## Abstract

We provide a full analysis of the distortion effects produced by the first and second order in-band dispersion of fiber Bragg grating based optical demultiplexers over analogue SCM (Sub Carrier Multiplexed) signals. Optical bandwidth utilization ranges for Dense WDM network are calculated considering different SCM system cases of frequency extension and modulation conditions.

© 2002 Optical Society of America

## 1. Introduction

Optical Add Drop Multiplexers (OADMs) based on fiber Bragg gratings (FBG) are a suitable technology for channel routing and switching tasks to be carried in the optical layer of future DWDM networks. Different works have addressed the limitations that the in-band dispersion [1–2] and the out-band dispersion [3–5] of the uniform apodised FBG produce over the dropped out (in-band case) or transmitted (out-band case) channel. Over digital transmission systems the impairments manifest as a penalty increase depending on the spectral location of the signal with respect to the center Bragg position of the FBG. For the in-band cases this penalty increase can be translated immediately into an effective optical bandwidth reduction that can deteriorate the tolerance of optical network to frequency deviations of lasers and demultiplexers respect to the standard values. The frequency tolerance reduction can be especially important in Dense WDM systems with channel spacing of 25 and 12.5GHz as it has been proposed in future optical networks.

The induced dispersion effects over an analogue SCM (Sub Carrier Multiplexed) signal (i.e typically a set of RF sub-carriers transporting multiple and different signals formats and applications), is due to the rising of distortion and intermodulation terms that can lead into a deleterious effect over an arbitrary interfered channel. These distortion and intermodulation terms have to be limited in amplitude to a maximum level depending of the signal format transported by the RF channels. In a general case moreover the number of interfering terms “falling” over an interfered channel can grow dramatically with the number of SCM channels limiting the system feasibility.

In this paper we deal with the in-band dispersion effects over analogue SCM signals, obtaining as final results a set of effective bandwidth utilization factors of the optical filter (OADM based on FBG), under very different and common system situations as: ITU channel spacing ( 100, 50, 25GHz), frequency extension of the SCM plan, modulation characteristics, etc. In order to accomplish the study we have completed the analytical model of the distortion calculations of [6] in order to include both combined first and second order dispersion effects. The FBG response and their dispersion characteristics have been numerically calculated solving their Coupled Wave Equations, and their fundamental parameters have been properly chosen to fit the ITU grid spacing maintaining also a low cross-talk ration between adjacent channels (<-35dB). Typically apodization windows have been employed also in order to obtain the closest similarity to a practical case.

## 2. Analogue SCM system definition and FBG dispersion description

We will assume the SCM system conveyed by an optical wavelength placed inside the band pass of an Uniform (non-chirped) and apodized fiber Bragg grating (UAFBG). We will follow the classical analysis carried out by Ih *et al*. [6], to obtain the values of intermodulation (IMD) and harmonic distortion (HD) in the system. As developed in [6] we will start from the modulated optical signal described by its electric field as:

where *m _{i}*, and

*m*are the intensity and frequency modulation indexes,

_{f}*w*is the optical carrier, and

_{o}*w*is the sub-carrier.Following a similar procedure as in [6] but in this case including also the second order dispersion term, the detected photocurrent at the receiver can be written as:

_{m}where (*C _{k}*)

^{2}= (

*C*)

_{ck}^{2}+ (

*C*)

_{sk}^{2}and ϕ

_{k}=arctan((

*C*)

_{ck}^{2}/(

*C*)

_{sk}^{2}). The coefficients

*C*

_{0},

*C*and

_{ck}*C*are formed by multiple terms given by:

_{sk}$${C}_{\mathit{ck}}=2{a}_{o}{a}_{k}\mathit{cos}\left[{k}^{2}{\phi}_{2}\right]\mathit{cos}\left[{k}^{3}{\phi}_{3}\right]-2{a}_{0}\phantom{\rule{.2em}{0ex}}{b}_{k}\mathit{sin}\left[{k}^{2}{\phi}_{2}\right]\mathit{sin}\left[{k}^{3}{\phi}_{3}\right]+$$

$$+\left(1/2\right){{\sum}^{k-1}}_{i=1}\left({a}_{i}{a}_{k-i}-{b}_{i}{b}_{k-i}\right)\mathit{cos}\left[\left({k}^{2}-2\mathrm{ik}\right){\phi}_{2}\right]\mathit{cos}\left[\left({k}^{3}-{3\mathrm{ik}}^{2}+{3\mathrm{ik}}^{3}\right){\phi}_{3}\right]+$$

$$+{{\sum}^{\infty}}_{i=1}\left({a}_{i}{a}_{i+k}+{b}_{i}{b}_{i+k}\right)\mathit{cos}\left[{\left({k}^{2}+2\mathrm{ik}\right)\phi}_{2}\right]\mathit{cos}\left[\left({k}^{3}+3{\mathrm{ik}}^{2}+3{\mathrm{ki}}^{2}\right){\phi}_{3}\right]-$$

$$-{{\sum}^{k-1}}_{i=1}{a}_{i}{b}_{k-i}\mathit{sin}\left[{\left({k}^{2}+2\mathrm{ik}\text{}\right)\phi}_{2}\right]\mathit{sin}\left[\left({k}^{3}-3{\mathrm{ik}}^{2}+3{\mathrm{ki}}^{2}\right){\phi}_{3}\right]-$$

$$-{{\sum}^{\infty}}_{i=1}\left({a}_{i}{b}_{i+k}+{b}_{i}{a}_{i+k}\right)\mathit{sin}\left[{\left({k}^{2}+2\mathrm{ik}\text{}\right)\phi}_{2}\right]\mathit{sin}\left[\left({k}^{3}+3{\mathrm{ik}}^{2}+3{\mathrm{ki}}^{2}\right){\phi}_{3}\right]$$

$${C}_{\mathit{sk}}=2{a}_{o}{a}_{k}\mathit{cos}\left[{k}^{2}{\phi}_{2}\right]\mathit{sin}\left[{k}^{3}{\phi}_{3}\right]+2{a}_{0}\phantom{\rule{.2em}{0ex}}{b}_{k}\mathit{sin}\left[{k}^{2}{\phi}_{2}\right]\mathit{cos}\left[{k}^{3}{\phi}_{3}\right]+$$

$$+\left(1/2\right){{\sum}^{k-1}}_{i=1}\left({a}_{i}{a}_{k-i}-{b}_{i}{b}_{k-i}\right)\mathit{cos}\left[\left({k}^{2}-2\mathrm{ik}\right){\phi}_{2}\right]\mathit{sin}\left[\left({k}^{3}-{3\mathrm{ik}}^{2}+{3\mathrm{ik}}^{2}\right){\phi}_{3}\right]+$$

$$+{{\sum}^{\infty}}_{i=1}\left({a}_{i}{a}_{i+k}+{b}_{i}{b}_{i+k}\right)\mathit{cos}\left[{\left({k}^{2}+2\mathrm{ik}\text{}\right)\phi}_{2}\right]\mathit{sin}\left[\left({k}^{3}+3{\mathrm{ik}}^{2}+3{\mathrm{ki}}^{2}\right){\phi}_{3}\right]+$$

$$+{{\sum}^{k-1}}_{i=1}{a}_{i}{b}_{k-i}\mathit{sin}\left[{\left({k}^{2}-2\mathrm{ik}\right)\phi}_{2}\right]\mathit{cos}\left[\left({k}^{3}-3{\mathrm{ik}}^{2}+3{\mathrm{ki}}^{2}\right){\phi}_{3}\right]+$$

$$+{{\sum}^{\infty}}_{i=1}\left({a}_{i}{b}_{i+k}+{b}_{i}{a}_{i+k}\right)\mathit{sin}\left[{\left({k}^{2}+2\mathrm{ik}\right)\phi}_{2}\right]\mathit{cos}\left[\left({k}^{3}+3{\mathrm{ik}}^{2}+3{\mathrm{ki}}^{2}\right){\phi}_{3}\right]$$

where *φ*
_{2}/*z*=(1/2)∙β_{2}(*W _{m}*)

^{2}is the phase shift produced by the first order dispersion and

*φ*/z=(1/6)∙β

_{3}_{3}(

*W*)

_{m}^{3}is due to the second order dispersion term, and the

*a*, and

_{i}*b*are the Fourier series coefficients of (1). In previous formulations we have considered only one sub-carrier in

_{i}*w*, however the method is immediately applicable to a multiple tone SCM system if we describe each sub-carrier frequency as an integer multiple of a common frequency (f

_{m}_{u}) (example below: f

_{1}=4∙f

_{u}(named as C1), f

_{2}=5∙f

_{u}(named as C2)). The Harmonic distortion (HD) and the Intermodulation Distortion (IMD) are evaluated as the relative levels of the distortion terms respect to the sub-carriers level.

The dispersion features β_{2}L_{g} and β_{3}L_{g} (L_{g} is the grating length) of the UAFBG under analysis are shown in Figs. 1–2, where we have assumed a proper apodisation profile of the coupling strength and device length to comply with the typically required parameters of the ITU DWDM grid spacing (Aυ_{C}=100GHz, 50GHz and 25GHz) in terms of 3dB optical bandwidth (Aυ_{B}=50GHz, 25GHz and 12.5GHz respectively) and cross-talk levels at the adjacent channel locations <-35 dB. Specifically the results shown in Figs. 1–2 correspond with a 50GHz bandwidth FBG The resulting required FBG lengths are under Lg=2.34 cm (narrowest case of 12.5GHz bandwidth).

The inset of Fig. 1 shows the apodisation functions evaluated. Also the relative location of the SCM system with respect to the center of the FBG band-pass Δ*v*, and the optical bandwidth Δ*v*
_{B} are related with the frequency detuning ratio y=Δ*v*/Δ*v*
_{B}.

In order to reduce the results into an approachable quantity we will employ for the rest of the paper a single apodization window as Hiperbolic Tangent because it is commonly used for FBG fabrication, and as we can see in Figs. 1–2 the dispersion properties of the different apodization functions do not differ strongly in their shape or magnitude. Nevertheless a precise calculation developed for a specific system must to take this parameter into consideration.

## 3. Results and Discussion

Figure 3 shows the compared results of distortion for the three ITU standards (100, 50 and 25GHz of channel spacing). These results reveal that the ‘effective optical bandwidth’ free of distortion is not the whole 3dB bandwidth of the FBG, but a reduced region inside of it. We can define this ‘effective optical bandwidth’ as the spectral range inside de band presenting distortion levels lower than a maximum fixed level (in our case taken as -60dBc which is the common maximum distortion level for analogue CATV applications).

For instance considering the IM2 terms (i.e. the distortion term generated at f_{1}+f_{2} ), and the ITU case of 50GHz channel spacing, the distortion level below -60 dBc can be only obtained for detuning ratios values (*y*) in between *y*=-0.3 and *y*=0.25 (55% of the 3dB total bandwidth). In the case of the 25GHz spacing the bandwidth available reduces to the 20% (i.e 12.5GHz∙0.2=2.5GHz). The strong increasing of the distortion level observed when the ITU spacing decreases, has a clear physical explanation. When the ITU spacing decreases (and also the optical bandpass bandwidth), the required physical length of the FBG is bigger and its dispersive behaviour increases proportionally (see Fig. 3). So in general, as observed from the results, the distortion level restriction from the FBG based OADMs employed in each time more and more Dense-WDM systems will represent a serious limitation.

As expected from the dispersion in-band features depicted in Figs. 1–2, the distortion levels increase rapidly as the optical carrier displace from the centre to the edges of the FBG spectrum. Moreover, the evolution of this distortion increase with the optical carrier location is not perfectly symmetric, yielding a notch near the centre of the FBG but displaced from it a quantity depending of the FBG optical bandwidth. This non-symmetric results is due to the combined effect or interaction between the first and second order dispersion phase terms that are all included in (3). In this way, it is interesting to point out how the results predicts a better distortion behaviour if we situate the carrier displaced from the centre. For example in the trace of 25GHz of spacing the optimum displacement is about of 9% of the total bandwidth.

The IM3 terms present smaller levels and of symmetric nature respect to FBG centre. Mainly they come dominated by the second order dispersion of the FBG (considering only the first order dispersion in (3) the IM3 values fall drastically). However, despite the low levels, they increase rapidly from -120 to -90 dBc (at the centre of the band if the ITU spacing decrease) and increase by 10dB from the centre to the edges of the spectrum.

The intermodulation results presented were concentrated on particular beating combinations (IM2(f_{1} + f_{2})/C2 and IM3(2f_{2} - f_{1})/C2) for sake of simplicity. In any case the method proposed is completelly indepented and applicable to other different terms as (f2-f1, or triple beating from three different carriers). The distortion level results of IM3 generated by three different frequencies present exactly the same evolution versus the detunning parameter (*y*) to that shown for IM3(2f_{2} - f_{1}), and their levels can differ between zero and almost +12 dB depending strongly of the specific frequency plan employed. In order to validate our analytical formulation incorporating second order dispersion we have made the numerical simulation to obtain the distortion values of Fig. 3 (dots), resulting in a perfect agreement.

We have to point out in this point a very important consideration that it has to be done for the proper extension of the distortion level results to a practical system. This consideration is that the distortion levels depicted have to be multiplied by the proper “number of distortion terms” that they could follow over a specified channel in a complete SCM frequency plan. For example the NTSC-79 frequency plan leads into N_{CSO}=70 (second order terms count) (i.e.+18dB) and N_{CTB}=2350 (the third order terms count) (i.e.+34dB)).

The radio frequency extension of the SCM system will affect strongly to the distortion level. Notice that the previous results correspond to f_{u}=100MHz and therefore the two tones composing the plan are of relatively low frequency (f_{1}=400MHz and f_{2}=500MHz). Also the type of modulation (direct or external modulation) will determinate the spectral composition and bandwidth and therefore the resultant distortion level.

We present in Fig. 4 a summary of distortion results as a function of the common frequency f_{u}, evaluated for different modulation sceneries which adjust with three different practical possibilities. Those are: 1) *m _{i}* = 5% and

*m*= 20% (case of direct modulation of a DFB laser, valid for low frequency plans), 2)

_{f}*m*=5% and

_{i}*m*=5% (case of externally modulated with a non-perfect balanced EOM (Electro Optical Modulator) or a MQW laser direct modulated), and finally 3)

_{f}*m*= 5% and

_{i}*m*= 0 (perfectly balanced dual-drive EOM). The 2° and 3° case allow the assumption of wider frequency plans due to the modulation bandwidths of the EOMs. So the the f

_{f}_{u}range of analysis have been extended up to 2 GHz, corresponding with the cases of systems transmitting SCM channels from 8 to 10GHz.

From Fig. 4 we can point out that for direct modulated systems (continuous traces) the IM2 distortion level grows up to near -60dBc for f_{u}=200MHz corresponding with f_{1}=800MHz and f_{2}= 1 Ghz. For this limit values of f_{u} the IM3 level is about -95dBc but we have to take into account that it has been obtained for the center of the FBG (better case). Looking the trace of 50GHz in Fig. 3, we can expect an increment of about 20dB from the previous values if we move the optical carrier from the center to y=0.2. A similar reasoning can be done for the external modulation cases. For m_{f}=5% (non perfectly-balanced EOM), the IM2>-60dBc and IM3>-95dBc are reached at f_{u}=600MHz (f_{1}=2.4GHz and f_{2}=3Ghz ). If the SCM system under consideration transport microwave tones extended between 8 and 10 GHz the distortion levels reach values higher than IM2>-30dBc and IM3>-60dBc, and as it was mention before, these results are obtained also for the center of the FBG band.

Previous results show a considerable source of distortion produced by the intrinsic in-band dispersive behavior of the FBG. The distortion results were represented fixing one of the main parameters and varying the other one (i.e common frequency, detuning parameter *y*). Also we could see that they depended strongly on the type of modulation and the optical bandwidth of the FBG employed. In order to provide a more complete view of the limitation, we present the results of Fig. 5 as a condensed summary of the dispersion induced distortion impact for a wide variety of system types and FBG parameters. To this objective we define the “free of distortion optical bandwidth” (BW_{fd}), as the frequency range where the distortion level is below a certain specified level. This BW_{fd} has been considered separately for second terms (IM2 and HD) and third order terms (IM3) since the IM3 in general presents lower limitations that the IM2 and HD terms. The limit value adopted for the presented results was - 60dBc for both second and third order terms.

From previous figures we can summarise the following conclusions: 1) As expected the BW_{fd} reduces strongly when the common frequency increases, but its decaying slope depends on the FBG optical bandwidth and modulation format. 2) The BW_{fd} considering only IM3 terms are higher than those obtained by considering only IM2 and HD due to the maximum distortion level was the same (-60dBc) for both. 3) The indexes of modulation from 2% to 16% and the degree of chirping of the modulation also strongly determine the limitation (comparison between (a) and (b)). 4) Finally the effect of the FBG optical bandwidth is considered if we compare the upper figures (Ultra-WDM, spacing of Δυ_{C}=25GHz, and Δυ_{B}=12.5GHz) and the bottom ones (D-WDM, spacing of Δυ_{C}=50GHz, Δυ_{B}=25GHz).

Figure 5 provides design limits for the maximum channel detuning, or equivalently, the maximum channel wavelength tolerance for optical filters and sources depending on the concrete application. The results in Fig. 5 show some impacting examples like those for systems operating over D-WDM(Δυ_{C}=50GHz), with common frequency equal to 300MHz (f_{1}=1.2GHz, f_{2}=1.5Ghz), and with medium/low chirped modulation and *m _{i}*=[4–8%]. These will be limited to around the

*y*=30% of the optical bandwidth. If the same system is operated over a U-WDM(Δυ

_{C}=12.5GHz), the optical bandwidth will be limited to

*y*<10%. This two cases translate into a catastrophic results is the high chirped modulation is considered so we have

*y*around 10% for D-WDM and

*y*=0 for U-WDM (i.e all the FBG bandwidth provide high distortion levels than the -60dBc fixed).

## 4. Conclusions

We have analyzed the distortion produced over SCM analogue systems by the combined first and second order dispersion inside the optical (reflection) band of the uniform FBGs. To do this, first we have developed an upgraded the closed analytical expression given in [6] to include the second order dispersion. We have established the maximum allowed detuning between the FBG center and the optical carrier containing the SCM system that maintain acceptable distortion levels. The results show clearly that in future Dense and Ultra-Dense WDM optical networks employing FBGs as demultiplexer, the distortion induced by dispersion will be a important limiting factor over the feasible analogue systems that they could be transported on it. Although the previous conclusions could set out a very dramatic scenery, they are applicable to standard apodised and non-chirp affected FBG, which represent the more common alternative used nowadays. Other more sophisticated FBG design based on a perfect design and fabrication of the strength and chirp along the grating have been demonstrated for narrow filtering with an almost free of dispersion FBG pass band [8].

## Acknowledgements

The authors wish to acknowledge the financial support of the Spanish CICYT projects TEL 99-0437, TIC2001-2969-C02-01 and TIC2001-2895-C02-01 and the European projects NEFERTITI IST-2001-32786 , INTAS 97-30748 and LABELS IST-2001-37435. Alfonso Martinez acknowledges the funding of a FPI grant from the Universidad Politécnica de Valencia.

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