1. Introduction

Optical frequency combs had a seminal impact on optical precision metrology during the past decade [1, 2]. In this paper, we present a novel approach for shifting the frequencies of an optical frequency comb enabling an unsurpassed degree of agile, arbitrary and highly precise manipulation of its frequency and phase, which will further expand the application fields of frequency combs.

A significant class of these applications is formed by widely tunable continuous wave (cw) optical frequency synthesizers (OFSs). A cw OFS provides a phase-coherent optical cw field whose frequency and phase can be arbitrarily adjusted within a certain tuning range and resolution while they are continuously traced back to the frequency and phase of a reference signal which may be either of optical (hundreds of THz) or electrical (microwave range) nature. Early concepts for OFSs have been presented in the 1990s exploiting a frequency shift of an optical reference signal inside a recirculation loop combined with suitable filtering [3]. Here, the frequency resolution is limited by the frequency shift acquired during one loop round trip. More recent OFS concepts rely on frequency combs providing a phase-coherent link between the OFS output field and its reference signal, thus achieving the high precision and small uncertainty of the output frequency just as it is well known from comb metrology. The cw output field is phase-coherently retrieved from the field of a single comb line using “cleanup”-techniques removing the other lines, either by phase-locking a cw cleanup laser to the comb line (optionally with a frequency-offset) or by filtering out the desired comb line.

Tuning of the OFS output frequency (phase) is then achieved by either tuning the frequency (phase) offset between the comb line and the cleanup-field [4, 5], or by tuning the frequency (phase) of the comb line [6].

Cw optical frequency synthesizers should not be confused with general optical arbitrary waveform generators [7] (although they can be regarded as cw optical arbitrary waveform generators [8]). Furthermore, mere optical frequency comb generators based on mode locked lasers are often referred to as optical frequency synthesizers [1]. However, comb generators are not cw OFSs, but rather multiline-OFSs with fixed frequency and phase relations between the individual lines. Although such optical frequency comb generators are commercially available since many years, there is no widely tunable cw OFS product on the market. This may be due to the drawbacks or constraints of the tunable cw OFS approaches as known hitherto. An impairment of the techniques based on frequency-offset phase locking a cw “cleanup-laser” to a comb line is an ambiguity at critical frequencies exactly on or exactly midway between neighboring comb lines which requires conditional decisions during frequency tuning. All other tuning approaches reported so far which do not exploit an offset phase-lock are limited either in their frequency tuning range or agility due to insufficient control elements for optical carrier frequencies. Tuning the carrier frequency by piezo-controlled macroscopic resonator length changes [6], for example, requires components to be mechanically moved which is slow and susceptible to acoustic perturbations.

A new approach to tunable OFSs is enabled by the frequency shifter presented in this paper. It is a carrier frequency shifter which exploits the properties of frequency combs and achieves unprecedented simultaneous tuning range and agility, i.e. control bandwidth.

It is applicable to frequency combs regardless of how they are generated, as long as they correspond to a periodic train of pulses. In particular, it does not require any intrusion into the resonator of the comb generator. Such a resonator external approach is more universal than existing resonator internal approaches and does not impair the comb generation process. Furthermore, it does not require conditional decisions and is thus ideally suited for implementations of agilely and widely tunable OFSs.

2. Principle of the frequency comb shifter

The principle of the frequency comb shifter is first described in a time-domain picture. Frequency shifting is fundamentally based on the relation between instantaneous frequency $\nu \left(t\right)$ and phase $\varphi \left(t\right):$

In the following, an optical frequency comb corresponding to a train of strictly time-periodic short pulses is considered. As shown in Fig. 1, its optical field is sent through an electro-optic modulator (EOM) whose refractive index depends on an applied voltage. By modification of the refractive index, the carrier phase of the light field transmitted through the EOM is changed. The short pulses sample the instantaneous refractive index of the EOM at sampling points of time $t_m=m\Delta T,$ where the time span between the pulses is given by the inverse of the pulse repetition rate: $\Delta T={1/f}_{\text{rep}}.$ It is remarkable that it does not matter how the EOM refractive index evolves during the time between the pulses, i.e. while the light is off and hence cannot sense the refractive index. For this reason, the method is not applicable to frequency combs which do not correspond to a pulse train, but e.g. to a cw signal with periodic short phase excursions instead.

 

Fig. 1 Basic building blocks of the frequency comb shifter.

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The pulses thus experience the EOM-induced phase shifts ${\phi }_m=\varphi \left(t_m\right)$ and the change of the carrier phase between subsequent pulses is given by

It is instructive to highlight the process alternatively in the frequency domain. In this picture, the frequency comb can be treated as a multitude of equally-spaced cw fields. Each of these cw fields experiences phase shifts according to Eq. (6), where for the moment we neglect the wavelength dependence of these phase shifts for a given voltage applied to the EOM. The cw fields sense the evolution of the EOM refractive index including the transients of the fly-back jumps, which leads to spurious signals for each cw field. The existence of darkness between the pulses is equivalent to the fact that the cw fields interfere destructively during that time. As a consequence also all spurious signals sensed by the cw fields between the pulses identically add up to zero.

Unlike assumed before, the carrier phase shift φ induced by an electric field E in a LiNbO₃ crystal of length L via the electro-optic effect in general is wavelength-dependent [9]:

Since a frequency-proportional phase shift corresponds to a temporal group delay, the pulse envelope experiences a timing jitter according to the instantaneous phase actuating values $\left|{\phi }_m\right|$ of Eq. (6). Assuming a peak amplitude of $\left|{\phi }_m\right|\le \pi$ as it is the case with the described technique using 2π-fly-back jumps, this leads to a peak pulse envelope timing deviation of $\approx 2.5\text{fs}\text{@}\nu =200\text{THz}\text{.}$ Pulse timing jitter rms-values of this order are typical for low-noise mode locked fiber lasers, as integrated over all Fourier frequencies above the kHz range. Hence, the timing jitter accompanying the carrier frequency shift can be tolerated in many cases.

The fact that the voltage-induced phase shifts are not wavelength-independent leads to spurious signals at optical frequencies for which fly-backs do not correspond to 2π. These spurs consist, amongst others, of the remaining unshifted lines of the input frequency comb. However, this does not imply any problem in single-frequency applications because the phase shift at one single and well-known carrier frequency can be easily held at its proper value by adjustment of the EOM driver gain via feed-forward techniques.

If actually necessary, simultaneous broadband operation requires more sophisticated approaches, e.g. introduction of an additional time-proportional phase shift mapped to a frequency-proportional phase shift using longitudinal spectral decomposition of the light pulses [11].

It should be noticed that if the frequency shifter shifts all comb lines in common without generation of spurs, the frequency shifter acts as a control element for the carrier-envelope frequency ${\nu }_{\text{CEO}}$ of the frequency comb [2]. In reference [9], the electro-optic effect is employed to control the carrier-envelope (CE) phase of short pulses. Due to the material dispersion of the LiNbO₃ crystal, the carrier and group phase shifts induced by a voltage applied to the EOM are not equal and thus a CE-phase shift given by their difference results. However, in that approach the induced phase shifts are still wavelength-dependent and hence both unwanted carrier and pulse timing phase shifts accompany the CE-phase servo control action. These phase shifts are correlated following an elastic tape relation [12–14]. Thus, at a specific fix point frequency, the different contributions cancel each other, i.e. no phase shift is induced. In the case of a frequency-proportional phase shift induced by a voltage in an EOM, i.e. if material dispersion is discarded, the fix point frequency is zero. Hence, in this case, there is no effect on the CE-phase, i.e., the phase shift at zero optical frequency. Taking material dispersion into account, the fix point frequency can be determined from Eq. (7) as ${\nu }_{\text{fix,}\text{EOM}}\approx +20\text{THz}\text{.}$ This means that + 1 rad CE- phase shift is accompanied by a correlated carrier phase shift of - 9 rad @ ν = 200 THz, which implies an optical carrier collapse [15], i.e., the modal power of a comb line becomes smaller than the power in the phase noise pedestal [16, 17] due to the quasi-random phase shifts with (9 mod 2π) rad amplitude accompanying the CE-phase shifts, with strong impact on phase-trackability.

Although several alternative approaches for comb-generator external frequency comb shifters have been published, our method has the potential to achieve high frequency and phase precision, traceability to a reference signal, high agility, wide tuning range, large optical bandwidth and low levels of spurious signals simultaneously.

Furthermore, most conventional frequency shifters do not allow what we call “endless” frequency shifting of comb lines: The pattern of the comb lines can be frequency-shifted endlessly, while the power of an individual line is determined by ist position inside the comb’s envelope. It increases during entering the envelope on one edge and vanishes upon leaving it on the other. One example for conventional frequency shifters are acousto-optical modulators, which are limited both in optical (~10 nm) and control bandwidth (~100 kHz), introduce higher-order chromatic dispersion and, at least in single-pass configuration, angular dispersion of the frequency-shifted beam. Another alternative are integrated suppressed-carrier single-sideband modulators [18], which are limited in tuning range (≤ 100 GHz) as well as in optical bandwidth, and their spur free dynamic range is sensitive to environmental conditions. The limited tuning range, however, could be circumvented using modulation frequencies modulo f_rep when applied to frequency combs (compare comment to Eq. (3)).

The novel method is related, but not equivalent to traditional serrodyne techniques, and should not be confused with them. In order to achieve tolerable spurious signals, conventional serrodyning schemes require the fly-backs to occur with a very small duty factor [19], or, when applied to mode locked lasers, to be synchronized to an integer multiple of the repetition rate [20]. In the latter case only the spectral envelope can be changed, in analogy to a blazed grating. Our method instead can be seen as a generalization of the serrodyne concept, using conditional fly-backs, which only occur when they do not cause spurious signals (e.g. when the light is off between the pulses). Thus it allows spur-free 'serrodyne' frequency shifting of a comb line.

A fundamental limit to the control bandwidth of the novel frequency shifter is the Nyquist-criterion: Since a frequency shift requires more than two subsequent pulses sampling the change of the EOM refractive index to be unambiguous, this limit is given by $\Delta {\nu }_{\text{max}}=f_{\text{rep}}/2N,$ where N is an oversampling factor. Even with oversampling, this results in more than 10 MHz control bandwidth for frequency combs with line spacings in the 50-500 MHz range. Within this limit, the maximum agility, i.e. change of the frequency shift over time is given by

3. Experimental setup

The 2π phase fly-backs characteristic for the frequency shifting method can be implemented with digital electronics. The central element of these electronics is a numerically controlled oscillator (NCO, the time- and amplitude-discrete part of a direct digital synthesizer, DDS [21, 22], implemented in an Analog Devices AD9958 evaluation board) clocked by the repetition rate of the frequency comb or a multiple thereof. The tuning word of the NCO determines the increment of its phase accumulator value per clock cycle, i.e. per pulse. The driving voltage at the EOM is generated by A/D conversion from the phase accumulator value where the conversion gain is adjusted such that the accumulator’s most significant bit (MSB) corresponds to a phase shift of π. Consequently, the 2π fly-back jumps mentioned above automatically take place during the overflow events of the phase accumulator. Thus, this scheme can be considered as an optical DDS, since the time-varying value of the digital accumulator is converted to the (optical) carrier’s phase, similar to its rf-counterpart. It is noteworthy that the NCO is frequency agile, i.e. the tuning word can in principal be changed every clock cycle with an immediate effect on the frequency shift. Furthermore, the scheme is robust against changes of the repetition rate, since the NCO clock is synchronized with it.

Without further provisions, the maximum clock rate of the digital electronics may pose the limit for the repetition rate of the pulse train in the general case. Current state of the art digital electronics reach clock rates up to the GHz range. However, using undersampling techniques like changing the EOM phase only every N-th pulse can in principle overcome this limit at the cost of increased spurious signal levels and of tuning speed.

For a first demonstration of the novel frequency comb shifter function, an experiment is set up providing both the original, unshifted frequency comb emitted by a mode-locked Erbium-doped fiber laser, and the frequency shifted comb. During the experiments, the frequency shift follows a target temporal evolution. This is implemented by sending a corresponding sequence of tuning words to the NCO in the EOM control electronics. For analysis, the temporal evolution of the shifted optical comb spectrum is tracked with respect to the unshifted comb using a scanning-cavity based optical spectrum analyzer. This provides a coarse overview at a moderate frequency resolution. In order to simultaneously analyze the temporal evolution as well as unwanted spurious signals with high frequency resolution, the spectrum of the beat note between the field of the shifted and unshifted frequency comb is monitored.

Figure 2 shows a schematic sketch omitting a distinction between free space and fiber optical parts of the optical setup. A mode locked Erbium-doped fiber laser providing 100 fs short pulses at a repetition rate of f_rep = 56.2 MHz and 1560 nm wavelength is used as optical frequency comb generator. By means of a grating-based prefilter, the comb spectrum is narrowed to ~100 GHz FWHM and subsequently amplified using an Erbium-doped fiber amplifier (EDFA). Such prefiltering reduces potentially detrimental dispersion effects of the mirrors [23] employed in the cavity for spectral analysis of the comb spectra.

 

Fig. 2 Experimental setup. See text for details.

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The light path is split in two paths with orthogonal linear polarizations using polarizing beam splitter PBS1. In path 1, the EOM introduces a time-dependent frequency shift which we want to demonstrate. The acousto-optical modulator (AOM) in path 2 is used for shifting the frequency of the beat-note between the fields in path 1 and 2, see below. The light pulses are recombined with orthogonal linear polarizations at PBS2.

The light then enters the V-shaped, folded scanning cavity consisting of coupling mirror M1 (R = 98%) and the end mirrors M2 and M3. Due to the reflection birefringence of M1 under oblique incidence, the V-shaped cavity resonances show a polarization splitting in dependence on the reflection angle. The orthogonally polarized light from paths 1 and 2 is coupled into the cavity polarized along its two principle axes, such that the polarization splitting does not mix the light from the two paths. The resonator length is scanned with M3 mounted on a piezo actuator which is driven by a triangular voltage. The free spectral range (FSR) of the cavity is adjusted to match to an integer multiple (factor 19) of the comb generator repetition rate, resulting in ${\nu }_{\text{FSR}}\approx 1.07\text{GHz}\text{.}$ The FWHM line width of the cavity resonances is $\Delta {\nu }_{\text{res}}\approx 8\text{MHz,}$ corresponding to a Finesse of $F\approx 130$ in accordance with the cavity losses. Each cavity resonance can select a single comb line, because the resonance line width is smaller than the comb line separation, $\Delta {\nu }_{\text{res}}\ll f_{\text{rep}}.$ Since the cavity free spectral range coincides with the 19-fold comb line separation, the cavity is resonant with every 19th comb line simultaneously. The result of linearly ramping the resonator length is that the temporal evolution of the power transmitted through the resonator corresponds to the convolution between the comb spectrum and the cavity resonance.

Both cavity output signals at the principal polarizations are coupled out by the 50:50 beam splitter BS, separated using PBS3 and detected with photodiodes PD1 and PD2. Their output signals are captured by an oscilloscope which is conditionally triggered by the piezo driving voltage and the signal from path 2, i.e. the comb which was not shifted with the EOM. In this way, the spectra of the combs from path 1 and path 2 are mapped to the oscilloscope traces. The amplitude of the piezo driving voltage is adjusted for a stroke larger than 10 f_rep and the scanning frequency is 80 Hz. The angle of the V-shaped cavity and thus its polarization splitting of resonance frequencies is set to match the AOM driving frequency, i.e. the resonance maxima of both signals coincide for zero frequency shift in path 1.

For the simultaneous high-resolution analysis, the beat note between both signals is characterized with an rf spectrum analyzer. To this end, the pulses from the two paths are temporally overlapped using a delay line not shown in Fig. 1, and the polarization planes of the signals reflected at the cavity are mixed by means of a polarizer Pol oriented at 45°. The resulting beat note is detected with PD3. Considering the output of path 2 as local oscillator (LO), the scheme enables heterodyne detection of the output signal of path 1. Owing to the LO frequency offset as given by the fixed AOM driving frequency of 80 MHz, it allows detection of unwanted spurious signals generated even at frequency shifts equal or close to zero and integer multiples of f_rep/2.

4. Experimental results

In the first experiment, we apply a target temporal evolution of the control variable, i.e. the desired frequency shift consisting of a sequence of 5 sinusoidal periods with a peak-to-peak amplitude of one comb line distance followed by a linear ramp over 10 comb lines, and subsequently a cubic time dependence with opposite sign as shown by the overlaid green solid line in Fig. 3(a).

 

Fig. 3 (a) Color-coded PD1 photocurrent showing the measured temporal evolution of the optical frequency comb. The overlaid green solid line shows the target temporal evolution of the frequency shift. (b) Video (Media 1) corresponding to (a), yellow: target frequency evolution, green: PD1, violet: PD2. A high-quality version of the video is available [24].

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For generation of the plot in Fig. 3(a), oscilloscope traces of the photocurrent of PD1, i.e., of the power transmitted through the scanned cavity, were recorded sequentially using the FastFrames mode of a Tektronix TDS 5104B oscilloscope. Each pixel column in the image corresponds to one oscilloscope trace in the recorded sequence. The photocurrent is color coded (blue corresponding to zero, red to maximum photocurrent). Small fluctuations and slight distortions of the experimental data can be observed, stemming from acoustics and pressure fluctuations in the cavity and from nonlinear scanning errors.

Before the start of the target sequence at t = 0, a representation of the optical frequency comb is visible as parallel horizontal lines separated from each other by one repetition rate. At t = 0, the target temporal evolution starts and is accurately followed by the frequency comb. At $t\approx 33\text{s},$ the sequence ends. In principle, much faster changes of the frequency shift are possible. However, for the purpose of visualization in a video (Fig. 3(b)) intended here, faster changes are not useful. Simultaneous with the recording of the oscilloscope data using the FastFrames mode, the oscilloscope screen was recorded with a video camera. The right pane in the video shows the oscilloscope screen, whereas in the left pane the synchronized target temporal frequency shift evolution is shown as a yellow solid line. The red bar and dot indicates the current position. The upper half of the oscilloscope screen shows about 16 comb lines (green: PD1 = shifted comb from path 1, violet: PD2 = comb from path 2), whereas the lower part is a zoom spanning four lines as indicated by the violet dashed lines in the upper half. In spite of the observable fluctuations due to acoustics and air pressure variations in the cavity, the video demonstrates in an intuitive way that the frequency comb follows the target frequency evolution.

The results demonstrated in Fig. 3 prove the correct operation of the novel frequency shifter. However, it does neither provide a high-resolution insight nor can spurious signals be analyzed. We define spurs as optical signal power at frequencies where signal power should be zero in the ideal case. One example is remaining power at the lines of the input frequency comb. In the ideal case all power of the input lines should be shifted to the lines of the shifted comb. In a real system this is not the case, because the required phase fly-back jumps are not exact integer multiples of 2π, e.g. for the reasons discussed in section 2. The time-averaged power of a spurious signal depends on the rate of occurrences of the causing event (e.g. inaccurate 2π fly-back), which in turn depends on the frequency shift. The highest rate of 2π fly-backs occurs at a frequency shift of $n{f}_{\text{rep}}/2,n\in \mathbb{Z}$ as the jump occurs for every second pulse.

Hence, these and nearby frequency shifts are susceptible to this type of spurs. There exist also other sources of spurs, such as nonlinearities or thermal effects in the EOM driver electronics. The spurs can be observed in the beat spectrum detected with PD3 and an electrical spectrum analyzer (ESA).

Figure 4 shows a frame from a video. It comprises a video of the beat spectrum on the ESA (left) along with the synchronized video of the oscilloscope trace (right) and target frequency evolution (lower pane). The target evolution is simply a high-resolution linear frequency sweep over a span of 2 f_rep. Before recording the video, we optimized both the trigger delay between digital electronics and temporal pulse position, as well as the amplitude of the EOM driving signals, for minimum spurs during a linear frequency sweep between two or more comb lines.

 

Fig. 4 Frame from video of a linear frequency sweep showing optical (right) and beat photocurrent spectrum (left) of the shifted and unshifted combs along with the target frequency shift evolution (lower pane) (Media 2). A high-quality version of the video is available [25].

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The beat spectrum on the ESA is shown over a span of one comb line separation f_rep = 56.2 MHz. Aliasing causes the presence of mirror frequencies, which renders the spectrum symmetric about the center frequency at 28.1 MHz. Small deviations from this symmetry result from the fact that the sweep time of the ESA is slightly too slow for the 30 Hz video frame rate. The frequency shifted comb line and its mirror counterpart are marked as (1) and (1'). The two prominent central spurs marked as (2) and (2') in Fig. 4 are the mentioned remaining lines of the unshifted comb from path 1 beaten with the comb shifted by 80 MHz in path 2. Hence, their frequencies are (80 – 56.2 = 23.8) MHz and the mirror frequency at 32.4 MHz. The spurs remain more than 22 dB below the beat notes between shifted comb lines and LO for any frequency shift in the video. Even in this first experimental demonstration, this spur-free dynamic range is comparable to that of dual-parallel-Mach-Zehnder interferometer based SC-SSB (suppressed-carrier single sideband) modulators, however at the benefit of a weaker sensitivity on environmental parameters such as temperature. The observed spurs could in principle be further reduced with more sophisticated driving electronics of the EOM, which avoid, e.g., thermal effects.

5. Conclusions

A novel approach to shift the frequencies of an optical frequency comb has been presented. This resonator-external approach relies on changes applied to the carrier phase within the time between subsequent pulses. Arbitrary time-dependent frequency shift evolutions have been experimentally demonstrated for the first time using this technique. A spur-free dynamic range of more than 22 dB under all circumstances was demonstrated in this first proof of principle.

The described endless frequency shifter for optical frequency combs is a key component for a precisely and agilely tunable optical frequency synthesizer with a broad range of applications, e.g. in precision spectroscopy, dimensional metrology, telecommunication, Terahertz science or quantum optics.