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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 7 — Apr. 7, 2014
  • pp: 8598–8616
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Analytical characterization of four wave mixing effect in direct-detection double-sideband OFDM optical transmission systems

Tiago M. F. Alves and Adolfo V. T. Cartaxo  »View Author Affiliations


Optics Express, Vol. 22, Issue 7, pp. 8598-8616 (2014)
http://dx.doi.org/10.1364/OE.22.008598


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Abstract

A closed-form expression for the variance of the four-wave mixing (FWM) induced in each subcarrier of a double sideband orthogonal frequency division multiplexing (OFDM) system employing direct detection is proposed and validated. Particularly, using a small signal analysis, equivalent transfer functions that characterize the frequency response of the FWM effect are derived taking into account the walkoff effect between the modulated pump waves and the FWM wave. The accuracy of the variance estimates provided by the closed-form expression is assessed for different sets of system parameters. The closed-form expression provides good variance estimates of the FWM-induced degradation caused by degenerate and nonsymmetric nondegenerate FWM components. For symmetric non-degenerate FWM components, the proposed expression provides reliable but pessimistic variance estimates, not exceeding the actual FWM variance in 1.5 dB for modulation indexes of interest.

© 2014 Optical Society of America

1. Introduction

The analysis developed in this work is focused on the distortion induced by the four-wave mixing (FWM) effect, which consists in the generation of a new optical field (FWM component) by energy transferring provided by the interaction between three optical channels, in DD DSB-OFDM systems. FWM effect in optical fiber communication systems was widely studied along the years [13

13. K. Hill, D. Johnson, B. Kawasaki, and R. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978). [CrossRef]

17

17. S. Song, C. Allen, K. Demarest, and R. Hui, “Intensity-dependent phase-matching effects on four-wave mixing in optical fibers,” J. Lightwave Technol. 17, 2285–2290 (1999). [CrossRef]

]. Historically, the physical layer impairments, as the FWM effect, are firstly modeled theoretically and, based on this modeling, new techniques are proposed to mitigate such nonlinear impairment [18

18. H. Song and M. Pearce, “Range of influence and impact of physical impairments in long-haul DWDM systems,” J. Lightwave Technol. 31, 846–854 (2013). [CrossRef]

]. This occurred, for instance, a few years ago for on-off keying (OOK) systems [19

19. C. Mahon, L. Olofsson, E. Bødtker, and G. Jacobsen, “Polarization allocation schemes for minimizing fiber four-wave mixing crosstalk in wavelength division multiplexed optical communication systems,” IEEE Photon. Technol. Lett. 8, 575–577 (1996). [CrossRef]

21

21. H. Batshon, I. Djordjevic, and B. Vasic, “An improved technique for suppression of intrachannel four-wave mixing in 40-Gb/s optical transmission systems,” IEEE Photon. Technol. Lett. 19, 67–69 (2007). [CrossRef]

] and, recently, for OFDM-based coherent long-haul systems [22

22. A. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express 15, 13282–13287 (2007). [CrossRef] [PubMed]

25

25. L. Du, M. Morshed, and A. Lowery, “Fiber nonlinearity compensation for OFDM super-channels using optical phase conjugation,” Opt. Express 20, 19921–19927 (2012). [CrossRef] [PubMed]

].

In this work, the theoretical modeling of the FWM in DSB-OFDM DD systems indicated for access networks is performed. A closed-form expression for the variance of the FWM generated by a set of modulated DSB OFDM signals is derived considering the inter-channel walkoff effect. The accuracy of the proposed formulation is validated by comparison with the performance evaluated numerically for an application scenario based on the distribution of DSB OFDM ultra wideband (UWB) radio signals along long-reach passive optical networks (LR-PONs) [12

12. T. Alves and A. Cartaxo, “Transmission of multiband OFDM-UWB signals along LR-PONs employing a Mach-Zehnder modulator biased at the quasi-minimum power transmission point,” J. Lightwave Technol. 30, 1587–1594 (2012). [CrossRef]

].

2. Theory

In this section, a closed-form expression for the variance of each OFDM subcarrier induced by the FWM effect is derived. The analytical characterization presented along this section is valid for DD systems employing DSB OFDM format and chirpless electro-optic converters. The general case, in which the FWM is generated by a set of optical channels with each channel comprising a continuous-wave (CW) component and a modulated signal, is considered. To allow for a simple scalar formulation, the theoretical analysis considers also that the signals are copolarized and assumes that the signal polarization is kept the same along fiber propagation.

2.1. Analytical model of FWM effect

The FWM consists in the generation of a new wave resulting from the mixing of three pump waves. The frequency of the new FWM component, ν1, is obtained from the frequencies of the three pump waves by ν1 = ν2 + ν3ν4 [16

16. R. Tkach, A. Chraplyvy, F. Forghieri, A. Gnauck, and R. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995). [CrossRef]

].

Figure 1 illustrates the spectrum (optical carrier and modulated DSB-OFDM signal) of three input pump channels of a generic wavelength division multiplexing (WDM) signal of the DSB-OFDM system considered in this work, and the generated FWM components at the fiber output. Each optical channel at the fiber input is obtained by intensity modulation of an independent laser source using a Mach-Zehnder modulator. Only the carrier-carrier and carrier-signal FWM components are presented at the fiber output, as it is considered that they are dominant compared with the FWM components generated through the carrier-signal-signal and signal-signal-signal beats. The coupled equations, describing the propagation of the three pump waves and the generated FWM wave involved in the FWM process, at a distance z from the fiber input, can be written as [20

20. A. Akhtar, L. Pavel, and S. Kumar, “Modeling and analysis of the contribution of channel walk-off to nondegenerate and degenerate four-wave-mixing noise in RZ-OOK optical transmission systems,” J. Lightwave Technol. 24, 4269–4285 (2006). [CrossRef]

]:
A1(z,t)z=12αA1(z,t)β1,1A1(z,t)tj13DγA2(z,t)A3(z,t)A4*(z,t)exp(jΔβz)
(1)
A2(z,t)z=12αA2(z,t)β1,2A2(z,t)t
(2)
A3(z,t)z=12αA3(z,t)β1,3A3(z,t)t
(3)
A4(z,t)z=12αA4(z,t)β1,4A4(z,t)t
(4)
where Ai(z, t) is the field envelope of the wave at frequency νi, α and γ are the fiber loss coefficient and nonlinearity parameter, β1,i is the first-order dispersion parameter calculated at the frequency νi, and D is the degeneracy factor. For nondegenerate FWM components (ν2ν3), D = 6, whereas in degenerate FWM components (ν2 = ν3), D = 3. As the degenerate FWM component is generated from only two optical fields, Eq. (1) can still be further simplified using A2(z, t)=A3(z, t). Equation (1) neglects the effect of the fiber nonlinearity on the pump waves. The propagation constant mismatch is given by Δβ = β1) + β4) − β2) − β3) where βi) is the propagation constant of the wave transmitted at the angular frequency Ωi = 2πνi. Using a Taylor series around a reference frequency Ωr, βi) can be approximated by β(Ωi)=β0(Ωr)+β1(Ωr)(ΩiΩr)+β2(Ωr)2(ΩiΩr)2+..., where β0r) and β2r) are the zero-order and second-order dispersion parameter calculated at the frequency Ωr [20

20. A. Akhtar, L. Pavel, and S. Kumar, “Modeling and analysis of the contribution of channel walk-off to nondegenerate and degenerate four-wave-mixing noise in RZ-OOK optical transmission systems,” J. Lightwave Technol. 24, 4269–4285 (2006). [CrossRef]

]. As our focus is the FWM, Eq. (1) neglects the self-phase modulation and cross-phase modulation. In addition, Eq. (1) takes into account the impact of the dispersion on the walkoff between the pump and the FWM waves (β1 term) and neglects the impact of the dispersion on the distortion of the FWM wave and pump waves.

Fig. 1 Illustration of the spectrum (optical carrier and modulated DSB-OFDM signal) of 3 optical channels of a WDM signal at the fiber input, and corresponding carrier-carrier and carrier-signal FWM components at the fiber output.

The field envelope of the pump waves at a distance z from the fiber input can be expressed as a function of the pump waves launched into the fiber by Ai(z,t)=Ai(0,tβ1,iz)exp(12αz), 2 ≤ i ≤ 4. The field envelope of the pump waves can still be written as:
Ai(z,t)=[Ai¯+ai(0,tβ1,iz)]exp(12αz)exp(jϕi)
(5)
where ϕi is the initial phase uniformly distributed between −π and π, and Ai¯ and ai (0, t) are the direct current (DC) and OFDM modulated real-valued components of the i-th pump wave. The real-valued nature of the two components is because DSB systems employing chirpless electro-optic converters are considered in the analysis. From Eq. (5) and assuming that the pump waves launched into the fiber can be represented by a small signal analysis, the product of the three pump waves included in the most right-hand side term of Eq. (1) can be approximated as:
A2(z,t)A3(z,t)A4*(z,t)=[A2¯A3¯A4¯+A2¯A3¯a4(0,tβ1,4z)+A2¯A4¯a3(0,tβ1,3z)++A3¯A4¯a2(0,tβ1,2z)]exp(32αz)exp(jϕg)
(6)
where ϕg = ϕ2 + ϕ3ϕ4. The derivation of Eq. (6) considers that the terms obtained from the double or triple beat between the modulated signals ( (Ai¯ajak) or aiajak) can be neglected. This assumption is valid for systems where the optical carrier power is much higher than the total power of the OFDM sidebands, as it is the case of DD DSB-OFDM systems with high CSPR. In this case, the FWM between the OFDM subcarriers is negligible and the main FWM components are generated due to the mixing between the optical carriers of two channels and the OFDM sidebands of a third channel, as illustrated in Fig. 1.

Substituting Eq. (6) into Eq. (1), Eq. (1) is easily solved in the frequency domain. Assuming that the FWM wave at the fiber input is null and the time reference of the FWM wave as time basis (t′ = tβ1,1z), and after integrating Eq. (1) over the fiber length, L, the field envelope of the FWM wave at the fiber output can be expressed as:
A1(L,t')=A¯FWM+i=24ai(0,t')*hi(L,t')
(7)
where ai (0, t′) = −1 {ãi (0, ω)}, hi (L, t′) = −1 {Hi (L, ω)} is the impulse response of the contribution provided by the beat between the carriers of two pump waves and the modulated signal of the i-th pump wave (carrier-signal beat) to the generated FWM wave, and AFWM is the field contribution to the generated FWM wave due to the beat of the carriers of the three pump waves (carrier-carrier beat). Equation (7) shows that, when the terms obtained from the double or triple beat between the modulated signals are neglected, the FWM effect can be modeled by an equivalent transfer function that describes the dependence of the FWM-induced distortion on the spectral content of each pump channel. The FWM contribution provided by the carrier-carrier beat is given by:
A¯FWM=j3Dγexp(jϕg)exp(12αL)A2¯A3¯A4¯jΔβα[exp(jΔβLαL)1]
(8)
Note that, in the degenerate FWM cases, Eq. (7) can still be simplified using a2 (0, t′)=a3 (0, t′) and h2 (L, t′)=h3 (L, t′). The power of the FWM field provided by the carrier-carrier beat represents the average optical power generated through the FWM process [13

13. K. Hill, D. Johnson, B. Kawasaki, and R. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978). [CrossRef]

], and has been used to evaluate the FWM-induced crosstalk in WDM systems [14

14. N. Shibata, R. Braun, and R. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” J. Quantum Electron. 23, 1205–1210 (1987). [CrossRef]

, 16

16. R. Tkach, A. Chraplyvy, F. Forghieri, A. Gnauck, and R. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995). [CrossRef]

, 17

17. S. Song, C. Allen, K. Demarest, and R. Hui, “Intensity-dependent phase-matching effects on four-wave mixing in optical fibers,” J. Lightwave Technol. 17, 2285–2290 (1999). [CrossRef]

]. Further analysis of this component of the FWM field is reported in Appendix A. Henceforth, tt′ for the sake of notation simplicity.

The transfer functions Hi (L, ω) (i = 2, 3, 4) of the contribution provided by the carrier-signal beat of the i-th pump wave to the generated FWM wave are given by:
Hi(L,ω)=j3Dγexp(jϕg)exp(12αL)Πm=2mi4Am¯j(Δβ+ωd1,i)α[exp(j(Δβ+ωd1,i)LαL)1]
(9)
where d1,i = β1,1β1,iDc (λr) × (λ1λi) represents the walkoff parameter between the FWM wave and the i-th pump wave. Dc is the chromatic dispersion parameter, λr = 2πcr is the reference wavelength (c is the speed of light in vacuum), and λ1 and λi are the FWM and the i-th pump wavelengths, respectively. The physical meaning of these transfer functions is explained in detail in Appendix A.

Assuming now that the generated FWM wave can be represented as an additive small perturbation of the field envelope being transmitted at the frequency ν1, the envelope of the total field can be represented as At (L, t) = As (L, t)+ A1 (L, t), where the signal field envelope is given by:
As(L,t)=[As¯+as(0,t)]exp(12αL)exp(jϕs)
(10)
ϕs is the initial phase, and As¯ and as (0, t) are the DC and the modulated components of the signal field transmitted at frequency ν1. Equation (10) neglects the impact of the dispersion on the field envelope being transmitted at the frequency ν1.

The photocurrent, it (t) = Rλ|At (L, t)|2, can then be written as:
it(t)=Rλ[|As(L,t)|2+As(L,t)A1*(L,t)+A1(L,t)As*(L,t)+|A1(L,t)|2]
(11)
where Rλ is the photodetector responsivity. The right-hand side of Eq. (11) consists of four terms. The first term is the signal component and comprises a DC contribution, the OFDM signal itself and the SSBI term. The second and the third terms represent the beat between the signal and the generated FWM field of frequency ν1. This beat is responsible by the FWM-induced degradation of the OFDM signal. The last term of Eq. (11) occurs due to the beat between the generated FWM field with itself. Part of this term occupies also the same frequency range as the signal As (t). However, for the situations of interest in which the power of the FWM wave is much lower than the power of the signal being transmitted at frequency ν1, the strength of the FWM-FWM beat is negligible when compared with the beat between the signal and the generated FWM fields. Using Eq. (7) and Eq. (10), the FWM contribution to the photocurrent can then be approximated by:
iFWM(t)Rλ[As(L,t)A1*(L,t)+A1(L,t)As*(L,t)]==Rλ[As¯+as(0,t)]exp(12αL)[A¯eq,FWM+i=24ai(0,t)*heq,i(L,t)]
(12)
where A¯eq,FWM=exp(jϕs)A¯FWM*+exp(jϕs)A¯FWM and heq,i(L,t)=exp(jϕs)hi*(L,t)+exp(jϕs)hi(L,t). To the best of our knowledge, Eq. (12) shows, for the first time, that the photocurrent due to the nonlinear FWM effect in DD OFDM systems can be modelled by a set of equivalent transfer functions. A similar modelling was also performed in the past to describe the effect of FWM in coherent optical OFDM systems [26

26. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16, 15777–15810 (2008). [CrossRef] [PubMed]

] and to describe the XPM effect [27

27. T. Chiang, N. Kagi, M. Marhic, and L. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14, 249–260 (1996). [CrossRef]

].

2.2. Variance of the FWM effect

The variance of the FWM effect at the photodetector output can be evaluated using Eq. (12). Considering that the mean value of the FWM current is null, the FWM variance is given by:
σFWM,p2=E[iFWM2(t)]=Rλ2exp(αL)E{[As¯2+2As¯as(0,t)+as(0,t)as(0,t)]××|A¯eq,FWM+i=24ai(0,t)*heq,i(L,t)|2}
(13)
where E[x(t)] is the mean value of x(t). A detailed analysis of Eq. (13) shows that the FWM variance can be written as a sum of three contributions:
σFWM,p2=σc,c2+σc,22+σs,s2
(14)
where σc,c2, σc,s2 and σs,s2 are the variance contributions resulting from the carrier-carrier beat (term dependent on As¯2 in Eq. (13)), the carrier-signal beat (term dependent on As¯as(0,t) in Eq. (13)), and the signal-signal beat (term dependent on as2(0,t) in Eq. (13)), respectively. The expressions for these variance contributions are derived in Appendix A, B and C. From Eq. (28), Eq. (33), Eq. (35) and Eq. (37), the total variance of the FWM component is given by:
σFWM,p2=Rλ2{exp(αL)As¯2+A¯eq,FWM|2++[n=15SFWM,n(L,f)]dt}
(15)
where
SFWM,1(L,f)=exp(αL)|A¯eq,FWM|2Ss(f)
(16)
SFWM,2(L,f)=exp(αL)As¯2ad,2S2(f)|Heq,2(L,f)|2
(17)
SFWM,3(L,f)=exp(αL)As¯2ad,3S3(f)|Heq,3(L,f)|2
(18)
SFWM,4(L,f)=exp(αL)As¯2ad,4S4(f)|Heq,4(L,f)|2
(19)
SFWM,5(L,f)=exp(αL)and,sAs¯Ss(f)[A¯eq,FWMHeq,4*(L,f)+A¯eq,FWM*Heq,4(L,f)]
(20)
In Eqs. (16)(20), ad,i and and,s are constants that allow distinguishing between the variance contributions for nondegenerate and degenerate FWM components:
ad,i={1fornondegenerateterms(ν2ν3)4fordegenerateterms(ν2=ν3)and i=20fordegenerateterms(ν2=ν3)andi=31fordegenerateterms(ν2=ν3)and i=4
(21)
and,s={2forsymmetricnondegenerateterms(ν1=v4)0fornonsymmetricnondegenerate(ν1ν4)anddegenerateterms(ν2=ν3)
(22)
In Eqs. (16)(20), Si (f) and Ss (f) represent the power spectral densities (PSDs) at the fiber input of the OFDM signal transmitted in the i-th pump wave and in the channel where the FWM is being generated, respectively, and Heq,i (L, f) = {heq,i (L, t)} is the FWM frequency response associated with the pump wave at frequency νi. The power of Āeq,FWM and the magnitude of the equivalent FWM transfer functions, |Āeq,FWM|2 and |Heq,i (L, f)|2, respectively, and their dependence on the phase mismatch factor and walkoff parameter are discussed in Appendix A.

The variance of the FWM current at the PIN output shown in Eq. (15) presents the sum of five integration terms. These terms quantify the contribution of the pump and signal waves to the FWM variance through the frequency response of the FWM effect (see Eqs. (17)(20)) and the power of Āeq,FWM (see Eq. (16)). In addition, the sixth frequency-independent term accounts for the contribution of the FWM to the DC component of the variance at the PIN output.

To obtain the variance of the FWM effect in each OFDM subcarrier, the PSDs at the PIN output of the different contributions to the FWM given by Eqs. (17)(20) must be converted to the output of the OFDM electrical receiver. This conversion can be represented by an equivalent filter that describes the conversion of the current at the PIN output to the k-th output of the OFDM demodulator associated with the k-th subcarrier. The transfer function of the equivalent filter of the OFDM receiver was already reported in [28

28. T. Alves and A. Cartaxo, “Semi-analytical approach for performance evaluation of direct-detection OFDM optical communication systems,” Opt. Express 17, 18714–18729 (2009). [CrossRef]

] (see Eq. (41)). The FWM variance of the k-th subcarrier at the output of a conventional OFDM-based system, including down-conversion, low-pass filtering, fast-Fourier transform (FFT) and equalization, is expressed as:
σFWM2[k]=Rλ2+n=15SFWM,n(f)|Heq,r(f,k)|2df
(23)
Equation (23) does not contain the DC term shown in Eq. (15) because it is removed by the equivalent filter of the OFDM receiver. In Eq. (23), Heq,r(f,k)=14[Heq,r,I(f,k)+jHeq,r,Q(f,k)]. The I or the Q components of this equivalent filter are given by Heq,r,(I,Q) (f, k) = Heq,1,(I,Q) (f + fRF, k) + Heq,2,(I,Q) (ffRF, k), where (I,Q) stand for the I or the Q component, and fRF is the RF carrier of the OFDM signal. Heq,1,(I,Q) (f, k) and Heq,2,(I,Q) (f, k) are given by [28

28. T. Alves and A. Cartaxo, “Semi-analytical approach for performance evaluation of direct-detection OFDM optical communication systems,” Opt. Express 17, 18714–18729 (2009). [CrossRef]

]:
[Heq,1,(I,Q)(f,k)Heq,2,(I,Q)(f,k)]=Hr(f)2exp[j2πft0]P(I,Q)TAeq[k]
(24)
where Hr (f) is the transfer function of the low-pass filter (LPF) of the down-converter and Aeq [k] is the amplitude response of the equalizer transfer function for the k-th subcarrier. P(I,Q) is a matrix that takes into account the effect of orthogonality between the I and Q branches of the OFDM receiver. The matrix T reflects the effects of the FFT operation and it represents the main frequency limitation of the amplitude response of the equivalent filter of the OFDM receiver. The expressions for the two matrices can be found in [28

28. T. Alves and A. Cartaxo, “Semi-analytical approach for performance evaluation of direct-detection OFDM optical communication systems,” Opt. Express 17, 18714–18729 (2009). [CrossRef]

].

Equation (23) allows evaluating the variance (or, equivalently, the power) of the k-th OFDM subcarrier at the output of the OFDM receiver due to the FWM component generated at the optical frequency ν1 = ν2 + ν3ν4. Thus, the total variance, σFWM,t2[k] of the k-th subcarrier of the OFDM signal transmitted in a given optical channel, due to all the new components generated by the FWM process at the frequency of that channel, is given by:
σFWM,t2[k]=i=1NFWMσFMW,i2[k]
(25)
where σFWM,i2[k] is the variance given by Eq. (23) evaluated for the i-th FWM component generated by the FWM and NFWM is the number of FWM components whose frequency coincide with the optical channel frequency in which the OFDM signal performance is being assessed.

Equation (25) provides the analytical formulation to evaluate the power of the fluctuation due to the FWM effect. The error vector magnitude (EVM) caused by the FWM effect in each subcarrier of a DD DSB OFDM system is given by the ratio between the power of the FWM fluctuations and the power of the corresponding subcarrier. Thus, the EVM of each subcarrier due to the FWM effect in a DD DSB OFDM system can be estimated from the total variance of the FWM-induced degradation by:
EV^M[k]σFWM,n2[k]=σFWM,t2[k]ps[k]
(26)
where ps [k] and σFWM,n2[k]; are the average power and the normalized FWM variance of the k-th subcarrier at the OFDM receiver output, respectively. The EVM due to the FWM effect is obtained by averaging the EVM of each subcarrier over all the OFDM subcarriers.

3. Analysis of equivalent FWM transfer functions

In this section, the power of Āeq,FWM and the magnitude of the frequency response of the FWM effect, used in Eq. (23) to evaluate the FWM variance, are analyzed. The expressions of those quantities are detailed in Appendix A. These expressions depend only on the optical carrier power, fiber parameters, channel spacing between optical channels and degeneracy factor.

The study is performed for a standard single-mode fiber (SSMF), whose dispersion parameter at 1550 nm is 17 ps/nm/km, the dispersion slope is 90 fs/nm2/km, the attenuation coefficient is 0.2 dB/km and the nonlinearity coefficient is 1.3 W−1km−1. Additionally, a WDM signal comprising three channels is considered. The central frequency of each channel is 193.075 THz (channel a), 193.1 THz (channel b) and 193.125 THz (channel c), and the average optical power at the fiber input is the same for all channels. The three in-band FWM waves (channel 1) generated by this 3-channel WDM signal are illustrated in Fig. 2. To simplify the illustration, the scheme represents each channel only by the optical carrier. Notice that the FWM waves generated at the frequencies of channel a and c are degenerate components, whereas the FWM wave generated at the frequency of the central channel b is a symmetric nondegenerate component.

Fig. 2 Illustration of the three in-band FWM waves generated by a 3-channel WDM signal. It is assumed that the spacing between channels a and b is the same as the spacing between channels b and c.

Figure 3 shows the amplitude response of the equivalent FWM transfer functions of the three in-band FWM waves generated by the WDM signal. The results are presented up to 30 GHz to show the frequencies at which the maximums of the FWM amplitude response occur. In a real WDM system using 25 GHz of channel spacing, it is not expected to have the modulated signal located at 25 GHz away from the optical carrier as, in that case, the modulated signal would be located at the frequency of the optical carrier of the adjacent channel.

Fig. 3 Amplitude response of the equivalent FWM transfer functions of the FWM-wave generated at the frequency of channel (a) a (b) b and (c) c. SSMF lengths of 75 km and 100 km are considered. The power of the optical carrier per channel at the fiber input is 12 dBm. ν0 is the central frequency of each optical channel and ν is the optical frequency.

Figure 4 shows results similar to the ones of Fig. 3(a) (FWM component that is generated at the frequency of channel a) but as a function of the fiber length, for four different frequencies and considering the FWM frequency responses normalized by the power loss introduced by the SSMF. Frequencies of 2 GHz, 3.7 GHz, 4.4 GHz and 12.5 GHz were chosen. The frequencies of 3.7 GHz and 4.4 GHz were chosen because they are in the frequency band of the OFDM-UWB signals used in section 4. The frequencies close to 2 GHz and 12.5 GHz correspond to minimums and maximums of the amplitude response shown in Fig. 3(a). Figs. 4(a)–4(c) show that the power of Āeq,FWM and the amplitude of the equivalent FWM transfer functions vary with the fiber length. This variation is attributed in part to the dependence of the FWM efficiency on the equivalent phase mismatch factor (Δβeq,i (f) L) and decreases as the fiber length increases due to the reduction of the effective fiber length. In addition, the inspection of Figs. 4(b) and 4(c) confirms that the amplitude of the equivalent FWM transfer functions are differently affected by the frequency. These results indicate that, despite the FWM effect is dominantly generated along the effective length of the fiber (21.5 km for a fiber length of 100 km), the FWM magnitude can still present noticeable fluctuations when the fiber length is higher than the effective length of the fiber.

Fig. 4 (a) Power of Āeq,FWM normalized to the power of the optical carrier at the fiber output as a function of the fiber length. (b) and (c) Amplitude response of Heq,2 (L, f) and Heq,4 (L, f), respectively, normalized by the power loss introduced by the SSMF, as a function of the fiber length. The FWM component is being generated at the frequency of channel a (see Fig. 2) and the power of the optical carrier per channel at the fiber input is 12 dBm. In (b) and (c): results obtained for a frequency of 2 GHz (dashed-dotted line), 3.7 GHz (continuous line), 4.4 GHz (dashed line) and 12.5 GHz (dotted line).

4. FWM variance induced in OFDM-UWB radio signals distributed in long-reach PONs

In this section, the application scenario considered to study the impact of the FWM on the performance of DD DSB OFDM systems is introduced. Additionally, the accuracy of the analytical model derived in section 2 to estimate the EVM induced by the FWM on the OFDM signal is analyzed by comparison with the EVM obtained by numerical resolution of Eq. (1). An extensive analysis of the validity range of the proposed analytical model is also performed.

4.1. System description

The application scenario used in this work is the transmission of DSB OFDM-UWB radio signals along LR-PONs employing DD [8

8. T. Alves, M. Morant, A. Cartaxo, and R. Llorente, “Transmission of OFDM wired-wireless quintuple-play services along WDM LR-PONs using centralized broadband impairment compensation,” Opt. Express 20, 13748–13761 (2012). [CrossRef] [PubMed]

,29

29. R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo, and J. Marti, “Ultra-wideband radio signals distribution in FTTH networks,” IEEE Photon. Technol. Lett. 20, 945–947 (2008). [CrossRef]

]. The OFDM-UWB signal is composed of 128 sub-carriers, the signal bandwidth is 528 MHz and the time interval of each OFDM-UWB symbol is 312.5 ns [30

30. High Rate UltraWideband PHY and MAC Standard, 2ECMA Int. (2007).

]. The simultaneous transmission of UWB band 2 (central frequency of 3.96 GHz) and 3 (central frequency of 4.49 GHz) is considered [8

8. T. Alves, M. Morant, A. Cartaxo, and R. Llorente, “Transmission of OFDM wired-wireless quintuple-play services along WDM LR-PONs using centralized broadband impairment compensation,” Opt. Express 20, 13748–13761 (2012). [CrossRef] [PubMed]

]. Only UWB bands 2 and 3 are transmitted as most of the UWB transceivers commercially available nowadays are operating in the first three UWB bands, and to avoid interference with the WiMAX signal whose frequency location is the same as UWB band 1. The electro-optic conversion is modeled by a chirpless Mach-Zehnder modulator (MZM) biased at quadrature point and without band limitations.

LR-PONs comprise a feeder fiber, connecting the central office to a remote node (RN), and distribution fibers, connecting the RN to the users’ premises. In a WDM LR-PON, where different users are served by different wavelengths, the RN comprises a wavelength router to select the optical channel delivered to each user through the distribution fiber [8

8. T. Alves, M. Morant, A. Cartaxo, and R. Llorente, “Transmission of OFDM wired-wireless quintuple-play services along WDM LR-PONs using centralized broadband impairment compensation,” Opt. Express 20, 13748–13761 (2012). [CrossRef] [PubMed]

]. This means that the FWM effect occurs only along the feeder fiber. Therefore, the study of the FWM effect developed in the following section is performed for fiber lengths typical of the feeder fiber reach (≤100 km). The feeder fiber is a SSMF with parameters presented in section 3. After photodetection, a rectangular LPF with 5 GHz of bandwidth is used to remove the out-of-band distortion components. The OFDM demodulation is performed by a conventional OFDM receiver including frequency synchronization, down-conversion, FFT and equalization [8

8. T. Alves, M. Morant, A. Cartaxo, and R. Llorente, “Transmission of OFDM wired-wireless quintuple-play services along WDM LR-PONs using centralized broadband impairment compensation,” Opt. Express 20, 13748–13761 (2012). [CrossRef] [PubMed]

]. The transfer function of the one-tap equalizer is estimated from the information provided by the channel response induced on the OFDM-UWB pilot subcarriers (UWB uses pilot subcarriers, spread along the OFDM signal band, for channel estimation rather than training symbols). A 4-th order least-squares regression method is then implemented to estimate the equalizer transfer function along the band of the transmitted subcarriers.

The accuracy of the FWM model proposed in this work is assessed considering noiseless optical and electrical amplifiers to focus the attention only on the performance degradation induced by the FWM effect.

4.2. Variance and EVM of the FWM-induced degradation

The numerical resolution of Eq. (1) is performed considering a fiber step size of 100 m and, for each run, random delays between optical channels, random binary sequences for each UWB signal of each optical channel and random initial phases of each pump and signal waves. The central channel of the WDM signal is located at the frequency of the ITU-T grid 193.1 THz. If the WDM signal consists in an even number of channels, the number of channels with central frequencies higher than 193.1 THz exceeds the number of channels with central frequencies lower than 193.1 THz. In the following studies, the EVM of the UWB bands transmitted in the optical channel of the WDM signal with lower wavelength is labeled as the EVM of channel 1.

In a first step, the dependence of the EVM due to the FWM induced in UWB bands 2 and 3, obtained from the numerical resolution of Eq. (1) (henceforth mentioned as NR results), on the number of runs was assessed. In each run, the OFDM-UWB signal comprises 8 OFDM symbols generated from random binary information. The EVM of a set of different runs is evaluated by averaging the EVM of each run over all runs. The objective of this analysis was to identify how many runs are needed to estimate accurately the FWM-induced EVM. The EVM estimated from NR is then used as a reference to assess the accuracy of the FWM variance provided by the proposed analytical model (henceforth mentioned as AM results).

To ensure reliable EVM estimates, the following NR results shown in this section are obtained considering 300 runs. This decision is supported by the reduced EVM fluctuation around the final value (does not exceed 0.2 dB) observed in the study of the EVM as a function of the number of runs. This conclusion held regardless the optical channel under analysis and the system parameters considered.

4.2.1. Dependence of FWM-induced degradation on the modulation index

Figure 5 depicts the EVM of UWB band 2 and 3 transmitted in each optical channel as a function of the modulation index. The modulation index is defined as the ratio between the root-mean-square voltage of the signal applied to the modulator arms and the switching voltage of the modulator. This analysis is useful to identify the validity range of the small signal analysis performed in section 2. It allows assessing if, for the modulation indexes of interest (the optimum modulation index for this kind of application scenario is lower than 20% [31

31. T. Alves and A. Cartaxo, “Distribution of double-sideband OFDM-UWB radio signals in dispersion compensated long-reach PONs,” J. Lightwave Technol. 29, 2467–2474 (2011). [CrossRef]

]), the FWM contributions obtained from the double or triple beat between the modulated signals ( Ai¯ajak or aiajak) can be neglected as assumed in the derivation of Eq. (6). The average optical power per channel at fiber input is 12 dBm and a channel spacing of 25 GHz is considered. In Figs. 5(a) and 5(b), the WDM signal comprises 3 channels and a SSMF length of 75 km, whereas a 4-channel WDM signal and a SSMF length of 90 km are used for the results of Figs. 5(c) and 5(d). EVM results obtained through NR and EVM estimates obtained considering the FWM AM of section 2 (see Eq. (26)) are presented. The EVM resulting from adding the analytical FWM EVM estimates with the EVM obtained through numerical simulation of the OFDM system in back-to-back operation (b2b) is also shown. These results are useful to provide insight on the weight of the EVM induced by the FWM effect and by the MZM distortion.

Fig. 5 EVM of each optical channel as a function of the modulation index for (a) and (c) UWB 2, and (b) and (d) UWB 3. The average optical power per channel at fiber input is 12 dBm and the channel spacing is 25 GHz. In (a) and (b): 3-channel WDM signal, SSMF length of 75 km. In (c) and (d): 4-channel WDM signal, SSMF length of 90 km.

Two main conclusions can be drawn from the inspection of Fig. 5. (i) In the case of the edge channels (Figs. 5(a), 5(b): channels 1 and 3; Figs. 5(c), 5(d): channels 1 and 4), the EVM obtained from the AM provides reliable estimates (EVM difference not exceeding 0.6 dB) of the EVM obtained by NR for modulation indexes up to 20%. For modulation indexes not exceeding 8%, the EVM difference is lower than 0.3 dB. A detailed analysis of the results of Fig. 5 shows also that the EVM of each UWB band is dominantly impaired by the FWM effect only up to modulation indexes of 10%. For higher modulation indexes, the MZM-induced distortion dominates the EVM. (ii) In the case of the center channels (Figs. 5(a), 5(b): channel 2; Figs. 5(c), 5(d): channels 2 and 3), the proposed analytical formulation of the FWM leads to a EVM discrepancy, when compared with the EVM obtained from NR, that achieves 1.5 dB for modulation indexes of 20%. For modulation indexes up to 8%, this discrepancy does not exceed 1 dB. Two different effects origin this EVM discrepancy in the central channels. (a) Assuming that the EVM mismatch induced by neglecting the double or triple beat between the modulated signals is similar regardless the location of the optical channels, the analysis of the edge channels accomplished before suggests that about one third (0.3 dB and 0.6 dB for a modulation index of 8% and 20%, respectively) of the EVM discrepancy is due to the neglected contribution of the double and triple beat terms. (b) The remaining part of the EVM discrepancy (0.7 dB and 0.9 dB for a modulation index of 8% and 20%, respectively) is due to a rough approximation performed in the evaluation of the FWM generated in the symmetric (f1 = f4) nondegenerate (f2f3) case. In this situation, one of the pump waves coincides in frequency with the generated FWM component. This means that, in Eq. (1), the field envelopes A1(z, t) and A4(z, t) correspond to the same channel and, consequently, the solution of Eq. (1) presented in section 2 is not rigorous. As the EVM estimates provided by the analytical model are slightly worse than the EVM obtained by NR, this EVM estimates can be used as worst case scenario while avoiding the far more complex formulation required to obtain the exact analytical solution of Eq. (1). The comparison between the EVM obtained from NR and the EVM obtained from the AM shows that the EVM discrepancy tends to decrease as the modulation index increases. However, this is because the distortion induced by the MZM starts to dominate the total EVM and not due to the reduction of the inaccuracy of the EVM estimates provided by the AM.

Figure 5 shows also that the discrepancy between the EVM obtained by NR and the EVM obtained by adding the analytical EVM estimates induced by the FWM with the EVM obtained in b2b operation is getting larger as the modulation index increases. This discrepancy increase occurs because the CSPR decreases when the modulation index increases, i. e., the power of the OFDM signal increases. Under such high modulation indexes, the contribution of the double and triple beat terms cannot be neglected when compared with the terms of Eq. (6). Nevertheless, for a modulation index of 20%, this EVM discrepancy does not exceed 0.5 dB.

Henceforth, to focus the attention on the analysis of the performance degradation due to the FWM, the EVM results shown for the NR case only consider the FWM-induced degradation, i. e., the EVM obtained in b2b is subtracted from the total EVM obtained through NR of Eq. (1).

4.2.2. Dependence of FWM-induced degradation on the fiber length

Figure 6 shows the EVM of UWB 2 and UWB 3 of each optical channel as a function of the fiber length for a WDM signal comprising 3 channels and a modulation index of 8%. Figures 6(a) and 6(b) show results for a channel spacing of 25 GHz and an average optical power per channel at fiber input of 12 dBm, whereas Figs. 6(c) and 6(d) consider a channel spacing of 50 GHz and an average optical power per channel at fiber input of 17 dBm. Figure 6 confirms that excellent EVM estimates are obtained from the AM for the FWM effect induced in the edge channels regardless the fiber length considered. In the case of the central channel and despite the EVM discrepancy between the AM and NR estimates may, for specific fiber lengths, slightly exceed 1 dB, the EVM discrepancy between the estimates obtained by the two methods does not, in general, exceed 1 dB. The cases, where a sightly higher EVM discrepancy is obtained, are attributed to the non-null uncertainty of the EVM convergence process associated with the finite number of initial random phases, optical delays and OFDM symbols considered.

Fig. 6 EVM of each optical channel as a function of the fiber length for (a) and (c) UWB 2, and (b) and (d) UWB 3. A 3-channel WDM signal and a modulation index of 8% are considered. (a) and (b) Channel spacing of 25 GHz and average optical power per channel at fiber input of 12 dBm; (c) and (d) Channel spacing of 50 GHz and average optical power per channel at fiber input of 17 dBm.

4.2.3. Dependence of FWM-induced degradation on the optical power

Figures 7(a) and 7(b) show the EVM of UWB band 2 and 3, respectively, as a function of the average optical power per channel launched into the fiber. A 4-channel WDM signal, a SSMF length of 100 km, channel spacings of 25 GHz and 50 GHz, and a modulation index of 8% are considered. The assessment of the EVM dependence on the optical power aims at evaluating the accuracy of Eq. (12) to describe the FWM current at the PIN output as that equation neglects the FWM-FWM beat term. The inspection of Figs. 7(a) and 7(b) reveals that very good agreement is obtained between the analytical EVM estimates and the EVM obtained numerically. This agreement shows that, for the optical power levels of interest, the FWM-FWM beat term is negligible when compared with the beat between the signal and the generated FWM fields.

Fig. 7 EVM of each optical channel as a function of the average optical power per channel at fiber input for (a) UWB 2 and (b) UWB 3. A WDM signal comprising 4 channels, a SSMF length of 100 km and a modulation index of 8% are considered.

4.2.4. Dependence of FWM-induced degradation on the optical channel

As a final analysis of the accuracy of the proposed analytical FWM model, Fig. 8 shows the EVM of UWB 3 as a function of each channel of a WDM signal comprising 5, 7 and 9 optical channels. Analytical and numerical EVM results are presented and the system parameters are: modulation index of 8%, SSMF length of 60 km, average optical power per channel at fiber input of 12 dBm and channel spacing of 25 GHz. Figure 8 presents results only for UWB band 3 as they are similar to the EVM results of UWB band 2. Figure 8 shows that the main conclusions drawn above for a WDM signal comprising 3 or 4 channels hold also for a higher channel count. Particularly, Fig. 8 shows that the excellent accuracy of the analytical EVM estimates obtained for the UWB signals transmitted in the edge optical channels and the reduced (not exceeding 1 dB) discrepancy between the analytical and numerical EVM for the inner optical channels (that include symmetric nondegenerate FWM contributions) remain valid when dozens of FWM components are generated, as it is the case when the number of channels of a WDM system increases.

Fig. 8 EVM of UWB 3 as a function of each channel of a WDM signal comprising 5, 7 and 9 optical channels. A modulation index of 8%, a fiber length of 60 km, an average optical power at fiber input of 12 dBm and a channel spacing of 25 GHz are considered.

5. Conclusion

The FWM-induced degradation of DSB OFDM signals in DD systems has been analytically characterized. A closed-form expression of the FWM variance of each OFDM subcarrier has been proposed and discussed. This expression uses equivalent FWM transfer functions to account for the dependence of the FWM efficiency on the propagation constant mismatch and walkoff between the pump waves and FWM wave. It has been shown that the walkoff leads to a frequency-dependent equivalent propagation constant mismatch.

The validity range of the proposed analytical expression of the FWM variance of each OFDM subcarrier has been assessed for different sets of parameters of an application scenario based on the distribution of OFDM-UWB radio signals along LR-PONs. It has been shown that the proposed model provides suitable variance estimates of the FWM-induced degradation due to degenerate and nonsymmetric nondegenerate components. The discrepancy between the variance estimated in these two cases and the actual performance does not exceed 0.6 dB for modulation indexes as high as 20%. In the case of symmetric nondegenerate FWM components, reliable but pessimistic variance estimates are achieved, with discrepancy not exceeding 1.5 dB for modulation indexes up to 20% when compared with the actual performance. Hence, it can be concluded that the proposed closed-form expression for the FWM variance can be employed as a powerful tool to provide fast and accurate performance estimates of DSB OFDM DD systems limited by the FWM effect.

A. FWM variance contribution due to carrier-carrier beat term

From Eq. (13) and considering that E[ai(0, t)] = 0, i. e., the mean value of the OFDM signal is null, the FWM variance contribution resulting from the carrier-carrier beat is given by:
σc,c2=Rλ2exp(αL)As¯2{|A¯eq,FWM|2+E[i=24ai(0,t)*heq,i(L,t)l=24al(0,t)*heq,l*(L,t)]}
(27)
We have to distinguish between degenerate and nondegenerate cases to further simplify Eq. (27).

A.1. Nondegenerate FWM

In the nondegenerate case, the pump waves are located at different frequencies and are independent. In that situation, E[ai (0, t) aj (0, t)] = E [ai (0, t)] E[aj (0, t)] = 0 for ij. Using this result and also the Wiener-Kintchine theorem, Eq. (27) is simplified to:
σc,c2=Rλ2exp(αL)As¯2{|A¯eq,FWM|2+i=24+Si(f)|Heq,i(L,f)|2df}
(28)
where |Āeq,FWM|2 = 2|ĀFWM|2, and |Heq,i (L, f)|2 = |Hi (L, − f)|2 + |Hi (L, f)|2. Si (f) is the PSD of the OFDM signal ai (0, t). The derivation of Eq. (28) uses the fact that the mean value of sinusoidal functions, whose total phase is a linear combination of independent and uniformly distributed phases with domain multiple of 2π (as the initial phases ϕi), is null. Assuming that the optical carrier power is similar in the three pump waves, the magnitude of the FWM transfer functions and the FWM field power due to the carrier-carrier beat are given by:
|Hi(L,f)|2=ηi(f)(Dγ3)2exp(αL)P¯2Leff2
(29)
|A¯FWM|2=ηi(0)(Dγ3)2exp(αL)P¯3Leff2
(30)
where is the optical carrier power at the fiber input and Leff=1exp(αL)α is the effective interaction fiber length. The efficiency of the generated FWM wave, ηi (f), is given by:
ηi(f)=α2α2+(Δβ+2πfd1,i)2{1+4exp(αL)sin2[12(Δβ+2πfd1,i)L](1exp(αL)2)}
(31)
Equation (31) illustrates the traditional dependence of the FWM on the fiber dispersion and channel spacing, expressed by the phase mismatch factor ΔβL [17

17. S. Song, C. Allen, K. Demarest, and R. Hui, “Intensity-dependent phase-matching effects on four-wave mixing in optical fibers,” J. Lightwave Technol. 17, 2285–2290 (1999). [CrossRef]

, 20

20. A. Akhtar, L. Pavel, and S. Kumar, “Modeling and analysis of the contribution of channel walk-off to nondegenerate and degenerate four-wave-mixing noise in RZ-OOK optical transmission systems,” J. Lightwave Technol. 24, 4269–4285 (2006). [CrossRef]

]. Equation (31) reveals also the impact of the walkoff on the FWM efficiency through the walkoff factor 2πfd1,i. Contrarily to the traditional FWM efficiency, the walkoff factor leads to a frequency-dependent FWM efficiency. When combined with the impact of the FWM effect on the photocurrent expressed by |Heq,i (L, f)|2 in Eq. (28), this dependence induces a maximum FWM efficiency that occurs at the modulation frequencies given by fc = ±Δβ/(2πd1,i).

A.2. Degenerate FWM

In the degenerate case, two of the pumps that origin the FWM process are the same wave, i. e., ν2=ν3 (or, equivalently, a2 (0, t)=a3 (0, t)) and heq,2 (L, t)=heq,3 (L, t). In that situation, E[a2 (0, t) a3 (0, t)] ≠ 0 and Eq. (27) can be expressed as:
σc,c2=Rλ2exp(αL)As¯2{|A¯eq,FWM|2+E{[a4(0,t)*heq,4(L,t)][a4(0,t)*heq,4*(L,t)]++4[a2(0,t)*heq,2(L,t)][a2(0,t)*heq,2*(L,t)]}}
(32)
Using a procedure similar to the one used in the nondegenerate case, Eq. (32) is simplified to:
σc,c2=Rλ2exp(αL)As¯2{|A¯eq,FWM|2++S4(f)|Heq,4(f)|2df++4+S2(f)|Heq,2(f)|2df}
(33)

B. FWM variance contribution due to carrier-signal beat term

From Eq. (13) and considering again that the mean value of the OFDM signal is null and that the OFDM signals from the three pump waves are independent, the FWM variance contribution resulting from the carrier-signal beat is given by:
σc,s2=2Rλ2exp(αL)As¯E{as(0,t)i=24[A¯eq,FWMheq,i*(L,t)+A¯eq,FWM*heq,i(L,t)]ai(0,t)}
(34)
As for the carrier-carrier term, Eq. (34) must be distinguished between degenerate and nonde-generate cases. In the case of nonsymmetric (ν1ν4) nondegenerate (ν2ν3) and degenerate terms, the OFDM signal of the i-th pump wave is independent of the OFDM signal transmitted in the channel where the FWM is being generated (E[as (0, t) ai (0, t)] = 0) and, consequently, the carrier-signal variance term expressed in Eq. (34) is null. In the symmetric (ν1 = ν4) non-degenerate (ν2ν3) FWM situation, the OFDM signal of one of the pump waves is the same as the OFDM signal transmitted in the channel where the FWM is being generated. In this situation, E[as (0, t) a4 (0, t)] ≠ 0 and E[as (0, t) a2 (0, t)]=E [as (0, t) a3 (0, t)] = 0 and, therefore, Eq. (34) can be simplified to:
σc,s2=2Rλ2exp(αL)As¯+Ss(f)[A¯eq,FWMHeq,4*(L,f)+A¯eq,FWM*Heq,4(L,t)]df
(35)
where Ss (f) is the PSD (at the fiber input) of the OFDM signal transmitted in the channel where the FWM is being generated and
A¯eq,FWMHeq,4*(L,f)+A¯eq,FWM*Heq,4(L,f)=2[A¯FWMH4*(L,f)+A¯FWM*H4(L,f)]
(36)

C. FWM variance contribution due to signal-signal beat term

Considering that E[as(0, t)as(0, t)ai(0, t)] = 0, and that E [as(0, t)as(0, t)ai(0, t)ai(0, t)] and E[as(0, t)as(0, t)as(0, t)] are negligible compared with E [as(0, t)as(0, t)], the FWM variance contribution resulting from the signal-signal beat expressed in Eq. (13) can be approximated by:
σs,s2=Rλ2exp(αL)|A¯eq,FWM|2POFDM
(37)
where we have used the Wiener-Kintchine theorem and POFDM=+Ss(f)df is the optical power of the OFDM signal at the fiber input. Equation (37) is valid for symmetric and nonsymmetric nondegenerate and degenerate FWM terms.

Acknowledgments

This work was supported in part by Fundação para a Ciência e a Tecnologia from Portugal under the project MORFEUS-PTDC/EEI-TEL/2573/2012. The project PEst-OE/EEI/LA0008/2013 is also acknowledged.

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T. Alves and A. Cartaxo, “Semi-analytical approach for performance evaluation of direct-detection OFDM optical communication systems,” Opt. Express 17, 18714–18729 (2009). [CrossRef]

29.

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo, and J. Marti, “Ultra-wideband radio signals distribution in FTTH networks,” IEEE Photon. Technol. Lett. 20, 945–947 (2008). [CrossRef]

30.

High Rate UltraWideband PHY and MAC Standard, 2ECMA Int. (2007).

31.

T. Alves and A. Cartaxo, “Distribution of double-sideband OFDM-UWB radio signals in dispersion compensated long-reach PONs,” J. Lightwave Technol. 29, 2467–2474 (2011). [CrossRef]

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(060.2330) Fiber optics and optical communications : Fiber optics communications

ToC Category:
Optical Communications

History
Original Manuscript: December 16, 2013
Revised Manuscript: February 14, 2014
Manuscript Accepted: February 22, 2014
Published: April 3, 2014

Citation
Tiago M. F. Alves and Adolfo V. T. Cartaxo, "Analytical characterization of four wave mixing effect in direct-detection double-sideband OFDM optical transmission systems," Opt. Express 22, 8598-8616 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-8598


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References

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