Calculation of power limit due to fiber nonlinearity in optical OFDM systems

Transcription

1 Calculation of power limit due to fiber nonlinearity in optical OFD systems rthur James Lowery, Shunjie Wang and alin Premaratne Department of Electrical & Computer Systems Engineering, onash University, Clayton, 38, ustralia bstract: We develop a simple formula for estimating the effect of Four- Wave ixing (FW) on received signal quality in coherent optical systems using Orthogonal Frequency Division ultiplexing (OFD) for dispersion compensation. This shows the nonlinear limit is substantially independent of the number of OFD subcarriers. Our analysis agrees well with full split-step Fourier method simulations, so allows the nonlinear limit of multi-span systems to be estimated without lengthy simulations. 7 Optical Society of merica OCIS codes: (6.48) odulation; (6.451) Optical communications References and links 1.. R. S. Bahai, B. R. Saltzberg and. Ergen, ulti-carrier Digital Communications: Theory and pplications of OFD, nd Edition, (Springer, New York, 4)... J. Lowery and J. rmstrong, Orthogonal-frequency-division multiplexing for dispersion compensation of long-haul optical systems, Opt. Express 14, (6). 3. W. Shieh, X. Yi, and Y. Tang, " Experimental Demonstration of Transmission of Coherent Optical OFD Systems," in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OS Technical Digest Series (CD) (Optical Society of merica, 7), paper OP B. J. Schmidt,. J. Lowery, and J. rmstrong, " Experimental Demonstrations of Gbit/s Direct- Detection Optical OFD and 1 Gbit/s with a Colorless Transmitter," in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OS Technical Digest Series (CD) (Optical Society of merica, 7), paper PDP S. L. Jansen, I. orita, N. Takeda, and H. Tanaka, " -Gb/s OFD Transmission over 4,16-km SSF Enabled by RF-Pilot Tone Phase Noise Compensation," in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OS Technical Digest Series (CD) (Optical Society of merica, 7), paper PDP J. Lowery, L. B. Y. Du, and J. rmstrong, Performance of optical OFD in ultralong-haul WD lightwave systems, J. Lightwave Technol. 5, (7). 7. R. W. Tkach,. R. Chraplyvy, F. Forghieri,. H. Gnauck, and R.. Derosier, Four-photon mixing and high-speed WD systems, J. Lightwave Technol. 13, (1995). 8. G. P. grawal, Nonlinear Fiber Optics (Optics and Photonics), 3 rd Edition, (cademic Press, San Francisco, 1) 9. W. Shieh and C. thaudage, Coherent optical orthogonal frequency division multiplexing, Electron. Lett. 4, (6). 1. K-D. Chang, G-C Yang, and W. C. Kwong, Determination of FW products in unequal-spaced-channel WD lightwave systems, J. Lightwave Technol. 18, (). 1. Introduction The widely-used radio communications modulation scheme Orthogonal Frequency Division ultiplexing (OFD) [1] can also be used to compensate for fiber chromatic dispersion in ultra-long haul communications links [], and has recently been demonstrated experimentally [3] at data rates of Gbit/s [4] over distances up to 416 km [5]. However, for long-haul systems, Kerr nonlinearity in the optical fiber limits the practical power per WD channel for a given received signal quality or Bit Error Ratio (BER) [6]. This is expected, as OFD (C) 7 OS 1 October 7 / Vol. 15, No. / OPTICS EXPRESS 138

2 transmits on hundreds of narrowly-spaced subcarriers, and it is well known that strong Four- Wave ixing (FW) [7] occurs between closely-spaced optical channels because dispersion does not reduce phase matching between FW products generated along the fiber. In this paper, we analyze a coherent optical OFD system [3], and show that the signal degradation due to FW of electrical signal quality at the receiver can be easily predicted by considering the accumulation of FW products along the link, then applying a simple statistical analysis. We show that the degradation is nearly independent of the number of OFD 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 OFD, which is a key parameter in determining the spacing of the optical amplifiers. It also confirms that FW theory is sufficient for estimating the nonlinear degradation in coherent optical OFD systems. coherent OFD system has been chosen for analysis as a direct-detection system will have additional nonlinear terms due to the transmission of a carrier [6].. Optical OFD system Figure 1 shows the coherent OFD 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, uaternary mplitude odulation (). cyclic prefix ensures that dispersion does not destroy the orthogonality [1]. The subcarriers are modulated onto an optical carrier, using a complex (I) 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 OFD (CO-OFD) [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 OFD 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. n nalog to Digital Converter (DC) 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 timewindow 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 (E) in the frequency domain using one complex multiplication per subcarrier. decoders then translate each subcarrier into binary data. Zeros Inverse FFT Cyclic Prefix Parallel to Serial Re DC Im DC Bias Bias-T Complex ZI Laser Bias-T Bias n 8-km Fiber Link Input Spectrum subcarriers Local Oscillator Laser Photoreceiver Output Spectrum carrier subcarriers DC Serial to Parallel Strip Cyclic Prefix Forward FFT E E f opt f opt FW Products Fig. 1. Coherent optical OFD system block diagram showing typical optical spectra at the input and output of the fiber link. (C) 7 OS 1 October 7 / Vol. 15, No. / OPTICS EXPRESS 1383

3 3. pproximate 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 () and degenerate (DG) four-wave mixing [7]. 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]; D ijk exp( α L) PPP i j k Pijk = ( γ Leff ) (1) 9 π c 1+ ( λi λk )( λj λk) D λα where D ijk is the degeneracy factor which equals 6 for products and 3 for DG products, and the nonlinearity coefficient of the fiber is: π n γ = () λ where: n is the nonlinear coefficient of the fiber material, eff is the effective core area of the fiber and the effective length of the fiber is: L = 1 e α L / α (3) eff eff 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 OFD systems is in the order of tens of Hz, the second term in the denominator of the final term of Equation 1 is negligible. lso if the fiber loss is compensated by an amplifier, the exponential term becomes unity. Thus the power of each FW product is approximately related to the power of a single OFD subcarrier, P SC, by: Dijk 3 Pijk ( γ Leff ) PSC (4) 9 The number of FW products,, depends on the number of subcarriers, N, at the fiber s input. The total number of FW products falling at all frequencies is exactly [7]: 3 = ( N N )/ (5) For an optical OFD system with 51 OFD subcarriers, evaluates to 66,977,79 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 FW products fall on the N OFD subcarriers equally, the average FW power falling on each subcarrier, P FW/SC, will be P = N P (6) FW / SC / ijk Random-walk theory can be applied to the optical field to find the statistics of the sum of the FW contributions, because each subcarrier is phase modulated with 4 different phases due to the uadrature mplitude odulation (). s the FW contributions have random relative phases, their powers add, rather than their fields. t 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 FW 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 (C) 7 OS 1 October 7 / Vol. 15, No. / OPTICS EXPRESS 1384

4 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 / ). Using the statistical properties of a -D random walk, the electrical signal quality is found to be: P P SC SC ( qelec ) = = (7) P N P / FW / SC ijk In OFD systems, N 3 >>N so N / from (5) and the number of degenerate products is insignificant. Using P total = N.P SC, the electrical signal quality is then approximately: 1 q (8) elec ( γ LP ) e total 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 OFD s digital signal processing algorithms (proportional to Nlog N) and the overhead of the cyclic prefix (proportional to 1/N) [6]. 4. ccurate formula for the numbers FW products falling on subcarriers Equation 8 is an approximate expression because: (i) some FW mixing products will fall outside the range of the subcarriers; (ii) it is likely that more FW 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 FW products that fall on the each of the OFD subcarriers. TLB was used to conduct an exhaustive search of all combinations of subcarrier frequencies (f i, f j, f k ) that generate FW products, and identified the frequency of each FW product. Figure plots the numbers of degenerate and non-degenerate FW products falling on and around 51 OFD subcarriers. Note the different scales for the nondegenerate,, and degenerate, 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 66,716,16 and the total number of degenerate products is 61,63. These numbers add to give the result of Eqn , Number of Degenerate products, DG DG OFD Subcarrier Band subcarrier Index, i 5, Fig.. number of degenerate and non-degenerate FW products falling on and outside the OFD subcarrier band for 51 subcarriers in the OFD subcarrier band. Number of Nondegenerate products, (C) 7 OS 1 October 7 / Vol. 15, No. / OPTICS EXPRESS 1385

5 We found that the following formulae fit the results of Fig. exactly for in-band products, and we also found they fit the exhaustive search when N = p (p integer): () i = N/ 1; = 3N 1N 4i i /8 (9) DG where i is the subcarrier index, from (N/ 1) to N/. For N= 64, neglecting all but the N term gives a count inaccuracy of 3% at the band edges, reducing to.8 % for N= 56. Thus, substituting (9) into (7) confirms that q elec is only weakly dependent on N. t the center of the band i=, giving: = ( 3 N + 1 N + 8 )/ N (1) i= t the band edges, i = (N/ 1) or N/, giving: = i N/ ( N 8N + 8 )/8.5N (11) = 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 : rms = ( 7. N )/ 8.335N (1) This agrees with Equation 9 of Reference 1. Equations 1 and 11 show that the number of non-degenerate products at the center of the subcarrier band is 1.5 the number at each edge of the band. Thus, (db) =.log 1 (q elec ) is reduced by 1.7 db at the center of the band compared with the edges. The worst-case subcarrier s will be.5 db below the average. 5. Comparison with simulated signal quality 5.1 versus subcarrier index Figure 3 plots the (db) versus the subcarrier index for a system using coherent detection with N = 64 and 51. The systems have five 8-km spans of -ps/nm/km.-db/km fiber with n = m /W, eff = 8 (µm), and a power of dbm into each span. The results were obtained by running the simulation for the duration of 51 separate OFD symbols then calculating q elec for each subcarrier averaged over the 51 symbols. The simulated results agree with the estimates for 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 is substantially independent of N (db) 16 (db) Subcarrier Index Subcarrier Index Fig. 3. Simulated (dots) and theoretical (lines) estimates of for each OFD subcarrier. Left: 64 subcarriers; Right: 51 subcarriers. 5. 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 8-km and the last span of 6 km. To achieve this, by summing the variances introduced by each span, we define an effective length of the whole system, L ES, as α L 1 e last LES = + ( s 1) Leff (13) α (C) 7 OS 1 October 7 / Vol. 15, No. / OPTICS EXPRESS 1386

6 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 of any multi-span system. Figure 4 compares the simulated, calculated from the average variance over all subcarriers for 3 OFD symbols, to the theoretical result from Equations 4, 7, 1 and 13 for a system with 16 ps/nm/km fiber and -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 s of around 1 db, showing that the formulae are useful predictors for systems design, and that FW is sufficient to explain the degradation due to fiber nonlinearity in coherent optical OFD systems. It is obvious from Equation 1 that will scale with nonlinearity (db) 3 1 dbm -1dBm -7.5dBm -5.dBm -.5dBm -1.dBm System Length (km) Fig. 4. ulti-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. 6. Conclusions We have developed an accurate model for the quality of coherent optical OFD signals when limited by fiber nonlinearity in a multi-span optical link. We have developed formulae for the number of FW 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 OFD 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 multispan coherent optical OFD systems. This model will speed the design of OFD systems as it provides an accurate upper bound for the transmission power: the lower bound is governed by amplifier noise [6]. Noninteger 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 -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. cknowledgements This research is supported under the ustralian Research Council s Discovery funding scheme (Grant DP77937). We should like to thank VPIphotonics, ( a division of VPIsystems, for the use of their simulator VPItransmissionaker WD V7.1. (C) 7 OS 1 October 7 / Vol. 15, No. / OPTICS EXPRESS 1387