1 Introduction

Chromatic dispersion is one of the main propagation problems, in addition to attenuation, in optical fiber communication systems. The attenuation problem has been solved by the development of optical amplifiers and especially, the erbium doped fiber amplifier (EDFA). Now, dispersion is the main issue to enhance the performance of optical fiber communication systems (Ibarra-Villalon et al. 2021). Chromatic dispersion is the pulse broadening due to the dependency of core refractive index on the propagating signal wavelength (Sonne 2021). This causes different wavelengths propagate with different velocities, which leads to group velocity delay (GVD). The broadened pulses overlap together and cause intersymbol interference (ISI) at the receiver. This leads to decrease the bit rate and the overall system performance (Dahir et al. 2020).

To overcome this, various chromatic dispersion compensation techniques have been developed. One of the most efficient techniques is the use of (FBG) (Mustafa et al. 2021).

In optical fiber communication systems, FBG is used to compensate chromatic dispersion at the transmitter side. FBG contains a number of small planes, gratings, with different refractive indices inside its core (Sahota et al. 2020). Based on planes distribution, FBG has two types: uniform FGB (UFBG) and chirped FBG (CFBG) (Sayed et al. 2020a, b). Many apodization functions are developed to enhance FBG performance. The main idea behind using FBG as a dispersion compensator is narrowing the full width at half maximum (FWHM) with an acceptable level of reflectivity (Dar and Jha 2017). To minimize FWHM, a chain of n-stages cascaded FBG is used. FBG has also many other applications in different fields, like irradiated polymers (Hamdalla and Nafee 2015) and in wavelength shift compensation under sea water (Mahran et al. 2009).

In this work, we use apodized uniform FBG (AUFBG) (Toba et al. 2019). In addition, the using of suitable and efficient modulation scheme leads to increasing the tolerance of chromatic dispersion, the spectral efficiency and the transmission distance. Therefore, advanced modulation schemes such as DB modulation and MDB modulation have been used. These advanced modulation schemes have many advantages. They have a narrow spectral width, high anti-nonlinear ability, better dispersion tolerance, good transmission performance and a simple and cost-effective configuration (Mishra et al. 2020).

The aim of this work is to enhance the performance of chromatic dispersion compensation in a standard single mode optical fiber. To achieve this, we propose a model of four-stages cascaded identical apodized UFBGs with DB modulation format. The model is evaluated in terms of Q-factor and BER. In the beginning, the most common apodization functions are investigated to determine the best function. Then, we examine different FBG connection schemes (i.e., pre, post and symmetrical schemes). Finally, DB modulation format is compared with MDB and NRZ modulation formats. Optisystem 17.0 is used to simulate and evaluate 10 Gbps single channel optical fiber communication system.

The remainder of this paper is organized as follows. Section 2 illustrates the basic model and analysis. The proposed model is explained in Sect. 3. Section 4 displays and discusses the simulation results. Section 5 is devoted to the main conclusions.

2 Basic model and analysis

In this section, the basic model and analysis are discussed. We start with discussing the modulation schemes. Then, the use of AUFBG in dispersion compensation is described.

2.1 Modulation schemes

2.1.1 NRZ modulation scheme

The NRZ modulation scheme is the simplest and most common modulation scheme (Gutiérrez-Castrejón et al. 2021). This is illustrated in Fig. 1.

Fig. 1
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NRZ transmitter

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In NRZ, the pulse is ON for the entire bit period. The NRZ is used because it is perceptive to laser phase noise, and it needs a moderately low electrical bandwidth for transmitter and receivers. The NRZ pulses have a narrow optical spectrum which improves the dispersion tolerance but it has the effect of ISI. However, it is not suitable in high bit rate long haul systems (Seraji and Kiaee 2017).

2.1.2 DB and MDB modulation schemes

The DB modulation scheme is an effective scheme which can increase the spectral efficiency (Kaur and Dewra 2014). In view of spectral efficiency and chromatic dispersion tolerance, DB modulation scheme outperforms the NRZ scheme. Due to the increased bandwidth requirement for high bit rates long haul optical fiber communication systems, DB scheme became a significant modulation scheme (Krishna and Tiwari 2015). The fundamental idea of DB modulation is adding a data sequence to a 1-bit delayed version of itself. The transmitted signal of DB modulation is given by Šalík et al. (2015).

$$x\left(t\right)=\sum\limits _{k=-\infty }^{\infty }{d}_{k}q\left(t-kT\right), {d}_{k}=\text{0,1}$$
(1)

where dk is the data bits, q(t) is the transmitted pulse and T is the bit period.

Figure 2 shows the DB transmitter (Kaur and Dewra 2014).

Fig. 2
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Duobinary transmitter (Kaur and Dewra 2014)

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For duobinary pulses, direct detection PIN photodetector is used. It is simple and hence the Mach–Zehnder modulator (MZM) for direct detection should be biased at nullpoint. The MDB modulation provides a compressed bandwidth (Kaur and Dewra 2014). Generation of MDB pulses is carried out by adding an extra delay as shown in Fig. 3.

Fig. 3
figure 3

MDB transmitter (Kaur and Dewra 2014)

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2.2 Apodized uniform FBG (AUFBG)

FBGs have emerged as an important element, mostly in optical fiber communications, especially in long haul networks.

The reflectivity of FBG has sidelobes, which can be minimized by applying different index profiles, called as apodization. The refractive index modulation along the fiber axis can be represented by Šalík et al. (2015)

$$\text{n}\left(\text{z}\right)={\text{n}}_{\text{c}\text{o}}+{\Delta }{n}_{o}\cdot\text{A}\left(\text{z}\right)\cdot{n}_{d}\left(z\right)$$
(2)

where nco is the core refractive index, Δno is the maximum index variation, A(z) is the apodization function, nd(z) is the index variation function.

There are many apodization functions which depend on the grating length, L. In this work, we focus on the most useful functions illustrated in Table 1 (Šalík et al. 2015).

Table 1 Apodization functions in the proposed model (Šalík et al. 2015)
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2.3 AUFBG dispersion compensation

The main idea of chromatic dispersion compensation by FBG in optical fiber communication system is inserting a wavelength-dependent time delay (Sayed et al. 2021). The time delay, τ(λ) for each wavelength along the AUFBG can be obtained by Mustafa et al. (2021)

$$\tau \left(\lambda \right)=\left({\lambda }_{B}-\lambda \right)\frac{2{n}_{eff}}{\varDelta \lambda c} L$$
(3)

where c is the speed of light in free space and Δλ is difference in wavelength between wavelengths corresponding to maximum and minimum value of induced refractive index along the fiber grating, and could be obtained by Mustafa et al. (2021)

$$\Delta \lambda = 2\Lambda ({\text{n}}_{{{\text{max}}}} - {\text{n}}_{{{\text{min}}}} )$$
(4)

3 Proposed model

We propose a model to achieve the aim of this study which is the chromatic dispersion compensation in single mode optical fiber. The proposed model is based on applying DB modulation scheme on 4-stages of cascaded identical AUFBGs. Each stage of the proposed model is connected to a standard optical circulator. Figure 4 illustrates the connections and operation of the cascaded AUFBGs.

Fig. 4
figure 4

Connections and operation of the cascaded AUFBG

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The input signal enters the first stage and the reflected signal is connected to the input of the second stage, and so on. The reflected signal of the last stage is considered as the output of the cascaded AUFBGs. The cascaded module is connected in a 10 Gbps WDM link. Through a comparative study, we evaluate the performance of the proposed model in terms of Q-factor and BER. Figure 5 shows the procedure of this comparison. It includes three cases based on which modulation scheme we use, NRZ, DB or MDB. In each case, different connection schemes (i.e. pre, post or symmetrical) are applied. Under each connection scheme, we apply, in one-by-one manner, all the apodization functions shown in Table 1.

Fig. 5
figure 5

Procedure of the comparison study

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The best configuration is the one having the highest Q-factor and the lowest BER. The proposed model is simulated and evaluated by Optisystem 17.0.

4 Results and discussion

This section addresses and discusses the results of the comparative study. Table 2 includes the simulation parameters of this study for the FBG, standard single mode fiber (SSMF) and the link. The proposed configuration to be acceptable must meet the minimum operational requirements in terms of Q-factor (> 6) and BER (< 10− 9) (Palacharla et al. 1995). The configuration with unreasonable results is rejected.

Table 2 Simulation parameters (Sayed et al. 2020a, b)
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4.1 Case one: NRZ modulation scheme

Figure 6 illustrates the structure of the proposed model in different compensation schemes with NRZ modulation scheme.

Fig. 6
figure 6

a Pre-compensation scheme of case one. b Post-compensation scheme of case one. c Symmetrical compensation scheme of case one

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The obtained results of NRZ in different compensation schemes do not meet the operational requirements. In most sub-cases, the output signal is totally lost where Q-factor = 0 and BER = 1. Only the sub-cases illustrated in Table 3 provide an output signal but it can’t be distinguished.

Table 3 Results for case one
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4.2 Case two: DB modulation scheme

4.2.1 Pre-compensation

The basic structure of the pre-compensation scheme in case two is displayed in Fig. 7.

Fig. 7
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Pre-compensation scheme of case two

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Table 4 shows the obtained results of the pre-compensation scheme in case two, all apodization functions, except Gaussian function, exceed the minimum operational requirements.

Table 4 Results of pre-compensation scheme in case two
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The tanh apodization function achieves the maximum acceptable results, Q = 8.964 and BER = 1.316 × 10–19. The other apodization functions results are rejected, where the obtained values of BER are unreasonable.

4.2.2 Post-compensation scheme

Figure 8 illustrates the basic structure of the post-compensation scheme in case two.

Fig. 8
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Post-compensation scheme of case two

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The obtained results shown in Table 5 reveal that all apodization functions are unreasonable.

Table 5 Results of post-compensation scheme in case two
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4.2.3 Symmetrical compensation scheme

The symmetrical compensation scheme of case two is illustrated in Fig. 9.

Fig. 9
figure 9

Symmetrical compensation scheme of case two

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In case of Gaussian apodization function, the output signal is totally lost. The other apodization functions lead to unreasonable values of BER. Table 6 summarizes the simulation results of symmetrical compensation scheme of case two.

Table 6 Results of symmetrical compensation scheme of case two
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4.3 Case three: MDB modulation scheme

4.3.1 Pre-compensation scheme

Figure 10 illustrates the pre-compensation scheme of case three.

Fig. 10
figure 10

Pre-compensation scheme of case three

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In this case, none of the apodization functions meet the minimum operational requirements. Table 7 shows that the output signal is indistinguishable in all apodization functions except in Gaussian, where it is totally lost.

Table 7 Results of pre-compensation scheme in case three
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4.3.2 Post-compensation scheme

The basic structure of the post-compensation scheme of case three is displayed in Fig. 11.

Fig. 11
figure 11

Post-compensation scheme of case three

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Table 8 summarizes the results of post-compensation scheme of case three. None of the apodization functions meet the minimum operational requirements.

Table 8 Results of post-compensation scheme of case three
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4.3.3 Symmetrical compensation scheme

The symmetrical compensation scheme of case three is displayed in Fig. 12.

Fig. 12
figure 12

Symmetrical compensation scheme of case three

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As shown in Table 9, only the tanh apodization function exceeds the minimum operational requirements. The output signal is totally lost in case of Gaussian apodization and is indistinguishable in the other functions. The tanh apodization function has the best results with a Q-factor of 6.39 and BER of 8.44 × 10− 11.

Table 9 Results of symmetrical compensation scheme of case three
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4.4 Summarized results

As mentioned above, all cases in NRZ modulation do not meet the operational requirements and the output signal is totally lost. Also, in MDB modulation the results do not meet the operational requirements but the output signal is not totally lost. We note that all connection schemes in DB modulation meet the operational requirements. It provides the maximum reasonable results in pre connection scheme with tanh apodization function, where Q-factor = 8.964 and BER = 1.3 × 10− 19. Figure 13 summarizes the obtained results, Q-factor, for all sub-cases which meet the minimal operational requirements.

Fig. 13
figure 13

Q-factor for all sub-cases which meet the minimal operational requirements

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The BER, of the sub-cases that meet the minimal operational requirements is illustrated in Table 10.

Table 10 BER of sub-cases that meet the minimal operational requirements
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From Table 10, it is clear that the tanh apodization function achieves the best reasonable results in terms of Q-factor and BER in the pre-compensation scheme under DB. Also, it is the only apodization function that meets the minimum operational criteria in symmetrical-compensation under MDB.

4.5 Related studies results

To declare the performance optimization of the proposed model, this section illustrates the previous studies results, see Table 11.

Table 11 Previous studies results
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5 Conclusion

In this paper, we proposed a 4-stages cascaded identical AUFBG model to minimize the chromatic dispersion and to enhance system performance. The proposed model is evaluated by Optisystem 17.0 in terms of Q-factor and BER. A 10 Gbps WDM link is simulated at distance of 100 km. A comparative study is performed based on connection schemes, apodization functions and modulation schemes. Three different modulation schemes, NRZ, DB and MDB are used. It is found that DB modulation scheme with tanh apodization in pre-compensation scheme provides the best acceptable results: a Q-factor of 8.964 and a BER of 1.3 × 10− 19. We do not recommend using the proposed model in the sub-cases with unreasonable results; shaded cells in Table 10. The percentage of improvement in the system performance over the related works is about 99%.