Main

The data capacity of the internet is expected to grow by a factor of two every year6, but present optical techniques are not able to meet the rising demand on the bandwidth of the undersea fibre network7. Techniques such as space-division multiplexing1, mode-division multiplexing8 and wavelength-division multiplexing (WDM)5 in combination with time-domain multiplexing are being investigated in order to exploit the existing network to its full capacity. Current WDM systems use an array of individually stabilized lasers, which are not phase locked to each other. For the next generation, a major shift in the paradigm—from multiple independent optical carriers to coherent optical frequency combs (OFCs)7, combined with real-time numerical calculation of the nonlinear pulse5—will be necessary. The advantage of OFCs is that they can be generated from a single laser, which could reduce the overall energy consumption of the system considerably. Furthermore, depending on the method of comb generation, OFCs can feature high phase and frequency stability, and may also offer tunability of the comb’s line spacing. Of particular interest for future WDM applications is the intrinsic phase lock between all comb lines, which allows the numerical counteracting of one of the main limitations—the nonlinear pulse distortion—in order to accept higher optical powers within the telecommunication fibres5. Commercial OFCs are at present based on femotosecond lasers9,10. However, over recent years, combs generated in microresonators via the third-order Kerr nonlinear four-wave mixing effect (whereby two pump photons are converted into a signal and idler photon)2,10,11,12 and cascaded second-order parametric downconversion13,14,15 have become more and more successful.

An alternative approach to comb generation is electro-optic modulation (EOM)16. This scheme exploits a second-order nonlinearity, whereby two continuous waves—for example, one in the optical range (the carrier wave) and one in the microwave range (the modulating wave)—are mixed within a nonlinear noncentrosymmetric crystal. In the past, overdriving of conventional electro-optic modulators has been explored for comb generation17,18. These EOM-based combs have unique advantages, such as tunable central frequency and comb-line spacing, which are crucial for applications from telecommunications to spectroscopy. However, these combs suffer from inherent phase instabilities and excess multiplication of microwave phase noise18, which limits their usage in long-distance data transfer5.

Here we report the experimental demonstration of efficient resonantly enhanced electro-optic comb generation. Such a fully coherent comb with intrinsic phase-noise suppression allows an increase in the information content and data rate in optical communication. The phase stability between the comb lines is inherent owing to the fixed interaction of two stable sources; high-quality optical resonators filter the fundamental electro-optic noise18. Our experiment draws on an in-depth theoretical description of the resonantly enhanced nonlinear interaction. We show in a first experiment the potential of this method, by describing a coherent comb with 8.9 GHz spacing and a span of almost 200 lines in the C-band (1,530–1,565 nm), using only 20 dBm of microwave modulation power. The results should spark a resurgence of interest in EOM-based comb generation, and show the potential of resonantly enhanced electro-optic modulation for the next generation of ultra-dense WDM.

The scheme we use for comb generation is based on the Pockels effect: an electric field applied to a noncentrosymmetric crystal, such as lithium niobate, results in a change in the optical refractive index of the material that is directly proportional to the applied voltage. As a consequence, light passing through the crystal encounters a varying optical path length and is thus phase modulated with the frequency of the applied voltage. This phase modulation generates sidebands that become separated from the optical carrier frequency by multiples of the modulation frequency and form a comb (Fig. 1). One can observe this behaviour in a standard EOM19, but the magnitude of the sidebands usually decays quickly with distance from the carrier. The length of the comb is determined by the modulation index, which scales with the amplitude of the applied electric field20. To boost the electric field and thus the efficiency of the process and to generate a wide-spanning comb, we embed a high-quality optical whispering gallery mode (WGM) resonator into a high-quality microwave resonator.

Fig. 1: Principles of generating WGM-based χ(2)-frequency combs.
figure 1

a, Diagram showing the setup for the creation of electro-optic second-order (χ(2))-frequency combs. A high-quality WGM optical resonator—a convex-shaped disk that guides light via total internal reflection along its inner surface, and machined from single-crystalline lithium niobate—is embedded in a microwave cavity made from copper. Light is coupled with a prism evanescently by frustrated total internal refection into the WGM. The microwave resonator is designed so that the resonantly enhanced microwave field has maximum overlap with the optical WGM. Microwave radiation is fed into the cavity through a microwave coupling pin. b, Illustration showing how the microwave field phase-modulates the light via the Pockels effect and a frequency comb is generated. The optical modes are separated by a nearly constant free spectral range (FSR), which approximately matches the frequency of the microwave tone, Ω, allowing for sum- and difference-frequency generation (SFG, DFG) of sidebands, indicated as the coloured vertical lines. Given the fixed microwave frequency, the comb lines are generated strictly equidistantly. This leads to a detuning of the comb lines from the optical modes (indicated by the Lorentzian-shaped black curve with grey shading), which are subject to dispersion, and eventually breaks the comb, as illustrated in Fig. 2.

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Another equivalent way to view the process is sum- and difference-frequency generation: the energy of the microwave photons is either added to or subtracted from the carrier photon, yielding new frequencies. These sidebands themselves interact with microwave photons and give rise to new frequencies, again by the same process. For efficient comb generation, the sidebands have to match an optical resonance. As a consequence, the microwave resonator needs to be designed such that its resonance frequency coincides with the free spectral range (FSR) of the optical resonator. Mathematically, the system is described by an infinite but simple set of linear equations, each representing an optical mode participating in the nonlinear process. It can be solved analytically under the condition of a constant FSR (see Supplementary Information and ref. 21), and for the time-dependent optical field amplitudes in reflection of the resonator, we find:

$$\frac{{A}_{{\rm{o}}{\rm{u}}{\rm{t}}}(t)}{{A}_{{\rm{i}}{\rm{n}}}(t)}=\frac{(2\gamma /\Gamma )-1-i(\Delta +2\sqrt{G}\cos (\Omega t))}{1+i(\Delta +2\sqrt{G}\cos (\Omega t))}$$
(1)

Here Ain and Aout stand for the input and output electric field; Γ = γ + γ′ is the half-linewidth of the optical modes, where γ and γ′ are the optical coupling rate and intrinsic field loss rate, respectively; the detuning of the optical carrier frequency, ω0 − ω, from its resonance frequency, ω0, is normalized to half of the optical linewidth, Δ = (ω0 − ω)/Γ; Ω is the microwave frequency, which we assume to be on-resonance; t is time; and i is the imaginary number \(\sqrt{-1}\). The figure of merit is \(G={n}_{\Omega }{g}^{2}/{|\Gamma |}^{2}\) , where nΩ is the number of microwave photons in the resonator and g is the single-photon coupling rate, which is a function of the electro-optic coefficient and the overlap between the optical and the microwave modes. G is often referred to as electro-optic cooperativity22 and represents the ratio between the nonlinear photon-conversion rate and the photon-decay rate, and is hence a measure of the strength of the nonlinear interaction. It is interesting to compare the modulation of a resonant EOM with a nonresonant EOM, which is simply described by Aout(t) = eiξ cos(Ωt)Ain(t), where the modulation index, ξ, is approximately \(4\sqrt{G}\). The nonresonant EOM does not contain any amplitude modulation; the resonant EOM, on the other hand, does. Only when the resonant EOM is strongly overcoupled (when γ is much greater than γ′) and for very small nonlinearities (G ≈ 0) the two systems are comparable. At critical coupling (γ = γ′) and increasing G, the resonant EOM starts generating pulses with a repetition rate of 1/2τ, where τ is the roundtrip time of the optical cavity. We show in  Supplementary Information that the pulse width, Δτ, can be approximated as \({\rm{\Delta }}\tau \approx \tau /(2{\rm{\pi }}\sqrt{G})\).

The power of the comb lines, P(ω ± kΩ), can also be analytically described as:

$$\frac{P(\omega \pm k\Omega )}{{P}_{{\rm{i}}{\rm{n}}}}=\frac{4{\gamma }^{2}}{{\Gamma }^{2}}{|\frac{{{\rm{e}}}^{-\beta (G)}}{\sqrt{G}+\sqrt{G}{{\rm{e}}}^{-2\beta (G)}}|}^{2}{{\rm{e}}}^{-2|k|\beta (G)}\,{\rm{f}}{\rm{o}}{\rm{r}}\,k\ne 0$$
(2)

where k represents the order of the sideband; Pin is the input optical power; e is Euler's number; and 2β(G) is the decay constant and can be approximated to be \(\beta (G)\approx 1/\sqrt{4G}\) in the limit of strong nonlinear interaction when 4G is much greater than 1 (see Supplementary Information). The resonant comb scales strictly exponentially, which is another difference to the nonresonant comb, where the comb line power is described by Bessel functions23. In Fig. 2a we plot equation (2) for different modulation strengths as dashed lines on a logarithmic scale. It is apparent that the slope of the exponential decay decreases with increasing G, leading to the generation of longer combs.

Fig. 2: Theoretical scaling of sideband power and dispersion-induced breakdown of the comb.
figure 2

a, Top, power of the sidebands as a function of sideband order for different modulation strengths (G). The dashed lines represent the analytical solution given by equation (2), and the coloured curves are numerical solutions that include the optical dispersion of lithium niobate (see Supplementary Information). Bottom, the dispersion leads to a detuning of the equidistantly generated comb lines from the optical resonance frequencies. When a certain detuning, Δ, of the comb lines from the optical resonance frequency is reached, the comb breaks down drastically. b, The dependence of the breakdown on the cooperativity, G, can be understood by the nonlinear induced broadening of the modes shown here: the yellow contour plot on the left combines 101 spectra of the resonances exemplified on the right into a plot of the optical coupling depth (indicated by the colour bar) versus optical detuning, Δ, and a function of the microwave power, \({\rm{log}}\left(\sqrt{G}\right)\). The measured mode splitting is a function of G, which is proportional to the square root of the microwave power sent to the system.

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To derive the previous analytical solution for the system, we neglected optical dispersion, which is intrinsic owing to geometric and material dispersion. As a consequence, the optical modes are not exactly equally spaced and the generated comb lines become increasingly detuned from the optical resonance frequencies with distance from the carrier mode (Fig. 1b). This eventually breaks the comb generation, as illustrated in Fig. 2a: the solid curves show the sideband power as function of the sideband order obtained from numerically solving the rate equations while including optical dispersion of lithium niobate (see Supplementary Information for details). Close to the pump frequency the power decreases exponentially (as expected from equation (2)) until a certain G-dependent sideband order, at which point it drops drastically. This can be intuitively understood from Fig. 2b: with increasing microwave power, the optical modes broaden and show a mode splitting of \((4\sqrt{G}-1.2)\Gamma \). The dispersion-induced detuning of the optical eigenfrequencies from the generated sidebands does not affect the comb generation greatly at the half-width of a cold cavity mode (Γ), but rather much later, at \(2\sqrt{G}-0.6\). Figure 2b shows that this intuitive picture is accurate. As soon as the detuning (Fig. 1b) equals the nonlinear broadening, the comb starts to break down. Nevertheless, with reasonable parameters we expect a few hundred lines, as shown in Fig. 2a.

To demonstrate our highly efficient scheme experimentally, we embedded a high-quality WGM optical resonator machined from single-crystalline lithium niobate into a microwave cavity made from copper. The optical resonator is a convex-shaped disk that guides light via total internal reflection along its inner surface, interfering with itself after each roundtrip24. Given that lithium niobate is quite transparent, the light can do many roundtrips before being absorbed, boosting the optical intensity by orders of magnitude and facilitating efficient nonlinear interactions25,26,27,28. The microwave resonator is a three-dimensional copper cavity enclosing the optical resonator, designed such that the resonantly enhanced microwave field has maximum overlap with the optical WGM. The cavity contains two protruded copper rings facing each other, which clamp the optical resonator when the cavity is closed. This ensures that the microwave field is focused on the rim of the optical resonator where the optical modes are located, maximizing the overlap between the two vastly different frequencies. Two holes are used to couple optical light into and out of the cavity. Apart from that, the copper cavity is closed to prevent the microwave mode from radiating into the far field, which would decrease its quality factor, Q. Inside the cavity, we use a silicon prism placed close to the optical resonator to couple the light evanescently into the lithium niobate disk. The WGM resonator has a radius of 2.45 mm and a thickness of 0.4 mm, resulting in a free spectral range of about 8.9 GHz at the used pump frequency ω0 = 2π × 193.3 THz.

On the basis of numerical simulations of the microwave mode (see Methods), we designed the copper cavity such that the microwave resonance approximately matches the optical free spectral range and added a fine-tuning mechanism to compensate for small fabrication tolerances. Our system has a single-photon coupling rate of g = 2π × 7.4 Hz; the optical and microwave quality factors are Qo = 1.4 × 108 and QΩ = 241.4, respectively. The optical resonator is critically coupled, resulting in γ = γ′ = 2π × 0.35 MHz, while the microwave resonator is undercoupled, with γΩ = 2π × 3.6 MHz and \({\gamma }_{\Omega }^{{}^{{\rm{^{\prime} }}}}=2{\rm{\pi }}\times 16.2\,{\rm{M}}{\rm{H}}{\rm{z}}\) . For convenient comparison with canonical electro-optic phase modulators, one can estimate the π-voltage from these parameters to be Vπ = 260 mV (see Supplementary Information). Typical nonresonant modulators have values of a few volts, showing the high efficiency of our system. This is traded, of course, by having a bandwidth of only a few megahertz, compared with the bandwidth of several gigahertz for nonresonant systems.

To observe the comb, we coupled 320 μW of optical power into a fundamental WGM and modulated it with 20 dBm of microwave power (with a transmission loss of 9.4 dB), corresponding to \(\sqrt{G}\approx 25\). The light emitted from the cavity was coupled to an optical fibre and measured with an optical spectrum analyser (OSA). Figure 3b shows that the generated comb has a span of about 11 nm, corresponding to 1.6 THz, and 90 visible comb lines in each direction. The comb is, as expected, symmetrical and decays approximately exponentially. According to our theory, we expect a dispersion-induced breakdown of the comb at a span of 320 lines, which we cannot observe owing to the noise floor of the OSA.

Fig. 3: Experimental realization.
figure 3

a, Measurement network. From top left, a tunable monochromatic coherent laser source (from DLPro) in the C-band of the telecommunication domain (λ = 1,549 nm) goes through a phase modulator, used for measuring the optical FSR and the linewidth of the modes. The polarization controller sets the gradient-index (GRIN)-lens output beam to transverse electric (TE) polarization; 1% of the light is used for power calibration (photodetector 1), while 99% is directed towards the WGM resonator. The silicon prism couples 30% of the light into the WGM and the remaining 70% are directly reflected and outcoupled into photodetector 2 to characterize the optical spectrum of the resonator. The comb lines, also outcoupled through the silicon prism, are detected on an OSA. The microwave tone (right) was transmitted through a cable to the pin coupler inside the cavity. This transmission line was also used to characterize the microwave mode with a vector network analyser (VNA). See Methods for more details. b, Spectrum of the frequency comb generated with 20 dBm microwave power (PΩ) at 8.9 GHz, spanning more than 180 comb lines and resulting in a width of about 1.6 THz. In blue is the spectrum of the pump laser without the added microwave tone.

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In summary, we have demonstrated that multiresonant electro-optic modulation can lead to the formation of a broadband frequency comb at very low electrical power consumption. We have presented a complete analytical solution for a dispersionless system that describes the comb formation thoroughly, including the phase and amplitude of the generated comb lines and their temporal behaviour at steady state. Our numerical simulations show that, even when taking into account optical dispersion, the comb can span hundreds of lines owing to nonlinear line broadening of the optical modes before breaking down. Because two stabilized sources are used to generate this electro-optic comb via a second-order nonlinear effect, the resulting comb lines have a fixed and predetermined phase relation to each other. The combination of compact, high-quality resonant enhancement with careful microwave-field engineering and a fast way to calculate nonlinear pulse propagation in real time could be key to power-efficient optical interconnects, and could extend the range of long-distance interconnects owing to the inherent phase relation of all the comb lines.

Methods

Theory

Theoretically we can describe the system of a strong microwave field coupled nonlinearly to two optical fields by their total energy, its Hamiltonian. Assuming no depletion of the microwave field (owing to the presence of both sum- and difference-frequency generation), the nonlinear interaction Hamiltonian is given by (see Supplementary Information):

$${ {\hat{H}} }_{{\rm{int}}}=\hbar \left|\alpha \right|g\left({{ {\hat{a}} }}_{1}{{ {\hat{a}} }}_{2}^{\dagger }+{{ {\hat{a}} }}_{1}^{\dagger }{{ {\hat{a}} }}_{2}\right)$$

This is also known as the beam-splitter Hamiltonian. Here \(\hbar \) is Planck’s constant;\({\left|\alpha \right|}^{2}\) is the number of microwave photons in the cavity; and \({{ {\hat{a}} }}_{1}\) and \({{ {\hat{a}} }}_{2}^{\dagger }\) are the standard quantum optical annihilation and creation operators of the two optical fields29, and can represent sum- as well as difference-frequency generation. This linearization of the Hamiltonian is also known as a zero-photon process, characterized by photons scattering from one mode into another without changing the total number of photons30. The single-photon coupling rate, g, is defined as:

$$g=2{\varepsilon }_{0}{\chi }^{(2)}\sqrt{\frac{\hbar {\omega }_{0}{\omega }_{{\rm{s}}}{\omega }_{\Omega }}{8{\varepsilon }_{0}{\varepsilon }_{{\rm{s}}}{\varepsilon }_{\Omega }{V}_{0}{V}_{{\rm{s}}}{V}_{\Omega }}}\int {\rm{d}}V{\psi }_{\Omega }{\psi }_{{\rm{s}}}^{\dagger }{\psi }_{0}$$

where χ(2) is the second-order nonlinear susceptibility; ε0 is the vacuum permittivity; and \({\omega }_{i}\), εi, Vi and \({\psi }_{i}\) are the frequency, relative permittivity, normalization volume and the field distribution of the optical pump (0), sideband (s) and microwave tone (Ω), indicated here by the index i. The integral over the fields is only nonzero if the so called phase-matching relation between the individual fields is fulfilled, which leads to the relation ms = m0 + mΩ, where m is the angular momentum number for the sideband, pump and microwave field, respectively (see Supplementary Information). In a system with multiple modes ωk, equidistantly separated from the optical pump field frequency ω0, a strong microwave field leads to a cascade of the sum- and difference-frequency generation. Assuming the same nonlinear coupling constant, g, between the optical and microwave fields, and noting that, because of phase matching, only spectrally adjacent modes can interact, the Hamiltonian can be written as:

$${ {\hat{H}} }_{{\rm{int}}}=\hbar \left|\alpha \right|g\sum _{k=-N}^{N-1}\left({{ {\hat{a}} }}_{k}{{ {\hat{a}} }}_{k+1}^{\dagger }+{{ {\hat{a}} }}_{k}^{\dagger }{{ {\hat{a}} }}_{k+1}\right)$$

where 2N + 1 is the number of optical modes involved in the system, and \({{ {\hat{a}} }}_{k}\) denotes the operator of the k participating optical modes. To describe their dynamics under the presence of a coherent microwave drive, we find the equations of motion using the Heisenberg picture to be:

$${\dot{\hat{a}}}_{k}=\frac{i}{\hbar }[{\hat{H}}_{{\rm{i}}{\rm{n}}{\rm{t}}},{\hat{a}}_{k}]=-i\sqrt{{n}_{\Omega }}g({\hat{a}}_{k-1}+{\hat{a}}_{k+1})$$

where nΩ is the number of microwave photons in the system. We introduce coupling and loss channels and solve these equations for the steady state in the rotating wave approximation and classical limit. Taking into account that we need to outcouple the fields from the resonator, we obtain equations (1), (2).

Numerical cavity design

The microwave cavity was designed with the help of numerical simulations using COMSOL Multiphysics version 5.4 (www.comsol.com). The goal was to maximize the overlap between microwave and optical fields while choosing the system geometry such that the optical FSR coincides with the mΩ = 1 microwave mode with a frequency of around 10 GHz. In compliance with these boundaries, we optimized the microwave coupling and allowed for an optical coupling port to the WGM resonator. We found a solution for efficient type-0 phase matching (TE + TE → TE) for which the electro-optic coefficient of lithium niobate is largest. We note that this hybrid system is very similar to our previous work on coherent microwave upconversion, showing the versatility of this approach26.

Experiment

We fabricate a lithium niobate WGM resonator (no = 2.21 and neo = 2.14 for ordinary and extraordinary polarization31) with major radius R = 2.45 mm, minor radius r = 1.5 mm, and height h = 0.4 mm, mounted within the copper cavity shown in Extended Data Fig. 1a, b (see also Supplementary Information). As the optical pump we use a narrow-band tunable laser (Toptica DLPro; λ ≈ 1,550 nm; linewidth 100 kHz) which goes through a polarization controller. The pump is split 99/1, with the weak port being sent to the detector to keep track of the pump power (Fig. 3). The other arm goes to the graded index lens for WGM-prism coupling to a transverse electric mode. We use a silicon prism (nSi = 3.47) mounted within the copper cavity. Two holes in the copper cavity allow the optical pump light to enter the cavity and the reflected light and the emitted sideband from the WGM resonator to leave. The signal and reflected pump are coupled to a fibre and sent to an InGaAs photodetector (PDA10CS; Thorlabs), and its output signal to the oscilloscope. We can use a signal generator to sweep the laser frequency, to scan over the optical modes of the cavity, and to measure the optical FSR and optical loaded Q using sideband spectroscopy32. The microwave signal generated by a microwave source (SMR20; Rohde and Schwarz) is coupled via a coaxial pin coupler introduced close to the WGM resonator inside the cavity. The same transmission line is used for reflection measurements with a vector network analyser (ENA E5072A; Keysight) to obtain the central frequency, coupling and loss rate of the microwave mode. A metallic tuning screw is used to perturb the microwave field for fine adjustment of its resonance frequency (Fig. 2a). The whole setup is thermally stabilized at 30 °C to the millikelvin level with a proportional-integral-derivative (PID) controller (TC200; Thorlabs) in combination with a temperature sensor (AD590; Analog Devices) and a thermoelectric element attached on the outer side of the closed copper cavity. Once the laser is set to the mode, we send the microwave tone and create the sidebands (Fig. 3b). These signals are outcoupled through the prism and coupled to the fibres. The sidebands together with the reflected optical pump are measured with an OSA (AQ6370C; Yokogawa).