Open Access
Add to BibSonomyAbstract
We introduce an all-optical, format transparent hash code generator and a hash comparator for data packets verification with low latency at high baudrate. The device is reconfigurable and able to generate hash codes based on arbitrary functions and perform the comparison directly in the optical domain. Hash codes are calculated with custom interferometric circuits implemented with a Fourier domain optical processor. A novel nonlinear scheme featuring multiple four-wave mixing processes in a single waveguide is implemented for simultaneous phase and amplitude comparison of the hash codes before and after transmission. We demonstrate the technique with single polarisation BPSK and QPSK signals up to a data rate of 80 Gb/s.
© 2013 Optical Society of America
1. Introduction
Detection and correction of mistaken data is a critical function in information systems. Therefore, numerous error management techniques have been developed, including Forward Error Correction (FEC), checksums, cyclic redundancy checks and more generally hash functions [1]. The underlying principle of any hash method is the computation of a number as a signature of a data packet mapping the original data to a value in a smaller mathematical space [2]. Identical blocks of data yield the same hash code, while different ones should have different signatures, heralding an error. In a transmission system, failure can be detected with high probability by comparing the hash before and after propagation. With a higher degree of complexity, FEC algorithms are able to repair corrupted data without re-transmission, and are in use in most current telecommunication networks [3] allowing coding gains of over 10 dB [4–6]. These integrity check methods imply the transmission of some redundant information (i.e. the hash codes) and usually result in increased latency and power consumption due to the required processing.
In the context of high bandwidth optical communications, these algorithms are part of the Digital Signal Processing (DSP) performed at both the transmitter and the receiver [7]. The intense processing operations required introduce constraints in terms of bandwidth, energy efficiency and latency [8,9]. An implementation of a hash function directly in the optical domain could significantly reduce these burdens. A recent report established the feasibility of a Semiconductor Optical Amplifiers (SOA) based hash encoder and decoder. However, the proposed scheme only works for On-Off Keying (OOK) signals of limited bandwidth [10,11] and cannot be generalised to advanced modulation formats which are now used extensively in communication systems [12].
Here we introduce and demonstrate for the first time an all-optical, format transparent, hash code verified transmission link comprising a low latency hash code generator and comparator. In this scheme, the hash code is generated using a Fourier domain optical processing operation on the data signal, while the hash code comparator is implemented via coherent optical subtraction based on multiple Four Wave Mixing (FWM) processes in a nonlinear medium. Data validation consists of comparing hash signals calculated before and after transmission. In contrast to most electronic methods, which incorporate the hash codes into the bitstream or packets, we transmit the codes on a separate wavelength channel thus avoiding the need for sophisticated bit-rate adjustment, check symbol insertion and segmentation schemes. Experiments are reported for a 40 Gb/s single channel Binary Phase Shift Keying (BPSK) signal and a 80 Gb/s single channel Quadrature Phase Shift Keying (QPSK) signal. Our approach can be easily scaled to higher symbol rates and modulation formats without modification. This work represents a key result for ultra-low latency optical networks.
2. Principle
Hash based error detection codes require transmission of redundant information along with the data. In most electronic systems it is concatenated to the payload of each data packet. In our method however, the original data channel remains unchanged while the hash code is transmitted on a parallel wavelength channel as depicted on Fig. 1. At the receiver side, integrity of the data is checked by comparing the hash code from before to the one after transmission via all-optical signal subtraction.
Fig. 1 Principle of the all-optical hash key verified link. Hash codes are generated inside the transmitter and communicated through the network along with the data in an adjacent channel. At the receiver side, hash codes are recalculated from the data channel and compared to the transmitted hash. On the diagram, red components indicate optical devices and green the electrical domain.
The functional diagram in Fig. 2 illustrates the successive transformations of the optical channels. The same data is encoded onto two phase locked WDM channels. One channel is kept as it is, while the other is used as a base for computing the hash signal (Hash 1) in a first hash generator device. Both channels are then propagated through the link. At the receiver, the comparison is performed by calculating the hash (Hash 2) from the received data channel using the same hash function and combining it with Hash 1 and two phase-locked pump waves. The four waves, occupying different wavelengths, are sent through a nonlinear medium to carry out the hash comparison. Upon nonlinear propagation, the two hash channels are translated to the same wavelength channel (Idler) by degenerate FWM with the two pump waves. If the relative phase between the hash channels is set to π, the wavelength conversion will result in a coherent subtraction, i.e. if the hash codes are the same, then destructive interference cancels the optical power at the translated wavelength. A transmission error causing the hash codes to differ would unbalance the interference term and trigger a spike at the idler wavelength, producing an error signal which can be extracted by bandpass filtering.
Fig. 2 Functional diagram. Arrows aligned with a same horizontal level represent signals at a same wavelength.
The hash code of an N-symbol packet is calculated by multiplying N successive symbols with custom weight and sign coefficients and coherently adding them. Such an operation can be realized using a multipath interferometric circuit (MIC) with tuneable phase offset and attenuation, represented on the right part of Fig. 5. The sign of the coefficients is controlled by the relative phase of the different interferometric arms, while the weight can be set via attenuation. We use a Fourier Domain Programmable Optical Processor (FD-POP) configured as a custom MIC by implementing the corresponding phase and amplitude transfer functions. Unlike previous methods [10, 13], this is compatible with advanced modulation formats and allows for easily reconfiguring packet length and the hash function without any physical change to the setup. Wavelength selective delay, phase control and attenuation are implemented in the FD-POP [14, 15], as detailed in section 3. The resulting hash signal takes the form of a multilevel phase- and amplitude-coded channel at same rate as the data. Note that, because the hash code is generated from N data symbols, its required bandwidth can be reduced to 1/Nth of this value by sampling, as further explained in section 4.
In contrast to previous systems that only allowed for comparing binary intensity coded signals, our hash comparator enables simultaneous comparison of multi-level phase and amplitude encoding. A more detailed explanation of the scheme is depicted in Fig. 3. The two phase-locked hash channels to be compared (Eqs (1) and (2)) are co-propagated with two pumps separated by half their spectral separation (Eqs (3) and (4)).
These four waves have phasors EH1, EH2, EP1 and EP2, amplitudes AH1, AH2 and AP1 = AP2 and wave vectors kH1, kH2, kP1 and kP2. As a consequence of the choice of frequency separations, first order FWM products of the pairs Hash 1 - Pump 1 and Hash 2 - Pump 2 appear at same frequency to add up coherently and have phasors EI1 = AI1eiϕI1 and EI2 = AI2eiϕI2. The phase matching condition of FWM [16] determines the wave vectors of the generated idlers kI1 = −kH1 + 2kP1 and kI2 = −kH2 + 2kP2. By developing the expression of the idler fields, one can deduce the phase relations of these idlers [17]:
Where the two phase locked pumps have been set with equal phases ϕP1 = ϕP2 = ϕP. In order to operate coherent subtraction, the hash signals are set with opposite phases ϕH2 = ϕH1 + π. Given that the degenerate FWM products amplitudes verify the relations and , the total idler field generated is
The absolute phase ϕP is not relevant for the error signal and can be omitted. The subtraction in the complex space implies that the total intensity at idlers frequency cancels only if AH1 = AH2. Any mismatch between the hash signals translates into an idler spike signalling an error. This requires Hash 1 and Hash 2 to be phase locked (which is automatically the case if the two initial copies of the signal come from the same broadband source) and Pump 1 and Pump 2 to be phase locked. Importantly no phase relation is required between the hash channels and the two pumps.Fig. 3 Principle of the all optical coherent signal comparator. Left: hash signals to be compared are copropagated with pumps through a third-order nonlinear medium after relative phase and amplitude adjustment. Inset: Two simultaneous FWM-based wavelength shifting processes occur inside the nonlinear medium. Two pumps with half the frequency spacing of the hash signals bring both idlers to the same wavelength. Initial configuration of the signals with opposite phases (represented by arrows facing in opposite directions) causes the idlers to interfere destructively so as to cancel the total idler product when both hash are equal. Mismatch between hash codes causes an idler spike. Bandpass filtering (BPF) of the total idler extracts the error signal.
3. Experiment and results
The experimental setup is shown in Fig. 4. The signal is generated from a mode-locked fibre laser that delivers a 40 GHz pulse train (2 ps full width at half maximum) at 1550 nm. The spectrum is broadened inside a highly nonlinear fibre (HNLF) and filtered to obtain a quasi-flat comb spectrum over a bandwidth of 6 nm [18]. The pulse train is then encoded using two successive phase modulators driven by two independent pseudo-random bit sequences (PRBS) of either 64 bits (Figs. 6 and 8) or 231 – 1 bits (Fig. 10) and biased to deliver a π and phase shift in order to generate an 80 Gb/s QPSK signal [19]. The 40 Gb/s BPSK signal is generated by bypassing the second modulator.
Fig. 4 Experimental setup. A wideband flat frequency comb is QPSK-encoded with two successive phase modulators (PM). Hash keys are calculated before and after transmission inside Fourier-domain programmable optical processors (FD-POP). Combination with a pump pulse train and wavelength-selective phase control by the second FD-POP conditions the signals for coherent hash comparison inside a 30 m section of highly nonlinear fibre (HNLF). PM: phase modulator ; BPF: bandpass filter.
The hash is calculated by coherent addition of N successive symbols with custom phase and amplitude offsets. This could be performed in an integrated MIC with calibrated phase shifts and attenuations on each arm illustrated on the right of Fig. 5. Here we use a more flexible solution based on a FD-POP. The effect of the linear MIC is fully represented by its spectral phase and amplitude transfer function programmed into the FD-POP device.
Fig. 5 Hash code generation by Fourier domain optical processing. Left: power and phase transfer functions programmed in the FD-POP. The spectral profiles applied to the data signal reproduce the characteristics of a multipath interferometric circuit (MIC) coherently adding successive bits. Blue, green and red traces correspond to different MIC configurations. The other channel is left untouched through a bandpass filter transfer function. Right: equivalent MIC implemented in the FD-POP.
The pass channel of the wavelength selective MIC is transmitted through a flat top bandpass filter with constant phase. The hash channel results from equal splitting of the field ES = ASeiϕS into N arms, each delayed by an integer number of symbol lengths ΔLj = (j − 1)ΔLsymbol (j indexes the arms). Before recombination of the arms, each path is attenuated by βj and offset in phase by ϕj according to the desired hash function equation. The field after propagation in arm j is . At the output of the device the delayed complex fields are passively combined resulting in:
with 0 ≤ βj ≤ 1. Let us define being the frequency offset associated to the delay j such that . The amplitude transfer function is written and the phase transfer function is The FD-POP allows for custom control of the spectral phase and amplitude with a resolution of 5 GHz. Implementation of the transfer functions of Eqs. (11) and (12) realizes the hash key function. Figure 5 shows examples of spectral transfer functions for various MIC configurations. The red trace corresponds to a 2-symbols packet with the hash definition HN = BN + BN+1eiπ, where HN and BN are the optical field amplitudes of the hash and data at symbol N. The green and blue traces correspond to 3-symbols packets with hash definitions respectively and . Other hash definitions can be obtained by choosing linear combinations of the BN with other coefficients and phases.The calculated hash keys using the above functions for 40 Gb/s BPSK and 80 Gb/s QPSK signals made of 64 symbols are shown on Fig. 6. Changing the hash parameters leads to different hash sequences for the same initial signal. In this experiment, hash key and data signal are transmitted through a short fibre link made of 50 m of SMF and 20 m of DCF. Note that the time traces and eye diagrams represent only the intensity of the hash channel. Information is also contained in its phase, which is not shown in the traces but is taken into account by the coherent signal comparator. Noise is mostly introduced in the signal generation by the phase modulator delivering the phase shift.
Fig. 6 Experimental intensity plots of the hash codes for BPSK (first and second columns) and QPSK (third and fourth columns) signals. Information encoded in the phase of the hash signals is not represented. Time traces (top row) and eye diagrams (bottom) of the 64 bits patterns are measured with a sampling oscilloscope. Blues traces show simulation results for the corresponding bit patterns.
In order to compare the hash key calculated from the transmitted signal to the input signal hash we use a second FD-POP at the end of the link. The FD-POP calculates the receiver hash key from the transmitted data channel using the same phase and amplitude transfer function as described for the transmitter, while leaving the transmitted hash channel untouched. The relative phase between the transmitted hash channel and the hash calculated from the transmitted data signal is adjusted to π by the FD-POP to result in coherent subtraction.
The pump pulse train is coupled to a second input port of the FD-POP and shaped to obtain a dual pump configuration with half the spectral separation of the hash signals. The hash and pump inputs are combined inside the FD-POP to form the input of the comparator. After amplification to a total optical power of 300mW and out-of-band noise filtering, nonlinear mixing occurs in a 30 m span of HNLF. The signals both have powers 10 dB lower than the pumps.
Figure 7 shows the output spectrum measured after the HNLF for three configurations: solid green: the two hash functions are identical, resulting in idler output power at zero level due to coherent subtraction (H1 – H2); dashed black: the hash functions are programmed differently at the receiver and the emitter, resulting in unbalanced coherent subtraction and therefore output pulses indicating errors ; dotted blue: the hash functions are equal, but one bit every 512 bits presents an error that triggers an idler spike at the comparator output. However, the effect on the spectrum is barely noticeable in the last case, due to the low duty cycle (1/512) of the error pulses.
Fig. 7 Experimental optical spectrum at the HNLF output. Insert: the total idler cancels out for identical hash and not if they differ. Solid: both hash signals equal ; dotted: one error every 512 symbols ; dashed: hash functions different for both signals.
Error detection is performed in the time domain, by narrow band (70 GHz) spectral filtering at the frequency of the first order FWM products (detailed in the inset), to extract the idler. After amplification, the hash verification signal is viewed on a sampling scope with 65 GHz bandwidth.
The amplitude of the error signal translates to the degree of mismatch between the hash keys and the degree of packet reliability within the limit of small errors. Indeed a small phase and amplitude offset applied to one symbol causes a proportional change in the hash function. Note however, that stronger perturbations or changes to multiple symbols have very small, but nonzero, probability of cancelling out in the hash calculation.
Figure 8 relates the comparator output to the spectrum of the extracted idler for an 80 Gb/s QPSK data. When both hash functions are generated using the same parameters before and after transmission, the mismatch signal after the comparator is constant zero, reporting no error (bottom left). If the hash signals are generated with different hash functions on both generators, the output is nonzero heralding transmission errors (bottom right) at every symbol.
Fig. 8 Time traces of the error signal after bandpass filtering of the idler. Top: Idler channel filtered out. Bottom left: both hash signals equal. Bottom right: hash signals different. Hash code mismatch is reflected by a nonzero error signal after optical subtraction.
Figure 9 shows a variation of the setup introduced in Fig. 4 in order to create localized errors in the data signal in the case of BPSK encoding with 231 – 1 PRBS data. The two replicate channels encoded with a first phase modulator (PM) are split onto two arms. One passes through a second phase modulator driven by a separate pattern generator operating a π phase shift to one bit every 512 symbols. The other arm is delayed and adjusted in phase with a phase shifter (PS) to exactly match the path lengths of the interferometer. Both paths are recombined inside the first FD-POP operating the hash calculation by assigning the two wavelength channels to different ports of the device. This structure keeps the two data channels identical, except for introducing a one symbol error every 512 symbols. The relative phase between the two paths is set manually via the phase shifter and phase locking is maintained by placing the interferometric structure inside an enclosure to isolate it from vibrations and thermal drift.
Fig. 9 Variation of the signal generator to create localized errors in a BPSK signal. The two replicate channels are split onto two arms, one being affected by a π phase shift of one bit period every 512 symbols. Both paths are recombined into the first FD-POP acting as a wavelength selective switch. PM: phase modulator ; PS: phase shifter.
A single bit error yield idlers mismatch in the comparator at the symbols that have different hash keys. For a hash function calculated for data packets made of N symbols, the error signal exhibits N spikes, due to the fact that all of the N hash keys involving the mistaken bit are altered. Figure 10 shows the comparator output for a single bit error when data packets contain 3 bits (N=3). Similarly, a single bit error during transmission in a real system would change the hash code of the data channel, leading to a spike at the comparator output.
Fig. 10 Time traces of the error signal after bandpass filtering of the idler in the case of a single error. Top: Idler channel filtered out. Bottom left: both hash signals equal (no error). Bottom right: a single error causes a spike in the error signal.
4. Discussion
4.1. Tolerance to long distance transmission
In order to verify the validity of the technique for propagation through a longer path, we implemented the system on a 40 Gb/s OOK link made of 20.72 km of SMF and a matching spool of DCF (340 ps/nm) as presented on the left diagram of Fig. 11. Residual dispersion was further compensated inside the end-of-link FD-POP by applying an additional quadratic spectral phase. The hash key channel was configured for 3-bit packets according to the equation .
Fig. 11 Hash key verified link of 20.72 km. Left: composition of the link. Right: Idler channel filtered out in case of both equal (green) and different (black) hash function definitions. The corresponding eye diagrams show the error signal in the time domain.
The only concern related to the addition of the mid-haul link was a slow scale timing drift between the transmitted signal and the pump (that follows another path) due to ambient temperature fluctuations. In our setup, the pump and signals were found to be significantly desynchronized within a period of about 10 s. This issue, however, would not show up in a real system where the pump pulse train would be generated based on a clock recovery module from the received signal. Importantly, the coherent comparator is not affected by the phase drift between hash and pump channels as mentioned in the principle section.
Figure 11 features the optical spectrum of the idlers and the corresponding eye diagrams for two cases. The solid green trace shows the destructive interference between idlers when transmitted and received hashes are equal and no error occurs. The dashed black trace marks the increase in total idler power when the comparator is unbalanced by configuring both hashes with different settings, which is equivalent to errors occurring at every bit. The corresponding time trace of the error signal, measured with a sampling oscilloscope, still comprises symbols apparently without error (error signal at zero), which is due to the fact that the data packets are formed of only three bits. Hence there is still a high probability of having equal hash keys for different signals.
4.2. Extension of the method to higher symbol rates
The potential for high baudrate operation enabled by the ultra fast reaction time of Kerr effect exploited in the comparator offers a solution for error management at symbol rates inaccessible to electronics.
Our implementation of the hash generator is particularly well-suited for ultra high baudrate OTDM signals. The delay of our FD-POP is limited to about 50 ps [20], which at 40 Gbaud only allows hash calculation for packets of up to 3 symbol (note that this still corresponds to a bit-length of 6 in case of QPSK). However, increasing the symbol rate to 640 Gbaud, for example, would enable processing of packets of up to 32 symbols. Also hash code based verification of an OTDM channel only requires one instance of the hash generation and comparison infrastructure, while WDM systems would need separate hash comparators for all channels.
4.3. Latency
The latency introduced by our system is primarily determined by the 150 ns propagation time through the 30 m of HNLF. This latency can be almost entirely removed by either unwinding the HNLF so that it is part of the transmission link or by replacing it by an on-chip nonlinearity [21].
We acknowledge the fact that DSP based error correction mechanisms provide much more sophisticated solutions. However, these introduce significant latency that can be removed by using the all-optical method proposed in this work.
4.4. Hash channel sampling for improved bandwidth efficiency
In the experiment described in this paper, the hash channel has the same bandwidth as the data channel because a hash code is generated at every symbol period. However, it can be downsampled to 1/N of the original baudrate to keep only one hash code per data packet made of N symbols, which is enough for error detection. This could be done for example, by sampling the hash with an electro-absorption modulator [22] followed by a narrow-band filter or, for ultra high bandwidth OTDM signals, by nonlinear optical sampling [23, 24]. The resulting lower bandwidth hash channel would be less subject to distortions and have better spectral efficiency.
4.5. Effect of dispersion
Residual chromatic dispersion can cause timing misalignment of the symbols in the data and hash channels transmitted at different wavelengths. To counter this, the FD-POP provides a flexible way to compensate for group velocity dispersion as well as higher orders of dispersion, at the same time as they apply the hash phase masks [20]. This is the method used in our 20.72 km link transmission to remove the residual delay due to imprecise dispersion compensation by the DCF section.
Also, solutions have been demonstrated for automatic compensation of fluctuations of multiple orders of dispersion using a FD-POP as tuneable compensator [25]. In presence of strongly varying dispersion, the FD-POP devices already used for the hash function generation can be feedback-controlled to cancel dispersion in real time.
4.6. Phase relations
The comparator requires the received data and hash signals to be phase locked. This condition is automatically verified provided that the hash is generated from the signal itself and that both channels propagate along the same path of fibres and amplifiers. Although the pumps involved in the comparator must also be phase locked, no phase condition is required between signals and pumps.
However, accidental desynchronization of the two channels or introduction of phase noise in long amplified links may degrade the action of the comparator. A perturbation Δϕ of the relative phase between hash and data channels would introduce a maximum power inaccuracy in the error signal of the order of (1 − cos(Δϕ)), where IADD is the maximum total idler power when the phase relation is set in the adder configuration (ϕH1 = ϕH2). A dependence of the total idler power of this type has effectively been measured when sweeping the relative phase over a 2π range (Fig. 12). In the case of our demonstration of a 20.72 km link reported in paragraph 4.1, no effect of phase noise could be observed and the extinction ratio of the coherent subtracter was similar to the one measured for the back-to-back transmission.
Fig. 12 Theoretical and experimental inaccuracy in the hash subtraction as functions of the phase perturbation between the two hash signals. The measurement was realized by feeding two equal signals in the coherent comparator and sweeping over their relative phase.
5. Conclusion
We have shown flexible generation of hash keys using a linear optical device for high bandwidth optical signals encoded with complex modulation formats. Comparison of hash keys before and after transmission is implemented by coherent all-optical subtraction via a novel dual FWM scheme. This scheme is to be considered as a compromise between the high latency introduced by DSP FEC operation and a latency free link without error detection. The method is demonstrated with an 80 Gb/s QPSK signal.
Acknowledgments
The authors acknowledge the Australian Research Council (ARC) Centres of Excellences Program (Project CE110001018), Laureate Fellowship (Project FL120100029), Discovery Early Career Researcher Award (DECRA) (Project DE120101329) and an ARC Linkage grant with Finisar Australia (Project LP120100661).
References and links
1. J. G. Proakis and M. Salehi, Digital Communications (McGraw-Hill Higher Education, 2008).
2. A. A. Bruen and M. A. Forcinito, Cryptography, Information Theory, and Error-Correction: A Handbook for the 21st Century (John Wiley & Sons, 2011).
3. D. Russell, The Principles of Computer Networking (Cambridge University Press, 1989).
4. T. Mizuochi, “Recent progress in forward error correction and its interplay with transmission impairments,” IEEE J. Select. Topics Quantum Electron . 12, 544–554 (2006). [CrossRef]
5. F. Chang, K. Onohara, and T. Mizuochi, “Forward error correction for 100 G transport networks,” IEEE Communications Magazine, 48–55 (2010). [CrossRef]
6. C. Xie, Y. Zhao, Z. Xiao, D. Chang, and F. Yu, “FEC for high speed optical transmission,” Proc. of SPIE-OSA-IEEE 8309, 83091R1 (2011).
7. K. Azadet, E. Haratsch, and H. Kim, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid-State Circuits 37, 317–327 (2002). [CrossRef]
8. M. Freiberger, D. Templeton, and E. Mercado, “Low latency optical services,” in “National Fiber Optic Engineers Conference,” p. NTu2E.1 (OSA, Washington, D.C., 2012).
9. Y. Miyata, K. Sugihara, W. Matsumoto, K. Onohara, T. Sugihara, K. Kubo, H. Yoshida, and T. Mizuochi, “A triple-concatenated FEC using soft-decision decoding for 100 Gb/s optical transmission,” in “Optical Fiber Communication Conference,” p. OThL3 (OSA, Washington, D.C., 2010).
10. M. Suzuki and H. Uenohara, “Investigation of all-optical error detection circuit using SOA-MZI-based XOR gates at 10 Gbit/s,” Electron. Lett. 45, 224–225 (2009). [CrossRef]
11. Y. Aikawa, S. Shimizu, and H. Uenohara, “Investigation of all-optical division processing using a SOA-MZI-based XOR gate for all-optical FEC with cyclic code,” Photonics in Switching 1, 3–5 (2010).
12. P. Winzer and R. Essiambre, “Advanced optical modulation formats,” Proceedings of the IEEE 94, 952–985 (2006). [CrossRef]
13. J. E. McGeehan, S. Kumar, and A. E. Willner, “Simultaneous optical digital half-subtraction and -addition using SOAs and a PPLN waveguide.” Opt. Express 15, 5543–5549 (2007). [CrossRef] [PubMed]
14. J. Schröder, O. Brasier, J. VanErps, M. A. F. Roelens, S. Frisken, and B. J. Eggleton, “OSNR monitoring of a 1.28 Tbaud signal by interferometry inside a wavelength-selective switch,” J. Lightw. Technol. 29, 1542–1546 (2011). [CrossRef]
15. J. Schröder, M. A. F. Roelens, L. B. Du, A. J. Lowery, S. Frisken, and B. J. Eggleton, “An optical FPGA: reconfigurable simultaneous multi-output spectral pulse-shaping for linear optical processing,” Opt. Express 21, 690–697 (2013). [CrossRef] [PubMed]
16. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2001).
17. J. Wang, Q. Sun, and J. Sun, “All-optical 40 Gbit/s CSRZ-DPSK logic XOR gate and format conversion using four-wave mixing,” Opt. Express 17, 12555–12563 (2009). [CrossRef] [PubMed]
18. T. Inoue and S. Namiki, “Pulse compression techniques using highly nonlinear fibers,” Laser & Photonics Review 2, 83–99 (2008). [CrossRef]
19. J. Yu and X. Zhou, “Multilevel modulations and digital coherent detection,” Optical Fiber Technology 15, 197–208 (2009). [CrossRef]
20. M. A. F. Roelens, S. Frisken, J. A. Bolger, D. Abakoumov, G. Baxter, S. Poole, and B. J. Eggleton, “Dispersion trimming in a reconfigurable wavelength selective switch,” J. Lightw. Technol. 26, 73–78 (2008). [CrossRef]
21. B. Eggleton, T. Vo, R. Pant, J. Schroder, M. Pelusi, D. Yong Choi, S. Madden, and B. Luther-Davies, “Photonic chip based ultrafast optical processing based on high nonlinearity dispersion engineered chalcogenide waveguides,” Laser & Photonics Reviews 6, 97–114 (2012). [CrossRef]
22. A. Ellis, J. Lucek, D. Pitcher, D. Moodie, and D. Cotter, “Full 10 × 10 Gbit/s OTDM data generation and demultiplexing using electroabsorption modulators,” Electron. Lett. 34, 1766–1767 (1998). [CrossRef]
23. P. Andrekson, “Picosecond optical sampling using four-wave mixing in fibre,” Electron. Lett. 27, 1440–1441 (1991). [CrossRef]
24. H. Hansen Mulvad, L. Oxenløwe, M. Galili, A. Clausen, L. Gruner-Nielsen, and P. Jeppesen, “1.28 Tbit/s single-polarisation serial OOK optical data generation and demultiplexing,” Electron. Lett. 45, 280–281 (2009). [CrossRef]
25. Y. Paquot, J. Schröder, J. Van Erps, T. D. Vo, M. D. Pelusi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Single parameter optimization for simultaneous automatic compensation of multiple orders of dispersion for a 1.28 Tbaud signal,” Opt. Express 19, 25512–25520 (2011). [CrossRef]
