Single-modulator, direct frequency comb spectroscopy via serrodyne modulation

Abstract

Traditional electro-optic frequency comb spectrometers rely upon the use of an acousto-optic modulator to provide a differential frequency shift between probe and local oscillator legs of the interferometer. Here we show that these modulators can be replaced by an electro-optic phase modulator which is driven by a sawtooth waveform to induce serrodyne modulation. This approach enables direct frequency comb spectroscopy to be performed with a single dual-drive Mach-Zehnder modulator, allowing for lower differential phase noise. Further, this method allows for more facile production of integrated photonic comb spectrometers on the chip scale.

Electro-optic frequency combs have been shown to be a powerful and flexible tool for spectroscopy, communications, and physical sensing (see Ref. [1] for a recent review paper). In most of these applications, an acousto-optic modulator (AOM) is placed in either the probe or local oscillator (LO) leg to induce a frequency shift. This shift ensures that positive- and negative-order comb teeth occur at unique optical heterodyne frequencies once combined and down-converted into the radiofrequency domain on a photodiode. Such AOMs exhibit narrow operating frequency ranges and exhibit high optical loss, while also requiring high radiofrequency power. Here we present an alternative approach in which the AOM is replaced by an electro-optic phase modulator (EOM) which is driven with a serrodyne (sawtooth) waveform [2–5]. This approach allows for efficient and frequency-agile tuning of the induced frequency shift while also enabling comb generation and shifting in a single dual-drive Mach-Zehnder modulator (DD-MZM).

A schematic of a traditional self-heterodyne electro-optic comb spectrometer [6–8] can be found in Fig. 1a. Light from a continuous-wave laser is split into probe and LO arms, with an EOM employed for comb generation on the probe arm. On the LO arm, an AOM is placed for frequency shifting of the resulting beat frequency away from DC. Both combs are passed into the device under test (DUT) in a symmetric (colinear) geometry [9]. We note that dual comb (multiheterodyne) spectroscopy can similarly be performed by placing an EOM on the LO arm as well [1, 10, 11].

Figure 1.

(a) Schematic of a typical self-heterodyne spectrometer. (b) Schematic of the single-modulator, direct frequency comb spectrometer presented herein. Abbreviations are direct digital synthesizer (DDS), external-cavity diode laser (ECDL), arbitrary waveform generator (AWG), electro-optic phase modulator (EOM), acousto-optic modulator (AOM), device under test (DUT), photodiode detector (DET), dual-drive Mach Zehnder modulator (DD-MZM).

When driving an EOM using a high-fidelity sawtooth waveform with a peak-to-peak amplitude that corresponds to a 2π phase shift, the input optical beam experiences a frequency shift given by the repetition rate of the sawtooth. This so-called serrodyne modulation [2–5] allows for the AOM found in Fig. 1a to be replaced by an EOM. This change allows for far higher frequency tunability and agility of the frequency shift while also allowing the entire system to be dramatically simplified.

To model the modified laser frequency caused by periodic sawtooth phase modulation, we consider a laser field $\tilde{E}\left(t\right)=e^{i\left(2\pi f_{0}t\right)}$ having a carrier frequency $f_0$, and unity amplitude. Upon phase modulation ${\Delta \varphi }_m\left(t\right)$, the time-dependent field becomes $e^{i(2\pi f_{0}t+{\Delta \varphi }_m\left(t\right))}$ with a power spectrum given by $P\left(f\right)={\left|ℱ\right\{e^{i(2\pi f_{0}t+{\Delta \varphi }_m\left(t\right))}\left\}\right|}^2$, where $ℱ$ and $f$ are the fast-Fourier transform operator and Fourier frequency, respectively.

Denoting the repetition rate and period of the phase modulation by $f_r$ and $T_r=f_r^{-1}$, respectively, we calculated a single cycle of the waveform as ${\Delta \varphi }_m\left(t\right)=2{\Delta \varphi }_a\left(f_{r}t/\beta -\frac{1}{2}\right)$ over the time interval $0\le t<\beta T_r$, and ${\Delta \varphi }_m\left(t\right)=2{\Delta \varphi }_a\left(\frac{1}{2}{-f}_{r}t\right)$ for $\beta T_r\le t<T_r$. Here ${\Delta \varphi }_a$ is half the peak-to-peak modulation amplitude (a constant), and $\beta$ is an asymmetry factor, which when equal to zero or ½ gives an ideal sawtooth or a triangle wave, respectively. When $\beta$ is slightly greater than zero, we can consider the effect of small deviations from an ideal sawtooth. Based on this analysis, the shape of the power spectrum is a function of $f$ and varies parametrically with ${\Delta \varphi }_a$ and $\beta$.

The calculated $P(f;{\Delta \varphi }_a,\beta )$ for $\beta$ = 0.02 and 0 are summarized in Fig. 2. Two different cases for ${\Delta \varphi }_a$ are shown: ${\Delta \varphi }_a=\pi$ which corresponds to the optimal serrodyne modulation and ${\Delta \varphi }_a=\pi /2$ which is representative of a non-optimized case. As expected, all spectra exhibit power only at integer multiples (orders), $m$, of the repetition rate, $f_r$. In general, both even and odd orders occur. When ${\Delta \varphi }_a=\pi /2$ the spectra are dominated by the $m$ = 0 and $m$ = −1 terms. For $\beta$ greater than zero, the spectra are asymmetric with power falling rapidly with increasing mode order. For the perfect sawtooth case, the amplitudes for $m$ = 0 and $m$ = −1 are identical, and the spectrum is symmetric about $m$ = −1/2.

Fig. 2.

Computed power spectra for sawtooth modulation of the laser phase corresponding to amplitudes of $\pi /2$ (upper panels), $\pi$ (lower panels) and indicated sawtooth symmetry factors, $\beta$. Here the x-axes represent normalized detuning about $f_0$ expressed as integer orders, $m=({f-f_0)/f}_r$.

When ${\Delta \varphi }_a=\pi$ the contrast between the $m$ = −1 and remaining orders is much larger than for the ${\Delta \varphi }_a=\pi /2$ case. With $\beta$ = 0.02, all other modes are more than three decades weaker than the $m$ = −1 mode and exhibit a weak frequency dependence. For an ideal sawtooth $(\beta =0)$ and modulation amplitude of $\pi$, the power spectrum reduces to a delta function located at $m$ = −1, corresponding to a pure shift of the laser frequency from $f_0$ to $f_0-f_r$. This result is not surprising given that $d{\Delta \varphi }_m/dt=-{2\pi f}_r$ at all times for this limiting case.

The efficient frequency shifting of an incident beam via serrodyne modulation allows us to replace the AOM commonly found in self-heterodyne spectrometers with an EOM. Further, instead of using a separate EOM on the probe and LO arms, we can use a single DD-MZM. This removes the need for bulk beamsplitters and the additional modulator. In addition to the lower system cost and complexity, this approach greatly reduces phase noise in the system by reducing the length of non-common paths within the interferometer.

A schematic of the DD-MZM-based direct frequency comb spectrometer used in the present measurements can be found in Fig. 1b. The light source was a 1550-nm external-cavity diode laser. The upper arm of the commercial DD-MZM was driven by a linear frequency chirp spanning DC to 1.2 GHz with a 10 MHz repetition rate which was produced by a direct digital synthesis chip [12]. This leads to the efficient generation of a 2.4 GHz wide optical frequency comb with a 10 MHz comb tooth spacing. A 60 kHz sawtooth waveform produced by a 5×10⁶ samples/s arbitrary waveform generator was then applied to the lower arm of the DD-MZM for serrodyne modulation. A constant DC voltage was applied to the phase control of the DD-MZM to control the relative phase between the two arms.

The measured power spectra of the serrodyne modulation showed good agreement with the modeling found in Fig. 2. However, in the $\beta =0$ case we observed weak harmonics of the serrodyne frequency which extended for hundreds of MHz (even when driven at ${\Delta \varphi }_a=\pi$). We attribute their occurrence to the finite bandwidth of the arbitrary waveform generator which produced the sawtooth waveform. Although these harmonics (generally six orders of magnitude weaker than the m = −1 tone) were not predicted by our model, we found that the use of an experimental sawtooth described by a small asymmetry factor (such as $\beta =0.02$) greatly reduced these features and led to improved agreement between the observed and calculated power spectra.

After the two arms were combined inside the DD-MZM they were passed into a circulator and subsequently sent into the DUT, which was a microscale, optomechanical accelerometer. This accelerometer, which has been described in detail previously [13, 14], consists of a 250 μm long, planoconcave Fabry-Pérot cavity in which the flat mirror is suspended by nanoscale beams. The device exhibits an optical finesse of 5430 and a mechanical resonance near 9.9 kHz with a quality factor of 98, and has exhibited an acceleration sensitivity of 314 nm∙s⁻²/Hz^1/2 [13]. The accelerometer was mounted on a commercial electromechanical shaker table for excitation. The light reflected from the accelerometer was passed back into the circulator and subsequently on to a 1 GHz bandwidth photodiode before digitization at 3×10⁹ samples/s. A typical self-heterodyne comb spectrum can be found in Fig. 3, showing the roughly 240 individual comb teeth.

Fig. 3.

Typical self-heterodyne comb power spectrum shown as an average of 500 individual power spectra with a total acquisition time of 0.5 s. As can be seen in the inset, the large positive- and negative-order comb teeth are separated by 120 kHz (i.e., twice the serrodyne frequency). The serrodyne harmonics are also visible as the lower magnitude comb teeth with a spacing of 60 kHz.

For each of the present measurements we recorded for 0.5 s, which corresponds to 1.5×10⁹ samples. This was then divided into 50 kS portions which were zero padded to 150 kS and subsequently Fourier transformed. This Fourier transform length corresponds to a 16.6 μs time resolution on the accelerometer motion. The comb tooth frequencies were then extracted from the spectrum and normalized against an averaged reference spectrum which was recorded with the comb bypassing the DUT. This resulted in spectra of the cavity mode such as that found in Fig. 4, in which relevant cavity parameters such as displacement, coupling, and finesse can be readily extracted and quantified. The offset and center of the cavity mode spectrum were determined using a fit to a Fano lineshape [15] for each spectrum with the other parameters fixed.

Fig. 4.

Typical spectrum of a given optical cavity mode of the optomechanical accelerometer as well as the corresponding Fano lineshape [15] fit. This spectrum was acquired in 16.7 μs with no averaging performed. Elevated noise is seen near zero detuning as this corresponds to DC in the self-heterodyne spectrum.

In the absence of shaker excitation, we observed a root-mean-square noise on the comb displacement measurement of 80 fm, leading to a displacement sensitivity of 300 am/Hz^1/2. This displacement sensitivity corresponds to an acceleration sensitivity of 1×10⁻⁶ m∙s⁻² /Hz^−1/2 (i.e., 1×10⁻⁷ g/Hz^1/2). We note that this sensitivity is a factor of 6 better than we previously demonstrated with a self-heterodyne approach with this accelerometer [8] (after accounting for the difference in optical power), and is similar to the sensitivity achieved with a far lower dynamic range and lower bandwidth side-locking approach [13]. Thus, the reduction in complexity by moving to the DD-MZM approach does not incur reductions in sensitivity, but rather leads to improved sensitivity likely due to a reduction in fiber phase noise.

The most significant disadvantage of this approach in comparison to a more traditional self-heterodyne approach appears to be a reduction of the measurement bandwidth, as in the present case we are limited by the serrodyne modulation frequency (i.e. to 16.6 μs). The use of a faster serrodyne modulation frequency, produced either from a higher bandwidth arbitrary waveform generator or through the use of a non-linear transmission line [16], could overcome this impediment and allow for higher bandwidth measurements.

Another difference of this method is that the optical signal contains harmonics of the serrodyne modulation frequency due to deviations from an ideal sawtooth waveform. Although these harmonics are suppressed by at least 20 dB, we obtained the best results by choosing a serrodyne frequency such that at least the first 100 serrodyne overtones do not overlap any comb-tooth frequencies.

To examine the DD-MZM readout for quantifying dynamic motion we drove the shaker table with an amplitude-modulated 200 Hz sine wave. As can be seen in Fig. 5, the cavity motion is captured over significant cavity detunings, larger than could readily be tracked with a traditional side-lock approach [13]. Based upon the previously described noise and the comb width of 2400 MHz we can calculate the readout dynamic range to be 6×10⁴, which is far higher than generally achieved with optomechanical accelerometers due to the limited range of traditional locking approaches [17].

Fig. 5.

Observed cavity mode detuning recorded with the DD-MZM spectrometer when the electromechanical shaker table was driven with an amplitude-modulated 200 Hz sine wave.

We have shown that serrodyne modulation allows direct frequency comb spectroscopy to be performed in a single DD-MZM, both enhancing sensitivity and reducing the complexity of the system. This also opens new applications. For example, dual comb spectroscopy could be performed in a single modulator by summing a second frequency chirp with the serrodyne waveform. These reductions in complexity are particularly important in moving toward integrated photonics packages where simultaneously producing EOMs and AOMs on a single chip is challenging. Thus, the serrodyne approach described herein provides a pathway toward rapid, high sensitivity spectroscopy in both macroscale and integrated photonics platforms for a wide range of physical and molecular sensing applications.

Data availability.

Data underlying the results presented in this paper are available at doi:/.