Time stretch transformation of wideband waveforms boosts the performance of analog-to-digital converters and digital signal processors by slowing down analog electrical signals before digitization. The transform is based on dispersive Fourier transformation implemented in the optical domain. A coherent receiver would be ideal for capturing the time-stretched optical signal. Coherent receivers offer improved sensitivity, allow for digital cancellation of dispersion-induced impairments and optical nonlinearities, and enable decoding of phase-modulated optical data formats. Because time-stretch uses a chirped broadband (>1 THz) optical carrier, a new coherent detection technique is required. In this paper, we introduce and demonstrate coherent time stretch transformation; a technique that combines dispersive Fourier transform with optically broadband coherent detection.
© 2013 Optical Society of America
Analog-to-Digital Converters (ADCs)  map signals from the physical world to the discrete realm of computing, where they can be analyzed, manipulated, and shared. The speed and accuracy (number of bits) at which ADCs capture analog signals are their key performance metrics. For example, high-throughput biomedical imaging and screening systems, such as flow cytometers, rely on high-speed, high-resolution ADCs to gather and process data in real-time [2, 3]. In military applications, wideband ADCs and real-time processing are integral components in electronic surveillance and counter measure systems . Furthermore, the surge of Internet traffic has led modern optical communication networks to employ higher data rates and multi-level advanced modulation formats [5–9]. Here, high-performance ADCs and DSP are required at the back-end of these networks to capture, equalize, and decode the high-speed data.
Photonic time-stretch enhanced recording (TiSER) extends the bandwidth and resolution of ADCs beyond the state-of-the-art by slowing down analog electrical signals before digitization [10–13]. At the back-end, the time-dilated signals can be captured by ADCs running at significantly slower speeds than the original Nyquist rates and with concomitantly higher bit depth and resolution. Stretching further reduces the effective aperture jitter, and avoids channel mismatches inherent in sample interleaving architectures . Previous implementations of TiSER have demonstrated record signal-to-noise ratio (SNR) of 45 dB and spurious free dynamic range (SFDR) of 52 dB over 10 GHz analog noise bandwidth . TiSER uses a linearly chirped broadband optical pulse train as a carrier, upon which is modulated an analog radio-frequency (RF) signal of interest [Fig. 1]. The chirped pulse source typically consists of a mode-locked laser followed by a dispersive element. Large chromatic dispersion further chirps the optical pulses after modulation, separating the spectral components and simultaneously stretching the modulated RF signal in time. The ratio of the optical pulse widths after and before the final dispersion determines the total stretch factor. Stretch factors of up to 250 have previously been demonstrated .
In optical communication links, two detection schemes can be employed: direct detection and coherent detection . In direct detection, a photodetector (PD) captures an optical signal and the power, proportional to the amplitude squared of the electric field, is converted to a current. Direct detection is the most straightforward receiver system and has been exclusively employed in TiSER thus far. Along with simplicity, however, come drawbacks. All phase information of the signal is lost, the non-linear squaring of the electric field by the PD leads to high-harmonic distortion, and the output signal lies at baseband where it is susceptible to interference from low frequency noise. In coherent detection receivers, the optical signal is mixed with a reference beam, termed the local oscillator (LO), before detection. The LO is designed to be stable and at a higher power than the signal. Since the PD responds to the optical electric field squared, a cross term is generated which is linearly proportional to the electric field of the signal and LO. Although more complicated to implement, coherent detection offers many advantages over direct detection. High-harmonic distortion due to squaring is avoided, amplitude and phase information can be extracted, the cross term can be upshifted away from low frequency noise, and the signal is effectively amplified by the strength of the LO. All of these benefits result in a receiver with up to 20 dB greater sensitivity . Finally, with coherent detection, advanced phase and amplitude modulation schemes with high spectral efficiency and robustness to distortions can be employed . For all of these reasons, coherent detection has become the universal standard in modern lightwave communications.
When considering coherent detection for TiSER, a unique challenge arises because of the chirped broadband optical carrier. Here we introduce the coherent dispersive Fourier transform (cDFT), an extension of the optical dispersive Fourier transform (ODFT) [19, 20], which enables coherent detection in TiSER. In cDFT, we interfere two chirped broadband optical pulses, one carrying the time-stretched RF signal and the other acting as an LO. A beat frequency is generated from the interference, from which we can recover the full optical phase and amplitude of the modulated signal. The full coherent TiSER (cTiSER) system promises to significantly improve bandwidth and resolution, as well as reduce power consumption. Additionally, when used in all-optical TiSER , this technique will enable time-stretching of advanced phase and amplitude modulated optical signals. A previous, preliminary work demonstrated phase recovery of time-stretched pulses, though at a lower bandwidth and without distortion mitigation . Phase recovery was also exploited in a different context to perform high-speed vibrometry . Here we demonstrate a full cTiSER prototype system, with an RF bandwidth of greater than 30 GHz (limited by the intensity modulator), and with a stretch factor of 24. We take advantage of the full complex electric field information to achieve digital dispersion compensation across the full bandwidth of the system, and employ it in equalization of a time-stretched 40 gigabit per second (Gbit/s) data signal.
2. Coherent detection
In conventional coherent detection systems, a stable continuous wave (CW) LO reference beam mixes with the signal of interest. We can represent the signal and LO electric fields in the time domain as Esig(t)exp(iωsigt) and ELOexp(iωLOt) respectively, where Esig(t) carries the information, ELO is the approximately time independent electric field amplitude of the LO source, and ωsig and ωLO are the optical carrier frequencies. The physical fields are the real portions of the complex fields written here. The optical intensity, converted to a current signal at the PD, is proportional to the amplitude squared of the sum of the electric fieldsEq. (1) is the cross term of interest, which scales linearly with the signal and LO electric fields, and which is upshifted to the IF. In homodyne detection, the signal and LO operate at the same frequency so the IF cancels to zero, leaving only phase offsets. In heterodyne detection the signal and LO operate at different optical frequencies, and the IF is some finite RF frequency given by the difference of the two. The plus and minus signs in front of the cross term represent the complementary outputs of a 2 × 2 interferometric mixer. Balanced detection is achieved by subtracting the two complementary outputs, leaving only the cross term.
Coherent detection offers many advantages over direct detection. With a sufficiently well characterized LO beam, the phase and amplitude of the signal electric field Esig(t) can be determined in DSP. This capability has spurred the recent advances in coherent lightwave technology, in which phase and amplitude modulated signal formats are employed for improved spectral efficiency and robustness to impairments . With the full complex field information, digital back propagation can be used to mitigate distortions from chromatic dispersion, non-linear amplitude modulation, and optical nonlinearities [24–26]. In heterodyne detection systems, the cross term is upshifted away from low frequency noise, allowing for improved resolution. Finally, with a high power LO, the signal is effectively amplified upon interference resulting in a mixing gain, which boosts the signal above the thermal noise limit without the need for additional amplifiers.
3. Coherent dispersive Fourier transformation
Developing a coherent detection system for TiSER poses a unique challenge. Unlike in conventional optical communication links, which use a narrow band CW beam as the carrier, the TiSER system uses a broadband, chirped optical pulse to carry the information. If we mix a broadband, chirped pulse with a narrowband LO, the IF of the mixing term will be chirped and span the full optical bandwidth of the pulse (>1 THz). The resulting bandwidth of the heterodyne signal will extend beyond the detection bandwidth of the backend PD and electronics.
To overcome this obstacle, we have developed the cDFT. To realize this coherent version of the time stretch transform, we use as the LO a second broadband chirped optical pulse, which is an unmodulated copy of the carrier pulse. When the two chirped optical pulses, relatively delayed with respect to each other, are mixed, the optical signals interfering at each point in time are offset by a constant frequency, resulting in a chirp-free IF modulation along the optical pulse [see Fig. 2]. The phase and amplitude of the modulated time-stretched RF signal are encoded in the IF modulation, similar to in heterodyne detection, and can be recovered by adapting standard analog and digital techniques from coherent lightwave communications .
The cDFT technique can be made clearer by noting the similarity with Fourier transform spectral interferometry . Indeed, cDFT is a single-shot spectral interferometer mapped to the time-domain via the ODFT. In conventional spectral interferometry, two optical pulses, relatively delayed, are mixed and the optical spectrum is measured. In the simplest case, one pulse is a well characterized LO pulse, with the goal being to ascertain the spectral amplitude and phase of the electric field of the second pulse. The optical spectral intensity measured is proportional toEq. (1) and Eq. (2) can be seen, the crucial difference being that Eq. (1) is a time-domain signal and Eq. (2) is a frequency domain signal. The cross term of Eq. (2) is proportional to the signal and LO electric fields, and upshifted by an IF. If the LO spectrum is known, the phase and amplitude of the signal electric field can be determined.
In cDFT, the mixed signal is not detected by an optical spectrum analyzer, but is instead dispersed and captured with a single pixel PD. The large dispersion performs a ODFT, mapping the optical spectrum into the time domain via ω→t/(β2L), where β2 is the dispersion parameter and L is the length of the fiber . The result is a transformation of the complex exponential in the cross term of Eq. (2)Fig. 3(a), and the IFs for various time delays are plotted in Fig. 3(b). A linear fit is well matched to the data, with a slope indicating a dispersion of 1090 ps/nm, in good agreement with the dispersion used.
In the cTiSER system, the LO pulse is an unmodulated copy of the signal pulse, and it is the RF time-stretched signal that is being characterized, not the inherent phase and amplitude of the carrier pulse electric field. The signal and LO spectral profiles from Eq. (2) after ODFT can be written as28] is insufficient to resolve spectral components on that scale, so indeed τ can be ignored in Eenv(t). After balanced detection, the signal is proportional to the cross term of Eq. (2), with the substitutions of Eq. (3) and Eq. (4),
4. Experimental setup
We implemented an optical fiber based cTiSER prototype system [Fig. 4]. An Erbium doped fiber mode locked laser (MLL) generates a 37MHz optical pulse train, which is band-pass filtered around 1570 nm to 10 nm 3 dB bandwidth. We pre-chirp the pulse train using a dispersive fiber of 45 ps/nm dispersive parameter, stretching the pulses to 450 ps width. An erbium doped fiber amplifier (EDFA) amplifies the pre-chirped pulse train to 20 dBm average power before a 1 × 2 splitter separates the pulses into signal and LO. The signal pulses are sent to a Mach-Zehnder modulator (MZM), which modulates the RF signal on top. The LO is sent through a variable optical attenuator (VOA), which matches the insertion loss of the MZM (~7 dB). It is important to match the powers of the signal and LO pulses at the receiver to maximize modulation depth of the interference fringes and to ensure the IF amplitude spans the full scale of the back end ADC. Analog balanced detection would obviate the need for this additional 7 dB attenuation of the LO, enabling a 3.5 dB mixing gain. The two pulse trains are counter-propagated through a second dispersive fiber of 1045 ps/nm dispersion imparting a total stretch factor of 24. Optical circulators (OC) at each end of the dispersive fiber direct the pulses into and out of the dispersive fiber accordingly. A delay line in one of the arms adds the relative time delay τ. In this demonstration we used a delay of 47 ps for an IF of 5.3 GHz. The signal and LO pulses are mixed in a 2 × 2 coupler, the two outputs being complementary. The polarizations of the two optical beams are aligned using a fiber coupled polarization controller in one of the arms [not shown in Fig. 4]. In lieu of a balanced detector (BD), we delay the complementary output pulses, staggering them in time, before combining and sending to a single pixel PD. The complementary pulses are realigned and subtracted digitally. If subtraction of complementary outputs were carried out before digitization, matching of signal and LO optical power would not be necessary and mixing gain could be achieved. Average power reaching the PD is approximately −6 dBm. A trans-impedance amplifier integrated on the PD amplifies the photocurrent, which is then digitized by an ADC sampling at 50 giga-samples per second (GS/s) and with an analog bandwidth of 16 GHz. All patch chords are single mode fiber of total length less than 10m and negligible dispersion (17 ps/nm-km).
5. Digital signal processing
DSP is implemented in MATLAB  to subtract the complementary outputs, down-convert the IF signal, divide out the spectral envelope, and recover the full complex optical field of the time-stretched signal. We perform balanced detection digitally by capturing both complementary outputs from the 2 × 2 interferometer in a single PD. The complementary outputs are aligned and subtracted, removing the common mode DC portion of the optical pulses. The subtracted signal is band-pass filtered around the IF to remove out of band noise, in particular low frequency noise. We perform a discrete Hilbert transform on each pulse and extract the instantaneous frequency of the IF signal and average over 1000 pulses . We then down convert the upshifted signal to baseband. Because the RF time-stretched signal is AC coupled and uncorrelated with the laser pulses, averaging of the 1000 pulses leaves only the background envelope. We divide out the background from each pulse to recover the full complex electric field of the stretched RF signal. If a dual output MZM is used, we could alternatively implement a differential cTiSER system, in which the background envelope is recovered on a pulse-to-pulse basis by adding the two complementary modulated pulses from the MZM. Recovering the envelope in this way avoids distortions due to pulse-to-pulse variations .
6. Dispersion compensation
As an illustration of the power of cTiSER, we digitally equalize the parasitic dispersion penalty. Dispersion acts by imparting a phase quadratic with optical frequency, i.e. exp(iβ2Lω2/2) . In an optical communication link using double sideband modulation and direct detection, this dispersion induced phase leads to an intensity transfer function proportional to cos2(β2LωRF2/2), limiting the bandwidth of the system . Conceptually one can understand this intensity attenuation as arising from destructive interference of the two sidebands modulated about the carrier, which accrue different dispersion induced phase shifts because of their different optical frequencies.
In TiSER, dispersion is exploited to our advantage to map the optical spectrum into time and stretch the RF modulated signal. However, the parasitic dispersive effect still manifests itself, though with an attenuated dispersion induced phase, i.e.12]. Two methods exist which allow TiSER to overcome this bandwidth limiting dispersion penalty: one employs single side-band modulation to avoid the interference at the detector [12, 34], and the second, termed phase diversity, uses maximum ratio combining of two outputs from a MZM, which have inherently complementary transfer functions . However, these dispersion penalty mitigation techniques are not compatible with the differential TiSER system, as they require specialized MZMs, which do not offer complementary dual-outputs. Additionally, with all-optical TiSER, for which the dispersion penalty is the dominant bandwidth limitation, the above techniques are not always applicable, depending on the modulation format of the optical signal. There is as yet no general technique to mitigate the dispersion penalty for an arbitrarily modulated input.
With the full complex optical field captured using cTiSER, we are able to digitally equalize the dispersion-induced phase in Eq. (6), independent of the modulation format of the input signal. For each captured pulse we take a 576-point discrete Fourier transform and multiply by the inverse of the exponential in Eq. (6). A set of 576 samples (sample rate is 50 GS/s) spans 11.5 ns, the approximate pulse width after stretching. We characterized the transfer functions for both conventional direct detection and broadband coherent detection back-ends by capturing and time-stretching single-tone RF signals ranging from 15 GHz to 43 GHz. We performed dispersion compensation on the coherent detection data, and plotted the intensities for each frequency [Fig. 5]. From Eq. (6) we calculate the dispersion-induced phase to be 25.6 ps2. The z-cut MZM imparted an additional chirp-induced phase to the sinusoidal transfer function, shifting the first dispersion null to a lower frequency and further limiting the RF bandwidth. A non-linear regression estimated the chirp parameter to be 0.61, and this effect was accounted for in the dispersion compensation . As illustrated, dispersion compensation mitigates the parasitic dispersive effects, boosting the bandwidth of TiSER.
To further demonstrate the dispersion compensation capabilities, we captured a 40 Gbit/s non-return to zero (NRZ) pseudo-random binary sequence (PRBS) signal and generated eye-diagrams for both direct detection and coherent detection back-ends. With direct detection, dispersion penalty distorts the eye-diagram due to high-frequency attenuation [Fig. 6(a)]. When applying dispersion compensation with the cTiSER system the distortion is mitigated [Fig. 6(b)]. As an indication of the increased bandwidth due to dispersion compensation, we recorded improvement in rise/fall times, measured at 10% and 90% levels, from 18.5/19.1 ps to 16.1/15.8 ps.
Photonic TiSER is a powerful technology that boosts the bandwidth and resolution of ADC technology, enabling many advanced applications, such as for military systems, biomedical diagnostics, and telecommunications. In this paper, we demonstrated a TiSER system with a coherent receiver back-end, which promises to offer to TiSER many of the benefits that coherent detection has brought to modern lightwave communications. In particular, we developed a fully functioning prototype cTiSER system with a stretch factor of 24 and analog bandwidth of >30 GHz, which captures the full amplitude and phase information of a time-stretched RF signal. Taking advantage of the full field information, we successfully performed digital dispersion compensation, equalizing the bandwidth reducing parasitic dispersive effect. Further studies will develop and characterize improvements in SNR from upshifting the signal away from low frequency noise, avoiding the non-linear distortion from square-law detection, and through implementing additional distortion corrections such as equalization of non-linear modulation. When combined with analog balanced detection, mixing gain can also be exploited to boost the signal power and improve sensitivity without additional optical or electrical amplifiers. Looking further ahead, application of this coherent detection technique in all-optical TiSER will extend the bandwidth beyond the dispersion penalty limit, as well as allow it to capture advanced phase and amplitude optical modulation formats. With all of the afforded benefits, we foresee broadband coherent detection to become the standard receiver system for time-stretch transformation.
The authors would like to acknowledge the partial support from National Science Foundation through CIAN NSF ERC under Grant No. EEC-0812072.
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