1. INTRODUCTION

Fourier transformation is a fundamental tool in data processing, but can require large computation resources, depending on the complexity of the signal under test. Therefore, the possibility to achieve analog real-time Fourier transformation (RTFT) of a given signal simply by linear propagation through a suitable dispersive line [1] has attracted a great deal of attention [2 –14]. This concept is based on frequency-to-time mapping (FTM) in second-order dispersive media: provided that a temporal far-field condition is satisfied, the variation of the signal intensity with time at the output of the dispersive medium is proportional to the power spectrum of the signal launched at the input. This property has been extensively used on optical waveforms for a wide range of applications, including real-time spectroscopy [2], interferometry/reflectometry [3], continuous imaging of rapidly changing events [4,5], and temporal magnification or compression of ultrafast optical waveforms for improved detection or generation tasks [6 –8,15]. Notice that optical RTFT has been implemented using mainly optical fibers or chirped fiber Bragg gratings, as these elements provide very large amounts of dispersion over broad spectral bandwidths (above the terahertz range) in practical, lightweight configurations.

Optical-dispersion-based RTFT has also proven to be particularly useful for processing radio-frequency (RF) signals [9 –13]: there is a lack of broadband highly dispersive elements in the RF domain, so the use of fiber-optics dispersive lines has enabled significant improvements in the capabilities of radio-wave analog processing systems [9]. These include fiber-based temporal imaging systems for time-domain magnification [10], compression of gigahertz (GHz)-bandwidth microwave signals [11], or schemes aimed at the generation of high-speed arbitrary RF waveforms [12,13].

However, dispersive-RTFT still suffers from important performance limitations, mainly imposed by the relatively limited amount of dispersion that can be typically achieved in available transparent media. Suppose a light pulse of duration $T$ propagates in a medium with a group-velocity dispersion ${\psi }_2=-L{\partial }^{2}k(\omega )/\partial {\omega }^2$, where $k(\omega )$ and $L$ are the wave vector and the length of the medium, respectively. The far-field approximation in the time domain holds when the following condition is satisfied: $T^2/|{\psi }_2|<\pi /2$. Under this condition, an RTFT process is observed. Considering that the signal temporal duration $T$ is inversely related to the signal’s minimum frequency resolution, the temporal far-field condition limits the frequency resolution of the RTFT process to $\simeq 1/\sqrt{|{\psi }_2|}$ [12]. To give a reference, the dispersion induced by a 1 km long section of a conventional single-mode fiber (SMF-28) is about $2\times {10}^{-5}{\mathrm{ns}}^2$ [16] (slope of the group delay as a function of radial frequency). Thus, the achievable frequency resolution using a 1 km long SMF-28 section exceeds $\sim 200\mathrm{GHz}$, while a frequency resolution of 1 GHz would require the use of a highly unpractical SMF-28 length of about 43,500 km.

Such severe restrictions inherently constrain the resolution of RTFT-based spectroscopy and imaging methods, significantly impairing their practical detection and sensing capabilities, as well as the maximum duration and complexity of the waveforms that can be processed and/or generated through these methods. The complexity of a given waveform is usually quantified using the time–bandwidth product (TBWP) parameter, namely, the ratio of the signal’s full spectral bandwidth to its frequency resolution. Assuming that the 1-km-fiber-based Fourier transformer can be used over the entire erbium-doped amplification band (1530–1565 nm, i.e., $\sim 4.5\mathrm{THz}$), the TBWP of the RTFT system would be limited to $<23$. The frequency-resolution limitation of dispersion-based RTFT schemes is particularly critical for use in photonic-assisted processing of RF signals. Because the bandwidth of RF signals is typically below a few GHz, their duration is usually above the nanosecond range, thus requiring a relatively large amount of group-velocity dispersion for RTFT-based processing operations, e.g., at least a few ${\mathrm{ns}}^2$ for a resolution in the sub-GHz regime [9]. In line with the estimates given above, this requirement is far beyond the capabilities of optical dispersive media.

It should be also mentioned that, similar to a thin-lens-based Fourier transformation system, the use of quadratic temporal phase modulation, namely, a time-lens process, in combination with dispersion, allows one to implement RTFT of time-domain signals without the need to satisfy the temporal far-field condition [14]. However, time-lens-based RTFT schemes ultimately suffer from similar frequency-resolution restrictions, as imposed by the limited temporal aperture of available linear (electro-optics) [15] or nonlinear optics [8] time-lens technologies.

In this paper, we introduce a new concept for RTFT of arbitrary waveforms capable of overcoming the mentioned intrinsic limitations of dispersion-based schemes. Our concept involves the overlapping of a set of signal replicas that are simultaneously shifted in time and in frequency. Provided a specific relationship is satisfied between the two (time and frequency) shifts, the output waveform maps, repeatedly in time, the frequency spectrum of the input signal. Contrary to techniques based on dispersive propagation in the far field, the newly proposed method realizes an exact FTM process of the signal of interest, in the sense that a discrete frequency input, not necessarily restricted to a prescribed limited time duration, results in a discrete time output, with a delay proportional to the input frequency. Moreover, we introduce here a simple technique for practical implementation of our RTFT concept on optical signals using a frequency-shifted feedback (FSF) laser [17]. This novel approach is found to exceed by orders of magnitude the resolution and the TBWP of dispersion-based optical RTFT schemes. Proof-of-concept experiments are reported, demonstrating a frequency resolution of $\sim 30\mathrm{kHz}$ over a TBWP exceeding 400, with the FTM process extending over a bandwidth above 20 GHz. Another key distinctive feature of the demonstrated method is that it provides the desired spectral information with an optimal latency (processing time delay from the input to the output) equal to the inverse of the obtained frequency resolution. This represents yet another important advantage over dispersion-based RTFT, where the latency is determined by the length of the dispersive medium used and is typically orders of magnitude larger than the optimal value associated with the achieved frequency resolution.

2. THEORETICAL DESCRIPTION

A. Frequency Response of an FSF Loop

To explain the fundamental mechanism behind our proposal, recall that a conventional FSF laser consists of a cavity (loop) with a round-trip time ${\tau }_c=1/f_c=2\pi /{\omega }_c$, which typically includes an acousto-optics frequency shifter (AOFS) inducing a frequency shift per round trip $f_s={\omega }_s/2\pi$ [Fig. 1(a), 18]. An optical amplifier is inserted in the loop to compensate for the cavity losses and to increase the photon lifetime, while a coupler enables seeding the loop and extracting a fraction of the intracavity light field. We assume that the optical power in the cavity is small enough to neglect any nonlinear or dynamic phenomenon in the loop and in the gain medium, and to justify a purely linear description of the propagation of the electric field in the cavity. We also neglect any group-velocity or higher order dispersion effects in the loop. As explained in what follows, when the conditions in the described FSF laser configuration are properly set, then the energy spectrum of the optical seed waveform is periodically mapped along the time domain at the laser system output.

 

Fig. 1. (a) Sketch of a generic FSF cavity. The AOFS is driven at frequency $f_s={\omega }_s/2\pi$, resulting in a frequency shift per round trip equal to $f_s$. The cavity round-trip time is ${\tau }_c=1/f_c=2\pi /{\omega }_c$. (b) Optical spectrum of the output of the FSF laser seeded by a monochromatic wave at angular frequency ${\omega }_0$. The envelope of the spectrum of the FSF frequency comb is the function $H$ (see text).

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To evidence the possibility of RTFT in FSF cavities, we calculate the time response of the FSF loop to a monochromatic seed in the form $E_0{e}^{-i{\omega }_{0}t}$. At each round trip in the cavity, the angular frequency is shifted by ${\omega }_s$, and the resulting light field consists of a periodic comb of individual frequencies starting at ${\omega }_0$ and regularly spaced by ${\omega }_s$ [Fig. 1(b)]. As detailed in [17,19], the electric field can be expressed as

Here we consider the specific case where $\varphi$ is a multiple of $2\pi$, i.e., the frequency shift per cavity round trip, $f_s$, is a multiple of the fundamental cavity round-trip frequency, $f_c$. Interestingly, this specific relationship between the time and frequency shifts of the electric field in the FSF loop was first studied by Kowalski, who used the scheme for generation of nanosecond pulses from a passive FSF cavity seeded with a CW He–Ne laser [20]. Under this condition, $e^{i\frac{n(n+1)}{2}\varphi }=1$. The Poisson summation formula for the function $H$ writes

B. Properties of the FSF-based RTFT Process

A direct, linear relationship exists between the input frequency and output time variables: a monochromatic seed (delta function in the frequency domain) is mapped into an optical pulse with a finite time width, $\delta \tau \sim 2\pi /\mathrm{\Delta }\omega$. This imposes a fundamental limitation in the resolution of the FSF-based RTFT process. Considering the FTM rate of the FSF laser system, the intrinsic frequency resolution of the FSF laser technique can be estimated as

In the general case of an input field of the form $E_{\text{in}}(t)=\int {\tilde{E}}_{\text{in}}(\omega )e^{-i\omega t}\mathrm{d}\omega$, the output field is given by the fundamental relationship

 

Fig. 2. (a) Representation of the response of the FSF cavity to a single input frequency. A single seed frequency generates a train of pulses at a repetition rate ${\omega }_s/2\pi$. (b) Principle of FTM in a FSF cavity: the temporal shape of the output pulses maps the spectrum of the input light field.

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Finally, it is also interesting to evaluate the latency of the newly proposed RTFT concept. In this context, latency refers to the processing time delay needed between the signal launched at the input of the system and the Fourier transformed waveform at the system output. This is a key parameter for applications aimed at obtaining the Fourier transformation in real time. Consider the practical case of a time-limited input signal with a frequency bandwidth smaller than $f_c$ and a time duration $T$, and assume that the input waveform is sufficiently long so that the leading edge of the signal has traveled through the whole spectral bandwidth of the FSF loop by the successive frequency shifts before the trailing edge of the input has been loaded into the loop (Fig. 3). Defining $N=\mathrm{\Delta }\omega /{\omega }_s$ as the corresponding number of round trips in the loop, this condition writes $T>N*{\tau }_c=N/f_c$ and simply corresponds to the case when the frequency resolution of the input spectrum (inverse of $T$) is smaller than the frequency resolution of our RTFT technique [as per the definition in Eq. (5)]. In this case, the steady-state regime—where the output pulse train maps the input spectrum—is reached after the input has achieved $N$ cavity round trips, which corresponds to a build-up time (or latency) of ${\tau }_L=N*{\tau }_c=2\pi \mathrm{\Delta }\omega /({\omega }_s{\omega }_c)$. We recover here the exact inverse of the frequency resolution given in Eq. (5), which makes this system optimal in terms of processing time delay or latency. As mentioned above, this constitutes another significant improvement as compared to conventional dispersion-based RTFT schemes, where the latency is set by the propagation time in the dispersive medium before reaching the far-field regime: typically, such latency is orders of magnitude larger than the inverse of the corresponding frequency resolution. Finally, after the trailing edge of the input time-limited signal has entered the loop, the output waveform will simply transition into a zero output signal.

 

Fig. 3. Numerical example of the latency of the FSF-RTFT technique. Left: time-limited input waveform of duration $T$ at the input of the FSF loop. The spectral content consists of five frequency components. Right: intensity at the output of the FSF loop. After a transient time ${\tau }_L$ (or latency), the output signal maps repeatedly the input spectrum.

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3. EXPERIMENTAL RESULTS

The fiber FSF laser we developed for our proof-of-concept experiments includes a free-space AOFS driven at $\sim 80\pm 5\mathrm{MHz}$, an Er-doped fiber amplifier (EDFA), an isolator, and a tunable bandpass filter (Fig. 4). The AOFS is designed to downshift the frequency of the photons. The loop is seeded though a $2\%$ Y coupler by a 1 mW power, narrow line-width (100 Hz) distributed feedback CW laser at 1550 nm, while a fraction of the intracavity optical field is extracted by a second $1\%$ Y coupler. The purpose of the intracavity bandpass filter is to prevent that the amplified spontaneous emission (ASE) generated at frequencies higher than the frequency of the seed laser is amplified in the loop, possibly inducing an undesired modeless operation regime [21]. The output spectrum consists of a comb of optical frequencies spaced by $f_s$. A high-resolution (5 MHz) optical spectrum analyzer shows that the power of the comb lines exceeds the ASE background by more than 20 dB. The spectral bandwidth of the comb can be adjusted by the bandpass filter up to $\simeq 200\mathrm{GHz}$. In the reported experiments, it has been purposely limited down to 30 GHz, in order to match the bandwidth of the output photodetection system (10 ps rise-time photodiode and 28 GHz bandwidth real-time oscilloscope). The length of the FSF loop is set to achieve a cavity round-trip frequency of $f_c=12.89\mathrm{MHz}$ and the frequency of the AOM is set to $f_s=6f_c=77.34\mathrm{MHz}$. When seeded with a single-mode laser, the intracavity light field consists of a train of transform-limited pulses at a repetition rate equal to $f_s$. With the current experimental parameters, the expected frequency resolution and TBWP of the RTFT setup are approximately equal to 30 and 400 kHz, respectively.

 

Fig. 4. Sketch of the fiber FSF laser. The loop is seeded with a phase-modulated CW laser. The AOFS is driven by an arbitrary function generator (AFG). An optical bandpass filter (OBPF) enables control of the spectral bandwidth of the laser and limits the ASE, and an optical isolator (OI) prevents backward laser operation in the loop. A heterodyne interferometer (bottom) enables measurement of the spectrum of the phase-modulated input (see text).

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To demonstrate the capability of the system to achieve optical RTFT, a phase modulation (PM) is applied to the CW seed laser to create sidebands in the input spectrum, by means of an electro-optics phase modulator (EOPM). In a first experiment, the modulation depth is kept low so as to generate only two sidebands. The output time trace consists of periodic sequences of three pulses, the most intense corresponding to the carrier frequency (“carrier” pulses), and the additional ones to the sideband frequency components (“sideband” pulses). As expected, the repetition rate of the sequence is equal to 77.34 MHz, and the time delay between the trains of pulses increases with the frequency of the PM [Fig. 5(a)].

 

Fig. 5. (a) Output time traces recorded for an input laser phase modulated by a sine wave at different frequencies (50, 450, and 1050 kHz). No averaging is performed. (b) Linearity of the frequency-to-time mapping: the time delay between the “carrier” and the “sideband” pulses is plotted versus the PM frequency offset from DC, 1, 10, 15, and 20 GHz. A linear frequency to time mapping curve has been numerically extrapolated over the whole PM frequency range, according to Eq. (4).

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We characterize the linearity of the technique by varying the frequency of the PM up to 20 GHz. The time traces are recorded and the variation of the time delay between the “carrier” and the “sideband” pulses is plotted as a function of the PM frequency. (Note that, for PM frequencies larger than a few GHz, the high-frequency sideband falls outside the transmission of the intracavity bandpass filter and is filtered out by the system: in this case, the output time signal consists of a sequence of only two trains of pulses, corresponding, respectively, to the carrier and the low-frequency sideband.) The experimental data are compared to the theoretical values of the delay, given by $\tau =f_m/6f_c^2$ according to Eq. (4). An excellent agreement is obtained, demonstrating the linearity of the FTM over 20 GHz [Fig. 5(b)].

To infer the experimental frequency resolution of the technique, we compare the output time traces obtained with different values of the PM frequency. As shown in Fig. 6, PM frequencies as close as $\sim 30\mathrm{kHz}$ can be resolved without ambiguity, matching the predicted value for the frequency resolution of the technique.

 

Fig. 6. (a) Experimental output time traces recorded with different values of the PM frequency $f_m$ around 1 GHz. No averaging is performed. (b) The time traces are overlapped to synchronize the “carrier” pulses. (c) Zoom on the “sideband” pulses, showing the capability of the technique to distinguish frequency shifts with the predicted frequency resolution of $\sim 30\mathrm{kHz}$.

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Finally, to demonstrate the use of our scheme for RTFT of optical waveforms with more complex spectra, we apply different PM formats and modulation depths to the seed laser, and we compare the output time trace with the spectrum of the optical seed after PM. The resolution of the optical spectra under analysis is well beyond the capabilities of conventional spectrometers, so an indirect interferometric method is used to obtain an experimental estimate of the relevant spectra. In particular, the input optical spectrum of each evaluated PM seed is measured by recording the heterodyne beating produced by the mixing on a photodiode of the PM seed with a reference auxiliary output from the seed laser itself (before PM) (Fig. 4). The RF spectrum of the photocurrent provides the power ratio of the different spectral sidebands, and the optical spectrum is then reconstructed by symmetry (Fig. 7, top traces). Note that, through this technique, the optical power of the carrier remains undetermined. The optical spectrum is compared to the time trace at the output of the FSF laser and a good agreement is observed in all cases (Fig. 7, bottom traces).

 

Fig. 7. Top: optical spectra of the input laser after two different PM formats, reconstructed from heterodyne beatings with the seed laser (linear scale). The zero-frequency lines are added manually (see text). The PM functions are sine waves at frequencies of 200 (left) and 100 kHz (right). Bottom: time traces recorded at the output of the FSF loop seeded by the corresponding signals (16 traces averaged). The time axis in the bottom plots has been scaled in relation to the frequency axis of the upper plots, according to the FTM scaling law given in Eq. (4).

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4. DISCUSSION AND PERSPECTIVES

For the first time, to our knowledge, we report here an optical RTFT scheme enabling a frequency resolution in the kHz range. In principle, it is possible to improve further this resolution: provided the response time of the detection system ${\tau }_d$ is matched to the spectral bandwidth of the FSF laser ${\tau }_d\approx 2\pi /\mathrm{\Delta }\omega$, the frequency resolution can then be estimated as $f_s{f}_c{\tau }_d$ [Eq. (5)]. The cavity length can be made arbitrarily longer and the frequency shift $f_s$ can be set in the kHz range or less, e.g., by using two AOFS with opposite orientations and slightly different frequencies. Hence, in principle, frequency resolutions at the hertz level or higher are realistically feasible. However, recall that the use of low values of $f_c$ restricts the acceptable bandwidth of the input spectrum. Considering the mentioned trade-offs, the design of an RTFT system based on a FSF laser can be optimized according to the maximum expected spectral bandwidth of the signals to be processed, $\mathrm{\Delta }{\omega }_{\mathrm{signal}}$. First, the value of the cavity round-trip frequency $f_c$ should be set to match the maximum expected signal bandwidth: $f_c=\mathrm{\Delta }{\omega }_{\mathrm{signal}}/(2\pi )$. Then, to increase the frequency resolution, $f_s$ needs to be minimized while the ratio $f_s/f_c$ remains an integer, which sets $f_s=f_c$. Thus, the optimal frequency resolution and TBWP of an FSF-based RTFT system designed to process a maximum radial-frequency bandwidth signal of $\mathrm{\Delta }{\omega }_{\mathrm{signal}}$ are ${(\mathrm{\Delta }{\omega }_{\mathrm{signal}})}^2/(2\pi \mathrm{\Delta }\omega )$ and $\mathrm{\Delta }\omega /\mathrm{\Delta }{\omega }_{\mathrm{signal}}$, respectively.

As discussed, the maximum bandwidth of the signal to be processed is limited by the cavity round-trip frequency, $f_c$, meaning that an unknown input optical spectrum can be inferred without ambiguity by the technique if its spectral bandwidth is smaller than $f_c$. Moreover, provided some a priori information is known on the input spectrum, spectral features can be detected over a 20 GHz bandwidth with a 30 kHz resolution. In practice, $f_c$ is limited up to a few hundreds of megahertz (MHz) with current fiber laser technologies. Reaching the GHz bandwidth range would require integrated technological solutions, combining in the same optical circuit a gain medium and a frequency shifter. An integrated FSF laser based on a doped lithium niobate waveguide has been previously demonstrated, and a value of $f_c$ larger than 700 MHz was reported [22]. It is expected that progress in miniaturization should enable further improvement of this value well into the GHz range. Ultimately, higher values of $f_c$, in the tens of GHz range, may be challenging to reach with present technological solutions. Also note that, in the case of values of $f_c$ in the GHz range, the frequency shift per round trip, $f_s$, should be also similarly increased. Fiber-optics FSF lasers with frequency spacing above 10 GHz have been successfully developed using surface acoustic waves or electro-optic single sideband modulation technologies for frequency shifting [23]. Assuming that a FSF laser could be created with a frequency shift and cavity round-trip frequency of 10 GHz, operating over the entire erbium-doped amplification band ($\sim 4.5\mathrm{THz}$), this would enable unambiguous RTFT of optical signals with a resolution of 20 MHz over bandwidths up to 10 GHz, corresponding to an operation TBWP exceeding 450. Note that, in this case, a temporal detection system with subpicosecond resolutions is needed, which can be practically implemented using nonlinear optical sampling or autocorrelation techniques.

5. CONCLUSIONS

We have proposed and demonstrated a new concept for realization of RTFT, i.e., frequency-to-time mapping (FTM), of temporal waveforms. We have also shown how the concept can be implemented on optical signals using a FSF laser, a realization that enables scaling the RTFT operation to achieve frequency resolutions that are simply not possible using conventional dispersion-based optical RTFT schemes, e.g., down to $\sim 30\mathrm{kHz}$ in the reported example. This would enable dramatic improvements in the performance (e.g., in terms of frequency, time, or spatial resolutions) of optical RTFT-based applications. For example, in spectroscopy, the possibility to record optical spectra in real time with such an unprecedented frequency resolution would enable instantaneous measurements of narrow line-width atomic and molecular transitions (e.g., homogeneous line-width or long lifetime transitions). In ranging and reflectometry, this work opens new avenues, potentially enabling, for instance, direct detection of distant targets moving at speeds as low as a few centimeters per second. Additionally, the FSF-based RTFT technique offers the potential for significant frequency-resolution enhancement in schemes aimed at detection, measurement, or generation of optical or RF waveforms. The FSF-based RTFT technique also provides a high TBWP, exceeding 400 in our reported experiments, while offering an optimal latency for computation of the signal Fourier transformation, limited only by the fundamental time-frequency uncertainty relationship.

FSF-based RTFT shows other important distinctive features. First, the output time-domain waveform, proportional to the input spectrum, is periodically repeated and not limited to a single time trace, even in the case of a nonperiodic input optical waveform. This fact could be potentially exploited to increase the output signal-to-noise ratio through averaging of consecutive time-domain spectra. Second, contrary to dispersion-based RTFT, where time windowing of the input signal is strictly required according to the frequency-resolution constraints of the dispersive line used, the RTFT method proposed here can be applied on infinitely long signals, without requiring any temporal truncation. This suggests the possibility of using the method for RTFT of incoherent or partially coherent light waves.

Finally, it should be also noted that the high frequency resolution achieved with the proposed method inherently implies a correspondingly narrow operation bandwidth, even if this benefits from a very high TBWP, as well. We have discussed potential alternatives to scale the technique in order to achieve operation bandwidths exceeding 10 GHz. This perspective should be particularly appealing for applications on RF signals, or to process broader optical bandwidths following spectral segmentation, e.g., using available wavelength-division multiplexing schemes. Furthermore, we also anticipate that the fundamental, general concept introduced here, involving linear interference of consecutive signal replicas simultaneously shifted in both time and frequency, might be implemented using other photonic designs, beyond the FSF laser scheme studied here. This could potentially allow reaching a broader range of performance specifications. Therefore, we expect FSF laser cavities, and more generally, the RTFT method introduced here, to play an important role in future optical and RF signal processing systems.