1. Introduction

The widely-used radio communications modulation scheme Orthogonal Frequency Division Multiplexing (OFDM) [1] can also be used to compensate for fiber chromatic dispersion in ultra-long haul communications links [2], and has recently been demonstrated experimentally [3] at data rates of 20 Gbit/s [4] over distances up to 4160 km [5]. However, for long-haul systems, Kerr nonlinearity in the optical fiber limits the practical power per WDM channel for a given received signal quality or Bit Error Ratio (BER) [6]. This is expected, as OFDM transmits on hundreds of narrowly-spaced subcarriers, and it is well known that strong Four-Wave Mixing (FWM) [7] occurs between closely-spaced optical channels because dispersion does not reduce phase matching between FWM products generated along the fiber.

In this paper, we analyze a coherent optical OFDM system [3], and show that the signal degradation due to FWM of electrical signal quality at the receiver can be easily predicted by considering the accumulation of FWM products along the link, then applying a simple statistical analysis. We show that the degradation is nearly independent of the number of OFDM subcarriers used in the system, but is strongly dependent on optical power and the nonlinear coefficient of the fiber. Our analysis agrees extremely well with numerical simulations using the split-step Fourier method [8]. It is useful for estimating the maximum power in a long-haul communication system using coherent optical OFDM, which is a key parameter in determining the spacing of the optical amplifiers. It also confirms that FWM theory is sufficient for estimating the nonlinear degradation in coherent optical OFDM systems. A coherent OFDM system has been chosen for analysis as a direct-detection system will have additional nonlinear terms due to the transmission of a carrier [6].

2. Optical OFDM system

Figure 1 shows the coherent OFDM system. The transmitter uses an inverse fast Fourier Transform (FFT) to generate several hundred orthogonal subcarriers. Each subcarrier is encoded with digital data modulated using, for example, Quaternary Amplitude Modulation (QAM). A cyclic prefix ensures that dispersion does not destroy the orthogonality [1]. The subcarriers are modulated onto an optical carrier, using a complex (IQ) optical modulator [4], [5], which creates an optical single-sideband (OSSB) spectrum with a suppressed optical carrier (left inset of Fig. 1). If the carrier is suppressed fully, this is known as coherent optical OFDM (CO-OFDM) [9]. The modulator’s output is boosted in power by an optical amplifier, then propagated through multiple spans of fiber, each followed by an optical amplifier.

In coherent optical OFDM systems [9], the output of a local oscillator laser must be added to the received signal with an identical polarization before photodetection; alternatively, a polarization diversity receiver can be used. An Analog to Digital Converter (ADC) samples the detected waveform and converts it to digital data. The cyclic prefix is stripped, then a forward FFT determines the phases of each electrical subcarrier. Provided the FFT’s time-window is aligned with the transmitted time-window, the received subcarriers will remain orthogonal. Owing to fiber dispersion, the subcarrier spectrum has a quadratically-increasing phase shift across it, which is easily equalized (EQ) in the frequency domain using one complex multiplication per subcarrier. QAM decoders then translate each subcarrier into binary data.

 

Fig. 1. Coherent optical OFDM system block diagram showing typical optical spectra at the input and output of the fiber link.

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3. Approximate estimate of the effect of fiber nonlinearity

Fiber Kerr nonlinearity along the transmission path causes intermixing of the optical subcarriers [8], shown in the right inset of Fig. 1. Two classes of intermixing are observed: non-degenerate (NDG) and degenerate (DG) four-wave mixing [7]. NDG involves three original optical frequencies, generating a fourth that may lie on top of an original frequency. DG involves two original frequencies, generating a third frequency that lies away from the original frequencies. For low-dispersion fiber with dense subcarrier spacing, the strength of a single mixing product, P_ijk due to three polarization aligned subcarriers with wavelengths λ_i, λ_j, λ_k, and optical powers P_i, P_j, P_k is given by [7];

where D_ijk is the degeneracy factor which equals 6 for NDG products and 3 for DG products, and the nonlinearity coefficient of the fiber is:

where: n ₂ is the nonlinear coefficient of the fiber material, A_eff is the effective core area of the fiber and the effective length of the fiber is:

where: L is the physical length of the fiber, α is its loss coefficient in Nepers/m of the fiber, and D is its dispersion coefficient. Because the subcarrier spacing in OFDM systems is in the order of tens of MHz, the second term in the denominator of the final term of Equation 1 is negligible. Also if the fiber loss is compensated by an amplifier, the exponential term becomes unity. Thus the power of each FWM product is approximately related to the power of a single OFDM subcarrier, P_SC, by:

The number of FWM products, M, depends on the number of subcarriers, N, at the fiber’s input. The total number of FWM products falling at all frequencies is exactly [7]:

For an optical OFDM system with 512 OFDM subcarriers, M evaluates to 66,977,792 products. Fortunately, the power per product, P_ijk, is low because the transmitted optical power is divided amongst the N subcarriers. For example, if we assume that the M FWM products fall on the N OFDM subcarriers equally, the average FWM power falling on each subcarrier, P_FWM/SC, will be

Random-walk theory can be applied to the optical field to find the statistics of the sum of the FWM contributions, because each subcarrier is phase modulated with 4 different phases due to the Quadrature Amplitude Modulation (QAM). As the FWM contributions have random relative phases, their powers add, rather than their fields.

At the receiver, the field of the local oscillator laser mixes with the field of each subcarrier to produce a baseband electrical signal with a phase that corresponds to the transmitted data symbol. The local oscillator field also mixes with the sum of the FWM contributions falling on each subcarrier, to produce a baseband error vector. Each electrical subcarrier thus has a signal vector and error vector which add to give a point on the complex plane. When the points of all subcarriers are plotted a constellation diagram is produced [6], with groups of points in each quadrant representing a pair of data bits. The electrical signal quality, q_elec, is defined by the voltage of the expected value of a symbol along one Cartesian coordinate, divided by the standard deviation of the symbols along that coordinate [6]. The Bit Error Ratio (BER) can be estimated from BER = ½erfc(q_elec/√2). Using the statistical properties of a 2-D random walk, the electrical signal quality is found to be:

In OFDM systems, N ³>>N ² so M ≈ N ²/ 2 from (5) and the number of degenerate products is insignificant. Using P_total = N.P_SC, the electrical signal quality is then approximately:

For an optical-power limited system Equation 8 suggests that, q_elec is independent of N. This result is desirable, as the choice of N becomes a simple trade-off between computational complexity for the OFDM’s digital signal processing algorithms (proportional to Nlog₂ N) and the overhead of the cyclic prefix (proportional to 1/N) [6].

4. Accurate formula for the numbers FWM products falling on subcarriers

Equation 8 is an approximate expression because: (i) some FWM mixing products will fall outside the range of the subcarriers; (ii) it is likely that more FWM products will fall on the center of this subcarrier band than at its edges; (iii) DG products are assumed to be insignificant. It is therefore desirable to find accurate numbers for the degenerate and nondegenerate FWM products that fall on the each of the OFDM subcarriers.

MATLAB^TM was used to conduct an exhaustive search of all combinations of subcarrier frequencies (f_i, f_j, f_k) that generate FWM products, and identified the frequency of each FWM product. Figure 2 plots the numbers of degenerate and non-degenerate FWM products falling on and around 512 OFDM subcarriers. Note the different scales for the nondegenerate, M_NDG, and degenerate, M_DG, products. The number of degenerate products is constant within the subcarrier band (and this holds for any number of subcarriers), so that all subcarriers are affected equally: the number of nondegenerate products is far higher, confirming the approximation used in Eqn. 8, and peaks at the center of the band. The total number of nondegenerate products in and out of band is 66,716,160 and the total number of degenerate products is 261,632. These numbers add to give the result of Eqn. 5.

 

Fig. 2. number of degenerate and non-degenerate FWM products falling on and outside the OFDM subcarrier band for 512 subcarriers in the OFDM subcarrier band.

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We found that the following formulae fit the results of Fig. 2 exactly for in-band products, and we also found they fit the exhaustive search when N = 2^P (p integer):

where i is the subcarrier index, from -(N/2 - 1) to N/2. For N= 64, neglecting all but the N ² term gives a count inaccuracy of 3% at the band edges, reducing to 0.8 % for N= 256. Thus, substituting (9) into (7) confirms that q_elec is only weakly dependent on N.

At the center of the band i=0, giving:

At the band edges, i = -(N/2-1) or N/2, giving:

The average of the variance across the band, which gives the average electrical signal quality, can be found from the r.m.s. value of M_NDG:

This agrees with Equation 9 of Reference 10. Equations 10 and 11 show that the number of non-degenerate products at the center of the subcarrier band is 1.5x the number at each edge of the band. Thus, Q(dB) = 20.log₁₀(q_elec) is reduced by 1.7 dB at the center of the band compared with the edges. The worst-case subcarrier’s Q will be 0.5 dB below the average Q.

5. Comparison with simulated signal quality

5.1 Q versus subcarrier index

Figure 3 plots the Q(dB) versus the subcarrier index for a system using coherent detection with N = 64 and 512. The systems have five 80-km spans of 2-ps/nm/km 0.2-dB/km fiber with n ₂ = 2.6×10^-20 m²/W, A_eff = 80 (μm)², and a power of 0 dBm into each span. The results were obtained by running the simulation for the duration of 512 separate OFDM symbols then calculating q_elec for each subcarrier averaged over the 512 symbols. The simulated results agree with the estimates for Q from Equations (7) and (9) with the effective length equal to five-times the effective length of a single span. The simulation result for N=64 is slightly poorer than theory because there are fewer symbols to estimate and correct the mean phase shift due to nonlinearity; however, these results show that Q is substantially independent of N.

 

Fig. 3. Simulated (dots) and theoretical (lines) estimates of Q for each OFDM subcarrier. Left: 64 subcarriers; Right: 512 subcarriers.

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5.2 Q versus system length for multispan systems

The theory can also be applied with systems with fractional spans, for example a system of six-spans: five of 80-km and the last span of 60 km. To achieve this, by summing the variances introduced by each span, we define an effective length of the whole system, L_ES, as

where L_last is the length of the last span of s spans. This can be substituted for L_eff of Equation 4 to find the Q of any multi-span system.

Figure 4 compares the simulated Q, calculated from the average variance over all subcarriers for 30 OFDM symbols, to the theoretical result from Equations 4, 7, 12 and 13 for a system with 16 ps/nm/km fiber and 0-dBm input power: additional curves show the fit is also valid over a range of powers. The fits are extremely good, especially for longer systems with realistic operating Q’s of around 10 dB, showing that the formulae are useful predictors for systems design, and that FWM is sufficient to explain the degradation due to fiber nonlinearity in coherent optical OFDM systems. It is obvious from Equation 1 that Q will scale with nonlinearity.

 

Fig. 4. Multi-span system simulation results (circles) compared with theory (lines) for a number of span input powers. Short spans at lower input powers have been omitted for clarity.

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6. Conclusions

We have developed an accurate model for the quality of coherent optical OFDM signals when limited by fiber nonlinearity in a multi-span optical link. We have developed formulae for the number of FWM products that fall on subcarriers that are valid for any number of subcarriers. We have derived the electrical signal quality for each subcarrier in terms of number of subcarriers, input power, system length and fiber nonlinearity. The model can predict the average quality over a band of OFDM subcarriers and the quality of each subcarrier. For power-limited systems, the electrical signal quality is approximately independent of the number of subcarriers. This analysis fits well extremely with numerical simulations of multi-span coherent optical OFDM systems.

This model will speed the design of OFDM systems as it provides an accurate upper bound for the transmission power: the lower bound is governed by amplifier noise [6]. Non-integer numbers of spans are also considered by modifying the equation for effective length. In dB terms, a 1-dB decrease in transmission power leads to a 2-dB increase in quality. Conversely, a doubling of the effective length of the whole system decreases the signal quality by 6 dB. Thus, doubling the system length requires a 3-dB decrease in fiber power.