1. Introduction

The strong fundamental vibration of molecules makes mid-infrared spectroscopy an important tool for qualitative and quantitative analysis of material composition. As excellent sources of mid-infrared spectrometers, frequency combs [1,2] earn extensive attention because of their extraordinary sensitivity, accuracy and acquisition speed. In the past a few years, various methods have been developed to generate mid-infrared frequency combs, such as ion-doped gain medium in the mode-locked laser [3,4], second-order nonlinear frequency conversion (including difference frequency generation (DFG) [5–17] and optical parametric oscillation (OPO) [18–20] and third-order nonlinear Kerr comb [21–23]. Among these methods, mid-infrared frequency comb generated by DFG is particularly attractive due to its long center wavelength, great flexibility and ease of implementation. Yet, it still faces the challenges of high pump-power demand, low conversion efficiency and low integration level.

Recently, the realization of electro-optic frequency combs [24] in a lithium-niobate (LN) microring resonator [25] based on integrated LNOI photonics platform [24–29] provides a new perspective on the generation of mid-infrared frequency comb. Compared with other integrated photonics platform, such as silica ($\rm SiO_{2}$) [30], silicon nitride ($\rm Si_{3}N_{4}$) [31], gallium arsenide (GaAs) [32] and aluminum nitride (AlN) [33], LN has the advantages of broad transparent window (350 nm to 4500 nm) [28], large second-order nonlinearity coefficient ($d_{33}=30$ pm/V) [34], large electro-optic coefficient ($r_{33}$=30 pm/V) [24]. Moreover, distinct from traditional diffusive LN waveguides or proton exchange LN waveguides [35], LNOI-based LN ridge waveguides have a better confinement of the light field because of the larger refractive index contrast between waveguide and surrounding medium ($\sim 1.2$), enabling the highest nonlinear conversion efficiency so far. Such a high nonlinear conversion efficiency combined with LNOI’s high integration level and low loss enable an efficient generation of mid-infrared frequency comb.

In this work, we theoretically demonstrate two novel designs for generating mid-infrared optical frequency combs based on LNOI platform. In contrast to conventional methods, whose mid-infrared frequency combs are generated via the DFG in a PPLN [28,36] with various poling periods between a near-infrared CW laser and a near-infrared frequency comb produced from femtosecond pulses [5–17], we reverse the comb generation process and the DFG process. As a result, our designs avoid using the PPLN with various poling periods designed for matching the momentum of all the comb teeth of a near-infrared frequency comb, thus significantly increasing the nonlinear conversion efficiency and obviating the need for femtosecond lasers. Through the DFG of two near-infrared CW lasers and the electro-optic phase modulation of a mid-infrared CW, both designs are able to produce a frequency comb centered near 4.3 $\mu$m with a tunable repetition rate and flat, nanowatt-power-level comb teeth in an ultra-high conversion efficiency. Significantly, our numerical results shows that the second design can generate a frequency comb with a pump power lower than 5 mW, which is two orders of magnitude lower than previous works, while maintaining a state-of-the-art conversion efficiency of 1800 %/W.

2. Overview and fundamentals

In this section, we delineate our two mid-infrared frequency comb generation schemes and the underlying physics. As a ferroelectric crystal, when applying an external electric field sufficient to overcome the coercive field (21 kV/mm) [36], the polarization direction of the LN ferroelectric domain will be reversed, which is numerically equivalent to the change of the sign of the nonlinear coefficient. Therefore, the periodic modulation of the LN nonlinear coefficient can be achieved through periodic polarization, thereby compensate for the phase mismatch of the pump wave, signal wave and idler wave caused by the linear dispersion. This process is also known as quasi-phase matching (QPM) [36,37]. Here, we use an x-cut LN wafer [37] to fully leveraging LN’s largest second-order nonlinear coefficient $d_\textrm {33}$. PPLN ridge waveguides fabricated by surface poling can convert near-infrared wave to mid-infrared wave through DFG process with ultra-high efficiency.

In our first design, as shown in Fig. 1(a), the mid-infrared wave generated by the PPLN ridge waveguide is directly coupled into the LN microring resonator, while sinusoidal modulation signals $f_\textrm {RF}$ are loaded on the electrodes on both sides of the microring to modulate the mid-infrared wave. When the modulation frequency is equal to integer times of the free spectral range (FSR) of the microring, the sidebands generated by electro-optic phase modulation are in resonance. Unlike on-chip Kerr combs relying on third-order nonlinearity, electro-optic frequency combs utilize the second-order nonlinear Pockels effect [24]. Furthermore, the microring resonator greatly enhances the interaction between the modulation signal and the optical field, thus enabling the generation of a mid-infrared frequency comb with a flat and broadband spectra. As shown in Fig. 1(c), this design generates a mid-infrared CW light as the optical source of the electro-optic modulator, avoiding the DFG between a near-IR frequency comb and a near-IR CW laser in a PPLN with various poling periods, whose nonlinear conversion efficiency is low.

 

Fig. 1. Designs and physical principle of mid-infrared frequency combs based on LNOI. (a) Schematic diagram of the first design based on x-cut LNOI platform. Two near-infrared pumps are injected into a PPLN ridge waveguide. The generated mid-infrared CW is transformed into a mid-infrared frequency comb through the electro-optic modulation in the microring resonator. (b) Schematic of the second design. The DFG process and the electro-optic modulation occur simultaneously in a microring resonator. (c) Nonlinear processes of the mid-infrared frequency comb.

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An alternative design shown in Fig. 1(b) integrates PPLN and electro-optic modulation in a single microring resonator, leading to higher conversion efficiency, greater complexity and challenge. Two waveguide couplers are used to couple in near-infrared sources and couple out mid-infrared frequency comb, respectively. Pump wave and signal wave can approach critical coupling [38] simultaneously by tuning the gap between couplers and microring, thus maximizing the conversion efficiency (detailed in Appendix 1). An extra temperature controller could be used to achieve the perfect phase-matching and resonance control. For a better explanation of the high efficiency in our designs, we define mid-infrared frequency comb conversion efficiency as $P_\textrm {comb}/(P_\textrm {pump}P_\textrm {signal})$, where $P_\textrm {comb}$ is the overall power of the generated mid-infrared comb and $P_\textrm {pump}$ and $P_\textrm {signal}$ are pump power and signal power, respectively. The conversion efficiency of the second design can be as high as 1800 %/W, which is one order of magnitude higher than the state-of-the-art in the previous works [5,6].

3. Method

3.1 DFG in ridge waveguide

Let us begin with the DFG process in a LN ridge waveguide. Figure 2(a) depicts the cross section of the ridge waveguide in our design. A 700 nm-thick x-cut LN film is deposited on the sapphire substrate and the ridge waveguide is formed by an etch depth of 450 nm. Here, lithium niobate on sapphire [39] avoids the strong absorption of traditional $\rm SiO_{2}$ substrate in the mid-infrared regime. By solving the coupled equations, the difference frequency power $P_\textrm {i}$ generated by a device of length $L$ is given by [40,41]

 

Fig. 2. Schematic diagram and simulated results of LNOI-based LN ridge waveguide. (a) Schematic cross section of the waveguide. Ridge width is 4 $\mu$m and etch depth is 450 nm. (b) Simulated overlap integral $S_\textrm {eff}$ versus the LN ridge width. (c) Simulated $\rm TE_{00}$ mode profiles of pump, signal and idler.

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In order to utilize x-cut LN film’s largest nonlinear coefficient $d_\textrm {33}$, we consider the fundamental TE modes of three interacting waves. Figure 2(c) shows the mode profiles of pump, signal and idler, respectively, simulated by finite element method (FEM) mode solver (Multiphysics, COMSOL, SE). According to Eq. (2) and the simulation results, an $S_\textrm {eff}$ around 3.88 $\rm \mu m^{2}$ can be obtained as shown in Fig. 2(b). Such a small effective area is more than one order of magnitude smaller than that of traditional diffusive waveguides or proton exchange waveguides [42]. Under the QPM condition $\Delta k=k_\textrm {p}-k_\textrm {s}-k_\textrm {i}-\frac {2\pi }{\Lambda }=0$, where $\Lambda$ is the poling period ($\rm \sim 6 \mu m$), the DFG conversion efficiency is maximal. When pump and signal powers $P_\textrm {p,0}=P_\textrm {s,0}=25\ \rm mW$ are injected into our device, the generated mid-infrared power is 2 mW. The DFG normalized conversion efficiency $\eta _\textrm {norm}=P_\textrm {i}/(P_\textrm {p,0}P_\textrm {s,0}L^{2})$ can reach up to 500 $\rm \%W^{-1}cm^{-2}$, which is the state-of-the-art theoretical DFG conversion efficiency so far [42]. The ultra-high conversion efficiency is attributed to the excellent mode confinement of LN ridge waveguide as a result of the large index contrast.

3.2 Design 1

In this section, we numerically analyse the comb generation in our first design. As calculated in section 3.1, a mid-infrared CW of 2 mW can be generated by a DFG process at a pump power of 25 mW. This mid-infrared CW is subsequently coupled into a microring resonator, as shown in Fig. 1(a). The input mid-infrared optical field propagates in the coupler can be expressed as $E_\textrm {i}(t)=\hat {E_\textrm {i}}\rm exp(\it i\omega _\textrm {i}t)$, where $|\hat {E_\textrm {i}}|^{2}$ is proportional to the power of the optical field $P_\textrm {i}$. Here, $P_\textrm {i}=2$ mW. After multiple cycles of sinusoidal phase modulation in the microring phase modulator, the output optical filed $E_\textrm {o}$ is formulated as [24,43]

 

Fig. 3. Power spectrum of the output mid-infrared frequency comb for modulation coefficient $\beta =1.2\pi$ in design 1 (Fig. 1(a)) with a mid-infrared input of 2 mW.

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3.3 Design 2

Different from the first design discussed in section 3.2, DFG and electro-optic modulation of design 2 occur in a single microring resonator simultaniously, as shown in Fig. 1(b). Such a design improves the conversion efficiency of nonlinear process, but induces the unwanted modulation of incident near-infrared beams at the same time. The modulation frequency $\omega _\textrm {m}$ has to match the FSR of the microring in mid-infrared region to maximize the sideband generation. However, as indicated in Fig. 4(a), the dispersion of waveguide structures leads to a modulation frequency offset $\Delta \omega _\textrm {m}$ between the modulation frequency and the FSR of the incident waves. Actually, a relatively large modulation frequency offset is favorable to decrease the modulation on the incident waves. We introduce phase offset $\Delta \phi _\textrm {m}=\Delta \omega _\textrm {m}T$. Substitute this offset into Eq. (4), then find the intra-cavity power of the incident wave (the 0-th comb tooth) considering an input power $P_\textrm {in}$ and the electro-optic modulation is

 

Fig. 4. Physical model and output spectrum of design 2. (a) Simulated effective refractive index and FSR of the ridge waveguide as a function of the idler wavelength. The FSR mismatches between idler and pump, signal are 3.1 GHz and 2.7 GHz respectively. (b) Schematic intra-cavity power of pump, signal and idler calculated by coupled mode theory. The grey area shows that the calculation is simplified to a non-depletion model when the idler intrinsic loss is high. (c) Power spectrum of the output mid-infrared frequency comb for modulation coefficient $\beta =0.4\pi$ in design 2 (Fig. 1(b)) with near-infrared input of 5 mW. The inset shows the power spectrum of the intra-cavity pump optical field with a modulation frequency offset $\phi _\textrm {m}=0.42\pi$ and the signal optical field with a modulation frequency offset $\phi _\textrm {m}=0.36\pi$.

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In the resonance state, the modulation frequency perfectly matches the FSR of the resonator in the mid-infrared regime. Equation (5) shows that almost all ($> 98\%$) of mid-infrared power transfer to electro-optic comb sidebands when phase offset $\Delta \phi _\textrm {m}$ equals to zero, and fortunately offsets between incident wave frequencies and modulation frequency maintain values closing to modulation coefficient $\beta$ due to the dispersion of the device. The phase offsets of pump and signal are $0.42\pi$ and $0.36\pi$, respectively. As a result, only a fraction of the energy leaks to a few primary near-infrared comb teeth, as shown in the inset of Fig. 4(c). Pump affected by electro-optic modulation retains a large proportion of intra-cavity energy compared to that in a non-modulation resonator when a modulation index of $0.4\pi$ is loaded. The physical mechanism of this process can be simply explained by round-trip phase model [24,45] and the existence of the p-th comb tooth should satisfy the dispersion relation as below

The intra-cavity power calculated by Eq. (5) does not consider the effect of nonlinear conversion. Although the modulation losses of input waves are effectively suppressed by dispersion, the modulation loss of idler wave is still huge compared with the loss in a non-modulation microring resonator. Hence we treat the energy leak caused by electro-optic modulation as an intrinsic loss of the microring. The intra-cavity power of three interactive waves can be calculated by the intra-cavity coupled-mode equation [33,46], as shown in Fig. 4(b). When the cavity possesses an ultra-high intrinsic loss for the idler wave, the coupling of three waves reduces to a non-depletion model. I.e., we can ignore the depletion of pump and signal in DFG process. The power calculated using Eq. (5) is approximately the actual intracavity power. Under such an approximation, the mid-infrared source generated in one round trip can be expressed as

This system of linear algebraic equations can be solved by a coefficient matrix. The result in Fig. 4(c) indicates that only 5-mW pump and signal are needed to generate a broadband (about three hundred comb teeth as far as the noise floor of the optical spectrum analyzer is lower than -75 dBm) mid-infrared frequency comb with nanowatt-level comb mode when the modulation coefficient is 0.4$\pi$. Furthermore, the frequency comb conversion efficiency can soar to 1800 %/W.

4. Feasibility

To further demonstrate the feasibility of our designs, we calculate the GVD of the device (Fig. 5(a)). We obtain the simulated total phase mismatch $\Delta \phi$ versus the idler wavelength using the round-trip phase model in Eq. (6). Figure 5(b) shows that the mid-infrared frequency comb possess a bandwidth of 500 nm when the modulation coefficient $\beta$ equals to $1.2\pi$. Such a broad mid-infrared spectrum can satisfy the basic requirement of molecular detection.

 

Fig. 5. Banwidth and controllability of the mid-infrared frequency comb. (a) Simulated dispersion of the ridge waveguide in our design (Fig. 1(a, b)). (b) The total phase mismatch $\Delta \phi$ as a function of the idler wavelength. The phase-matching condition shows that a mid-infrared frequency comb with a 500 nm bandwidth can be generated when $\beta =1.2\pi$. (c) Power spectra of the output mid-infrared frequency combs for different optical frequency offsets $\Delta \omega _\textrm {o}$ when $\beta =1.2\pi$.(d) Power spectra of the output mid-infrared frequency combs for different modulation frequency offsets $\Delta \phi _\textrm {m}$ when $\beta =1.2\pi$.

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Furthermore, electro-optic frequency comb has the advantages of flexibility and robustness against the effect of environment and fabrication imperfections. Due to thermo-optic effect [47] and fabrication imperfections, the input optical frequency might mismatch the resonant frequency of the cavity, corresponding to a non-zero optical frequency offset $\Delta \phi _\textrm {o}$. Then, the power of the p-th comb tooth can be expressed as below

It is thus clear that only an extra offset phase $\Delta \phi _\textrm {o}$ is added compared to Eq. (4). Using Eq. (10), we compute the power spectra of the mid-infrared frequency comb under different optical and modulation frequency offsets $\Delta \phi _\textrm {o}$ and $\Delta \phi _\textrm {m}$ (Fig. 5(c, d)). For a small optical frequency offset, the comb spectrum remains virtually unchanged. Only when the offset $\Delta \phi _\textrm {o}$ approaches the modulation coefficient $\beta$, the comb frequency span reduces dramatically. These rules also apply to the non-resonant modulation condition which has already been mentioned in section 3.3. When the modulation frequency offset $\Delta \phi _\textrm {m}$ is small, the flatness of the frequency comb is not affected (Fig. 5(d)). However, a relative large offset decreases the comb frequency span dramatically. The electro-optic frequency comb’s tolerance to optical frequency offset and modulation frequency offset have already been verified experimentally [24].

5. Discussion

LNOI-based ridge waveguide’s ultra-low propagation loss (0.027 dB/cm) [25] and the corresponding high quality factor ($10^{7}$) [25,27] of the microring resonator build the experimental foundation for the high conversion efficiency of mid-infrared frequency comb in our theoretical proposal. Moreover, the flexibility in the ridge waveguide’s geometry makes the dispersion engineering possible.

Another on-chip frequency comb generation approach — most Kerr combs provide repetition rates from a few tens of GHz to a few THz [1], which are too high for precise gas-phase spectroscopy. In contrast, electro-optically modulated microring can achieve a resolution of 10 GHz [24]. Such a resolution could be further improved by cascading a resonator-based electro-optic comb and a traveling-wave phase modulator [48]. Furthermore, we can sample an object’s spectrum with different repetition rates by fully taking advantage of the tunable repetition rate of an electro-optic comb. In this way, we could obtain several different discrete spectra with different spacings, from which we may better reconstruct the original continuous spectrum by using certain post-processing algorithms, such as compressed sensing.

Our second design is more efficient than the first one. Compared to the straight waveguide, microring structure significantly enhances the input power of the DFG process. Moreover, the second design generate the idler wave in the microring resonator, thus obviates the coupling of input idler beam from the coupler to the microring, which further enhancing the conversion efficiency. However, the modulation on pump wave and signal wave in the second design limits the value of modulation coefficient $\beta$. A relatively small modulation coefficient induces a faster decrease in the comb teeth and a smaller comb span. To overcome this drawback, we would like to choose a longer target mid-infrared wavelength or increase the modulation frequency offset between modulation signal and idler wave through dispersion engineering.

6. Conclusion

In conclusion, we theoretically demonstrated two novel designs of on-chip mid-infrared frequency comb. We believe that the combination of PPLN and electro-optic modulation based on LNOI integrated photonics platform can significantly increase the comb generation efficiency. Our theoretical results showed that our designs can lead to nanowatt-level comb teeth with ultra-high efficiency and an input power lower than 5 mW. Moreover, we may able to reconstruct an object’s continuous spectrum from several discrete spectra with different spacings obtained by tuning the repetition rate of the electro-optic modulator. Our designs pave the avenue of on-chip mid-infrared spectroscopy with ultra-low power consumption and ultra-high resolution.

Appendix 1. Critical coupling design

In the second design (Fig. 1(b)), the critical coupling of both pump and signal can efficiently enhance the intra-cavity energy, thus improve the nonlinear conversion efficiency. Here, as shown in Fig. 6(a), instead of conventional straight waveguide, we choose a single pulley waveguide [28] due to its strong coupling strength. The coupling between pulley waveguide and microring can be modeled using a unitary coupling matrix [38]

(11)$$\begin{bmatrix} b_{1}\\b_{2} \end{bmatrix}=\begin{bmatrix} t & \kappa\\-\kappa^{*} & t^{*} \end{bmatrix}\begin{bmatrix} a_{1}\\a_{2} \end{bmatrix},$$

where $|\kappa |^{2}+|t|^{2}=1$. Here, $\kappa$ is the coupling coefficient between pulley waveguide and microring, $t$ is the single-pass transmission, $a_{i}$ and $b_{i}$ (i=1,2) represent the energy of the optical modes in coupler and microring resonator.

 

Fig. 6. Critical coupling design for pump wave and signal wave. (a) Schematic diagram of the pulley waveguide. (b) Simulated coupling quality factor $Q_\textrm {c}$ of pump wave and signal wave versus width of gap. The yellow box indicates a region where both pump and signal can approach critical coupling. (c) Simulated electric field profile in the coupling region.

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Approaching the resonant state, the transmissivity $T_{0}$ can be expressed as

(12)$$T_{0}=\frac{|b_{1}|^{2}}{|a_{1}|^{2}}=\frac{(|t|-e^{-\frac{\alpha}{2}L})^{2}}{(|t|e^{-\frac{\alpha}{2}L}-1)^{2}},$$

where $\alpha$ denotes the propagation loss and $L$ is the circumference of the resonator. Under the critical coupling condition, input waves are totally confined in the resonator. Hence, the single-pass transmission $t$ equals to the round-trip residue $\rm exp(\it -\frac {\alpha }{2}L)$. I.e., the intrinsic quality factor of resonator $Q_\textrm {int}$ equals to the coupling quality factor $Q_\textrm {c}$,

(13)$$Q_\textrm{c}=\frac{2\pi n_\textrm{eff}L}{\lambda|\kappa|^{2}},$$

where $n_\textrm {eff}$ is effective refractive index and $\lambda$ is wavelength. $Q_\textrm {int}$ is determined by the intrinsic absorption, scattering and bending loss of the waveguide, which depend on the fineness of fabrication. On the contrary, $Q_\textrm {c}$ relates to the coupling between waveguide and resonator, which changes with coupling gap, width of coupler, etching depth and so on. Therefore, for a specific $Q_\textrm {int}$, we can approach critical coupling for both pump and signal by optimizing the gap between waveguide and resonator. Figure 6(c) shows the electric field distribution in the coupling region, which is simulated by 2.5D variational FDTD (varFDTD) solver. The simulation results in Fig. 6(b) indicates that a 300-nm gap can make both pump and signal approach critical coupling with a $Q_\textrm {int}$ of two million.

Appendix 2. Resonant condition control

Another essential precondition for comb generation is to maintain the resonant states of three different wavelengths. The resonant condition can be roughly written as

(14)$$m\lambda=n_\textrm{eff}L,$$

where $m$ is azimuthal mode numbers. In other words, triple-resonant system requires the phase-matching between three modes, which means

(15)$$m_\textrm{p}=m_\textrm{s}+m_\textrm{i}.$$

Here, $\rm p, s, i$ indicate the pump, signal and idler, respectively. Apparently, we can theoretically find three different azimuthal mode numbers corresponding to a given circumference $L$ (the integer multiples of the lowest common multiple of three different $\lambda /n_\textrm {eff}$), thus all three wavelengths are in resonance. In experiment, for a resonator with a particular $L$, we can satisfy the triple-resonant condition by firstly fine-tune the wavelengths of pump and signal waves. When the pump and signal are in resonance, we can achieve the resonance of idler wave via the perfect phase matching in PPLN. Furthermore, utilizing the thermo-optic effect of lithium niobate provides us an alternative approach for resonance adjustment. Triple-resonant system for nonlinear conversion has already been accomplished in whispering-gallery-mode resonators for sum-frequency generation [46,49].

Funding

National Natural Science Foundation of China (62075114).

Acknowledgments

The authors gratefully thank Professor Cheng Wang, Rongjin Zhuang, Yifan Qi for helpful discussions on electro-optic frequency comb.

Disclosures

The authors declare no conflicts of interest.

References

1. N. Picqué and T. W. Hänsch, “Frequency comb spectroscopy,” Nat. Photonics 13(3), 146–157 (2019). [CrossRef]  

2. A. Schliesser, N. Picqué, and T. W. Hänsch, “Mid-infrared frequency combs,” Nat. Photonics 6(7), 440–449 (2012). [CrossRef]  

3. B. Bernhardt, E. Sorokin, P. Jacquet, R. Thon, T. Becker, I. T. Sorokina, N. Picqué, and T. W. Hänsch, “Mid-infrared dual-comb spectroscopy with 2.4 µm cr 2+: Znse femtosecond lasers,” Appl. Phys. B 100(1), 3–8 (2010). [CrossRef]  

4. M. A. Solodyankin, E. D. Obraztsova, A. S. Lobach, A. I. Chernov, A. V. Tausenev, V. I. Konov, and E. M. Dianov, “Mode-locked 1.93 µm thulium fiber laser with a carbon nanotube absorber,” Opt. Lett. 33(12), 1336–1338 (2008). [CrossRef]  

5. D. L. Maser, F. C. Cruz, G. Ycas, T. Johnson, A. Klose, F. Giorgetta, L. C. Sinclair, I. Coddington, N. R. Newbury, and S. A. Diddams, “Dual-comb spectroscopy with difference-frequency-generated mid-infrared frequency combs,” in Frontiers in Optics, (Optical Society of America, 2015), pp. FTu2E–3.

6. F. C. Cruz, D. L. Maser, T. Johnson, G. Ycas, A. Klose, L. C. Sinclair, I. Coddington, N. R. Newbury, and S. A. Diddams, “Mid-infrared optical frequency combs based on difference frequency generation for dual-comb spectroscopy,” in CLEO: Science and Innovations, (Optical Society of America, 2015), pp. SW1G–8.

7. A. Sell, R. Scheu, A. Leitenstorfer, and R. Huber, “Field-resolved detection of phase-locked infrared transients from a compact er: fiber system tunable between 55 and 107 thz,” Appl. Phys. Lett. 93(25), 251107 (2008). [CrossRef]  

8. C. Erny, K. Moutzouris, J. Biegert, D. Kühlke, F. Adler, A. Leitenstorfer, and U. Keller, “Mid-infrared difference-frequency generation of ultrashort pulses tunable between 3.2 and 4.8 µm from a compact fiber source,” Opt. Lett. 32(9), 1138–1140 (2007). [CrossRef]  

9. F. Keilmann and S. Amarie, “Mid-infrared frequency comb spanning an octave based on an er fiber laser and difference-frequency generation,” J. Infrared, Millimeter, Terahertz Waves 33(5), 479–484 (2012). [CrossRef]  

10. A. Ruehl, A. Gambetta, I. Hartl, M. E. Fermann, K. S. Eikema, and M. Marangoni, “Widely-tunable mid-infrared frequency comb source based on difference frequency generation,” Opt. Lett. 37(12), 2232–2234 (2012). [CrossRef]  

11. A. Gambetta, R. Ramponi, and M. Marangoni, “Mid-infrared optical combs from a compact amplified er-doped fiber oscillator,” Opt. Lett. 33(22), 2671–2673 (2008). [CrossRef]  

12. M. Yan, P.-L. Luo, K. Iwakuni, G. Millot, T. W. Hänsch, and N. Picqué, “Mid-infrared dual-comb spectroscopy with electro-optic modulators,” Light: Sci. Appl. 6(10), e17076 (2017). [CrossRef]  

13. S. A. Meek, A. Poisson, G. Guelachvili, T. W. Hänsch, and N. Picqué, “Fourier transform spectroscopy around 3 µm with a broad difference frequency comb,” Appl. Phys. B 114(4), 573–578 (2014). [CrossRef]  

14. M. Gubin, A. Kireev, A. Konyashchenko, P. Kryukov, A. Shelkovnikov, A. Tausenev, and D. Tyurikov, “Femtosecond fiber laser based methane optical clock,” Appl. Phys. B 95(4), 661–666 (2009). [CrossRef]  

15. E. Baumann, F. R. Giorgetta, W. C. Swann, A. M. Zolot, I. Coddington, and N. R. Newbury, “Spectroscopy of the methane ν 3 band with an accurate midinfrared coherent dual-comb spectrometer,” Phys. Rev. A 84(6), 062513 (2011). [CrossRef]  

16. P. Maddaloni, P. Malara, G. Gagliardi, and P. De Natale, “Mid-infrared fibre-based optical comb,” New J. Phys. 8(11), 262 (2006). [CrossRef]  

17. S. M. Foreman, A. Marian, J. Ye, E. A. Petrukhin, M. A. Gubin, O. D. Mücke, F. N. Wong, E. P. Ippen, and F. X. Kärtner, “Demonstration of a hene/ch 4-based optical molecular clock,” Opt. Lett. 30(5), 570–572 (2005). [CrossRef]  

18. N. Leindecker, A. Marandi, R. L. Byer, and K. L. Vodopyanov, “Broadband degenerate opo for mid-infrared frequency comb generation,” Opt. Express 19(7), 6296–6302 (2011). [CrossRef]  

19. N. Leindecker, A. Marandi, R. L. Byer, K. L. Vodopyanov, J. Jiang, I. Hartl, M. Fermann, and P. G. Schunemann, “Octave-spanning ultrafast opo with 2.6-6.1 µm instantaneous bandwidth pumped by femtosecond tm-fiber laser,” Opt. Express 20(7), 7046–7053 (2012). [CrossRef]  

20. K. Vodopyanov, E. Sorokin, I. T. Sorokina, and P. Schunemann, “Mid-ir frequency comb source spanning 4.4–5.4 µm based on subharmonic gaas optical parametric oscillator,” Opt. Lett. 36(12), 2275–2277 (2011). [CrossRef]  

21. C. Y. Wang, T. Herr, P. Del’Haye, A. Schliesser, J. Hofer, R. Holzwarth, T. Hänsch, N. Picqué, and T. J. Kippenberg, “Mid-infrared optical frequency combs at 2.5 µm based on crystalline microresonators,” Nat. Commun. 4(1), 1345 (2013). [CrossRef]  

22. P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107(6), 063901 (2011). [CrossRef]  

23. Y. Okawachi, K. Saha, J. S. Levy, Y. H. Wen, M. Lipson, and A. L. Gaeta, “Octave-spanning frequency comb generation in a silicon nitride chip,” Opt. Lett. 36(17), 3398–3400 (2011). [CrossRef]  

24. M. Zhang, B. Buscaino, C. Wang, A. Shams-Ansari, C. Reimer, R. Zhu, J. M. Kahn, and M. Lončar, “Broadband electro-optic frequency comb generation in a lithium niobate microring resonator,” Nature 568(7752), 373–377 (2019). [CrossRef]  

25. M. Zhang, C. Wang, R. Cheng, A. Shams-Ansari, and M. Lončar, “Monolithic ultra-high-q lithium niobate microring resonator,” Optica 4(12), 1536–1537 (2017). [CrossRef]  

26. Y. Qi and Y. Li, “Integrated lithium niobate photonics,” Nanophotonics 9(6), 1287–1320 (2020). [CrossRef]  

27. B. Desiatov, A. Shams-Ansari, M. Zhang, C. Wang, and M. Lončar, “Ultra-low-loss integrated visible photonics using thin-film lithium niobate,” Optica 6(3), 380–384 (2019). [CrossRef]  

28. J. Lu, J. B. Surya, X. Liu, A. W. Bruch, Z. Gong, Y. Xu, and H. X. Tang, “Periodically poled thin-film lithium niobate microring resonators with a second-harmonic generation efficiency of 250,000%/w,” Optica 6(12), 1455–1460 (2019). [CrossRef]  

29. J. Lin, B. Fang, Y. Cheng, and J. Xu, “Advances in on-chip photonic devices based on lithium niobate on insulator,” Phys. Lett. 116, 101104 (2020).

30. X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, “Soliton frequency comb at microwave rates in a high-q silica microresonator,” Optica 2(12), 1078–1085 (2015). [CrossRef]  

31. T. Herr, K. Hartinger, J. Riemensberger, C. Wang, E. Gavartin, R. Holzwarth, M. Gorodetsky, and T. Kippenberg, “Universal formation dynamics and noise of kerr-frequency combs in microresonators,” Nat. Photonics 6(7), 480–487 (2012). [CrossRef]  

32. T. H. Stievater, R. Mahon, D. Park, W. S. Rabinovich, M. W. Pruessner, J. B. Khurgin, and C. J. Richardson, “Mid-infrared difference-frequency generation in suspended gaas waveguides,” Opt. Lett. 39(4), 945–948 (2014). [CrossRef]  

33. X. Guo, C.-L. Zou, and H. X. Tang, “Second-harmonic generation in aluminum nitride microrings with 2500%/w conversion efficiency,” Optica 3(10), 1126–1131 (2016). [CrossRef]  

34. K. Abdelsalam, T. Li, J. B. Khurgin, and S. Fathpour, “Linear isolators using wavelength conversion,” Optica 7(3), 209–213 (2020). [CrossRef]  

35. M. Bazzan and C. Sada, “Optical waveguides in lithium niobate: Recent developments and applications,” Appl. Phys. Rev. 2(4), 040603 (2015). [CrossRef]  

36. L. Chang, Y. Li, N. Volet, L. Wang, J. Peters, and J. E. Bowers, “Thin film wavelength converters for photonic integrated circuits,” Optica 3(5), 531–535 (2016). [CrossRef]  

37. A. Boes, B. Corcoran, L. Chang, J. Bowers, and A. Mitchell, “Status and potential of lithium niobate on insulator (lnoi) for photonic integrated circuits,” Laser Photonics Rev. 12(4), 1700256 (2018). [CrossRef]  

38. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000). [CrossRef]  

39. C. J. Sarabalis, T. P. McKenna, R. N. Patel, R. Van Laer, and A. H. Safavi-Naeini, “Acousto-optic modulation in lithium niobate on sapphire,” APL Photonics 5(8), 086104 (2020). [CrossRef]  

40. R. W. Boyd, Nonlinear optics (Academic press, 2019).

41. L. Gui, “Periodically poled ridge waveguides and photonic wires in linbo3 for efficient nonlinear interactions,” University of Paderborn (2010).

42. X. Wang and C. K. Madsen, “Design of a hybrid as 2 s 3-ti: Linbo 3 optical waveguide for phase-matched difference frequency generation at mid-infrared,” Opt. Express 22(22), 27183–27192 (2014). [CrossRef]  

43. K.-P. Ho and J. M. Kahn, “Optical frequency comb generator using phase modulation in amplified circulating loop,” IEEE Photonics Technol. Lett. 5(6), 721–725 (1993). [CrossRef]  

44. M. Abramowitz and I. A. Stegun, “Handbook of mathematical functions with formulas, graphs, and mathematical table,” in US Department of Commerce, (National Bureau of Standards Applied Mathematics series 55, 1965).

45. B. Buscaino, M. Zhang, M. Lončar, and J. M. Kahn, “Design of efficient resonator-enhanced electro-optic frequency comb generators,” J. Lightwave Technol. 38(6), 1400–1413 (2020). [CrossRef]  

46. D. V. Strekalov, A. S. Kowligy, Y.-P. Huang, and P. Kumar, “Optical sum-frequency generation in a whispering-gallery-mode resonator,” New J. Phys. 16(5), 053025 (2014). [CrossRef]  

47. U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microring resonator,” Appl. Phys. Lett. 88(11), 111107 (2006). [CrossRef]  

48. A. Shams-Ansari, C. Reimer, N. Sinclair, M. Zhang, N. Picqué, and M. Loncar, “Low-repetition-rate integrated electro-optic frequency comb sources,” in CLEO: Science and Innovations, (Optical Society of America, 2020), pp. STh1O–2.

49. Z. Hao, J. Wang, S. Ma, W. Mao, F. Bo, F. Gao, G. Zhang, and J. Xu, “Sum-frequency generation in on-chip lithium niobate microdisk resonators,” Photonics Res. 5(6), 623–628 (2017). [CrossRef]