Calculation of power limit due to fiber nonlinearity in optical OFDM systems Arthur James Lowery, Shunjie Wang and Malin Premaratne Department of Electrical & Computer Systems Engineering, Monash University, Clayton, 3800, Australia [email protected] http://www.ecse.monash.edu.au Abstract: We develop a simple formula for estimating the effect of Four- Wave Mixing (FWM) on received signal quality in coherent optical systems using Orthogonal Frequency Division Multiplexing (OFDM) for dispersion compensation. This shows the nonlinear limit is substantially independent of the number of OFDM 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. ©2007 Optical Society of America OCIS codes: (060.4080) Modulation; (060.4510) Optical communications References and links 1. A. R. S. Bahai, B. R. Saltzberg and M. Ergen, Multi-carrier Digital Communications: Theory and Applications of OFDM, 2nd Edition, (Springer, New York, 2004). 2. A. J. Lowery and J. Armstrong, “Orthogonal-frequency-division multiplexing for dispersion compensation of long-haul optical systems,” Opt. Express 14, 2079-2084 (2006). 3. W. Shieh, X. Yi, and Y. Tang, " Experimental Demonstration of Transmission of Coherent Optical OFDM Systems," in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper OMP2. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-OMP2 4. B. J. Schmidt, A. J. Lowery, and J. Armstrong, " Experimental Demonstrations of 20 Gbit/s Direct- Detection Optical OFDM and 12 Gbit/s with a Colorless Transmitter," in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper PDP18. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-PDP18 5. S. L. Jansen, I. Morita, N. Takeda, and H. Tanaka, " 20-Gb/s OFDM Transmission over 4,160-km SSMF Enabled by RF-Pilot Tone Phase Noise Compensation," in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper PDP15. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-PDP15 6. A. J. Lowery, L. B. Y. Du, and J. Armstrong, “Performance of optical OFDM in ultralong-haul WDM lightwave systems,” J. Lightwave Technol. 25, 131-138 (2007). 7. R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841-849 (1995). 8. G. P. Agrawal, Nonlinear Fiber Optics (Optics and Photonics), 3rd Edition, (Academic Press, San Francisco, 2001) 9. W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42, 587-588 (2006). 10. K-D. Chang, G-C Yang, and W. C. Kwong, “Determination of FWM products in unequal-spaced-channel WDM lightwave systems,” J. Lightwave Technol. 18, 2113-2122 (2000). 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 #84465 - $15.00 USD Received 27 Jun 2007; revised 10 Sep 2007; accepted 23 Sep 2007; published 28 Sep 2007 (C) 2007 OSA 1 October 2007 / Vol. 15, No. 20 / OPTICS EXPRESS 13282 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. Q Q Local E A Re Bias n × 80-km Oscillator A Data Data Q M Fiber Link Laser M Bias-T Photoreceiver DAC ADC Zeros Complex Im MZI Inverse FFT Inverse Forward FFT Q Cyclic Prefix Laser Q E Parallel to to Serial Parallel Bias-T Serial Parallel to A PrefixStrip Cyclic A Input Spectrum Output Spectrum Data DAC Data Q M carrier M Bias subcarriers subcarriers fopt fopt FWM Products Fig. 1. Coherent optical OFDM system block diagram showing typical optical spectra at the input and output of the fiber link. #84465 - $15.00 USD Received 27 Jun 2007; revised 10 Sep 2007; accepted 23 Sep 2007; published 28 Sep 2007 (C) 2007 OSA 1 October 2007 / Vol. 15, No. 20 / OPTICS EXPRESS 13283 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, Pijk due to three polarization aligned subcarriers with wavelengths λi, λj, λk, and optical powers Pi, Pj, Pk is given by [7]; 2 DLPPP2 exp()−α = ijkγ i j k PLijk() eff (1) 9 ⎛⎞2π c 2 1+−−⎜⎟()λλ() λ λD ⎝⎠λα2 ik jk where Dijk is the degeneracy factor which equals 6 for NDG products and 3 for DG products, and the nonlinearity coefficient of the fiber is: 2π n γ = 2 (2) λ Aeff where: n2 is the nonlinear coefficient of the fiber material, Aeff is the effective core area of the fiber and the effective length of the fiber is: Le=−()1 −α L /α (3) 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 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, PSC, by: 2 D 2 P ≈ ijk ()γ LP3 (4) ijk9 eff SC 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]: MNN=−( 32)/2 (5) For an optical OFDM system with 512 OFDM subcarriers, M evaluates to 66,977,792 products. Fortunately, the power per product, Pijk, 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, PFWM/SC, will be = () PFWM/ SCMNP/ ijk (6) 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).