Generating an optical frequency comb using an electro-optic modulation in an optical parametric oscillator

Abstract

An optical frequency comb generator may be provided. The optical frequency comb generator may include an optical parametric oscillator cavity configured to generate a signal and an idler from an optical pump field. The optical frequency comb generator may also include an electro-optic modulator configured to provide an amplitude modulation and phase modulation on at least one of the optical pump field, signal, or the idler to generate a frequency comb.

Description

[0001] Attorney Reference No.390351-995471 GENERATING AN OPTICAL FREQUENCY COMB USING AN ELECTRO-OPTIC MODULATION IN AN OPTICAL PARAMETRIC OSCILLATOR CROSS-REFERENCE TO RELATED APPLICATIONS[1] This application claims priority to U.S. Provisional Application No. 63 / 639,290 filed on April26, 2024, which is hereby incorporated by reference in its entirety. FIELD[2] This disclosure relates to generating an optical frequency comb by using electro-opticmodulation in an optical parametric oscillator. BACKGROUND[3] An optical frequency comb is discrete spectrum made of equally spaced spectral lines. Theoptical frequency comb may function like a ruler to measure the frequencies of an electromagnetic radiation. The optical frequency comb has been an indispensable tool in spectroscopy and metrology, with emerging applications in fields such as laser ranging, communications, quantum computing, etc. An example use of the optical frequency comb is performing wavelength division multiplexing in telecommunications, where each spectral line is used to encode a stream of data. Current technology for generating frequency comb needs bulky and expensive equipment. Due to these limitations, the current frequency comb generation technology is not applicable for applications such as transceivers in telecommunications. Other miniaturized systems to generate optical frequency comb also need technical improvements. SUMMARY[4] In some embodiments, an optical frequency comb generator may be provided. The opticalfrequency comb generator may include an optical parametric oscillator cavity configured to generate a signal and an idler from an optical pump field. The optical frequency comb generator may also include an electro-optic modulator configured to provide an amplitude modulation and phase modulation on at least one of the optical pump field, signal, or the idler to generate an optical frequency comb.[5] In some embodiments, a method of generating an optical frequency comb may be provided. Themethod may include generating, by an optical parametric oscillator cavity, a signal and an idler from an optical pump field. The method may also include generating, by an electro-optic modulator, an optical frequency comb by providing an amplitude modulation and phase modulation on at least one of the optical pump field, the signal, or the idler. Attorney Reference No.390351-995471 BRIEF DESCRIPTION OF DRAWINGS[6] FIG.1A shows an illustrative electro-optic comb optical parameter oscillator (ECPO) configuredto generate an optical frequency comb, according to example embodiments of this disclosure.[7] FIG. 1B shows an illustrative Ikeda map of the ECPO shown in FIG. 1A, according to exampleembodiments of this disclosure.[8] FIG.2 shows an example method of generating an optical frequency comb, according to exampleembodiments of this disclosure.[9] The figures are for purposes of illustrating example embodiments, but it is understood that thepresent disclosure is not limited to the arrangements and instrumentality shown in the drawings. In the figures, identical reference numbers identify at least generally similar elements. DESCRIPTION

[0010] Embodiments disclosed herein may generate optical frequency combs using integrated opticalchips. In some embodiments, an electro-optic comb optical parameter oscillator (ECPO) may use one or more χ(2)crystals an optical parametric oscillator and electro optic modulator to generate the optical frequency combs. The χ(2)crystals may be integrated into an optical cavity in an optical chip. The electro optic modulator may perform both phase modulation and intensity modulation within the optical cavity. That is, when pumped with a high frequency, the ECPO generates a signal and the idler, and the electro- optic modulator generates the optical frequency comb using at least one of the signal and the idler. The ECPO may not cause the same efficiency loss as with the conventional systems and also may generate a flatter comb spectra. Additionally, a broader bandwidth—i.e., with more comb lines—may be achieved because of the gain in the ECPO cavity. Furthermore, because the ECPO may be non- degenerate and singly resonant, i.e., the signal may only be resonant without the idler being resonant, the ECPO can be pumped at any frequency to generate frequency combs in a wider spectral range. Therefore, a relatively unstable laser may be used for pumping and the noise of the laser may not be imparted to the signal.

[0011] FIG. 1A shows an illustrative ECPO 100 configured to generate an optical frequency comb,according to example embodiments of this disclosure. As shown, the ECPO 100 may generate an optical frequency comb 150. In some embodiments, the ECPO 100 may include singly resonant, optical parametric oscillator 102, which may include a non-linear optical crystal 104 and an electro-optic modulator 106. In some embodiments, the non-linear optical crystal 104 may include a χ(2)crystal.

[0012] The ECPO 100 may include a laser pump 108 that may drive the non-linear optical crystal 104at a pump angular frequency ^^^^while the optical parametric oscillator 102 cavity may resonate at the angular frequency ^^^^of the signal 110. The non-linear optical crystal 104 may therefore mediate a threewave interaction ^^^^ ^ ^^^^ + ^^^^, where ^^^^ may be the angular frequency of the idler 112. Theelectro-optic modulator 106 may introduce both phase modulation and intensity modulation (also Attorney Reference No.390351-995471 referred to as amplitude modulation) within the optical parametric oscillator 102 cavity and cause a stable formation of the frequency comb 150, with a first frequency comb 152 at signal angular frequency ^^^^and a mirrored second frequency comb 154 at idler angular frequency ^^^^. That is, the electro-optic modulator 106 may generate and broaden the optical frequency comb 150 by redistributing power among neighboring comb lines while the parametric gain provided by the optical parametric oscillator 102 compensate for cavity round trip. Therefore, the net round trip loss of the ECPO 100 may be zero.

[0013] FIG. 1B shows an illustrative Ikeda map of the ECPO 100, according to example embodimentsof this disclosure. The Ikeda map is used to explain the principles of operations of the ECPO 100, particularly the pulse dynamics within the optical parametric oscillator 102.

[0014] As shown in the Ikeda map, the state of the optical parametric oscillator 102 cavity may beencoded in a propagating field ^^(^^) of the signal 110, which may satisfy a T-periodic boundary condition where T may be the repetition rate. A round trip in the optical parametric oscillator 102 cavity may be decomposed into a sequence of the following discrete operations: 1. Optical parametric amplification due to propagation through the non-linear opticalcrystal 104. Because the optical parametric oscillator 102 cavity is singly resonant, this may be the only operation where the laser pump 108 and the idler 112 participate. 2. Partial reflection off the out-coupling mirror 114.3. Electro-optic modulation by the electro-optic modulator 106.4. Group delay dispersion, in addition to any applicable higher-order dispersion terms.Mirror reflection (operation #2) and group delay dispersion (operation #4) may generally be trivial. Optical parametric amplification and electro-optic modulation are described in detailed below.

[0015] For optical parametric amplification, difference frequency generation between the laser pump108 ^^(^^) and signal 110 ^^(^^) may lead to signal amplification and generation of idler 112 ^^(^^). The pulse dynamics within the optical parametric oscillator 102 cavity may be governed by the non-linear pulse propagation equations: thesignal 110 ^^(^^) resonates with high finesse, its pulse shape will not change much on a given round trip. Moreover, above a threshold, the signal amplitude may usually be much larger than the laser pump 108or idler 112 amplitude, i.e., |^^| ≫ |^^|, |^^|. This may be because given round trip, only a fraction ^^ ∼1 / ℱ of the signal 110 ^^(^^)) light may escape, whereas all the laser pump 108 ^^(^^) and the idler 112 Attorney Reference No.390351-995471 ^^(^^) may escape. If the conversion efficiency is an ^^(1) constant, the resonant signal may generally scale as |^^|2 ∼ ℱ to maintain energy conservation. Below and slightly above the threshold, the laserpump 108 may be approximately undepleted. In this case, the dynamics of ^^(^^) may be given by linear dispersion, where ^^(^^) may become a constant for a continuous wave pump. This undepleted-pump model may calculate squeezing spectra below the threshold and approximate the pulse shapes of models slightly above the threshold.

[17] In a high-finesse cavity, the nonlinear interaction may affect the signal 110 ^^(^^) perturbatively.Therefore, an approximation (^^) may be made as follows: signal may be facilitating optical frequency conversion between the laser pump 108 and the idler 112, a process that may be linear in fields (^^, ^^). When (^^, ^^, ^^) and (^^, ^^, ^^) are computed from equations (2) and (3), theresults may be plugged back into equation (1) to obtain the perturbation to ^^(^^).

[18] idler 112frequency conversion equations may be (where ^^ may be the coupling coefficient): The In ≈1 ^^|^^22^^^^| ^^. The sinusoidal|^^|dependence may result from pump depletion.

[19] In a generic case, mode mixing between ^^^^, ^^^^, and ^^^^ may be considered, where ^^ = ^^ + ^^.Let ^^^^ = ^^^^(^^Ω), ^^^^ = ^^^^(^^Ω), and ^^^^ = ^^^^(^^Ω) be the propagation constants. Ignoring the indicesand applying the substitutions ^^ = ^^^^^^^^^^^̃^, ^^ = ^^^^(−^^^^+ ^^^^+^^^^)^^ / 2^^ , and ^^ =following vector equation may be derived for (^^ , ^̃^): Attorney Reference No.390351-995471 This vector equation may be visualized as a rotation on a Bloch sphere, where the state may be initializedalong the −^^ axis (^^ = 0) with a rotation along the axis ^⃗^ = ^^ (−^^^^[^^]^̂^ + ^^^^[^^]^̂^) +1 2 ∆^^^̃^. Therefore, a 100% laser pump 108 depletion may be possible matching may correspond to a 180orotation about an axis may no longer be along the equator, so the pump depletion is imperfect and the gain may be reduced. The solution then may be order in theperturbation expansion may go as ^⃗^ → (^^ + ∆^̃^)^^^^^^^^^^.

[0020] For the undepleted laser pump be constant (e.g., because of continuous wave pumping) and equations (1) and (2) may be solved for (^^, ^^) as follows: dynamics may be a competition between gain ^^^^ and detuning ∆^^. The signal gain may be found analytically using: 1.This case may be encountered in high-finesse resonators. The gain may become a power series in pump ^^: resonators. The gain may become a power series in the phase mismatch ∆^^^^: Attorney Reference No.390351-995471 that appeared in the small gain case. The reason may be that, for a large-gain optical parametric oscillator 102, the signal 110 pulse may be pulled toward the idler 112 pulse, where the gain is larger and the optical parametric amplification may occur more efficient. Because the signal 110 and the idler 112 need each other to produce gain, the signal 110—idler 112 pair may propagate at the average group 1 index 2(^^^^,^^ + ^^^^,^^). This may be a nondegenerate analog of the simulton acceleration observed inparametric oscillators and may occur even in absence of pump depletion.

[0022] optical parametric amplifier is phase matched, the peak gain may occur at ^^ = 0. To thegiven orders in the perturbation expansion, the gain relative to this peak value is quantified by a differential gain ^^ and phase ^^: To

[0023] For the electro-optic modulation, the electro-optic modulator 106 on each round trip may phase-modulate the waveform in time and frequency space, and this modulation may correspond to a convolution: Attorney Reference No.390351-995471 be the reflectivity of the out-coupling mirror (plus the effect of any other losses). Residual dispersion of optical parametric oscillator 102 cavity components may be captured with a mode-dependent phase ^^^^.The overall mode-dependent phase may then be ^^^^ = ^^^^ + ^^^^, which may generally be linear in ^^.

[0025] The oscillations within the ECPO 100 may be modeled as a linear gain model. To the mode, the optical parametric oscillator 102 may be be at the threshold, where the phasematched gain equals loss: (^^^^^^)2 = ^^. The Ikeda map may then be written as an eigenvalue equation: around the point ^^ = 0. For real valued ^^^^ and ^^^^, equation (39) is the Schrodinger equation for aparticle in a harmonic potential, for which the ground state solution may be a Gaussian: velocity dispersion-broadened) Gaussian pulse: where delay dispersion and the chirped pulse width may be ^^^^ℎ = √^^2 + ^^2 / ^^2. From the coefficients inequation (39), the limit a large chirp ^^ ≫ ^^2, it may be As may pump108 depletion is sufficiently small. However, it is desired that the optical parametric oscillator 102 be Attorney Reference No.390351-995471 operated in the regime of large laser pump 108 depletion, where the conversion efficiency may be high. All the eigenmodes above the threshold may share the same chirped Gaussian profile: general . near oscillator 102 above the threshold (at least because the gain mechanism may be local in ^^), a Wentzel– Kramers–Brillouin (WKB) like ansatz for the field may be chosen: where ^^

[0027] threefactors: (i) Optical parametric amplification gain as shown in equation (10). Here a slowly varyingenvelope ansatz ^^(^^) = ^^(^^)^^−^^Φ(^^) may be made, which may assume that the idler 112 imparts a phaseopposite the signal 110. The phase of the laser pump 108 may have to be independent of the signal 110such that ^^(^^) = ^^(^^); (ii) Cavity dispersion given by ^^ → (1 + ^^^^(^^^^^^))^^. If the optical parametricoscillator 102 cavity is path-length matched, the group velocity dispersion may generally dominate:^^(^^) =122 ^^2^^ ; (iii) Modulation: It may be advantageous to use a chirped modulator, where amplitudeis A modulator transfer function may be def ^^^^(^^)−^^(^^) / 2 ined as: ^^ → ^^ ^^, where both^^ and ^^ may be periodic in time. Combing all the three factors, the for ^^(^^) may be:

[0028] into equations for ^^(^^) and Φ(^^): Attorney Reference No.390351-995471 where the slowly varying envelope approximation may have been invoked and a replacement is made−^^^^^^ → Φ′(^^), |^^| → ^^(^^) to get to:

[0029] ^^, whileA may affect ∆Φ through ^^ as well. However, the coupling between A and Φ may be relatively weak. The optical parametric oscillator 102 may have phase matching may cover approximately the wholesignal bandwidth, so ∆β ≪ κA. In such a case, an approximation of ^^ ≈ ^^^^ may be made, therebyremoving the Φ dependence from equation (50). Furthermore, the rightmost term of equation (51) maybe proportional to (κcL)2(∆βL) times an O(1) term. Because (κcL)2 ≾ ^^ and ∆βL ≾ 1, the term(κcL)2(∆βL) may become ≾ ^^ in general. If the optical parametric is high finesse, ^^ ≪ 1, while ^^ ∼ ^^(1). the rightmost term in equation may removingthe equation’s A dependence. The result may be two decoupled equations for the amplitude and the phase.

[0030] It follows from these decoupled equations that that the amplitude may be determined by thesteady-state equation of a continuous wave optical parametric oscillator 102 (although the loss may depend on ^^ in some embodiments through the modulator chirp term ^^(^^)), while the phase may bedetermined by the constant phase condition δ(Φ′(^^)) + ^^(^^) = 0. That is, for a stabled pulse operation,the round-trip phase, which may be a sum of electro-optic modulator 106 phase and dispersion, may be independent of ^^. Therefore, the pulse shape may equilibrate so that the dispersive phase may compensate the dispersive phase of the electro-optic modulator 106. Because the phase of the electro- optic modulator 106 may ramp with ^^, a stable pulse may have a chirped waveform.

[0031] To model the mechanism and stability of ECPO 100, let ^^(^^, ^^), ^^(^^, ^^), and ^^(^^, ^^) be signal110, idler 112, and laser pump 108, respectively, normalized so that |^^|2, |^^|2, |^^|2may be the corresponding photon fluxes. The pulse propagation equations therefore may be:Here, basis^^^^(^^) → ^^^^(^^) − ^^^^(0) − ^^^′^(0), in which ^^^^(^^) may have quadratic and higher-order terms. Because Attorney Reference No.390351-995471^^0 may be pumped at CW, most of the processes may be of the form ^^0 ↔ ^^^^ + ^^−^^. For simplicitylet’s denote ^^ → ^^^^, ^^ → ^^−^^, ^^ → ^^0, as well as ^^^^ → ^^^^(^^Ω), ^^^^ → ^^^^(−^^Ω), ^^^^ → ^^^^(0).

[0032] At the high-finesses limit, which may be likely applicable in the thin film lithium niobateimplementations, |^^| ≫ |^^|, |^^|. Therefore, on a just beperturbative. The phase matched CW solution (^^^^ = ^^^^ = ^^^^ = 0) may be: ^^ =^^^^(^^^^+ ^^^^+^^^^)^^ / 2^̃^ may be applied, leading to the equation: =− − may ^^ = √(Δ^^ 2)2 + (^^^^^^^^)2, the solution may be: The = ^^^^^^ − =^^^^^^may pump may strongerthan the laser pump 108 provided that the efficiency is ^^(1).

[0033] Taking in the near threshold limit ^^^^^^ → 0, the following threshold amplitude may be derived^^ℎ(Δ^^) = √^^ / (^^^^^^^^^^^^(Δ^^^^)). This threshold amplitude may be lowest at phase matching: |^^^^ℎ(0) = relative to the threshold therefore may become: Attorney Reference No.390351-995471 of ^^:In

[0034] may increase for larger amplitudes of the laser pump 108.

[0035] In some embodiments, the pulse evolution may be governed by three forcing terms: (i) gain andloss, (ii) pulse modulation, and (iii) group delay dispersion. Gain and loss may be responsible forpumping energy into the pulse and depleting it on the tails to prevent spillover. Phase modulation maychirp the pulse, shifting the carrier frequency of the tails to create a frequency modulated constant wave waveform. Finally, group delay dispersion may stabilize the pulse waveform by pushing the chirped waves away from the pulse center, so as to prevent the chirp from building up indefinitely. The group delay dispersion may also have the effect of stabilizing the overall waveform even if parts of the pulse are not locally stable, as unstable oscillations get pushed away from the pulse center and damped out before they can grow to appreciable amplitudes.

[0036] An ansatz for a frequency modulated continuous wave signal may be modeled as a waveform^^(^^) where the amplitude ^^(^^) and the carrier angular frequency ^^(^^) may vary slowly in time as follows: Attorney Reference No.390351-995471with round-trip index ^^. The goal is to obtain a Lugiato-Lefever equation for ^^(^^, ^^), ^^(^^, ^^), where ^^ =^^^^ is a slow time and ^^ is he period of the electro-optic modulator 106 drive. Using the slowly varyingenvelope approximation and treating ^^(^^+1) − ^^(^^) → ^^ ^^^^⁄ ^^^^ , the following field equations may bederived for ^^ and ^^: of =− may^^(^^) may be the round-trip phase shift of electro-optic modulator 106. A round-trip intensity modulation is also introduced, so that the loss in the optical parametric oscillator 102 cavity ^^(^^) may be time- dependent as well. Equations (73) can then be reduced to a system of ordinary differential equationsusing the method of characteristics, where (^^, ^^) may be integrated along characteristics curves(^^, ^^(^^)) as follows:The This has a has a saddle point at the potential maximum and a center at the potential minimum. Stable operation of the ECPO 100 may be obtained when the gain maximum overlaps with the saddle point, and the steady-state solution branches out from that point along the separatrix. This saddle point may be located at:

[0037] Within the phase-matching envelope, the comb’s center frequency may be tuned by adjustingthe electro-optic modulator 106 radio frequency drive frequency ^^ = 1⁄ ^^ . This may be convenientbecause waveform generators may be tunable using length and tuning. The tuning rate maybe given by ∆^^ =∆^^=−∆^^. For2^^2^^2^^2 ^^2 ∼ 1000 fs(1 cm of LiNbO3) and ^^=10 GHz, the frequencyamplification factor may be ≈ 106, or around∆^^ 12 nm / MHz. Using equation (75), a solution can be derived from ^^(^^) and then find a solution Attorney Reference No.390351-995471

[0038] The steady-state solution may be pumped in the high-gain region near ^^ = 0, while chirp andgroup delay dispersion push power out to the tails. Therefore, an example usable characteristic may be the separatrix, which may be the solution of equation (75) with ^^= 0.

[0039] For modulation, a Mach-Zehnder modulator (MZM) may be asymmetrically driven with phases^^1(cos(Ωt) − 1) / 2 and ^^2(cos(Ωt) − 1) / 2. Here (^^1, ^^2) may be the peak-to-peak phase shifts of theconstituent phase modulators that make up the MZM. This may yield both a phase modulation and an amplitude modulation: where Using these expressions, ^^(^^) and ^^(^^) may be extracted as follows: The ^^ to a continuous variable.

[0040] The below expressions consider dispersion only up to the second order:The reason term ^^ may ignored as it may lead to a global phase shift, ^^1may be eliminated byshifting the center frequency of the signal window ^^ → ^^1 / ^^2. Thus, truncation of higherdispersion term, there may be no loss of generality in equation (80).

[0041] Combining equations (75) and (79)-(80), the following time-frequency chirp may be found alongthe separatrix: The . the quasi-static limit, where the left-hand side can be ignored and the quasi-static field (denoted as ^^0) may be obtained bysolving ^^ (^^0, ^^(^^)) = ^^(^^) and using equation (68). Equation (82) may have a sharp cutoff when ^^^^^^^^ (2∆^^(^^(^^))^^) = ^^(^^) / (^^0^^2). Beyond this point,net gain may not be possible at ^^(^^) and ^^0 = 0. Attorney Reference No.390351-995471

[0042] Provided that ^^′(^^) is small, equation (77) may be solved using a perturbation expansion^^ = ^^0 + ^^^^, where the equation for ^^^^ may be given by:

[0043] (ii) assume that the electro-optic modulator 106 is driven by a single frequency, i.e., equation (78) holds such that^^(^^) = −^^^^^^^^^^2(Ω^^ / 2). In this case, equations (74) may take the form: benoted. The resemblance may be made more exact by transforming into normalized coordinates. oscillator 102: ^^^^may govern the comb bandwidth, while the unitless ^^^^may govern the characteristic slow time, number of round trips that may needed to build up a stable comb. For parameters ^^22=2000 fs, ^^2=3.2 rad, ^^0=2 20 GHz, which may be typical for a bulk Periodically Poled LithiumNiobate (PPLN) implementation (about 2 cm LN), the calculation may be ^^^^ / 2^^=4.5 THz and ^^^^=200.

[0044] In normalized variables, the dynamics in equations (85) may reduce to: which may be the inverted terms of ^^ may take the form of elliptic functions, which may trace out constant-phase curves: For the = curve, may Attorney Reference No.390351-995471 The Frequency Modulated Wave (FMCW) solution may follow the separatrix, with power emanating from the saddle point = ^^ ̅ = 0 , where the gain may be maximum. The bandwidth ofthe curve may be ∆^̅^ = 4 in dimensionless units, thereby giving an upper bound on the comb bandwidth: This

[0045] For analyzing bandwidth operation and tunability of the ECPO 100, the analysis may begin withthe optical parametric oscillator 102 gain that is governed by both the power of laser pump 108 and phase matching as shown in equation (68) the laser pump 108 and signal 110 wavelengths, Δβ(ω) may be the mismatch term for the process ^^^^→(^^, ^^^^ − ^^) with gain maximized at Δβ = 0. If ^^^^ is the target frequency. Expanding about ^^^^, i.e.,^^ = ^^^^ + Δβ, the following expression may be found: either be zero or a minimum of ^^(^^) for the gain to be maximized at ^^^^. The group velocity mismatchof signal 110-idler 112 Δβ1 = ^^1,^^ − ^^1,^^ may have to minimized or set to zero for maximizing thephase-matching bandwidth. In mean signal 110-idler 112 group velocity dispersion̅ ^̅^2 ̅ =1 2(^^2,^^ + ^^2,^^) may have to be minimized. By Taylor expanding ^^(^^) about the center frequency ^̅^ =1 2(^^^^ + ^^^^), the following expression may be derived: , Attorney Reference No.390351-995471

[0047] The 3-dB gain bandwidth may be obtained by solving equation (113) for Δ^^(Δ^^^^^^^^) = Δ^^^^^^^^,where Δ^^^^^^^^ = (2 ^^) ^^^^^^^^−1 (1 √2) ≈ 2.8 / ^^ per equation (110). When group velocities match, it may bebecause signal the idler 112 may be at approximately of the zero-(group)-dispersionwavelength, so ^^^^^^^^. Assuming an approximately perfect phase-matching at ^^^^, thebandwidth limit may be as follows: matching may have a bandwidth of Δ^^^^^^^^^^ = 100 nm and 270 nm for signal 110 at O-band (1.3 µm)and C-band (1.5 µm), respectively.

[0049] The ECPO 100 may be tuned using RF frequency. As described above in relation to equation(74), the ECPO 100 may form an FMCW pulse shaped by phase space dynamics: and ^^(^^) may the phase of electro-optic modulator 106. The round trip dispersion ^^(^^) and the ^^(^^) may lead to group delay and chirp, respectively. The pulse within the ECPO 100 may center around a saddle pointwhere ^^′(^^) = ^^′(^^) = 0, which may be engineered to be near the maximum of the gain spectrum.Detuning the electro-optic modulator 106’s RF frequency ^^^^^^^^relative to the optical parametricoscillator 102 free spectral range by Δ^^^^^^^^ may be equivalent to an added group delay Δ^^ = Δ^^rep / ^^r2ep,which may add a linear term to the cavity dispersion: ^^(^^) → ^^(^^) + ^^Δ^^. Expanding ^^(^^) to secondorder, it may be found that the frequency center may shift by: Therefore, increasing the RF frequency may redshift the comb, while decreasing the RF frequency may blueshift the comb. To first order, this relation is linear, and the RF-to-optical frequency-shift multiplierΔ^^ / ΔΩ (here Ω = 2^^^^rep ) may be: Attorney Reference No.390351-995471 bandwidth and Δ^^BW= 2√2^^^^ / ^^2. The result may then depend on the phase and the number of comb lines: optic modulator 106may the inverted pendulum model (no higher-order dispersion), the following expressions may be derived: As by re-expressing ^^ in terms of Δ^^(^^) = ^^(^^) − Δ^^ / ^^2, i.e., the deviation from the steady-state centerfrequency. The equations may become: ofequation(122). This saddle may be shifted relative to the gain maximum:This Thismay set an upper limit on the detuning sweep rate, and using equation (117) the following result may be arrived at:

[0051] temperaturetuning. A formula relating the pump tuning range Δ^^^^to the signal 110 / idler 112 tuning range Δ^^^^,^^may be derived by Taylor-expanding the phase mismatch Δ^^ about the degeneracy point, and assumingphase-matching at degeneracy ^^ + ^^ ↔ 2^^: Attorney Reference No.390351-995471 pump frequency shift ^^^^^^, corresponding to the signal 110 / idler 112 group-velocity matched case. This maximum may occur when: is defined as twice the signal perturbation (because the range may be [−^^^^^^, +^^^^^^], covering bothsignal and idler), the following expression may be reached: This mostappropriate.

[52] In some embodiments, the pump may be amplitude modulated rather than the intracavity field.The amplitude modulation may be beneficial in small, high-finesse cavities, where it may be difficult to achieve amplitude modulation with high contrast at a reasonable voltage.

[53] FIG. 2 shows an example method 200 of generating an optical frequency comb, according toexample embodiments of this disclosure. It should be understood that the steps of the method 200 are merely intended as examples and should not be considered limiting. That is, methods with additional, alternate, or fewer number of steps should be considered within the scope of this disclosure.

[54] At step 210, an optical parametric oscillator cavity may generate a signal and an idler from apumped signal. The optical parametric oscillator cavity may be a part of an ECPO, where the ECPO may be fabricated on a single optical chip using thin film lithium niobate.

[55] At step 220, an electro-optic modulator may generate a frequency comb by providing anamplitude modulation and phase modulation on at least one of the signal and the idler. In some embodiments, the electro-optic modulator may generate the frequency comb based on just the signal. In some embodiments, the electro-optic modulator may generate the frequency comb using both the signal and the idler. Attorney Reference No.390351-995471

[0056] As disclosed in the example embodiments above, the bandwidth of the EO Comb OPO (like anyOPO) may be limited by its gain spectrum. Gain bandwidth may be broadened with dispersionengineering-for example, signal-idler group-velocity matching ^^1,^^ = ^^1,^^ and GVD cancellation ^^2,^^ ≈−^^2,^^. But in some situations, dispersion engineering alone may not produce a sufficiently wide gain window. In this case, extra bandwidth may be obtained by chirping the QPM crystal. Chirp may trade gain for bandwidth, and it can also be used to flatten the gain spectrum, a technique that may help stabilizing the ECPO against mode-hop instability. Embodiments disclosed below describe formalism for using chirped gratings in the EO Comb OPO and describes the effect of using them.

[0057] The formalism may begin with the field propagation equations (Eqs. (1-3)).^̇^ = ^^^^ (^^)^^ + ^ ( ) ∗^^ ^^,^^ ^^ ^ ^^ ∑   ^^^^+^^^^^^ , ^̇^^^^^

[0058] Eqs. (144) may be kept general: both the dispersion relations ^^^^, ^^^^ , ^^^^, and the nonlinearcoefficient ^^ can depend on the axial coordinate ^^. In this case, the focus may be on QPM systems with a poling-period chirp, where only the first-order QPM process may be relevant to the dynamics. In this case, the following replacement may be made ^^(^^) → ^^^^^^^^qpm(^^) (145)where Λ(^^) = 2^^ / |^^ ′ qpm (^^)| is thethe corresponding

[0059] Here, the focus may be on CW gain, so pump / signal / idler triplet (^^^^, ^^^^, ^^^^+^^) may be picked.Defining ^^^^ = ^^^^,^^, ^^^^ = ^^^^,^^, and ^^^^ = ^^^^,^^+^^, the following equations of motion may be obtained:^̇^ = ^^^^ −^^^^^^^^ + ^^^^ qpm^^∗^^, ^̇^ = ^^^^ −^^^^^^^^ + ^^^^ qpm^^∗^^, ^̇^ = ^^^^ ^^^^^^^^ − ^^^^ qpm^^^^ (146)

[0060] The a 4th-order Runge-Kutta stepper. However, there are three common cases where simplified solutions are possible:

[0061] In a high-finesse cavity with an undepleted pump, the gain may be a simple integral over^^^^Δ^^(^^′), which can be efficiently evaluated with an oscillatory Riemann sum. Attorney Reference No.390351-995471

[0062] In a low-finesse cavity with an undepleted pump, the pump may be approximately constant (upto a phase), and the signal-idler propagation may be governed by linear ODEs, which may be efficiently evaluated when the poling period is piecewise-constant.

[0063] In a high-finesse cavity with a depleted pump, the signal may be approximately constant (up toa phase), and the pump-idler propagation is governed by linear ODEs, which may be easy to evaluate by cascading ^^^^(2) matrices. High-Finesse, Undepleted Pump

[0064] Here, the SHG / DFG terms do not affect the pump and signal, so these remain constant to firstorder: ^^(^^) = ^^in ^^ ^^^^^^(^^), ^^(^^) = ^^in ^^ ^^^^^^(^^), where ^^^^ = ∫ ^^^^ d^^, ^^^^ = ∫ ^^^^ d^^. The signal equationmay be then integrated to give: ^^(^^) = ^^^^^^ ′i∗^^^^n^^in^^^^(^^) ∫(^^ ) ′0 d^^ (147)where Δ^^ = ^^^^ − ^^^^ − ^^^^ − the signal gain is equal to the number of idler photons generated, which is: ^^ =^out^^out^^ 2 |^ |2 | |2 2 1^^Δ2− 1 =2= |^^^^in^^||∫   ^^^^(^^)d^^|(148) ^^

[0065] This can be toa perturbation to ^^(^^) when integrating Eq. (146), giving: ^^ ^^ ^^ = ^^ ^^^^^^(^^) 1 + |^^ |2^^(Δ^^(^^)−Δ^^(^^′)) ′out [ ^^in ∫    ∫   ^^d^^ d^^] ^^in (149)

[0066] The gain case is the real part of the double integral. Taking the real part simplifies the integral, yielding Eq. (148).

[0067] The integrand in Eq. (148) may be rapidly oscillatory, so naively integrating it may require avery fine grid. However, for most phase-matching profiles, the QPM period and other waveguideparameters change gradually, so Δ^^(^^) = Δ^^′(^^) is a slowly-varying function even though ^^^^Δ^^(^^) mayhave rapid oscillations. Therefore, Eq. (148) may be numerically integrated by as a piecewise-constant function, so while Δ^^(^^) may be piecewise linear. The integral becomes a Riemann- like sum: 1 ^^ ^^−1 ^^^^+Δ^^ / 2 ^^Δ^^1+1^^Δ^^1^^Δ^^ ^^^^ ^^d^^ Δ^^. accurate convergence even when the phase term ^^^^Δ^^(^^)is rapidly oscillatory. Attorney Reference No.390351-995471

[0068] For an unchirped grating, the exact gain may be achieved using only a single element in Eq.(150)(^^ = 1); in this case, the gain spectrum may take the familiar form ^^ = |^^^^in ^^|2sinc2(Δ^^^^ / 2)that was derived previously (e.g., Eq. (8) with the limit |^^in | → 0 ).

[0069] In this case, only the pump may remain constant: ^^(^^) = ^^in^^^^^^^^(^^). Substituting into Eqs. (146), the equations for ^^(^^) and ^^(^^) may become: ^̇^ = ^^^^^^^^ + ^^^^in^^ ^^(^^^^−^^qpm)^^∗, ^̇^ = ^^^^ ^^^^ in^^∗(151) Making the ,Eqs. (151) may be oscillatory d ^̃^−^^Δ^^ / 2 ^^^^[ d^^ ^̃^∗] = [ in^̃^ ^^^^i∗n ^^Δ^^ / 2] [^̃^∗] (153)

[0070] As before, Δ^^(^^) may as piecewise constant. Each may a × ^^^^,with phase-adjustment terms ^^(^^), ^^(0) at the end to convert between (^^, ^^) and (^̃^, ^̃^) :0 ^^ [out^^∗out ] = ^^(^^) ( ∏    ^^ −1^^in^^) ^^ (0) [^^∗in ] (154)^^=^^−1 where the phase-adjustment matrix is ^^(^^) = [^^^^(^^^^(^^)+Δ^^(^^) / 2)( 0] (155)^^−^^ ^^^^(^^)+Δ^^(^^) / 2)and the form of each ^^ ^^^^inand Δ^^ / 2 : Δ^^ Δ^^sin (^^ ) ^^^^ Δ ( ) ìécos (^^^^) − ^^ ^^ ^^ in^^sin ^^^^ù

[0071] Considering the high-finesse case, where the signal stays constant: ^^(^^) = ^^in^^^^^^^^(^^), and induces mixing between the pump and idler fields: ^̇^ = −^^^^ ∗ −^^(^^^^^^ + ^^^^in ^^ ^^+^^qpm)^^, ^̇^ = −^^^^ ^^(^^^^^^ − ^^^^in^^ ^^+^^qpm)^^ (157)

[0072] The ^^(^^) = ^^^^(^^^^(^^)+Δ^^(^^) / 2)^̃^(^^), ^^(^^) = ^^^^(^^^^(^^)−Δ^^(^^) / 2)^̃^(^^) (158)to convert Eqs. Attorney Reference No.390351-995471 d−^^Δ^^ / 2 ^^^^∗ ind^^[^̃^] = [ ] [^̃^^̃^ −^^^^in^^Δ^^ / 2] (159)^̃^

[0073] As before, Δ^^(^^) may bemay be modeled as a cascade of 0 ^^out^^ [ ^^] = ^^(^^) ( ∏    ^^^^) ^^−1(0) [ inin ] (160)where each ^^ may (161)0 ^^Linearly Chirped

[0074] Considering the case of a linearly chirped grating with length ^^ and chirp rate ^^′, centered aboutpoling period Λ , so that ^^ (^^) = 2^^ / Λ + ′0 qpm 0 ^^ (^^ − ^^ / 2). The phase mismatch is then Δ^^(^^) = ^^ +^^′(^^ − ^^ / 2), where ^^ is poling period Λ0. This embodiment startswith a calculation of the gain spectrum and bandwidth of a chirped grating. In the simplest case (undepleted, high-finesse), an analytic expression may be derived using the steepest-descent and stationary-phase methods, which limits to a constant gain spectrum in the case of an infinite chirped grating. The other cases (lowfinesse or pump depleted) may also yield analytic results in the limit of an infinitely long crystal, as their dynamics can be mapped to the Landau-Zener problem. The pump- depleted case may give analytic formulas for the pump vs. efficiency relation, which differ from the standard OPO.

[0075] Once properties of CW gain have been understood, a study of the modulation instability withrespect to perturbation modes may be performed. The insights derived here will help inform the design of chirped-gain ECPOs in order to avoid the pitfalls of mode-hop instability. Gain Bandwidth

[0076] As discussed before, there are three regimes in which the gain may be analyzed: high-finesse / undepleted, low-finesse / undepleted, and high-finesse / depleted.

[0077] (I) High-Finesse, Undepleted Pump. In this regime, the gain is related to the poling pattern by aFourier integral, Eq. (148), which can be expressed as Γ(^^) = ^^ / |^^^^in^^|2. Up to a phase, the relationship is: Γ(^^) =1 ^^ / 2 ^^(^^^^+^^′^^2 / 2)^^∫   ^^d^^ (163)−^^ / 2 Attorney Reference No.390351-995471

[0078] Integrals of the form ∫^^ ^^ ^^^^^^(^^)d^^, where ^^^^^^(^^)is rapidly oscillating, can be solved by the methodof stationary phase. Classically, the expectation is that the grating to amplify signals in the range ^^ ∈[−^^′^^ / 2, +^^′^^ / 2]. Defining ^^ = 2^^ / ^^′ = ^^ / ^^ ′ 2max and ^^ = ^^ ^^ / 4 = ^^max^^ / 2 The result may be:^^sin (^^) = √^^ ^^^2 / 2^^(1 − |^^|)1^^^) − ^^cos (^^^^)Γ(^^^^^^− ^^1 2(164) 2^^^^− ^^

[0079] The amountlarger than the enhancement factor. As Eq. (164 ) shows, the gain is the sum of two terms: the first is due to thestationary point at ^^ ′0 = ^^ / ^^ = (^^ / 2)^^, which may yield a constant, flat gain for all frequencies insidethe gain window (for frequencies |^^| > 1 outside the gain window, the stationary point is outside theintegration range of Eq. (163) , so does not contribute. The second is due to the endpoints, an artifact of the instantaneous turn-on of the nonlinear interaction during an entry into / exit from the crystal. The interference between these terms may create a spectrum that, although flat on average, may be noisy gain with sharp oscillations near the edges.

[0080] The stationary-point and endpoint contributions to Eq. (164) may enter with different powers of^^. In the limit ^^ → ∞ (large chirp bandwidth enhancement), the former term is dominant. Neglectingthe endpoint contribution may cause to arrive at a constant |Γ(^^)| ≈ √^^ / 2^^, which may give anundepleted gain of: ^^|^^^^i^^|2 2^^|^^^^ |2 ^^0≈n=in(165) ^^′

[0081] Because ^^0 ∝ 1 / ^^ ∝ 1 / ^^ bandwidth product ^^0^^maxrelatively independent of ^^.

[0082] (II) Low-Finesse, Undepleted Pump.-The calculations may be much more involved in the low-finesse, high-gain case. Here, the linearized equations Eq. (153) may resemble those of a Landau-Zener avoided crossing, but with a non-Hermitian coupling term that leads to exponential gain near the zero- phase-mismatch point. An approximate power gain (ignoring any ripples due to edge effects) may be given by the Rosenbluth formula: ^^ = ^^2^^|^^^^in|2 / |^^′|(166)

[0083] This formula may be exact for an infinitely long, linearly-chirped grating. That is, Eq. (153) maybe related to to the Landau-Zener problem and an analytic continuation may be performed. Landau, Zener, Stückelberg, and Majorana showed that the transition matrix for the avoided crossing with a linear transition rate +∞ ^^ ^^ −^^ −^^^^ Ω Attorney Reference No.390351-995471 has an analytic form. Specifically, if a particle is initialized to the |0^ state, then its amplitude in this state after the transition may be ^0|^^(+∞, −∞)|0^ = ^^−^^Ω2 / 2|^^|(168) where ^^^^ may be the time-varying parameter initiating the avoided crossing and Ω may be the splitting.Eq. (168) may often be reported in terms of the transition probability ^^ = ^^−^^Ω2 / |^^|, i.e. the probabilitythat the state diabatically transitions from the ground state to the excited state; where for ^^ → 0 thetransition is slow, ^^ → 0.

[0084] Comparing Eq. (153) to Eq. (168), and taking a linear QPM chirp Δ^^(^^) = ^^′^^ as before, thecorrespondences ^^ → ^^′ / 2 and Ω → −^^|^^^^in | may be made (considering without loss of generality^^in ∈ ℝ ). The latter correspondence may break the Hermiticity of the model, but Eq. (168) may be exactand analytic, so it can be continued to complex-valued Ω. Making these substitutions, the transitionprobability formula ^^ = −^^Ω2 / |^^| may yield Eq. (166).

[0085] (III) High-Finesse, Depleted Pump. Here, a gain calculation may be performed in the presenceof pump depletion. This case can also be analyzed by analogy to Landau-Zener - and the analogy is closer, because the linearization gives signal-mediated DFG (a Hermitian process that conserves photon number) rather than pump-mediated OPA (a non-Hermitian process with gain). Starting with Eqs. (169), reported below, ^̇^ = −^^^^ ∗ −^^(^^^^^^ + ^^^^in ^^ ^^+^^qpm)^^, ^̇^ = −^^^^^^^^ − ^^^^in^^ ^^(^^^^+^^qpm)^^ (169) Zener form. Now performing the final transformation ^̂^ = ^^^^Δ^^(^^) / 2^̃^, ^̂^ = ^^−^^Δ^^(^^) / 2^̃^, which may leadback to the form of Eq. (159): d [^^ ^^^^^̃^] = [−^^^^′^^ / 2 ^^^^|^^in | ^^^^^^^̃^ ^^^^ ^^ ^^^^ ] [ ] (171)where a substitution ^^(^^) = and the crystal assumed to extend to ^^ = ±∞. Eq. (171) now has a Landau-Zener form with ^^ =^^′ / 2, Ω = ^^|^^in |, which may give a conversion efficiency that approaches 100% exponentially as thesignal power is increased: | 2 ^^ −^^out| −^^−2^^|^^^^in |2 ′ Attorney Reference No.390351-995471

[0086] The gain ^^ = |^^out |2 / |^^in|2may also be calculated. It may be helpful to relate this to theundepleted gain ^^0 from Eq. (165) and the pump relative to threshold, ^^ = |^^in / ^^th |, where the thresholdmay be given by ^^th = √|^^′|^^ / 2^^^^2 (so ^^ 20 = ^^ ^^ due to the ^^0 ∝ |^^in |2dependence): |^^in|2−2^^|^^^^ |2 ′−2 | |2 / |^^′| 2 1 − ^^ in / |^^ | ^^ = 1 − ^^^^ ^^^^in2 ( ) = ^^ ^^2 ′(173) |^^in| 2^^|^^^^in| / |^^ |

[0087] Setting ^^ = arelation between the −log (1 − ^^)^^2= (174) ^^

[0088] This function may be monotonic (unlike in the standard OPO, where the efficiency peaks at ^^ =^^ / 2 and drops at higher pump powers) and may be approximated by an exponential: ^^ ≈ 1 − ^^−4(^^−1) (175)

[0089] It may prove useful to obtain the ^̂^ ^̂^(^^) during the transition (rather than in thelimit ^^ → ±∞ ), in order to calculate the gain that leads to mode hopping. Zenerobtained an exact solution based on Weber functions, but this approach is very involved, and calculating the Weber functions on the complex plane may be numerically slow. Alternatively, one can get a good approximate solution using a Markov approximation, and despite being approximate, the result producesthe exact transition probabilities at ^^ → ∞. However, when an exact solution is desired, method ofnumerical integration may be used as follows.

[0090] The first step is to change the integration variable to ^^ = tan−1 ((^^′^^ / 2) / (^^|^^in |)), giving:d [^^ ^^^^^̃^] = ^^2|^^^^in|2 sec3 −sin (^^) ^′(^^) [ (176)^̃ ^^ ^̃^

[0091] The Eq.(176) may be written in the form (d / d^^)^⃗^ (^^) = −^^^^(^^)^⃗^ (^^) and transformed into a basis of eigenstatesof ^^ (often called dressed states), as follows: ⃗^^ (^^) =1 [1 1 ] [ cos (^^ / 2) sin (^^ / 2)]^⃗^(^^) (177)

[0092] This gives: d^⃗^ 2|^^ |2 =− † † ′^^in 3 1 010 1d^^^^^^ ^^ ^^ − ^^ ^^ ^⃗^ = ^^ − ^⃗^ but may not be stable near the endpoints ^^ = ±^^ / 2 due to the large sec3 (^^) coefficient. It was foundthat postulating a piecewise-constant ^^^^(^^), and computing the matrix exponent ^^−^^^^^^(^^)Δ^^for each step, is an accurate and stable integration method. It can be shown that the pump power |^^|2and coherentamplitude (Re[^̂^], Im[^̂^]) as the field passes may through crossings with the adiabaticity parameter: Attorney Reference No.390351-995471 |^^ |2 ^^ ≡^^in′ (179) |^^| It can further be shown that the adiabaticitywith efficiency ^^ = 0.09 ) and ^^ = 0.20(^^ .by sharper oscillations in the pump field (both in power and phase), although the phase always asymptotes to zero at ^^ → ∞.Modulation Instability

[94] In the pump-depleted regime, pump depletion may reduce the signal gain from ^^0 > ^^ to ^^, sothat gain is balanced by loss in steady state. However, as explained above, this solution may be susceptible to modulation instability (MI), where perturbation signals with ^^MI > ^^ grow from vacuumand disrupt the steady-state field.

[95] To analyze modulation instability, Eqn. (144) may be converted into an interaction frame(^^ ^^^^ (^^^^(^^) = ^̂^^^(^^)^^ ^^,^^ ), ^^ ^^^^^^(^^) = ^̂^^^(^^)^^ ^^,^^(^^), ^^^^(^^) = ^̂^^^(^^)^^^^^^^^,^^(^^)) the signal, idler, and pump of the current oscillating CW field. As discussed above, Eqs. (180) may be perturbed to derive linear ODEs for the perturbation modes ^^^̂^^^, ^^^̂^^^, ^^^̂^^^, and to assumethat ^^^^^^ ≈ 0 because of the large phase mismatch for detuned pump fields (a function of the largepump-signal group-velocity mismatch). The MI equations may become: (d / d^^)^^^̂^^^ = ^^^̂^0^^^^Δ^^^^,−^^^̂^∗^^ , ( d / d^^)^^^̂^^^ = ^^^̂^0^^^^Δ^^^^,−^^^̂^∗^^(181)

[96] These can finesse) regime, the result simplifies because the signal field ^̂^^^(^^) ≈ ^^^^, in is approximately aconstant. The gain, which proportional to the idler field generated, is then given by the integral (compare Eq. (148)): the gain by integration. However, Eq. (182) may be casted into a more instructive form by recalling from Eqs. (180) that the pump ^̂^0 and idler ^̂^0 are related by (d / d^^)^̂^0 = ^^^^^^Δ^^0,0^̂^0^^i∗n . Substituting ^̂^0: ^^ ^̂^^^ Δ^^ −Δ^^d^̂^0

[0098] Examining the term Δ^^^^,−^^(^^) − Δ^^0,0(^^) = (^^^^,0(^^) − ^^qpm(^^) − ^^^^,^^^^ − ^^^^,−^^^^) − (^^^^,0(^^) − ^^qpm(^^) − ^^^^,0^^ − ^^^^,0^^) Attorney Reference No.390351-995471where ^^^^ = Δ^^^^ − Δ^^0 may be the difference in phase mismatch between the perturbation mode andthe oscillating mode. Thus, Eq. (183_ becomes the Fourier transform of d^̂^0 / d^^ : ^^ ^̂^^^, out = ∫    ^^^^^^^^^^( d^̂^0 / d^^)d^^ (185)

[0099] This may work for ^^ =Theorem of Calculus (assuming =.2 ^^ 2 ^^    ^^^^^^^^^ =    ^^^ |∫( d^̂^0^^ ( d^̂^0 / d^^)d^^|^ ^^^^^^^0 / d^^) = ^^ ×^^2(186)where ^^MI is theinstability can ,nothappen in an unchirped when oscillating in the phase-matched configuration: in this case, ^̂^(^^) grows monotonically and its phase stays constant, meaning that the numerator integral is smaller than the denominator. However, this may change when chirped poling is added, because pump amplification (and idler generation) may be highly oscillatory.

[0100] It was found numerically that ^^MI is a step function, limiting to the undepleted gain ^^pre 2^^^^(Eq. (165)) for ^^′^^ < 0 while the gain for ^^′^^ > 0 sees a depleted pump (|^^|2 = |^^in |2(1 − ^^) atsteady-state, implying ^^post = 2^^^^^^−2^^^^) : 2^ ′^^MI = { ^^^ ^^ ^^ < 02^^^^^^−2^^^^ ^^′^^ >(187)

[0101] Eq. (187) may hold in the gain long before (or after) pump depletion has occurred. However, the transition appears to be sharp, and in numerical investigations, it appears indistinguishable from a step function. This behaviour islikely due to the highly oscillatory form of ^̂^(^^) in the limits ^^ → ±∞. Rapidly oscillating tails can givea function a discontinuous Fourier transform, even if the function itself is smooth (such as the sinc function). Inverse Design of Optimal Chirp Profiles

[0102] Having developed the general theory of chirped QPM structures and found an exact solution tothe ideal case of an infinitely long grating with linear chirp, now it is time to find the optimal chirp profile for a finite-length grating, numerically. A linearly-chirped crystal gives a gain profile that is only flat on average, with large oscillations arising from the instantaneous turn-on (and turn-off) of the nonlinearity. This embodiment describes two techniques that can be used to obtain flatter gain spectra: (1) apodization near the crystal edges, and (2) inverse design using a gradient-based numerical optimizer. Attorney Reference No.390351-995471

[0103] (1) Apodization. When designing a flat-spectrum chirped grating, the effort may be to optimizetwo competing figures of merit: (a) the flatness of the spectrum ^^(^^) within some region |^^| < ^^max,and (b) the gain within this region. The gain in the high-finesse, undepleted regime takes the form ^^ =|^^^^in ^^|2^̃^(^^), where: 1 2 ^̃^ ^^ / (^^max^^) |^^| < ^^maxtarget (^^) = { 0 |^^| > ^^max(189)

[0105] This spectrum may This is a common problem is to apodize the filter, e.g., by convolving it with a kernel function to smooth out the sharp edges. Once there is a smoothly-varying ^̃^targetone can find an approximation to the corresponding chirp profile analytically using the stationary-phase method. Expanding the exponential in Eq. (188) about the stationary-phase point ^^0(^^): Γ≈1 ^^^^(^^^^0+Δ^^(^^0)+Δ^^′′(^^0) / 2)^^^^(^^^^0+Δ^^(^^0))2^^^^ = ^̃^target , at the point where ^^ = −Δ^^(^^0). Substituting ^^, we rewrite this as a differential equation for Δ^^: Δ^^′(^^) = ±2^^ targetΔ^^(^^))^^2(191) ^̃^ (−

[0106] This ODE may be exactly integrable. If there is a definition of ^^(^^) = (^^ / 2^^) ∫^^ −∞ ^̃^target(^^)d^^,normalized to ensure ^^(+∞) = 1. Eq. (191) integrates to the following form:^^(−Δ^^) = ±^^ ^^+ const (192)and because ^^(^^) has a range of two solutions: Δ^^ = ^^−1(^^ / ^^) and Δ^^ =^^−1(1 − ^^ / ^^). These correspond to the two chirp directions (red-blue or blue-red), which yield identicalgain spectra. (Note that flipping the sign of Δ^^′(^^) in the middle of the crystal is not allowed; this may technically satisfy Eq. 190) but breaks the smoothness assumptions that underlie the stationary-phase method). Here are a few examples:

[0107] A Lorentzian gain spectrum ^̃^target (^^) = (2 / ^^0^^) / (1 + (^^ / ^^0)2) is implemented with a tangentchirp profile ^^(^^) = ^^0tan (^^(^^ + 1 / 2)). Attorney Reference No.390351-995471

[0108] The gain spectrum ^̃^target (^^) ∼ tanh ((^^ + ^^max) / ^^0) − tanh ((^^ − ^^max) / ^^0), which limitsto a Eq. (190) when ^^0 ≪ ^^max, is implemented with ^^′(^^) = (^^0 / 2)log [sinh (2^^max^^ / ^^0) / sinh (2^^max(1 − ^^) / ^^0)].

[0109] A box-shaped spectrum with Lorentzian “tails” on both ends of the box can be implemented witha chirp profile that is linear in the center of the crystal and follows a tangent function near the edges: 1− ^^^^ 2^^ ì tan [ ( ^^ 2^^(1 − ^^) − 1)] −^^ 22^^ / ^^ − 1 < −^^Here, ^^ is the for linear . a guess at the chirp profile, Eq. (193) is reasonably accurate.

[0110] (2) Gradient-Based Inverse Design. Even with an apodized chirp, the gain spectrum may stillhave significant oscillations (although they are greatly reduced from the simple linear chirp). However, because the result is qualitatively similar to the target spectrum, it can be used as an initial guess for a numerical gradient-based optimizer. Having gradients may allow quick convergence, but numerically differentiating Eq. (148) is trivial in principle, and can be done easily in software (e.g., with 10-20 lines of Python code) and an automatic differentiation package (e.g., autograd). However, highly chirped QPM waveguides will lead to rapidly oscillating integrands, which may require a very fine grid to compute ^^(^^) accurately. And many variables take a lot of iterations, even if gradients are provided. Because of this, some C code was written to calculate the spectrum and its derivative (using the more accurate modified Riemann sum, Eq. (150)) and speeds up the algorithm with OpenMP parallelism and some use of trigonometric identities to reduce the number of trig-function calls from ^^(^^^^^^^^) to^^(^^^^) + ^^(^^^^).

[0111] It was found that the following cost function performs satisfactorily at obtaining chirp profiles:^^^^^ ∣ ^̂^^2 ^^ ^^ = −^̂^− ^^^^^ ^^^ ^^^ ≡ ^^

[0112] window of relevance ^^ ∈ [^^, ^^], while the second term favors solutions with higher overall gain. Therelative strength of these terms is set by the hyperparameter ^^. With a reasonably good initial condition and cost function Eq. (194), it was found that the L-BFGS-B solver can usually find optimal chirp profiles quickly (e.g., in under a second).

[0113] While comparing the numerically-optimized solutions to the ones obtained from analyticheuristics within a window of interest, one can accurately imitate a wide range of gain profiles, including Attorney Reference No.390351-995471 flat-top spectra, linearly ramped spectra, and double-peak spectra. Often, this programmability may come at a cost of small wiggles in the gain spectrum away from the region of interest, but this may not necessarily matter for the embodiments disclosed herein because the gain of those modes may be toolow to be relevant. Furthermore, a comparison of the chirp profile Δ^^(^^) and integrated phase Δ^^(^^) =∫^^ 0 Δ^^(^^′)d^^′of the numerical and analytic (linear / apodized) solutions was performed. While the numerical optimum may largely follow the apodized QPM profile, one can also see some phase wiggles. These phase wiggles, whose form may be derived using numerical optimization, may compensate for the wiggles in ^^(^^), leading to well-matched gain spectra.

[0114] Numerical optimization is a robust way to find chirp profiles for a wide range of spectra. Whenthe numerically optimized chirp profiles for many flat-top gain bandwidths is compared, one can observe a linear tradeoff between the maximum gain and the FWHM, with most solutions following the curve: 5.2 ^^max≈2Δ^^|^^^^in^^(195) FWHM^^|

[0115] The chirp profile, while Some of the QPM structures with narrower seem shift).

[0116] Additionally, there may be an additional degree of freedom: the poling duty cycle. Controllingthe duty cycle lets us control the amplitude in Eq. (188), not just the phase: ^̃^(^^) = |1 2 ^^∫ ^^^^^^^^^^(^^)d^^| , |^^(^^)| ≤ 1 (196)

[0117] Compared to the over gain profile Eq. (196) almost always leads to a solution where |^^(^^)| = 1, i.e., the optimal solutionmay always force the amplitude to its maximum. Suppose that one wants to optimize an overlap integral of the following form: ^^′[^^, ^̂^] = − ∫ √^^(^^)^̂^(^^)d^^ (197)

[0118] This cost function may not necessarily be used for optimization because Eq. (194) may do muchbetter; however, Eq. (197) may produce the right general behavior: when constrained by thenormalization of ^^(^^), ^^′ is minimized when ^^(^^) ∝ ^̂^(^^). Make the substitution^̃^(^^) = |Γ(^^)| = m^a^x  Re[^^^^^^Γ(^^)] =^^m^ ≤1 Re[^^(^^)∗Γ(^^)] (198)minimization of m  Re ^^∗ ^^ ^^^^^^^^ d^^

[0119] If (^̂^, ^̂^) is the problem where ^^ → ^̂^is held constant, i.e., m ^a^x {Re ∬ ^̂^∗(^^)^^(^^)^^^^^^^^ d^^ d^^}. This is a linear problem with convexconstraints, so the maximum is found on the boundary |^^(^^)| = 1. Attorney Reference No.390351-995471

[0120] Nevertheless, it was empirically found that maxima almost always occur when |^^(^^)| = 1, withexceptions usually arising when the solver has not run enough iterations to find the ground state. EO Comb OPO with a Chirped Grating

[0121] Chirped QPM gratings may improve the bandwidth of the EO Comb OPO. For example,significant bandwidth enhancements are possible, especially for NLO processes that are not ideally dispersion-engineered.

[0122] Embodiments disclosed herein describe three case studies that highlight the potential advantagesof a chirped EO Comb OPO, ordered by increasing comb bandwidth:

[0123] 1. Achieving a broadband comb in presence of strong signal-idler group-velocity mismatch. Fora 1550 nm signal and 1064 nm pump in bulk PPLN, we increase the comb bandwidth by 5 × from 20nm to 100 nm. Thus, chirp can be a substitute for dispersion engineering when the latter is not feasible. Using this model system, one can also study the mod-hop instability in the chirped ECPO.

[0124] 2. Enhancing the bandwidth in the dispersion-engineered case (group-velocity matching). For a1300 nm signal, pumped at the GVM point (approx.890 nm pump, 2800 nm idler), the comb bandwidthmay be increased by 2 × from 170 nm to 350 nm , showing that chirp can work in tandem with dispersionengineering to give performance that may not necessarily be achievable with either technique alone.

[0125] 3. Achieving an octave-spanning EO Comb OPO, with significant power at the fundamentalharmonic in order to facilitate efficient ^^ − 2^^ locking.Fixing Dispersion: 1550 nm Comb with 1064 nm pump in Bulk PPLN

[0126] One application of the ECPO is for generating telecom light as a substitute for a WDM laserbank. Because comb-line power is an important concern here, it is desired to pump the OPO at a wavelength like 1064 nm where high-power lasers are readily available. However, at this pump wavelength, the signal and idler (3393 nm) have a significant group-velocity mismatch Δ^^1=0.95ps / cm. The 3 − dB bandwidth formula BWgain = 5.6 / |Δ^^1^^| (see Eq. (114) ), Δ^^max^^ = 2.8 for 3 − dB) gives a gain bandwidth of 7 nm. One can get slightly more than this in thedue to the chirping effect of the EO modulator, which redistributes power from the center frequencies (which experience net gain) to the tails (which experience net loss). For example, a large round-trip GDD of 100,000fs2may be used to keep the comb confined to the narrow gain windowwhen a moderate PM phase ^^^^ = 3.2 is applied (reducing the GDD does not increase the combbandwidth, but leads to mod-hop instability effects). The largest bandwidth I could get is around 20 nm, about 3 × the 3 − dB gain window.

[0127] While a 20 nm comb covers about 60% of the C-band, or around 50 DWDM channels on theITU grid with 50 − GHz spacing, even more bandwidth may be desired in the future. To that end, aswapping may be performed in in a chirped gain crystal, whose QPM profile is inverse-designed to give Attorney Reference No.390351-995471a flat gain spectrum in the 100 nm window around 1.55^^ m. Keeping the EOM drive the same, one canreduce the cavity GDD by 50 × to 2,000fs2, yielding a stable 100 nm comb, a 5 × increase in bandwidthover the unchirped case.

[0128] The chirped crystal may need to be pumped harder (roughly 8.8 × ) to achieve the 5 × bandwidthincrease. It is always possible to increase the phase-matching bandwidth by using a shorter crystal and pumping harder. However, an optimized comb spectrum for an equivalent unchirped OPA segment may be obtained using the same pump power as in the chirped-case. With the higher pump power, one can use a shorter crystal ( 3.4 mm vs.1 cm ), which increases the comb bandwidth to about 40 nm. However,even on this equal-power basis, the chirped crystal may still yield a comb with 2.5 × broader bandwidth.Mode-Hop Instability in Chirped EO Comb OPOs

[0129] At first glance, it may appear that chirped QPM offers a clean solution to the mode-hopinstability: by flattening the gain spectrum, one might expect to suppress mode-hopping, because all frequencies within the comb bandwidth experience equal gain. The truth is more subtle than this, owing to the dynamics of chirped OPA, where the chirp may imposes a "unidirectional" pump-depletionphenomenon. In one case, a crystal is roughly linearly chirped from Λ = 31.10^^ m (phase-matched to1.6^^ m signal) to 29.95^^ m (phase-matched to 1.5^^ m ), which means that longer wavelengths areamplified before shorter wavelengths. The oscillations in the intermediate gain profile ^^(^^, ^^) areevidence of the Landau-Zener nature of this phenomenon, as discussed in the embodiments above.

[0130] The OPA may be turned into an OPO, introducing a steady-state oscillating CW field (round-trip loss ^^ = 0.2 ) and use the MI equations in Eqn. (181) to calculate the gain of the perturbation fields.The oscillating field may suppress gain for shorter wavelengths ^^MI < ^^osc , but may not have an effectwhen ^^MI > ^^osc , and the gain dynamics may indicate why: the longer wavelengths have already beenamplified before the pump is depleted, because their gain region (in this crystal) is upstream of the oscillating field. Because pump depletion is unidirectional, all these oscillations may be unstable to MI, and despite its broad, flat gain spectrum, the CW chirped OPO may stably oscillate at its longestwavelength, around 1.6^^ m. (If the crystal is chirped in the opposite direction, all these results may bereversed, but the conclusion is qualitatively the same).

[0131] Unlike a CW OPO, the ECPO truncates the MI gain to the finite time in which perturbationsreside in the gain region which allows us to suppress mode-hopping. The perturbations to the a portion of the oscillating field experience net gain - but are then swept away by the phase-space dynamics. Boththe frequency ^^rep and the cavity dispersion (represented by DCF = GDDOPA+EOM / GDD, which is theratio of OPA+EOM GDD to total cavity GDD) affect the dwell time in the MI gain region, and thus contribute to mode-hop instability. This may illustrates the tradeoff described above, where attempts to increase the comb bandwidth or number of comb lines may bring the system to the brink of instability. Attorney Reference No.390351-995471 Increasing the EOM phase, by contrast, may increase the bandwidth without instability, but EOM phase is related to RF power, which may be a finite resource.

[0132] Considering MI figure of merit ^^MI = Δ^^Δ^^^^ / 4^^^^^^, ^^MI ≳ 20 may generally imply instability.For example , instability tends to set in around Δ^^ = 100^^ m, ^^rep = 20GHz, ^^^^ = 3.2, with an averageMI gain of Δ^^ = ^^MI ≈ 0.07 therefore giving ^^ ≈ 7. This is slightly lower than the value in theembodiments described above, but the MI gain profile is also different, which may contribute to the lower threshold for instability.

[0133] Another example strategy to mitigate mode-hop instability without sacrificing bandwidth is topump closer to threshold, which may lead to an efficiency tradeoff. Because the worst-case MI gain is equal to the OPO's undepleted gain, one can expect that working near threshold should make any ECPOmore stable against mode-hopping. To see this mathematically, ^^ ∼ ^^2^^ in the high-finesse limit (Eq.68) ), which implies Δ^^ ≈ 2^^(^^ − 1). As to the efficiency, ^^ ≈ ^^SE(^^ − 1) where ^^SE = d^^ / d^^ is theslope efficiency Substituting these into the expression for MI gain: Δ^^^^^^^^ ^^ ^^^^ ^^ = =combMI(200)

[0134] This may set an upper limit ^^max ^^comb<SE(201) ^^^^

[0135] For instance, taking ^^max = 7, we get ^^comb ≲ 700. At ^^rep = 25GHz, this implies a bandwidth limit of 17.5 THz or 130 nm.

[0136] One from Eq. (201) is that the maximum efficiency scales inversely with the numberof comb lines (and vice versa). This is the tradeoff as the conventional resonant EO comb, where^^comb < 4log (2) / ^^. The prefactors, however, are different, and the tradeoff is "better" in the EO CombOPO by the factor: (^^comb^^)EO+OPO^^ =max^^^^^^SE(202)

[0137] For the ECPO, the bandwidth-efficiency tradeoff is not fundamental, as it depends on variousengineering parameters (loss, EOM phase), which can be tuned. Considering the values of ^^^^, ^^SE, ^^max,and ^^ used above, Eq. (202) gives a factor of 50 ×.Enhanced Bandwidth of 1300 nm Comb at GVM Point

[0138] As described in the above embodiments, a chirp may compensate for a lack of dispersionengineering by broadening the gain spectrum. But chirp can also supplement dispersion engineering: if one starts with a dispersion-engineered OPA crystal, adding chirp can still increase the gain bandwidth beyond what the dispersion-engineered ECPO could achieve alone. Attorney Reference No.390351-995471

[0139] To see this, the case of an EOPO targeting a 1.3^^ m signal wavelength may be considered. Tomaximize the gain bandwidth, one can operate at the group-velocity matching point (approx. 890 nm pump, 2800 nm idler). A perfectly phase-matched 1-cm PPLN crystal may have a gain bandwidth of around 130 nm , while applying a slight phase mismatch can expand this window by a factor of√2 toabout 170 nm . This may be significantly narrower than the gain bandwidth possible for a 1.55^^ mcomb, due to the latter being closer to degeneracy and the scaling law BWgain ∝ |^^^^ − ^^^^|−1(Eq.114). For example, a 340 − nm stable comb at 1.55^^ m, which could probably be made even broader ifnot for loopback effects.

[0140] When comparing two unchirped 1.3^^ m ECPOs with a chirped design, the following differencescan be seen. Without chirp, one can have only a single free design parameter, the phase mismatch, withthe optimum being around Δ^^0^^ = −2.5, which may yield a double-peaked gain profile. As describedin the embodiments above, an inverse-designed apodized chirp may give a flat and broad gain spectrum.This allows for stable combs of up to 350 nm bandwidth, or 2 × the value without chirp.

[0141] As before, controlling mode-hop instability may be desired for getting broad combs. For thechirped cases, phase mismatch Δ^^(^^) may be locally quadratic in ^^ due to the group-velocity matching. Displacing the GVM point slightly from the comb center may provide the most stable combs. As before, the stability is also a function of pump, exhibiting a tradeoff between efficiency and stability. Octave-Spanning Comb

[0142] In some embodiments, OPA chirp may be used to obtain a stable, octave-spanning comb withsufficient power to achieve ^^ − 2^^ locking. Such a comb may function as an optical-to-RF frequencydivider, a potentially key component in optical clocks and other applications. One can design a chirpedOPA that supports a flat gain spectrum all the way from 1.3^^ m to the degeneracy point. With the pumpat 0.918^^ m, this may give a combined signal+idler gain region all the way out to 3.1^^ m, over anoctave. Next, with appropriate cavity dispersion compensation, it may be possible to design stable combswhose signal spectra fill this gain window (to degeneracy, not to 3^^ m ). For example, cavity dispersionmay be set by 2 cm of LiNbO3, plus some amount of GVD and TOD compensation, but no compensationfor higher dispersion orders. For ^^^^ = 6 radians, this may leads to a separatrix that stretches all the wayfrom 1.25^^ m to a maximum excursion of 2.1^^ m, with the loopback occurring at around 1.85^^ m,right at degeneracy. Stable comb formation with a 600 nm spectrum, may be observed in this regime.

[0143] In this embodiment, the signal and idler combs may collectively span an octave. Moreover, for^^^^ > 5, these combs overlap at the degeneracy point. The resulting spectrum is a dual-comb consistingof two equally-spaced EO combs with unknown carrier offsets ^^^^and ^^^^. To find these frequencies, theoutput may be sent through an SHG crystal (phase-matched to frequency-double 3^^^^ → (3 / 2)^^^^,roughly 2.76^^ m → 1.38^^ m ) followed by a dichroic. Attorney Reference No.390351-995471

[0144] Spectral band A, centered at the SHG second harmonic (3 / 2)^^^^, gives the beatnote between thesignal and the frequency-doubled idler, ^^^^ − 2^^^^. This signal is proportional to the product of the signaland (squared) idler fields ^^b(^^FH)^^^^(^^SH)1 / 2, where the most critical factor is the idler power at the FH, which gets frequency doubled. This power may not be at the tail of the pulse or even at a dispersive- wave peak, as in the soliton case. Because the FH idler power is large, achieving a strong SHG self- referencing signal is easier.

[0145] Spectral band B, centered at the degeneracy point, may give the beatnote between the signal andidler, without frequency doubling, ^^^^ − ^^^^. Only the comb tails pass through the degeneracy point, butthe power is still considerable, and because frequency-double is not needed, this should be a fairly strong signal, and easy to detect.

[0146] Combining these signals (either in analog with mixers or digitally after sampling), one canseparately find the reference frequencies ^^^^and ^^^^. Furthermore, by adjusting the DC bias on the ECPO's phase modulator, one can tune ^^^^, locking either the signal or idler comb to a fixed reference. (To lock both combs simultaneously, one needs feedback to the pump laser, because the relative phase of pump, signal, and idler is locked).

[0147] There may be many other ways to perform the comb locking that are functionally equivalent.For example, one could put the SHG crystal after the filter, or one could put it in the cavity to benefit from resonant enhancement of the signal (if the signal is the longer wavelength, this would increase the SHG power by ^^−2). One could only use a single detector (assuming the detector is optically broad enough) and separate out the two beatnotes in postprocessing.

[0148] On the optics side, one could also improve this scheme by engineering an optimal loopback.Loopbacks may lead to spectra a lot like dispersive-wave emission, with a peak at the maximum- excursion point. If this peak lines up with the degeneracy point, it might lead to a larger beatnote. Similarly, if both orders of dispersion can be controlled, one can in principle create a two-sided loopback (i.e. loopbacks on both red and blue ends of the spectrum) which may further improve the bandwidth or relax other requirements of the device. Finally, in the same way that one can compress the signal output to an ultrashort pulse with dispersion compensation, the idler output may be compressed before SHG to increase the SHG beatnote, if desired. Additional examples of the presently described method and device embodiments are suggested according to the structures and techniques described herein. Other non-limiting examples may be configured to operate separately or can be combined in any permutation or combination with any one or more of the other examples provided above or throughout the present disclosure.

[0149] It will be appreciated by those skilled in the art that the present disclosure can be embodied inother specific forms without departing from the spirit or essential characteristics thereof. The presently Attorney Reference No.390351-995471 disclosed embodiments are therefore considered in all respects to be illustrative and not restricted. The scope of the disclosure is indicated by the appended claims rather than the foregoing description and all changes that come within the meaning and range and equivalence thereof are intended to be embraced therein.

[0150] It should be noted that the terms “including” and “comprising” should be interpreted as meaning“including, but not limited to”. If not already set forth explicitly in the claims, the term “a” should be interpreted as “at least one” and “the”, “said”, etc. should be interpreted as “the at least one”, “said at least one”, etc. Furthermore, it is the Applicant's intent that only claims that include the express language "means for" or "step for" be interpreted under 35 U.S.C. 112(f). Claims that do not expressly include the phrase "means for" or "step for" are not to be interpreted under 35 U.S.C.112(f).

Claims

Attorney Reference No.390351-995471 CLAIMS What is claimed is:

1. An optical frequency comb generator comprising:an optical parametric oscillator cavity configured to generate a signal and an idler from a an optical pump field; and an electro-optic modulator configured to provide an amplitude modulation and phase modulation on at least one of the optical pump field, the signal, or idler to generate a frequency comb.

2. The optical frequency comb generator of claim 1, where the optical parametric oscillator cavityis singly resonant.

3. The optical frequency comb generator of claim 1, where the optical parametric oscillator cavityis non-degenerate.

4. The optical frequency comb generator of claim 1, wherein the optical parametric oscillator cavitycomprises a first χ(2)crystal configured to provide a non-linear interaction between the optical pumped field, the signal, and the idler.

5. The optical frequency comb generator of claim 1, wherein the electro-optic modulator comprisesa second χ(2)crystal.

6. The optical frequency comb generator of claim 1, wherein the electro-optic modulator isconfigured to provide the amplitude modulation on the optical pump field and the phase modulation on the idler to generate the frequency comb.

7. The optical frequency comb generator of claim 1, wherein the electro-optic modulator comprisesa Mach-Zehnder modulator (MZM).

8. The optical frequency comb generator of claim 1, wherein the optical parametric oscillator cavityand the electro-optic modulator are fabricated on a single chip.

9. The optical frequency comb generator of claim 1, wherein at least one of the optical parametricoscillator cavity and the electro-optic modulator are formed by thin film lithium niobate.

10. The optical frequency comb generator of claim 1, wherein a phase matching condition of theoptical parametric oscillator cavity is chirped or inverse designed.Attorney Reference No.390351-99547111. The optical frequency comb generator of claim 1 configured to be tuned using radio frequency.

12. A method of generating an optical frequency comb, the method comprising:generating, by an optical parametric oscillator cavity, a signal and an idler from an optical pump field; and generating, by an electro-optic modulator, a frequency comb by providing an amplitude modulation and phase modulation on at least one of the optical pump field, the signal, or the idler.

13. The method of claim 12, where the optical parametric oscillator cavity is singly resonant.

14. The method of claim 12, where the optical parametric oscillator cavity is non-degenerate.

15. The method of claim 12, wherein the optical parametric oscillator cavity comprises a first χ(2)crystal, the method further comprising: providing, by the first χ(2)crystal, a non-linear interaction between the pumped signal, the signal, and the idler.

16. The method of claim 12, wherein the electro-optic modulator comprises a second χ(2) crystal.

17. The method of claim 12, further comprising:providing, by the electro-optic modulator, the amplitude modulation on the optical pump field and the phase modulation on the idler to generate the frequency comb.

18. The method of claim 12, wherein the electro-optic modulator comprises a Mach-Zehndermodulator (MZM).

19. The method of claim 12, wherein the optical parametric oscillator cavity and the electro-opticmodulator are fabricated on a single chip.

20. The method of claim 12, wherein at least one of the optical parametric oscillator cavity and theelectro-optic modulator are formed by thin film lithium niobate.

21. The method of claim 12, further comprising:chirping, a phase matching condition of the optical parametric amplifier.

22. The method of claim 12, further comprising:tuning the frequency comb using radio frequency.

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