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We investigate the underlying mechanism that allows OFDM subcarriers in an all-optical OFDM system to be assigned to any optical frequency using an optical filter, even if that frequency is not generated by the comb-line source feeding the filters. We confirm our analysis using simulations, and present experimental results from a 252-subcarrier system that uses a mode-locked laser (MLL) as the comb source and a wavelength selective switch. The experimental results show that there is no correlation between the programmed frequency offset between a subcarrier and nearest comb line, and the received signal quality. Thus, subcarriers could be inserted into unused portions of an optical transmission system’s spectrum without restriction on their particular center frequencies. Any percentage of cyclic prefix can be added to the OFDM symbol simply by reprogramming the optical filter to give wider subcarrier frequency spacing than the comb line spacing, which is useful for tailoring the CP to the dispersion of various optical transmission paths, to maximize the spectral efficiency. Finally, the MLL’s center frequency need not be locked to a system reference.
© 2014 Optical Society of America
1. Introduction
Optical OFDM promises to provide high spectral efficiencies by packing together wavelength channels (subcarriers in OFDM) so that they spectrally overlap [1–3]. The overlapped subcarriers can be demultiplexed without crosstalk using the properties of a Fourier transform applied to a windowed portion of the received waveform, provided that the data symbols of every subcarrier are time aligned [4].
An efficient way to generate optical OFDM subcarriers is by combining modulated optical signals with different wavelengths, known as all-optical OFDM [5]. There are two general techniques to do this: one takes a comb-line source, such as a mode-locked laser (MLL) and demultiplexes its lines into monochromatic tones; each tone is then modulated using an optical modulator before being recombined with all of the other tones using a non-frequency selective optical coupler [6, 7]. The second method, shown in Fig. 1, is more akin to an RF OFDM system in that it uses an inverse Fourier transform (IFT) to generate a superposition of subcarriers, all with data-modulated phases and amplitudes; in optics, this can be achieved by splitting the output of the MLL into several paths, all with the same spectral content, then modulating each path [8–10], or modulating then combining the outputs of separate lasers [4]. The paths then form the inputs to an optically-implemented IFT, which generates the superposed subcarriers. The comb lines of the MLL are phase-locked, as this is implicit when producing regular pulses; however, this phase locking is not strictly necessary for optical OFDM because each subcarrier can be independently phase tracked at the receiver if the phase fluctuations are sufficiently slow. Both of these techniques produce a signal that can be demodulated orthogonally, as proposed by Chang [11], who did not use a Fourier transform in his early work on OFDM; thus, both techniques can be called optical OFDM.
Fig. 1 All-optical OFDM system using an IFT at the transmitter. C-MZI: Complex Mach-Zehnder Interferometer modulator. Rx: coherent receiver.
We have previously proposed that the transmitter in Fig. 1 could be implemented using an Arrayed Waveguide Grating Router (AWGR) as the IFT, if the AWGR is designed for uniform loss across all paths [9]; soon after, this was demonstrated experimentally by Takiguchi et al. [12]. An AWGR design was shown to be less sensitive to phase errors than coupler-based Fourier transforms [13] by Lim and Rhee [14]. This use of an AWGR is similar to earlier work implementing optical code-division multiplexing (OCDM) using AWGRs [15, 16], as recently clarified by Cincotti [17]. We have also shown that the technique in Fig. 1 demands less bandwidth of the optical modulators than placing the modulators after the wavelength demultiplexer [9]; this is because the optical inputs to the modulators in Fig. 1 are short pulses, which gives the modulators the majority of the OFDM symbol period to transition from one state to the next, in other words the optical pulses sample the modulator state at the symbol. A cyclic prefix (CP) can also be added optically, using a simple modification to the AWGR design, to further reduce the impact of optical and electrical bandwidth limitations and to increase the system’s tolerance to fiber dispersion [18].
Recently, we have implemented the system in Fig. 1 using a programmable wavelength-selective switch (WSS) [19–21]. The WSS can be thought of as a bank of parallel filters with separate inputs, whose outputs are combined [22]. If the filters’ impulse responses have a rectangular envelope, they will generate the sinc spectra required of OFDM systems, and so the WSS is implementing a Fourier transform. Interestingly, we found that the center frequencies of the filters need not align with the comb lines generated by the MLL, nor the internal frequency grid of the WSS; the spectrum of each subcarrier is determined only by the programmable optical filter profile of the WSS. For example, this flexibility enables a group of OFDM subcarriers (an OFDM superchannel) to be assigned to any wavelength in an optical transmission system. Another advantage is that the center frequency of the MLL need not be locked to the frequency grid of the transmission system, simplifying the design of the MLL. Also, a CP of any percentage of the OFDM symbol duration can be applied, simply by spacing the center frequencies of the filters for the subcarriers wider than the spacing of the comb lines. Thus it is possible to allocate a long CP to links with high dispersion, or shorter CPs to low dispersion links, enabling the spectral efficiency to be maximized for a given link.
This paper uses simple explanations to show how the subcarrier frequencies can be allocated to any output wavelength, based on time and frequency dualities, such as multiplication and convolution. It shows the effect of leaving some IFT inputs un-modulated. The explanations are confirmed by computer simulations and analysis of experimental results from our 252-subcarrier system carrying 10 Tbit/s.
2. Spectra in optical OFDM systems
An OFDM subcarrier is modulated by a series of adjacent data symbols, each of duration TOFDM. There is a transition in phase and/or amplitude at the end of each data symbol to the start of the next data symbol. An OFDM signal is a superposition of many modulated subcarriers, with their transitions aligned at the end of each OFDM symbol. The transmitted spectrum in an optical OFDM system is often described as being composed of overlapping sinc spectra, with each sinc spectrum representing the energy in a subcarrier. In the case of no CP, the frequency spacing of the sinc spectra equals the inverse of the OFDM symbol duration, so the main spectral peak of a given subcarrier coincides with the nulls of all other subcarriers. This statement is often used to imply that the subcarriers are orthogonal (have no inter-subcarrier interference). However, a more complete explanation is that each OFDM symbol is processed separately at the receiver. The Fourier transform assumes that the signal within the block represents one period of an infinite waveform. As each subcarrier is arranged to have an integer number of cycles within a block, each subcarrier transforms into a Dirac line; that is, there is no interference between the subcarriers. In optics, this processing can be achieved with a fractional Fourier Transform implemented all-optically [23].
It is insightful to study how the spectra of OFDM signals arise, so we can demonstrate that subcarriers can be assigned to frequencies not in the MLL comb spectrum. We will present this study in stages, using simplified time and frequency representations. In Section 2.1 we present an analysis of a MLL spectrum, as this is the comb source in our experiments. In Section 2.2 we explore the expected spectrum when the MLL pulses form a direct input to the IFT, without modulation. We show that each input of the IFT will create a set of spectral lines if the IFT frequency is misaligned with the comb source. These lines will fall upon the MLL’s comb frequencies. In Section 2.3 we use the Wiener-Khinchin(e) theorem to show that modulating the MLL pulses leads to an almost-white spectrum. Section 2.4 shows that the IFT can form sinc-spectrum subcarriers from the almost-white spectrum. Section 2.5 then illustrates how a CP of any proportion of the OFDM symbol can be added simply by retuning the center frequencies of the subcarriers.
2.1 Waveform and spectrum of a mode-locked laser (MLL)
Generating the OFDM subcarriers from a MLL implies that the symbol duration of an OFDM symbol, TOFDM, must be equal to the pulse spacing of the MLL. In this paper, we presume that the data symbol rate, R, is fixed, regardless of CP duration. The spectrum can be predicted by using a series of convolution and multiplication arguments in the time and frequency domains. Figure 2 shows the sequence of operations in the analysis.
Fig. 2 Steps to analyze a train of optical pulses, as would be output from a MLL. * represents convolution.
First we consider the pulses at baseband (the optical carrier is ignored). In the analysis, a train of pulses can be constructed in the time domain by convolving a regular train of Dirac pulses with the shape of a single MLL pulse, as shown in the first three rows of the left-hand column of Fig. 2. The dual of convolution in the time domain is multiplication in the frequency domain, shown in the right-column of Fig. 2. Thus, the spectrum of the train of pulses is a comb of Dirac lines, R = 1/TOFDM apart, under an envelope that is the spectrum of the single pulse. Ideally the envelope of this spectrum will cover the C-band and/or other transmission bands. If the output of the MLL is a train of unchirped Gaussian pulses each of width tpulse (Full Width Half Maximum, FWHM), the spectral envelope will be 0.44/tpulse. For 2-ps wide pulses, this means an envelope bandwidth of only 220 GHz FWHM. Fortunately, it is possible to significantly broaden the spectrum of the pulses using the Kerr nonlinearity [7]. Another very promising source are frequency combs based on micro-resonators, which have been demonstrated to yield very stable pulse trains with a spectral extent of >100 nm [24].
The optical carrier can be included in the analysis by multiplying a CW carrier in the time domain. The result is shown in the last row of Fig. 2. Because the carrier within each pulse is a very high relative frequency, it is shown as shading of the pulses.
2.2 Effect of inverse Fourier transform (IFT) on unmodulated pulses
Here we consider the effect of connecting a MLL directly to the input of an optical IFT. We have previously shown that the optical IFT can be used to shape the spectra of modulated optical pulses to generate one or more optical OFDM subcarriers [19, 20]. For a single input, the optical IFT is essentially an optical filter: for multiple inputs, it is multiple filters and a combiner, spectrally shaping each subcarrier and multiplexing them together to form an optical OFDM signal.
First we consider the case of unmodulated pulses, shown in Fig. 3. The output spectrum of the filter can be obtained by multiplying the input spectrum with the filter’s frequency response, as shown in the right-hand column of Fig. 3. Since these are linear filters, no new frequencies are created. In the special case where the peak of the sinc response lies on a comb line and the duration of the impulse response of the filter equals the pulse spacing, the nulls of the sinc function will align with all but one of the comb lines. Thus the output will be a single line. However, if the sinc is misaligned, or its nulls not spaced at the comb line spacing, then multiple lines will be output with only a single input, as shown on the bottom row of Fig. 3. If multiple inputs are of the IFT excited, each output comb line may be a combination of lines from multiple inputs. In the time domain (left column), the convolution of the filter’s impulse response with the MLL pulses produces train of identical symbols, that slightly overlap due to the finite width of the MLL pulses.
Fig. 3 Creating an OFDM subcarrier from unmodulated pulses using an IFT. When the desired center frequency of a subcarrier is offset from a comb line, it will have multiple lines in its spectrum, because the comb lines do not align with the nulls of the sinc spectrum. The convolution causes a slight overlap of the IFT impulse response caused by the finite width of the pulses. This will result in an amplitude ripple where the impulses responses overlap (darker shading).
2.3 Spectrum of modulated pulses
For the system to carry any information, the pulses must be modulated with data. For good sensitivity, Quadrature Phase Shift Keying (QPSK) can be used. Alternatively, for higher spectral efficiency, higher-order Quadrature Amplitude Modulation (QAM) is suitable.
If random data is carried on the amplitude and/or phase of successive MLL pulses, the Auto-Correlation Function (ACF) can be used to predict the spectrum using the Wiener–Khinchin(e) theorem. First taking the modulated pulses to be Dirac pulses, the ACF of a (wide-sense-stationary random) pulse sequence of infinite duration will be a delta function, and therefore, the spectrum is flat (as it is the Fourier transform of the ACF [25]). This is shown in the first row of Fig. 4.
The time-domain envelope of a continuous sequence of pulses will be a convolution of this randomly-modulated Dirac pulse sequence with the envelope of a single pulse. Therefore, the baseband spectrum of a sequence of Gaussian pulses with random data modulation measured over a long time tends towards a wide continuous spectrum with a Gaussian envelope. To include the carrier frequency into the analysis, we modulate the pulses with the carrier to produce a passband spectrum. It is this continuous spectrum that is fed into the IFT to obtain an OFDM subcarrier in Section 2.4.
2.4 Effect of inverse Fourier transform (IFT) on modulated pulses
This section brings together the ideas of the previous sections to examine the operation of the schematic in Fig. 1. The IFT in Fig. 1 takes a pulse train at each of its inputs and applies a filter function to each input, then combines the filtered pulse trains. Figure 5 shows the operation for a single input to the IFT. The IFT impulse response is a carrier wave with a square envelope: this has an equivalent frequency response of a sinc function around a carrier frequency. Each ‘filter’ convolves the randomly modulated pulse train with a rectangular impulse response, which spreads each symbol over the entire MLL pulse spacing, TMLL. These near-rectangular-envelope pulses have phase and amplitude transitions between the symbols, with some overlap due to the finite widths of the MLL pulses. In the frequency domain, the broad spectrum of the randomly-modulated pulses (Section 2.5) is multiplied by the sinc-response of the ‘filter’.
The sinc filter may be centered anywhere within the broad spectral envelope of the MLL, it does not have to be centered on a comb line. This is extremely useful for filling unused portions of the fiber’s transmission window; conversely, the MLL need not be tuned to the output wavelength. In effect, the IFT allocates its inputs to any set of wavelengths. If the IFT is made from a reconfigurable optical switch, then the frequency allocations of its inputs are completely reprogrammable. Also, because there is no electro-optical conversion and the ROADM (Reconfigurable Optical Add Drop Multiplexer) can be made from LCOS (Liquid-Crystal on Silicon) technology, which is electrically capacitive, so only uses power during reconfiguration.
The greatest spectral efficiency is obtained if the sinc responses of adjacent subcarriers are offset by 1/TOFDM. In this case, the nulls of one sinc spectrum will fall on the peaks of the other subcarriers’ sinc spectra. This is equivalent to having no CP, and so the spectrum will not have a ripple between its peaks.
2.5 Effect of adding a cyclic prefix (CP)
In most practical systems a CP is essential to maintain orthogonality; a CP helps overcome many system imperfections, including residual dispersion and the limited bandwidths of electrical components [18]. Without a CP, these effects will cause adjacent symbols to spread into each other [26]. Adding a CP also allows the subcarrier spacing and the symbol rate to be decoupled; the receiver can then strip the CP to regain the orthogonality of the signals.
It is trivial to add a CP using a WSS: reprogramming will allow any percentage CP to be added without hardware modifications [20] or having to alter the transmitted data rate. A CP is created by increasing the frequency spacing of the subcarrier allocations, as shown in Fig. 6. The MLL pulses are at the desired data symbol rate, so the total OFDM symbol duration must be equal to the pulse spacing of the MLL, that is TOFDM = TMLL. CP insertion (fourth row) therefore requires the OFDM symbols to be shortened to T’OFDM, so that T’OFDM + TCP = TMLL. This exact effect is achieved by reprogramming the WSS that implements the IFT to increase the frequency separation of the subcarriers to be 1/T’OFDM. The frequency separation of the subcarriers (1/T’OFDM) is now wider than the frequency separation of the MLL lines (1/TMLL) causing a ripple on the combined spectrum of many subcarriers.
Fig. 6 Effect of adding a CP and the important waveforms (left) and their spectra (right). When a CP is added, the duration of the OFDM symbol, TOFDM, is reduced to accommodate the CP, to T’OFDM. The frequency spacing of the subcarriers correspondingly increases, but their individual widths remain constant. The symbols θ1,2,3 indicates the phase of the modulation applied to each impulse. Amplitude modulation is also illustrated. The carrier waves have been greatly reduced in frequency to illustrate their phase discontinuities.
Alternatively, if the duration of the impulse response of the IFT filters is reduced in line with the OFDM symbol duration, to T’OFDM, there will be a gap between each OFDM symbol, rather than a CP. This scheme will generate a ‘zeroed’ temporal Guard Interval (GI), which gives inferior performance compared to CP, if residual CD or finite bandwidth components are used (see Fig. 7 of [18]).
Fig. 7 Spectra of an unmodulated (red) and modulated (blue) subcarrier with a 21% offset from a MLL line. No CP. The broadening of the unmodulated lines is due to the 500-MHz resolution of the spectrum analyzer model.
3. Simulation results
Simulations were performed with VPItransmissionMaker, to confirm the analyses. The simulation combined the outputs of four subcarrier generators to form an OFDM symbol; each generator produced a QPSK-modulated subcarrier with instantaneous transitions and an adjustable center frequency. The modulation rate of 10 Gsymbols/s equaled the MLL rate, and so the MLL’s comb spacing was 10 GHz. 1024 OFDM symbols were simulated and the spectra were calculated over all of these symbols using Fourier transforms. Figure 7 (—) shows the spectrum for a modulated single subcarrier that is offset by 21% from a MLL comb line (100% is the comb line spacing). This is a sinc spectrum offset from the comb lines, as predicted in Section 2.4. The spectrum without modulation (—) shows that the amplitudes of the comb lines follow the amplitudes of the modulated case’s sinc spectrum, as predicted in Section 2.2. Thus, if the subcarrier’s center frequency was not offset from a comb line, then the spectrum would only have one visible comb line, as all others would have fallen on the nulls of the sinc function.
Figure 8 shows the spectrum when four independently modulated subcarriers are combined without a CP. The carriers are all offset by 21% from the MLL frequency grid. The subcarriers combine to give a flat-top spectrum, as shown in Section 2.5. When the subcarriers are unmodulated, the spectrum consists only of the comb lines. The amplitudes of the comb lines depend on the relative phases of the subcarriers. These phases can be extracted using the forward Fourier transform in the receiver.
Fig. 8 Spectra of four unmodulated (red) and modulated (blue) subcarriers with a 21% offset from a MLL line. No CP.
The effect of adding a 22% CP is shown in Fig. 9. The comb-line spacing remains equal to the symbol rate ( = 1/TMLL), but the filters are tuned to have a spacing of 1.22/TOFDM. The increased spacing of the subcarriers means that the combined spectrum no longer has a flat-top, as predicted in Section 2.5. The unmodulated spectrum appears to have 5 dominant subcarriers, though there are only 4 inputs to the IFT. This is a result of increasing the frequency spacing between the subcarriers to greater than the comb line spacing.
Fig. 9 Spectra of un-modulated and modulated subcarriers when a 22% CP has been added, with the symbol rate kept at 10 Gsymbols/s.
4. Experimental verification
To experimentally verify these predictions, we re-analyzed the data set from a 252 × 40-Gbit/s all-optical OFDM superchannel experiment [20], to identify any signal degradation due to the misalignment of the subcarriers and comb lines. As shown in Fig. 10, this experiment used a 10-GHz Ergo MLL as a 2-ps pulse source, spectrally broadened using a highly-nonlinear optical fiber (HNLF). The repetition rate was locked to the data baud rate. The spectrum was flattened using a single input WSS, which also limited the signal bandwidth to 3 THz. A pseudo-random bit sequence generator (PRBS), with a delay between its data and data-bar outputs, was used to modulate QPSK data at 10 Gbaud onto the pulses, using a complex Mach-Zehnder interferometer (C-MZI) modulator. The single polarization output of the modulator was input at 45-degrees to a polarization beam-splitter (PBS) to generate two polarizations. These were delayed by 50 symbols relative to one another and combined using a polarization beam-combiner (PBC) to produce a polarization-multiplexed (polmux) signal at 40 Gbit/s. This signal was split into four paths, and {0,100,200,300}-symbol relative delays inserted into each path. The four resulting decorrelated signals, each a stream of polmux-QPSK-modulated pulses, were fed into the four input ports of a Finisar Waveshaper WSS, programmed to be an optical inverse Fourier transform (OIFT) generating sixty-three optical OFDM subcarriers from each input, that were interleaved with the subcarriers of the other inputs (i.e. ABCDABCD…), to form a 2.4-THz wide spectrum carrying 10.08 Tbit/s.
Fig. 10 Experimental setup to identify the impact of tuning the optical subcarriers away from the comb lines of the mode-locked laser (MLL).
A 10% CP was used to overcome the limited bandwidth of the coherent receiver. The CP was added by programming the WSS to space the center frequencies of the subcarriers 11-GHz apart. Therefore, only every tenth subcarrier was centered on a line of the MLL comb. Illustrative optical spectra along the system are given in [21]. The signal was sent over six recirculations of a loop containing two spans of standard single-mode fiber (S-SMF), two amplifiers and a WSS used as a gain flattener. The use of the CP conveniently demonstrates how the performance of the subcarriers is not affected by the misalignment of these subcarriers with the MLL lines, as will be demonstrated later.
At the receiver, another 4-port WSS was used to split the signal up into 66-GHz bands spaced 33-GHz apart. A coherent receiver, with a carrier wave local oscillator, was used to down-convert the optical signal to electrical baseband. A 4 × 80 GS/s real-time digital sampling oscilloscope (DSO) digitized the signal for offline signal processing. The signal was down-sampled to 60 GS/s before CD compensation was performed to prevent leakage from neighboring correlated subcarriers from improving the measured signal quality [21]. A 25-tap 1/6-spaced TD-equalizer (TDE) demultiplexed the subcarriers and compensated for polarization mode dispersion (PMD) and residual CD. Using six-times oversampling allows three subcarriers to be demultiplexed simultaneously without penalty, after shared CD compensation [27]. Finally, Viterbi-Viterbi carrier phase recovery was used. The quality of the signal, Q, was calculated from the Cartesian spreads of the constellations as in [28]. This means that a Q (sometimes known as Q2) of 9.8 dB equates to an error rate of 10−3, assuming the data points have a circularly-symmetric complex Gaussian distribution.
To identify the effect of the offset of a subcarrier’s center frequency from a MLL line on the signal quality, we grouped the subcarriers in terms of their offsets from the nearest MLL line. We then averaged the linear signal qualities of the subcarriers within a particular group, before conversion to a dB scale. Figure 11 shows the result of this processing, with the X and Y polarizations plotted separately. As an example, an offset of 5 GHz means this group of subcarriers is half way between two MLL lines and 0 GHz means the subcarrier group is aligned with a MLL line. The results in Fig. 11 show that there is no obvious correlation between the Q of a subcarrier group and its offset. Therefore, we conclude that the performance of subcarriers is not systematically affected by their frequency offsets from the comb lines of the MLL, which confirms the arguments in Section 2.
Fig. 11 Signal quality Q of a group of subcarrers versus the offset of an individual carrier from the closest comb line of the MLL.
5. Conclusions
This paper has analyzed the generation of optical OFDM subcarriers by modulating then Fourier transforming optical pulses from a MLL. The analysis confirms that it is possible to assign any modulated pulse sequence to any optical frequency within the bandwidth of the MLL, even away from an MLL comb line. Furthermore, we have shown that this flexibility provides a very simple method of prepending a CP to OFDM symbols. The analysis steps have been confirmed by numerical simulations and analysis of experimental results.
This technique of optical processing offers great flexibility for optical communications systems. For example, unused spectrum in an existing system can be filled with new superchannels of several OFDM subcarriers at any wavelength. The proportion of CP (hence the spectral efficiency) can be adjusted to suit the dispersion characteristics of the link. Furthermore, a set of inputs to a WSS can be assigned to subcarriers in any wavelength order, without the need for electronic switching. Additionally, the system can be used to generate a broad-spectrum of subcarriers for test purposes, because each input can be assigned to multiple outputs. Finally, the MLL’s center frequency need not be locked to a reference.
Acknowledgments
We thank VPIphotonics (www.vpiphotonics.com) for the use of their simulator, VPItransmissionMakerWDM V8.5. This work is supported under the Australian Research Council’s Discovery funding scheme (DP1096782) and CUDOS – ARC Centre of Excellence for Ultrahigh bandwidth Devices for Optical Systems (CE110001018). Arthur Lowery is also supported by an ARC Laureate Fellowship (FL130100041). Jochen Schröder is funded by an ARC Discovery Early Career Researcher Award (DE120101329).
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