## Abstract

Arrayed-Waveguide Grating Routers (AWGR) can be used as multiplexers and demultiplexers in optical OFDM systems, as they provide both the serial-to-parallel converter and the optical Fourier transform in one component. This paper shows how the design of the AWGR at the transmitter can be modified to insert a cyclic prefix or postfix (CP). We use simulations of a 4-subcarrier system to compare systems without the CP, with a guard-interval, and with a CP. We show that the CP greatly improves the orthogonality of the subcarriers and resilience to timing errors. Furthermore, the CP allows for uncompensated fiber dispersion, especially if the relative timing of the subcarriers upon transmission is adjusted.

©2012 Optical Society of America

## 1. Introduction

Implementing Fourier transforms using optical circuits [1] has recently become extremely topical [2], because optical orthogonal frequency division-multiplexed (O-OFDM) communications systems [3,4] require Fourier transforms at their receivers, and sometimes at their transmitters. O-OFDM itself is of interest, as it offers high spectral efficiencies, because the spectra of individual subcarriers are allowed to overlap, as they can be demultiplexed without interference using the Fourier transform. Many optical circuits have been proposed to implement the Fourier transform on serial optical waveforms, including ‘unwrapped’ fast Fourier transforms (FT) formed with couplers, phase-shifters and delays [2] and, more recently, modified designs of Arrayed-Waveguide Grating Routers (AWGRs) [5–8]. All designs share a common purpose of outputting a phase-weighed sum of a time-series of input samples for each demultiplexed channel, and this can be achieved by many equivalent optical circuits.

A common feature of many *electronic* implementations of O-OFDM transmitters that use digital processing to generate and decode the OFDM symbol [4], is the addition of a cyclic prefix (or, equivalently, cyclic postfix) to each OFDM symbol. An advantage of the CP is that the OFDM symbol can suffer chromatic dispersion (where each subcarrier receives a different delay along the fiber), without destroying the orthogonality of the subcarriers when demultiplexed by the receiver’s FT. This is because, with a sufficiently long CP, each subcarrier will remain periodic within the receiver’s FT window, which is equal to the duration of an OFDM symbol [9]. An OFDM symbol is formed from a sum of modulated subcarriers, each with an integer number of periods within the duration of the symbol. As shown in Fig. 1
, the CP extends the symbol by pre- or post-fixing additional cycles of each of the subcarriers to the symbol, so it effectively becomes longer. Note that the *next symbol* carries different data on all of its subcarriers, so has a waveform unrelated to the first symbol. In digital implementations of OFDM, adding a CP simply means reusing some of the sample values of an OFDM symbol and pre-pending them to the OFDM symbol to extend it.

Cyclic Pre- or Post-fixes (CP) can also be added to ‘all-optical’ implementations of O- OFDM, though early designs used guard-interval-less optical OFDM [3]. In addition to accommodating some dispersion in the link, a CP makes the system more tolerant to the practical limitations of real components in the system, such as electronic timing errors. At the receiver, optical implementations of FTs only produce a valid output during a fraction of an OFDM symbol, usually the duration of the symbol, *T _{symb}*, divided by the number of the OFDM subcarriers. This means a fast sampler (optical or electronic) is required after the FT [10], with a similar performance to that of a sampler used in a time-division multiplexed system. Adding a CP increases the width of the valid region by the duration of the CP,

*T*. A downside of a CP is that it reduces the data throughput of the system, as each subcarrier’s baud rate is reduced to$1/\left({T}_{symb}+{T}_{CP}\right)$.

_{CP}Although an optical circuit with a response at least approximating a FT is required at the receiver, to give close to zero inter-subcarrier interference, all-optical OFDM systems do not necessarily require an FT at the transmitter. Instead, the data modulators each modulate a separate optical wavelength, before all of the modulators’ outputs are combined using an optical coupler, as shown in Scheme 1 of Fig. 1 in Reference 6. A caveat is that the data streams should be synchronized so that the subcarriers of a given OFDM symbol all arrive in unison at the receiver. The optical wavelengths can come from separate lasers [11], although they are commonly obtained by demultiplexing an optical comb source (such as a mode-locked laser fiber) using narrow filters [12]. Because the comb lines are themselves narrow, the exact shape of the optical filter is unimportant, as is its phase response.

Lee, Thai and Rhee [10], Huang *et al*. [13], and Wang *et al*. [8], have studied systems using an inverse FT at the transmitter, following the modulators (Scheme 2 in Fig. 1 of Reference 6). Recently, Lowery and Du [6] have shown that this arrangement is beneficial as lower-bandwidth modulators can be used, because the modulators are fed with optical pulses, so that they only have to have the correct phase state during a small fraction of the OFDM symbol. Another benefit of feeding the modulators with pulses is that phase modulators can be substituted for complex optical modulators in the case of 4-QAM (QPSK) [13]. This is because, for a phase modulator illuminated with a CW source, the 3π/2 transition path – around ¾ of a circle in the complex plane – would cause a substantial frequency shift or chirp, so a complex optical modulator is conventionally used to make this transition cross the origin to remove the chirped portion of the signal. On the other hand, the illumination of a phase modulator with pulses forces the transitions to cross the origin.

An open question is how a cyclic pre- or post-fix can be added to an optical transmitter’s FT in the case where the FT is after the modulators. In this paper, we show that the problem is more complex than extending the data’s bit period by *T _{CP}*, as would work for transmitters where the modulators follow the optical filter. This paper shows how an AWGR can be modified to insert a CP. We use simulations to show that a CP is advantageous for all-optical networks as it increases the dispersion tolerance and decreases the sensitivity to the exact sampling point.

## 2. AWGR with cyclic prefix

#### 2.1 Initial design

Figure 2
shows a possible layout of an integrated-optic circuit designed to perform an optical inverse Fourier transform on modulated inputs, followed by a parallel-to-serial conversion to produce one OFDM symbol per input pulse. Also shown is the equivalent signal-flow diagram, which can be used to describe the digital implementation of this function. From left to right, the first slab coupler simply divides an input pulse to feed four parallel data modulators. The modulators are driven so that the data arrives at the input to the second slab coupler in synchronism, and may modulate the phase and/or amplitude of a given input pulse. The outputs of the modulator are effectively the Fourier coefficients of the OFDM waveform. The second slab coupler performs an inverse Fourier transform by forming a weighted sum of its inputs (*m* = 0, 1, 2, 3) for each of its outputs (*n* = 0, 1, 2, 3), the weights being phase shifts dependent on *m* and *n* [6]. The wanted outputs appear simultaneously (that is, in parallel) at the right-hand side of the second slab coupler, so need parallel to serial conversion. This is accomplished by applying a different delay to each output, then summing the delayed outputs with a third slab coupler. The delays are imposed by the array of grating waveguides. Note that the first and third slab couplers are simply a 1:*N* splitter and a *N*:1 couplers, so they could be implemented with different technologies. The second slab coupler is an *N*:*N* coupler with specific phase shifts, and is usually designed using Rowland circle methods to obtain the desired phase shift values [5].

The waveguide in green is an additional waveguide designed to create a second copy of the *n* = 0 output of the second slab coupler, then delay it, so that it becomes the last sample of the waveform. In this way, it forms a cyclic postfix. Unfortunately, splitting the power of the *n* = 0 output means that the first sample of the OFDM waveform is halved in power, and the CP is also halved in power. This could be compensated for by redesigning the tapered regions of the waveguides to double the power at the *n* = 0 output. A second problem is that the CP requires the longest delay, so the arrayed waveguide has to take a serpentine path, which may increase its losses. That said, this design is conceptually very close to the digital signal processing approach, shown as a signal flow diagram below the slab waveguides; of course, reusing values in digital signal processing does not reduce their magnitude, but it does in optical signal processing.

#### 2.2 Simplified design

Figure 3
shows a more elegant implementation of the transmitter with CP. The design is identical to Fig. 2, up to the output of the second slab coupler. An additional output (*n* = 4) has been added to the second slab coupler, and this passes through the longest waveguide before reaching the third slab coupler. In this way, it forms a cyclic postfix, so that every input pulse forms five output pulses of the transmitter: four for the OFDM symbol and one for the postfix.

This design avoids splitting the power of the *n* = 0 power and the associated problems of balancing the powers. This is because the *n* = 4 output can be designed to give identical phase shifts to the *n* = 0 output, if differences that are multiples of 2π are ignored. Thus, if the input waveguides of the second slab coupler are designed with appropriate tapers, so that they illuminate all outputs near-uniformly, the magnitude of the postfix should match the magnitude of all samples of the OFDM symbols, as is required. Alternatively, it may be possible to use attenuating regions in the waveguides and slabs to balance the powers. The positioning of the longest waveguide is also more convenient, as it follows a monotonic progression in waveguide lengths. Note that this is not simply an AWGR design with five arrayed waveguides, because the additional waveguide is placed strategically at the output of the second slab coupler, beyond the position that most AWGR designs would use.

## 3. System design process

The design process of the AWGR is closely linked to the system design. For example, the following process would give the major functional requirements for the AWGR, before the AWGR itself is designed:

- • Decide on a subcarrier spacing, Δ
*f*. This is limited by the bandwidth of the optical modulators; obviously more low-bandwidth modulators could be traded against fewer higher-bandwidth modulators. Recent work has shown that the optimum spacing in terms of fiber nonlinearity is dependent on the total transmission distance and in the order of a few GHz [14]._{SC} - • Decide on the number of data-carrying subcarriers,
*N*. This depends on the data rate,_{SC}*R*, to be carried by a wavelength channel (a group of subcarriers managed as one wavelength), the modulation format used for each subcarrier and whether polarization-multiplexing should be used. For example, for systems without a CP using*m*-QAM and a single polarization, the payload data rate will beAdditional subcarriers will be required if a cyclic prefix is used, as the data rate per subcarrier will be reduced.

- • Decide on the number of nulled subcarriers,
*N*, around the data-carrying subcarriers. These can be used as frequency guard-bands around a wavelength channel. The guard-bands carry the tails of the subcarrier’s sinc spectra, and so are necessary for the subcarriers to remain orthogonal. Typically, two nulled subcarriers either side of the data-carrying subcarriers would be sufficient; that is,_{null}*N*= 4._{null} - • The total spectral bandwidth of the wavelength channel, Δ
*f*, can be estimated from_{λ}The spectral efficiency of a multi-wavelength system will depend on whether the wavelengths are combined to overlap (which is possible with OFDM), or a frequency guard-band is used between the wavelength channels. A guard-band makes conventional wavelength management, such as add-drop multiplexing, easier, but reduces the spectral efficiency by a factor of ${N}_{SC}/\left({N}_{SC}+{N}_{null}\right)$.

- • Decide on the duration of the CP,
*T*. This depends on several factors as the CP fulfills several purposes. The CP should be at least the fiber’s differential delay across all data-carrying subcarriers (including their tails), so is governed by the link length, fiber type, and use of dispersion-compensating modules. A reasonable estimate for a link with an uncompensated dispersion of_{CP}*DL*(s/m) operating at a wavelength*λ*, would beThe CP’s duration could also be increased to lengthen the duration of the valid sampling window at the receiver, to allow for slow sampling gates and for polarization-mode dispersion.

- • The duration of the CP is constrained to be an integer multiple,
*N*, of the differential delay between the AWGR’s arrayed waveguides, Δ_{CP}*T*. The differential delay between the arrayed waveguides is_{wg}In transmitters using an optical comb source rather than a pulse source, the duration of the CP is not constrained to be an integer multiple of Δ

*T*._{wg} - • After adding a CP of duration
*N*. Δ_{CP}*T*, the duration of the OFDM symbol increases to_{wg}Note that, because the OFDM symbol is limited in duration by the minimum subcarrier spacing, Δ

*f*, that can be achieved using AWGRs with a reasonable footprint, the duration of the CP will become unreasonable in links without optical dispersion compensation._{SC} - • The pulse repetition frequency,
*PRF*, of the optical source feeding the modulators (which is equal to the data rate of the modulator drives), is simplyIn contrast, the repetition rate in designs that use the MLL as a spectral comb source that is demultiplexed to feed the modulators is not affected by the CP, and is simply equal to the subcarrier spacing.

At this stage, there is enough information to calculate the key parameters the AWGR itself.

- • Without a CP, the number of arrayed waveguides is simply
*N*._{FT} - • The differential delay between the waveguides is equal to Δ
*T*._{wg} - • The slab region of the AWGR has to be designed so optical paths across the input slab, from inputs to the arrayed waveguides, have the appropriate phase delays to implement the Fourier transform and the cyclic prefix [5].
- • The addition of the CP simply requires
*N*additional arrayed waveguides, either longer (postfix), or shorter (prefix) than the waveguides in a design without a CP. The differential delays between these new waveguides are identical to the differential delays between the non-CP waveguides._{CP}

The full design of the AWGR, to mask level, would require simulations using planar waveguide simulators. The AWGR design for an FFT requires that the losses from any input to any output are substantially uniform [5]. This could be achieved by designing the waveguide tapers to produce a substantially uniform illumination of the waveguides at the opposite side of the second slab coupler, though this would increase the loss of the system, due to power spilling beyond the output waveguides.

## 4. System simulations

#### 4.1 System without CP

An OFDM system carrying four data-carrying subcarriers (*N _{SC}* = 4) was simulated using VPItransmissionMaker v8.6, to assess the effectiveness of adding the cyclic prefix optically. Considering the simulations without a CP, the subcarriers are spaced at Δ

*f*= 10 GHz, giving an OFDM symbol duration of 100 ps. 4-QAM (QPSK) was modulated onto each subcarrier, giving a data capacity of 80 Gbit/s per polarization, though only a single polarization was simulated. To allow for spectral guard bands, the AWGR was designed with 8 guides, to give a free-spectral range of 80 GHz, so

_{SC}*N*= 4 and

_{null}*N*= 8. The differential delay between the adjacent arrayed waveguides is (100/8) ps = 12.5 ps. The sample rate of the simulation was set to be 3.125 ps, giving a simulation bandwidth of 320 GHz.

_{FFT}The pulsed laser was simulated as a 3.25-ps pulse at a 10 GHz repetition rate. These pulses are modulated by complex optical modulators fed with QPSK data. The outputs of the modulators feed a model of the transmitter AWG, which is a network of splitters, delay lines, phase shifters and star-couplers designed to implement the signal-processing representation of the transmitter circuit of Fig. 3. The output of the transmitter is bandlimited with an optical filter to 80-GHz bandwidth, within the free-spectral range of the AWGR. This represents the wavelength multiplexers at the transmitter and receiver. The output of the filter is passed to a linear model of a dispersive standard single-mode optical fiber (*D* = 16 ps/nm/km), whose length can be parametrically swept. The output of the fiber is amplified and passed to a model of the receiver Fourier transform circuit (including the serial to parallel converter) [5]. The outputs of this circuit are passed to coherent receivers with a 25-GHz electrical bandwidth and then sampled using an electrical sampler, then equalized by training the system with known data, to form a constellation diagram. The main performance parameter was the quality, *Q*, of the electrical signals after the coherent receiver [15].

#### 4.2 System with CP

This is similar to the system without a CP except that two extra arrayed waveguides are added to the transmitter AWGR. These have delays of 100 ps and 112.5 ps, relative to the first guide, so extend the length of the OFDM with CP symbol to 125 ps. The outputs of these guides receive the same phase shifts as the first and second guides (with delays of 0 ps and 12.5 ps) as they traverse the phase shift network. The pulse repetition frequency of the laser also has to be reduced to 8 GHz, to accommodate the extended symbol length. Furthermore, the extended symbol length means that the phases of the local oscillators at the receiver advance from symbol to symbol, so the equalizers need to add a phase shift of *np*π/2 per symbol, where *n* = −1,0,1,2 is the receiver index and *p* is a integer up-counter incremented every OFDM symbol. The receiver’s AWGR is identical to the system without the CP.

#### 4.3 System with guard interval (GI)

The system with a CP can also be operated as a system without a CP, but with a (time-domain) guard interval (GI) between the symbols. In wireless OFDM, the term *guard interval* is used to convey an interval where no power is transmitted: this definition is used herein. A system with a GI uses the same the design parameters as for an AWGR with a CP, but the additional waveguides for the CP are deleted from in the mask layout. In the simulation, this simply means blocking the additional waveguides in the model, while keeping the lowered repetition rate at the source, and the evolving phase at the equalizer.

## 5. Simulation results

#### 5.1 System with zero dispersion, no CP nor a GI

Figure 4
plots the spectrum, eye diagrams for the lowest-frequency and next lowest subcarrier, and the *Q* versus sample point, for a noiseless back-to-back system with no dispersion. The spectrum is almost flat where the four subcarriers overlap, and has sidebands which are the superposition of the tails of the sidebands. The eye for the lowest-frequency channel (Channel 1) shows some closure. The *Q* versus sample point has a single peak reaching 23 dB for the outer channel and 24 dB for the inner. Although acceptable for low constellation sizes, even with the addition of noise, if the subcarriers were orthogonal, the peak should be much higher. Simulations with an 80-GHz bandwidth receiver identified that this reduction is due to the band-limiting due to the optical filter and the electronic sampler. The peak value reduces by around 3 dB for one sample point (3.125 ps) either side of the peak, showing that timing is critical in this system.

#### 5.2 System with zero dispersion, with GI

Figure 5
plots the spectrum, eye diagrams, and the *Q* versus sample point, for a noiseless back-to-back system with no dispersion, but with a guard interval added after each OFDM symbol. As there is no power in this interval, the spectral envelope is similar to the system without the guard interval. The eye openings are wider than before because there is an extra 25 ps per symbol for the transitions. The eye traces look more deterministic; the individual traces from one symbol to the next can be distinguished. The *Q* versus sample point has a single peak reaching 27 dB for the inner channels, but there is a similar sensitivity to the timing point as before, if measured 3-dB below the peak, though the range of timings with an acceptable *Q* is broadened.

#### 5.3 System with zero dispersion, with CP

Figure 6
plots the spectrum, eye diagrams, and the *Q* versus sample point, for a noiseless back-to-back system with no dispersion, but with a cyclic postfix added after each OFDM symbol. Because the postfix is coherent with the signal within the OFDM symbol, there is some interference, causing nulls and peaks in the spectrum. The eye diagrams have very flat upper and lower rails. These flat rails are a result of choosing the optical bandwidth to be 80 GHz; increasing the bandwidth makes the rails have deterministic ripples, as each Channel becomes carried by several interfering subcarriers spaced at the free-spectral range of the AWGR, so the sampling point becomes critical. Compared with Fig. 4 and Fig. 5, the *Q* versus sample point has a broadened peak, meaning timing is much less critical, which is expected from the flatness of the eyes. The high value of *Q* indicates the channels are nearly orthogonal. Thus, adding a CP is far preferable to adding a guard interval, and offers the same spectral efficiency as with a guard interval.

#### 5.4 Effect of fiber dispersion

One advantage of cyclic pre/postfixes is that they allow the subcarriers to pass through a dispersive link without losing orthogonality. This is because the receiver’s Fourier transform only samples over the OFDM symbol length (without the CP), rather than the OFDM plus the CP. Thus, if properly time aligned it will not sample any leakage from adjacent symbols, which is caused by the differential group delay between subcarriers. The duration of the CP should be chosen to accommodate slightly more than the maximum differential group delay across the OFDM band, but given that the tails of the subcarriers’ spectra theoretically extend to infinity, the width where the majority of the power lies should be considered. In our case the CP is 25-ps long, and given a 40-GHz signal bandwidth (0.32 nm), the maximum length of uncompensated single-mode fiber will be 25/(0.32 × 16) km, that is 4.9 km.

Figure 7
plots the *Q* versus the uncompensated fiber length for all three systems. In each case, the *Q* for a given subcarrier is calculated assuming that the sample point can be swept to find the optimum eye opening for that subcarrier, rather than having a common sample time for all subcarriers. This can improve the *Q* by several dB because the eyes of the subcarriers become shifted relative to one another differential group delay of the fiber. The system without a guard interval or a CP is the worst performing, with a limited *Q* for no fiber, a 10-dB penalty for 5-km of fiber and the *Q* drops below 9.8 dB (equivalent to a BER of 10^{−3}) at around 10 km of fiber. Adding a guard interval improves the *Q* at all of these points by over 3 dB, though the data payload rate is reduced to 64 Gbit/s. Adding the CP greatly improves the back-back performance of the system, as it allows for the effects of bandlimiting in the optical link and in the electrical receiver. An uncompensated fiber length of 30 km could be envisaged, though there would be no margin for other forms of degradation such as amplifier noise.

#### 5.5 Dispersion precompensation by retiming the data modulators

One interesting observation while performing the simulations in Section 5.4 was that the optimum sample time for each subcarrier is different. This is because each subcarrier receives a different group delay from the fiber, so does not arrive synchronized with the other channels at the receiver. This de-synchronisation also causes the ‘OFDM condition’ [9] to be broken if the differential delay exceeds the duration of the CP. Figure 8 shows an eye for a system with 15-km of uncompensated dispersion, which is beyond what the CP can accommodate; the randomness of the paths (compared with Fig. 5) indicates that there is significant inter-symbol interference between the OFDM symbols.

A possible solution to the desynchronization is to delay the transmission of higher-frequency channels at the transmitter relative to the frequency channels [16]. The fiber will delay the lower-frequency channels so that all channels arrive in synchronism at the receiver. By sweeping the precompensation delay for a 10-km link, we found that the *Q* over the four subcarriers could be improved by 1 dB, from 18.4 dB to 19.5 dB. The optimum precompensation was to advance the transmission of the subcarriers (1, 2, 3, 4) by (9.375, 3.125, −3.125, −9.375) ps, respectively. The main effect was to increase the performance of the outer subcarriers (1, 4) by 3 dB, so they no longer limited the overall *Q*. For a 20-km link, the optimum timing advances were double those for 10-km, and the improvement in the worst subcarrier of each set was 0.7 dB. Thus, precompensation only gives a small improvement, probably because each subcarrier has wide spectral tails which suffer from large group delays.

## 6. Conclusions

This paper has demonstrated that a CP can be easily added to an all-optical OFDM transmitter based on an AWGR placed after the modulators, by adding one or more extra waveguides from the output of the slab coupler. Importantly, these waveguides need to be placed so they receive identical signals (apart from a phase shift which is a multiple of 2π) to the shortest waveguides. In this way, the layout of the transmitter’s AWGR is much simplified. If a variable-duration CP is required, or a guard-interval is preferred, one or more of these waveguides could be switched out of the optical circuit, for example, using variable attenuators.

Interestingly, the pulse source also has to be modified so that the pulse repetition rate is lower than the frequency spacing of the subcarriers. This is in contrast with systems using separate CW sources for the subcarriers, where the pulse repetition rate equals the subcarrier spacing, even when a CP is added. Although a variable-length CP is more easily added to systems using CW sources, by lowering the data rate driving the modulators, these systems require higher-electrical-bandwidth modulators to maintain signal orthogonality [6].

This paper has defined a design process for the system and transmitter and receiver AWGRs, then use simulations to verify that the CP has a beneficial effect on the performance of a dispersion-limited system, far exceeding the performance improvement when a simple (signal power free) guard interval is added between the OFDM symbols. This paper also shows that precompensation can be applied to the transmitters, by delaying the relative transmission times of the subcarriers, to provide some mitigation of the effect of dispersion.

## Acknowledgments

I should like to 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).

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