## Abstract

We provide an analytical study on the propagation effects of a directly modulated OOFDM signal through a dispersive fiber and subsequent photo-detection. The analysis includes the effects of the laser operation point and the interplay between chromatic dispersion and laser chirp. The final expression allows to understand the physics behind the transmission of a multi-carrier signal in the presence of residual frequency modulation and the description of the induced intermodulation distortion gives us a detailed insight into the diferent intermodulation products which impair the recovered signal at the receiver-end side. Numerical comparisons between transmission simulations results and those provided by evaluating the expression obtained are carried out for different laser operation points. Results obtained by changing the fiber length, laser parameters and using single mode fiber with negative and positive dispersion are calculated in order to demonstrate the validity and versatility of the theory provided in this paper. Therefore, a novel analytical formulation is presented as a versatile tool for the description and study of IM/DD OOFDM systems with variable design parameters.

© 2013 OSA

## 1. Introduction

Orthogonal frequency division multiplexing (OFDM) is a multi-carrier format widespread used in wireless communications and DSL applications [1, 2]. Since 2005 optical-OFDM (OOFDM) has also attracted lots of attention for a great variety of optical communication scenarios, from long-reach to short-reach optical links, or communications systems based on single mode, multimode and plastic optical fiber [3–8]. In the context of access networks, passive optical networks have emerged as a cost-efficient architecture to provide advanced and speed rate demanding communications applications. In this cost-sensitive scenario, OOFDM is a promising format/multiplexing technique, what has led to a high number of research results published by several investigation groups [9, 10]. Besides its versatility, the potentials of OOFDM have contributed as well to this interest: it may offer high spectral efficiencies by using high quadrature-amplitude modulation formats, robustness to frequency roll-off of transceiver electronics, dispersion equalization complexity easily scalable with transmission rate or subcarrier granularity for the provision/sharing of the services/resources, and the possibility of using subcarrier power allocation is also useful to combat, at some extent, the carrier suppression effect due to the transmission of an intensity modulated double sideband signal through a dispersive link at those frequencies where the attenuation is not too high. The proposed transceiver architecture varies for each scenario depending on the required trade-off between system performance/cost. Thus, we may find architectures which employ optical intensity or field modulation for the transmission of the data through the optical fiber. When the optical intensity is used, a simple pin photodetector is appropriate for the opto-electronic conversion to recover the transmitted information, what is called intensity modulated/directly detected (IM/DD) system. This variant is more appropriate for medium and short reach applications where the cost is of great concern, such as metro and access optical networks. In this sense, direct modulation of a laser is more convenient than external modulation to reduce the complexity of the system, and, thus, the cost. On the other hand, optical field modulation has been proposed in combination with both direct and coherent detection (CO-OOFDM) [10, 11]. CO-OOFDM represents the ultimate architecture in system performance, at the expense of a more sophisticated transceiver architecture.

Although impressive experimental results in terms of signal line rates in OOFDM systems with direct modulation (e.g. [10]), there is a lack of theoretical results which enlighten the physics behind the performance of such systems. We can find in the literature some IM/DD OOFDM analyses which study the system performance. However, they focus on the influence of the transceiver electronics design parameters but not on the direct modulation process and fiber transmission effects [13], or based on not general mathematical assumptions [14]. In [15], laser rate equations are considered to describe the directly modulated optical signal, but the resulting expression for the optical signal at the fiber input is based on a small-signal approach. In this paper, we present a theoretical analysis using a detailed mathematical treatment to study the effects of the laser nonlinearity and the global impact of the laser chirp and chromatic dispersion with regard to the system performance. The approach taken in this paper allows the evaluation of an IM/DD OOFDM single mode fiber link under different conditions and the separate computation of the different intermodulation distortion (IMD) penalties.

The contents of the paper are structured as follows: in Section II, the OOFDM system is described. Section III presents the mathematical formulation performed to obtain an analytical expression of the detected photocurrent. Section IV presents the results obtained by evaluating the previous expression and we compare them to those provided by commercial software simulations for different system parameter values with the aim of theoretical validation. Finally, in section V, we outline the conclusions and future work.

## 2. System description

The optical OFDM communication system is basically composed of an OFDM transmitter, the directly modulated laser (DML), the dispersive fiber link, a photodetection stage, and, finally, the OFDM receiver, as depicted in Fig. 1.

In the OOFDM system, the information binary data is mapped into M-QAM complex symbols and a block of the resulting complex symbols is fed to an IFFT processor. In order to obtain a real-valued signal at the output, the original information complex symbols and their corresponding conjugate values are arranged with hermitian symmetry at the IFFT input. The obtained real-valued discrete signal is composed of *N* subcarriers, each of them modulated by an information complex symbol. After parallel-to-serial conversion, a cyclic prefix is added to each OFDM symbol in order to combat inter-symbol interference due to transceiver electronics and optical fiber dispersion. The signal is interpolated, low-pass filtered and its amplitude swing is normalized to [−1, 1]. The analog OFDM signal may be expressed as:

*X*= |

_{k}*X*|

_{k}*e*

^{j·φXk}are the information complex symbols, Ω

*are the discrete value frequencies of which the OFDM is composed of and*

_{k}*N*its number. The value of the discrete frequency Ω

*is given by*

_{k}*k*·ΔΩ, where ΔΩ is the spacing angular frequency between consecutive subcarriers, determined by the particular OFDM parameters employed in the system. The analog OFDM signal is then scaled to yield a certain value of peak current and a dc value is added to operate the laser. The resulting signal,

*i*(

*t*), is fed into a distributed feedback laser to modulate the optical intensity.

The direct modulation of the laser is governed by the next equations [16, 17]:

*p*(

*t*),

*n*(

*t*) are the photon and carrier densities in the laser active region, respectively,

*ϕ*(

*t*) is the phase of the output optical signal, Γ is the confinement factor,

*v*is the group velocity,

_{g}*a*is the linear material gain coefficient,

_{g}*n*is the transparency carrier density,

_{t}*ε*is the nonlinear gain coefficient,

_{nl}*τ*is the photon lifetime,

_{p}*ζ*determines the fraction of spontaneous emission that is emitted into the fundamental mode of the laser,

*V*is the volume of the active region,

*i*(

*t*) is the driving current fed into the laser,

*e*is the electron charge,

*A*is the non radiative-recombination coefficient,

*B*is the radiative-recombination coefficient and

*C*is the Auger recombination coefficient. This DFB model takes into account a wide range of nonlinear effects such as longitudinal-mode spatial hole-burning, linear, and nonlinear carrier recombination and nonlinear gain. It is assumed that the laser chirp conversion to intensity after propagation dominates while the effects of laser phase noise are negligible [14,17]. The optical field at the output of the DML laser is given by:

*P*(

*t*) is the output optical intensity,

*ω*

_{0}is the laser central wavelength and

*ϕ*

_{1}(

*t*) is the small signal approximation of the time-varying phase determined by Eq. (2), given by:

*m*is the frequency modulation index at the k-th subcarrier and

_{k}*φ*

_{mk}its corresponding phase. Its value is determined by several laser parameters, such as the linewidth enhancement factor [18]. Higher order terms of the frequency chirp have thus not been included. The transfer response of the single-mode fiber is given by:

*β*(

*ω*) is the propagation constant of the fiber,

*β*

_{0}its value at

*ω*=

*ω*

_{0},

*β*

_{1}and

*β*

_{2}are its first and second derivatives evaluated at

*ω*

_{0}and

*L*is the length of the fiber.

*β*

_{2}is related to the dispersion parameter

*D*of the fiber through ${\beta}_{2}=-D\frac{{\lambda}_{0}^{2}}{2\pi \cdot c}$, where

*c*denotes the speed of light in vacuum. Therefore, the mathematical analysis performed is applicable to OOFDM systems operating in the third window where the first order dispersion parameter dominates and the coefficient of the second order dispersion is small.

Since the fiber link is considered as a linear medium, the output field can be calculated as the inverse Fourier Transform (FT) of the output field in the frequency domain:

*I*, in order to show the degradating effects due to laser nonlinearity, laser chirp, chromatic dispersion and square-law photodetection.

_{ph}## 3. Mathematical formulation

Our goal is to obtain a mathematical expression of Eq. (7), since it is a much more useful tool for the understanding of directly modulated OOFDM systems than the observation of the final results in numerical simulations. The laser modulating signal, *i*(*t*), is the result of scaling the signal *s*(*t*) in Eq. (1) to drive the laser with a certain peak current (Δ*i*) and adding a certain bias value for laser operation (*i*_{0}). Thus, it is given by:

*, and*

_{k}*m*is the modulation index which yields a determined value for Δ

*i*. As a result of the multitone modulation, multiple harmonics and intermodulation products appear at the output optical intensity. An analytic expression for the modulus of the optical field can be obtained by means of a perturbative analysis of Eq. (2) [19] which renders the first and second order optical intensity versus driving current transfer functions,

*H*

_{p}_{1}(Ω

*),*

_{k}*H*

_{p}_{2}(Ω

*),*

_{k}*H*

_{p}_{11}(Ω

*, Ω*

_{k}*). The explicit form of these are given in Appendix 1. As the reader can observe, the transfer functions are completely determined by the laser parameters involved in Eq. (2) and the steady state values of the carrier and photon densities.*

_{l}The optical intensity may thus be approximated by:

*p*·

_{k}*e*

^{j·φpk},

*p*

_{2}

*·*

_{k}*e*

^{j·φp2k},

*p*·

_{kl}*e*

^{j·φpkl}and

*p*

_{k_l}e^{j·φpk_l}are the first and second order complex coefficients of the intensity at frequencies Ω

*, 2Ω*

_{k}*, (Ω*

_{k}*+ Ω*

_{k}*) and (Ω*

_{l}*− Ω*

_{k}*), respectively, whose values are determined by the transfer responses*

_{l}*H*

_{p}_{1}(Ω

*),*

_{k}*H*

_{p}_{2}(Ω

*),*

_{k}*H*

_{p}_{11}(Ω

*, Ω*

_{k}*) and the input driving current:*

_{l}We remark that the mathematical approximation used to describe the optical OFDM signal, though the assumptions made, offers us a powerful tool to take into account a) the laser filtering effects on the intensity optical modulated signal through the transfer function *H _{p}*

_{1}, b) the laser chirp effects through Eq. (4), which includes both transient and adiabatic types of laser chirp, and c) the laser nonlinearity through the second order transfer responses

*H*

_{p}_{2}, and

*H*

_{p}_{11}.

Unlike the analysis in [14] and [15] where a small argument approximation was used to study the time-varying output phase from the laser, we use the Jacobi-Anger identity [20] to deal with the approximated output optical phase *ϕ*_{1}(*t*). Although the analysis of the optical signal propagation through the dispersive link becomes much more complicated, the exact mathematical treatment of laser chirp *ϕ*_{1}(*t*) offers us a greater accuracy of the laser chirp effects on the detected signal.

The field at the fiber output is calculated as Eq. (6). Given the fact that the magnitude of *P*_{0} is much higher than that of *P*_{1}(*t*), *P*_{2}(*t*) and *P*_{11}(*t*) because of the high PAPR of OFDM signals, it is reasonable to only consider the beats *P*_{0}x*P*_{0}, *P*_{0}x*P*_{1}, *P*_{0}x*P*_{2}, *P*_{0}x*P*_{11} and *P*_{1}x*P*_{1} for the calculation of the photocurrent as in Eq. (7). The resulting expression is simplified by using a theorem for the summation of Bessel functions [20], and the photocurrent is approximated by:

*φ*=

_{k}*φ*

_{mk}−

*φ*

_{pk}, ${\theta}_{k}=-\frac{{\beta}_{2}}{2}{\mathrm{\Omega}}_{\mathit{imp}}{\mathrm{\Omega}}_{k}L$, being ${\mathrm{\Omega}}_{\mathit{imp}}={\sum}_{k=1}^{N}{n}_{k}\cdot {\mathrm{\Omega}}_{k}$, and

*μ*= 2

_{k}*m*·

_{k}*sin*(

*θ*).

_{k}In Eq. (14) we can observe different components which arise as result of the beating of intensity components considered in Eq. (9) and those which come from the interplay between the laser phase modulation and the chromatic dispersion of the fiber. Firstly, the DC term, labeled as *T*0, due mainly to the electrical bias added for laser operation *i*_{0}. Secondly, the term which contains the useful part for the transmission of information, proportional to 2*ab* and labelled as *T*1. Thirdly, the components due to the second order laser nonlinearity, affected by the laser chirp and chromatic dispersion, with labels *T*2 and *T*3. Lastly, harmonic and intermodulation terms which are generated because of the imbalance caused by the chromatic dispersion, with labels *T*4 and *T*5. As expected, the terms which are generated by the beat between sideband components and the optical carrier are affected by the carrier suppression effect caused by chromatic dispersion.

The mathematical treatment performed to deal with the laser chirp using the Jacobi-Anger identity allows us to calculate any order of intermodulation which results of the conversion of the laser chirp to intensity. Effectively, the different intensity terms in Eq. (9) and transmitted through the dispersive link interfere with those due to the laser chirp, what causes nonlinear distortion in the detected photocurrent. The particular contribution of each term (*T*0, *T*1,...*T*5) at different frequencies is obtained by particularizing for different values of *n*_{1}, *n*_{2},...,*n _{N}*, being its order equals to
${\sum}_{i=1}^{N}\left|{n}_{i}\right|$[18]. As example, the contribution of

*T*0 at Ω = Ω

_{4}can be calculated for an order equals to 2 by imposing

*n*

_{2}= 2,

*n*= 0 with

_{k}*k*∈ [1, 3...

*N*], or

*n*

_{1}= 1,

*n*

_{3}= 1,

*n*= 0 with

_{k}*k*∈ [2, 4, 5,...

*N*], or

*n*

_{5}= 1,

*n*

_{1}= −1,

*n*= 0 with

_{k}*k*∈ [2, 3, 4, 6, 7,...

*N*]. In general, we refer to these type of harmonic and intermodulation distortion as 2

*A*,

*A*+

*B*and

*A*−

*B*, respectively, since the concrete values of each possible combination of values for the indices

*n*

_{1},

*n*

_{2},...,

*n*must be different to fall at the different discrete frequencies Ω

_{N}_{1}, Ω

_{2}, ...,Ω

*. Proceeding in this way, we can calculate the contribution of*

_{N}*T*0,

*T*1,...

*T*5 for different combinations of values for

*n*

_{1},

*n*

_{2},...,

*n*and, thus, higher orders of intermodulation (which can be referred in general as 3

_{N}*A*,

*A*+

*B*+

*C*,

*A*+

*B*−

*C*, 4

*A*,..).

Equation (14), together with the definitions given by Eq. (10) and the transfer function expressions provided by Eq. (17) constitute the central result of this paper, that is, a completely analytical method to describe the dispersive propagation of directly modulated/detected OOFDM systems. As compared to standard numerical techniques based on the solution of Eq. (2), our model offers two main advantages. First of all, faster computation speed as it only requires the evaluation of closed formulas instead of numerical algorithms. Secondly, and perhaps more important, it provides the possibility of gaining a deeper insight into the impact that the different design parameters have on the performance of directly modulated/detected OOFDM systems. This knowledge is very useful to propose and design impairment pre/post compensation techniques, such as, for instance, those focused on nonlinear distortion cancellation techniques [15]. It also may be used for optimization procedures of system parameters in sensitivity studies in order to get more efficient designs and higher transmission capacities. We thus believe that the model provided is a step forward in order to obtain the maximum potential of directly modulated/detected OOFDM systems.

Given the complex derivation procedure of the analytical formulation, one is left with the uneasy question of whether it may be correct and accurate. In order to validate it we have compared the results provided by Eq. (14), with those provided by commercial simulation packages based on the numerical integration of the laser rate equations.

Finally, we would like to point out that although Eq. (14) has been derived for the transmission of an OFDM signal with no RF upconversion, the model can be easily adapted, as shown in Appendix 2, to cover the case of RF upconversion which is useful for avoiding low-frequency intermodulation products.

## 4. Numerical results

#### 4.1. Model parameters

For the OOFDM system in Fig. 1, where the complex information symbols are arranged with hermitian symmetry at the IFFT input, the electrical available bandwidth BW is divided into *FS*/2, being *FS* the size of the (I)FFT, yielding a certain value of spacing angular frequency between subcarriers, ΔΩ. Subcarrier at DC-frequency is set to zero. A certain number of subcarriers at highest frequencies are null too in order to avoid filtering border effects. The remaining subcarriers are used to transmit QAM information symbols. The samples are serialized and cyclic prefix and postfix are added to the discrete OFDM symbol (*T _{pre}* and

*T*expressed as fraction of the duration of the original OFDM symbol). The total duration of the prefix plus the postfix is 25% of the original OFDM symbol. An interpolation by a factor of 16 is employed to mimic the digital-to-analog conversion. Once the amplitude of the signal is normalized to [−1, 1], the signal is accommodated to modulate the optical intensity of a laser. The values of the laser parameters are listed in Table 1. The dispersion of the fiber,

_{pos}*D*, is set to 17

*ps*/(

*nm*·

*km*). The responsitivity of the photodetector, , is equal to 1. Low-pass photodiode filtering effects are not included in simulations.

#### 4.2. Obtained received constellations

In order to demonstrate the accuracy and versatility of the theory provided in this paper to describe IM/DD OOFDM systems, numerical results obtained by evaluating the analytical expression (14) have been obtained for different system parameter values. Terms *T*0 and *T*1 of Eq. (14) are evaluated up to an order of intermodulation,
${\sum}_{i=1}^{N}\left|{n}_{i}\right|$, equals to 3, whilst the rest of terms are evaluated up to an order of 2. With the aim of validation, the OOFDM system has been also simulated by using Matlab (version R2011b) and VPI (version 8.7) commercial software: the electrical OFDM signal is generated by using Matlab, whereas direct laser modulation, optical fiber propagation and intensity detection are carried out in VPI; the resulting photocurrent is translated back into Matlab for subsequent receiver signal processing. In Fig. 2 the received 4-QAM complex symbols before equalization obtained by the evaluation of Eq. (14) and through the simulation of the IM/DD OOFDM system are shown. We have chosen to show the complex symbols before equalization to demonstrate how accurately the analytical model can follow the physics behind the whole process of transmission and direct detection of directly modulated OOFDM signals. Together with the graphical representations, an averaged value of the next metric:

*k*is the number of subcarrier,

*Y*and

_{rec}*Y*are the complex symbols before equalization from the numerical and the analytical models, respectively, and 〈·〉 is the averaging operator.

_{theo}In general, we can observe a very good agreement between the results obtained by simulations and those obtained by evaluation of Eq. (14). The developed theory is able to accurately describe the effects of the accumulated dispersion, as we can see when changing the fiber length, e.g. in Fig. 2(a) and 2(b). Constellation Fig. 2(a) is that obtained for a back-to-back configuration, and thus the only intermodulation distortion is due to laser nonlinearities (which have been theoretically considered through the terms *P*_{2}(*t*) and *P*_{11}(*t*) in Eq. (11)).

The operation point of the laser and the modulation index of the driving signal have important effects on the signal recovered at the receiver. Bias point operation and/or amplitude swing of the laser driving signal have been changed in Fig. 2(c), 2(d), 2(e) and 2(h). A variation of the bias point induces a change of the dynamics of the laser because of the different value of the steady-state photon density value, whereas a higher value of the modulation index yields a stronger nonlinearity impact. We have also changed the bandwidth of the signal, and the theory provided is also able to take into account the different filtering effects due to the direct modulation laser as result of a higher occupied bandwidth (constellation in Fig. 2(d)). In order to check whether the theory is able to describe more intensive laser frequency modulation effects, the linewidth enhancement factor is set to 10 in constellation Fig. 2(f), which confirms the validity of the expression for such conditions.

The analytical formulation is also applicable to OOFDM signals with different number of tones, since the theory calculates accurately the different intermodulation combinations as we can observe when *N* is changed from a value equal to 14 in Fig. 2(a)–2(f) to a value of 55 in Fig. 2(g)–2(h). The nonlinear gain coefficient of the laser is also changed to a value of 3×10^{−23} m^{3} in Fig. 2(h). This parameter determines the gain saturation of the laser, changing the value of the intensity modulation transfer function at the resonant frequency and having a large impact on the modulation bandwidth of the laser. We can observe that the evaluation of Eq. (14) still gives accurate results when internal laser parameters are changed.

Regarding the numerical deviation between simulation and analytical results, Fig. 2(a), 2(b), 2(c), 2(g) and 2(h) show an error below 1%. On the other hand, for the conditions in Fig. 2(d), 2(e) and 2(f) the error values are higher, concretely 7.6%, 8.75% and 13.13%, respectively. This coincides with the increase on the value of *BW* to 5.5GHz for a (I)FFT size of 32, and is due to the effects of inter-symbol and inter-carrier interference arising from the linear convolution when the OFDM signal passes through the whole communication system. It is therefore important to be aware on the effects of these impairments and consider the effects in combination with the theoretical expression Eq. (14). Nevertheless, when the number of tones is increased, and, thus, the absolute duration of the guard interval, this effect is reduced and low error values are obtained, as it can be seen in Fig. 2(g) and 2(h).

The theory provided is also flexible on the type of modulation format employed. Figure 3 shows the constellations of the 32-QAM symbols received before equalization and the averaged value of *Err _{y}* for IM/DD OOFDM systems with different nonlinear gain coefficient and fiber dispersion values.

Once again we obtain low error values, demonstrating the great accuracy of the theory provided in Section 3. The theoretical results obtained by evaluation of Eq. (14) under different working conditions in this paper show a good agreement with the simulations obtained by Matlab and VPI, what validates the analytical formulation presented in this paper.

#### 4.3. Carrier to interference power ratio

As a result of the harmonic and intermodulation distortion, the information transmitted on each subcarrier of the OOFDM signal is impaired at a different level depending on its frequency value. Carrier to interference power ratio is a useful metric to compare the power of the useful part of the distorted signal with the power of the interference:

At the presence of weak nonlinearity, the interference power is small and this ratio achieves high values, what means that the BER performance (if not impaired by other sources of degradation, such as noise) is high.

In order to further compare the results obtained with the evaluation of Eq. (14), we calculate the power ratio between the information and the intermodulation and harmonic distortion terms in Eq. (14), for all possible combinations up to an order (
${\sum}_{i=1}^{N}\left|{n}_{i}\right|$) equals to 3 (2*A*, *A* ± *B*, *A* ± *B* ± *C*, 2*A* ± *B*,...) for T0 and T1 and up to an order equals to 2 (2*A* and *A* ± *B*) for T2, T3, T4 and T5.

The system conditions in the simulations were *BW* =5.5GHz, *FS*=128, *N*=55, *T _{pre}*=

*T*=0.125,

_{pos}*i*

_{0}=60mA, Δ

*i*=0.01mA,

*α*= 4 and 4-QAM as modulation format. The theoretical expression allows us to separately calculate the different contributions to total intermodulation. The corresponding carrier to interference power ratio are evaluated following Eq. (14) and plotted in Fig. 4, where the total carrier to interference ratio, according to simulations, are also presented with the aim of validation.

We can observe that the total carrier to interference ratio of both the simulated and numerical models agree for different distances. Because of the influence of laser chirp and chromatic dispersion, a spectral null occurs at certain frequencies for distances equal to 40, 60 and 80 kilometers, and a significant reduction of the carrier to interference power ratio occurs as result. The separate calculation of different intermodulation products allows us to understand their contribution to the total nonlinear distortion, as well as its source. For these particular conditions, the second order modulation is the dominant source of intermodulation distortion, whilst the third order intermodulation distortion is negligible, except the product A−B+C of *T*1 which is significant. As we can see, *T*1, apart from containing the useful part for the information transmission, yields intermodulation distortion in presence of laser chirp and chromatic dispersion. The intermodulation products A+B and A−B of *T*1 are the most impairing distortion products. We can also observe that as the distance increases, the distortion imposed by *T*0 increases, which is due exclusively to the beat between different phase-to-intensity converted components. The high power allocated on the optical carrier, given by the *a* factor in Eq. (12), yields a significant value of *T*0, and thus the interplay between laser phase modulation and chromatic dispersion may degrade considerably the signal quality. Laser nonlinearity, represented by *T*3, is, in this case, more important for short distances. For a distance of 20km, the second intermodulation distortion *A* + *B* and *A* − *B* caused by *T*3 reach close values to the total intermodulation curve, but other terms get more important as the distance is increased.

#### 4.4. Bit-error-rate

Once we have an estimation of the global impact of the intermodulation and harmonic distortion in the directly modulated OOFDM signal transmitted through a dispersive optical fiber, the effective signal-to-noise ratio can be calculated, considering also thermal and shot noises at the receiver. The analytical BER for a particular QAM modulation format is then directly determined [13]. We compare in this section the BER obtained by simulation of the system depicted in Fig. 1, and the BER obtained using expression (14) to estimate the signal and interference powers. The equation employed for BER determination can be consulted in [21]. The system parameters values are: *BW* =5.5GHz, *FS*=128, *N*=55, *T _{pre}*=

*T*=0.125,

_{pos}*i*

_{0}=60mA, Δ

*i*=0.01mA,

*α*= 4 and 32-QAM as modulation format. For BER computation, we need to introduce hard clipping to maintain constant the optical modulation efficiency. We have employed a clipping ratio of 13.8 dB. At the receiver-end side, the optical signal is attenuated, and thermal and shot noises are added to the detected photocurrent. The spectral density of the thermal noise is ${10}^{-12}A/\sqrt{Hz}$ The BER has been calculated for distances equal to 10 km, 20 km, 30 km and 40 km, and for both positive and negative values of dispersion. The results are shown in Fig. 5.

We can observe that the signal quality in terms of BER calculated by means of Eq. (14) fits perfectly with the simulation results. The BER obtained with negative dispersion fiber is lower than that obtained with standard dispersion fiber, according to previously reported results [22]. The BER for positive dispersion fiber tends to increase with distance due to a higher accumulated dispersion, and thus, a greater intermodulation distortion and a greater carrier suppression effect due to chromatic dispersion. Similarly to what happens with propagation of chirped gaussian pulses through negative dispersion fibers, the BER of an OOFDM signal reduces with distance. In this case, the smaller BER is well explained by a higher effective signal-to-noise ratio of each OOFDM subcarrier. Both phenomena are well described by the theory provided, and the system performance is perfectly determined with Eq. (14).

## 5. Conclusions

We theoretically investigated the effects of laser direct modulation and chromatic dispersion in an IM/DD OOFDM system. The laser was modelled using a second order approximation for the optical intensity, and the time-varying output optical phase was approached by a first order approximation. The effects of the fiber chromatic dispersion have been also included in order to obtain a detailed expression of the photocurrent at the receiver side, what represents a helpful tool for the design of IM/DD OOFDM systems. The results obtained by using the theory derived in this study were validated with software numerical simulations. First, as we have seen by comparing the constellations of the symbols received, the theory is able to describe IM/DD OOFDM systems with different transmission conditions. Secondly, the total intermodulation distortion along the information signal frequency band has been also compared, showing very good agreement. In the results shown in this paper, second order intermodulation distortion caused by the beat between the information signal and phase-to-intensity converted components have an important degradating effect on the quality signal. The analysis and penalty quantification of the separate intermodulation distortion products allow the design of techniques for its overcoming. Lastly, we compared the information transmission quality in terms of bit-error-rate obtained by simulations and by using the derived theoretical expression, obtaining good results for different distances. Consequently, we demonstrate the validity of the analytical formulation presented in this paper to describe the system performance of IM/DD OOFDM systems. Finally, the model presented here may be suitable to describe OOFDM systems with different transceiver architectures due to the rigorous description of the transmitted optical signal magnitudes, intensity and phase, and subsequent treatment of transmission and detection effects. The study of the effects of system design parameters, such as clipping in the transmitter, not sufficiently length of cyclic pre/posfix or DML optimization is suggested as future work. The improvement of the analytical formulation through the consideration of more sophisticated system component models is also a possible field of future study.

## Appendix 1: Laser output power/ input current transfer functions

In order to find the linear laser transfer functions *H _{p}*

_{1}(Ω

*) and the nonlinear laser transfer functions*

_{k}*H*

_{p}_{2}(Ω

*),*

_{k}*H*

_{p}_{11}(Ω

*, Ω*

_{k}*), we perform a perturbative analysis of the rate equations in Eq. (2), which follows the general procedure presented in [19]. The resulting expressions for*

_{l}*H*

_{p}_{1}(Ω

*),*

_{k}*H*

_{p}_{2}(Ω

*),*

_{k}*H*

_{p}_{11}(Ω

*, Ω*

_{k}*) are:*

_{l}*n*

_{0}and

*p*

_{0}the carrier and photon density steady-state values, respectively, and

*C*the photon-to-optical intensity conversion factor.

_{p}## Appendix 2: Photocurrent for frequency up-converted signal

In the case frequency up-conversion is performed, the quadrature and in-phase components of the complex baseband signal can be transmitted by using an IQ modulator. The baseband complex OFDM signal is given by:

*k*has changed, but the definition of the discrete Ω

*remains the same, Ω*

_{k}*=*

_{k}*k*·ΔΩ. The laser driving modulating signal after up-conversion at frequency Ω

*can be expressed as:*

_{rf}*m*is the frequency modulation index at Ω_{k}+ Ω_{rf}and_{k}*φ*its phase._{mk}*p*·_{k}*e*^{j·φpk},*p*_{2}·_{k}*e*^{j·φp2k},*p*·_{kl}*e*^{j·φpkl}and*p*_{k_l}e^{j·φpk_l}are the first and second order complex coefficients of the intensity at frequencies Ω+ Ω_{rf}, 2(Ω_{k}+Ω_{rf}), (Ω_{k}+Ω_{rf}+Ω_{k}+Ω_{rf}) and (Ω_{l}+ Ω_{rf}− Ω_{k}− Ω_{rf}), respectively._{l}- ${\mathrm{\Omega}}_{\mathit{imp}}={\sum}_{k=-N/2}^{N/2-1}{n}_{k}\cdot \left({\mathrm{\Omega}}_{rf}+{\mathrm{\Omega}}_{k}\right)$ and ${\theta}_{k}=-\frac{{\beta}_{2}}{2}{\mathrm{\Omega}}_{\mathit{imp}}\left({\mathrm{\Omega}}_{rf}+{\mathrm{\Omega}}_{k}\right)L$.

## References and links

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