## Abstract

Optical Orthogonal Frequency Division Multiplexing (O-OFDM) systems use electronic digital computation to provide dispersion compensation that can be rapidly adapted to changes in the optical link or optical network. Recent demonstrations have shown compensation of several thousand kilometers. Earlier simulations and analysis showed better sensitivities than non-return to zero systems; however, they assumed optical filters with very narrow bandwidths and narrow-linewidth lasers. This paper explores the effect of the optical filter bandwidths and laser linewidths for both coherent and direct-detection systems using analysis and simulations.

© 2008 Optical Society of America

## 1. Introduction

Electronic Dispersion Compensation (EDC) of long-haul optical fiber communications systems is gaining in popularity [1]–[4]. Recently, we have proposed [5] and demonstrated [6] EDC using Orthogonal Frequency Division Multiplexing (OFDM), combined with optical-single-sideband transmission and direct detection (DD-O-ODFM) with almost limitless dispersion compensation ability. Others have proposed and demonstrated optical OFDM with a coherent receiver (CO-OFDM) [7]–[11]. Initial simulations, using an ideal optical filter, showed that direct-detection optical OFDM has a better sensitivity than non-return to zero (NRZ) modulation [12]. However, the effect of filter bandwidth on receiver sensitivity has yet to be reported, and no simple analytical formula has been developed.

This paper provides a simple analysis for the noise limit, then simulation results for the receiver sensitivity of coherent and direct-detection optical OFDM systems. These show that, for ideal receivers, coherent OFDM is far more sensitive than direct-detection OFDM. However, the difference in performance is reduced when non-zero linewidth lasers are used. The key difference between this and earlier analysis of optically-amplified systems [13] is that the OFDM system is treated as having a rectangular band of OFDM subcarriers, whereas the signal in earlier analyses is usually treated as a single frequency. This is important as the design rules for filter bandwidth depend critically details of the optical signal spectrum.

## 2. Optical OFDM and ASE spectra

Optical OFDM uses tens to hundreds of closely-spaced subcarriers [12]. These are generated digitally in the electrical domain using an inverse Fast Fourier Transform, then mapped onto the optical domain using several different schemes of optical modulator. The result of modulation is a band of optical subcarriers with or without an optical carrier, corresponding to coherent or direct-detection optical OFDM. For coherent O-OFDM an optical carrier regenerated by a local oscillator is added at the receiver. This has to be the same polarization as the incoming signal from the fiber link; alternatively, a polarization-diverse receiver can be used in which the signal is split into two orthogonal polarizations which are independently mixed with two versions of the local oscillator [11]. The direct-detection system assumes that the optical carrier receives the same polarization rotation as the signal, so mixes with the subcarriers with good efficiency, though again a polarization diversity receiver can be used for high polarization mode dispersion fibers [14]. In direct-detection systems, the power of the optical carrier is optimally equal to the summed power of all OFDM subcarriers, unless the optical signal to noise ratio (OSNR) is high [15]. In coherent systems, the optical carrier is injected at the receiver and so can be beneficially far stronger then the subcarriers without affecting the power along the fiber path.

The ASE spectra of both systems are assumed to be white but band-limited by an optical filter directly before the receiver. Practically, this could be the WDM demultiplexer, designed to separate WDM channels at 50-GHz or 100-GHz spacing, although narrower filters will provide better receiver sensitivity. The ASE noise is assumed to be unpolarized; that is, there is equal power in the polarization aligned to the signal and its orthogonal component. This is a reasonable assumption for analysis, although polarization dependent gain in the amplifiers and polarization-dependent loss along the system may complicate matters. The coherent receiver is assumed to be a polarization-sensitive heterodyne receiver.

## 3. Electrical signal and noise spectra

The receiver’s photodiode (or photodiodes in a balanced coherent receiver) can be modeled as producing a photocurrent proportional to the square of the sum of all optical fields in an arbitrarily-chosen polarization, plus the square of the sum of all optical fields in the orthogonal polarization. The optical spectrum at the receiver input is shown in Fig. 1 (left). The ASE noise extends from *B _{NL}* below the carrier (

*f*) to

_{c}*B*above it. The subcarriers have a bandwidth

_{NH}*B*and there is a gap,

_{SC}*B*, between the carrier and the subcarriers. The middle spectra indicate the components of the optical spectrum that mix on detection to produce the electrical mixing products on the right.

_{gap}The electrical spectra (a)–(e) are:

*Carrier*×*subcarriers*. This is the desired electrical OFDM signal, bandwidth*B*._{SC}*Subcarriers*×*subcarriers*. This produces a band of unwanted tones close to DC, bandwidth*B*. In direct detection systems, these are arranged to fall away from the wanted tones by placing a spectral guard band with_{SC}*B*>_{gap}*B*between the optical carrier and the OFDM subcarrier band._{SC}*Carrier*×*ASE*(one polarization). The carrier and all ASE noise falling under the subcarrier band, in the same polarization, mixes into the bandwidth of the electrical OFDM signal. This is known as “real beat noise” [16], shown in green. In an ideal system, this would determine the noise limit of the system. Similarly, noise in the lower side band could also mix into the electrical subcarriers bandwidth to give “image beat noise”, shown in blue-green. If*B*<_{NL}*B*none of the image noise contribution will fall within the electrical subcarrier band. Thus, the gap is useful for rejecting image noise. A coherent image-rejection receiver can also remove image noise [16], as can a coherent homodyne receiver._{gap}*Subcarriers*×*ASE*noise (in same polarization). This is more complex than for the*carrier*×*ASE*noise, because the subcarriers occupy a finite bandwidth. Thus, all combinations of subcarrier frequency and ASE frequency that result in noise falling within the electrical OFDM signal’s bandwidth have to be considered. Using convolution in the frequency domain to calculate the effect of multiplication in the time domain, all subcarriers mixing with all noise below them produces the brown electrical spectrum: all mixing between tones and noise above them produces the red electrical spectrum. There is no situation where the ‘brown’ tones fall outside the signal bandwidth; however, if*B*<2._{NH}*B*, then none of the ‘red’ tones will fall in the subcarrier band. This requires an infinitely-sharp cut-off optical filter if_{gap}*B*_{gap}=*B*._{SC}*ASE*×*ASE*(either polarization). As in NRZ systems, this becomes significant at low OSNRs with wide optical filter bandwidths. This contribution is maximum close to DC and falls to zero at a point equal to the bandwidth of the noise.

In coherent systems with a strong local oscillator, noise mechanism (c) dominates, as this is proportional to the carrier power. Furthermore, mechanisms (d) and (e) can be eliminated using a balanced coherent receiver [17].

## 4. Analysis of electrical signal to noise ratio

The situation where the optical filter is sufficiently narrow and the OSNR sufficiently high that noise mechanism (e) is insignificant is analyzed for balanced coherent and direct detection systems. The power spectral density of the optical noise at the receiver, *PSD _{ASE}*, is:

where; *P _{subcarriers}* is the total signal power at the photodiode integrated over the subcarrier sideband,

*P*is the power at the photodiode of any transmitted carrier (zero for a coherent system) and

_{DDcarrier}*P*is the ASE power in both polarizations integrated over the OSNR measurement bandwidth,

_{ASE}*B*. In the RF domain the detected electrical signal power of a single subcarrier,

_{m}*P*, is related to the optical power in a single subcarrier,

_{SC,RF}*P*, by:

_{SC}where: *P _{carrier}* is the carrier power at the photodiode provided from the transmitter in a DD system, or from the local oscillator in a coherent system;

*R*is the photodiode responsivity, and

*R*is the load resistance. Similarly the noise power in the RF domain, over the bandwidth of an OFDM subcarrier, Δ

_{L}*f*, is:

Note that only half of the ASE power mixes with the carrier, because the ASE is unpolarised, so half its power is orthogonal to the carrier. The factor *k _{Car}* is 1 if image noise is out of band or is rejected; otherwise it increases with filter bandwidth up to 2 as shown in Fig. 1(c). The factor

*k*is zero for a balanced coherent receiver, as

_{SC}*ASE*×

*subcarriers*mixing is rejected, but is between 1 and 2 for the worst subcarriers of DD receivers, depending on the filter’s bandwidth, as shown in Fig. 1(d). The electrical signal to noise ratio is (2) divided by (3):

For *N* subcarriers, *N*.*P _{SC}*=

*P*and Δ

_{subcarriers}*f*=

*B*/

_{SC}*N*. Thus for a balanced coherent receiver with image noise rejection:

For a 9.8 dB electrical SNR (giving BER=10^{-3} for 4-QAM), a 12.5-GHz measurement bandwidth and a 5-GHz subcarrier band, an OSNR of 2.82 dB is required. For a direct-detection system with equal powers in the carrier and sideband, Eq. (4) gives a SNR up to 9-dB poorer than a coherent receiver, neglecting the additional degradation of *ASE*×*ASE*.

## 5. Simulations

To enable all noise mechanisms to be considered, the effect of optical filter bandwidth on coherent and direct-detection optical OFDM systems was simulated using Monte-Carlo simulations of ASE. The simulations were performed using VPItransmissionMaker™ version 7.1. The ASE was represented as Gaussian white noise, filtered with a brickwall optical filter. This was added to a 10-Gbit/s optical OFDM signal using 4-QAM modulation with 1024-bits per OFDM symbol (512 subcarriers) each carrying pseudo-random data. The OFDM waveform was modulated onto an optical carrier using an MZI biased at null so that its output optical field is proportional to the OFDM voltage. This gives an optical spectrum with no leakage outside the subcarrier band, which has a bandwidth of 5-GHz. An optical carrier was added after the modulator with the same optical power as summed over all subcarriers. A 5-GHz gap between the carrier and the subcarriers was used. The Optical Signal to Noise Ratio (*OSNR*) was defined in the usual way as the mean signal power (including carrier), divided by the ASE noise in both polarizations falling within a 12.5-GHz bandwidth, *B _{m}*. The filter was centered on the subcarrier band for the coherent systems, and on the lower-edge of this band for the direct detection systems. All lasers had zero linewidth at this stage of the simulations.

The electrical SNR was estimated from the mean distance from a constellation cluster to a Cartesian coordinate, *μ _{x}*, and the standard deviation of the points in the cluster in that coordinate direction,

*σ*, where SNR=(

_{x}*μ*/

_{x}*σ*)

_{x}^{2}. An SNR of 9.8 dB gives a BER of 10

^{-3}for 4-QAM. Averaging was performed over 100 OFDM symbols (1024K bits). Figure 2 shows the simulation results for: (a) polarization-diverse balanced homodyne coherent receivers with 7-dB OSNR; (b) direct-detection receivers with OSNRs of 7, 10, 13 dB. The coherent receivers will work with an ASE filter bandwidth down to 5 GHz, though this would be difficult to achieve in reality. The direct-detection receiver requires at least 10 GHz filter bandwidth to allow the carrier to pass, unless a double-peaked filter response is used.

As expected, the best performance for 7-dB OSNR is obtained with the coherent receiver. For a 5-GHz filter its performance is as predicted in Section 4. Filter bandwidths greater than 25 GHz will give an additional penalty for the lowest-frequency subcarriers, due to mixing of LSB ‘image’ beat noise into the frequency range of the electrical subcarriers. At 35-GHz, there is an additional 3-dB penalty for all subcarriers because of this mechanism.

The direct-detection receivers give an SNR 5-dB less than the coherent receivers for a 10-GHz filter bandwidth and an OSNR of 7 dB: Eq. (4) predicts 6-dB less but this is for the worst subcarriers as they are affected by mechanism (d) most. Increasing the filter bandwidth beyond 10-GHz (as is necessary practically), reduces the SNR due to further *ASE*×*subcarrier* mixing (d). Increasing the filter bandwidth beyond 20-GHz introduces image noise due to *ASE*×*carrier* mixing (c). Beyond 30-GHz, all electrical subcarriers suffer equally from mechanisms (c) and (d); the SNR then reduces gradually due to *ASE*×*ASE* mixing (e). For a 25-GHz filter, the SNR DD receiver is 9-dB worse than a homodyne/heterodyne coherent receiver, in agreement with Eq. (4). Because *ASE*×*ASE* mixing has less effect for higher OSNRs, a 13-dB OSNR DD receiver has similar performance to a heterodyne coherent receiver with an OSNR of 7 dB at wide filter bandwidths.

#### 5.1 Effect of laser linewidth

The above simulations used zero-linewidth lasers. However, coherent receivers require narrow-linewidth lasers to get good performance with OFDM, which is very sensitive to phase noise [18]. Figure 3 shows the simulated degradation in SNR due to laser linewidth, averaged over 50 OFDM symbols. Each OFDM symbol was corrected in phase by subtracting the mean-phase error of all the subcarriers from every subcarrier. A coherent system with a 3.5-dB OSNR and 512 subcarriers () gives only 9.2 dB SNR for 100-kHz linewidth lasers at the transmitter and receiver. Wider linewidth lasers can be compensated by increasing the OSNR (), but this is wasteful because the decrease in SNR is much more rapid. Alternatively, shorter symbols with 128 subcarriers can be used (, ) so that the phase deviation across a symbol is reduced, but this increases the overhead of the cyclic prefix. Another method is to insert a strong RF pilot tone [10] and use this to calculate and correct the phase error waveform within a symbol. In contrast, a direct-detection receiver () is insensitive to linewidth, and does not require frequent phase correction after initial training.

## 6. Discussion and conclusions

Reported sensitivities for early experimental optical OFDM systems are noticeably poorer than the theoretical limits given here. For example, the 8 Gbit/s 4-QAM coherent system [9], back to back, required a 9.2 dB ONSR for BER=10^{-3}, which is 7.4 dB worse than Eq. (5). Similarly, a 12.5 GBit/s 4-QAM system [19] required 8.4 dB OSNR for BER=10^{-3}, which is a penalty of 4.6 dB. Direct-detection receivers with a 200-GHz optical bandwidth required 17.2 dB for the same BER [6], which is 5.36-dB worse than Eq. (4) (*k _{Car}*=

*k*=2,

_{SC}*P*=

_{carrier}*P*=

_{DDcarrier}*P*) and 4.4-dB worse than predicted using an appropriate simulation. Although all of these results are poorer than theory, all systems were built using off-the-shelf components, were at an early stage of development, and were not optimized from an electrical signal path point of view. Thus there could be multiple reasons for these power penalties, including the limited bandwidth of the digital to analog converters, and the performance of the electrical amplifiers, modulators, receivers and analog to digital converters. The fact that the theory does not include noise mechanism (e) could also be a factor with low OSNRs.

_{subcarriers}In conclusion, the noise processes of optical OFDM receivers have been studied using a combination of decomposition of the mixing processes of signals and noise, analysis of the dominant noise mechanisms and numerical simulations. This paper has given the performance bounds of coherent and direct-detection receivers for a variety of optical filter bandwidths and laser linewidths. Coherent receivers give the ultimate performance for poor OSNRs, but only if narrow-linewidth lasers, short OFDM symbols, or a strong pilot tone [10] are used.

## Acknowledgments

I should like to thank VPIphotonics (www.vpiphotonics.com) for the use of their simulator, VPItransmissionMaker™WDM V7.1. This work is supported under the Australian Research Council’s Discovery funding scheme (DP 0772937).

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