1. Introduction

Acquirement of radio frequency (RF) or microwave spectrum information is fundamental in many fields such as wireless communication, radar, and electronic warfare. Real time operation is critical for military applications, which is now widely obtained by high-speed digitalization and following fast Fourier transform (FFT). However, the observation bandwidth is typically limited by analog to digital converter (ADC) as well as digital processing capacity, so people turn to photonics-assisted spectrum analysis approaches for further extended bandwidth and acquisition frame rate [1]. For example, huge-bandwidth ADC can be achieved by optical ultrafast sampling [2] or time stretch [3], but massive data still consume intolerant digital processing delay. The photonic compressive sampling cuts down the measurements, which can only deal with spectrally sparse signals [4,5]. Mapping the frequency information to other physical quantity for direct and more convenient measurement attracts many interests, for example, the frequency-to-power mapping (FPM) [6–8]. In order to correctly map multiple input RF carrier, an optical channelizer is usually employed before mapping [9,10]. Since optical-filter-based channelization has limited resolution (~GHz), further improvement needs additional implementation such as digitalization and processing [11,12].

Frequency to time mapping (FTM), also called analog real-time Fourier transform (RTFT), is an effective RF spectrum acquirement method [13–18]. The variation of optical intensity with time at the output of optical RTFT is proportional to the power spectrum of RF signal launched at the input, so that broadband spectrum can be read directly by high-speed ADC or modern oscilloscope, saving large computation resources. FTM is typically realized by second-order chromatic dispersion. Through electro-optic modulator, a long, chirped pulse is frequency-shifted by input RF signal. The dispersion, on one hand, shifts the pulse delay according to its frequency shift, and, on the other hand, de-chirps and compresses the pulse so that pulses with different delays can be distinguished by oscilloscope [the principle is shown in Fig. 1(a)]. Though photonic devices, such as optical fiber and fiber Bragg gratings (FBGs), have provided much larger amounts of dispersion over broader spectral bandwidth than RF ones, dispersive RF FTM still suffers from limited dispersion value achieved in practical transparent media. For example, in order to distinguish two RF tones where frequency difference is 25 MHz, the dispersion should be as large as 1 × 10⁵ ps/nm, corresponding to more than 5800-km standard single mode fiber (SSMF), if the de-chirped pulse is 20 ps long (which is almost the highest time resolution of commercial real-time oscilloscope). A few approaches were proposed to solve the problem. In [19], broadband optical source and optical dispersion were used to equivalently achieve large microwave dispersion. In [20], without enlarged dispersion, ultra-short optical pulses after de-chirping were employed while auto-correlation detection was used to distinguish them instead of conventional oscilloscope. Time stretch was also proposed for superb time-resolved detection after regular-dispersion-based FTM [21].

 

Fig. 1 Comparison between pulse de-chirping by continuous dispersion and by discrete dispersion. In discrete one, quadratic-phase-distributed but intensity-sliced frequency response results in periodic output of de-chirped pulse.

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Reference [22] demonstrated an inspired solution for equivalent ultra-large dispersion and kilohertz RTFT resolution. By an active fiber cavity with frequency shifted feedback (FSF), a set of optically-carried signal replicas are simultaneously shifted in time and in frequency. At the cavity output all the replicas are summed together, which maps the frequency spectrum of the input signal, following exactly the mathematical definition of Fourier transform. Extremely large equivalent dispersion was achieved to shift the output optical pulse by around 30 ps when RF carrier was under deviation as small as 30 kHz. However, the active cavity contains lossy frequency shifter and optical amplifier, which limits its free spectral range (FSR) as well as unambiguous RTFT bandwidth. Besides, the in-loop accumulated amplified spontaneous emission (ASE) noise may show impact on RTFT performance.

In this paper we propose a novel spectrally-discrete chromatic dispersion media for real-time Fourier transform. Such media has periodical ON/OFF intensity frequency response, while its phase response is the same or approximately the same as the traditional like fiber. We show in theory that discrete dispersion can also de-chirp and compress the matched chirped optical pulse, while in time domain periodically outputting transform-limited pulses. FTM can then be achieved during each period. Then we propose that such discrete dispersion can be implemented simply by cascaded ring resonators. FTM performance is studied numerically. In experiment, one passive fiber ring, with FSR around 400 MHz, is employed and FTM with resolution around 25 MHz is achieved. Highly-linear and unambiguous FTM is observed within 400-MHz bandwidth under 20 GHz. Possible advantages over continuous dispersion as well as active FSF loop are discussed.

2. Principle

Figure 1(a) shows the continuous-dispersion-based RTFT. Time-limited RF signal, $x\left(t\right)$, is firstly carried by a chirped optical carrier [which could be achieved, in practice, by carrier-suppressed single-side-band (CS-SSB) modulation], then passes dispersive media defined by its frequency response, $\text{exp}\left(i{\beta }_2{\text{Ω}}^2/2\right)$, where $\text{Ω}$ is the angular frequency offset from the center carrier. When the optical carrier chirp matches dispersion as $\text{exp}\left(i{t}^2/2{\beta }_2\right)$, the output optical waveform is then proportional to $X$, the Fourier transform of $x$, since

In this paper, we replace the above continuous dispersion by its spectrally-discrete version, as shown in Fig. 1(b). Mathematically, the original pure-phase frequency response is periodically intensity-modulated by ${\displaystyle \sum }_{k}P\left(\text{Ω}-k\cdot 2{\text{πFSR}}_{\text{DD}}\right)$ where ${\text{FSR}}_{\text{DD}}$ is the free spectral range of comb filter, and $P\left(\text{Ω}\right)$ describes one passband. Expression inside F ^{-1$〈\cdot 〉$} in the first line of Eq. (3) describes the frequency response of discrete dispersion. Such discretization results in periodic appearance of $X$ in time domain, since

Note in an ideal discrete dispersion described by Eq. (3), each passband inside still follows the dispersive phase response. For easy implementation, we further propose that the discrete dispersion could be approximated by a comb filter with quadratic phase retardation distribution among separated passbands, since $\text{exp}\left(i{\beta }_2{\text{Ω}}^2/2\right)\cdot {\displaystyle \sum }_{k}P\left(\text{Ω}-k\cdot 2\pi {\text{FSR}}_{\text{DD}}\right)\approx {\displaystyle \sum }_k\text{exp}\left[i{\beta }_2{\left(k\cdot 2\pi {\text{FSR}}_{\text{DD}}\right)}^2/2\right]\cdot P\left(\text{Ω}-k\cdot 2\pi {\text{FSR}}_{\text{DD}}\right)$. That is, phase retardation at each passband center is designed, instead of detailed dispersive curve in each passband. Assume

Advantage of our proposal is the much smaller volume and possible integrated implementation of large chromatic dispersion, because the discrete phase response as shown in Eq. (3) could be approached without real dispersion. We here propose that such comb filter could be achieved by ring combinations. In the simplest case, i.e. $M=1$, the discrete dispersion degenerates into a pure-intensity comb filter with uniform phase response ${\phi }_k\equiv 0$ among each channel, which could be an optical ring resonator with FSR of $B_{\text{FTM}}$. When $M>1$ phase retardation in Eq. (5) will be repeated every $M$ channels since $\text{exp}\left(i{\phi }_{k+M}\right)=\text{exp}\left(i{\phi }_k\right)$. As a result, such response could be obtained by several combined ring resonators. All rings have the same FSR as $B_{\text{FTM}}$, while their resonant frequency and phase retardation are controlled independently: for the $m$^th group of channels, $m\in [1, M]$, the center is at $f_m=m/M\cdot B_{\text{FTM}}$, and all have uniform phase retardation as $mM\cdot \pi +\pi m^2/M$. The rings can be either in parallel or in series. Figure 2 shows a design example where $2M$ resonators are employed and the $m$^th couple is used to realize the $m$^th group of channels.

 

Fig. 2 A comb filter example based on optical ring resonators which could approximate the desired discrete dispersion by individual phase control instead of huge-volume real dispersion. f_m is the resonant frequency of the corresponding ring.

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As mentioned, we design the discrete phase retardation at each passband center while let the detailed dispersive curve in each passband alone, so that comb filter by rings is an approximation to ideal sliced dispersion in Eq. (3). Single channel, $P\left(\text{Ω}\right)$, should then have narrow enough bandwidth so that the comb filter is able to precisely approach the target under given input. In other words, though the whole comb functions as dispersion, its single channel deviates far away from each slice [as shown in Fig. 3(b)], so that signal spectrum variation within channel bandwidth cannot be distinguished even though a higher $\text{RBW}$ is calculated under extra optical bandwidth, $B_{\text{O}}$, according to Eq. (2). As a result, channel bandwidth, defined as $\text{RBW}$ here, sets the best available resolution of the proposed FTM, which could be achieved when $B_{\text{O}}$ and duration of input chirp pulse, $T_{\text{O}}$, are larger than

 

Fig. 3 (a) Simulated power transmission spectrum of the proposed ring-based discrete dispersion shown in Fig. 2. (b) Blue line: the corresponding phase response; red dot: wrapped phase retardation at each transmission peak; red dotted line: wrapped quadratic fit.

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As a numerical example, here we design an FTM with $B_{\text{FTM}}=5 \text{GHz}$ and $\text{RBW}=50\text{ MHz}$. Forty rings are combined as Fig. 2 so that $M=20$. According to Eq. (5) the equivalent dispersion is about 1.27 × 10⁵ ps², corresponding to SSMF of 5800 km around. The required bandwidth of input chirped optical pulse is 25 GHz, according to Eq. (6), and its duration is 20 ns. A rectangle time window is employed. For such bandwidth, the chirping could be achieved electronically through frequency multiplexing or broadband arbitrary waveform generator (AWG) directly, which is then followed by CS-SSB opto-electronic modulation on a continuous-wave (CW) light. Such generation is easy to implement and the resulted chirped pulse could have full duty-cycle under long duration, which may benefit RTFT of infinitely-long input signal [22].

We assume rings are loss-free, and loop delay is 200 ps for all. Cross power ratio of all couplers between ring and straight bus waveguide is 4.8%, and round-trip phase of each ring and phase shifter in each pair are adjusted so that all transmission peaks are correctly located both in frequency and in phase according to Eq. (5). The ring combination is simulated, and its frequency response within $B_{\text{FTM}}$, i.e. of successive $M$ channels, is shown in Fig. 3(a). 3-dB bandwidth of each channel, i.e. the designed $\text{RBW}$, is 50 MHz. In Fig. 3(b), though nonlinear phase responses are found in all channels, which are individually far from the wanted dispersion, the discrete phase retardation at each peak fits the desired quadratic curve well.

In simulation, input pulse is firstly frequency shifted randomly within $[0, B_{\text{FTM}}]$, and then passes through the rings. Output waveforms in time domain are plotted together and shown in Fig. 4. The black line in Fig. 4(b) corresponds to non-shifted input. Periodic output with power fading on both sides is observed, and the period is $M/B_{\text{FTM}}=4 \text{ns}$ as predicted by Eq. (3). One period is zoomed in and plotted in Fig. 4(c), where each color corresponds to a specific frequency shift. Nearly uniform intensity is obtained. Output delay offsets from non-frequency-shifted one are calculated, and the frequency-dependent time delay is plotted in Fig. 4(c). The delays are in precise proportion to the preset 16 random frequency shifts from 0 to $B_{\text{FTM}}$, and the slope is 0.8 ps/MHz. At the minimum shift of 50 MHz, the output waveform is shown in Fig. 4(a), together with the non-shifted one. The pulse interval from peak to peak is exactly 40 ps while each pulse duration is also 40 ps around (i.e. $1/B_{\text{O}}$). We then conclude that the resolution of FTM is the preset value of 50 MHz.

 

Fig. 4 Simulated FTM through the proposed ring-based FTM. (a) Output waveform under zero (black line) and 50-MHz frequency shift (dark blue line). (b) Periodic FTM output. (c) Zoom in of one period; waveform with different color corresponds to specific frequency shift. (d) The calculated frequency to delay mapping (colorful dots) and theory prediction (red line).

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3. Experiment result

A proof-of-concept experiment is carried out based on a single ring, i.e. $M=1$. In this case the comb filter has constant phase retardation among passbands. FTM is explained by Eq. (3): the comb filter could be seen equivalently as a continuous-dispersion media with ${\beta }_2=1/\left(2\pi \cdot B_{\text{FTM}}^2\right)$, which is then cascaded by the same constant-phase comb filter. The former performs the FTM and the latter periodically outputs the result. Experiment setup is shown in Fig. 5, and key performance, such as mapping slope, unambiguous bandwidth, as well as mapping resolution is measured.

 

Fig. 5 Experiment setup. e-AWG: electronic AWG; e-LO: electronic local oscillation; RF amp: RF amplifier; PM: phase modulator; OC: optical coupler; OSC: real-time oscilloscope; red line: optical fiber; blue line: electronic cable. Inset shows the measured power transmission of the passive fiber loop.

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The passive loop is a 0.5-meter-long single mode fiber with delay around 2.5 ns, which is connected with optical in- and output through two 90/10 2 × 2 optical couplers. Accordingly, the unambiguous FTM bandwidth is $B_{\text{FTM}}\approx 400 \text{MHz}$, and the achieved dispersion is ${\beta }_2\approx {10}^6 {\text{ps}}^2$, equivalent to ~4.6 × 10⁴-km SSMF. The measured 3-dB bandwidth of single transmission peak is $\text{RBW}\approx 25\text{ MHz}$. Based on Eq. (6), the required optical bandwidth, $B_{\text{O}}$, for input chirping should be larger than 6.4 GHz, duration, $T_{\text{O}}$, should be larger than 40 ns, and its chirping rate is 160 MHz/ns. In our setup, chirping, centered at $f_c=7 \text{GHz}$, is provided by an electronic AWG (Tektronix, AWG70001A), and the output is $\propto \text{Reexp}\left(-i2\pi f_{c}t+i{t}^2/2{\beta }_2\right),-20\text{ ns}\le t\le 20\text{ ns}$. Sampling rate is 45 GS/s. AWG output is up-converted to 24 GHz by an electronic mixer and local oscillation (LO), power amplified, and phase modulates a CW lightwave (AOI, D5572XOPB16, linewidth is about 30 kHz). The desired chirped optical pulse is then obtained after a single-sideband filter (Finisar Waveshaper, 1000S). The chirped pulse is power-divided, and one part is CS-SSB modulated by RF signal to be Fourier-transformed, which is coupled with the other un-modulated part under proper power. The latter, without frequency shift, offers a zero-frequency reference after FTM. The coupled lightwave, passing through single-loop FTM, is opto-electronically converted by a high-speed photo detector (PD, DSC30S) and is received by a real-time oscilloscope (Agilent, DSO91204A).

The desired frequency-dependent time delay is observed by oscilloscope when single RF tone with four different frequencies (20.16 GHz, 20.08 GHz, 20.00 GHz, and 19.92 GHz, respectively) is input. The results are shown in Fig. 6(a), without average. Under each input, the output waveform contains periodic sequence of two pulses. The stronger one corresponds to the un-modulated input (“reference” pulse), while the weaker one corresponds to the frequency-shifted part (“FTM” pulse). Period of the two pulses is 2.5 ns, consistent with our theory. Within one period, time interval between two pulses increases when the driving frequency changes monotonically, which illustrates FTM. Note that without any active control, the total phase retardation of long fiber loop is unstable, which randomly drifts between 0 to 2π. As a result, the output waveform drifts too in time domain. However, “reference” pulse acts as benchmark so that the input frequency is mapped to delay offset from “reference” pulse rather than absolute time position. Within 400 MHz, the mapping is plotted in the last line of Fig. 6(b) without ambiguity. Excellent linearity is found with slope of 6.25 ps/MHz, which agrees with the designed ${\beta }_2$.

 

Fig. 6 (a) Output waveforms when single RF tone with four different frequencies around 20 GHz is input. “Reference” pulses are aligned in time. No averaging is performed. (b) Measured driving-frequency-dependent time delay (dots) of the “FTM” pulse offset from “reference” pulse. Line: theoretical prediction. Four sub-plots show FTM at different unambiguous 400-MHz bandwidth.

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If input frequency is out of the 400-MHz window, the tracked “FTM” pulse also moves out of the 2.5-ns time window but a new “FTM” pulse enters from the other side. Such periodic FTM could be observed within the whole RF-optic modulation bandwidth. Figure 6(b) shows FTMs around 20 GHz, 15 GHz, 10 GHz, and baseband 0 GHz. Four sub-plots show FTM at different unambiguous 400-MHz bandwidth. In each frequency window, one can observe the same mapping slope and good linearity. Note at low frequency our CS-SSB modulation does not work well and two sidebands are input into FTM loop at the same time. As a result, the unambiguous bandwidth is reduced by half (i.e. 200 MHz).

In Fig. 7(a), three driving frequency as close as 25 MHz (10.125 GHz, 10.15 GHz and 10.175 GHz) are input in turn, and the corresponding output waveforms are plotted together. Note all “reference” pulses are aligned. One can see that duration of each “FTM” pulse is around 150 ps ($1/B_{\text{O}}=156.25 \text{ps}$), while their separation is 156.25 ps according to linear fitting of Fig. 6(b). Three “FTM” pulses are clearly distinguished, and the resolution of our FTM is around 25 MHz, matching the theoretically predicted value ($\text{RBW}$, single channel bandwidth). Such resolution can be found at each unambiguous window, for example, at 15.3 GHz, 18.1 GHz, and 20.1 GHz, as shown in Figs. 7(b, c) and 7(d), respectively.

 

Fig. 7 Output waveforms when three driving frequencies, separated by 25 MHz, are input individually. “Reference” pulses are aligned in time. No averaging is performed.

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4. Conclusion

In summary, we proposed a novel RF-to-time mapping based on comb filter with spectrally-discrete dispersion. Experimentally, with a single fiber ring resonator, we achieved real-time FTM under 400-MHz unambiguous bandwidth, high linearity of 6.25 ps/MHz, and 25-MHz resolution. Compared with traditional continuous-dispersion-based FTM, our proposal provides much larger dispersion with compact size, e.g. ${\beta }_2\approx {10}^6 {\text{ps}}^2$ (equivalent to ~4.6 × 10⁴-km SSMF) with only 0.5-meter-long fiber loop in our proof-of-concept experiment. Instantaneous bandwidth or equivalent dispersion could be further extended by cascading more ring resonators. A numerical example was then studied, where compact property could be preserved since phase retardation instead of true time delay is controlled. Compared with existing equivalent dispersion scheme such as that by frequency shift feedback loop, our proposal employs simple passive cavity so that more compact device could be achieved, avoiding accumulated in-loop noise. Further, capability to integrate lots of rings could reduce the required optical bandwidth as well as the difficulty during final real-time observation.