## Abstract

All-optical OFDM uses optical techniques to multiplex together several modulated lightsources, to form a band of subcarriers that can be considered as one wavelength channel. The subcarriers have a frequency separation equal to their modulation rate. This means that they can be demultiplexed without any cross-talk between them, usually with a Discrete Fourier Transform (DFT), implemented optically or electronically. Previous work has proposed networks of optical couplers to implement the DFT. This work shows that the topology of an Arrayed Grating Waveguide Router (AWGR) can be used to perform the demultiplexing, and that the AWGR can be considered as a serial-to-parallel converter followed by a DFT. The simulations show that the electrical bandwidths of the transmitter and receiver are critical to orthogonal demultiplexing, and give insight into how crosstalk occurs in all-optical OFDM and coherent-WDM systems using waveforms and spectra along the system. Design specifications for the AWGR are developed, and show that non-uniformity will lead to crosstalk. The compensation of dispersion and the applications of these techniques to ‘coherent WDM’ systems using Non-Return to Zero modulation is discussed.

© 2010 OSA

## 1. Introduction

Orthogonal Frequency Division Multiplexing (OFDM) [1] allows the spectra of many subcarriers to be partially overlapped, so that a very high spectral efficiency can be achieved [2]. Optical implementations of OFDM using digital signal generation divide the data amongst many tens of subcarriers, using Fourier transforms to generate then detect these subcarriers [3–6]. All-optical OFDM [7], or Coherent WDM [8], use optical couplers to combine a few modulated subcarriers, with the subcarrier spacing set to be equal to the modulation rate of each subcarrier, and the data symbols synchronized in time [9]. At the receiver, these subcarriers have been demultiplexed using optical filters [8, 10], extremely-high sampling rate oscilloscopes followed by digital processing [11], or optical processing [12, 13].

Chandrasekhar and Liu [9] have recently investigated the design requirements for systems using a few subcarriers per wavelength, with electronic signal processing to separate the subcarriers. Their experiments confirmed that the OFDM condition (the modulation baud rate, *when no cyclic prefix nor guard bands are used*, equals the subcarrier spacing and the modulation of the subcarriers is time-aligned) gives a local optimum for received signal quality whilst maintaining a high spectral efficiency. To obtain low cross-talk between the demultiplexed subcarriers, the receiver required four-times oversampling and fractionally-spaced equalizers. A five subcarrier system was also studied, using a standard 0.5-nm optical band-pass filter for subcarrier separation. This arrangement led to a floor in the Bit Error Ratio (BER) versus Optical Signal to Noise Ratio (OSNR) characteristic.

Chandrasekhar and Liu’s work leads to the open question of whether a custom-designed optical filter may provide improved performance; that is, ideally zero penalty due to adjacent-channel crosstalk, as obtained when using electronic Fourier transforms. Other groups have approached the issue by designing optical-analogs of the Fourier transform. In 2002, Sanjoh, Yamada and Yoshikuni [14] proposed a network of a splitter, an array of delays and phase shifts, a coupler and a gate (sampler) to demultiplex one orthogonal channel of an orthogonal wavelength-division multiplexed system, and demonstrated a 2-channel system using an unbalanced Mach-Zehnder Interferometer. In 2008, Lee, Thai and Rhee [13] simulated a multi-channel all-optical discrete Fourier transform processor and demonstrated that perfect demultiplexing of four 25-Gbit/s QPSK channels is possible. Their system proposed a network of delay lines, optical splitters, phase shifts and optical combiners to implement the discrete Fourier transform (DFT), with individual outputs for each subcarrier. Although these optical arrangements provide a continuous output, based on the values of the input signal at discrete equally-separated times, this output needs to be sampled to recover the data [14]. In their paper, Lee, Thai and Rhee suggest that an Arrayed Waveguide Grating Router (AWGR) could be used to implement their proposed DFTs, with the addition of “*time delay waveguides at the single wavelength ports of an AWG WDM and a power combiner…*” [13]. Other authors have proposed and demonstrated systems using an optical time-lens for Fourier transformation [15]; optical fiber Bragg gratings (FBGs) at the transmitter to form an OFDM waveform from short pulses [16], or FBGs at the transmitter and receiver to form then demultiplex an OFDM signal [17]; an optical coupler to implement a 2-point transform [18]; or a silica Planar Lightwave Circuit (PLC) to implement a Cooley-Tukey [19] Fast-Fourier Transform topology that uses fewer phase shifters and couplers [12, 20]. Demultiplexing of a 160-Gbit/s polarization-multiplexed OFDM signal using an 8-point FFT was experimentally demonstrated in 2010 by Takiguchi *et al.* [21]. Simultaneously, Hillerkuss *et al*. have reported a rearranged combination of a parallel-to-serial converter and a 4-point optical Fast Fourier Transform (FFT) to demonstrate real time processing of optical OFDM waveforms at 392 Gbit/s [22]. The same group has recently experimentally implemented one channel of an 8-point optical FFT to demultiplex 18 Gbaud/s channels from 10 Tbit/s line rates [23] and give a detailed analysis of their methods including a general proof of how a 2* ^{n}*-point FFT can be “unraveled” for easier implementation as a planar circuit [24].

This paper concentrates on how to design OFDM demultiplexers for closely-spaced subcarriers using Arrayed-Waveguide Grating Router (AWGR) [25, 26] topology. The aim is to be able to demultiplex the subcarriers without cross-talk; to make maximum use of the orthogonality of the transmitted carriers. The simulations in this paper show that an AWGR could be used to achieve this perfect demultiplexing. The AWGR has the advantage that it may be simpler to fabricate and phase-control than an arrangement of couplers as used in above DFT and FFT implementations, particularly if all channels are to be demultiplexed at once. The simulations show that the electrical bandwidth of the modulation and the receivers is critical to maintaining orthogonality; this is illustrated using eye diagrams which show the mechanism for crosstalk. The effect of non-uniformity of the AWGRs is investigated, and then the bandwidth requirements for a system carrying 120 Gbit/s are simulated. Finally, the compensation of dispersion and the applications of these techniques to coherent-WDM systems are discussed.

## 2. Theory of optical OFDM

Figure 1 shows one wavelength channel of a typical all-optical OFDM system [9]. This comprises a number of lasers; each is modulated with a separate optical modulator, before the outputs of the modulators are combined with a coupler. An alternative to discrete lasers is to use a single laser source that is then strongly modulated to form a comb of laser lines [27]. These are then demultiplexed, and each line becomes the input to a separate modulator. This ‘comb’ approach has the advantage that the lines of the comb are phase locked to one another (though this is not necessary for OFDM) and so lie on a well-controlled frequency grid, which can be locked to the modulating data’s rate. Ellis and Gunning [8] have shown that phase locking between the subcarriers is desirable in Non-Return-to-Zero (NRZ) systems; however, Chandrasekhar and Liu [9] show this is unimportant in QPSK-modulated systems. This discrepancy will be considered in the discussion in Section 5. The combined signal is transmitted over the fiber plant, which may include optical amplifiers and dispersion-compensation modules. At the receiver, a demultiplexer filter is used, and the demultiplexed subcarriers are fed to separate receivers. These receivers can use differential QPSK detection, or can be coherent, so the phase modulation appears as inphase and quadrature waveforms. Coherent receivers capture the full optical field [28], so digital signal processing can be used to compensate residual dispersion, or the full dispersion of the link, including PMD [29].

#### Conditions for orthogonality

Chandresekhar and Liu [9] have shown that an important condition for minimizing cross talk between the subcarriers is to have the data waveforms into each modulator time-aligned, though this was probably an implicit assumption in previous work. They also showed that crosstalk is also minimized by adjusting the frequency-spacing of the subcarriers to equal the symbol rate of each data channel. This is called the “OFDM” condition. Yang, Shieh and Ma, showed that the OFDM condition also applies when multiplexing several bands of subcarriers [30]. Their work also showed that if the OFDM condition (or a multiple of it) is not met, a guard-band with a width several times the subcarrier rate is required to reduce the crosstalk penalty to < 1 dB. Obviously this would give a low spectral efficiency, compared with meeting the OFDM condition.

The conditions for orthogonality of subcarriers (the OFDM condition) impose requirements on the transmitter and the receiver. The problem is one of designing the transmitter waveform so that the receiver can separate the sub-carriers without interference. The most common system is for each subcarrier to have an integer number of periods within an OFDM symbol [31]. This is achieved when the subcarriers are each centered on a frequency grid, with the grid spacing equal to the inverse of the OFDM symbol’s duration (neglecting cyclic prefixes that are added to the symbol).

To send useful data, the phase (and/or amplitude) of each subcarrier must be modulated. This modulation produces phase and/or amplitude transitions between the OFDM symbols, as shown in Fig. 2 , which presents the real part of the modulation for two subcarrier frequencies. The subcarriers are represented as separate waveforms for the purposes of this illustration, though they are combined for transmission to produce a complicated non-sinusoidal waveform. They have also been frequency-shifted to baseband, to reduce the number of cycles in each waveform. Many all-optical OFDM systems consider the transmitter in great detail, but use a simple optical filter to demultiplex the subcarriers. Chandrasekhar and Liu simulated and experimentally evaluated the performance of such systems. The use of a simple optical filter can be partially mitigated using electronic signal processing, but this requires multiple samples per bit to achieve effective correction.

The method by which the signal is received is also important if the transmitted signal’s orthogonality is to be useful in eliminating crosstalk between subcarriers. Continuing with Fig. 2, if the inphase and quadrature components of the received waveform are sampled *N* times during each OFDM symbol, it is possible to completely distinguish between *N* subcarriers, and also detect each subcarrier’s phase and amplitude. For example, in Fig. 2, consider four equally-spaced samples, A, B, C, D of the first symbol. These four samples form a *sampling window*. The lower (‘red’) subcarrier would produce samples with + + − − values: the upper (‘teal’) subcarrier would produce samples with + − + − values. Thus by weighting the samples, then adding them, it is possible to differentiate between the subcarriers; for example, the sum P_{teal} = A − B + C − D would detect the ‘teal’ subcarrier of the first symbol, completely rejecting the ‘red’ subcarrier. Looking at the second OFDM symbol in the left graph, P_{teal} would be zero, because the samples fall on the zero-crossings of the teal waveform. Fortunately, the when the same sampling instants are applied to the quadrature (or imaginary-component) of the teal subcarrier (not shown), they produce a non-zero result, indicating a phase of 90°.

Figure 3
can be used to illustrate some important requirements for the OFDM condition. For example, if the samples were delayed in time by one sample, so that sample D becomes sample C, and sample D now falls at the beginning of the second symbol, giving sample values for the red-waveform of + − − −, then the sum P_{teal} = A − B + C − D is no longer zero when applied to the red waveform. This means that information in the red waveform has leaked into the detector for the teal waveform, so the waveforms are no longer orthogonal. This introduces the condition that: “*the sampling window must not include a phase transition*”. This allows some drift of the sampling instants: if the phase transitions are instantaneous, this drift can be up to half the period between samples.

Another important point is that the *phase transitions must be fast enough so as not to affect the value of the samples* (particularly samples A and D). Figure 3, Right, shows an example using our simplified waveforms, but shifted 90-degrees. The top waveform is for the red subcarrier with a fast transition: The sum P_{teal} = A − B + C − D will produce a null output, as desired, because the magnitudes of all of the samples are equal. The bottom waveform is for a slow transition, which has lowered the magnitude of sample D. The sum P_{teal} = A − B + C − D now produces a non-zero output, indicating leakage between channels. In this example, the transition must be substantially complete between two consecutive samples, if the sampling is instantaneous. In Fourier transform theory, it is well known that non-uniform windowing of waveforms (reducing the magnitude of some samples) will cause leakage between adjacent frequency components.

#### Forming the sum of the samples

The examples in Figs. 2 and 3 used weights of + 1 and −1 when forming the sum of the samples. More generally, these weights are phase shifts applied to the inphase (‘real’) and quadrature (‘imaginary’) components of the complex received signal waveform, *V _{in}*. The sum of the samples is typically written as a discrete Fourier transform (DFT), giving a complex-valued output,

*V*, carrying the amplitude and phase subcarrier

_{sc,k}*k*:

*V*is the value of the complex input waveform at sample points

_{in}(m)*m*= 1,2…

*N*-1, where

*N*is the number of subcarriers. Because the samples are being continually updated as the input waveform arrives at the receiver, the sum of the samples, hence the estimate of

*V*, is also being updated, that is, it is a continuous waveform. If the interval between samples is Δ

_{sc,k}*T*, Eq. (1) can be rewritten as:

This received signal waveform, *V _{in}*, could be derived from the baseband signal obtained from a homodyne optical coherent receiver. Equation (2) can be implemented in the optical or electrical domains. In the electrical domain, it could be formed using heterodyne detection (using an RF mixer) and an integrator based on an analog electrical filter, or an

*integrate and dump*circuit. More commonly Eq. (2) is evaluated at discrete times [14], using digital computation, which processes discrete samples of the input waveform. As outlined in the Introduction, many authors have suggested optical networks to form the sum of the samples.

## 3. The AWGR as a DFT

Lee, Thai and Rhee’s [13] suggestion that an Arrayed Waveguide Grating Router [26] could be used to implement their proposed DFTs, with the addition of “*time delay waveguides at the single wavelength ports of an AWG WDM and a power combiner…*” neglects an important equivalence between a standard AWGR (followed by samplers) and the DFT, which is illustrated in Fig. 4
. The top of this Fig. shows an AWGR [26] with, from left to right, an input slab coupler, an array of grating waveguides, an output slab coupler and optical (or electronic) samplers. The lower part of Fig. 4 shows the equivalent signal processing representation of the AWGR with the functions of, left to right, a splitter, an array of time delays, a matrix of phase shifts surrounded by splitters and couplers, and samplers. The equivalences will now be described left to right. The notation of Madsen and Zhao [32] is used.

#### AWGR input slab coupler

The AWGR input slab serves as the power splitter in the signal processing equivalent. With correct placement and design of the input waveguide, the input waveguide will illuminate all the grating waveguides with an equal phase. By designing the waveguide tapers of the waveguides using simulation, the coupling of power to from the input to each grating waveguide will be reasonably uniform [33]. The required degree of uniformity will be discussed later. Alternatively, a multi-mode interference (MMI) coupler [34] could be used.

#### AWGR grating waveguides

The grating waveguides provide time delays which increment by a constant amount, due to their increasing lengths [26]. In signal processing terms [32], the input slab coupler (*splitter*) and the grating waveguides perform as a *serial-to-parallel converter*. This is because each grating waveguide receives a replica of the input waveform, which is time-shifted by the waveguide. Thus at any one instant, the outputs of the grating waveguides are four successive samples of the input signal. This is illustrated in Fig. 5
. The difference in length, Δ*L*, between two successive grating waveguides is approximately related to the time interval between samples, Δ*T*, by [32]:

*c*is the speed of light and

*n*is the index of the grating waveguides.

_{g}#### AWGR output slab coupler

The outputs of the grating waveguides, label *m*, are presented to the output slab coupler, which implements a matrix of phase shifts, surrounded by splitters and couplers. The phase shifts implement the exponential term in Eq. (1); the couplers implement the summation for each subcarrier frequency, and present the result for each subcarrier at the outputs of the AWGR. Ideally all paths from any grating waveguide to any output have the same loss. Again, by proper design of the waveguide tapers, it is possible to make the loss reasonably uniform. Alternatively, a multimode-interference coupler could be used [35].

Many AWGR designs [26] use a Rowland Circle (RC) configuration [36] for the slab couplers (see Fig. 4). This means the arrayed grating waveguides terminate on a circular boundary of the slab, radius, *R*, while the output waveguides terminate on a circular boundary of radius *R*/2 (the Rowland Circle). The RC circle is co-tangential with the large circle at the centre of the arrayed waveguides, called the ‘pole’. The distance between any input (from the arrayed waveguides) and output, *AP*, is given by a rather complex expression (Eq. (9) of [36]), which can be approximated to the following expression ([*F*
_{1}] of [36]) if the spacing between the chosen grating waveguide and the pole* _{,} w,* is small compared with

*R*:

*θ*is the angle from the pole to the output waveguide and

*r*is the distance from the pole to the chosen output waveguide. This path length introduces a phase delay dependent on the originating arrayed waveguide and the receiving output waveguide. Because

*r*and θ are fixed for any chosen output guide, and

*d*increments from one arrayed waveguide to the next, it is possible to implement a set of linearly-decreasing phase shifts between any arrayed waveguide and any output. Poguntke and Soole [37] provide a simpler analysis than [36] by defining the problem to two dimensions, which is appropriate for integrated optics.

Madsen and Zhao [32] provide the following equation (Chapter 4, Eq. (28), lower) to describe the second slab coupler of the AWGR as a matrix of transmission coefficients between the outputs of the arrayed waveguides (label *m*) and the outputs of the AWGR (label *n*, which carries the signal of a subcarrier, *k*). A similar expression is also provided by Gholipour and Faraji-Dana [38]. The phase factors for each element of the matrix are:

*n*is the effective index of the slab,

_{s}*d*is the spacing (pitch) of the grating waveguides as they enter the output slab,

*λ*is the centre wavelength of the system,

*R*is the focal length of the grating array as used in the RC derivation (e.g. from the output of any grating waveguide to the central output waveguide),

*d*

_{o}is the pitch of the output waveguides at the output-side of the output slab.

By comparing Eq. (5) to the phase term (the argument of the exponential) in Eq. (1), we can choose appropriate values for the design of the AWGR. Note that the negative sign in Eq. (1) can be neglected because the phase factors in Eq. (5) are delays, so should be negative. For example, for *d* = *d _{o}* = 25 μm, the focal length should be around 2.5 mm in Silica.

#### Samplers

Because each output of the AWGR is the weighted sum of four successive samples of the input, it implements Fourier transform operating on discrete samples in time; however, the inputs and outputs are continuous waveforms. Thus, it is a sliding (or running) Discrete Fourier Transform (Eq. (2) where the time increments between successive transforms are infinitesimally small. As discussed in Section 2, the output will only be valid if all samples are within the same OFDM symbol. This requires that the output of the AWGR be sampled [14], either in the optical or electrical domains, for example, at time E as shown in Fig. 5. In the following simulations, the sampling is assumed to be performed electrically. The sampling is assumed instantaneous, though the effect of any baseband bandwidth limitation (such as a sampler with a slow gate) will be investigated in Section 5.

## 4. System example

The four-subcarrier system shown in Fig. 1 was simulated to illustrate the performance of the method and the effect of various practical limitations using VPItransmissionMaker v. 7.6. Each subcarrier was generated by modulating a separate zero-linewidth laser with QPSK data, using a model of a complex optical modulator made from two Mach-Zehnder modulators. The lasers were at frequencies of 193.10, 193.11, 193.12 and 193.13 THz. The modulators were modeled as being linear (voltage to optical field), which can be approximated in reality using very low-level drive signals. Electrical filters (4th-order Bessel) were placed before the modulators to represent their limited bandwidths. The simulation bandwidth was 160 GHz, giving 16 simulation time samples per OFDM symbol. The data rate per subcarrier was 20 Gbps, and the baud rate was 10 Gbaud. Neither a cyclic prefix nor a guard band was used. This gives a total payload rate of 80 Gbps, or 160 Gbps if polarization-division multiplexing were to be used [39]. Higher rates can also be obtained by using amplitude and phase coding of the data, such as 16-QAM [40].

The demultiplexer as implemented as in Fig. 4, but with the optical samplers replaced by models of balanced-homodyne coherent receivers followed by electrical samplers. The time-delay increment between two adjacent arrayed waveguides (ΔT) was 25 ps, or one-quarter of the FFT window duration. Each output of the slab coupler was simulated by combining outputs of the arrayed waveguides with appropriate phase shifts (Eq. (5) and loss (where specified). The coherent receivers have local oscillator lasers at frequencies equal to the transmitter lasers’ frequencies, and thus produce baseband inphase (I) and quadrature (Q) signals at their outputs. These were band-limited, using 4th-order Bessel filters then sampled, with the sampling point optimized to give the highest quality outputs signal for each simulation. The I, Q sample plots can be plotted as and X,Y graph to give a constellation or their statistic analyzed to give a signal quality, *Q*, value from which the Bit Error Ratio (BER) can be estimated.

#### Optical spectra along the system

Figure 6 plots the transmitted optical spectrum of a single channel (red, point ‘a’ in Fig. 1), of all channels (green, point ‘b’) and of a demultiplexed channel (blue, point ‘c’). These were averaged over 16 simulations of 1024 OFDM symbols and a 1-GHz resolution bandwidth applied (so the power is per GHz). All subcarriers are normalized to 1 mW. The transmitted spectrum for a single channel is a typical QPSK spectrum, though the high bandwidth of the modulators means that the sidelobes are retained. The combined spectrum of the four transmitters has an almost rectangular profile, formed by the overlapping spectra of the four channels. The demultiplexed channel, Ch. 4, has a dominant main lobe, but also has strong leakage from the edges of Ch. 1 and Ch. 2. Fortunately, in OFDM, this leakage does not cause cross-talk unless the bandwidth for the receivers is limited. This is best illustrated in the time-domain, using eye diagrams; as will be shown in the next section.

#### Eye diagrams at the outputs of the coherent receivers

Figure 7 shows the Inphase (I, red) and Quadrature (Q, yellow) eye diagrams out of the coherent receiver for Channel 1. The left-most plots are for when only Ch. 1 is being transmitted. The maximum eye opening occurs over approximately 20% of the one OFDM symbol period. This opening corresponds to when all four samples of the FFT are within one OFDM symbol. The middle plot shows the effect of adding Ch. 2, 3 and 4. The eye is again open for approximately 20% of the OFDM symbol; however, at other times there is random interference from the other channels. The eye diagram on the right shows the effect of transmitting Ch. 2, 3 and 4, but not the desired channel. The flat-lines, where the eye was most open in the left and middle plots, confirms that Ch. 2, 3 and 4 produce no interference on Ch. 1 during approximately 10% of the symbol.

#### Sampling of the outputs of the coherent receivers

To convert the I and Q waveforms into data symbols, they must be sampled once per OFDM symbol. The sampling window must only be ‘open’ during the open part of the eye. The output of a sampler can be plotted on a constellation diagram, shown in Fig. 8
. This plots the imaginary (Q) sample against the real (I) sample on a Cartesian graph. The plotted samples lie in one of four quadrants for 4-QAM (Quadrature Amplitude Modulation). The degradation of the signal can be estimated by the spread of the sample points. A quantitative measure is *q*, where *q* = mean-value-squared/variance in either the x- or the y-coordinate. The samplers are then followed by thresholders, to produce digital bits representing the I and Q signals. These can then be decoded into 2-bits per channel per OFDM symbol, typically using a Gray code to minimize errors from a single threshold error. If the spread along one axis is Gaussian, the Bit Error Ratio (BER) after thresholding can be estimated from BER = (0.5)erfc(*q*/√2). More often a dB quality value is used, where *Q*(dB) = 20.log_{10}(*q*) using these definitions. For example, when *Q* = 9.8 dB the BER will be 10^{−3}.

Figure 8 shows typical constellations. The constellation on the left is for a system with 40-GHz 4th-order Bessel filters before the modulator and after the photodiodes. The location of the bandwidth limitation is unimportant in a linear system, such as this. In practice it will be the product of all of the transfer functions of all of the baseband devices along the system, including modulator drivers, modulators, photodiodes, preamplifiers and the input stages of the samplers. The *Q* value is >40 dB, indicating negligible cross-talk between subcarriers and an extremely-low BER. For comparison, the *Q* obtained when only one channel was transmitted was 60 dB. As a *Q* of greater than 40 dB would also require an optical signal to noise ratio of the same order, the crosstalk in these systems would have a negligible impact on systems performance in all but the shortest of systems. Such systems could support a very-high QAM constellation, allowing at least 8-bits per symbol to be transmitted. The constellation on the right is for a system with 40-GHz 4th-order Bessel filters before the modulator and after the photodiodes. The new optimum sample time is changed by the delay of the filters: the optimum was found by sweeping the sampling instant across the eye. The *Q*-value has dropped to 13 dB for the outer channels. This would only just support an 8-QAM system with 3-bits/symbol, but realistically, added amplifier noise would reduce the *Q* further.

As the constellations indicate, lowering the transmitter and/or the receiver bandwidths increases crosstalk between the channels. This is because the interference occurring before the open parts of the eye is spread into the open eye by the long transient responses of the electrical filters. This can be explained by taking the transmitter as an example, though because the system is linear, this same effect will occur due to any bandwidth restrictions at any point along the system before the sampling. Figure 9
shows eye for 10-GHz and 30 GHz electrical bandwidths (applied at the transmitters and receivers). There is a large closure in the eye for a 10-GHz electrical bandwidth, giving a *Q* of 13 dB. The eight-vertical mini-eyes between the main eyes indicate that the interference is deterministic; that is, it is caused by the cross-talk between the channels. Noise-driven eye closure would not have this distinct feature. The information in the mini-eyes also suggests that the main eye could be opened by electronic signal processing including sampling of the mini-eyes, which is well known in communications engineering. Increasing the electrical bandwidth to 30-GHz opens up the main eye, giving a *Q* of >35 dB if the sampling is instantaneous. Interestingly, the higher-bandwidth also allows sharper and higher peaks between the open eyes. It is these peaks that are smeared into the eye for the lower-bandwidth system. Electrical bandwidths of 20-GHz give a respectable *Q* of 20 dB. Chen *et al.* [17] have shown that a cyclic prefix will extend the duration of the open eye and Hillerkuss *et al.* [22] have used this to mitigate bandwidth-limited modulation.

#### Effect of AWGR non-uniformity

AWGRs for demultiplexing are generally designed so that the pass-band loss is low. This requires that the array of grating waveguides is wide enough to capture most of the input power traversing the first slab coupler. Thus the far-field of the input waveguide must be narrower than the array of waveguides. This implies that the outer guides of the array of waveguides will receive less power than the central guides, as it is impossible to achieve a beam with a sharp transition from its peak to its null [32]. For our FFTs to maintain orthogonality, the power transmission from AWGR input to AWGR output via any of the waveguides should be equal. For a 4-waveguide device, the only design specification for uniformity, assuming some symmetry, is the difference between the powers traversing the AWGR via the outer waveguides relative to the power in the central waveguides. The following simulations show how this non-uniformity affects the *Q* of the outer and inner channels, due to crosstalk (a reduction in orthogonality).

Figure 10
plots the *Q’*s for the four channels against AWGR power non-uniformity for 3 electrical bandwidths. Each bandwidth produces two sets of lines, corresponding to the inner (2, 3) and outer (1, 4) channels. The inner channels generally suffer the most crosstalk, whether caused by reduced electrical bandwidths or AWGR non-uniformity. From this plot, a non-uniformity of 1 dB would be acceptable for all but the highest constellation sizes. To confirm that crosstalk is the main cause of signal degradation when the grating is not uniform, the simulations were re-run with only one transmitter active. The *Q* was extremely high (>60 dB) for all values of grating non-uniformity.

#### Effect of limiting the total optical bandwidth

For wavelength management in a WDM system, it is likely that groups of subcarriers, each group carrying 120 Gbit/s for example, would be wavelength-multiplexed using standard optical filters, with a guard-band between each group of carriers to remove crosstalk between the groups. The following simulation investigates the effect of the bandwidth of the standard optical filter on a group. Figure 11
plots the received signal quality verses the FWHM bandwidth of a 5th-order Gaussian-shaped demultiplexing filter. Three subcarriers were transmitted, which would give a data rate of 60 Gbps per polarization using 4-QAM. Consider the top 3 traces, which are for wide-bandwidth (40 GHz) electrical filters. With a 50-GHz demultiplexer filter, the central channel ( + ) suffers the most degradation, because it collects crosstalk from both its neighbors: with a 30-GHz demultiplexer filter, the outer channels (Δ, ○) are most affected, because their spectra are truncated. The same cross-over occurs when the electrical bandwidths are reduced to 20-GHz ( × , ∇, ○); however, the *Q* is reduced overall. The results in Fig. 11 indicate that a system with high-bandwidth transmitters and receivers would give good performance with cascades of wavelength-selective switches, which typically have an optical bandwidth of around 35 GHz.

## 5. Discussion

These simulations have shown that a standard AWGR, albeit with fewer waveguides and a narrower free-spectral range, can be used to perfectly demultiplex subcarriers within an optical-OFDM system, provided that the electrical bandwidths within the transmitter and receiver are comparable with the overall bandwidth of the combined subcarriers. The AWGR acts as a serial to parallel converter, followed by a matrix of phase shifts, which is exactly what is used in a digital-electronics implementation of a DFT. The transmitter bandwidth could be artificially improved by overdriving the modulators, so they square-up signals, and receiver bandwidth could be lowered by using optical sampling. The AWGR also needs to be designed to have reasonably uniform transmission, whichever path the light takes through it.

#### Compensation of dispersion

One might suspect that fiber chromatic dispersion should be compensated before the AWGR demultiplexer, otherwise the four channels’ phase transitions would become misaligned due to the differential group delay of the fiber, so the OFDM condition would be broken. Because the system is linear, this is not the case, and the dispersion compensation can be moved to beyond the demultiplexer. Furthermore, because the coherent receivers perform a linear translation of optical field to electric field, the dispersion compensation can be performed in the RF domain, using, for example analog filters. If the signal is sampled many times per symbol, then the bandwidth of the optical signals is preserved, and the compensation can be performed in the digital domain [7]. Unfortunately, sampling only once per symbol causes aliasing, which means the outer edges of the signal spectrum are folded back into the main body of the spectrum. Because the outer edges suffer greater group delays than the body of the spectrum, then, after aliasing, no single value of group delay could be applied to a particular frequency that will compensate dispersion both in the outer edges and the body, as they are now indistinct. Thus dispersion compensation will be imperfect with 1-bit/sample for OFDM.

#### Application to NRZ formats

Further simulations showed that this design is also applicable to NRZ modulation formats as using in “Coherent WDM” systems [8], which use direct-detection receivers. Interestingly, there is a paradox between systems optical-OFDM (or Coherent WDM) that use QPSK (“all-optical OFDM” [10] or “Zero Guard Band OFDM” ^{7}) and NRZ systems (“Coherent WDM” [41]). This is that phase-control of each subcarrier is critical for the performance of NRZ systems, whereas the phase of each subcarrier is modulated in the QPSK systems, so cannot be optimized, except within a 90° range. Simulations using NRZ modulation showed that inter-subcarrier interference produces intensity peaks that can be moved from close to the region of the open eye, to between the open eyes, by adjusting the subcarriers’ phases. In systems with reduced electrical bandwidths, it is desirable to advance these interference peaks in time, so that even after the spreading of their energy due to the transient response of the electrical filters, their energy will not encroach on the open part of the eye. Thus there will be an optimum relative phase between the subcarriers as demonstrated by Ellis and Gunning [8].

## 6. Conclusion

This paper has shown that to maintain orthogonality between subcarriers in an optical OFDM system, the receiver demultiplexer needs to be designed with the theory of OFDM in mind. Typically, this requires that a phase-weighted sum of several samples of the optical signal should be formed with the demultiplexer, with the phase-values of the weights determining the subcarrier being decoded. Previous work has proposed networks of optical couplers to implement an appropriate demultiplexer. By comparing with a signal-processing representation of the optical path through the demultiplexer, this work has shown that the topology of an Arrayed Grating Waveguide Router (AWGR) can be used to perform the demultiplexing. Interestingly, the AWGR, when followed by a sampler, can be considered as a serial-to-parallel converter followed by a discrete Fourier transform. System simulations show that the electrical bandwidths of the transmitter and receiver are critical to orthogonal demultiplexing, and give insight into how crosstalk occurs in all-optical OFDM and coherent-WDM systems using waveforms and spectra along the system. Design specifications for the AWGR are developed, and show that non-uniformity will lead to crosstalk.

## Acknowledgements

I should like to thank VPIphotonics (www.vpiphotonics.com) for the use of their simulator, VPItransmissionMakerWDM V7.6. This work is supported under the Australian Research Council’s Discovery funding scheme (DP1096782).

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