrip effects within the two mirrors, due to the large propagation loss of our system (~\(0.2\) dB/lattice point). Improving the quality factor of our TFLN rings ~\({10}^{6}\) (this work) to ~\({10}^{7}\) [33] and further improve the microwave driving power may lead to the realization of a frequency domain cavity [34]. The constructive/destructive interference redistributes the trapped optical energy within the two mirrors, which could be useful for frequency-specific engineering of the frequency spectrum, while avoiding energy leakage to other frequencies.

In summary, we have shown the reflection and trapped state of light in the frequency domain by introducing a mirror, that is, a defect, inside a frequency crystal. Our investigation utilizes the polarization mode-coupling and coupled-resonator both realized using the TFLN platform. We show the reflection and trapped state can be formulated as the reflection of Bloch waves due to defect scattering and be tuned by varying the wavevectors of Bloch waves. Introducing periodic mirrors via multiple-additional resonators with lower propagation loss may lead to the realization a frequency domain photonic crystal [35, 36]. Furthermore, the ability to control the distribution of light in the frequency synthetic dimension provides an advantageous way to manipulate the light frequency. For example, the trapped state in the synthetic frequency crystal can be used to generate flat slope EO combs with better energy confinement in frequency domain, which is important for applications in spectroscopy, astronomy (astro-comb), and quantum frequency comb [37–41]. Finally, realizing frequency domain scattering beyond the reflection and transmission, using a high-dimensional frequency crystal [1, 18, 21, 25] or other crystal structures [20, 22], could pave ways to investigate high-dimensional geometrical phases and topologies. Our approach could form a basis for controlling the crystal lattice structure, band structure, and energy distributions in frequency domain.

Note added: In the process of writing this manuscript another group reported on the observation of frequency boundaries in the fiber-cavity system [42].