Abstract
Mirrors are ubiquitous in optics and are used to control the propagation of optical signals in space. Here we propose and demonstrate frequency domain mirrors that provide reflections of the optical energy in a frequency synthetic dimension, using electrooptic modulation. First, we theoretically explore the concept of frequency mirrors with the investigation of propagation loss, and reflectivity in the frequency domain. Next, we explore the mirror formed through polarization modesplitting in a thinfilm lithium niobate microresonator. By exciting the Bloch waves of the synthetic frequency crystal with different wave vectors, we show various states formed by the interference between forward propagating and reflected waves. Finally, we expand on this idea, and generate tunable frequency mirrors as well as demonstrate trapped states formed by these mirrors using coupled lithium niobate microresonators. The ability to control the flow of light in the frequency domain could enable a wide range of applications, including the study of random walks, boson sampling, frequency comb sources, optical computation, and topological photonics. Furthermore, demonstration of optical elements such as cavities, lasers, and photonic crystals in the frequency domain, may be possible.
Introduction
Synthetic dimensions, typically formed by a set of atomic^{1,2} or optical modes^{3,4,5,6,7,8,9}, allow simulations of complex structures that are hard to do in real space, as well as highdimensional systems beyond threedimensional 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^{3,10,11,12}. Exploration of synthetic dimensions using optics has been of particular interest in recent decades, leveraging several of degrees of freedom of light, including space^{4,5,6,13}, frequency^{14,15,16,17,18,19,20,21,22,23,24,25}, time^{9,26}, and orbital angular momentum^{22,27}.
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 nonHermitian 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 (DOS) of frequency crystals up to four dimensions^{16}, realization of two synthetic dimensions in one cavity^{7}, dynamical band structure measurement^{15}, topological windings^{8} and braiding^{28} in nonHermitian bands, spectral longrange coupling^{17}, highdimensional frequency conversion^{19}, frequency diffraction^{24}, and Bloch oscillations^{25,29,30}. With a few exceptions^{16,31}, investigations of synthetic frequency dimensions on photonic chips have not been extensively studied. In particular, one of the most fundamental phenomenathe reflection of light by synthetic mirrorshas 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., a onedimensional frequency crystal, caused by frequencydomain mirrors introduced in such a frequency crystal. The lattice points of the frequency crystal are formed by a set of frequency modes inside a thinfilm lithium niobate (TFLN) microresonator, and the lattice constant is determined by the free spectral range (FSR) of the resonator (for a single spatial mode)^{16}. Applying a continuouswave (CW) electrooptic phase modulation to the optical resonator (Fig. 1a), at a frequency equal to the FSR (microwavefrequency range), results in coupling between adjacent frequency modes. Photons injected into such crystals can hop from one lattice site to another, leading to a tightbinding crystal^{11,16}. 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 \({{\Omega}}=\frac{\beta }{2\pi }{{{{{\rm{FSR}}}}}}\) (in a conventional modulator, the relationship between \(\beta\) and the driving voltage \(V\) is \(\beta=\pi \frac{V}{{V}_{\pi }}\), in which \({V}_{\pi }\) is the voltage required to achieve a \(\pi\) phase shift)^{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 a 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 traversemagnetic (TM) and traverseelectrical (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.
Results
Theory
We first theoretically investigate the frequency crystal dynamics for a reflection. For a conventional electrooptic frequency crystal without mirrors^{16}, the Hamiltonian is described by \(H={\sum }_{{{{{{\rm{j}}}}}}={{{{{\rm{N}}}}}}}^{{{{{{\rm{N}}}}}}}\left({\omega }_{j}{a}_{j}^{{{\dagger}} }{a}_{j}+\Omega \,{{{{{\rm{cos }}}}}}\, {\omega }_{m}t \left({a}_{j}^{{{\dagger}} }{a}_{j+1}+h.c.\right)\right)\) where \({a}_{j}\) represents each frequency mode and \({\omega }_{m}\) is the modulation frequency that equal to the \({{{{{\rm{{FSR}}}}}}}\). In the rotating frame of each mode \({a}_{j} \to {a}_{j} {e}^{i({\omega }_{L}+j {\omega }_{m}) t}\), they are all frequencydegenerate with a tightbinding coupling, i.e., \(H={\sum }_{{{{{{\rm{j}}}}}}={{{{{\rm{N}}}}}}}^{{{{{{\rm{N}}}}}}}\left(\frac{\varOmega }{2}\left({a}_{j}^{{{\dagger}} }{a}_{j+1}+h.c.\right)\right)\). As a result, Lorentzian resonances of the resonator are broadened and have a profile corresponding to the DOS of the crystal (Fig. 2a)^{16}. 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 (\(E=\hslash \Delta\) 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:
where \(a={{{{{{\rm{FSR}}}}}}}\) 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 \({\phi }_{p}={k}_{\pm }\times a=\pm 0.5\pi\) 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 frequencydegenerate 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:
where \(\xi \approx \frac{1}{1+\left(1+G\right)u}\). The parameter \(G=4{\mu }^{2}/{\kappa }_{b}\kappa\) is used to qualify the strength of the mirror (\(G \sim 2004300\) in this work. See Methods for details), and we assumed \(u\equiv \frac{\kappa }{\Omega }\ll 1\) (see details in Methods). This leads to interference between the forward propagating and the reflected waves (Fig. 1b) resulting in the final state:
where \(k={k}_{\pm }+{{{{{\rm{i}}}}}}\frac{\alpha }{2}\), \({x}_{{mr}}\) 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 \(\varOmega\) and linewidth of the resonator \(\kappa\):
In our TFLN platform we estimate the propagation loss \({L}\) is \(0.1{{{{{\rm{dB}}}}}}\; {{{{{\rm{per}}}}}}\; {{{{{\rm{lattice}}}}}}\; {{{{{\rm{point}}}}}}\) with \(u=0.024\) (Fig. 2c), which is low enough to observe the interference and trapped state effects. With the above expression for the final state \(\psi (x)\), 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 HeisenbergLangevin equation, we numerically show that constructive/destructive interference in the frequency domain (see Methods) leads to trapped states using multiple mirrors (Fig. 2e). The mirror provides a sharp cutoff to the propagation and a \({25.9{{{{\rm{dB}}}}}}\) power drop after passing the mirror. The mirror reflectivity is 0.994.
Experiment
The first approach that we use to realize frequency domain mirrors is based on polarization mode coupling inside a dispersionengineered TFLN microresonator. This requires both refractive index and frequency degeneracy of TElike and TMlike modes (from here on referred to as TE and TM modes, respectively) propagating inside the ring (Fig. 3a, b). The group index degeneracy provides large \(\mu\) while frequency degeneracy leads to mode splitting. Note that lithium niobate is a material with birefringence. As a result, for the optical modes that propagate along the y and zdirection crystal axes of xcut TFLN, the TE (ypropagation and zpropagation) modes have different indices \({n}_{o,{TE}}\) and \({n}_{e,{TE}}\), respectively. The TM (ypropagation and zpropagation) indices are both \({n}_{o,{TM}}\). Therefore, by optimizing the crosssection of an xcut lithium niobate microresonator, 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). The TE mode can have different average indices in the range of \({n}_{e,{TE}}\) to \({n}_{o,{TE}}\) at different spatial points within the bending region 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 is 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 polarizationsplitting is used, propagating wave are reflected by polarizationsplitting induced mirror, and interference between the two waves leads to a constructive/destructive pattern at every other lattice point due to the propagation phase of \({\phi }_{p}=\pm 0.5\pi\). 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). This polarization mirror shows a power cutoff of 16 dB (15.2 dB in simulation) with a reflectivity of 0.94. The reflectivity that the polarization crossing approach provided is limited since this coupling originates from fabrication imperfectioninduced perturbation.
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 multimirrors configuration. Such control and strong mirror reflectivity can be achieved in TFLN using the coupledresonator platform (Fig. 4a). In our design, a long racetrack cavity (cavity 1) with a FSR_{1} = 10.5 GHz is used to generate the frequency crystal through electrooptic modulation, while a small squareshaped cavity (cavity 2) with a FSR_{2} = 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 quite large, even comparable with the FSR_{1}, 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 twomodesplitting but instead results in dispersive interactions that gradually reduce FSR_{1} in the frequency range around the resonances of cavity 2 (Fig. 4b). Indeed, the transmission spectrum of the device shows that the FSR_{1} gradually varies from ~10.5 GHz to ~8.5 GHz and back to ~10.5 GHz at a wavelength around 1628.8 nm (Fig. 4c), corresponding to a 20% variation of the FSR_{1}. To verify that this large change of FSR_{1} originates from the formation of multihybrid modes due to the presence of cavity 2, we measured the wavelengthdependence of FSR_{1,} and found that it is periodic with a period equal to FSR_{2} (Fig. 4d). We extracted the coupling strength, finding it to be \(6.8{{{{{\rm{GHz}}}}}}\). This further verifies the existence of multihybrid modes since this system has \(2\mu\) > FSR_{1} where \(2\mu\) represents the conventional twomode splitting. Such a strong coupling strength gives a mirror reflectivity of 0.999914. Finally, with the existence of multiple frequency mirrors, we verified the trapped state with constructive/destructive interference at every other lattice point in the coupledresonator device (Fig. 4e). The strong mirror provides a cutoff of >30 dB for energy propagation in the frequency crystal (44 dB in simulation, measurement limited by noise floor). Despite the strong cutoff 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 (~\(0.15\) dB/lattice point). Constructive/destructive interference redistributes the trapped optical energy within the two mirrors, which could be useful for frequencyspecific engineering of the frequency spectrum, while avoiding energy leakage to other frequencies. A list of all the relevant parameters of the polarization and coupledresonator frequency mirrors are in Table S1.
Discussion
Note that group velocity dispersion can lead to a gradually accumulated frequency detuning between the frequency comb line and the corresponding resonance frequency. Therefore, light that is spread over the frequency domain can eventually reach a “soft” frequency “boundary” when detuning becomes large. However, our platform^{32} can generate several hundreds of lattice points without reaching this “soft” dispersion “boundary”. Since this “soft boundary” does not result in a sharp discontinuity of the propagation, and introduces Bloch oscillations^{33}, it does not serve exactly as a frequency mirror. This effect has been previously observed in our TFLN system by applying a microwave detuning to effectively create large dispersion^{16}.
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 modecoupling to form mirrors in a single cavity and uses the coupledresonator platform on TFLN^{34} to achieve much stronger mirrors. We show that the reflection and trapped state can be formulated as the reflection of Bloch waves due to defect scattering and, can be tuned by varying the wavevectors of Bloch waves. Note that the TFLN platform features the lowest propagation loss in the frequency domain to date. The loss in this work (0.076 dB/lattice for the polarization system and 0.15 dB/lattice for the coupledresonator system) is still higher, however, than the spatial propagation loss of light. Improving the quality factor of our TFLN rings from ~\({10}^{6}\) in this work to ~\({10}^{7}\)^{35} with the same driving microwave power can reduce the propagation loss in the frequency domain, which is determined by \(u=\kappa /\Omega\) (\(\kappa\) is determined by quality factor and \(\Omega\) is determined by microwave voltage), to \(0.015\) dB/lattice point, yielding a mirror reflectivity of >0.99999. Therefore, coupled resonators on TFLN can be promising to investigate multiroundtrip dynamics in the frequency synthetic dimension, which may lead to the realization of a frequency domain cavity^{36}. Introducing periodic mirrors via multiple additional resonators with lower propagation loss may lead to the realization a frequency domain photonic crystal^{37,38}. 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 a synthetic frequency crystal can be used to generate flatslope EO combs with better energy confinement in frequency domain, which is important for applications in spectroscopy, astronomy (astrocomb), and quantum frequency combs^{39,40,41,42,43}. Finally, realizing frequency domain scattering beyond reflection and transmission, using a highdimensional frequency crystal^{1,16,19,23} or other crystal structures^{18,20}, could pave ways to investigate highdimensional geometrical phases and topologies. Specifically, with the ability to introduce defects in the frequency domain, the recently emerged topological frequency comb^{44} might can be combined with our coupledresonator platform. TFLN can generate frequency combs using both EO and Kerr nonlinearity simultaneously^{32}, leading to the realization of a spatialfrequency topological frequency comb. Our approach could form a basis for controlling the crystal lattice structure, band structure, and energy distributions in frequency domain.
Methods
Device fabrication
The devices are fabricated on a xcut lithium niobate wafer (NANOLN). The wafer contains a 600 nm LN layer, 2 \({{\upmu }}{{{{{\rm{m}}}}}}\) buried oxide, and a 500 \({{\upmu }}{{{{{\rm{m}}}}}}\) Si handle. The optical layer is patterned using electronbeam lithography and the pattern is transferred to the lithium niobate layer using Ar^{+}based reactive ion etching with an etch depth of 350 nm. Metal electrodes are defined using optical lithography and deposited with electronbeam evaporation. The electrodes consist of 15 nm of Ti and 300 nm of Au. The oxide cladding is deposited using plasmaenhanced chemical vapor deposition. The heater layer (15 nm of Ti and 200 nm Pt) is also patterned by photolithography followed by electronbeam evaporation and bilayer liftoff.
Measurement
Telecommunicationwavelength light from a fibercoupled tunable laser passes through a polarization controller and is coupled to the LN chip using a lensed fiber. The output is collected using an aspheric lens and an optical spectrum analyzer is used to characterize the output frequency spectrum. The microwave signal is generated and amplified before sending it to the electrode of the device using an electrical probe.
Theory and simulation of the frequencymirrorinduced reflection and the trapped states
Normal crystal dynamics without mirrors
We first consider the crystal dynamics without frequency mirrors. The optical frequency crystal generated using electrooptic modulation on a single cavity can be described by its Hamiltonian in the following form:
where \({\omega }_{j}\) is the frequency of each mode, \(\Omega\) is the coupling rate induced by the microwave modulation, and \({\omega }_{m}\) is the frequency of the microwave modulation. The total number of frequency modes that are coupled are labeled by number −N to N. Using the HeisenbergLangevin equation, we derived the equations of motion of each mode \({a}_{j}\) in the frequency crystal:
where the \({\kappa }_{e}\) is the coupling rate between the waveguide and cavity, \(\kappa\) is the total loss rage of \({a}_{j}\), \({\alpha }_{{in}}\) is the pump power, \({\omega }_{L}\) is the laser frequency, and \({\delta }_{j,0}\) is used to denote the pumped mode. The frequency of each mode \({a}_{j}\) is \({\omega }_{j}={\omega }_{0}+j\times {{{{{\rm{FSR}}}}}}\) with \({\omega }_{0}\) representing the 0th mode. Such a set of equations can be solved by changing the rotating frame for each mode \({a}_{j}\to {a}_{j}{e}^{i{\omega }_{L}t}{e}^{{i\; j}{\omega }_{m}t}\) and doing the Fourier transformation to solve this equation in frequency domain:
where \(\delta={\omega }_{m}{{{{{\rm{FSR}}}}}}\) and \(\Delta={\omega }_{L}{\omega }_{0}\) are the microwave detuning and laser detuning, respectively.
In the case of \(\Delta=\delta=0\), the equation for each mode \({a}_{j}\) is:
Note that the equation for the frequency mode \({a}_{N}\) with a mode number of N is:
which leads to \({a}_{N}=i\frac{\Omega }{\kappa }{a}_{N1}\). As a result, the equation for \({a}_{N1}\) is:
Therefore, we obtained another relation \({a}_{N1}=i\frac{\Omega }{\kappa (1+\frac{{\Omega }^{2}}{{\kappa }^{2}})}{a}_{N2}\). The relation for an arbitrary mode can be obtained using iteration:
where we have the total number of the factor \(\frac{{\Omega }^{2}}{{\kappa }^{2}}\) as \(Nl\). Using this relation for an arbitrary mode, the equation of motion for the pump mode (0th mode) can be obtained as:
This equation is equivalent to:
in which \({\kappa }_{{{{{{\rm{MW}}}}}}}=\kappa \times 2\left(\,{f}_{n}1\right)\) with:
When N is very large, the limit of \({f}_{n}\) can be used: \(\mathop{{{{{{\rm{lim}}}}}}}\nolimits_{n\to \infty }{f}_{n}=f\). As a result, we have \(f=1+\frac{\frac{{\varOmega }^{2}}{{\kappa }^{2}}}{f}\), which can be used to solve the final expression for \(f\) and \({\kappa }_{{{{{{\rm{MW}}}}}}}\):
The \({\kappa }_{{MW}}\) is the effective loss rate for the pump mode that generated by the microwave modulation^{32}. With the expression of \(f\), we can simplify the relation between \({a}_{l}\) and \({a}_{l1}\) as:
where \(u\equiv \kappa /\Omega\) and \(f=\frac{1+\sqrt{1+4/{u}^{2}}}{2}\).
This gives the propagation loss\(:\)
Frequency crystal with frequency mirrors
The frequency mirror can be introduced by coupling additional modes \({b}_{k}\) (\(k={{{{\mathrm{1,2}}}}},\ldots\)) to the frequency crystal. To derive the reflection and transmission coefficient for the mirror, we consider a problem that a single additional mode \(b\) is coupled to a frequency mode \({a}_{{mr}}\) with a coupling strength \(\mu\) and \(b\) is frequencydegenerate with \({a}_{{mr}}\). Further, we consider the case that the pump source is far away from the mirror in frequency domain, i.e., the pump frequency is far away from the frequency of \(b\). The equation of motions for \({a}_{{mr}}\) and \(b\) are:
Note that \(b\) is in a rotating frame of \({a}_{10}\) since they are frequency degenerate. The steadystate solution of the above equations for each frequency mode and additional modes gives the simulated energy distribution of frequency crystals with mirrors.
The mirror mode splits frequency space into two different regions: region that contains both input and reflected waves and the region that contains only the transmitted wave. Therefore, the region of frequency space that contains the transmitted wave should follow the normal dynamics of the frequency crystal, i.e., free propagation without reflection. As a result, we conclude that \({a}_{{mr}}\) and \({a}_{mr+1}\) will obey the relation:
At the same time, we have the following equations:
With the Eqs. (21)–(23) we find:
where \(u\equiv \kappa /\Omega\), \(f=\frac{1+\sqrt{1+4/{u}^{2}}}{2}\), and \(G=4{\mu }^{2}/{\kappa }_{b}\kappa\) is the parameter used to qualify the strength of the mirror.
For the region that contains both the input and reflected waves, due to the interference, we have:
where \({A}_{0}\) represents the input wave amplitude at the position of mirror.
The transmission and reflection coefficients are:
Using Eqs. (24)–(27), the reflection coefficient is:
with \(=\frac{1}{{u}^{2}f\left(1+G\right)+1}\). In the regime that \(u\,\ll \,1\), we have \(\xi \,\approx \frac{1}{\left(u+\frac{{u}^{2}}{2}+\frac{{u}^{3}}{8}\right)\left(1+G\right)+1}\approx \frac{1}{1+\left(1+G\right)u}\).
Finally, we have:
To simulate the frequency crystal with arbitrary additional mirrors, i.e., several additional modes \({b}_{k}\) (\(k={{{{\mathrm{1,2}}}}},\ldots\)) coupled to the crystal, we use the equation:
where \({a}_{{mr},k}\) is the mode that is frequency degenerate with \({b}_{k}\). We combine the above equations with the equations for modes that are not coupled to the mirrors:
to numerically simulate the frequency crystals with mirrors.
Note added to proof
In the process of writing this manuscript another group reported the observation of frequency boundaries in a fibercavity system^{45}.
Data availability
The datasets generated and analyzed during the current study are available from the corresponding authors on reasonable request.
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Acknowledgements
This work is supported by ARO W911NF2010248 (Y.H.), NSF QuIC TAQS OMA2137723 (Y.H.), DARPA LUMOS HR001120C0137 (M.Y., R.C., and M.L.), AFRL Quantum Accelerator FA95502110056 (N.S.), ONR N0001422C1041 (M.Y. and R.C.), NASA 80NSSC21C0583 (M.Y. and R.C.), NIH 5R21EY03189502 (M.L.), Harvard Quantum Initiative (D.Z.), Research Grants Council, University Grants Committee (CityU 11212721) (C.W.). N.S. acknowledges support from the AQT Intelligent Quantum Networks and Technologies (INQNET) research program.
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Y.H. conceived the idea, developed the theory, performed the simulation of frequency mirrors. M.Y. performed the dispersion simulations and carried out the measurement of the polarization mirrors. M.Y. and Y.H. measured the coupledresonator mirrors. Y.H. and R.C. fabricated the device. Y.H. wrote the manuscript with contributions from all authors. N.S., D.Z., and C.W. helped with the project. M.L. supervised the project.
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M.L. are involved in developing lithium niobate technologies at HyperLight Corporation. The remaining authors declare no competing interests.
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Nature Communications thanks Thuy Hoang, Mohammad Hafezi and the other anonymous reviewer(s) for their contribution to the peer review of this work. Peer review reports are available.
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Hu, Y., Yu, M., Sinclair, N. et al. Mirrorinduced reflection in the frequency domain. Nat Commun 13, 6293 (2022). https://doi.org/10.1038/s4146702233529w
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DOI: https://doi.org/10.1038/s4146702233529w
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