Abstract
Controlling the temporal and spectral properties of light is crucial for many applications. Current stateoftheart techniques for shaping the time and/or frequencydomain field of an optical waveform are based on amplitude and phase linear spectral filtering of a broadband laser pulse, e.g., using a programmable pulse shaper. A wellknown fundamental constraint of these techniques is that they can be hardly scaled to offer a frequency resolution better than a few GHz. Here, we report an approach for userdefined optical field spectral shaping using a simple scheme based on a frequency shifting optical loop. The proposed scheme uses a single monochromatic (CW) laser, standard fiberoptics components and lowfrequency electronics. This technique enables efficient synthesis of hundreds of optical spectral components, controlled both in phase and in amplitude, with a reconfigurable spectral resolution from a few MHz to several tens of MHz. The technique is applied to direct generation of arbitrary radiofrequency waveforms with time durations exceeding 100 ns and a detectionlimited frequency bandwidth above 25 GHz.
Introduction
The customized control of the time and/or frequencydomain field of an optical waveform, usually referred to as optical arbitrary waveform generation (OAWG), is fundamental for many important applications, e.g., for the coherent control of quantum processes^{1,2,3,4,5}, implementation of critical functionalities in telecommunication systems^{6} and microwave photonics applications^{7,8,9,10,11,12}. Key performance parameters in an OAWG system are the overall range of frequencies that can be synthesized (bandwidth) and the spectral resolution with which one can manipulate the optical spectrum. The former defines the time resolution of the generated optical waveforms and the later determines their maximum time duration. The ratio of these two parameters (bandwidth/frequency resolution) is known as the timebandwidth product (TBP) and often used as a main performance specification to evaluate the complexity of the waveforms that can be generated with a given system.
Among the numerous approaches proposed to date for OAWG^{13,14}, the Fourier synthesis method^{7} remains the most popular one, and in fact, programmable pulseshaping devices based on this principle are commercially available. This method is based on userdefined filtering of the amplitude and phase profiles of a broadband phasecoherent optical spectrum, e.g., a short optical pulse from a modelocked (ML) laser. This scheme has proven suitable for the synthesis of relatively complex ultrabroadband optical waveforms, with a large timebandwidth product (TBP > 100). However, the frequency resolution offered by this solution is usually poorer than ~10 GHz, which constrains the duration of the synthesized waveforms to below the subnanosecond regime^{15}. The combination of highresolution spatialdomain pulse shapers with high repetition rate ML lasers has enabled spectral linebyline shaping of the corresponding optical frequency comb (periodic set of discrete frequency lines that defines the spectrum of a repetitive optical pulse train) with a resolution of only a few GHz^{16,17}. However, this has been achieved at the expense of large insertion losses, lower TBP, and other technical challenges. Other approaches using the Fourier synthesis concept have also been proposed, including solutions based on acoustooptic programmable dispersive filters^{18}, spacetotime mapping^{19}, or through a coherent combination of mutually phaselocked lasers^{20}. However, these approaches may still require the use of a broadband ML laser^{18,19}, and most importantly, they are very challenging to scale for the synthesis of complex waveforms (with a large TBP).
Recently, to overcome the difficulties associated to the use of a ML laser, OAWG solutions have been developed that employ spectral shaping of optical frequency combs directly generated from a single continuous wave (CW) laser. The comb can be generated by nonlinear interactions in a microresonator^{21}, or by electrooptic modulation^{22}. In these solutions, the comb frequency spacing can be adjusted to match the shaper’s resolution, enabling efficient linebyline spectral shaping^{23,24,25,26,27}. However, the number of spectral lines in these schemes, i.e., the TBP, is still limited to ~100^{24}. Moreover, these methods still suffer from the frequency resolution limitations of Fourier filtering techniques, thus being unsuited for generation of arbitrary optical waveforms with durations beyond the subnanosecond regime.
A different approach for optical field spectral shaping is based on the direct generation of the target complex spectrum by temporally modulating a CW laser^{28}. The frequency resolution of this technique is only constrained by the spectral linewidth of the CW laser. However, increasing the number of frequency components remains a technological challenge, and the performance (e.g., frequency bandwidth) of the method is ultimately limited by the electronic arbitrary waveform generation system (AWG) driving the modulators^{29}. Extensions of this method to enable broader bandwidth operations have been demonstrated through joint spectral filtering and linebyline temporal modulation, but these systems are inherently very complex and still require the use of a broadband ML laser^{30,31}.
In this paper, we demonstrate a concept for reconfigurable OAWG, where an arbitrary broadband optical spectrum, with a userdefined amplitude and phase spectral profile, is synthesized directly from a CW laser seeding a recirculating frequency shifting loop (FSL). Beforehand, the CW laser is temporally modulated, in amplitude and phase, by a lowbandwidth (typically < 100 MHz) input electrical signal. After multiple rounds through the FSL, the instantaneous optical spectrum of the light is reshaped proportionally to the temporal shape of the modulation input signal (in amplitude and phase), i.e., a timetofrequency mapping process is achieved. This simple scheme enables the synthesis of arbitrary optical spectra, with hundreds of frequency components simultaneously controlled both in amplitude and in phase. Most importantly, this is achieved with a spectral resolution that is reconfigurable from a few MHz to several tens of MHz, and over an optical bandwidth (BW) that can potentially exceed the hundreds of GHz range. To showcase the unique capabilities offered by this new concept for OAWG, we report here direct generation of arbitrary broadband RF waveforms, both baseband and on an RF carrier, with a TBP exceeding 300, with a frequency resolution as narrow as ~9.5 MHz (time duration > 100 ns) or a spectral BW as large as 25 GHz (frequency resolution: ~85 MHz), limited by the available detection bandwidth. Additionally, the carrier frequency of the generated RF waveforms can be arbitrarily controlled, from DC to a few tens of GHz. The unprecedented flexibility and performance of this simple and lowcost platform should fulfill the requirements for numerous applications in physics, telecommunications, photonics, and microwave engineering.
Results
Theoretical description
The OAWG concept described here is based on a recirculating frequency shifting loop (FSL) seeded by a narrowlinewidth monochromatic (CW) laser^{32,33,34}. Recall that a FSL comprises an acoustooptic frequency shifter (AOFS) and an amplifier with moderate gain, to compensate for the losses in the loop. A tunable optical bandpass filter (TBPF) is also inserted in the FSL, both to control the spectral bandwidth and to limit the amplified spontaneous emission (ASE) generated by the amplifier. We define f_{s} as the frequency shift per roundtrip induced by the AOFS, and τ_{c} as the propagation time in the loop.
An optical coupler enables to seed the FSL with a CW laser (frequency: f_{0}), temporally modulated in amplitude and phase by means of an acoustooptic modulator (AOM). We define N as the ratio between the optical bandwidth of the TBPF, and f_{s}: N corresponds to the maximum number of roundtrips of the input light in the loop. The output optical signal is obtained by extracting a fraction of the light from the loop by means of a second optical coupler (Fig. 1). For characterization purposes, the measurement of the instantaneous optical spectrum of the output waveforms is carried out by recombining the output signal with a fraction of the seed laser on a photodiode (see Methods). The timedomain heterodyne beating is acquired and processed (e.g., Fourier transformed) offline by means of a digital sampling oscilloscope (DSO). When the reference beam is blocked, the detection system provides a direct measurement of the temporal intensity waveform at the FSL output.
Let us assume that the seed laser is modulated by an electrical signal e_{in}(t) with a duration τ > τ_{c}, and with a bandwidth narrower than 1/τ_{c}. Said other way, the input modulation signal is approximately constant at the time scale of one roundtrip in the loop. We define \(E_{\mathrm{in}}(t)=e_{\mathrm{in}}(t)e^{i2\pi f_{0}t}\) as the optical field injected in the FSL (analytical representation), where e_{in}(t) is the field temporal complex envelope. The optical field circulating in the loop is constructed from the coherent addition of replicas of the input field, shifted along the time and the frequency domains by τ_{c} and f_{s}, respectively. In the case where the power of the EDFA is adjusted so as to compensate for the losses of the loop, neglecting the amplified spontaneous emission (ASE) emitted by the amplifier, the output optical field can be expressed as follows^{35}:
where γ is a complex term that depends on the transmission coefficient of the couplers, and the gain and losses in the FSL. The simultaneous shifts along the temporal and spectral domains induce a quadratic phase term: \(e^{i\pi n\left( {n + 1} \right)f_{\mathrm{s}}\tau _{\mathrm{c}}}\). Notice that this quadratic phase term induces a temporal Talbot effect when f_{s}τ_{c} is equal to a ratio of two mutually prime integers^{33}. It also enables the generation of reconfigurable optical chirped waveforms when f_{s}τ_{c} is close, but not equal, to an integer value^{36}. In the following, f_{s} is chosen such that f_{s}τ_{c} is an integer; under this condition, this quadratic phase term reduces to unity, and the output optical field rewrites:
In the case when no modulation is applied to the seed laser (e_{in} is constant), the output optical field E_{out}(t) consists of a comb of phaselocked optical frequencies starting at f_{0} and separated by f_{s}, over a total bandwidth Nf_{s}. This set of optical frequencies exhibits a linear phase profile, \(e^{i2\pi nf_{0}\tau_{c}}\), simply corresponding to an overall time delay of the resulting waveform. In particular, in the temporal domain, the output signal is a train of Fourier transformlimited pulses, repeating at a rate of f_{s}. When the modulation signal is applied to the seed laser, the comb lines are slightly broadened by the modulation function e_{in}(t). As illustrated in Fig. 1 and according to the mathematical description in Eq. (2), at each roundtrip, the input modulation waveform is simultaneously shifted in time and in frequency by τ_{c} and f_{s}, respectively. After a number R of roundtrips, the signal at the output of the FSL is composed by the sum of R time and frequencyshifted replicas of the input signal. As such, the output waveform at any given instant of time can be described as a superposition of R consecutive temporal segments of the input signal, each with a duration of τ_{c}, and each shifted in frequency with respect to each other by f_{s}. When the overall number of roundtrips is such that the entire input waveform is already within the FSL cavity, that is when Rτ_{c} > τ (total time duration of the input waveform), the time segments of the input signal that are summed up to form the corresponding instantaneous output waveform will cover the full input signal duration. As each of these time segments is shorter than the fastest time feature of the input waveform (recall that the signal bandwidth is narrower than 1/τ_{c}) and these time segments are also consecutively shifted along the frequency domain, the described process will effectively induce a mapping of the entire input temporal waveform into the frequency spectrum at the FSL output, see illustrations in Fig. 1 for the case e (long input signal) at the observation time t_{2}.
The intuitive picture of the operation principle shown in Fig. 1 highlights the timetofrequency mapping process, as well as the main design tradeoffs in the system. In particular, the scaling law of the implemented timetofrequency mapping process is simply defined by the ratio f_{s}/τ_{c}. The spectral bandwidth of the output signal is set to Nf_{s} by the TBPF; as such, the parameter N determines the maximum number of roundtrips that a signal can go through the FSL cavity. This implies that for a full timetofrequency mapping, the duration of the input modulation signal must be set to satisfy the following condition: τ < Nτ_{c}. Moreover, the mapping of the entire temporal input modulation signal to the instantaneous output spectrum effectively holds after the input signal has completely entered the FSL, i.e., after an R (<N) number of roundtrips such that Rτ_{c} > τ, and until the leading edge of the input signal is filtered out by the TBPF (Fig. 1), i.e., until the signal undergoes N roundtrips through the FSL cavity.
A rigorous mathematical derivation of the concept is given in the Methods section. Briefly, from Eq. (2), it can be shown that the resulting timedomain signal at the output of the FSL consists of a train of consecutive waveforms separated by 1/f_{s}, here labelled by m (=1, 2, 3 …) (Fig. 1). The baseband Fourier spectrum of the optical waveform (i.e., Fourier spectrum of the corresponding field temporal complex envelope) emitted at time m/f_{s}, extending over a duration of 1/f_{s}, can be written as
for 0 < f < Nf_{s}, where f is the optical baseband frequency (i.e., with respect to f_{0}). In practice, the optical waveform emitted at time m/f_{s} can be extracted from the output pulse train by time gating (e.g., by means of a temporal intensity modulator). The spectral truncation realized by the TBPF implies also that \(\tilde e_{{\mathrm{out}}}\left( {f,m} \right) = 0\) for f < 0 or f > Nf_{s}. Equation (3) highlights the mapping of the input complex temporal modulation signal from the time to the frequency domain. Notice also that the central frequency of the baseband output spectrum evolves linearly with m, i.e., with the evaluation time m/f_{s}. This feature can be easily understood by considering that the waveform emitted at m/f_{s} + τ_{c} has gone through one more roundtrip (i.e., one more frequency shift) than the one emitted at m/f_{s}.
The spectral bandwidth of the output waveform is equal to f_{s}/τ_{c} × τ, i.e., proportional to τ, the duration of the input modulation signal. Recall that the bandwidth of the input modulation signal is assumed to be narrower than 1/τ_{c}, which translates into a temporal resolution of the modulation waveform longer than ~τ_{c}. This corresponds with a minimum spectral resolution of the synthesized output optical signal that is approximately equal to the frequency shift induced by the AOFS, ~f_{s}. As a result, the maximum number of spectral components in the synthesized output waveform (i.e., the maximum timebandwidth product, or TBP) is given by τ/τ_{c}. As discussed above, this ratio is limited by N, the maximum number of roundtrips that can be achieved in the loop. Experimentally, this parameter is determined by the spectral bandwidth of the incavity TBPF, and can reach values above 1000^{36}. Notice also that the bandwidth of the output optical waveform (~τf_{s}/τ_{c}) is about τf_{s} times, i.e., orders of magnitude larger than the bandwidth of the input modulation signal (<1/τ_{c}). This enables generation of broadband optical waveforms (e.g., with a bandwidth up to hundreds of GHz^{36}) at the FSL output from lowbandwidth electronic input signals (typ. <10 MHz). Finally, it is also important to recall that in the case where the product f_{s}τ_{c} is not an integer, an additional quadratic component would be imprinted onto the spectral phase of the output optical waveforms, as indicated in Eq. (1), potentially providing an additional degree of control on the generated output spectra^{35,36}.
Demonstratio n of arbitrary optical field spectral shaping
To demonstrate the validity of the proposed concept for highresolution arbitrary optical field spectral shaping, we first implemented a fiber FSL, with a roundtrip propagation time τ_{c} = 105 ns, which incorporated two AOFS providing a net frequency shift equal to \(f_{\mathrm{s}} = 1/\tau _{\mathrm{c}} = 9.482\,{\mathrm{MHz}}\) (see Methods). The maximum number of roundtrips in this first demonstrated system is N ~300. The loop is injected with a CW laser temporally modulated by an AOM. The latter is driven by a sine wave at f_{m} = 80 MHz, modulated in amplitude and phase by the input RF signal, as detailed in the Methods section. f_{0} is defined as the carrier frequency of the light wave at the input of the FSL. As predicted, the signal at the output of the FSL consists of a train of waveforms spaced temporally by 1/f_{s} (Fig. 2a). For an accurate characterization of this output optical waveform, the resulting optical spectrum is downconverted to the RF domain by heterodyning the output signal with a fraction of the seed CW laser (Fig. 1)^{34}. The resulting light field is sent to a photodetector and the obtained photocurrent is recorded by a 3.5GHzbandwidth oscilloscope. A single waveform is numerically extracted from the recorded signal: its Fourier transform (FT) corresponds to the complexfield optical spectrum of the selected waveform downconverted to baseband (i.e., with respect to f_{0}). This setup allows us to measure the Fourier spectrum (amplitude and phase) of the individual optical waveforms at the system output. In the example shown in Fig. 2, the CW laser is modulated in amplitude by a 2µs long squarelike (flattop) signal (in light blue). The output waveforms after recombination are recorded (dark blue), and the spectra are numerically calculated for each of the individual traces (Fig. 2c). As expected, all waveforms share the same flattop spectral shape. Notice that the high relative level of the lowfrequency components in the experimentally recovered spectra of the output waveforms is due to the relatively low power of the CW laser used in the heterodyne recombination process. The waveforms generated before the whole input signal has entered the FSL (e.g., around the time t = 0.66 μs in the shown examples) exhibit a frequency spectrum that maps a truncated version of the input square temporal modulation signal, i.e., a narrower flattop. Consistently with Eq. (3), when the whole input signal has entered the FSL, the width of the spectral shape remains unchanged and equal to ~180 ± 10 MHz, in good agreement with the value predicted by the timetofrequency scaling law f_{s}τ/τ_{c} = 181 MHz. Moreover, as predicted, the absolute central frequency of consecutive waveforms increases as the number of roundtrips in the FSL is also increased, with a frequency increment of f_{s} every roundtrip (τ_{c}), equivalent to a frequency increase of 1/τ_{c} between consecutive individual waveforms (separated by 1/f_{s}).
In a second set of experiments, we demonstrated the versatility of the concept for generation of arbitrary optical spectra with userdefined amplitude and phase profiles. Plots in Fig. 3a–d are obtained when the FSL is set with the design conditions defined above. In this case, as expected, arbitrary optical spectra can be synthesized with a frequency resolution equal to f_{s} (~9.5 MHz). Plot a shows the instantaneous spectrum (measured in a temporal window of duration 1/f_{s}) in the case of an unmodulated seed laser. In this case, the TBPF has been set such that about N = 1000 spectral components are generated over a bandwidth exceeding 10 GHz. Therefore, theoretically the maximum TBP of the technique exceeds 1000. However, in practice, due to the imperfect flatness of the spectrum, the maximum TBP directly achievable (i.e., without compensation of the spectral magnitude) is about ~300. In Fig. 3e, the FSL is now set in a different configuration, where \(f_{\mathrm{s}} = 9/\tau _{\mathrm{c}} = 84.91\,{\mathrm{MHz}}\), as described in Methods. In this case, the system enables the synthesis of arbitrary optical spectra with a relatively poorer frequency resolution (equal to f_{s}) but over a broader bandwidth, about nine times larger than in the highresolution scheme. In all cases, there is an excellent agreement between the experimentally recovered amplitude and phase spectral shapes of the synthesized optical waveforms and those theoretically expected according to the amplitude and phase temporal modulation functions imprinted on the input CW laser.
Baseband RFAWG
The possibility of optical field spectral shaping with a MHz resolution makes this technique particularly attractive for direct application to radiofrequency AWG (RFAWG). The proposed concept enables generation of baseband RF arbitrary waveforms in a very straightforward fashion, namely, by simple direct photodetection of the temporal intensity waveform at the output of the FSL (i.e., without heterodyning with the CW laser). In this case, according to the calculations given in Methods, when the whole input signal has entered the FSL, the photocurrent produced by the waveform m can be simply written as:
where \(\tilde e_{{\mathrm{in}}}\left( f \right)\) is the Fourier transform of the complexmodulation function e_{in}(t) that is imprinted on the input CW light. Equation (4) indicates that the generated RF waveforms are not dependent on the observation time (parameter m), or said other way, the target waveform is produced in a repetitive fashion, with a repetition period fixed by the inverse of the frequency shift f_{s}. Additionally, the number of properly shaped waveforms is ultimately limited by the total number of roundtrips in the loop (N). In this case, the input modulation signal to be applied on the input CW laser in order to achieve a target waveform is determined by the inverse FT of the square root of the desired output intensity waveform. In Fig. 4a, we plot the input temporal modulation signal (amplitude and phase) enabling the generation of output waveforms mapping an arbitrary signal (here, Belledonne mountain, near Grenoble, see bottom trace). In this case, the system is designed for a frequency shift of f_{s} = 84.91 MHz, and the output bandwidth exceeds 25 GHz, limited again by the detection device. Other examples of arbitrary signals in the same experimental conditions are given in Fig. 4g–i. When f_{s} = 9.482 MHz, the FSL enables direct generation of waveforms with durations exceeding ~100 ns (corresponding to a frequency resolution of f_{s}), and with a frequency bandwidth limited by the detection bandwidth to ~3.5 GHz, see results shown in Fig. 4d–f. The TBP of the synthesized waveforms in this case exceeds 300.
OAWG and RFAWG on a carrier
We now consider the full optical field at the output of the FSL. As detailed in Methods, the complex amplitude of the output optical waveform emitted at a time m/f_{s}, when the entire input signal has entered the FSL, can be written as:
Recall that m refers to the evaluation time, in steps determined by the inverse of the fundamental frequency shift induced in the FSL, 1/f_{s}. Equation (5) implies that the complex envelope (amplitude and phase) of the generated output waveforms is readily set by the FT of the input modulation function, and the carrier frequency evolves linearly with m; as described above, the carrier frequency increases by f_{s} per slot of time corresponding to τ_{c}, or equivalently, the carrier frequency increases by an amount of 1/τ_{c} from one waveform to the following one (temporally spaced by 1/f_{s}). The maximum value of the carrier frequency corresponds to the waveform having experienced the maximum number of roundtrips N, and is approximately given by f_{0} + Nf_{s}.
A key advantage of the proposed setup is the intrinsic mutual coherence of the output optical waveforms with the CW seed laser. This property can be exploited to achieve direct generation of RF arbitrary waveforms on a carrier by simply heterodyning the output optical field with the seed laser. This is similar to the technique used above for characterization of the highresolution complexfield optical spectra at the FSL output. In particular, the resulting heterodyne beating term in the photocurrent can be expressed as (see Methods):
which corresponds to an RF waveform, whose temporal variations of amplitude and phase are set by the FT of the input modulation signal. The carrier frequency of the mth waveform is equal to m/τ_{c} and is directly proportional to m/f_{s}, the evaluation time of the waveform.
To demonstrate this capability, we set the FSL system to work in the configuration with f_{s} = 9/τ_{c} = 84.91 MHz (τ_{c} = 106 ns). The required input modulation signal to be applied on the CW light wave through the AOM is determined by calculating the FT of the desired output temporal waveform (or the reversed one, depending on the sign of f_{s}). After heterodyning with a fraction of the seed laser, the output train of waveforms is measured with a fast photodiode and oscilloscope (25GHz detection bandwidth). Figure 5 shows a few examples of arbitrary RF waveforms on a carrier synthesized through the proposed method, particularly, data pulse sequences arbitrarily modulated in phase (according to a quadrature phaseshift keying format, QPSK), as well as in amplitude and phase (according to a quadratic amplitude modulation format, QAM). Each of the synthesized data sequences is ~2 ns long.
Discussion
We have demonstrated a new concept for arbitrary broadband complexfield optical spectral shaping, based on timetofrequency mapping in a frequency shifting loop (FSL). The system, which involves a single CW laser, a lowbandwidth RF generator, and standard fiberoptics components, is particularly simple to implement and is easily reconfigurable. Contrary to most optical spectral shaping systems, it does not require ML lasers or fast electronics. Moreover, the proposed method offers a frequency resolution that is ordersofmagnitude higher than that of conventional optical pulseshaping techniques. This new feature of frequency shifting loops adds to a set of previously reported capabilities of these systems, including realtime integer/fractional Fourier transformation^{35,37}, arbitrary frequency chirp generation^{36}, or generation of pulse trains with arbitrary repetition rates^{32,38}.
Using the experimental setup demonstrated here, we report a frequency resolution that can be adjusted by the value of the frequency shift in the FSL, from ~10 MHz to ~80 MHz. We have reported two practical applications of this system: RF AWG on a carrier, which provides arbitrary signals on an RF carrier whose frequency increases waveform to waveform in the generated output train, and baseband RF AWG, which can provide repetitive identical waveforms. The bandwidth and the TBP of the generated waveforms exceed 25 GHz and 300, respectively (both values limited by the detection equipment). It is noteworthy that our system bridges the gap between electronicbased RF AWG methods, which show high frequency resolution but limited bandwidth (<10 GHz), and conventional optical pulseshaping techniques, which provide ultrabroad bandwidth (>THz), but poor frequency resolution (>GHz). Notice that highspeed (wideband) RF waveforms with durations above the nanosecond regime, corresponding to MHz resolutions, have been synthesized through conventional spectral optical shaping combined with frequencytotime mapping induced by chromatic dispersion^{39}. These methods are, however, strictly limited to synthesizing RF waveforms, i.e., just the temporal intensity profile of the resulting optical waveforms. The pulseshaping scheme proposed here allows the direct synthesis of both complexfield optical signals and RF waveforms with similar specifications to those of the dispersive method, e.g., in terms of frequency resolution and bandwidth, while entirely avoiding the need for a ML laser and bulky chromatic dispersion devices.
The parameters of the demonstrated scheme could be also easily modified to match the specifications of different potential applications. For instance, the frequency resolution could be improved to nearly arbitrary values by reducing the value of f_{s}, and correspondingly increasing the travel time τ_{c}. On the other hand, increasing the value of f_{s} would enable the synthesis of larger bandwidths (>100 GHz), and this could be practically achieved by use of singlesideband electrooptics modulators in the loop, instead of acoustooptic frequency shifters^{38}. In our present experimental setup, the TBP is close to 300 (limited by the detection bandwidth). Ultimately, by precompensating the slight unflatness of the generated spectral components, the system could be designed to achieve a TBP above 1000, consistently with previous demonstrations of FSL systems aimed at other different applications^{36}. However, increasing the frequency shift or increasing the number of light roundtrips in the FSL, N (i.e., the waveform TBP), both require expanding the bandwidth of the TBPF in the FSL cavity. In this case, the ASE emitted by the amplifier might become a limiting factor. As detailed in the Methods section, a convenient figure of merit of our OAWG concept, in regard to noise performance, is the ratio between the power of the frequency comb produced when the FSL is seeded with a CW laser, and the total ASE power over the whole TBPF bandwidth. Since this signaltonoise ratio (SNR) scales as the inverse of the TBPF bandwidth (see Methods), increasing N or f_{s} in the FSL system would inherently degrade the SNR of the synthesized waveforms. This effect could, however, be mitigated by decreasing the passive losses of the FSL, and/or through the use of noiseoptimized amplifiers.
In conclusion, owing to its flexibility and unique set of features, the optical waveform shaping technique introduced herein should fulfil the stringent requirements for a wide range of applications in fundamental physics, telecommunications, microwave photonics, etc.
Methods
Timetofrequency mapping in injected FSLs
Here we provide a mathematical proof of the mapping of the complex temporal signal at the input of the loop (duration: τ), to the optical field spectrum measured at its output. For analysis of the output signal, we define w_{m}(t) as a temporal window function centered at time m/f_{s} (m = 1, 2, 3, …) and with a duration ~1/f_{s}. In other words, the parameter m defines the evaluation time of the outcoming signal, in steps of 1/f_{s}.
We define e_{in}(t) as the complex RF modulation signal imprinted by the AOM onto the monochromatic laser, so that the electric field at the input of the loop can be written as: E_{in}(t) = e_{in}(t)e^{i2πf}_{0}^{t} (analytical representation, with e_{in}(t) being the temporal complex envelope). As derived in the main text (Eqs. (1) and (2)), when the product f_{s}τ_{c} is equal to an integer (p), the electric field at the output of the FSL is given by the following expression:
The number of replicas of the input is set by the spectral bandwidth of the TBPF, Nf_{s}. The slowly varying envelope of the output field e_{out}(t), defined by \(E_{\mathrm{out}}(t)=e_{\mathrm{out}}(t)e^{i2\pi f_{0}t}\), is given by:
The envelope of the output field along the temporal window w_{m} is simply: e_{out}(t, m) = w_{m}(t) × e_{out}(t). Consequently:
We have assumed that the temporal variations of e_{in} are slower than τ_{c}, which means that e_{in}(t) can be considered as constant over the duration of the window function w_{m} (i.e., over 1/f_{s} = τ_{c}/p). Therefore:
For simplicity reasons, the exponential term f_{0}τ_{c} in Eq. (10) can be ignored, as this simply represents a change in the origin of time. The baseband spectrum measured at time m/f_{s} is: \(\tilde e_{{\mathrm{out}}}\left( {f,m} \right) = {\mathrm{FT}}(e_{{\mathrm{out}}}(t,m))\), where FT is the Fourier transform operator, defined as: \({\mathrm{FT}}\left( {s\left( t \right)} \right) = \tilde s\left( f \right) = {\int}_{  \infty }^{ + \infty } s (t)e^{  i2\pi ft}dt\). Owing to the convolution theorem of the FT:
Once more, since the frequency bandwidth of e_{in}(t) is smaller than 1/τ_{c}, \(e_{{\mathrm{in}}}\left( {\frac{{m  f\tau _{\mathrm{c}}}}{{f_{\mathrm{s}}}}} \right)\) shows negligible variations when f varies by f_{s}, i.e., along the width of \(\tilde w_m(f)\). Then:
The term in brackets depends only on the choice of the window function. For convenience of analysis, by properly designing the specific window shape, this term can be approximated by a flattop function extending along the frequency range 0 < f < Nf_{s}, i.e., this is a constant term that vanishes when f < 0 or f > Nf_{s}. Therefore, the instantaneous output spectrum at time m/f_{s} can be expressed as follows:

for f < 0 or f > Nf_{s}
$$\tilde e_{{\mathrm{out}}}\left( {f,m} \right) = 0$$(13)

for 0 < f < Nf_{s}
$$\tilde e_{{\mathrm{out}}}\left( {f,m} \right) \propto e_{{\mathrm{in}}}\left( {\frac{{m  f\tau _{\mathrm{c}}}}{{f_{\mathrm{s}}}}} \right)$$(14)
This latest equation corresponds to the mapping of the input temporal complexmodulation signal into the optical field spectrum of the output signal, as illustrated in Fig. 1. The timetofrequency scaling coefficient is equal to f_{s}/τ_{c}.
Electric field at the output of the FSL
Suppose now that the whole input signal has entered the FSL, i.e., m > f_{s}τ. In this case, the output electric field envelope within the temporal window w_{m} is given by the inverse FT of Eq. (14):
This expression shows that the envelope of the output signal is simply equal to the spectrum of the input envelope mapped into the temporal domain, and multiplied by a linear phase term. The output electric field describing the waveform emitted at m/f_{s} can then be written as
and the corresponding photocurrent (at the output of the photodetector) is:
Suppose now that the train of waveforms, generated in the FSL, is recombined with a fraction β of the seed laser (see Fig. 1), and sent to the photodetector. The photocurrent during the temporal window w_{m} can be expressed as follows:
where the superscript “h” stands for “heterodyne”. Assuming that the intensity of the seed laser is much larger than that of the output waveform, the photocurrent expression can be rewritten as
ASE and signaltonoise ratio
In the following, we provide a simple model of an FSL in the steadystate, in order to provide an estimation of the SNR of the proposed OAWG process. The main source of noise in the system is the amplified spontaneous emission (ASE) emitted by the amplifier placed inside the FSL, because similarly to the seed signal, the ASE is repeatedly frequencyshifted and amplified in the loop. To get a better understanding of the effects of ASE without excessive complexity, we address the steadystate case, where the FSL is seeded with a CW laser (frequency: f_{0}, power: P_{0}). The light in the FSL is a comb of N optical frequencies, starting at f_{0}, and spaced by f_{s} (Fig. 6). We assume that the spectral transfer function of the TBPF in the FSL exhibits a flattop profile with a total bandwidth of Nf_{s}. We define P_{n}, the power of the comb line at frequency f_{0} + nf_{s}, and a_{n} the power of the ASE integrated in a frequency band of width f_{s}, centered around f_{0} + nf_{s}.
We also define G as the power amplification coefficient of the amplifier, and T as the singlepass transmission coefficient of the FSL, excluding the amplifier. P_{n} and a_{n} satisfy the following recurrence relationships:
where a_{0} = n_{sp}hf_{0}(G − 1)f_{s} is the ASE power generated by the amplifier in a single polarization state (here, we are using polarizationmaintaining fibers), and a frequency bandwidth of f_{s}. In this latest expression, h is the Planck constant, and n_{sp} is the spontaneous emission factor of the amplifier (n_{sp} > 1)^{40}. As per the design conditions of the proposed OAWG concept, we consider the case in which the amplifier just compensates for the losses in the FSL cavity, i.e. when GT ≈ 1. A convenient estimation of the SNR of the OAWG process can be given by the ratio between the total comb power (signal), divided by the total ASE power in the FSL (noise). The total comb power (P_{comb}) and the ASE power (P_{ASE}) are, respectively, equal to:
and the signaltonoise ratio is then: SNR ≈ 2P_{0}/(Na_{0}). Interestingly, the SNR scales as the inverse of the spectral bandwidth Nf_{s}. In practice, typical values of P_{0} range from a few µW to a few tens of µW. Stronger injection results in an exponentially decreasing comb envelope, due to gain saturation of the amplifier^{41}. For P_{0} = 1 µW, f_{s} = 80 MHz, n_{sp} = 2, G = 1/T = 10, and N = 300, the SNR is about 35. Notice that the SNR could be substantially improved by increasing the seed power (i.e., by increasing the saturation power of the amplifier), by reducing the losses of the loop (i.e., increasing T so as to reduce G), and/or by using a low noise optical amplifier.
Architecture of the FSL
The frequency shifting loop (FSL) comprises an erbiumdoped fiber amplifier (EDFA, Pritel, PMFA15), a tunable bandpass filter (TBPF, Yenista, XTM50), and one or two fiber acoustooptic frequency shifters (AOFS, AA Optoelectronic, MT80IIR30Fio) (Fig. 1). The role of the EDFA is to compensate for the passive losses of the FSL. In practice, the EDFA gain is about ~10 dB. In a first configuration, two AOFS are inserted in the loop: the first one generates a positive frequency shift (+80 MHz ± 5 MHz)), and the second one, a negative frequency shift (−80 MHz ± 5 MHz). The frequencies driving the two AOFS are set in such a way that the net frequency shift per roundtrip is f_{s} = 1/τ_{c} = 9.482 MHz (The FSL roundtrip propagation time is: τ_{c} = 105 ns). The loop also contains an optical isolator to prevent backreflection of the light. All fiberoptics components are polarizationmaintaining. The role of the TBPF is twofold: first to control the spectral bandwidth of the light field in the loop (i.e., the maximum number of roundtrips N), and second, to limit the influence of the ASE from the EDFA. A narrowlinewidth (<0.1 kHz) CW laser (OEwaves, HIQ™ 1.5 Micron Laser), delivering 10 mW of optical power at 1550.0 nm is split by means of a 3 dB coupler. The first port is sent to an acoustooptics modulator (AA Optoelectronic, MT80IIR30Fio, the AOM, in Fig. 1), driven by an arbitrary function generator (Keysight, 33600A). The modulation signal e_{in}(t) is defined according to the desired signal at the output of the FSL, and multiplied by a singlefrequency tone with a frequency f_{m} = 80 MHz. Recall that the AOM induces a frequency shift of f_{m} on the CW laser. A variable optical attenuator (not shown in Fig. 1) is used to control the power injected in the FSL. The second port is used as a reference arm (local oscillator) and can be recombined with the FSL output (selfheterodyning). The light circulating in the loop is extracted by means on a 2% Ycoupler. The intensity is detected by a fast photodiode (20 ps risetime), and recorded by a 3.5 GHz, or by a 25 GHzbandwidth realtime digital oscilloscope. All measured waveforms reported in this article are singleshot time traces (no averaging is performed).
This first configuration is particularly well suited to measure the amplitude and phase of the optical spectrum of the output waveforms, since heterodyning with the CW seed laser induces a downconversion from the optical to the RF domain. The first configuration is also used for generating long (~100 ns) arbitrary waveforms. In a second configuration, the same setup is used, except that a single AOFS is inserted in the FSL. It is driven at f_{s} = 9/τ_{c} = 84.91 MHz (τ_{c} = 106 ns). This configuration is used to demonstrate high bandwidth AWG.
Data availability
The data that support the findings of this study are available on request from the corresponding author H.G.C.
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Acknowledgements
This research was supported by the Agence Nationale de la Recherche (grant number ANR14CE320022). We acknowledge Tektronix France for the loan of a 25 GHzbandwidth realtime digital oscilloscope.
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J.A. and H.G.C. initiated the project, C.S. carried out the experiments, C.S. and H.G.C. carried out the theoretical analysis, with feedback from J.A. All authors contributed to the manuscript writing.
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Correspondence to Hugues Guillet de Chatellus.
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Schnébelin, C., Azaña, J. & Guillet de Chatellus, H. Programmable broadband optical field spectral shaping with megahertz resolution using a simple frequency shifting loop. Nat Commun 10, 4654 (2019) doi:10.1038/s41467019126883
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