## 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

Communication networks have experienced a huge traffic demand increase along the last years with current forecasts predicting an overall traffic increase by a factor of 100 in the next 10 years [1]. As a consequence, next generation optical fiber networks must accommodate several dozens of Tb/s and a remarkable upgrade of current (core, metropolitan and access) optical networks must be addressed in the next few years to ensure such capacity capabilities.

Orthogonal frequency division multiplexing (OFDM) has been widely proposed as a powerful modulation format to provide the desired capacity, spectral efficiency, granularity, and switching capabilities required by the next generation optical fiber networks. The advantages of OFDM-based optical systems include the sharp roll-off of OFDM spectrum, resilience to linear fiber effects, flexibility to fine tune the information rate of each OFDM signal and to provide multiple access [2–4]. In the last years, OFDM was proposed not only for long-haul systems [5, 6], but also for metropolitan [7], access [4], and radio-over-fiber networks [8]. Despite OFDM-based systems benefit from a set of worthwhile OFDM advantages, they are also impaired by the weak tolerance of OFDM signals to fiber nonlinearity [3].

With the capacity demand foreseen for the next years, the weak tolerance of OFDM to fiber nonlinearity will be an issue not only for core networks, but also for shorter reach optical networks due to the capacity increase envisioned for these networks. In this work, the attention is dedicated to short-reach double sideband (DSB) OFDM systems employing direct-detection (DD). OFDM systems employing DSB and DD are desired for short reach networks to keep the cost of optical transmitters and receivers as low as possible [9, 10]. DD-OFDM systems can or can not employ a frequency gap to accommodate the signal-signal beat interference (SSBI) [8,9, 11]. Although the frequency gap provides high tolerance to SSBI, it leads to spectral efficiency reduction and might not be allowed in systems where the spectral allocation of OFDM signals is standardized and cannot be managed. This is the case of some wireless signals delivered to customers using radio-over-fiber [8]. DD-OFDM systems that do not use the frequency gap can also present a high tolerance to SSBI degradation by using digital signal processing (DSP) algorithms to partially remove the SSBI [9] or by managing the system parameters to reduce the SSBI degradation [8]. Common to these non-gapped DD DSB-OFDM systems is the use of a large carrier-to-signal power ratio (CSPR) to ensure that the carrier-signal beat term after photodetection is dominant over the SSBI term. As reference, the optimum CSPR identified in [9], [10] and [12] for non-gapped DD DSB-OFDM systems is higher than 10 dB.

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 [13–17]. 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]. This occurred, for instance, a few years ago for on-off keying (OOK) systems [19–21] and, recently, for OFDM-based coherent long-haul systems [22–25].

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].

## 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, *ν*_{1}, is obtained from the frequencies of the three pump waves by *ν*_{1} = *ν*_{2} + *ν*_{3} − *ν*_{4} [16].

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]:

*A*(

_{i}*z*,

*t*) is the field envelope of the wave at frequency

*ν*,

_{i}*α*and

*γ*are the fiber loss coefficient and nonlinearity parameter,

*β*

_{1,i}is the first-order dispersion parameter calculated at the frequency

*ν*, and

_{i}*D*is the degeneracy factor. For nondegenerate FWM components (

*ν*

_{2}≠

*ν*

_{3}),

*D*= 6, whereas in degenerate FWM components (

*ν*

_{2}=

*ν*

_{3}),

*D*= 3. As the degenerate FWM component is generated from only two optical fields, Eq. (1) can still be further simplified using

*A*

_{2}(

*z*,

*t*)=

*A*

_{3}(

*z*,

*t*). Equation (1) neglects the effect of the fiber nonlinearity on the pump waves. The propagation constant mismatch is given by Δ

*β*=

*β*(Ω

_{1}) +

*β*(Ω

_{4}) −

*β*(Ω

_{2}) −

*β*(Ω

_{3}) where

*β*(Ω

*) is the propagation constant of the wave transmitted at the angular frequency Ω*

_{i}*= 2*

_{i}*πν*. Using a Taylor series around a reference frequency Ω

_{i}*,*

_{r}*β*(Ω

*) can be approximated by $\beta ({\mathrm{\Omega}}_{i})={\beta}_{0}({\mathrm{\Omega}}_{r})+{\beta}_{1}({\mathrm{\Omega}}_{r})({\mathrm{\Omega}}_{i}-{\mathrm{\Omega}}_{r})+\frac{{\beta}_{2}({\mathrm{\Omega}}_{r})}{2}{({\mathrm{\Omega}}_{i}-{\mathrm{\Omega}}_{r})}^{2}+\mathrm{...}$, where*

_{i}*β*

_{0}(Ω

*) and*

_{r}*β*

_{2}(Ω

*) are the zero-order and second-order dispersion parameter calculated at the frequency Ω*

_{r}*[20]. 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 (*

_{r}*β*

_{1}term) and neglects the impact of the dispersion on the distortion of the FWM wave and pump waves.

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
${A}_{i}(z,t)={A}_{i}(0,t-{\beta}_{1,i}z)exp\left(-\frac{1}{2}\alpha z\right)$, 2 ≤ *i* ≤ 4. The field envelope of the pump waves can still be written as:

*ϕ*is the initial phase uniformly distributed between −

_{i}*π*and

*π*, and $\overline{{A}_{i}}$ and

*a*(0,

_{i}*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:

*ϕ*=

_{g}*ϕ*

_{2}+

*ϕ*

_{3}−

*ϕ*

_{4}. The derivation of Eq. (6) considers that the terms obtained from the double or triple beat between the modulated signals ( $(\overline{{A}_{i}}{a}_{j}{a}_{k})$ or

*a*) 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.

_{i}a_{j}a_{k}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′* = *t* − *β*_{1,1}*z*), 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:

*a*(0,

_{i}*t′*) =

*ℱ*

^{−1}{

*ã*(0,

_{i}*ω*)},

*h*(

_{i}*L*,

*t′*) =

*ℱ*

^{−1}{

*H*(

_{i}*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

*A*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:

_{FWM}*a*

_{2}(0,

*t′*)=

*a*

_{3}(0,

*t′*) and

*h*

_{2}(

*L*,

*t′*)=

*h*

_{3}(

*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], and has been used to evaluate the FWM-induced crosstalk in WDM systems [14, 16, 17]. Further analysis of this component of the FWM field is reported in Appendix A. Henceforth,

*t*≡

*t′*for the sake of notation simplicity.

The transfer functions *H _{i}* (

*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:

*d*

_{1,i}=

*β*

_{1,1}−

*β*

_{1,i}≈

*D*(

_{c}*λ*) × (

_{r}*λ*

_{1}−

*λ*) represents the walkoff parameter between the FWM wave and the

_{i}*i*-th pump wave.

*D*is the chromatic dispersion parameter,

_{c}*λ*= 2

_{r}*πc*/Ω

*is the reference wavelength (*

_{r}*c*is the speed of light in vacuum), and

*λ*

_{1}and

*λ*are the FWM and the

_{i}*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 *ν*_{1}, the envelope of the total field can be represented as *A _{t}* (

*L*,

*t*) =

*A*(

_{s}*L*,

*t*)+

*A*

_{1}(

*L*,

*t*), where the signal field envelope is given by:

*ϕ*is the initial phase, and $\overline{{A}_{s}}$ and

_{s}*a*(0,

_{s}*t*) are the DC and the modulated components of the signal field transmitted at frequency

*ν*

_{1}. Equation (10) neglects the impact of the dispersion on the field envelope being transmitted at the frequency

*ν*

_{1}.

The photocurrent, *i _{t}* (

*t*) =

*R*|

_{λ}*A*(

_{t}*L*,

*t*)|

^{2}, can then be written as:

*R*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

_{λ}*ν*

_{1}. 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

*A*(

_{s}*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

*ν*

_{1}, 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:

#### 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:

*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: where ${\sigma}_{c,c}^{2}$, ${\sigma}_{c,s}^{2}$ and ${\sigma}_{s,s}^{2}$ are the variance contributions resulting from the carrier-carrier beat (term dependent on ${\overline{{A}_{s}}}^{2}$ in Eq. (13)), the carrier-signal beat (term dependent on $\overline{{A}_{s}}{a}_{s}(0,t)$ in Eq. (13)), and the signal-signal beat (term dependent on ${a}_{s}^{2}(0,t)$ 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:

*a*and

_{d,i}*a*are constants that allow distinguishing between the variance contributions for nondegenerate and degenerate FWM components:

_{nd,s}*S*(

_{i}*f*) and

*S*(

_{s}*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

*H*(

_{eq,i}*L*,

*f*) =

*ℱ*{

*h*(

_{eq,i}*L*,

*t*)} is the FWM frequency response associated with the pump wave at frequency

*ν*. The power of

_{i}*Ā*and the magnitude of the equivalent FWM transfer functions, |

_{eq,FWM}*Ā*|

_{eq,FWM}^{2}and |

*H*(

_{eq,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 *Ā _{eq,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] (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:

*H*

_{eq,r,(I,Q)}(

*f*,

*k*) =

*H*

_{eq,1,(I,Q)}(

*f*+

*f*,

_{RF}*k*) +

*H*

_{eq,2,(I,Q)}(

*f*−

*f*,

_{RF}*k*), where (I,Q) stand for the I or the Q component, and

*f*is the RF carrier of the OFDM signal.

_{RF}*H*

_{eq,1,(I,Q)}(

*f*,

*k*) and

*H*

_{eq,2,(I,Q)}(

*f*,

*k*) are given by [28]:

*H*(

_{r}*f*) is the transfer function of the low-pass filter (LPF) of the down-converter and

*A*[

_{eq}*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].

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 *ν*_{1} = *ν*_{2} + *ν*_{3} − *ν*_{4}. Thus, the total variance,
${\sigma}_{FWM,t}^{2}[k]$ 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:

*i*-th FWM component generated by the FWM and

*N*is the number of FWM components whose frequency coincide with the optical channel frequency in which the OFDM signal performance is being assessed.

_{FWM}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:

where*p*[

_{s}*k*] and ${\sigma}_{FWM,n}^{2}[k]$; 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 *Ā _{eq,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/nm^{2}/km, the attenuation coefficient is 0.2 dB/km and the nonlinearity coefficient is 1.3 W^{−1}km^{−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.

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.

The different curves presented in Figs. 3(a), 3(b) or 3(c) are associated with the contribution strength of each one of the three pump waves to the FWM component. The power of *Ā _{eq,FWM}* normalized to the optical carrier power of each channel is also shown as reference. The analysis is performed for a fiber length of 75 km and 100 km, and an optical carrier power at the fiber input of 12 dBm. Figures 3(a) and 3(c) show that the equivalent FWM frequency response of the FWM-wave generated at the optical frequency of channels

*a*and

*c*are similar. This similarity occurs because one of the optical channels that generate the degenerate FWM wave in the two situations is the same (channel

*b*) and the other channel (channel

*c*or channel

*a*, when the FWM wave is generated at the optical frequency of channel

*a*or channel

*c*, respectively) is symmetric with respect to the central channel, as shown in Fig. 2. Figures 3(a) and 3(c) show that the magnitude of the equivalent FWM transfer functions increase with frequency till a maximum. The maximum FWM contribution occurs for the modulation frequency at which the equivalent propagation constant mismatch (Δ

*β*(

_{eq,i}*f*) = Δ

*β*+ 2

*πfd*

_{1,i}), is zero. We note that the equivalent propagation constant mismatch comprises the influence of the walkoff and modulation frequency on the propagation constant mismatch between the pump and FWM waves. When the frequency spacing between one of the pump waves and the generated FWM wave increases, the frequency, where the maximum efficiency of FWM occurs, decreases due to the walkoff increase. On the other hand, the inspection of Fig. 3(b) shows that the term associated with the contribution of the modulated signal of channel

*b*(in this case, related with the equivalent FWM transfer function

*H*

_{eq,4}(

*L*,

*f*)) to the generation of the FWM component is frequency-independent. This is because the walkoff between the pump (channel

*b*) and the FWM wave is null as they are located at the same frequency. As a conclusion, Fig. 3 shows that the impact of the FWM on the performance of a OFDM signal is remarkably dependent on the modulation frequency due to the frequency dependence of the FWM efficiency induced by the walkoff effect. This dependence can lead to an increase of the amplitude response of the FWM effect as high as 15 dB.

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 *Ā _{eq,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 (Δ

*β*(

_{eq,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.

## 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 [8,29]. 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]. The simultaneous transmission of UWB band 2 (central frequency of 3.96 GHz) and 3 (central frequency of 4.49 GHz) is considered [8]. 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]. 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]. 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]), the FWM contributions obtained from the double or triple beat between the modulated signals (
$\overline{{A}_{i}}{a}_{j}{a}_{k}$ or *a _{i}a_{j}a_{k}*) 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.

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 (*f*_{1} = *f*_{4}) nondegenerate (*f*_{2} ≠ *f*_{3}) 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 *A*_{1}(*z*, *t*) and *A*_{4}(*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.

### 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.

### 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.

## 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[*a _{i}*(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:

## A.1. Nondegenerate FWM

In the nondegenerate case, the pump waves are located at different frequencies and are independent. In that situation, E[*a _{i}* (0,

*t*)

*a*(0,

_{j}*t*)] = E [

*a*(0,

_{i}*t*)] E[

*a*(0,

_{j}*t*)] = 0 for

*i*≠

*j*. Using this result and also the Wiener-Kintchine theorem, Eq. (27) is simplified to:

*Ā*|

_{eq,FWM}^{2}= 2|

*Ā*|

_{FWM}^{2}, and |

*H*(

_{eq,i}*L*,

*f*)|

^{2}= |

*H*(

_{i}*L*, −

*f*)|

^{2}+ |

*H*(

_{i}*L*,

*f*)|

^{2}.

*S*(

_{i}*f*) is the PSD of the OFDM signal

*a*(0,

_{i}*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 2

*π*(as the initial phases

*ϕ*), 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:

_{i}*P̄*is the optical carrier power at the fiber input and ${L}_{eff}=\frac{1-exp(-\alpha L)}{\alpha}$ is the effective interaction fiber length. The efficiency of the generated FWM wave,

*η*(

_{i}*f*), is given by:

*βL*[17, 20]. Equation (31) reveals also the impact of the walkoff on the FWM efficiency through the walkoff factor 2

*πfd*

_{1,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 |

*H*(

_{eq,i}*L*,

*f*)|

^{2}in Eq. (28), this dependence induces a maximum FWM efficiency that occurs at the modulation frequencies given by

*f*= ±Δ

_{c}*β*/(2

*πd*

_{1,i}).

## A.2. Degenerate FWM

In the degenerate case, two of the pumps that origin the FWM process are the same wave, i. e., *ν*_{2}=*ν*_{3} (or, equivalently, *a*_{2} (0, *t*)=*a*_{3} (0, *t*)) and *h _{eq}*

_{,2}(

*L*,

*t*)=

*h*

_{eq}_{,3}(

*L*,

*t*). In that situation, E[

*a*

_{2}(0,

*t*)

*a*

_{3}(0,

*t*)] ≠ 0 and Eq. (27) can be expressed as:

## 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:

*ν*

_{1}≠

*ν*

_{4}) nondegenerate (

*ν*

_{2}≠

*ν*

_{3}) 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[

*a*(0,

_{s}*t*)

*a*(0,

_{i}*t*)] = 0) and, consequently, the carrier-signal variance term expressed in Eq. (34) is null. In the symmetric (

*ν*

_{1}=

*ν*

_{4}) non-degenerate (

*ν*

_{2}≠

*ν*

_{3}) 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[

*a*(0,

_{s}*t*)

*a*

_{4}(0,

*t*)] ≠ 0 and E[

*a*(0,

_{s}*t*)

*a*

_{2}(0,

*t*)]=E [

*a*(0,

_{s}*t*)

*a*

_{3}(0,

*t*)] = 0 and, therefore, Eq. (34) can be simplified to:

*S*(

_{s}*f*) is the PSD (at the fiber input) of the OFDM signal transmitted in the channel where the FWM is being generated and

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

Considering that E[*a _{s}*(0,

*t*)

*a*(0,

_{s}*t*)

*a*(0,

_{i}*t*)] = 0, and that E [

*a*(0,

_{s}*t*)

*a*(0,

_{s}*t*)

*a*(0,

_{i}*t*)

*a*(0,

_{i}*t*)] and E[

*a*(0,

_{s}*t*)

*a*(0,

_{s}*t*)

*a*(0,

_{s}*t*)] are negligible compared with E [

*a*(0,

_{s}*t*)

*a*(0,

_{s}*t*)], the FWM variance contribution resulting from the signal-signal beat expressed in Eq. (13) can be approximated by:

## 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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