Simple method for mid-infrared optical frequency comb generation with dynamic offset frequency tuning

We present a simple method for fully-stabilized mid-infrared optical frequency comb generation based on single-pass femtosecond optical parametric generation that is seeded by a continuous-wave laser. We have implemented the method in a periodically poled lithium niobate crystal that produces a frequency comb tunable across 3325 - 4000 nm (2380 - 3030 cm-1). The method generates the mid-infrared (idler) comb with known and stabilized Carrier-Envelope Offset (CEO) frequency without the need to directly detect it. The idler CEO is continuously tunable for almost half of the repetition rate and can be modulated, while maintaining its central frequency. Together with the high output power (up to 700 mW) and low intensity noise (0.018% integrated in 10 Hz - 2 MHz bandwidth) this makes the demonstrated mid-infrared frequency comb promising for many applications such as high-precision molecular spectroscopy, frequency metrology and high harmonic generation.


I. Introduction
Since their first development about 20 years ago, Optical Frequency Combs (OFC) have advanced science and technology in many different ways.By creating a phase-coherent link between optical and radio frequencies 1 , OFCs have enabled accurate measurements of optical frequencies with a simple and elegant implementation leading to a rapid progress in the development of new optical frequency standards 2 .The establishment of OFCs has only become possible thanks to the invention of novel techniques for precise measurement and control of Carrier-Envelope Offset (CEO) frequency -a key parameter for ensuring pulse-to-pulse coherence in femtosecond Mode Locked Lasers (MLL) 3,4 .Consequently, OFCs have found applications in a variety of different fields [5][6][7] including optical atomic clocks 8 , optical arbitrary waveform generators 9 , precise ranging, telecommunications and molecular spectroscopy 10,11 .
Despite the fact that MLL technology opened up a new path to high-precision molecular spectroscopy, a major challenge has been to access the Mid-Infrared (MIR) spectral region, where a variety of strong molecular fingerprints can be measured 10,12 .Although many alternative techniques for MIR OFC generation were proposed, including direct MIR OFC generation by quantum cascade lasers 13 , in general it requires the nonlinear frequency conversion processes to be involved.The two most common methods for MIR OFC generation based on nonlinear conversion are Difference Frequency Generation (DFG) and Synchronously-Pumped Optical Parametric Oscillation (SPOPO).In most cases, DFG requires either two fully stabilized Near-Infrared (NIR) OFCs for the input 14 , or one NIR OFC and its extended version via Raman downshifted soliton in supercontinuum (SC) generation [15][16][17][18] .While the second option of DFG allows one to use a single NIR comb for the pump and signal, this implementation leads to the cancelation of CEO (it is always 0 for the MIR comb), and thus the CEO cannot be easily changed.SPOPO does not make the CEO tuning easier, since for a singly resonant SPOPO one needs to precisely control the pump CEO and the cavity length at the same time to tune the MIR comb CEO, and determination of the exact CEO frequency requires an additional measurement setup 12,19,20 .In doubly resonant degenerate SPOPO 21 the CEO tuning can be performed just by tuning the CEO of the pump laser, but this scheme implies the random choice between two possible CEO values for the MIR comb 20,[22][23][24] , complicating its CEO determination.In addition, SPOPOs generally require precise cavity locking 25 and careful engineering of the group delay dispersion 22 .
In the present work, we demonstrate a simple approach for the generation of fully stabilized MIR OFCs with CEO tuning and modulation.The approach is based on single-pass femtosecond Optical Parametric Generation (OPG) seeded by a continuous-wave (CW) laser, and it can be applied to any OPG.Importantly, OPG on its own does not directly lead to the generation of stable OFCs for the signal and idler, since the process starts from noise, and thus the CEO is random for two subsequent signal and idler pulses.In order to fully stabilize the output MIR comb, we use a CW laser phase-locked to the pump laser to seed the signal comb.Several important advantages arise here compared to DFG and SPOPO.First, the idler CEO is defined by the radio-frequency (RF) local oscillator (LO) used for phase-locking of the seed laser, which means that there is no need to measure the CEO, since it is always known.Second, the CEO of the MIR comb is continuously tunable by simply changing the RF LO frequency.It makes the system highly versatile combined with easily tunable repetition rate thanks to the cavity free design.Third, the idler CEO can also be modulated at relatively high frequencies precisely maintaining its CEO central frequency.Moreover, the method is inherently free of any optical self-referencing techniques like f-2f 4,5 and 2f-3f 26 interferometry meaning that the CEO of the pump MLL can be left free running.We believe that the presented simple method for the generation of fully-stabilized MIR OFCs has major advantages compared to the existing methods, especially in applications that require precise control of the CEO and repetition rate -such as cavity enhanced spectroscopy 27 , dual-comb spectroscopy [28][29][30] , comb-assisted spectroscopy 31,32 and high harmonic generation 33,34 .Additionally, the setup demonstrated here can be used as a pump source for another OPG, DFG or SPOPO 35 , since it has a large wavelength tuning range and high output power.The high output power is also vital for certain spectroscopic methods including OFC photoacoustic spectroscopy 36,37 and background-free absorption spectroscopy 38 .In this article, we demonstrate the proof of concept implementation of the method, including rigorous characterization of the system.

II. Principle
Usually, in OPG, high-energy pulses are used to pump a nonlinear crystal, which in nondegenerate case leads to the generation of signal and idler pulses.The generation of the new pulses is governed by the quantum noise amplification.This makes the process inefficient and leads to loss of pulse-to-pulse coherence of the produced signal and idler pulse trains.The situation changes if the OPG is seeded by an additional light source in the signal or idler spectral region, which is similar to Optical Parametric Amplifiers (OPA).First, the seeding may lead to a significant reduction of the OPG threshold 39 .Second, it improves the pulse-to-pulse coherence provided that the seeding source is coherent 40 .Third, it reduces the relative intensity noise (RIN) of the system 41 , which is vital for many spectroscopy applications.In addition, it was demonstrated that two OPG setups seeded by the same CW seed laser produce mutually coherent output NIR combs (signal), which is useful for dual-comb spectroscopy 42 .Despite all the benefits, to the best of our knowledge, the possibility of using CW-seeded OPG to produce a fully-stabilized MIR OFC source has not been demonstrated before.
Numerous OPG and OPA experiments have been reported using different nonlinear materials including BBO 43 , PPKTP 44 , GaAs 45 , LiTaO3 46 .Of particular interest is the OPG in MgO doped periodically poled lithium niobate (MgO:PPLN) because this crystal has a high nonlinear coefficient, high damage threshold and large transparency range that extends to the 3 µm CHstretching vibrational region, which is useful for molecular spectroscopy.Importantly, when the pump source has a pulse duration in the range of hundreds of femtoseconds or shorter, the interaction length may become an issue.Due to the difference in group velocities between the pump, signal and idler pulses, the interaction length can be so short that the OPG threshold is never reached without the risk of damaging the crystal.Fortunately, several research groups demonstrated that MgO:PPLN supports OPG where the interaction length can be as long as 50 mm for femtosecond pump pulses leading to the conversion efficiencies over 50% with low threshold 47,48 .A distinctive feature of this particular case is the convenient pump wavelength of 1030 -1064 nm that coincides with the mature technology of high power Nd:YAG solid state and Yb-doped fiber MLLs.When launched into the MgO:PPLN in the given spectral range, the highenergy pump generates the signal and idler pulses, whose group velocities have opposite signs and equal absolute values leading to a so-called pulse trapping effect 49 .
In general, each optical frequency component νn of a frequency comb can be represented as a combination of two radio frequencies using the following formula 5 where fr is the repetition rate, fCEO is the CEO frequency, and n is the mode number of the given optical frequency component (comb tooth).We will use CEO instead of fCEO further in the text.
As usual, fr is directly detectable using a fast photodiode, and thus easy to stabilize.On the other hand, CEO cannot be directly detected without an additional measurements setup, such as an f-2f interferometer 4 .Luckily, in OPG there is a way to determine, stabilize and freely tune the idler CEO without directly detecting it.Considering the principle of conservation of energy, the CEO equation for the OPG process can be written as follows: where CEOp, CEOs, CEOi are the offset frequencies of the pump, signal and idler combs, respectively.In regular OPG without seeding, even if the CEOp is stabilized, CEOs and CEOi are random.On the contrary, if one seeds the signal comb with a coherent source, the CEOs follows the seed source, which consequently leads to a change in CEOi to satisfy Eq. ( 1).We can use this feature in our favor to stabilize the CEOi.Let us first assume that CEOs = CEOp; according to Eq.
(1) it leads to CEOi = 0, which is the case of DFG mentioned above 15,16  if the CEOp is stabilized or not because the CEOi will always remain zero thanks to Eq. ( 1).Next, let us set the CEOs in a different way: where ∆f is a frequency offset; in this case we have two states: CEOi = ∆f, or CEOi = -∆f, respectively for Eq. ( 2) and Eq. ( 3), which are easy to distinguish in our setup.Hence, if one can precisely set ∆f, it means that the CEOi is known and freely tunable.
In practice, the abovementioned CEOi stabilization can be performed in the four steps schematically shown in Fig. 1.First, the output beam of the pump MLL is split into two arms.Note that in order to produce fully stabilized MIR comb, repetition rate of the pump MLL should be stabilized (CEOp can be left free running).Second, the first arm is used to generate SC that reaches the CW seed laser wavelength (see "Supplementary note 1: Supercontinuum generation" for details).Third, the CW seed laser frequency is phase-locked to the generated SC.When the phase-locking is performed, the frequency offset ∆f is defined by an RF LO, which is why we denote ∆f = ∆fLO; see "Supplementary note 2: Phase-locking" for more information.Since ∆fLO can be easily changed, the CEOi is continuously tunable.On the last step, the second arm of the pump MLL and the phase-locked CW seed laser are combined in a nonlinear crystal and used to generate the signal and idler combs via seeded OPG.
By phase-locking the CW seed laser to the SC produced from the pump MLL, we transfer all the changes and fluctuations of the CEOp to CEOs, which according to Eq. ( 1) makes the CEOi stable.Note that in Fig. 1 we only consider the case of Eq. ( 2) that corresponds to CEOi = ∆fLO.The states in Eq. ( 2) and Eq. ( 3) can be distinguished in the phase-locking process by simply monitoring the behavior of the RF beat note between the CW seed laser and the SC.The state of Eq. ( 2) is realized when prior to locking the RF beat note frequency decreases while increasing the optical CW laser frequency (decreasing wavelength), if the opposite is true then the state corresponding to Eq. ( 3) is realized.All the envelope spectra in Fig. 1 (a,b,d) are examples of real measured spectra.The spectra are shown in the linear scale with arbitrary units except for the SC spectrum in Fig. 1 (b), which is shown in the decibel scale for clarity (see "Supplementary note 1: Supercontinuum generation" for details).We will call the wavelengths used in Fig. 1 (1530 nm signal and 3400 nm idler) "reference point" throughout the text.

III. Experimental setup
Simplified schematic of our experimental setup is depicted in Fig. 2. Note that only essential components are shown in Fig. 2, and components with secondary importance such as half-wave plates, alignment mirrors, polarizing beam splitters etc. are omitted for simplicity.In our experimental setup we use a commercial Yb-doped fiber MLL (MenloSystems GmbH, Orange comb FC1000-250) as the pump source.It has 100 fs pulse duration Full-Width at Half-Maximum (FWHM), 250 MHz repetition rate locked to an RF source, <200 kHz comb tooth linewidth (FWHM), 10.5 W maximum output average power and free running CEOp.For the SC generation we use an 80 cm PCF with two zero dispersion wavelengths (NKT Photonics, NL-PM-750).In order to generate the SC with the optical bandwidth shown in Fig. 1 (b), our PCF required 200 mW of average pump power with a coupling efficiency of 50% (so only 100 mW reaches the PCF).The SC is optically filtered using a diffraction grating at 1530 nm with a bandwidth of 5 nm (FWHM) and combined with the CW seed laser (also set to 1530 nm) for the phase-locking.See "Supplementary note 1: Supercontinuum generation" for more information.The CW seed laser is a commercial ECDL (Toptica Photonics, CTL 1550) that is locked to the SC using a PID controller (Toptica Photonics, mFALC 110) and an RF LO (Rohde & Schwarz, SME03).In addition, all the Fig. 2 Simplified experimental setup of the CW seeded OPG.The pump MLL is split into two arms: One is used to produce SC in a PCF and another is used to pump the OPG in MgO:PPLN.The output of the ECDL is split into two arms for the seeding and phase-locking.The SC is optically filtered using DG and combined with the CW seed laser to produce an RF beat note on the PD for phase-locking.The phaselocking is performed using PID and RF LO; BS -beam splitter, FC -fiber collimator, DG -diffraction grating, DM -dichroic mirror, L -lens, PD -photodiode, PID -Proportional-Integral-Derivative controller, ECDL -External Cavity Diode Laser, PCF -Photonic Crystal Fiber.electronic instruments used in the experiments are referenced to a 10 MHz GPS-disciplined frequency reference (MenloSystems GmbH, GPS-8), which is not shown in the Fig. 2.
The MLL is used to pump a 50 mm long 5% MgO doped PPLN fanout crystal (HC Photonics) with quasi phase-matching (QPM) periods of 26.5 -32.5 µm.We use a focusing parameter of ξ = 2.17 close to an optimum 2.84 according to Boyd-Kleinman theory 50 , which resulted in 84 µm waist diameter inside the MgO:PPLN crystal.The crystal is placed in an oven with the temperature set to 75 ºC to reduce the possibility of photorefractive damage.The operating wavelengths can be continuously tuned by translating the crystal in the laser beam path so that different QPM periods of the fanout structure are used.The seed CW laser beam is delivered to the OPG crystal via an optical fiber, followed by a fiber collimator for free space transfer through a dichroic mirror to the crystal.The seed laser waist size and position in the crystal are matched with those of the generated signal.

IV. Results
Operating wavelengths, thresholds and output powers.Fig. 3 (a) shows the average pump powers required to reach the OPG threshold for different operating wavelengths without seeding (red) and with 5 mW of CW seed power (blue).Due to the limited tuning range of our seed laser (1510 -1630 nm), we could not seed the OPG for signal wavelengths shorter than 1510 nm.At Fig. 3 OPG power measurements.(a) Input average pump power vs. operating wavelengths for non-seeded OPG (red squares) and 5 mW CW seeded OPG (blue circles).Note that the threshold increases at short and long operating wavelengths because of the differences in group velocities between the pump, signal and idler pulses.(b), (c) Total converted OPG power (signal + idler, CW seed power excluded) at low and high CW seed powers, respectively, when the average pump power is fixed at 2.7 W (below the non-seeded OPG threshold).
the reference point, the OPG requires 3.5 W of average pump power or 2.37 GW/cm 2 peak intensity to reach the threshold without CW seeding, and 2.4 W (1.63 GW/cm 2 ) with 5 mW CW seed power, which is about 32% threshold reduction.Using 5.4 W (3.66 GW/cm 2 ) of pump power and 5 mW of seed power at the reference point, the average idler power can be as high as 700 mW (1.5 W of average signal power) with an optical bandwidth of more than 100 nm (FWHM).The damage threshold of our crystal was experimentally determined to be 6.8 W (4.61 GW/cm 2 ), which led to the grey tracking effect.The effect of CW seed power on the total converted OPG power (signal + idler) is demonstrated in Figs. 3 (b,c).In these measurements, the input average pump power was fixed at 2.7 W, which is below the threshold of non-seeded OPG.As can be seen in Fig. 3 (b), the OPG threshold is still reached even at 100 µW of CW seed power.In addition, one can significantly increase the OPG output power by simply using higher CW seed power (see Fig. 3 (c)).Nevertheless, for the measurements described further in the text, we used 5 mW of CW seed power, which is attainable with most semiconductor lasers at the telecom wavelengths.
Idler CEO stability and tuning.Let us start the discussion of CEOi stability from the CW seed laser phase-locking.Using the reference point, we first locked the seed laser to the state governed by Eq. ( 3) where ∆f = ∆fLO = 25 MHz. Figure 4 (a) shows the corresponding RF beat note between the seed laser and the SC when the phase-locking is established.Next, we verified the phase-locking quality by measuring the double-sideband phase-noise of the beat note shown in Fig. 4 (a) using the phase-detector method 51 .The integrated phase noise (100 Hz -2 MHz bandwidth) is as low as 16.5 mrad meaning that the CW seed laser is tightly locked to the SC.In addition, we performed frequency counting of the same RF beat note.The result can be seen in Fig. 4 (c) that shows the standard deviation of 34 mHz confirming high quality of the phaselocking.The noise in the frequency counting experiment averages out as the white noise, It is worth emphasizing that if one wants to make sure that the CEOi is stabilized, the data shown in Fig. 4 (a-d) can be used as a reference.In order to prove this statement, we measured the CEOi and verified its stability, which we discuss next.The most straightforward way to determine the CEO and measure its stability is f-2f interferometry.This would require an octave spanning SC spectrum generated directly in MIR, which is a challenging task.Instead, we emulated f-2f interferometry method by comparing the idler OFC to another OFC with a known and stabilized CEOref, which we will refer to as a reference comb.The reference comb is a commercial Er-doped fiber MLL (MenloSystems GmbH, Blue comb FC1500-250-WG) that has an inbuilt SC output spanning across 1050 -2100 nm, with the CEOref stabilized to 20 MHz.The reference comb does not directly overlap with the idler comb, hence we had to frequency double the idler comb using another MgO:PPLN crystal to produce an additional comb centered at 1700 nm.When frequency doubled, the CEO of the idler also doubles, meaning that the idler second harmonic (SH) offset frequency (CEOiSH) is equal to twice the CEOi.We combined the idler SH with the reference comb and set the repetition rates of both combs to 250 MHz.Using a simple delay line to overlap the optical pulses in time, we measured a single beat note with the following value: One can see a stable beat note with the expected value of 70 MHz in Fig. 4 (e).It proves that the phase-locked CW seed laser transfers the instabilities of the free running CEOp to the signal comb (CEOs = CEOp + ∆fLO), thus making the CEOi stabilized according to Eq. ( 1).We used the same beat note for a frequency counting experiment that demonstrates the long term stability of CEOi (see Fig. 4 (f)).The noise in Fig. 4 (f) is limited by the mutual instability (timing jitter) between the pump comb (hence the idler comb and its SH) and the reference comb.The pump and reference combs are locked to two different RF signal generators that are both referenced to the same 10 MHz GPS-disciplined crystal oscillator.The specified relative instability of the reference oscillator is 5 × 10 −12 in 1 s.In order to demonstrate the CEOi tuning, we swept the RF LO frequency from 10.5 MHz to 120 MHz with a step of 0.5 MHz and continuously measured the beat note frequency using an inbuilt peak finding function of the RF spectrum analyzer.The result can be seen in Fig. 4 (h).As evident from the figure, the idler CEO is continuously tunable for almost half of the idler repetition rate without any interruptions.For RF LO frequencies close to 125 MHz the phase-locking becomes unstable because the adjacent RF beat note from the next nearest comb tooth starts to a b overlap with the beat note under consideration.Nevertheless, when the 125 MHz point has passed, the CEOi can be further tuned without interruptions.Idler CEO modulation.Next, we modulated the RF LO frequency ∆fLO with a maximum modulation frequency of 20 kHz available in our RF LO.In addition, we chose the largest modulation amplitude of ±2 MHz that allowed for stable phase-locking without changing any locking parameters of the PID controller used in the previous experiments.The corresponding RF beat notes with modulation off and on are depicted in Fig. 5 (a,b) for comparison, respectively.When the modulation is on, the shape of the RF beat note is modified to the typical modulated pattern with two peaks located at the extremes of the modulation.Note that here we consider the idler SH, thus the modulation amplitude is doubled and results in ±4 MHz (see Fig. 5 (b)).When modulated, the central frequency of the RF beat note is precisely maintained, which would not be possible without the tight phase-locking.The CEOi modulation experiment is a good demonstration of the system's versatility.In practice this can be used in experiments where a dynamic control of the CEO is required.For instance, one could lock the comb to an external cavity for cavity-enhanced spectroscopy, in which case both the repetition rate and CEO often need to be precisely controlled 52 .Importantly, in our method these two parameters are independently adjustable, which is not necessarily the case with MLLs.
Idler linewidth and intensity noise.With regard to spectroscopy applications, there are two additional parameters of great importance -the comb tooth linewidth and the RIN.First, we estimated the comb tooth linewidth by another RF beat note measurement using a MIR CW HeNe laser (Research Electro-Optics, Model 30545) at 3390 nm.We do not have the exact information about the HeNe laser linewidth, thus this measurement only sets an upper limit for the comb tooth linewidth.The result can be seen in Fig. 6 (a) that shows an RF beat note with a 177 kHz FWHM (note the linear scale in Fig. 6 (a)).This is a narrow linewidth considering the fact that our pump MLL repetition rate is locked to an RF source, not to an ultra-stable CW-laser.We would like to emphasize that all the RF beat notes shown in Fig. 4 (e), 5 (a) and 6 (a) are detectable only if the OPG is seeded by the CW source.Without CW seeding, the CEOi is random and no RF beat note can be detected.See "Supplementary note 3: Pulse-to-pulse coherence" for more information.The RIN measurements were performed using the reference point with two different input powers that correspond to saturated and non-saturated OPG regimes (Fig. 6 (b)).At saturation we used an input pump power of 4.2 W that resulted in 450 mW of average output idler power.For the non-saturated regime, we used 3.2 W of average pump power that produced 170 mW of average idler power.The integrated RIN at the saturation is lower than for the non-saturated a b regime; we determined it to be 0.018% and 0.053% (integrated from 10 Hz to 2 MHz) for saturated and non-saturated regimes, respectively.The result is in agreement with the theoretical predictions as well as experimental observations by another research group 53 .

V. Conclusions
We have demonstrated a simple method that allows for the generation of stabilized MIR OFCs using femtosecond OPG with CW seeding.The CW seed laser plays a key role in the method, since it establishes the idler CEO stability.Moreover, the CW seed laser makes the system highly versatile allowing one to dynamically tune and modulate the idler CEO directly in the OPG process.Our setup does not have any cavities, which means that the repetition rate can also be freely tuned.We performed all the measurements required to prove the idler CEO stability, hence the usually challenging step of CEO determination can be omitted.An additional benefit of the scheme is its inherent insensitivity to relative timing jitter between the pump and seed, since the seed source is not pulsed.This is in contrast with DFG that relies on downshifted Raman solitons and requires a careful control of the temporal overlap between pump and signal pulses for intensity noise minimization 54 .Along with the already mentioned DFG and SPOPO methods, there are some examples of MIR OFCs that have been produced by spectral broadening in Si3N4 55,56 and LiNbO3 57 waveguides.These systems show a great promise in the miniaturization of the MIR combs, but unfortunately the output powers are quite limited in the MIR region.However, combining our method with nonlinear waveguide technologies could open up a new path to efficient and integrated MIR combs with dynamic CEO control.

Supplementary material
See supplementary material for details on the supercontinuum generation, phase-locking procedure and pulse-to-pulse coherence of the described OPG light source.

Fig. 1
Fig. 1 Schematic of the CW seeded OPG for stabilized MIR OFC generation.(a) Pump envelope spectrum.(b) SC envelope spectrum combined with CW seed laser.(c) CW seed laser (black line) locked to SC (blue lines) in the state corresponding to Eq. (2) -CEOs = CEOp -∆fLO.(d) Generated signal and idler envelope spectra using CW seeded OPG (above) and the schematic of the corresponding laser modes (below).

Fig. 4
Fig. 4 CEOi stability measurements and tuning.(a) RF beat note of the CW seed laser locked to the SC.(b) Double-sideband phase-noise (PSD) of the RF beat note (shown in (a)) when the seed laser is locked to the SC.(c), (d) Frequency counting of the beat note shown in (a) and its Allan deviation, respectively.(e) RF beat note between the reference comb and the idler SH. (f), (g) Frequency counting of the RF beat note shown in e and its Allan deviation, respectively.(h) CEOi tuning demonstration; note that the CEOi is continuously tunable for almost 125 MHz, which is half of the idler repetition rate.

Fig. 5
Fig. 5 Idler CEO modulation experiment.(a), (b) RF beat note between the reference comb and the idler SH without and with modulation, respectively.

Fig. 6
Fig. 6 Idler linewidth and RIN.(a) RF beat note between MIR HeNe laser and idler comb (red) (note the linear scale) and a Gaussian fit to the curve with servo sidebands excluded (black).(b) RIN of the idler comb in the saturated regime (blue) and non-saturated regime (red).