Optical filtering in directly modulated/detected OOFDM systems

C. Sánchez; B. Ortega; J. L. Wei; J. Capmany

doi:10.1364/OE.21.030591

1. Introduction

The increasing demand for higher transmission information capacities has fuelled the research towards more efficient technologies and solutions to accommodate them, ranging from more efficient optical devices to advanced coding techniques and modulation formats [1].

Advanced modulation formats offer higher spectral efficiencies as compared to traditional intensity format on-off keying and better robustness to channel impairments. These features are very attractive in a cost-sensitive scenario such as access and metro networks, where the use of bandwidth-limited optoelectronics devices is a requisite for cost-effective operation. Among the available modulation techniques, orthogonal frequency division multiplexed (OFDM) features an effective use of the available bandwidth by a judicious combination of high order quadrature amplitude modulation (QAM) formats and subcarrier granularity [2, 3].

The direct modulation of the laser in the OFDM transmitter brings additional advantages in terms of cost reduction, compactness, low power consumption and high optical output power. Nevertheless, its performance is severely limited by distortions arising from the laser and photodetector nonlinearities and the propagation of the chirped signal through the dispersive link connecting the central office and the optical network unit. To overcome these one can rely, for instance, on electronic techniques for the cancellation of the nonlinear distortions [4, 5], or use an optical filter to improve the system performance by suitable conversion of the inherent frequency modulation at the laser transmitter output into additional intensity modulation at the end of the link [6].

The optical filtering technique is only qualitatively understood and further work is necessary to grasp the theoretical foundations which can provide the design criteria for optimum operation. For instance a 7dB improvement of power budget has been reported in [7] but neither a thorough explanation for this improvement nor the directions for further exploit the potential of this technique are well understood [8].

This work addresses precisely this point by providing an end-to-end analytical model to describe the operation of a directly modulated/detected (DM/DD) Optical OFDM (OOFDM) system when an arbitrary optical filter is inserted in the dispersive link. This is done by extending the analysis presented in [9] for the description of DM/DD OOFDM systems. The paper is structured as follows: in Section 2, we provide a theoretical description of the optically filtered OOFDM system which includes the derivation of the mathematical formulae. We also describe mathematically the transfer functions of the different optical filtering structures which are later employed in the simulation and analytical results. Section 3 presents the results obtained by evaluating the expressions derived and those obtained through numerical simulations for the sake of validation. We then illustrate the optimization of the filter parameters by computing the effective signal-to-noise ratio and the power budget improvements are evaluated, paying special attention to the impact of the clipping ratio. Finally, in Section 4, we outline the conclusions and remaining future work.

2. Optically filtered DM/DD multicarrier signals: theory

2.1. Analytical formulation

The optically filtered DM/DD OOFDM system considered in this paper consists of an OFDM transmitter, the directly modulated laser (DML), the dispersive fiber link, the optical filter, the photodetection stage, and, finally, the OFDM receiver, as shown in Fig. 1.

Fig. 1 Schematic illustration of the simulated OOFDM system.

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The main operations in the OFDM transmitter include data mapping using 16-QAM, inverse fast Fourier transform (IFFT), cyclic prefix insertion, OFDM symbol serialization, clipping and digital-to-analog conversion (DAC). The generated analog OFDM signal at the transmitter is given by:

s (t) \propto \sum_{k = 1}^{N} | X_{k} | \cdot \cos (Ω_{k} t + φ_{X_{k}})

where X_k = |X_k|e^j·φ_{X_k} are the information complex symbols, Ω_k is the angular frequency of the kth subcarrier and N the number of subcarriers the OFDM signal is composed of.

The analog OFDM signal is then scaled by a factor m to operate the laser within a certain amplitude range, Δi, and a dc value, i₀, is added to operate the optical source. The laser driving OFDM signal is given by:

i (t) = i_{0} + \sum_{k = 1}^{N} 2 i_{k} \cdot \cos (Ω_{k} t + φ_{i_{k}})

where

i_{k} \cdot \exp (j φ_{i_{k}}) = \frac{m}{2} \cdot | X_{k} | \cdot \exp (j φ_{X_{k}})

is the driving current coefficient at frequency Ω_k.

The DML model, reported in [9], takes into account a wide range of nonlinear effects such as longitudinal-mode spatial hole-burning, linear, nonlinear carrier recombination and nonlinear gain. In order to derive a mathemathematical expression of the laser output signal which enables us to continue with the analytical study of the system, we carry out a perturbative analysis of the equations governing the photon and carrier densities, as well as the output optical phase. Since the direct modulation of the laser is a nonlinear process, in order to obtain an accurate approximation the perturbative analysis accounts for the second order nonlinear distortion. The relationship between the laser driving signal i(t) and the output optical signal can be expressed in function of the first order laser intensity and phase transfer functions (H_p₁ (Ω_k) and H_ϕ₁ (Ω_k), respectively) as well as the second order laser intensity and phase transfer functions (H_p₂ (Ω_k), H_p₁₁ (Ω_k, Ω_l) and H_ϕ₂ (Ω_k), H_ϕ₁₁ (Ω_k, Ω_l), respectively). The output optical signal is:

E (t) = \sqrt{P (t)} \cdot \exp (j \cdot ϕ (t)) \cdot \exp (j \cdot ω_{0} t)

The optical intensity is given by P(t) = P₀+P₁(t)+P₂(t)+P₁₁(t) where P₀ is the average optical power and

\begin{array}{l} P_{1} (t) = \sum_{k = 1}^{N} 2 p_{k} \cdot \cos (Ω_{k} t + φ_{p_{k}}); P_{2} (t) = \sum_{k = 1}^{N} 2 p_{2 k} \cdot \cos (2 Ω_{k} t + φ_{p_{2 k}}) \\ P_{11} (t) = \sum_{k = 1}^{N} \sum_{l = 1}^{k - 1} 2 p_{k l} \cdot \cos ((Ω_{k} + Ω_{l}) t + φ_{p_{k l}}) + \sum_{k = 1}^{N} \sum_{l = 1}^{k - 1} 2 p_{k_l} \cdot \cos (((Ω_{k} - Ω_{l}) + φ_{p_{k_l}})) \end{array}

being p_k · exp(jφ_{p_k}) the first order complex coefficient and p₂_k ·exp(jφ_{p_2k}), p_kl · exp(jφ_{p_kl}), p_{k_l} · exp(jφ_{p_{k_l}}) the second order complex coefficients of the optical intensity, given by:

\begin{matrix} p_{k} \cdot \exp (j \cdot φ_{p_{k}}) = H_{p 1} (Ω_{k}) \cdot i_{k} \cdot \exp (j \cdot φ_{i_{k}}) \\ p_{2 k} \cdot \exp (φ_{p_{2 k}}) = H_{p 2} (Ω_{k}) \cdot i_{k}^{2} \cdot \exp (j \cdot 2 \cdot φ_{i_{k}}) \\ p_{k l} \cdot \exp (j \cdot φ_{p_{k l}}) = H_{p 11} ((Ω_{k}, Ω_{l})) \cdot i_{k} \cdot i_{l} \cdot \exp (j \cdot (φ_{i_{k}} + φ_{i_{l}})) \\ p_{k_l} \cdot \exp (j \cdot φ_{p_{k_l}}) = H_{p 11} ((Ω_{k}, - Ω_{l})) \cdot i_{k} \cdot i_{l} \cdot \exp (j \cdot (φ_{i_{k}} - φ_{i_{l}})) \end{matrix}

The output optical phase ϕ(t) is given by:

ϕ (t) = \sum_{k = 1}^{N} m_{k} \cdot \sin (Ω_{k} t + φ_{m_{k}})

being m_k the frequency modulation index for the kth subcarrier and φ_{m_k} its corresponding phase, calculated as [10]:

\begin{array}{l} m_{k} \cdot \exp (j \cdot φ_{m_{k}}) = 2 j \cdot H_{ϕ_{1}} (Ω_{k}) \cdot i_{k} \cdot \exp (j \cdot φ_{i_{k}}) + \\ 2 j \cdot H_{ϕ_{2}} (Ω_{r}) \cdot i_{r}^{2} \cdot \exp (j \cdot 2 \cdot φ_{i_{r}}) + \\ 2 j \cdot H_{ϕ_{11}} (Ω_{m}, Ω_{n}) \cdot i_{m} \cdot i_{n} \cdot \exp (j \cdot (φ_{i_{m}} + φ_{i_{n}})) + \\ 2 j \cdot H_{ϕ_{11}} (Ω_{p}, Ω_{q}) \cdot i_{p} \cdot i_{q} \cdot \exp (j \cdot (φ_{i_{p}} - φ_{i_{q}})) \end{array}

Please note that we have included in the definition of m_k · exp(j · φ_{m_k}) the second order harmonic and intermodulation distortion of the laser chirp in order to get a more accurate model than that obtained by approximating the laser chirp with a linear function of the output optical power [9, 11]. In order to study the effects of the fiber dispersion and the optical filtering onto the optical signal, we need to get an approximation of the optical field. By using $\sqrt{1 + x} \approx 1 + \frac{x}{2} - \frac{x^{2}}{8}$ , the complex electrical field at the output of the laser source is approximated as:

E (t, z = 0) = (\sqrt{P_{0}} + \frac{P_{1} (t)}{2 \sqrt{P_{0}}} + \frac{P_{2} (t)}{2 \sqrt{P_{0}}} + \frac{P_{11} (t)}{2 \sqrt{P_{0}}} - \frac{P_{1} (t) \cdot P_{1} (t)}{8 P_{0}^{\frac{3}{2}}}) \cdot \exp (j \cdot \sum_{k = 1}^{N} m_{k} \cdot \sin (Ω_{k} t + φ_{m_{k}}))

Then, the OOFDM signal propagates through a single-mode fiber span of length L, which transfer function is given by:

H_{fib} (ω) = \exp (- j \cdot β (ω) \cdot L) = \exp (- j \cdot (β_{0} + β_{1} \cdot (ω - ω_{0}) + \frac{1}{2} β_{2} \cdot {(ω - ω_{0})}^{2}) \cdot L)

where β(ω) is the propagation constant of the fiber, β₀ its value at ω = ω₀, β₁ and β₂ are its first and second derivatives evaluated at ω₀. β₂ is related to the dispersion parameter D of the fiber through

β_{2} = - D \frac{λ_{0}^{2}}{2 π \cdot c}

, where c denotes the speed of light in vacuum. The field at the output of the fiber can be easily calculated through the inverse Fourier transform of the product of the spectra of the signal in Eq. (8) and the fiber transfer function in Eq. (9):

\begin{array}{l} E (t, z = L) = {FT}^{- 1} {E (Ω, z = 0) \cdot H_{fib} (Ω = ω - ω_{0})} = \\ \sum_{n_{1}, \dots, n_{N} = - \infty}^{\infty} E (n_{1}, \dots, n_{N}, z = L) \cdot e^{j (- β_{0} \cdot L + Ω_{imp} \cdot (t - β_{1} \cdot L) + \sum_{i = 1}^{N} n_{i} \cdot φ_{m_{i}} - \frac{β_{2}}{2} Ω_{imp}^{2} \cdot L)} \end{array}

being

Ω_{imp} = \sum_{k = 1}^{N} n_{k} \cdot Ω_{k}

. Note that the constant phase shift e^−jβ₀·L is cancelled out upon square-law detection and the time shift β₁·L is not of importance under proper time synchronization at the receiver, such that t′ = t − β₁·L. The effects of the optical filtering onto the optical signal can be easily taken into account using the well-known digital filtering theory [12, 13]. The impulse response and corresponding transfer function of the optical filter are given by:

h_{fil} (t^{'}) = \sum_{κ = 0}^{Ord - 1} h_{κ} \cdot δ (t^{'} - τ_{κ}) \overset{FT}{\leftrightarrow} H_{fil} (Ω) = \sum_{κ = 0}^{Ord - 1} h_{κ} \cdot e^{- j \cdot τ_{κ} \cdot Ω}

where Ord is the order of the filter. Equation (11) is obviously valid for optical filters with finite impulse responses, and gives reasonably accurate results in the case of optical filters with infinite impulse responses provided that the value of Ord is set to a sufficiently high value. The field at its output is calculated as

E_{fil} (t^{'}, z = L) = \sum_{κ = 0}^{Ord - 1} h_{κ} \cdot E (t^{'} - τ_{κ}, z = L)

Finally, the photocurrent is calculated as the squared modulus of the field

I_{ph} (t^{'}) = ℜ \cdot {| E_{fil} (t^{'}, z = L) |}^{2}

being ℜ the responsivity of the photodetector. For the sake of simplification, we only consider the most relevant beats and we use the Graf’s theorem for the sum of Bessel functions [14]. After a lengthy mathematical manipulation, I_ph(t′) can be expressed as (see Appendix I):

\begin{array}{l} I_{ph} (t^{'}) \approx ℜ \cdot \sum_{κ = 0}^{Ord - 1} \sum_{ε = 0}^{Ord - 1} h_{κ} h_{ε}^{*} \cdot \sum_{n_{1} \dots n_{N} = - \infty}^{\infty} (T 0 (n_{1} \dots n_{N}, κ, ε) + T 1 (n_{1} \dots n_{N}, κ, ε) + \\ T 2 (n_{1} \dots n_{N}, κ, ε) + T 3 (n_{1} \dots n_{N}, κ, ε) + T 4 (n_{1} \dots n_{N}, κ, ε) + T 5 (n_{1} \dots n_{N}, κ, ε)) \cdot \\ \exp (j (Ω_{imp} t^{'} + \sum_{i = 1}^{N} n_{i} (\frac{τ_{κ} + τ_{ε}}{2} + φ_{m_{i}} + \frac{π}{2}))) \end{array}

T0 and T1 contain the information component, which can be extracted by setting one of the indices n₁, n₂,...n_N to 1, and the rest to 0, as well as nonlinear distortion due to the laser chirp, the expressions of which are obtained by particularizing the indices n₁, n₂,...n_N such that

\sum_{1}^{N} | n_{k} | > 1

. T2, T3, T4 and T5 are essentially terms due to nonlinear distortion which stem from the laser nonlinearities (T2 and T3) and the imbalance caused by the chromatic dispersion and optical filtering on the optical field (T4 and T5) [9]. The expressions for T0,...T5 can be found in Appendix I.

As we can see in Appendix I, the dispersion-induced phase delay at Ω_r, equals to $- β_{2} / 2 \cdot Ω_{r}^{2} \cdot L$ , is increased by that due to the optical filter to result in $θ_{r, κ, ε} = - (\frac{τ_{κ} - τ_{ε}}{2} + \frac{β_{2}}{2} Ω_{r}^{2} L)$ . Moreover, from Eq. (14), the different spectra components are weighted by the product of the filter coefficients $h_{κ} \cdot h_{ε}^{*}$ and are also affected by the average phase delay (τ_κ + τ_ε)/2.

Because of the complexity of the expressions provided, we have obtained a simplified version which gives reasonable good results: provided that the argument of the Bessel functions is sufficiently small, they can be approximated by the first term of their series expansion [14]; besides, within each sum, only the most significant contribution at the rth subcarrier is retained. With these simplifications and making use of the transfer function of the optical filter Eq. (11), the expression for the signal information component can be expressed as:

\begin{array}{l} {T 0 |}_{n_{r} = 1} + {T 1 |}_{n_{r} = 1} = \\ (\frac{H_{fil}^{*} (0) \cdot H_{fil} (Ω_{r})}{2} (H_{p 1} (Ω_{r}) \cdot e^{- j \cdot Ω_{r}^{2} \frac{β_{2}}{2} L} + 2 \cdot P_{0} \cdot H_{ϕ_{1}} (Ω_{k}) \cdot e^{- j \cdot Ω_{r}^{2} \frac{β_{2}}{2} L}) + \\ \frac{H_{fil} (0) \cdot H_{fil}^{*} (- Ω_{r})}{2} (H_{p 1} (Ω_{r}) \cdot e^{j \cdot Ω_{r}^{2} \frac{β_{2}}{2} L} - 2 \cdot P_{0} \cdot H_{ϕ_{1}} (Ω_{k}) \cdot e^{j \cdot Ω_{r}^{2} \frac{β_{2}}{2} L})) \cdot \frac{m}{2} \cdot X_{r} \end{array}

whilst the expressions of the second order intermodulation distortion are shown in Appendix II. From expression Eq. (15) we note that:

Since we have only considered the photocurrent terms due to the beat with the optical carrier, the photocurrent is null when the optical carrier vanishes, which occurs when H_fil(0) = 0.
When no optical filter is used (H_fil(Ω) = 1), the expression reduces to that reported in [9]. Similar results are obtained when a symmetrical filtering with respect to the optical carrier (H_fil(Ω) = H_fil(−Ω)) is employed.
In the case some asymmetry is introduced, information components that otherwise would be counteracted, start to spring up, increasing thus the signal information detected at the receiver and the system performance. This is the working principle of optical filtering techniques aimed to increase the system performance.
Note that we have not taken into account inter-carrier interference (ICI) and inter-symbol interfence (ISI) effects due to the impulse response of the channel once the optical filter is inserted. The previous analysis can not account for these effects, since it begins from the assumption that the RF-waves in Eq. (2) are non-finite. However, ICI and ISI effects can be easily quantified once the linear channel transfer function of the whole system Eq. (15) is obtained and can be used with the equations reported in [10].

2.2. Optical filtering approaches

2.2.1. Supergaussian filter

Optical filters with Supergaussian profiles have been studied in [7, 8] and [15]. The transfer function modulus is given by:

| H_{fil} (Ω = ω - ω_{0}) | = \exp (\ln (\frac{1}{\sqrt{2}}) \cdot {(\frac{Ω - Ω_{c}}{(B W_{3 d B} / 2)})}^{2 \cdot q})

where Ω_c is the central angular frequency, BW₃_dB is the 3-dB bandwidth and q is the order of the Supergaussian filter. Its phase response is approximated by the finite summation [8]:

ϕ_{fil}^{*} (ω_{i}) = \frac{Δ ω}{π} \sum_{r = - R}^{R} \frac{ω_{i}}{ω_{i}^{2} - ω_{r}^{2}} \cdot \ln (| H_{fil} (ω_{r}) |)

which is based on the assumption that the optical filter satifies the causality condition and, thus, the real and imaginary parts of H_fil(Ω) are related by the Hilbert’s transform [16]. In Eq. (17) R is an integer value for the truncation of the spectra calculation, Δω the frequency step and ω_r = r · Δω.

2.2.2. Mach-Zehnder interferometer (MZI)

A simple implementable optical filter is given by the MZI, constructed simply by connecting the two output ports of a coupler to the two input ports of another coupler by means of two delay lines with different lengths, as depicted in the inset (c) of Fig. 1. Depending on the output port, its impulse response is given by:

h_{fil, 1} (t) = j \cdot \frac{1}{2} (\exp (j \cdot ϕ_{low}) + \exp (j \cdot ϕ_{upp}) \cdot δ (t - τ_{M Z I}))

if the output port 1 is selected, and:

h_{fil, 2} (t) = \frac{1}{2} (\exp (j \cdot ϕ_{low}) - \exp (j \cdot ϕ_{upp}) \cdot δ (t - τ_{M Z I}))

if the output port 2 is selected.

2.2.3. Uniform fiber-Bragg grating (UFBG)

For the case of a UFBG, the coupled mode differential equations describing the evolution of the two counter-propagating modes in the periodically perturbed refractive index structure can be solved analytically. For a FBG with grating period Λ, effective modal index n_eff and zero-to-peak amplitude of the refractive index modulation δn, the amplitude reflection and transmission coefficients can be expressed as [17]:

r (f) = \frac{- j \cdot \frac{κ}{γ} \cdot \sinh (γ \cdot L_{F B G})}{\cosh (γ \cdot L_{F B G}) + j \frac{δ}{γ} \cdot \sinh (γ \cdot L_{F B G})}

t (f) = \frac{1}{\cosh (γ \cdot L_{F B G}) + j \frac{δ}{γ} \cdot \sinh (γ \cdot L_{F B G})}

where L_FBG is the length of the grating, κ = π/(2 · n_eff · Λ) ·δn is the coupling coefficient, δ = 2π · n_eff ·f/c −π/Λ is the detuning from the grating resonance and γ² = κ² − δ². The grating bandwidth Δλ_FBG, defined as the wavelength range between the first zeros at each side of the Bragg peak is given by [17]:

Δ λ_{F B G} = λ_{b} \frac{δ n}{n_{eff}} {(1 + {(\frac{λ_{b}}{δ n \cdot L_{F B G}})}^{2})}^{\frac{1}{2}}

where λ_b = 2·n_eff ·Λ is the Bragg wavelength.

2.2.4. Balanced configurations

There are two interesting aspects that deserve consideration: (i) the transmitted optical signal is composed of two sidebands which are conjugated one another, that is, it presents some information redundancy, condition which is needed to obtain a real-valued signal to drive the laser intensity at the transmitter, and (ii) the improvement observed when an optical filter is used is not due to the elimination of some spectral signal content, but from introducing some degree of imbalance in such a way the spectral information components are mixed in a more efficient way upon photo-detection. These two aspects lead us to propose the balanced configurations shown in Fig. 2.

Fig. 2 Balanced filtered configurations. a) both output ports of a MZI are used and b) both reflected and transmitted optical signals are used for information detection.

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In both configurations, the reciprocal transfer function is used to detect the otherwise wasted sideband and the OFDM receiver processing on each branch is performed as usual. The information symbols from the two branches after FFT are combined by using a maximum ratio combiner:

\bar{Y} [r] = \frac{H_{upper}^{*} [r] \cdot Y_{upper} [r] + H_{lower}^{*} [r] \cdot Y_{lower} [r]}{{| H_{upper} [r] |}^{2} + {| H_{lower} [r] |}^{2}}, r = 1, 2, \dots N

where Y_upper[r] and Y_lower[r] are the received complex symbols at the r-th subcarrier of the upper and lower branch, respectively, H_upper[r] and H_lower[r] are the rth coefficient of the estimated transfer function corresponding to the upper and lower branch, respectively, and Y [r] is the resulting equalized symbol.

In the balanced configuration with a MZI, signals from both output ports are used, rather than only one of them, Fig. 2(a), and with a FBG, both the reflected and transmitted signals from the grating, Fig. 2(b), are detected. Furthermore, the proper combination of the two signals leads to some improvement, as shown in the following section.

3. Results

3.1. Default system parameters

The next default system parameters are used to obtain the presented results: the available electrical bandwidth (BW) is equal to 5.5GHz; the information binary stream is mapped into 16-QAM complex symbols, which are arranged with a Hermmitian symmetry at the input of an inverse fast Fourier transform (IFFT) processor of size FS = 256; N = 110 subcarriers are used for information transmission; a cyclic pre- and a post-fix of 16 samples (N_pre = N_pos = 16) are appended for the mitigation of ICI and ISI effects; the resulting signal rate is equal to 16.8Gbits/s; a high clipping ratio (CR = 13.8dB) is used in order to limit the amplitude swing of the OFDM signal. For the modeling of the digital-to-analog conversion, quantization noise due to a limited bit resolution is not taken into account and a square-root raised-cosine filter is employed. The obtained analog signal is adapted for laser driving by scaling the analog signal to yield a peak-to-peak value of Δi = 10mA and adding a DC-offset i₀ = 60mA. Note that we have set the clipping ratio to a relatively high value, though the employment of a smaller value may lead to some system performance improvement because of a higher modulation efficiency even at the expense of some clipping noise [18]. Nevertheless, in subsection 3.4 we study the impact of different clipping ratio values used at the transmitter in order to give power budget curves closer to those obtained in a practical scenario.

The fiber chromatic dispersion D is set to 17ps/(km·nm), its attenuation coefficient is 0.2dB/km and its length is equal to 25 km. After photodetection, shot and thermal noises [19] have been considered, being $10 pA / \sqrt{H z}$ the thermal noise spectral density. The process at the receiver is essentially the inverse to that at the transmitter. Training symbols are used to obtain a channel estimation and perform one-tap equalization of the detected information symbols.

3.2. Supergaussian filter approach

3.2.1. System performance

The analysis of the system performance starts with the study of the linear effects on the detected information signal, given by the expression Eq. (15). In Fig. 3 we show some results obtained when a Supergaussian optical filter is inserted, but without considering its phase response. The transfer function of the optical filter obtained through evaluation of Eq. (16) is shown in Fig. 3(a) to illustrate the influence of the change of its parameters (BW₃_dB, Ω_c and q). The modulus of the OOFDM system transfer function (|H(Ω_r)| = |T0|_{n_r=1} + T1|_{n_r=1}|/|X_r|) calculated through Eq. (15) for different values of Ω_c, BW₃_dB and q is shown in Figs. 3(b)–3(d), respectively. Simulation results are also shown for the sake of validation.

Fig. 3 a) Optical filter transfer function. OOFDM system transfer function with b) q = 2, BW₃_dB = 2π · 10×10⁹ rad/s, c) q = 2, Ω_c = 2π·3.75×10⁹ rad/s, and d) BW₃_dB = 2π · 10×10⁹ rad/s, Ω_c = 2π · 3.75×10⁹ rad/s.

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With respect to the OOFDM system transfer function, the theory provided gives reasonably accurate results when the central frequency, the optical bandwidth or the Gaussian order is varied as shown in Figs. 3 (b)–3(d), respectively. The variation of these parameters allows to control the degree of imbalance introduced between both sidebands and how much the optical carrier is attenuated. Both factors finally determine the system total transfer function. The results show that Eq. (15) gives us accurate results for the evaluation of the linear effects on the transmitted complex symbols in an DM/DD OOFDM system.

In Fig. 4 we compare the optical power budget obtained when a) a Supergaussian filter is employed, but without taking into account its phase, and b) when a Supergaussian filter is employed, and its phase is given by Eq. (17). In both figures we have included the power budget curve obtained when no optical filter is employed for comparison purposes. The parameters of the Supergaussian filters used to obtain the results shown are listed in Table 1. Simulation results and those derived from the analytical formulation are shown in Fig. 4 for the sake of validation.

Table 1. Supergaussian filter parameters

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Fig. 4 BER as a function of the received average optical power. a) Only the modulus of the Supergaussian filter transfer functions is considered, b) both modulus and phase of the Super-gaussian filter transfer functions are considered.

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First of all, we must remark the reasonably good accuracy provided by the analytical formulation. The values from the analytical formulation match greatly with those from simulations for the optical filters H_gauss_,1 and H_gauss₄, as we can observe from Figs. 4(a) and 4(b). When the optical filters H_gauss_,2 and H_gauss₃ are employed, the BER values obtained from the analytical formulation are slightly undervalued compared to those from simulations. The reason for such difference seems to be the assumptions made for the simplification of the theory: as we explained in the subsection 2.1, we approximated the Bessel functions by the first term of its series expansion in order to derive Eq. (15) and the expression for the nonlinear distortion in Appendix II. Since the argument of the Bessel functions is affected by the optical filtering, this approximation may be misleading. Therefore, for those interested on more accurate results from the mathematical expressions at the expense of little bit more complexity, it would be enough to include more terms of the expansion, e.g., by approximating J₀(x) ≈ 1 − x²/4.

For the conditions studied, we can observe that the insertion of an optical filter reports a significant improvement in terms of power budget. A BER equals to 10⁻³ can be achieved with a received power equals to −20.4dBm when |H_gauss_,2(Ω)| is used for filtering, which represents a 5.2dB improvement compared to the filter-free case. The resulting curves with |H_gauss_,1(Ω)|, |H_gauss_,3(Ω)| and |H_gauss_,4(Ω)| report similar power budget improvements: 4.2dB, 4.6dB and, 4.8dB, respectively.

From Fig. 4(b) it is clear that, for some cases, the optical filter group delay has a detrimental impact on the system performance, as it was observed in [8]. Effectively, as it can be observed by comparing Figs. 4(a) and 4(b), once the phase has been considered, the power budget is reduced when the optical filters H_gauss_,2(Ω) and H_gauss_,3(Ω) are employed. In the case H_gauss_,2(Ω) is used, the received power needed to assure a BER lower than 10⁻³ increases slightly to −20.2dBm, which means a small penalty of 0.2dB with respect to |H_gauss_,2(Ω)|. The most striking case occurs when H_gauss_,3 is used, in which such a case we need a received power equals to −17.75dBm, which means a penalty of 2.25dB compared to the case in which the phase is not considered. The mathematical model derived in Section 2 can help to understand the reason of such impact.

3.2.2. System performance limiting factors

The driving factors which may be behind this degradation are (i) the decrease on the transfer function in Eq. (15) when the phases of H_fil(Ω_r) and $H_{fil}^{*} (- Ω_{r})$ are considered, (ii) the ICI and ISI due to the optical filter, and (iii) the nonlinear distortion products which appear as result of the imbalance caused by the optical filter (Eqs. (31)–(39)). In Fig. 5 we show the power spectral density of the optical signal after optical filtering, as well as several metrics with the aim of exploring the different components which affect the system performance. The results have been obtained for a received power equals to −20dBm.

Fig. 5 a) Power spectral density of the OOFDM signal after filtering, b) OOFDM system transfer function, c) variance of ISI and ICI, d) variance of the intermodulation distortion.

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The optical spectral density of the OOFDM signal changes accordingly to the optical filter parameters used as we can observe in Fig. 5(a) when the filter order q or the bandwidth BW₃_dB are changed. The system transfer function in Fig. 5(b) has been obtained through the evaluation of Eq. (15). It is clear that, in this case, the magnitude of the transfer function does not change considerably after the phase of the filter is considered. We can also observe that the highest transfer function magnitude is obtained with the optical filter H_gauss_,3, whereas the lowest is obtained with H_gauss_,1.

The variance of the distortion due to ISI & ICI, $σ_{I S I + I C I}^{2}$ , has been calculated as reported in [10]. We can observe that the ISI & ICI effects increase with the order of the filter; however, the consideration of the phase of the filter only causes an increase of ISI & ICI effects for H_gauss_,3: once the phase is considered, the ISI & ICI effects increase by 2–8dB, a substantial difference which may contribute to the degradation observed in Figs. 4(a) and 4(b).

Finally, we have computed the variance of the intermodulation distortion, $σ_{I M D}^{2}$ , through the evaluation of Eqs. (31)–(39). Comparing the magnitude of Figs. 5(c) and 5(d), we can conclude that when an optical filter is used the nonlinear distortion is an important issue to be considered. Generally, the intermodulation distortion increases once the phase of the filter is included, and, for the particular conditions here studied, it is the most significant reason for the degradation observed in Fig. 4.

We group the different impairing phenomena into the effective signal-to-noise ratio, defined as the ratio ${| H (Ω) |}^{2} / (σ_{noise}^{2} + σ_{I S I + I C I}^{2} (Ω) + σ_{I M D}^{2} (Ω))$ and denoted as SNR_eff (Ω). In Fig. 6 we show the values obtained for SNR_eff (Ω_r), r = 1, 2,..N through the evaluation of the analytical formulation and that obtained through simulations. The results have been obtained again for a received power equals to −20dBm. We have also plotted a red line indicating the signal-to-noise ratio needed to obtain a BER equals to 10⁻³ (log₁₀(SNR_eff) = 1.6543) when 16-QAM is employed [20].

Fig. 6 Obtained effective SNR with a Supergaussian filter when a) only the modulus of the optical filter transfer function is considered, and b) both modulus and phase of the optical transfer function are considered.

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First of all, we can observe from Fig. 6 that, apart from the expected overestimation, the results obtained through the evaluation of the analytical formulation match well with those provided by simulations of the DM/DD OOFDM system. As expected from the results shown in Fig. 5, the consideration of the phase for the optical filter H_gauss_,3 yields a significant SNR decrease as result of the enhancement of the nonlinear distortion, as it can be observed by comparing the results in Figs. 6(a) and 6(b).

We have identified the factors with relevant impact on the system performance of an optically filtered DM/DD OOFDM system and we have checked that the effective signal to noise ratio derived from the analytical formulation can help us to estimate it.

3.3. MZI and UFBG filtering approaches

In this section we explore the system performance which can be achieved using two well-known filtering structures: a MZI, and a uniform FBG. By varying a couple of their parameters (τ_MZI and ϕ_upp for the MZI, Δn and Λ for the uniform FBG), we are able to tune them and change their transfer functions. For the sake of clarity, we have plotted in Fig. 7 the modulus of the transfer functions with different filter parameters. In Fig. 7(b) we have defined as Λ₀ = c/(193.1 × 10¹² · 2 · n_eff).

Fig. 7 Modulus of the transfer function of a) a MZI and b) a uniform FBG.

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As it can be observed in Fig. 7(a), the phase shift ϕ_upp allows us to control the location of the maximum of the transfer function, whilst the variation of τ_MZI allows us to control the period between the transfer function maximums. Regarding Fig. 7(b), both the period Λ and the refractive index variation δn are changed simultaneously to locate the optical carrier in the filter transition band; it is worth to observe that the secondary lobes which appear at the lower sideband increase with higher δn values, what necessarily affects the system performance as Eq. (15) indicates.

The obtained effective signal-to-noise ratio values with both filtering structures are shown in Fig. 8, which have been obtained once again for a received power equals to −20dBm. In Fig. 8(b) we have defined as $Δ f_{F B G} = c \cdot Δ λ_{F B G} / λ_{b}^{2}$ .

Fig. 8 Effective SNR using a) a MZI and b) a UFBG as optical filter.

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We can see in Fig. 8(a) that with a simple MZI we have been able to get an effective SNR around 19dB, which is obtained when ϕ_upp = π/2. This value for the phase shift entails a tradeoff point of the system, mainly determined by the optical carrier attenuation and the degree of imbalance introduced. When the value for ϕ_upp is changed to π/4 or 3π/4 the effective SNR gets worse as a result of a decrease of the system transfer function (Eq. (15)). We can also observe that for ϕ_upp = π/2, the delay τ_MZI can be varied over a broad range and good values of SNR_eff are still obtained (in Fig. 8, τ_MZI = [1/f_s–1.5/f_s]), which is advantageous for a practical implementation. Slightly lower values of effective SNR are achieved when a uniform FBG is used: values of δn equals to 1.8 × 10⁻⁴, 2 × 10⁻⁴ and 2.2 × 10⁻⁴ give us values of effective SNR around 18dB, provided the FBG is appropriately tuned by changing the period Λ.

3.4. Analysis and discussion: impact of clipping

Once we have explored the range of parameter which lead to reasonably good effective SNR values, now we determine the power budgets when a MZI, a uniform FBG and the balanced configurations explained in subsection 2.2.4 are used. Together with the numerical simulation results, we also show those derived from the analytical formulation when a MZI and a UFBG are employed. The derivation of the same results when the balanced configurations are employed is more challenging, since we need to finely know how the different impairments from both branches interact in amplitude, rather than in power. For this reason, its study is more appropriate within a detailed investigation on the optimization of these type of configurations.

For the MZI we set τ_MZI to 1.5/f_s and ϕ_upp to π/2, and for the uniform FBG we set Δn = 2× 10⁻⁴ and Λ = Λ₀ · 0.99992. The same parameter values are used in the balanced configurations. Given the importance of the clipping ratio used at the transmitter on the system performance [18], apart from the curves with the default clipping ratio (CR=13.8dB), some curves with smaller values of clipping ratio have been also plotted for each filtering structure. The results are shown in Fig. 9.

Fig. 9 BER as a function of the received average optical power when no-optical filter is used and when a) the Supergaussian filter |H_gauss_,2|, b) a MZI, c) a UFBG or d) the proposed balanced configuration is used as optical filter.

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First of all, the results obtained from the analytical formulation match well with those obtained from simulations when a high clipping ratio (13.8dB) is used, what once again confirms the validity of the theory presented. The results obtained with the clipping ratio value are explained as follows: when the clipping ratio is decreased, a higher optical modulation efficiency is achieved at the expense of some clipping noise and a higher nonlinear distortion, and, thus, the improvement/penalty observed when the clipping ratio is decreased depends on the relative impact of the different components which affect the system performance. For example, when no optical filter is used, the system is mainly limited by the receiver noise, and the system performance can be significantly increased by reducing the clipping ratio. Only when the clipping ratio is significantly reduced and the impact of the resulting clipping noise and nonlinear distortion dominates to that due to the receiver noise, the received power needed to obtain a certain value of BER may start to increase. For the particular conditions used to obtain the results in Fig. 9, this inflection point happens when the clipping ratio is around 8dB. When no optical filter is used, a received power equals to −15.2dBm is needed to achieve a BER equals to 10⁻³ when CR=13.8dB, whilst a received power equals to −17.6dBm is needed for CR=8dB.

We have seen in previous sections that the system transfer function and the nonlinear distortion are altered when an optical filter is inserted into the dispersive link, and this variation depends on the optical filter characteristics. As result, in an optically filtered system, the system performance does not change with a reduction of the clipping ratio in the same way as that for a non-optically filtered system. We can observe from Fig. 9(a) that a received power equal −20.4dBm is needed to obtain a BER equals to 10⁻³ when |H_gauss_,2| is employed with CR=13.8dB, whilst a received power equals to −21.1dBm is needed when CR=10dB. In terms of power budget improvement, it means a difference of 3.5dB compared to the non-optically filtered system. A similar situation occurs when a MZI is used, for which a received power equals to −20.9dBm is needed to obtain a BER equals to 10⁻³ when CR=13.8dB, whilst it is reduced to −22.1dBm when CR=10dB, as it can be observed in Fig. 9(b). Thus, with a MZI as optical filter, we are able to get an improvement equals to 4.5dB compared to the non-optically filtered system.

On the other hand, when a UFBG is used, a reduction of the clipping ratio from CR=13.8dB to CR=11dB does not lead to a system improvement, and the received power needed in both cases is equal to −20.6dBm, Fig. 9(c), what means an improvement of 3dB. This indicates that when a uniform FBG is used the system performance is more strongly impaired by the nonlinear distortion.

The balanced detection proposed in subsection 2.2.4 yields to some power budget improvement, as we can see in Fig. 9(d). When a MZI is used, a power budget improvement equals to 5.5dB is obtained for a clipping ratio equals to 10dB, whereas a power budget improvement equals to 4.2dB is achieved when a UFBG is used with a clipping ratio equals to 12dB.

4. Conclusions

By extending the theory in [9], we have provided an analytical description of a filtered DM/DD OOFDM system which accounts both for the information part of the detected signal and also the nonlinear distortions which impair the system performance. Using a simplified version of the model we have been able to verify that the linear part of the detected signal is spectrally umbalanced as a consequence of the filtering process and this translates into a higher magnitude of the overall system transfer function. The filtering process also affects the nonlinear distortion, the ISI and the ICI effects, potentially limiting the system performance.

The evaluation of the system performance by means of the effective signal-to-noise ratio has proved to be an extremely useful tool to predict the system performance and the impact of the filter design parameters, in particular we have considered two commonly employed optical filters: a MZI and an uniform FBG.

The performance improvement obtained by inserting the optical filter in the OFDM is highly dependent on the clipping ratio: when no optical filter is used, a significant power budget improvement can be obtained by reducing the clipping ratio, whilst this reduction results in a more limited improvement in the distortion-limited power budget when an optical filter is used. Simulation results predict improvement figures of 3.3 and 3dB when a MZI or a UFBG re respectively employed.

Finally, we have proposed the use of optical balanced detection for further enhancement of optically filtered DM/DD OOFDM system performance. With this configuration, an improvement of 5.5dB is achieved by using a MZI. Balanced detection opens the door to more efficient filter designs for higher cancellation of the impairment effects. The fabrication flexibility and transfer function tunability provided by FBGs may be of great interest to obtain customized filter transfer functions.

Appendix I: Derivation of the photocurrent expression

Following the same procedure as in [9], the expression of the photocurrent I_ph can be written in function of the five different terms T0, T1,...T5, which expressions are:

T 0 (n_{1} \dots n_{N}, κ, ε) = J_{n_{1}} (μ_{1, κ, ε}) \cdot \cdot J_{n_{N}} (μ_{N, κ, ε}) (P 0 + \frac{1}{2 P_{0}} \sum_{k = 1}^{N} p_{k}^{2} \cdot \cos (2 θ_{k, κ, ε}))

\begin{matrix} T 1 (n_{1} \dots n_{N}, κ, ε) = J_{n_{1}} (μ_{1, κ, ε}) \cdot \cdot J_{n_{N}} (μ_{N, κ, ε}) \\ (\sum_{k = 1}^{N} \frac{p_{k}}{J_{n_{k}} (μ_{k, κ, ε})} \cdot \cos (θ_{k, κ, ε}) \cdot (J_{n_{k} + 1} (μ_{k, κ, ε}) \cdot e^{j (φ_{m_{k}} - φ_{p_{k}} + \frac{π}{2})} + J_{n_{k} - 1} (μ_{k, κ, ε}) \cdot e^{- j (φ_{m_{k}} - φ_{p_{k}} + \frac{π}{2})})) \end{matrix}

\begin{matrix} T 2 (n_{1} \dots n_{N}, κ, ε) = J_{n_{1}} (μ_{1, κ, ε}) \cdot \cdot J_{n_{N}} (μ_{N, κ, ε}) \\ (\sum_{k = 1}^{N} \frac{p_{2 k} \cdot \cos (2 θ_{k, κ, ε})}{J_{n_{k}} (μ_{k, κ, ε})} (J_{n_{k} + 2} (μ_{k, κ, ε}) \cdot e^{j (2 φ_{m_{k}} - φ_{p_{2 k} + π})} + J_{n_{k} - 2} (μ_{k, κ, ε}) \cdot e^{- j (2 φ_{m_{k}} - φ_{p_{2 k}} + π)})) \end{matrix}

\begin{matrix} T 3 (n_{1} \dots n_{N}, κ, ε) = J_{n_{1}} (μ_{1, κ, ε}) \cdot \cdot J_{n_{N}} (μ_{N, κ, ε}) \\ (\sum_{k = 1}^{N} \sum_{l = 1}^{k - 1} \frac{p_{k l}}{J_{n_{k}} (μ_{k, κ, ε}) J_{n_{l}} (μ_{l, κ, ε})} \cos (θ_{k, κ, ε} + θ_{l, κ, ε}) (J_{n_{k} + 1} (μ_{k, κ, ε}) \cdot J_{n_{l} + 1} (μ_{l, κ, ε}) \cdot e^{j (φ_{m_{k}} + φ_{m_{l} - φ_{p_{k l}} + π})} + \\ J_{n_{k} - 1} (μ_{k, κ, ε}) \cdot J_{n_{l} - 1} (μ_{l, κ, ε}) \cdot e^{- j (φ_{m_{k}} + φ_{m_{l}} - φ_{p_{k l}} + π)}) + \\ \sum_{k = 1}^{N} \sum_{l = 1}^{k - 1} \frac{p_{k_l}}{J_{n_{k}} (μ_{k, κ, ε}) J_{n_{l}} (μ_{l, κ, ε})} \cos (θ_{k, κ, ε} - θ_{l, κ, ε}) (J_{n_{k} + 1} (μ_{k, κ, ε}) \cdot J_{n_{l} - 1} (μ_{l, κ, ε}) \cdot e^{j (φ_{m_{k}} - φ_{m_{l}} - φ_{p_{k_l}})} + \\ J_{n_{k} - 1} (μ_{k, κ, ε}) \cdot J_{n_{l} + 1} (μ_{l, κ, ε}) . e^{- j (φ_{m_{k}} - φ_{m_{l}} - φ_{p_{k_l}})})) \end{matrix}

\begin{matrix} T 4 (n_{1} \dots n_{N}, κ, ε) = J_{n_{1}} (μ_{1, κ, ε}) \cdot \cdot J_{n_{N}} (μ_{N, κ, ε}) \frac{1}{4 \cdot P_{0}} \\ (\sum_{k = 1}^{N} \sum_{\begin{array}{l} l = 1 \\ l \neq k \end{array}}^{N} \frac{p_{k} \cdot p_{l}}{J_{n_{k}} (μ_{k, κ, ε}) J_{n_{l}} (μ_{l, κ, ε})} (J_{n_{k} + 1} (μ_{k, κ, ε}) \cdot J_{n_{l} + 1} (μ_{l, κ, ε}) \cdot e^{j (φ_{m_{k}} - φ_{p_{k}} + φ_{m_{l}} - φ_{p_{l}} + π)} (\begin{matrix} - \cos (θ_{k, κ, ε} + θ_{l, κ, ε}) + \\ e^{j (θ_{k, κ, ε} - θ_{l, κ, ε})} \end{matrix}) \\ + J_{n_{k} - 1} (μ_{k, κ, ε}) \cdot J_{n_{l} - 1} (μ_{l, κ, ε}) \cdot e^{- j (φ_{m_{k}} - φ_{p_{k}} + φ_{m_{l}} - φ_{p_{l}} + π)} (\begin{matrix} - \cos (θ_{k, κ, ε} + θ_{l, κ, ε}) + \\ e^{- j (θ_{k, κ, ε} - θ_{l, κ, ε})} \end{matrix})) + \\ \sum_{k = 1}^{N} \sum_{\begin{array}{l} l = 1 \\ l \neq k \end{array}}^{N} \frac{p_{k} \cdot p_{l}}{J_{n_{k}} (μ_{k, κ, ε}) J_{n_{l}} (μ_{l, κ, ε})} (J_{n_{k} + 1} (μ_{k, κ, ε}) \cdot J_{n_{l} - 1} (μ_{l, κ, ε}) \cdot e^{j (φ_{m_{k}} - φ_{p_{k}} - φ_{m_{l}} + φ_{p_{l}})} (\begin{matrix} - \cos (θ_{k, κ, ε} - θ_{l, κ, ε}) + \\ e^{j (θ_{k, κ, ε} + θ_{l, κ, ε})} \end{matrix}) \\ + J_{n_{k} - 1} (μ_{k, κ, ε}) \cdot J_{n_{l} + 1} (μ_{l, κ, ε}) \cdot e^{- j (φ_{m_{k}} - φ_{p_{k}} - φ_{m_{l}} + φ_{p_{l}})} (\begin{matrix} - \cos (θ_{k, κ, ε} - θ_{l, κ, ε}) + \\ e^{- j (θ_{k, κ, ε} + θ_{l, κ, ε})} \end{matrix})) \end{matrix}

\begin{matrix} T 5 (n_{1} \dots n_{N}, κ, ε) = J_{n_{1}} (μ_{1, κ, ε}) \cdot \cdot J_{n_{N}} (μ_{N, κ, ε}) \\ \frac{1}{4 \cdot P_{0}} (\sum_{k = 1}^{N} p_{k}^{2} \frac{1 - \cos (2 θ_{k, κ, ε})}{J_{n_{k}} (μ_{k, κ, ε})} (J_{n_{k} + 2} (μ_{k, κ, ε}) \cdot e^{j 2 (φ_{m_{k}} - φ_{p_{k}} + \frac{π}{2})} + J_{n_{k} - 2} (μ_{k, κ, ε}) \cdot e^{- j 2 (φ_{m_{k}} - φ_{p_{k} + \frac{π}{2}})})) \end{matrix}

where

θ_{k, κ, ε} = - (\frac{τ_{κ} - τ_{ε}}{2} + \frac{β_{2}}{2} Ω_{imp} Ω_{k} L)

, and μ_k = 2m_k · sin(θ_k,κ,ε).

Appendix II: Simplified expression of the second order nonlinear distortion

In this appendix we give the expressions of the second order intermodulation distortion, derived from Eqs. (24), (25), (27) and (28). After some mathematical manipulation, the expressions obtained for the second order intermodulation falling on the rth subcarrier of the type A + B and A − B, where A and B stand for two arbitrary subcarriers, are:

Intensity modulation nonlinearity $I_{p, D M L}^{A + B} [r] = \sum_{l = 1}^{⌈ r / 2 ⌉ - 1} H_{p_{11}} (Ω_{l}, Ω_{r - l}) \cdot \frac{1}{2} (\begin{matrix} H_{fil} (0) \cdot H_{fil}^{*} (- Ω_{r}) \cdot e^{- j θ_{r}} \\ + H_{fil} (Ω_{r}) \cdot H_{fil}^{*} (0) \cdot e^{j θ_{r}} \end{matrix}) \cdot i_{l} \cdot i_{r - l} \cdot e^{j (φ_{i_{l}} + φ_{i_{r - l}})}$ $I_{p, D M L}^{A - B} [r] = \sum_{l = r + 1}^{N} H_{p_{11}} (Ω_{l}, - Ω_{l - r}) \cdot \frac{1}{2} (\begin{matrix} H_{fil} (0) \cdot H_{fil}^{*} (- Ω_{r}) \cdot e^{- j θ_{r}} \\ + H_{fil} (Ω_{r}) \cdot H_{fil}^{*} (0) \cdot e^{j θ_{r}} \end{matrix}) \cdot i_{l} \cdot i_{l - r} \cdot e^{j (φ_{i_{l}} - φ_{i_{l - r}})}$
Phase modulation nonlinearity $I_{ϕ, D M L}^{A + B} [r] = P_{0} \cdot 2 j \sum_{l = 1}^{⌈ r / 2 ⌉ - 1} \cdot H_{ϕ_{11}} (Ω_{l}, Ω_{r - l}) \cdot \frac{1}{2} (\begin{matrix} H_{fil} (0) \cdot H_{fil}^{*} (- Ω_{r}) \cdot e^{- j θ_{r}} \\ - H_{fil} (Ω_{r}) \cdot H_{fil}^{*} (0) \cdot e^{j θ_{r}} \end{matrix}) \cdot i_{l} \cdot i_{r - l} \cdot e^{j (φ_{i_{l}} + φ_{i_{r - l}})}$ $I_{ϕ, D M L}^{A - B} [r] = P_{0} \cdot 2 j \sum_{\begin{array}{l} l = \\ r + 1 \end{array}}^{N} P_{0} \cdot H_{ϕ_{11}} (Ω_{l}, - Ω_{l - r}) \cdot \frac{1}{2} (\begin{matrix} H_{fil} (0) \cdot H_{fil}^{*} (- Ω_{r}) \cdot e^{- j θ_{r}} \\ - H_{fil} (Ω_{r}) \cdot H_{fil}^{*} (0) \cdot e^{j θ_{r}} \end{matrix}) \cdot i_{l} \cdot i_{l - r} \cdot e^{j (φ_{i_{l}} - φ_{i_{l - r}})}$
Dispersion-imbalanced intensity/phase components $\begin{array}{l} I_{p / ϕ, β_{2}}^{A + B} [r] = (- 2 j) \cdot [\sum_{l = 1}^{⌈ r / 2 ⌉ - 1} \frac{H_{p_{1}} (Ω_{l}) H_{ϕ_{1}} (Ω_{r - l})}{4} \cdot \\ (\begin{matrix} H_{fil} (0) \cdot H_{fil}^{*} (- Ω_{r}) \cdot e^{- j θ_{r}} - H_{fil} (Ω_{r}) \cdot H_{fil}^{*} (0) \cdot e^{j θ_{r}} \\ - H_{fil} (Ω_{r - l}) \cdot H_{fil}^{*} (- Ω_{l}) \cdot e^{j (θ_{r - l} - θ_{l})} + H_{fil} (Ω_{l}) \cdot H_{fil}^{*} (- Ω_{r - l}) \cdot e^{- j (θ_{r - l} - θ_{l})} \end{matrix}) \cdot i_{l} \cdot i_{r - l} \cdot e^{j (φ_{i_{l}} + φ_{i_{r - l}})} \\ + \frac{H_{p_{1}} (Ω_{r - l}) H_{ϕ_{1}} (Ω_{l})}{4} \cdot \\ (\begin{matrix} H_{fil} (0) \cdot H_{fil}^{*} (- Ω_{r}) \cdot e^{- j θ_{r}} - H_{fil} (Ω_{r}) \cdot H_{fil}^{*} (0) \cdot e^{j θ_{r}} \\ + H_{fil} (Ω_{r - l}) \cdot H_{fil}^{*} (- Ω_{l}) \cdot e^{j (θ_{r - l} - θ_{l})} - H_{fil} (Ω_{l}) \cdot H_{fil}^{*} (- Ω_{r - l}) \cdot e^{- j (θ_{r - l} - θ_{l})} \end{matrix}) \cdot i_{l} \cdot i_{r - l} \cdot e^{j (φ_{i_{l}} + φ_{i_{r - l}})}] \end{array}$ $\begin{array}{l} I_{p / ϕ, β_{2}}^{A - B} [r] = (- 2 j) \cdot [\sum_{\begin{array}{l} l = \\ r + 1 \end{array}}^{N} \frac{H_{p_{1}} (Ω_{l}) H_{ϕ_{1}}^{*} (Ω_{l - r})}{4} \cdot \\ (\begin{matrix} H_{fil} (0) \cdot H_{fil}^{*} (- Ω_{r}) \cdot e^{- j θ_{r}} - H_{fil} (Ω_{r}) \cdot H_{fil}^{*} (0) \cdot e^{j θ_{r}} \\ - H_{fil} (- Ω_{l - r}) \cdot H_{fil}^{*} (- Ω_{l}) \cdot e^{- j (θ_{l - r} + θ_{l})} + H_{fil} (Ω_{l}) \cdot H_{fil}^{*} (Ω_{l - r}) \cdot e^{j (θ_{l - r} + θ_{l})} \end{matrix}) \cdot i_{l} \cdot i_{l - r} \cdot e^{j (φ_{i_{l}} - φ_{i_{l - r}})} \\ + \frac{H_{p_{1}}^{*} (Ω_{l - r}) H_{ϕ_{1}} (Ω_{l})}{4} \cdot \\ (\begin{matrix} H_{fil} (0) \cdot H_{fil}^{*} (- Ω_{r}) \cdot e^{- j θ_{r}} - H_{f i l} (Ω_{r}) \cdot H_{fil}^{*} (0) \cdot e^{j θ_{r}} \\ + H_{fil} (- Ω_{l - r}) \cdot H_{fil}^{*} (- Ω_{l}) \cdot e^{- j (θ_{l - r} + θ_{l})} - H_{fil} (Ω_{l}) \cdot H_{fil}^{*} (Ω_{l - r}) \cdot e^{j (θ_{l - r} + θ_{l})} \end{matrix}) \cdot i_{l} \cdot i_{l - r} \cdot e^{j (φ_{i_{l}} - φ_{i_{l - r}})}] \end{array}$
Dispersion-imbalanced phase components $\begin{array}{l} I_{ϕ, β_{2}}^{A + B} [r] = - 4 P_{0} \cdot \sum_{l = 1}^{⌈ r / 2 ⌉ - 1} \frac{H_{ϕ_{1}} (Ω_{l}) H_{ϕ_{1}} (Ω_{r - l})}{4} \cdot \\ (\begin{matrix} H_{fil} (0) \cdot H_{fil}^{*} (- Ω_{r}) \cdot e^{- j θ_{r}} + H_{fil} (Ω_{r}) \cdot H_{fil}^{*} (0) \cdot e^{j θ_{r}} \\ - H_{fil} (Ω_{r - l}) \cdot H_{fil}^{*} (- Ω_{l}) \cdot e^{j (θ_{r - l} - θ_{l})} - H_{fil} (Ω_{l}) \cdot H_{fil}^{*} (- Ω_{r - l}) \cdot e^{- j (θ_{r - l} - θ_{l})} \end{matrix}) \cdot i_{l} \cdot i_{r - l} \cdot e^{j (φ_{i_{l}} + φ_{i_{r - l}})} \end{array}$ $\begin{array}{l} I_{ϕ, β_{2}}^{A - B} [r] = - 4 P_{0} \cdot \sum_{l = r + 1}^{N} \frac{H_{ϕ_{1}} (Ω_{l}) H_{ϕ_{1}}^{*} (Ω_{l - r})}{4} \cdot \\ (\begin{matrix} H_{fil} (0) \cdot H_{fil}^{*} (- Ω_{r}) \cdot e^{- j θ_{r}} + H_{fil} (Ω_{r}) \cdot H_{fil}^{*} (0) \cdot e^{j θ_{r}} \\ - H_{fil} (- Ω_{l - r}) \cdot H_{fil}^{*} (- Ω_{l}) \cdot e^{- j (θ_{l - r} + θ_{l})} - H_{fil} (Ω_{l}) \cdot H_{fil}^{*} (Ω_{l - r}) \cdot e^{j (θ_{l - r} + θ_{l})} \end{matrix}) \cdot i_{l} \cdot i_{l - r} \cdot e^{j (φ_{i_{l}} - φ_{i_{l - r}})} \end{array}$
Dispersion-imbalanced intensity components $\begin{array}{l} I_{p, β_{2}}^{A + B} [r] = - \frac{1}{P_{0}} \cdot \sum_{l = 1}^{⌈ r / 2 ⌉ - 1} \frac{H_{p_{1}} (Ω_{l}) H_{p_{1}} (Ω_{r - l})}{4} \cdot \\ (\begin{matrix} H_{fil} (0) \cdot H_{fil}^{*} (- Ω_{r}) \cdot e^{- j θ_{r}} + H_{fil} (Ω_{r}) \cdot H_{fil}^{*} (0) \cdot e^{j θ_{r}} \\ - H_{fil} (Ω_{r - l}) \cdot H_{fil}^{*} (- Ω_{l}) \cdot e^{j (θ_{r - l} - θ_{l})} - H_{fil} (Ω_{l}) \cdot H_{fil}^{*} (- Ω_{r - l}) \cdot e^{- j (θ_{r - l} - θ_{l})} \end{matrix}) \cdot i_{l} \cdot i_{r - l} \cdot e^{j (φ_{i_{l}} + φ_{i_{r - l}})} \end{array}$ $\begin{array}{l} I_{p, β_{2}}^{A - B} [r] = - \frac{1}{P_{0}} \cdot \sum_{l = r + 1}^{N} \frac{H_{p_{1}} (Ω_{l}) H_{p_{1}}^{*} (Ω_{l - r})}{4} \cdot \\ (\begin{matrix} H_{fil} (0) \cdot H_{fil}^{*} (- Ω_{r}) \cdot e^{- j θ_{r}} + H_{fil} (Ω_{r}) \cdot H_{fil}^{*} (0) \cdot e^{j θ_{r}} \\ - H_{fil} (- Ω_{l - r}) \cdot H_{fil}^{*} (- Ω_{l}) \cdot e^{- j (θ_{l - r} + θ_{l})} - H_{fil} (Ω_{l}) \cdot H_{fil}^{*} (Ω_{l - r}) \cdot e^{j (θ_{l - r} + θ_{l})} \end{matrix}) \cdot i_{l} \cdot i_{l - r} \cdot e^{j \cdot (φ_{i_{l}} - φ_{i_{l - r}})} \end{array}$ where $θ_{k} = θ_{k, 0, 0} = - \frac{β_{2}}{2} Ω_{imp} Ω_{k} L$ .

Acknowledgments

The authors wish to acknowledge the financial support given by the Spanish Ministry of Economy and Competitiveness under the project TEC2011-26642 “NUEVA GENERACION DE TECNICAS OPTICAS DE TRANSMISION OFDM PARA FUTURAS REDES WDM-PONS” (NEWTON) and by the Generalitat Valenciana under the Research Excellency Award Program GVA PROMETEO 2013/012 Next Generation Microwave Photonic technologies.

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	q	BW₃_dB(rad/s)	Ω_c(rad/s)		q	BW₃_dB(rad/s)	Ω_c(rad/s)
H_gauss_,1	1	2π·5.5×10⁹	2π·2.5×10⁹	H_gauss_,3	3	2π·5.5×10⁹	2π·2.5×10⁹
H_gauss_,2	2	2π·5.5×10⁹	2π·2.5×10⁹	H_gauss_,4	2	2π·10×10⁹	2π·4.6×10⁹

Optical filtering in directly modulated/detected OOFDM systems

Abstract

1. Introduction

2. Optically filtered DM/DD multicarrier signals: theory

2.1. Analytical formulation

2.2. Optical filtering approaches

2.2.1. Supergaussian filter

2.2.2. Mach-Zehnder interferometer (MZI)

2.2.3. Uniform fiber-Bragg grating (UFBG)

2.2.4. Balanced configurations

3. Results

3.1. Default system parameters

3.2. Supergaussian filter approach

3.2.1. System performance

3.2.2. System performance limiting factors

3.3. MZI and UFBG filtering approaches

3.4. Analysis and discussion: impact of clipping

4. Conclusions

Appendix I: Derivation of the photocurrent expression

Appendix II: Simplified expression of the second order nonlinear distortion

Acknowledgments

References and links

Cited By

Figures (9)

Tables (1)

Equations (39)

Optics Express