The measurement and control of the spectral properties of optical frequency comb synthesizers [1–3], namely, the carrier-envelope offset (CEO) frequency and comb frequency spacing (repetition rate), are a fundamental experimental aspect for high-resolution and precision spectroscopy [4–6] and extreme light generation [7–10], especially when enhanced by the use of passive high-finesse optical cavities [11]. Although the comb repetition rate frequency can be easily measured using a simple fast photodetector, the measurement of the CEO frequency requires dedicated complex experimental methods. Usually, the CEO frequency can be detected using a so-called f–2f interferometer [12, 13] via a beat note between the higher-frequency end of the comb spectrum and the frequency-doubled lower-frequency end, if the optical spectrum covers an optical octave. Other methods that are less demanding in terms of spectral width of the comb spectrum make use of higher-order non-linear processes, such as 2f-to-3f schemes [14, 15]. However, non-linear interferometry requires the generation of a broadband supercontinuum and high-order harmonics that may be difficult to achieve with a frequency comb with a large mode spacing higher than 10 GHz as in microresonator and quantum cascade combs [16, 17], due to the very low peak power of the pulses. Several research groups have proposed alternative techniques that do not involve non-linear interferometry to measure the CEO frequency. For example, multi-heterodyning methods (or dual-comb methods) were developed in 2011 but at the price of increasing the complexity of the experimental setup using an additional CEO-stabilized frequency comb with a different repetition rate [18]. In 2015, the combination and manipulation of different beat note signals between the optical frequency comb and an additional reference CW laser have been proposed to extract the contribution of the CEO frequency [19], while spectrally and spatially resolved multiple-path interferometry [20] and Fabry–Perot interference transmission pattern analysis [21] demonstrated a precision of a few MHz only in determining the CEO value, but without providing any information on the CEO noise spectrum.

Here, we propose a spectral interferometric method for measuring the CEO frequency of an optical frequency comb based on an external optical resonator and the asymmetric Pound–Drever–Hall technique. The measurement of the asymmetric Pound–Drever–Hall signal allows the determination of the CEO frequency with a fractional precision of 0.87% of the comb repetition rate, as confirmed by the experimental comparison with respect to the traditional f–2f non-linear interferometer. The paper first presents the detailed theory of the proposed method and then describes the experimental validation using a commercial Er-doped fiber frequency comb operating at 1.5 μm and equipped with a non-linear f–2f interferometer module.

The technique presented in this work relies on a variation of the well-known Pound–Drever–Hall (PDH) technique [22] called asymmetric PDH (a-PDH). In particular, a mode-locked laser oscillator is coupled to an external enhancement optical cavity, and its reflected beam is exploited to retrieve an electric signal, which in turn is proportional to the carrier-envelope offset (CEO) of the laser. For this reason, we start our theoretical discussion by modeling the coupling between a mode-locked laser oscillator and an enhancement optical cavity.

In the time domain, a mode-locked laser produces a train of pulses separated by time t _rt. In the frequency domain, these pulses produce a comb of frequencies equally spaced by the repetition rate f rep = t tr − 1 . The corresponding electric field can be written as [23] E ν = ∑ m = 0 ∞ S ν m ∫ − ∞ ∞ e i 2 π ν − ν m t + i ϕ d t , (1) where S ν m is the power spectrum of the laser at the frequency ν _m and ϕ is a generic phase term. It can be demonstrated that the frequencies of the laser modes ν _m can be written in terms of an integer m as [12, 24] ν m l a s e r = m f rep + f ceo . (2) The uncompensated oscillator’s intracavity dispersion causes a phase displacement between one pulse and the following, usually called carrier-envelope phase (CEP) slippage ϕ _cep [25]. The CEP induces a frequency shift of the whole comb structure of f ceo = ϕ cep 2 π f rep , where the subscript CEO stays for the carrier-envelope offset. As far as the external optical cavity is concerned, modes resonate when the circulating and incoming fields are in phase [26]. For the fundamental transverse Gaussian mode TEM₀₀, this relation is given by [27] k L − 2 ϕ G + ϕ D ν = 2 π n , (3) where n is an integer, k = 2πν/c is the wave vector, L is the total cavity length, ϕ _G is the Gouy phase, and ϕ _D is an additional phase term due to the dispersion of the elements of the cavity (e.g., air and mirror dielectric coatings). By expanding the phase term in Eq. 3 around the frequency ν n 0 , the resonant modes of the cavity can be written as [28] ν n c a v = c L + c 2 π ϕ D ′ ν n 0 n − k cav . (4) The term c L + c 2 π ϕ D ′ ( ν n 0 ) − 1 represents the free spectral range (FSR) of the enhancement cavity, while k _cav is a constant term that depends on the phase of the modes. In this discussion, we neglect the non-linear components of the phase change, as we assume that these are much smaller than the mode width. Defining f _cav ≡FSR ⋅ k _cav, we can also write the cavity modes in terms of the integer n as ν n c a v = n F S R + f cav , (5) so that the laser and the enhancement cavity mode frequencies have the same comb-like structure. The resonance condition between the laser and cavity is kept in time exploiting the well-known PDH technique [22], which consists of frequency modulating the laser field at a frequency Ω and then demodulating the cavity-reflected field at the same frequency Ω. In a few algebraic steps, it can be shown that the field A (ν) reflected from the cavity can be expressed as [29] A ν = r ν r * ν − Ω − r * ν r ν + Ω , (6) where r(ν) is the reflectivity of the cavity at the frequency ν, expressed as the ratio between the input and the reflected fields. The PDH error signal ϵ _pdh corresponding to each laser mode ν _m can be written as [29] ϵ pdh ν m = 1 2 Re A ν m cos Φ + 1 2 Im A ν m sin Φ , (7) where Φ is the phase shift between the signal and the local oscillator. We obtain the total error signal by adding together the contributions given by all the comb modes, weighted over their spectral power: ε tot = ∑ m = 0 ∞ S ν m ϵ pdh ν m . (8) Let m ₀ be the index of the laser mode corresponding to the barycenter of the spectrum S(ν), such that ∑S (ν _m ) (m − m ₀) = 0. If we want to lock the comb of the laser to the comb of the cavity, the highest coupling is achieved when m ₀ = n ₀, so that the highest number of modes is matched. For small frequency variations around ν _m , the function ɛ _tot is proportional to the detuning between the laser and the cavity frequencies. Therefore, we can write ɛ _tot from Eq. 8, using Eqs 2–5, as ε tot ∝ ∑ m = 0 ∞ S ν m ( c a v ) ν m ( l a s e r ) − ν m ( c a v ) = ∑ m = 0 ∞ S ν m ( c a v ) m f rep − F S R + f ceo − f cav . (9) The PDH technique acts to minimize the mode detuning, and this happens when ɛ _tot = 0. Typically, a piezoelectric (PZT) actuator serves this purpose: ɛ _tot = 0 is processed by a proportional-integrator-derivative (PID) controller and then applied to a PZT-actuated mirror of the laser cavity, so that f _rep is modified. Alternatively, electro-optical modulators can be exploited for this scope, when a larger control bandwidth is required. Since m ₀ is the barycenter of the spectrum S (ν _m ), ɛ _tot = 0 if ν m o ( l a s e r ) = ν m 0 ( c a v ) ; thus, f off + m 0 f rep − F S R = 0 , (10) where we have introduced the relative offset f _off = f _ceo − f _cav. If f _off = 0, the highest cavity–laser coupling is achieved for f _rep = FSR (Eq. 10), and the laser modes perfectly overlap with the cavity modes for each m. On the other hand, if f _off ≠ 0, each mth mode has a detuning Δν _c which depends on its distance from the spectrum barycenter δm = m − m ₀. This last condition is shown in Figure 1, where both the laser and the cavity combs are reported.The condition of Eq. 10 is satisfied, and the detuning Δν _c can be written as Δ ν c δ m = F S R − f rep m 0 + δ m − f off = − f off m 0 δ m . (11) Therefore, the presence of the offset f _off distances the lateral modes, resulting in less-efficient coupling [30]. We can conveniently rewrite Eq. 11 in terms of frequency detuning δ ν = ν m − ν m 0 as δ m = − δ ν f rep , (12) Δ ν c f off , δ ν = − f off m 0 δ ν f rep , (13) where ν m 0 is the frequency of the m ₀ mode. With this notation, the spectrum function S (δν) is centered at δν = 0, as shown in Figure 1.

At this point, we describe the effect of f _off on the PDH error signal. The cavity reflectivity r in Eq. 6 can be written as a function of a generic detuning Δν between a laser and a cavity mode as r f off , Δ ν = − R 1 + 1 − R 1 R R 1 1 − R e i ϕ , (14) where R = ∏ i N R i is the total reflectivity of N mirror cavity, each of reflectivity R _i , and ϕ = 2π Δν(f _off, δν)/FSR. The total error signal results in ε tot f off = ∑ δ ν S δ ν ϵ pdh f off , δ ν . (15) Note that, actually, ɛ _tot (f _off) = 0 for every value of f _off due to the PDH stabilization. In particular, the leftmost teeth with respect to m ₀ have a positive (or negative) detuning, while the rightmost teeth have a negative (or positive) detuning: the opposite contributions to the PDH are cancelled. For this reason, to retrieve a signal sensitive to f _off, it is sufficient to perform spectral filtering of the PDH signal, so that the symmetrical compensation in the PDH signal is broken: we refer to this signal as asymmetric PDH (a-PDH). In the literature, similar spectral selections on the PDH signal have been used by Jones et al. [31] for a different purpose, namely, to lock the CEO of a mode-locked laser. Moreover, in [31], both the right and left wings of the spectrum were exploited simultaneously. If we perform a single-side spectral selection, the PDH signal of Eq. 15 becomes ε A f off = ∑ δ ν S δ ν M δ ν ϵ pdh f off , δ ν , (16) where M(δν) is a generic filtering function, which is asymmetric with respect to the center of the spectrum (e.g., Heaviside step). ɛ _A changes depending on M(δν), as explained in Section 3, where we compare theoretical and experimental trends.

In the previous model, we have neglected the effect of the group delay dispersion (GDD). Now, we extend the theory of the a-PDH signal to take into account a non-negligible GDD. In particular, the dispersion in the enhancement cavity is taken into account by including an additional phase term ϕ _D in the round trip. When the laser and the cavity are coupled with the PDH technique, this term can be written in terms of the detuning δν and it is given by ϕ D δ ν = 1 2 ∑ i 2 π δ ν 2 G D D i , (17) where GDD _i is the dispersion of the ith cavity elements, defined as the derivative of the group delay. As mentioned previously, the phase term of Eq. 17 causes a further frequency shift between the laser and cavity modes of Δ ν D δ ν = F S R 2 π ϕ D δ ν = π F S R ∑ i G D D i δ ν 2 . (18) It is clear from Eq. 18 that Δν _D is an even function with respect to δν, and thus, the effect of the GDD translates every tooth of the comb in the same direction. From Eqs 13–18, the total detuning Δν results are obtained as Δ ν = ν m ( l a s e r ) − ν m ( c a v ) = f off + m f rep − F S R = Δ ν c + Δ ν D + f off + m 0 f rep − F S R , (19) Δ ν δ ν = α δ ν + β δ ν 2 + γ , (20) where α = − f off m 0 f rep , β = π FSR∑ _i GDD _i , and γ = f off + m 0 f rep − F S R . Even in this case, the PDH feedback system sets ɛ _tot to 0, so that Eq. 9 is still valid, and in the presence of GDD, it becomes ε tot ∝ ∑ δ ν ± ∞ S δ ν ν m ( l a s e r ) − ν m ( c a v ) = ∑ δ ν ± ∞ S δ ν γ + α δ ν + β δ ν 2 = 0 . (21) Note that, in this case, the term γ cannot be 0; thus, the cavity is not locked to the barycenter of the spectrum. To find γ and then obtain Δν from Eq. 20, we impose the conditions ∑ δ ν ± ∞ S δ ν α δ ν = 0 , (22) and ∑ δ ν ± ∞ S δ ν β δ ν 2 + γ = 0 . (23) Eq. 22 refers to the case of negligible GDD, while Eq. 23 gives us the displacement of the modes when the laser and the cavity are locked in the case of non-negligible GDD. It results in γ = − β ∑ δ ν ± ∞ S δ ν δ ν 2 = − β Δ 2 , (24) where Δ² corresponds to the variance of the spectrum S (δν). Finally, from Eqs 20, 24, the displacement Δ ν f off , δ ν results in Δ ν f off , δ ν = α δ ν + β δ ν 2 + Δ 2 . (25) Using the reflectivity in Eq. 14 and the displacement in Eq. 25, the trend of the a-PDH signal can be obtained as a function of the offset f _off, following the same calculations shown up to Eq. 16. As explained in Section 3, the effect of the GDD unbalances the a-PDH signal. In particular, looking at Eq. 21, if f _off = 0, such that α = 0, the signum of the error signal ɛ _A (0) only depends on the sign of the total cavity dispersion ∑ _i GDD _i .

Lastly, we have shown the mathematical treatment to describe the a-PDH signal, which depends on the mode offset f _off = f _ceo − f _cav between a laser and enhancement cavity combs, and we have shown the effect of the presence of GDD. Finally, by exploiting the linear region of ϵ _a , it is possible to invert the function using this signal to retrieve f _off and, in turn, the CEO of the laser. As we show in the following sections, this can be performed by properly calibrating the a-PDH signal and by calculating the offset f _cav of the cavity comb from the cavity geometry. The limitation to this technique is not tracking the phase of the CEO and therefore not providing a phase-coherent link between the optical oscillation and the microwave frequency. However, this does not represent an issue for many applications, such as stabilization [31].

In this section, we describe the experimental setup used to validate the technique described in Section 2 and to measure the CEO of a mode-locked laser.

The setup used in this experiment is schematically shown in Figure 2.We used an M-comb Er-fiber mode-locked laser from Menlo Systems, whose spectrum is centered at 1,550 nm, with a repetition rate of 250 MHz. A computer allows the control of the f _rep and the CEO laser through a motorized translator and an intracavity wedge, respectively. The output laser beam is first spectrally selected with a 12 nm FWHM bandwidth, centered at 1,530 nm, with an interferometric filter (IF), and then frequency-modulated by an acoustic–optical modulator (AOM) at Ω = 1.54 MHz (AOM carrier frequency 81.274 MHz, and frequency modulation of ±1 MHz around the carrier) obtained from Gooch & Housego. The diffracted beam from the AOM (first order) is finally coupled to a two-mirror optical cavity with 1 GHz FSR. The input mirror is flat with a power reflection coefficient of 99.0%, while the other mirror is curved and with high reflectivity ( > 99.99 % ) , with a radius of curvature (ROC) = 175 mm. A polarized beam splitter (PBS) and a quarter-wave plate (QWP) allow the collection of the signal reflected from the cavity. A beam splitter (BS) divides such a beam into two arms: a part of the signal ends up in the photodetector PD1 (Thorlabs APD430C/M, bandwidth 400 MHz), while the other ends up in the photodiode PD2 (Thorlabs PDB450C, bandwidth 4 MHz). This second signal is spectrally modulated, thanks to a diffraction grating (DG) and a movable slit (MS) of adjustable aperture. The transmitted signal from the resonator is measured by the photodetector PD3 (Thorlabs PDA20CS-EC, bandwidth 500 kHz) behind the cavity. PD1 allows the stabilization of the f _rep of the oscillator to the cavity, using the PDH technique. The signal collected by PD1 is demodulated and processed by a digital controller DigiLock 110 (Toptica), which works also as a PID servo, and sent to the piezoelectric actuator on the laser cavity mirror that acts on f _rep (bandwidth nearly 10 kHz). The signal collected by PD2 for the a-PDH is instead demodulated through a mixer. Using the MS, different spectral regions contributing to the a-PDH can be selected. All the spectra displayed are acquired using an E4445A spectrum analyzer from Agilent.

In this section, we experimentally characterize the a-PDH model as described previously, showing its dependence on the selected optical spectrum and the GDD. Then, we apply our technique to measure the CEO of a mode-locked laser, and we compare the results with the measurement obtained using an f–2f interferometer.

We presented a theoretical description of the a-PDH signal as a function of the laser–cavity comb offset f _off, showing its dependence on the selected spectrum and the group delay dispersion of the cavity. We experimentally proved the trends in different setup configurations, where both these parameters were varied. Then, exploiting a properly calibrated a-PDH signal, we measured the laser CEO of a commercial Er-doped fiber mode-locked laser, ranging between −70 MHz and 150 MHz, and validated the results with precise f–2f interferometer measurements. We obtained CEO values with a fractional precision of 0.87% with respect to the laser repetition rate. Because this technique does not require any non-linear frequency conversion, it has clear advantages in terms of the complexity of the experimental setup.