## Abstract

We propose and demonstrate a fiber-optic incoherent signal processing scheme to achieve extraordinary dispersion amounts on arbitrary microwave signals with bandwidths over tens of GHz. Using this new scheme, we experimentally achieve microwave dispersion values approaching 24 ns/GHz (equivalent to the dispersion induced by a section of standard single-mode fiber with a length of ~185,000 km). The scheme is used for real-time Fourier transformation (linear frequency-to-time mapping) of nanosecond-long microwave signals, including a square-like waveform, a sinusoidal pulse and a double pulse waveform, with bandwidths over 20 GHz.

©2010 Optical Society of America

## 1. Introduction

A wide range of important time-domain signal processing operations are based on the use of large amounts of chromatic dispersion, usually first-order dispersion (corresponding to a linear group delay), over temporal waveforms. Dispersion-engineering has been extensively used for applications such as real-time Fourier transformation (RTFT) [1–5], real-time reflectometry and interferometry [6,7], pulse repetition rate multiplication [8], temporal imaging [9,10] etc. Similar concepts have also proved useful in other frequency regions, including the microwave domain. The lack of proper devices or media capable of introducing large amounts of dispersion on microwave signals, particularly over bandwidths in the GHz range, led to the use of photonic dispersive lines for processing microwave-domain information [11,12]. Relevant examples of the latter are fiber-based temporal imaging systems for time-domain magnification [11] or compression [12] of GHz-bandwidth microwave signals, e.g. of interest for high-speed analog-to-digital conversion or broadband arbitrary waveform generation. While photonic dispersive lines (e.g. optical fibers or linearly chirped fiber Bragg gratings, LCFGs [2]) offer operation bandwidths easily in the THz range, i.e. well beyond those needed for operations on microwave signals, they are however more limited in what concerns the amount of dispersion that they can provide (e.g. the dispersion introduced by a 120-km long section of standard single-mode fiber, expressed as the group-delay slope as a function of the *radial* frequency variable, is ~2.5 × 10^{−3} ns^{2}) [13]. It should be noted that generally speaking, the dispersion required for most of the above mentioned applications, e.g. RTFT, scales up with the square of the time duration of the signal to be processed. Thus, microwave waveforms, with typical durations in the sub-nanosecond to nanosecond regime, require the use of dispersion values (up to a few ns^{2}) significantly larger than those needed for processing optical waveforms. Solutions to this problem have been recently sought in the microwave domain: Dispersive lines based on the concept of chirped electromagnetic bandgap (EBGs), the microwave counterpart of an LCFG, have been recently created and demonstrated for processing nanosecond-long microwave waveforms [14–16]. These devices can provide dispersion values significantly higher than their photonic counterparts, still typically limited to <0.3 ns^{2}, but they offer smaller operation bandwidths, typically <10 GHz. Thus, in the microwave domain there is still a lack of a suitable device or medium capable of inducing a large amount of group delay dispersion, namely up to a few ns^{2}, over a wide frequency range (0 ~50 GHz).

In this communication, we propose and demonstrate a fiber-optic incoherent signal processing scheme to achieve extraordinary dispersion amounts on microwave signals with bandwidths over tens of GHz. The proposed scheme employs a highly dispersive fiber-optics medium and its operation induces a large amount of linear dispersion, with a value up to several orders of magnitude higher than that of the optical dispersive line, on the output microwave signal. The proposed system, which can be explained and designed using the time-spectrum convolution (TSC) concept [17], is practically implemented by temporally modulating a broadband incoherent light source driven by the microwave signal to be processed and subsequently applying chromatic dispersion through a suitable linear optical dispersive medium followed by self-interference in a dispersion-unbalanced interferometer. Using this new scheme, we demonstrate here microwave dispersion values approaching 3.8 ns^{2} (equivalent to the dispersion induced by a section of standard single-mode fiber of ~185,000 km) using a 10-m long LCFG providing a dispersion equivalent to ~120 km of single-mode fiber. The scheme is indeed used for RTFT (linear frequency-to-time mapping) of practically relevant nanosecond-long microwave signals, including a square-like waveform, a sinusoidal pulse and a double pulse waveform, with bandwidths over 20 GHz. Hence, our proof-of-concept experiments illustrate the outstanding performance offered by the proposed approach, clearly overcoming the critical limitations of present dispersive line solutions for microwave signals in terms of both dispersion values and operation bandwidths.

## 2. Operation principle

A schematic of the proposed incoherent photonic processing scheme for implementing ultrahigh dispersion on microwave waveforms is shown in Fig. 1(a) . The proposed setup, which resembles a general architecture previously used for discrete-time [18] and continuous-time [19,20] microwave photonic filtering, can be better understood using the TSC concept recently discussed in Ref [17]. This concept is practically implemented by temporally modulating a specially filtered broadband incoherent light source with the input microwave waveform to be processed, followed by linear propagation through a suitable first-order dispersive medium. The average intensity waveform obtained at the output of this system (e.g. after fast photo-detection) is determined by the convolution of the input microwave signal temporal profile and a time-domain version of the incoherent source energy spectrum, scaled along the time-domain according to the group-delay curve of the optical dispersive medium. Mathematically,

where ⊗ denotes a convolution and the symbol ∝ holds for proportionality,*I*(

_{out}*t*) is the average temporal intensity waveform at the system output (

*t*is the time variable),

*I*(

_{mw}*t*) is the temporal intensity profile of the modulation waveform,

*S*(

*ω*) is the spectral energy density of the incoherent broadband light source after amplitude filtering (

*ω*is the base-band radial frequency variable referred to the central frequency of the light source), and

*D*

_{0}is the dispersion of the optical dispersive line, defined as the slope of the (linear) group delay, (

*T*) as a function of the radial frequency variable

_{g}*ω*, i.e. (∂

*T*/∂

_{g}*ω*) Eq. (1) holds as long as the full-width frequency bandwidth of the intensity modulation waveform Δ

*ω*is sufficiently small so that the waveform is not affected by propagation through the optical dispersive line; this can be approximately estimated using the following inequality [17]:Otherwise, the time-domain Fresnel integral of the input microwave waveform should be considered in Eq. (1) [17], [19] (for a sufficiently large optical dispersion, the Fraunhofer approximation of this integral holds [21]). The idea to implement a dispersive line using the convolution process described by the general Eq. (1) is based on the fact that linear propagation of a given signal through a dispersive medium is mathematically described by the convolution of the time-domain signal with the medium’s temporal impulse response [2]. In the case of a first-order dispersion medium (i.e. a ‘transparent’ medium inducing a linear group delay over the entire bandwidth of the propagating signal), this impulse response can be mathematically expressed as a sinusoidal variation with a quadratic temporal phase profile, in particular ${h}_{mw}(t)\propto \mathrm{cos}\left({\omega}_{0}t+\left[1/2{D}_{mw}\right]{t}^{2}\right)$ where

_{mw}*ω*

_{0}is the central operation frequency of the microwave dispersive line (

*ω*

_{0}= 0 for an operation bandwidth centered at DC) and

*D*is the first-order dispersion value of this dispersive device (also defined as the slope of the group-delay as a function of the radial frequency). Hence, in order to achieve the desired impulse response using the describe TSC scheme, one needs to re-shape the energy spectrum of the incoherent broadband light source to follow the desired quadratic-phase variation. In practice, this can be implemented by linearly filtering a uniform-spectrum broadband incoherent light source with a dispersion-unbalanced optical interferometer, i.e. an interferometer in which different amounts of dispersion are introduced in its two arms, see illustration in Fig. 1(b).

_{mw}Let us assume an optical interferometer characterized by a relative time delay τ between its two arms and a first-order dispersion difference between its arms given by $\mathrm{\Delta}D={D}_{1}-{D}_{2}$, where *D*
_{1} and *D*
_{2} are the first-order dispersion values respectively introduced in the first and second arm of the interferometer. It can be easily inferred that the intensity spectrum at the output of such an interferometer when broadband light with an intensity spectrum *G*(*ω*) is launched at the interferometer’s input is given by $S(\omega )\propto G(\omega )\cdot \left[1+\mathrm{cos}\left(\tau \omega +\left[\mathrm{\Delta}D/2\right]{\omega}^{2}\right)\right]$. Ideally, the incoherent light spectrum should be approximately uniform over its full-width frequency bandwidth, $\mathrm{\Delta}{\omega}_{opt}$. The time-domain scaled version of this intensity spectrum, $S\left(\omega =t/{D}_{0}\right)$, resembles the ideal impulse response of a first-order microwave dispersive line, ${h}_{mw}(t)$, except for an additional nearly constant background. In particular, using the frequency-to-time mapping law determined by the optical dispersive line, $\omega =t/{D}_{0}$, one can derive that the resulting impulse response is proportional to that of an ideal first-order microwave dispersive line, $S\left(\omega =t/{D}_{0}\right)\propto 1+\mathrm{cos}\left({\omega}_{0}t+\left[1/2{D}_{mw}\right]{t}^{2}\right)$, with the following equivalent central frequency and dispersion values:

*N*, whose magnitude can be made much larger than 1 by properly decreasing the dispersion unbalance in the used interferometer. Dispersion values several orders of magnitude larger than the nominal dispersion amount introduced by the optical dispersive line can then be achieved on the propagating input microwave modulation waveform using the proposed scheme. The obtained microwave dispersive line is spectrally centered at ${\omega}_{0}$ defined in Eq. (3a), i.e. this can be spectrally centered at DC by fixing the relative delay in the interferometer to

*τ*= 0. Notice also that the limited time duration Δ

*t*of the resulting impulse response, $\mathrm{\Delta}{t}_{h}\approx \left|{D}_{0}\right|\mathrm{\Delta}{\omega}_{opt}$, translates into a limited operation bandwidth approximately given by

_{h}*N*| times narrower than the optical bandwidth of the incoherent light source. Nonetheless, the condition on the microwave bandwidth given by the inequality in Eq. (2) is typically more restrictive than the condition in Eq. (4). A larger microwave operation bandwidth can be achieved by use of an optical dispersive line with a lower dispersion.

As proved by our proof-of-concept experiments, the dispersion unbalance in a practical optical interferometer can be designed to be sufficiently low so that to achieve a very high equivalent microwave dispersion (see Eq. (3).b)), namely up to a few ns^{2}, using a relatively modest optical dispersion in the system, which in turn enables processing a relatively high microwave bandwidth (see Eq. (2)), in the tens of GHz range. In this way, one can achieve sets of microwave dispersion – bandwidth values that otherwise are not possible.

In the experiments reported here, we have used the newly introduced photonic-based microwave dispersive line concept to implement real-time Fourier transformation (RTFT) of GHz-bandwidth microwave signals [14,15]. RTFT is based on mapping the spectral-domain information of an incoming time-limited waveform along the time domain by simple linear propagation through a first-order dispersive element [1–7]. It is well known that in order to achieve RTFT the dispersion must be sufficiently large to satisfy the following approximate condition [2]: $\left|{D}_{mw}\right|>>\mathrm{\Delta}{t}_{mw}^{2}/8\pi $, where Δ*t _{mw}* is the full-width time duration of the microwave waveform to be processed. In our specific case, if this RTFT condition is satisfied and the frequency content of the input microwave signal is within the operation bandwidth of the created microwave dispersive filter, the average time-domain intensity profile measured at the system output will be approximately given by the following expression:

## 3. Experiments

Figure 1(b) shows a schematic of the experimental setup used in our proof-of-concept demonstrations. Incoherent broadband CW light centered at 1528 nm was generated using a superluminescence diode (SLD-6716-11748.5.A02, Covega inc.) with 2 mW average power and 80 nm bandwidth followed by a semiconductor optical amplifier (SOA) (BOA-3876, Covega inc.). The amplified output power was 28 mW. Fiber-optic isolators were used at the input and the output port of the SOA. The amplified CW light was modulated in intensity using an electro-optic Mach-Zehnder modulator, MZM (10Gbps, SDL inc.) driven by the microwave temporal waveform to be processed, which was generated by different means depending on the specific experiment. The modulated light was subsequently dispersed through a highly dispersive fiber-optics medium: (i) an LCFG operated in reflection (*D*
_{0} ~−2549 ps^{2} over a total bandwidth of ~42 nm centered at 1548 nm, Proximion inc. [22]) for the first experiment (processing of a microwave square pulse), and (ii) a dispersion compensation fiber (DCF) (*D*
_{0} ~491.6 ps^{2} at 1560 nm, Pureform DCF3000B, Avanex inc.) for the second and third experiments (processing of a sinusoidal pulse and of a double pulse waveform).

In our experimental setup, the spectral shaping has been performed after the modulation-dispersion process in order to implement a balanced detection scheme in which the two outputs from the Mach-Zehnder interferometer (MZI) spectral shaper are subtracted to cancel out the common background light (including common noise terms) [23], [24]. Our dispersion-unbalanced MZI consisted of two fiber arms with different dispersion characteristics: one arm was composed of a 100-m long section of non-zero dispersion-shifted fiber, NZDSF (LEAF, Corning inc.) with a total dispersion of ~0.2 ps/nm (*D*
_{1} ~0.25 ps^{2}), and the other arm was composed of a 100-m long section of single-mode fiber (SMF-28e, Corning inc.) with a total dispersion of ~1.7 ps/nm (*D*
_{2} ~2.16 ps^{2}) and a variable free-space optical delay line. The dispersion unbalance in the MZI was then fixed to Δ*D* ~1.91 ps^{2}. The spectrum measured at the output of the MZI is shown in
Fig. 2
. The delay line was used to adjust the optical path lengths in order to set the relative group delay at the source center wavelength (~1528nm) to zero, i.e. τ ~0, resulting in a microwave dispersive filter centered at DC. One can, otherwise, induce an offset of the group delay to tune the central frequency of the microwave processing bandwidth. The powers between the arms were almost equally balanced by adjusting the power coupling efficiency of the free-space optical delay line. The two output intensities out of the MZI were detected using a balanced photoreceiver (25 GHz bandwidth, DSC-R410, Discovery Semiconductor inc.) consisting of two fast photodiodes and a differential amplifier. The DC background expected for the output temporal intensity profile, see Eq. (5), was thus efficiently canceled out in our experiment using the described differential amplification mechanism, see measured output waveforms in the experiments reported below. This simple strategy also allowed us to achieve a significant improvement in the signal-to-noise ratio (SNR) of the measured DC-free output waveform, drastically reducing the number of required averaging events (<10) to display the signal with a reasonable quality in our oscilloscope. This represents an important achievement considering that the poor noise performance of incoherent photonic signal processors, such as the one employed here, has been identified as one of their most critical limitations for practical applications [25].

In our first experiment, RTFT of a square-like microwave signal was demonstrated. The MZM was directly driven by an arbitrary waveform generator (AWG710B, Tektronix inc) generating an electrical square pulse. The peak power at the modulator output was measured to be 6.2 mW using a calibrated photo-receiver. The input square-pulse intensity modulation signal was measured using a single-ended photo-receiver (80C01, Tektronix inc.) and is shown in Fig. 3(a)
. Figure 3(b) shows the output waveform acquired at the balanced detection output using a real-time oscilloscope (DPO70804, Tektronix inc.). Sampled waveform intensity was significantly reduced due to the large amount of dispersion introduced by the LCFG resulting in a poor SNR. Nonetheless, in all the cases reported here, the output time profile was clearly displayed in the oscilloscope by averaging only less than 10 consecutive waveforms. As expected from our theoretical predictions, a frequency-chirped waveform with a sinc-like temporal envelope was obtained at the system output. Half of this waveform (corresponding to the negative frequency side) is shown in Fig. 3(b). The full waveform, which corresponds to the full frequency band of the input microwave spectrum, is also shown in the inset and is compared with the numerical Fourier transform amplitude of the measured input temporal signal in Fig. 3(a). There is a fairly good agreement between the two curves; the observed truncation (on the right) was caused by the limited bandwidth of the LCFG. It can be also observed that there is a certain asymmetry between the positive (increasing frequencies along the sampled time) and the negative sides (decreasing frequencies along the sampled time). In fact, the negative-frequency side follows more precisely the corresponding RF spectrum. This was mainly caused by AC-coupling and the non-ideal impulse response characteristics of the photo-receiver and the RF amplifier since the microwave operation bandwidth extended over a wide frequency range (−2 to 2 GHz). On the basis of our experimental measurement (i.e. by directly comparing the equivalent time and frequency scales), we estimate that the microwave dispersion introduced by our setup in this experiment was *D _{mw}* ~3.8 ns

^{2}(~23.9 ns/GHz). This agrees fairly well with the dispersion that can be estimated from Eq. (3b), namely ${D}_{mw}=N\cdot {D}_{0}$, where $N={D}_{0}/\mathrm{\Delta}D$ ~1,335 and the optical dispersion is

*D*

_{0}~−2549 ps

^{2}. As anticipated, the optical dispersion introduced in the system has been effectively multiplied by a factor >1,335. This outstanding dispersion value was achieved over a full operation bandwidth >4 GHz.

The processing bandwidth of our scheme is ultimately limited by the employed optical dispersion: As expressed by the condition in Eq. (2), a larger microwave processing bandwidth can be achieved by decreasing the optical dispersion value. In the two following reported experiments, we used a dispersive optical fiber with a lower dispersion than that of the LCFG with the aim of increasing the processing bandwidth of our setup. RTFT of a high-frequency (~6.2 GHz) sinusoid pulse was demonstrated using the same processing platform, except for the dispersive medium. The microwave pulse waveform was prepared using an optical chirped-pulse interference technique [24]. In particular, an optical coherent double pulse, with a relative inter-pulse time delay of ~5 ps, was first generated from a passively mode-locked fiber laser followed by a bulk-optic Michelson interferometer where the input and output optical signals were coupled out of/into optical fibers, respectively. The double-pulse sequence was temporally stretched by an LCFG (Teraxion Inc.) operating in reflection with a dispersion of ~100 ps/nm over a reflection bandwidth of ~9 nm. A periodic interference pattern was generated along the time domain within the stretched pulse duration. This temporal interference was directly converted into an electrical signal using a fast photo-detector (1024, Newfocus Inc.) with a FWHM impulse response of ~12 ps followed by a 10-GHz RF amplifier (H301-110, JDSU Inc.). The optical intensity waveform directly measured at the output of the MZM, i.e. input microwave signal, is shown in Fig. 4(a)
. Instead of the LCFG previously used as the optical dispersive medium, we employed now the DCF (*D*
_{0} ~491.6 ps^{2}). According to the theoretical estimation in Eq. (2), the full-width operation bandwidth should be <35 GHz, even though the tested signal bandwidth was limited by the MZM bandwidth (half-width ~10 GHz). The waveform measured at the output of the balanced detector is shown in Fig. 4(b) and clearly displays the frequency-separated signal’s components corresponding to the microwave carrier around ~6.2 GHz and the slow-varying Gaussian-like envelope around DC. The full measured waveform is shown in the inset of Fig. 4(b) and is compared with the numerical Fourier transform amplitude of the input microwave waveform shown in Fig. 4(a). There is a certain disagreement in the frequency scale of the two curves, particularly towards the positive frequency side. This can be attributed to the relatively high dispersion slope (−1.3 ps/nm^{2}) of the DCF, causing a slightly non-linear frequency-to-time mapping. On the basis of the obtained results, the microwave dispersion in this experiment was estimated to be *D _{mw}* ~0.138 ns

^{2}(~0.87ns/GHz) over a full-width operation bandwidth larger than 20 GHz. This represents an increase over the optical dispersion of

*N*~281, fairly close to the theoretically expected value, $N\approx {D}_{0}/\mathrm{\Delta}D$.

In the final experiment, as a counterpart of the previous experiment, a microwave double pulse was dispersed into the RTFT regime. In particular, since the two pulses were ‘in-phase’, a constructive microwave interference pattern is expected at the output of the dispersive filter. The double pulse was dispersed using the same processing platform as for the second experiment. For preparing the electrical double-pulse waveform, an optical coherent double pulse, with a relative inter-pulse time delay of ~350 ps, was first generated using the procedure described above. The optical double-pulse waveform was then directly converted into an electrical signal using the fast photodetector followed by the RF amplifier. The measured optical intensity waveform after modulation (input microwave signal) is shown in Fig. 5(a) and the balanced detection output is shown in Fig. 5(b), clearly displaying the anticipated periodic modulation of the signal envelope, in excellent agreement with the numerical Fourier transform of the input signal.

## 4. Conclusions

In conclusion, we have shown that a simple fiber-optic incoherent signal processing architecture can be used to induce extraordinary amounts of first-order dispersion, i.e. up to a few ns^{2}, on microwave signals with tens-of-GHz bandwidths. The proposed scheme is based on temporally modulating a properly re-shaped incoherent light spectrum with the microwave waveform to be processed followed by optical dispersion. In particular, the incoherent light source should be spectrally re-shaped by filtering through a dispersion-unbalance optical interferometer. The principle of operation, design and performance specifications of this scheme have been discussed and derived based on the so-called time-spectrum convolution concept. The dispersion induced on the propagating microwave signal using this setup is *N* times larger than the optical dispersion, where *N* is determined by the ratio between the optical dispersion and the dispersion unbalance in the interferometer. The processing bandwidth is increased for a smaller optical dispersion. By properly fixing the dispersion unbalance in the interferometer, the system can provide an equivalent dispersion several orders of magnitude larger than that of the optical dispersive line. Using this new scheme, we have experimentally demonstrated microwave dispersions sufficiently large to implement RTFT (frequency-to-time mapping) of a variety of nanosecond-long microwave-domain waveforms, including a square-like pulse, a sinusoid pulse and a double pulse, with processing bandwidths broader than 20 GHz.

The technique demonstrated here would enable implementing a wide range of important dispersion-based signal processing methods, including RTFT-based operations, temporal imaging, tunable delay lines etc., on broadband microwave signals.

## Acknowledgments

This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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