Performance enhancement of optical communication system with cascaded FBGs of varying lengths

In this paper, an optical communication system constructed with cascaded fiber Bragg gratings or fiber Bragg gratings (FBGs) of varying lengths capable of operating in the C-band is presented. Here, initially, a passive optical device formed with four cascaded FBGs of varying lengths is proposed and analyzed. Subsequently, the proposed device is kept in the optical communication system to reduce the spectral width of the source, thereby enhancing the system’s performance. Analytical formulation based on the transfer matrix method of an optical device is incorporated. The effect of such a device on the system performance at various operating distances is discussed. Further, the system performance is studied with the apodized FBGs in the passive optical device. At the end, we investigated the effect of incorporating the cascaded FBG structure in the system utilizing four-level pulse amplitude modulation or PAM-4 modulation . As per the simulation results, the proposed device has given a maximum reflectivity of 98.39 % and a minimum FWHM of 0.175 nm for uniform FBGs. But with the apodized FBGs, an FWHM of 0.07 nm with a reflectivity of 59.81 % has been achieved. Simulation results reveal that the system formed with the proposed device has given better performance up to a distance of 105 km compared to the absence of the device. At the maximum operating distance of the system with apodized FBGs, the Q-factor and bit error rate (BER) are recorded as 6.712 and 9.5321×10−12, respectively. Apart from the Q-factor and BER estimation, eye height is also used to estimate the system performance at various operating distances.


Introduction
The fiber Bragg gratings or FBGs are the periodic refractive index (r.i.) modulations over some intended length of the fiber core [1,2].Such r.i.modifications inside the core of the fiber have led to the formation of many optical devices for sensing and communications purposes [3][4][5][6][7][8][9][10][11].Although FBGs have been extensively used for different purposes, utilizing them to improve the optical transmission link performance has revolutionized the telecommunication industry [12][13][14][15][16][17].Hence, nowadays, researchers are using cascaded FBG structures in optical communication systems to enhance its performance by reducing the spectral width of the laser source, thus reducing the dispersion [18,19].Therefore, it is noteworthy to study the spectral attributes of a cascaded FBG structure and its impact on the optical communication system.
For filtering purposes in optical communications, a cascaded FBG arrangement containing four uniform FBGs is reported in [20].Such a cascaded arrangement has given narrow bandwidth with reduced strength of side lobes in the reflected spectrum.The reflectivity of such a structure decreases stage-by-stage.So, to address this problem, a four stage cascaded uniform FBG structure with different lengths gives improved reflectivity is proposed in [21].The performance of a wavelength division multiplexing (WDM) system enhanced with a four stage cascaded FBGs of the same lengths is presented in [22].In such a system, the introduced cascaded structure is used to get the narrow linewidth of a continuous wave (CW) laser.Thus, improving the system performance is discussed.Likewise, a cascaded apodized FBG structure for post dispersion compensation in WDM networks had been developed in [18].The developed cascaded structure is used to get the spectral reduction of the source in the system.The performance of such a system up to a distance of 100 km had been analyzed in terms of bit error rate (BER) and Q-factor.Similarly, the performance of an optical communication system can be improved with a cascaded Tanh apodized FBGs combined with the maximum time-division multiplexing transmission technique is mentioned in [19].
This paper introduces a four stage cascaded FBG structure of varying lengths in the optical communication system to estimate its performance.In the proposed system, the output of the CW laser source is given as input to the proposed cascaded FBG structure.The reflected optical signal from the structure is utilized as input to the Mach-Zehnder (MZ) modulator.The spectral width reduction of a CW source attained with the cascaded structure reduces the dispersion in the system and thus enhances its performance.In this paper, initially, the analytical formulation for the introduced structure has been performed using coupled-mode theory (CMT) and the transfer matrix method (TMM).Subsequently, the performance of the system is estimated in terms of BER and Q-factor.
The rest of the paper is organized as follows.Section 2 describes the analytical formulation of the single uniform FBG structure and the proposed cascaded FBG structure.It also covers the experimental set-up used for the proposed structure and the dispersion compensation with the aid of an FBG in the system.An optical communication system formed with a cascaded FBG structure of varying lengths to enhance the system performance is discussed in section 3. Simulation results supported the analytical theory, and the suggested system is comprised in section 4. Finally, we have concluded the paper in section 5.

Single uniform FBG structure
As depicted in figure 1, FBG can be assumed as a four port device with four fields.P 01 and R M1 represents the input fields whereas Q 01 and S M1 denotes the output fields.Here, we have assumed that the considered FBG is operated in reflective mode and is produced in single mode fiber (SMF).The incident and reflected fields in transfer matrix form can be expressed as [5,7,23,24], where γ = √ κ 2 − σ2 , κ denotes 'ac' coupling coefficient between the waves travelling forward and backward, σ is the 'dc' self-coupling coefficient and L denotes the length of an uniform FBG.For simplicity equation ( 1) is designated as, For a reflective grating, the input field amplitude P 01 is normalized to unity whereas the amplitude of the reflected field S M1 at the output of the grating is considered zero.The reason for taking S M1 as zero is due to no perturbation further from the end of the grating.Hence, the transmitted field amplitude from the FBG can be estimated as R M1 = 1/T 11 and the reflected field amplitude Q 01 obtained using R M1 as The reflectivity of a uniform FBG structure shown in figure 1 can be represented as [5,25,26],  where r(L, λ) is the reflectivity and λ is the operating wavelength.The resonance peak obtained at a wavelength called Bragg wavelength (λ B ) due to the r.i.variation inside the fiber core along the z direction of a FBG can be determined as λ B = 2n eff Λ.Where n eff is the effective r.i. of the fiber core and Λ is the grating pitch.The reflected peak from an FBG has a bandwidth or full width at half maximum (FWHM) is given by [27], where ∆n represents maximum r.i.perturbation of an uniform FBG and M denotes the number of periods, and α is equals to 0.5 for weak grating whereas a strong grating structure allows a value of one.

Proposed cascaded uniform FBG structure of varying lengths
Refer to figure 2, which depicts a four stage cascaded uniform FBG structure of different lengths with P 0j and R Mj as the input signal field amplitudes while Q 0j and S Mj denotes the output or reflected signal field amplitudes of a respective grating.Where j represents the stage number taken in the range 1-4 as the number of the stages considered is four.Cascading of FBGs have been performed by connecting the reflected signal output of the jth stage FBG (available at Q 0j ) utilized as input to the (j + 1)th stage FBG (available at P 0( j+1) ) results a condition P 0( j+1) = Q 0j .The reflected optical signal obtained at Q 04 is the intended optical signal.In the proposed structure, the length of each FBG has been varied linearly and considered as a = (L + ( j − 1)) mm.The spectral response of each FBG in the structure can be obtained using CMT and TMM as discussed in section 2. Hence, the reflectivity offered by each FBG in a four stage cascaded FBG structure is given by, The reflectivity of a four stage cascaded FBG arrangement can be expressed as [18,28], Since FBG 4 receives the reflected optical signal from FBG 3, then equation ( 6) can be described as, Similarly, when FBG 3 receives the reflected optical signal from FBG 2 then equation ( 7) can be estimated as, Since FBG 2 receives the reflected optical signal from FBG 1 then the above equation is expressed as, (9) Rearranging the above equation (i.e.equation ( 9)) results as, Equation (10) represents the reflectivity of the proposed cascaded structure.To get the spectral attributes of each FBG and the proposed structure we have employed equation (5) and equation (10), respectively to write the in-house developed code in MATLAB.The obtained spectral profiles are designed at a λ B of 1550.00 nm are shown in figure 3. It has been observed from figure 3 that as the number of cascaded FBGs increases then the FWHM of the reflected spectral profile decreases from 0.202 nm (shown in grey color) to 0.175 nm (represented in brown color).Hence, a narrow bandwidth with reduced strength of side lobes can be accomplished with the proposed structure.Similarly, the recorded reflectivity has come down to 98.39 % (mentioned in brown color) as the number of cascaded FBGs increases.The reason for considering the four stages is that the side-lobe suppression ratio (SLSR) threshold value is achieved at the fourth stage.As per the literature, the SLSR must be greater than or equal to 45 dB for better performance of an FBG-based device.Also, the reflectivity and the FWHM are greater than 50% and less than or equal to 0.2 nm, respectively [29,30].Hence, in the fourth stage, we can achieve the required values.Further consideration of stages will give us no significant improvement in reflectivity and FWHM.Also, increases the complexity and cost of the system.The detailed values of spectral characteristics can be seen from table 1. Performance comparison of a single FBG with the cascaded FBG structure of four stages can be found from table 2. For the simulation we have used the L = 10 mm for first FBG, ∆n = 2 × 10 −4 and λ B = 1550.00nm.It has been observed from table 2 that an FWHM of 0.11 nm can be attained with the optimized single FBG (or FBG apodized with Hyperbolic Tangent function [24]).But with the use of cascaded FBG structure, the value can be narrow down to 0.07 nm.Similarly, the corresponding side-lobe suppression ratio or SLSR can be attained to 22.67 dB for single FBG whereas with cascaded structure, the value can be enhanced to 89.41 dB.It is also deduced from the table 2 that compared to the uniform FBG, apodized FBG has given as better spectral characteristics.On the other hand, compared with the apodized FBG, cascaded apodized FBG structure has given us good performance.Hence, to simulate the considered optical system, we have used cascaded apodized FBGs of varying lengths and the related performance metrics are mentioned in table 6.However, the reason for using cascaded FBG structure over the optimized single FBG is to get the narrow-bandwidth and maximum SLSR.On the other hand, as a preliminary study to validate the results or the code, we have simulated a chirped FBG (approximated using a piecewise uniform approach) with the design parameters mentioned in the experimental study [24].The design parameters are : Grating length L g = 4.5 cm, Λ = 524 nm, ∆n = 7.5 × 10 −4 and chirp rate of +6.73 nm cm −1 .From the simulation, the estimated value of λ B obtained is approximately at 1550 nm while the same value has been reported in [24].
Similarly, the reflectivity we have achieved with the simulation is 0.8798, while the reflectivity of chirped FBG reported in [24] is 0.7702.Since our code does not consider the measuring equipment insertion loss, there is a little discrepancy in the reflectivity.Therefore, one can conclude from the above discussion that the simulated results are corroborate the experimental study.Hence, we have used the same code for simulating the proposed cascaded FBG structure in this work.As per the above discussion, one can conclude that the proposed cascaded structure reduces the spectral width of the optical pulse launched.However, the extent of pulse broadening usually termed as 'dispersion' in the fixed fiber length L f depends on the spectral width of the pulse launched .One can conclude that the proposed cascaded structure reduces the spectral width ∆λ of the optical pulse launched.However, the extent of pulse broadening ∆T usually termed as 'dispersion' in the fixed fiber length L f depends on the spectral width of the pulse launched.The same can be confirmed using ∆T = DL f ∆λ, where D is the dispersion parameter [31].Hence, it is important to consider the effect of source spectral width on the system performance.As different spectral components of the pulse will receive the other end of the fiber at different time instants, the optical pulse launched gets broadened resulting in inter-symbol interference (ISI), thus reducing the performance of the system.Therefore, placing the proposed structure in the optical communication system reduces the source spectral width and thus improves system performance.

Proposed experimental setup
For a practical perspective, one can employ the experimental setup depicted in figure 4. As illustrated, four FBGs are connected using four broadband optical circulators.The input optical signal generated from a C-band amplified spontaneous emission (ASE) source is given to terminal 1 of circulator 1.The output from circulator 1 (refer to port 2) will excite FBG 1.Since the FBG will reflect only a narrow-band of light, one cannot collect the entire band of the input light at terminal 3 of circulator 1.That means most of the light will pass through FBG 1.However, the reflected signal (port 3 of circulator 1) will retain its spectral purity assuming very strong spectral  isolation among the ports of the circulator.To visualize the transmitted spectrum of the FBG, an optical spectrum analyzer (OSA) can be connected to the output terminal of FBG 1.
The reflected signal from FBG 1 is then transmitted to FBG 2 through circulator 2. The final reflected signal will be received at terminal 3 of circulator 4 as a result of an operation similar to that earlier mentioned.Also, optical amplifiers can be used to compensate for the circulator insertion losses while using the proposed device in the optical system.

Dispersion compensation using FBG
To bare the parameters include insertion loss, non-linear effects and expensiveness, one can use FBGs for dispersion compensation purposes instead of dispersion compensation fiber (DCF) [13,16].Particularly, a chirped FBG or CFBG can be used for dispersion compensation purposes [32,33].Generally, the grating pitch of a CFBG is not constant rather it is linearly varied as per the relation Λ N = Λ 1 + bz N , where b in terms of (nm/cm) represents the linear chirp parameter while N indicates the number of uniform FBG sections used to piecewise approximate the CFBG [7,22,34].Λ 1 , Λ N are the grating pitch of the first and N th uniform FBG section, respectively.In a linearly increased chirp (LIC) FBG, the low-frequency components of a pulse through the CFBG get delayed more due to the increasing optical pitch in the grating region.Such a LIC FBG provides an anomalous group velocity dispersion (GVD).And, the same can be used to nullify the normal GVD.The dispersion parameter D g of a LIC FBG of length L g can be written as, D g = T Rt /(L g × ∆λ).Where T Rt is the round trip time in the grating region of the fiber equals to 2n eff L g /c, c is the velocity of light in vacuum, and ∆λ denotes the bandwidth of the CFBG [31].Therefore D g can be reshaped and made equal to 2 × n eff /(c × ∆λ).

Optical communication system with proposed cascaded FBG structure
In this section, the proposed structure discussed in section 2.2 has been kept in the optical communication system to estimate its performance.The same can be verified by looking at figure 5.The blue solid or dotted lines indicate the electrical signal path while the green color refers to the optical signal path.The system consists of a transmitter, communication channel and receiver.The purpose of the optical transmitter is to convert the electrical signal into an optical signal.In this case, the launched optical pulses from the transmitter reached the receiver through a communication channel mentioned as optical fiber cable (OFC).The role of a receiver is to convert the received optical signal from the other end of the OFC into the original electrical signal.
In figure 5, one can identify the transmitter components as a pseudo-random bit sequence (PRBS) generator, CW laser, non-return to zero (NRZ) pulse generator, proposed cascaded FBG structure, and a MZ modulator.The output of a PRBS generator contains the sequence of ones and zeros emanating at a bit rate of 12.5 Gbps and is connected as an input to the NRZ pulse generator.The NRZ pulse generator is responsible for creating a sequence of non-return to zero electrical pulses that are coded by a digital signal format with a duty cycle of 0.5.The electrical pulses from the pulse generator are given as one of the two inputs of the MZ-modulator.Similarly, a continuous type output from a CW laser with a frequency of 193.41 THz (wavelength of 1550.00 ,nm) and power level of +5 dBm connected as an input to the proposed cascaded structure (particularly given as input signal to FBG 1).The proposed structure is responsible for reducing the spectral width of the source or the given input signal (as discussed in section 2).Hence, the reflected optical signal from FBG 4 of the proposed structure has reduced spectral width and is connected as an input to the other terminal of MZ-modulator.The extinction ratio of +30 dB of the modulator can perform the phase modulation.The phase-modulated output from the modulator reached the receiver through the OFC.
The L f of the OFC for the constructed system has been considered as 105 km.Also, the attenuation offered and the dispersion parameter D of the fiber are taken as 0.2 dB km −1 and 16.75 ps/(km-nm), respectively.The optical signal travelled through the OFC cable must be amplified using an optical amplifier.Typically, the reason for using optical amplifiers is to compensate for the fiber losses for proper recovery of the signal at the receiver.Hence, erbium-doped fiber amplifiers (EDFAs) with a gain (G) of 10 dB and noise figure (NF) of 4.9 dB are used for the optical amplification purposes before and after the chirped fiber Bragg grating (CFBG).However, a CFBG of length 13 mm, linear chirp parameter of 1 nm cm −1 and ∆n of 1 × 10 −4 had been used as a dispersion compensation element in the system to enhance its performance.After the post-amplification of an optical signal from the CFBG, it is given to the receiver or Photodetector (PD).The used PD in the system is considered a PIN photodiode with a responsivity of 0.69 A W −1 and has a dark current of 10 nA.Using PD in the system is to convert the optical signal into an electrical signal.So, the electrical signal from a PIN is connected as an input to the low pass filter (LPF) of the Bessel type with a cut-off frequency of 0.75 times the bit rate.Finally, the output from the LPF can be visualized using a BER analyzer.The key parameters of the equipment used in the transmission system for the simulation are depicted in table 3.

Simulation results and discussion
First we have constructed a cascaded FBG structure and observed its spectral characteristics using MATLAB as depicted in figure 3. Next, we have kept the same proposed structure in an optical communication system to reduce the spectral width of the source.The performance of the system in terms of BER, Q-factor, and eye height is visualized using OptiSystem software.In this work, each FBG of a cascaded structure is built on SMF-28 fiber.And, the design parameters for the proposed cascaded structure and the system are considered as per the earlier discussion in sections 2 and 3, respectively.In figures 6 and 7, black and pink colors are designated to estimate the performance of the system without and with the proposed structure, respectively.It is evident from figure 6(a) that the system operated on C-band (1530-1565 nm) at a bit rate   that as the value of Q increases, the value of BER decreases, escalating the system's performance.However, a mathematical approximation of BER attained using Q-factor can be made equal to 0.5erfc(Q/ √ 2) [35,36].Apart from the Q-factor and BER, the performance of the system is analyzed in terms of eye height ,which reflects the quality of the signal being noted as 0.00069 a.u. and 0.00071 in the absence and presence of the cascaded structure, respectively.Hence, it is evident from the above discussion that the values like Qfactor, BER, and eye height achieved with the proposed cascaded structure are better than those without a cascaded structure in the system.
In the preceding discussion, we have considered the L f as 105 km for the system.And, the proposed system can be used for WDM applications.Now, we are varying the L f and keeping all the design parameters same for the system shown in figure 5 to check its performance.It has been observed from the table 4 that as the L f changes from 60 km to 130 km, the value of Q decreases from 26.758 to 02.401 and from to 02.426 in the absence and presence of a cascaded structure, respectively.As we already know that decrease in a Q affects the increase in the BER value.Accordingly, the value of BER changes from 0 to 8.1564 × 10 −03 and 0 to 7.4628 × 10 −03 for without and with the presence of the cascaded structure, respectively.Similarly, the eye height decreases from 0.01088 a.u. to −0.00007 a.u. and from 0.01090 a.u. to −0.00006 a.u.for the absence and of the cascaded structure respectively.Hence it is emphasized from the discussion that the performance of the system is degraded as the L f increases.
It is also important to note that one cannot use the system at distances greater than 105 km for WDM applications as the maximum Q becomes smaller than 6.00 and the minimum BER will go with greater than 10 −9 .As a result, this is the maximum distance that can be covered while maintaining adequate Q and BER.In addition to the Q and BER estimation, a simulated model of an eye-diagram analysis regarding the eye height is shown in figure 7. It is observed from figure 7(a) that in the absence of cascaded structure one can get the eye height as 3.43 × 10 −3 a.u. at L f of 80 km.Likewise, in the presence of a cascaded structure, we can achieve an eye height of 3.47 × 10 −3 a.u.which is better in value than compared to without the presence of the cascaded structure is shown in figure 7(b).

Performance estimation of the system with the apodized FBGs in the structure
In the earlier discussion one can note that the performance of the system is improved with the cascaded structure.Further improvement in the system performance can be attained with the aid of apodized FBGs in the cascaded structure.And, the same can be observed by looking at the performance metrics of the system from tables 5 and 6, respectively.Here, we have considered two different functions for the apodization of FBGs, namely the Gaussian and hyperbolic tangent [30,37,38].To get the optimum response of the system the value of Gauss and tanh parameters are considered as 0.5.Therefore, it is evident from tables 5 and 6 that the value of Q changes from 27.361 to 02.471 for Gaussian apodized FBGs whereas the value of Q switches from 27.372 to 02.485 for hyperbolic tangent apodized FBGs in the structure.The corresponding BER values vary from 0 to 6.5721 × 10 −03 for the case of Gaussian apodized FBGs while the values are altered from 0 to 6.3781 × 10 −03 for hyper tangent apodized FBGs.An eye height variation from 0.01091 a.u to 0.00043 a.u. and from 0.01092 a.u. to 0.00044 a.u are being observed up to a distance 110 km for the Gaussian and tanh apodized FBGs, respectively.But, there is no further enhancement in eye height beyond the distance of 110 km compared to uniform FBGs in the structure.However, at the maximum operating distance (i.e.105 km) the values of Q are obtained as 06.601 and 06.712 for Gaussian and hyperbolic tangent apodized FBGs, respectively.The Hence, one can confirm from the above discussion that the presence of a cascaded uniform FBG structure has given better performance than the single uniform FBGs of varying lengths in the system.And the same is true for the proposed structure with Gaussian and Hyperbolic tangent apodized FBGs.

Performance test of the system with multilevel modulation format
So far, we have used the NRZ modulation format and the cascaded FBG structure in the proposed system to transmit data over the fiber.The performance of the proposed system has also been tested for multilevel modulation format.In this case, we have used four-level pulse amplitude modulation (PAM-4).Usually, such a modulation format doubles the data rate of the system compared to the NRZ modulation format.And, one does not need any expensive coherent optical  receiver to detect the signal.Hence such a modulation format can be used for data center applications to increase the serial line rate of the NRZ link.However, the generation of such an optical PAM-4 signal using two NRZ signals is demonstrated in [39].Here, we have initially checked the system's performance without a cascaded structure using PAM-4 modulation for different data rates and fiber lengths.Later we used a cascaded FBG structure in the system to reduce the source spectral width, thereby enhancing the system performance is checked.After that, we studied the system's performance by considering different apodized FBGs in the cascaded structure.structure have given us better performance than all the cases we discussed earlier.

Comparative study
A comparative study of various recently reported cascaded structures and their performances on the optical transmission systems with the present work is reported in table 8.In all the listed cascaded structures [18,20,40,41], authors used the same type and same length of FBGs except the one presented in [7].In the present work, we have used the same type of FBGs in the structure with varying lengths and tested its performance in the optical link.The reason for using a varying length cascaded structure over the same length structure is discussed in sections 1 and 2 receptively.It is evident from the table that the presented cascaded structure with four uniform FBGs [20] of the same type and same length has given an FWHM of 0.160 nm.However, this value has been improved to 0.008 nm with the apodized FBGs in the structure.Similarly, a cascaded structure formed using Cauchy's apodized FBGs has given us an FWHM of 0.102 nm [18].This structure has been used to reduce the source spectral width of the optical link, which is operated at a bit rate of 10 Gbps and tested up to 100 km and has given a minimum BER of 4.24 × 10 −10 at 30 km.To improve the performance by reducing the chromatic dispersion of an optical link of 100 km, a cascaded Hyperbolic tangent apodized FBG structure of the same lengths in symmetrical compensation mode along with the modified duobinary modulation scheme is introduced in [40].The introduced transmission system is tested at a bit rate of 10 Gbps and has given a BER of 8.44 × 10 −11 .Further improvement in the BER to 2.59 × 10 −13 can be achieved using four identical types of LIC FBGs of the same lengths used in 70 km link operated with a bit rate of 10 Gbps along with the differential phase shift keying or DPSK modulation is presented in [41].However, a cascaded CFBG structure formed with one LIC and one linearly decreased chirp (LDC) FBGs has given an FWHM of 1.56 nm is mentioned in [7].On the other hand, a proposed four-stage cascaded FBG structure formed with hyperbolic tangent apodized FBGs has given better spectral characteristics (narrow FWHM of 0.07 nm and SLSR of 89.41 dB) than the structures in [18,20,40,41] is listed in the table as present work.The structure's performance is tested on a 105 km optical link operated at a bit rate of 12.5 Gbps, giving a BER of 9.53 × 10 −12 .Therefore, one can deduce from the above discussion that the introduced optical system with the proposed structure can carry more information over a longer distance than the existing systems with acceptable BER (generally considered less than 10 −9 for telecommunications).

Conclusions
In this paper, we have proposed an optical communication system formed with a cascaded FBG based passive optical device operating in the C-band.Such an optical device constructed using four FBGs of varying lengths is demonstrated.A suitable analytical formulation for such a device based on TMM is incorporated.Simulation results corroborating the analytical formulation are included.As per the simulated results, the proposed device has a maximum reflectivity of 98.39% and a minimum FWHM of 0.07 nm.Simulation results confirm that in the presence of the optical device, the system performance (in terms of Q-factor, BER, and eye height) has been increased compared to the absence of the device.Also, at an operating distance of 105 km, the estimated values of Q-factor and BER are recorded as 6.712 and 9.5321 × 10 −12 , respectively.An experimental study to implement and validate the theoretical findings at the device and system level is the future aspect of the present work.However, based on the simulated findings, one can infer that the system formed with a cascaded structure also supports the PAM-4 modulation for short-distance applications.Furthermore, a comparative study of the system performance for various operating distances with uniform and apodized FBGs in the optical device is furnished.Based on such a study, we found that hyperbolic tangent apodized FBGs in the structure have given us better performance than uniform and Gaussian apodized FBGs.

Figure 1 .
Figure 1.Schematic of single uniform FBG structure: (a) Block diagram representation, (b) r.i.variation inside the fiber core along the grating length L = 10 mm with period Λ = 0.534 µm.

Figure 2 .
Figure 2. Schematic of proposed four stage cascaded FBG structure of different lengths.

Figure 4 .
Figure 4. Experimental setup for four stage cascaded FBG structure of varying lengths .

Figure 5 .
Figure 5. Performance enhancement of optical communication system with the proposed cascaded FBG structure.

Figure 6 .
Figure 6.Performance estimation of optical communication system: (a) Q-factor without cascaded structure.(b) Q-factor with cascaded structure.(c) Logarithm value of BER without cascaded structure.(d) Logarithm value of BER with cascaded structure.

Figure 7 .
Figure 7.A model of eye diagram analysis with regard to eye height at L f of 80 km: (a) without cascaded (b) with cascaded structure.

Figure 8 .
Figure 8. Eye diagram analysis using PAM-4 modulation format for different data rates of the proposed system in the absence of the cascaded structure.

Figure 9 .
Figure 9. Eye diagram analysis using PAM-4 modulation format for different data rates of the proposed system in the presence of the cascaded structure.
It is evident from figure8that as the data rate increases from 5 Gbps to 40 Gbps, the BER getting increases for different L f 's (1 km, 5 km, 10 km, 20 km, and 25 km) of the OFC.Therefore the opening of the eye is getting reduced.The red line in each eye diagram indicates the minimum BER curve.An improvement in the system performance in terms of BER achieved with the cascaded FBG structure in the system is depicted in figure9.Compared with figures 8 and 9 has given us better eye diagram.Therefore one can say that the proposed system also supports the multilevel modulation format.Similarly, as we already know that apodized FBG will give us better performance than the normal or uniform FBG.Hence we have considered Gaussian and hyperbolic tangent apodized FBGs in the cascaded structure and used them along with PAM-4.It can be demonstrated from figures 10 and 11 that the system has given improved performance in terms of BER or opening of an eye for the Gaussian and hyperbolic tangent apodized FBGs compared to without or with uniform FBGs in the structure.In particular, hyperbolic tangent apodized FBGs in the

Figure 10 .
Figure 10.Effect of system for different data rates by considering Gaussian apodized FBGs in the cascaded structure along with PAM-4 modulation format.

Figure 11 .
Figure 11.Effect of system performance for different data rates by considering hyperbolic tangent apodized FBGs in the cascaded structure along with PAM-4 modulation format.

Table 1 .
Spectral characteristics of cascaded FBGs at λ B of 1550.00 nm.

Table 2 .
Performance comparison of a single FBG with the cascaded FBGs of varying lengths.

Table 3 .
Key parameters of the equipment used in the optical transmission system.
the same decision instant, and one can observe the same from figure 6(b).The corresponding minimum BER values ( plotted on a logarithmic scale with base 10) for these cases (with and without proposed structure) are marked as −10.111 and −10.589 (equals to 7.7358 × 10 −11 and 2.5713 × 10 −11 on a regular scale) in figures 6(c) and (d), respectively.It indicates

Table 4 .
Performance estimation of the system with uniform FBGs in the structure for different lengths of the fiber.

Table 5 .
Performance estimation of the system with Gaussian apodized FBGs in the structure for different lengths of the fiber.

Table 6 .
Performance estimation of the system with hyperbolic tangent apodized FBGs in the structure for different lengths of the fiber.Performance estimation of the system with single and cascaded FBG structure at the maximum operating distance.In particular, hyperbolic tangent apodized FBGs given better performance compared to the Gaussian apodized FBGs in the structure and the same can be confirmed from table 7. On the other hand, as the length of the single FBG changes from 10 mm to 12 mm (depicted in table 7), the value of Q alters from 06.410 to 06.470 for the case of uniform FBGs, whereas the value of Q switches from 06.501 to 06.545 for Gaussian apodized FBGs.The corresponding BER values vary from 7.2291 × 10 −11 to 4.8685 × 10 −11 for the case of uniform FBGs while the values are changes from 3.9634 × 10 −11 to 2.9551 × 10 −11 for Gaussian apodized FBGs.Similarly, the values of Q for a single hyperbolic tangent apodized FBG in the system vary from 06.570 to 06.600, and the corresponding BERs are altered from 2.4991 × 10 −11 to 2.0420 × 10 −11 .However, with the proposed structure, the value of Q has been improved to 06.566 and 06.601 for uniform and Gaussian apodized FBGs of varying lengths (lengths are varied as discussed in section 2.2, with the first FBG of length 10 mm).In contrast, the value Q for hyperbolic tangent apodized FBGs of varying lengths in the structure is recorded as 06.712.The corresponding values of BER are recorded as 2.5713 × 10 −11 and 2.0281 × 10 −11 for uniform and Gaussian apodized FBGs, whereas the value of BER for hyperbolic tangent apodized FBGs is achieved as 9.5321 × 10 −12 .

Table 8 .
Performance comparison of recently reported cascaded structures on the optical transmission system .