- Email: [email protected]
A numerical method for pulse propagation in nonlinear fiber Bragg grating with ternary stability nature
Optical Fiber Technology 54 (2020) 102075
Contents lists available at ScienceDirect
Optical Fiber Technology journal homepage: www.elsevier.com/locate/yofte
A numerical method for pulse propagation in nonlinear fiber Bragg grating with ternary stability nature
T
⁎
Elham Yousefia,b, , Mohsen Hatamic a
Faculty of Science, Persian Gulf University, Bushehr 7516913817, Islamic Republic of Iran Atomic and Molecular Group, Faculty of Physics, Yazd University, Yazd, Islamic Republic of Iran c Physics Department, Shiraz University of Technology, Shiraz, Islamic Republic of Iran b
A R T I C LE I N FO
A B S T R A C T
Keywords: All-optical devices Chalcogenide fiber Bragg grating Nonlinear optics Optical signal processing Pulse propagation
Optical bistability and multistability are often produced in nonlinear media. Although continuous wave (CW) bistability and multistability have widely studied, solution of the nonlinear pulse propagation equations is rarely studied and practiced in bistable and ternary stable systems. In such systems, the history of the pulse should be considered for equation solving. In the present paper, the pulse propagation through the ternary nonlinear chalcogenide fiber Bragg grating is numerically investigated. A novel numerical method is proposed for solving the above mentioned problem using Fourier and inverse Fourier transform as well as dividing the Gaussian input pulse into two intervals which leads to the simplicity of hysteresis and time domain problems. As a result, an algorithm is suggested for solving nonlinear pulse propagation equation in bistable and ternary stable systems and the output pulse is obtained. This method can also be applied on various pulses of different shapes travelling through the other types of Fiber Bragg grating like chirped, phase-shifted, etc. Optical multistability systems are the main building blocks for designing all-optical pulse processing, all-optical memory and all-optical switching.
1. Introduction Fiber Bragg gratings (FBGs) are an attractive subject for linear applications not only in optical communication systems such as wavelength division multiplexing (WDM), optical filters [1–3], and pulse dispersion compensation [4], but also in other scientific fields such as strain sensors [5], pressure sensors [6],temperature sensors [7], acoustic sensors [8] and optical sensing [9,10]. Actually, through past decades because of showing pulse compression [11–13], high-speed optical switching [14–16], optical bistability [17] and optical multistability [18,19] effects in nonlinear fiber Bragg gratings, some ideas have been formed about all-optical communication systems [20–22]. Such systems can be controlled just only by light, hence; the prominent advantage of these systems is that their operation does not require converting the optical signals into the electrical ones or vice versa by complex electrical circuits. Moreover, the light in fiber transfers noisefree unlike the electrical signal in wires and also the speed of light is comparatively higher than the speed of an electron [1]. In recent works on FBG development, it has been concluded that FBG can show the bistability [23,24] and the ternary stability [18] characteristics in continuous wave (CW) propagation regime. Because of these nonlinear characteristics, FBG in optical systems can perform
⁎
like a binary or ternary element in digital electrical systems. Therefore, it can be utilized in all-optical switching systems [25,26], all-optical memories [21,27] and also in all-optical signal processing [18,23]. It is worth mentioning that FBG can be made up of silica or chalcogenide glasses where the chalcogenide glasses demonstrate about 200 times more nonlinearity effects than the silica glasses [28,29]. So, chalcogenide FBGs show optical switching in lower input intensities i.e. lower laser intensities can be used to fire the chalcogenide FBGs to show multistability effects respect to the glasses one [18,30]. In our latest works, we have investigated numerically the quasipulse propagation through the nonlinear chalcogenide FBGs with bistability [18] and the ternary stability [23] features. In the present study, the pulse propagation through the nonlinear FBG is studied numerically. Although some valuable efforts are done on solving the soliton solution of nonlinear Schrodinger equations (spatial equations) [31–35] there is no enough effort has been made to solve the nonlinear coupled mode equations (temporal equations) in the time domain with multistable nature of medium recently [22,36,37]. In this paper, firstly the pulse propagation equations through the nonlinear chalcogenide-glass-made FBG are expressed in detail. Secondly, the ternary stability of this system will be numerically studied. Finally, the new method is proposed and used for solving the pulse
Corresponding author. E-mail address: elhamyousefi@stu.yazd.ac.ir (E. Yousefi).
https://doi.org/10.1016/j.yofte.2019.102075 Received 19 April 2018; Received in revised form 25 October 2019; Accepted 12 November 2019 1068-5200/ © 2019 Published by Elsevier Inc.
Optical Fiber Technology 54 (2020) 102075
E. Yousefi and M. Hatami
propagation equations in the nonlinear FBG with multi-scale characteristics, especially with ternary stability nature. Here, Gaussian pulse is utilized as an input pulse and since the multi-stability is a hysteresis problem, the division of input pulse in time domain helps to consider the history of the pulse. This method utilizes Fourier and the inverse Fourier transform as well. The novel proposed method will be thoroughly explained in seven steps in section five with an illustrative flow chart. 2. Theory and numerical calculations The nonlinear periodic refractive index can be shown as [38]:
2π n (z ) = n 0 + n1 (z ) cos ⎛ z + ϕ (z )⎞ + n2 |E|2 + n4 |E|4 + n6 |E|6 ⎝Λ ⎠
(1)
where E is the electric field in FBG, Λ is the grating period, n 0 , n1 are the linear refractive index and refractive index modulation amplitude respectively. n2 , n4 , and n6 denote the third order, fifth order, and seventh order nonlinear refractive index coefficients, respectively. The forward and backward waves in FBG form the electrical wave like Eq. (2) as reported in [38]:
Fig. 1. A typical ternary stability curve. The odd branches are stable and the even ones are unstable. The considered trajectories and ternary stability interval are also pointed out.
1
E (r , t ) = 2 F (x , y ) [Af (z , t ) eiβB z + Ab (z , t ) e−iβB z] e−iω0 t + c. c.
stability interval, the forward path, and the backward path in the next section.
(2)
where ω0 is the carrier frequency, Af and Ab are the slowly varying amplitudes of forward and backward waves, respectively. F (x , y ) represents the transverse variations of two counter-propagating waves. Substituting (1) and (2) into the Maxwell wave equations and applying slowly varying amplitude approximation, the nonlinear coupled mode equations for both forward and backward wave amplitudes are obtained [30]:
∂Af ∂z
+ β1
∂Af ∂t
+
3. Optical ternary stability curve and trajectories By As mentioned in [18] ternary stability can be observed in chalcogenide FBGs because of their higher nonlinearity characteristics. Fig. 1 depicts one type of this curve, which is obtained from the parameter values reported in Table 1. It has five branches which are highlighted by their numbers. “Odd” branches, i.e. branches No. 1, 3, and 5 are stable and “even” branches, i.e. branches No. 2 and 4 are unstable. Ternary stability curves have an input intensity interval that is that is the result of overlapping between branches No. 1, 3 and 5. This interval is depicted in Fig. 1 by the words: “Ternary Stability Interval”. Because of this interval, these kinds of curves are called ternary stability curves. In this curve by increasing the input intensity from point A (about 0 MW/cm2 values) to point F (about 37 MW/cm2 value), the FBG passes the ABCDEF path in the ternary stability curve. These kinds of paths, which originate from lower intensities to the higher ones in FBG stability curves in CW-regime, are referred to as “forward path” in the present paper. There are two jumps in the ternary stability forward path. One jump can be observed from point B to point C (jump No. 1) and the other, from point D to point E (jump No. 2). In reverse, by decreasing input intensity from point F to point A, the curve does not follow up the previous path, but it passes the FGHIJA path in the ternary stability curve. Again, to refer to this kind of paths- which originate from higher intensities to the lower ones in the stability FBG curves- they are called “backward path” in the present paper. There are also two jumps down in the backward path. The first one is the jump from the point G to point H (jump No. 3) and the other from the point I to point J (jump No. 4 The length of both jumps in the backward path is almost smaller than the length jumps No.1 and No. 2 in the forward path. Because of the existence of these two different paths, i.e. forward and backward paths between the same points A and F, the pulse propagation investigations in FBGs that are characterized by these kinds of curves are not simple. However, we show that by some delicate
iβ2 ∂2Af α + Af = iδAf + iκAb + iγ1 (|Af |2 +2|Ab |2 ) 2 ∂t 2 2 Af + iγ2 (|Af |4 +6|Af |2 |Ab |2 +3|Ab |4 ) Af + iγ3 (|Af |6 +12|Af |4 |Ab |2 +18|Af |2 |Ab |4 +4|Ab |6 ) Af ,
−
(3)
iβ ∂2Ab α ∂Ab ∂A + β1 b + 2 + Ab = iδAb + iκAf + iγ1 (|Ab |2 +2|Af |2 ) 2 ∂t 2 2 ∂z ∂t Ab + iγ2 (|Ab |4 +6|Ab |2 |Af |2 +3|Af |4 ) Ab , +iγ3 (|Ab |6 +12|Ab |4 |Af |2 +18|Ab |2 |Af |4 +4|Af |6 ) Ab
(4)
whereα is the FBG loss which is negligible in the case studied in the present paper, δ and κ are the detuning from the Bragg wavelength and coupling coefficient, respectively. γ1, γ2 , and γ3 account for third order, fifth order and seventh order nonlinear parameters, respectively. One common way of investigating the pulse propagation is to solve Eqs. (3) and (4) numerically in the frequency domain [38]. It is performed by replacing i (∂/ ∂t ) into (ω-ω0), and the coupled mode equations in continuous wave (CW) regime (i.e. with one frequency) are obtained. The Runge-Kutta method is used for numerical calculations of equations, that obtaining in the frequency domain, from end to the beginning with special boundary conditions and given frequency [18,22]. A typical and correct result of output intensity versus input intensity is presented in Fig. 1 which is shown ternary stability curve; terminology is borrowed from the concept of bistability. In bistability curves there exist two points of stable output intensity for a given input intensity. In a similar way, in ternary stability there exist three points of stable output for a given input intensity and we called it ternary stability curves. For becoming more clear our strategy which is expressed in Section 5, it is necessary to introduce some important concepts like ternary
Table 1 Typical data which are applied in this article[18]
2
λB = 1550 nm
n 0 = 2.54
n2silicon = 0.273 × 10−19m2/ w [40]
λ = 1550.015 nm κL = 7
n1 = 1.5 × 10−4 n2 = 220 n2 silicon [39]
n6 = 7.2 × 10
n 4 = - 7.8 × 10
- 31
- 45
m4 /W2 [28]
m6/W3 [28]
Optical Fiber Technology 54 (2020) 102075
E. Yousefi and M. Hatami
Fig. 3. Input pulse intensity shape in the negative part of the time axis.
Fig. 2. Input intensity versus of the time with a typical peak intensity about 225 MW/cm2 and t0 = 0.173 ns.
numerical calculations and by knowing some tips that are presented in our latest reports [18,23]; these kinds of calculations can be easily performed. 4. Gaussian pulse as an input pulse For the simulation, the following Gaussian pulse is considered as an input pulse to the FBG [39]:
A exp(−t 2/2t02),
(6)
where A is the peak intensity and t0 is the halftime width. The input intensity is sketched in Fig. 2 with t0 = 0.173 ns. Short pulse has a wide spectrum with continues range of frequencies. To simulate the propagation of such pulses, Fourier transform should be utilized to convert the pulse shape into the pulse spectrum. The pulse spectrum is divided into a certain number of frequencies with the same interval. FBG coupled mode equations are solved numerically for each component of divided frequency. By using the results and our proposed strategy the pulse propagation through FBG will be simulated. The proposed strategy is completely expressed in seven steps in the next section.
Fig. 4. Input pulse intensity shape in the positive part of the time axis.
5. The simulation method for pulse propagation through the FBG In this section, the proposed novel method for simulation of pulse propagation through the FBG with ternary nature effects is explained in seven steps. Step 1- The input pulse intensity is divided into two intervals: the intensities which are located on the negative part of the time axis and the intensities which are on the positive part of the time axis as depicted in Figs. 3 and 4 respectively. Step 2- In this step, the square root of pulse intensity is being done to obtain the pulse shape field from two intervals of the intensities in step 1 (from Figs. 3 and 4). The pulse is decomposed into two parts, i.e. negative and positive parts. The pulse shape which is in negative time is called “negative part” and the pulse shape which is in positive time is called “positive part” in this paper. Step 3- The Fourier transform (FT) of each pulse shape is calculated which is shown in Fig. 5. Evidently, because of the FT traits, both pulse shapes have the same pulse spectrum. Step 4- Afterwards, the pulse spectrum is divided into many frequency intervals and the stability curves of each frequency are obtained. Some of these curves are shown in Fig. 6. As it can be noticed in this figure, a few curves have bistability characteristic which we have already addressed in [23]. However, many curves have ternary
Fig. 5. Fourier transform of each two pulse shapes.
stability shapes. Step 5-In order to simulate the pulse propagation, the bistability and ternary stability curves are needed for each frequency. It is because we need to obtain the output intensities related to each input intensity of each divided frequency. It is initially performed for the negative part of the pulse that enters at the FBG earlier than the positive part. So, the “forward path” in bistability and ternary stability curves has to be used for obtaining the output intensity (output No. 1). On the other hand, the “backward path” is used for 3
Optical Fiber Technology 54 (2020) 102075
E. Yousefi and M. Hatami
Fig. 8. Output No. 2 versus wavelength.
Fig. 6. The bistability and ternary stability curves which are obtained from coupled mode equations simulated for different frequencies.
obtaining the output intensity of positive part of the pulse that enters the FBG later (output No. 2). Finally, the intensity spectrum of the output pulses is calculated with respect to the wavelength. The shape of “output No. 1”and “output No. 2” for each wavelength (or each frequency) are shown in Figs. 7 and 8 respectively. Step 6- Both calculated outputs results, i.e. outputs No. 1 and No. 2 are summed up to obtain the total output in wavelength (frequency) domain. The output spectrum is depicted in Fig. 9. Step 7- In this step which is the last one, the “total output” is converted back into the output pulse shape in the time domain using inverse Fourier transform. The intensity of the “output pulse shape” is depicted in Fig. 10. Fig. 9. The summation of two output results (output No. 1 and output No. 2) in the frequency domain.
Fig. 11 illustrates a comparison between input and output intensities. As it was expected some delays in output intensity with respect to the input intensity can be observed. The output pulse has a bit deformation with respect to the input pulse. The output pulse width is almost the half of the width of the input pulse, so the output pulse is compressed during its propagation through the ternary stability FBG. Also, the peak of the output intensity is lower than the peak of the input intensity. Fig. 12 illustrates the proposed approach for numerical calculation of simulating the pulse propagation through the nonlinear FBGs with bistability and ternary stability feature.
Fig. 10. The output (transmitted) intensity versus time, which leaves from the end of the FBG.
6. Conclusion Fiber Bragg gratings (FBGs) which are very sensitive to the wavelength of incident waves are strong optical instruments for designing all-optical systems because of their wide properties in both linear and nonlinear regimes. In our recent works [12,17], the quasi-pulse wave propagation through the nonlinear FBGs with bistability and ternary stability have been numerically investigated. In this paper the new method is proposed to investigate the pulse propagation through the
Fig. 7. Output No. 1 versus wavelength. 4
Optical Fiber Technology 54 (2020) 102075
E. Yousefi and M. Hatami
ternary stability FBGs for first time. In this method, the history of the pulse is considered by dividing it into two pulses because of the multi stability nature of the nonlinear FBG and by using Fourier and inverse Fourier transforms the output pulse is simulated numerically. An illustrative flowchart is presented to clearly explain the method of solving coupled mode equations for pulse propagation in a ternary stable medium. At last, the transmitted pulse is obtained from ternary stability FBG and there are some delays on it. The shape of the pulse is a little different form input pulse i.e. some distortion can be seen, it is because the pulse is transmitted through the fiber Bragg grating that has strongly nonlinearity effect (like ternary stability). Also the width and the peak of the output pulse are almost half of the width and the peak of the input pulse that makes FBG as a pulse compressor. It is noteworthy that such studies introduce nonlinear FBGs as the main building block for all-optical signal processing systems and all-optical integrated circuits that can have their role in the development of future all-optical computers.
Fig. 11. A comparison between the input intensity (dashed line) and the output intensity (solid line).
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] A. Othonos, K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing, ArtechHouse, 1999. [2] X. Han, J. Yao, Bandstop-to-bandpass microwave photonic filter using a phaseshifted fiber Bragg grating, J. Lightwave Technol. 33 (2015) 5133–5139. [3] M.M. Ali, K.S. Lim, A. Becir, M.H. Lai, H. Ahmad, Optical gaussian notch filter based on periodic microbent fiber Bragg grating, IEEE Photonics J. 6 (2014) 1–8. [4] A.B. Dar, R.K. Jha, Chromatic dispersing compensation techniques and characterization of fiber Bragg grating for dispersion compensation, Opt. Quant. Electron. 49 (2017) 108. [5] L. Yuan, Y. Zhao, Sh. Sato, Development of a low-cost and miniaturized fiber Bragg grating strain sensor system, Jpn. J. Appl. Phys. 56 (2017) 052502. [6] F. Urban, J. Kadlec, R. Vlach, R. Kuchta, Design of a pressure sensor based on optical fiber Bragg grating lateral deformation, Sensors 10 (2010) 11212. [7] V. Kothari, A. Bedi, S. Kumar, and L. Singh, Fiber Bragg Grating based Temperature Sensor for Biomedical applications, in Frontiers in Optics 2017, OSA Technical Digest (online) (Optical Society of America, 2017), paper JTu2A.59 (2017). [8] D.C. Betz, G. Thursby, B. Culshaw, W.J. Staszewski, Acousto-ultrasonic sensing using fiber Bragg grating, Smart Mater. Struct. 12 (2003). [9] D. Barrera, J. Madrigal, S. Sales, Tilted fiber Bragg gratings in multicore optical fibers for optical sensing, Opt. lett. 42 (2017) 1460–1463. [10] G. Alvarez-Botero, F.E. Baron, C.C. Cano, O. Sosa, M. Varon, Optical sensing using fiber Bragg gratings: fundamentals and applications, IEEE Instrum. Measure. Mag. 20 (2017) 33–38. [11] T. Manjo Divagar, S. James Raja, S. Ilayaraja, Optical pulse compression and its effects using fiber Bragg grating, Int. J. Adv. Res. Comput. Eng. & Technol. (IJARCET) 3 (2014) 1497. [12] W.S. Al-Younis, M.H. Bataineh, Pulse compression using Fiber Bragg gratings, 2017 4th International Conference on Electrical and Electronic Engineering (ICEEE), Ankara, 2017, pp. 240–245. [13] Julijanas Želudevičius, Rokas Danilevičius, Kęstutis Regelskis, Optimization of pulse compression in a fiber chirped pulse amplification system by adjusting dispersion parameters of a temperature-tuned chirped fiber Bragg grating stretcher, J. Opt. Soc. Am. B 32 (2015) 812–817. [14] Z. Yu, W. Marguilis, O. Tarasenko, H. Knape, P.Y. Fonjallaz, Nanosecond switching of fiber Bragg grating, Opt. Express 15 (2007) 14948. [15] Ľ. Scholtz, L. Ladányi, J. Müllerová, Numerical modeling of all optical self switching in chalcogenide fiber Bragg grating, 2015 16th IEEE International Symposium on Computational Intelligence and Informatics (CINTI), Budapest, 2015, pp. 123–127. [16] H. Zoweil, J.W.Y. Lit, Bragg grating with periodic non-linearity as optical switch, Opt. Commun. 212 (2002) 57. [17] E. Yousefi, M. Hatami, S. Dehghani, Optimization of Bistability in nonlinear chalcogenide fiber Bragg grating for all optical switch and memory applications, Int. J. Opt. Photon. (IJOP) 11 (2017) 49–55. [18] E. Yousefi, M. Hatami, A. Torabi Jahromi, All-optical ternary signal processing using uniform nonlinear chalcogenide fiber Bragg gratings, J. Opt. Soc. Am. B 32 (2015) 1471. [19] S. Pawar, S. Kumbhaj, P. Sen, P.K. Sen, Optical multistability in nonlinear fiber Bragg grating, Optoelectron. Adv. Mater. 6 (2012) 25. [20] A. Melloni, M. Chinello, M. Martinelli, All optical switching in phase-shifted fiber Bragg grating, IEEE Photonics Technol. Lett. 12 (2000) 42. [21] M. Karimi, M. Lafouti, A.A. Amidiyan, J. Sabbaghzadeh, All-optical flip-flop based on nonlinear effects in fiber Bragg gratings, Appl. Opt. 51 (2012) 21.
Fig. 12. The flowchart of our proposed method to investigate the pulse propagation through the multistability FBGs.
5
Optical Fiber Technology 54 (2020) 102075
E. Yousefi and M. Hatami
(2019) 88–94. [32] A. Biswas, J. Vega-Guzman, M.F. Mahmood, S. Khan, Q. Zhou, S.P. Moshokoa, M. Belic, Solitons in optical fiber Bragg gratings with dispersive reflectivity, Optik 182 (2019) 119–123. [33] A. Biswas, M. Ekici, A. Sonmezoglu, M.R. Belic, Optical solitons in fiber Bragg gratings with dispersive reflectivity for parabolic law nonlinearity by extended trial function nonlinearity, Optik 183 (2019) 595–601. [34] A. Biswas, J. Vega-Guzman, M.F. Mahmood, M. Ekici, Q. Zhou, S.P. Moshokoa, M.R. Belic, Optical solitons in fiber Bragg graings with dispersive reflectivity for parabolic law nonlinearity using undetermined coefficients, Optik 185 (2019) 39–44. [35] A. Biswas, M. Ekici, A. Sonmezoglu, M.R. Belic, Optical solitons in fiber Bragg gratings with dispersive reflectivity for quadratic cubic nonlinearity, Optik 185 (2019) 50–56. [36] F. Emami, M. Hatami, A. Keshavarz, A.H. Jafari, A heuristic method to simulate the pulse propagation in nonlinear fiber Bragg gratings, Opt. Quant. Electron. 41 (2009) 429–439. [37] E. Yousefi, M. Hatami, Unique solution of short pulse propagation in nonlinear fiber Bragg grating, Int. J. Opt. Photonics (IJOP) 7 (2013) 85–90. [38] G.P. Agrawal, Applications of Nonlinear Fiber Optics, Academic Press, 2001. [39] G.P. Agrawal, Nonlinear Fiber Optics, Academic, 2013. [40] J.M. Harbold, F.O. Ilday, F.W. Wise, Highly nonlinear As-S-Se glasses for all-optical switching, Opt, Lett. 27 (2002) 119.
[22] A.F.G.F. Filho, J.R.R. de Sousa, A.F. de Morais Neto, J.W.M. Menezes, A.S.B. Sombra, Periodic modulation of nonlinearity in a fiber Bragg grating: a numerical investigation, J. Electromagn. Anal. Appl. 4 (2012) 53. [23] E. Yousefi, M. Hatami, All optical self signal processing using chalcogenide nonlinear fiber Bragg grating, Optik 125 (2014) 6637. [24] L. Scholtz, J. Müllerová, Numerical studies on wavelength-selective all-optical switching using optical bistability in nonlinear chalcogenide FBGs, 2015 17th International Conference on Transparent Optical Networks (ICTON), Budapest, 2015, pp. 1–4. [25] H. Lee, G.P. Agrawal, Nonlinear switching of optical pulses in fiber Bragg gratings, IEEE J. Quantum Electron. 39 (2003) 508–515. [26] Ľ. Scholtz, L. Ladányi, J. Müllerová, Cross phase modulation switching in chalcogenide fiber Bragg grating, 2016 ELEKTRO, Strbske Pleso, 2016, pp. 632–636. [27] Y.L. Kim, J.H. Kim, S. Lee, D.H. Woo, S.H. Kim, T.H. Yoon, Broad-band all optical flip-flop based on optical bistability in an integrated SOA/DFB-SOA, IEEE Photonics Technol. Lett. 16 (2004) 398. [28] Y. Yosia, P. Shum, Optical bistability in periodic media with third fifth- and seventh- order nonlinearities, J. Lightwave technol. 25 (2007) 875. [29] A. Zakery, S.R. Elliott, Optical Nonlinearities in Chalcogenide Glasses and Their Applications, Springer, 2007. [30] Y. Yosia, S. Ping, Double optical bistability and its application in nonlinear chalcogenide-fiber Bragg gratings, Phys. B 394 (2007) 293. [31] A. Biswas, M. Ekici, A. Sonmezoglu, M.R. Belic, Solitons in optical fiber Bragg gratings with dispersive reflectivity by extended trial function method, Optik 182
6
Contents lists available at ScienceDirect
Optical Fiber Technology journal homepage: www.elsevier.com/locate/yofte
A numerical method for pulse propagation in nonlinear fiber Bragg grating with ternary stability nature
T
⁎
Elham Yousefia,b, , Mohsen Hatamic a
Faculty of Science, Persian Gulf University, Bushehr 7516913817, Islamic Republic of Iran Atomic and Molecular Group, Faculty of Physics, Yazd University, Yazd, Islamic Republic of Iran c Physics Department, Shiraz University of Technology, Shiraz, Islamic Republic of Iran b
A R T I C LE I N FO
A B S T R A C T
Keywords: All-optical devices Chalcogenide fiber Bragg grating Nonlinear optics Optical signal processing Pulse propagation
Optical bistability and multistability are often produced in nonlinear media. Although continuous wave (CW) bistability and multistability have widely studied, solution of the nonlinear pulse propagation equations is rarely studied and practiced in bistable and ternary stable systems. In such systems, the history of the pulse should be considered for equation solving. In the present paper, the pulse propagation through the ternary nonlinear chalcogenide fiber Bragg grating is numerically investigated. A novel numerical method is proposed for solving the above mentioned problem using Fourier and inverse Fourier transform as well as dividing the Gaussian input pulse into two intervals which leads to the simplicity of hysteresis and time domain problems. As a result, an algorithm is suggested for solving nonlinear pulse propagation equation in bistable and ternary stable systems and the output pulse is obtained. This method can also be applied on various pulses of different shapes travelling through the other types of Fiber Bragg grating like chirped, phase-shifted, etc. Optical multistability systems are the main building blocks for designing all-optical pulse processing, all-optical memory and all-optical switching.
1. Introduction Fiber Bragg gratings (FBGs) are an attractive subject for linear applications not only in optical communication systems such as wavelength division multiplexing (WDM), optical filters [1–3], and pulse dispersion compensation [4], but also in other scientific fields such as strain sensors [5], pressure sensors [6],temperature sensors [7], acoustic sensors [8] and optical sensing [9,10]. Actually, through past decades because of showing pulse compression [11–13], high-speed optical switching [14–16], optical bistability [17] and optical multistability [18,19] effects in nonlinear fiber Bragg gratings, some ideas have been formed about all-optical communication systems [20–22]. Such systems can be controlled just only by light, hence; the prominent advantage of these systems is that their operation does not require converting the optical signals into the electrical ones or vice versa by complex electrical circuits. Moreover, the light in fiber transfers noisefree unlike the electrical signal in wires and also the speed of light is comparatively higher than the speed of an electron [1]. In recent works on FBG development, it has been concluded that FBG can show the bistability [23,24] and the ternary stability [18] characteristics in continuous wave (CW) propagation regime. Because of these nonlinear characteristics, FBG in optical systems can perform
⁎
like a binary or ternary element in digital electrical systems. Therefore, it can be utilized in all-optical switching systems [25,26], all-optical memories [21,27] and also in all-optical signal processing [18,23]. It is worth mentioning that FBG can be made up of silica or chalcogenide glasses where the chalcogenide glasses demonstrate about 200 times more nonlinearity effects than the silica glasses [28,29]. So, chalcogenide FBGs show optical switching in lower input intensities i.e. lower laser intensities can be used to fire the chalcogenide FBGs to show multistability effects respect to the glasses one [18,30]. In our latest works, we have investigated numerically the quasipulse propagation through the nonlinear chalcogenide FBGs with bistability [18] and the ternary stability [23] features. In the present study, the pulse propagation through the nonlinear FBG is studied numerically. Although some valuable efforts are done on solving the soliton solution of nonlinear Schrodinger equations (spatial equations) [31–35] there is no enough effort has been made to solve the nonlinear coupled mode equations (temporal equations) in the time domain with multistable nature of medium recently [22,36,37]. In this paper, firstly the pulse propagation equations through the nonlinear chalcogenide-glass-made FBG are expressed in detail. Secondly, the ternary stability of this system will be numerically studied. Finally, the new method is proposed and used for solving the pulse
Corresponding author. E-mail address: elhamyousefi@stu.yazd.ac.ir (E. Yousefi).
https://doi.org/10.1016/j.yofte.2019.102075 Received 19 April 2018; Received in revised form 25 October 2019; Accepted 12 November 2019 1068-5200/ © 2019 Published by Elsevier Inc.
Optical Fiber Technology 54 (2020) 102075
E. Yousefi and M. Hatami
propagation equations in the nonlinear FBG with multi-scale characteristics, especially with ternary stability nature. Here, Gaussian pulse is utilized as an input pulse and since the multi-stability is a hysteresis problem, the division of input pulse in time domain helps to consider the history of the pulse. This method utilizes Fourier and the inverse Fourier transform as well. The novel proposed method will be thoroughly explained in seven steps in section five with an illustrative flow chart. 2. Theory and numerical calculations The nonlinear periodic refractive index can be shown as [38]:
2π n (z ) = n 0 + n1 (z ) cos ⎛ z + ϕ (z )⎞ + n2 |E|2 + n4 |E|4 + n6 |E|6 ⎝Λ ⎠
(1)
where E is the electric field in FBG, Λ is the grating period, n 0 , n1 are the linear refractive index and refractive index modulation amplitude respectively. n2 , n4 , and n6 denote the third order, fifth order, and seventh order nonlinear refractive index coefficients, respectively. The forward and backward waves in FBG form the electrical wave like Eq. (2) as reported in [38]:
Fig. 1. A typical ternary stability curve. The odd branches are stable and the even ones are unstable. The considered trajectories and ternary stability interval are also pointed out.
1
E (r , t ) = 2 F (x , y ) [Af (z , t ) eiβB z + Ab (z , t ) e−iβB z] e−iω0 t + c. c.
stability interval, the forward path, and the backward path in the next section.
(2)
where ω0 is the carrier frequency, Af and Ab are the slowly varying amplitudes of forward and backward waves, respectively. F (x , y ) represents the transverse variations of two counter-propagating waves. Substituting (1) and (2) into the Maxwell wave equations and applying slowly varying amplitude approximation, the nonlinear coupled mode equations for both forward and backward wave amplitudes are obtained [30]:
∂Af ∂z
+ β1
∂Af ∂t
+
3. Optical ternary stability curve and trajectories By As mentioned in [18] ternary stability can be observed in chalcogenide FBGs because of their higher nonlinearity characteristics. Fig. 1 depicts one type of this curve, which is obtained from the parameter values reported in Table 1. It has five branches which are highlighted by their numbers. “Odd” branches, i.e. branches No. 1, 3, and 5 are stable and “even” branches, i.e. branches No. 2 and 4 are unstable. Ternary stability curves have an input intensity interval that is that is the result of overlapping between branches No. 1, 3 and 5. This interval is depicted in Fig. 1 by the words: “Ternary Stability Interval”. Because of this interval, these kinds of curves are called ternary stability curves. In this curve by increasing the input intensity from point A (about 0 MW/cm2 values) to point F (about 37 MW/cm2 value), the FBG passes the ABCDEF path in the ternary stability curve. These kinds of paths, which originate from lower intensities to the higher ones in FBG stability curves in CW-regime, are referred to as “forward path” in the present paper. There are two jumps in the ternary stability forward path. One jump can be observed from point B to point C (jump No. 1) and the other, from point D to point E (jump No. 2). In reverse, by decreasing input intensity from point F to point A, the curve does not follow up the previous path, but it passes the FGHIJA path in the ternary stability curve. Again, to refer to this kind of paths- which originate from higher intensities to the lower ones in the stability FBG curves- they are called “backward path” in the present paper. There are also two jumps down in the backward path. The first one is the jump from the point G to point H (jump No. 3) and the other from the point I to point J (jump No. 4 The length of both jumps in the backward path is almost smaller than the length jumps No.1 and No. 2 in the forward path. Because of the existence of these two different paths, i.e. forward and backward paths between the same points A and F, the pulse propagation investigations in FBGs that are characterized by these kinds of curves are not simple. However, we show that by some delicate
iβ2 ∂2Af α + Af = iδAf + iκAb + iγ1 (|Af |2 +2|Ab |2 ) 2 ∂t 2 2 Af + iγ2 (|Af |4 +6|Af |2 |Ab |2 +3|Ab |4 ) Af + iγ3 (|Af |6 +12|Af |4 |Ab |2 +18|Af |2 |Ab |4 +4|Ab |6 ) Af ,
−
(3)
iβ ∂2Ab α ∂Ab ∂A + β1 b + 2 + Ab = iδAb + iκAf + iγ1 (|Ab |2 +2|Af |2 ) 2 ∂t 2 2 ∂z ∂t Ab + iγ2 (|Ab |4 +6|Ab |2 |Af |2 +3|Af |4 ) Ab , +iγ3 (|Ab |6 +12|Ab |4 |Af |2 +18|Ab |2 |Af |4 +4|Af |6 ) Ab
(4)
whereα is the FBG loss which is negligible in the case studied in the present paper, δ and κ are the detuning from the Bragg wavelength and coupling coefficient, respectively. γ1, γ2 , and γ3 account for third order, fifth order and seventh order nonlinear parameters, respectively. One common way of investigating the pulse propagation is to solve Eqs. (3) and (4) numerically in the frequency domain [38]. It is performed by replacing i (∂/ ∂t ) into (ω-ω0), and the coupled mode equations in continuous wave (CW) regime (i.e. with one frequency) are obtained. The Runge-Kutta method is used for numerical calculations of equations, that obtaining in the frequency domain, from end to the beginning with special boundary conditions and given frequency [18,22]. A typical and correct result of output intensity versus input intensity is presented in Fig. 1 which is shown ternary stability curve; terminology is borrowed from the concept of bistability. In bistability curves there exist two points of stable output intensity for a given input intensity. In a similar way, in ternary stability there exist three points of stable output for a given input intensity and we called it ternary stability curves. For becoming more clear our strategy which is expressed in Section 5, it is necessary to introduce some important concepts like ternary
Table 1 Typical data which are applied in this article[18]
2
λB = 1550 nm
n 0 = 2.54
n2silicon = 0.273 × 10−19m2/ w [40]
λ = 1550.015 nm κL = 7
n1 = 1.5 × 10−4 n2 = 220 n2 silicon [39]
n6 = 7.2 × 10
n 4 = - 7.8 × 10
- 31
- 45
m4 /W2 [28]
m6/W3 [28]
Optical Fiber Technology 54 (2020) 102075
E. Yousefi and M. Hatami
Fig. 3. Input pulse intensity shape in the negative part of the time axis.
Fig. 2. Input intensity versus of the time with a typical peak intensity about 225 MW/cm2 and t0 = 0.173 ns.
numerical calculations and by knowing some tips that are presented in our latest reports [18,23]; these kinds of calculations can be easily performed. 4. Gaussian pulse as an input pulse For the simulation, the following Gaussian pulse is considered as an input pulse to the FBG [39]:
A exp(−t 2/2t02),
(6)
where A is the peak intensity and t0 is the halftime width. The input intensity is sketched in Fig. 2 with t0 = 0.173 ns. Short pulse has a wide spectrum with continues range of frequencies. To simulate the propagation of such pulses, Fourier transform should be utilized to convert the pulse shape into the pulse spectrum. The pulse spectrum is divided into a certain number of frequencies with the same interval. FBG coupled mode equations are solved numerically for each component of divided frequency. By using the results and our proposed strategy the pulse propagation through FBG will be simulated. The proposed strategy is completely expressed in seven steps in the next section.
Fig. 4. Input pulse intensity shape in the positive part of the time axis.
5. The simulation method for pulse propagation through the FBG In this section, the proposed novel method for simulation of pulse propagation through the FBG with ternary nature effects is explained in seven steps. Step 1- The input pulse intensity is divided into two intervals: the intensities which are located on the negative part of the time axis and the intensities which are on the positive part of the time axis as depicted in Figs. 3 and 4 respectively. Step 2- In this step, the square root of pulse intensity is being done to obtain the pulse shape field from two intervals of the intensities in step 1 (from Figs. 3 and 4). The pulse is decomposed into two parts, i.e. negative and positive parts. The pulse shape which is in negative time is called “negative part” and the pulse shape which is in positive time is called “positive part” in this paper. Step 3- The Fourier transform (FT) of each pulse shape is calculated which is shown in Fig. 5. Evidently, because of the FT traits, both pulse shapes have the same pulse spectrum. Step 4- Afterwards, the pulse spectrum is divided into many frequency intervals and the stability curves of each frequency are obtained. Some of these curves are shown in Fig. 6. As it can be noticed in this figure, a few curves have bistability characteristic which we have already addressed in [23]. However, many curves have ternary
Fig. 5. Fourier transform of each two pulse shapes.
stability shapes. Step 5-In order to simulate the pulse propagation, the bistability and ternary stability curves are needed for each frequency. It is because we need to obtain the output intensities related to each input intensity of each divided frequency. It is initially performed for the negative part of the pulse that enters at the FBG earlier than the positive part. So, the “forward path” in bistability and ternary stability curves has to be used for obtaining the output intensity (output No. 1). On the other hand, the “backward path” is used for 3
Optical Fiber Technology 54 (2020) 102075
E. Yousefi and M. Hatami
Fig. 8. Output No. 2 versus wavelength.
Fig. 6. The bistability and ternary stability curves which are obtained from coupled mode equations simulated for different frequencies.
obtaining the output intensity of positive part of the pulse that enters the FBG later (output No. 2). Finally, the intensity spectrum of the output pulses is calculated with respect to the wavelength. The shape of “output No. 1”and “output No. 2” for each wavelength (or each frequency) are shown in Figs. 7 and 8 respectively. Step 6- Both calculated outputs results, i.e. outputs No. 1 and No. 2 are summed up to obtain the total output in wavelength (frequency) domain. The output spectrum is depicted in Fig. 9. Step 7- In this step which is the last one, the “total output” is converted back into the output pulse shape in the time domain using inverse Fourier transform. The intensity of the “output pulse shape” is depicted in Fig. 10. Fig. 9. The summation of two output results (output No. 1 and output No. 2) in the frequency domain.
Fig. 11 illustrates a comparison between input and output intensities. As it was expected some delays in output intensity with respect to the input intensity can be observed. The output pulse has a bit deformation with respect to the input pulse. The output pulse width is almost the half of the width of the input pulse, so the output pulse is compressed during its propagation through the ternary stability FBG. Also, the peak of the output intensity is lower than the peak of the input intensity. Fig. 12 illustrates the proposed approach for numerical calculation of simulating the pulse propagation through the nonlinear FBGs with bistability and ternary stability feature.
Fig. 10. The output (transmitted) intensity versus time, which leaves from the end of the FBG.
6. Conclusion Fiber Bragg gratings (FBGs) which are very sensitive to the wavelength of incident waves are strong optical instruments for designing all-optical systems because of their wide properties in both linear and nonlinear regimes. In our recent works [12,17], the quasi-pulse wave propagation through the nonlinear FBGs with bistability and ternary stability have been numerically investigated. In this paper the new method is proposed to investigate the pulse propagation through the
Fig. 7. Output No. 1 versus wavelength. 4
Optical Fiber Technology 54 (2020) 102075
E. Yousefi and M. Hatami
ternary stability FBGs for first time. In this method, the history of the pulse is considered by dividing it into two pulses because of the multi stability nature of the nonlinear FBG and by using Fourier and inverse Fourier transforms the output pulse is simulated numerically. An illustrative flowchart is presented to clearly explain the method of solving coupled mode equations for pulse propagation in a ternary stable medium. At last, the transmitted pulse is obtained from ternary stability FBG and there are some delays on it. The shape of the pulse is a little different form input pulse i.e. some distortion can be seen, it is because the pulse is transmitted through the fiber Bragg grating that has strongly nonlinearity effect (like ternary stability). Also the width and the peak of the output pulse are almost half of the width and the peak of the input pulse that makes FBG as a pulse compressor. It is noteworthy that such studies introduce nonlinear FBGs as the main building block for all-optical signal processing systems and all-optical integrated circuits that can have their role in the development of future all-optical computers.
Fig. 11. A comparison between the input intensity (dashed line) and the output intensity (solid line).
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] A. Othonos, K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing, ArtechHouse, 1999. [2] X. Han, J. Yao, Bandstop-to-bandpass microwave photonic filter using a phaseshifted fiber Bragg grating, J. Lightwave Technol. 33 (2015) 5133–5139. [3] M.M. Ali, K.S. Lim, A. Becir, M.H. Lai, H. Ahmad, Optical gaussian notch filter based on periodic microbent fiber Bragg grating, IEEE Photonics J. 6 (2014) 1–8. [4] A.B. Dar, R.K. Jha, Chromatic dispersing compensation techniques and characterization of fiber Bragg grating for dispersion compensation, Opt. Quant. Electron. 49 (2017) 108. [5] L. Yuan, Y. Zhao, Sh. Sato, Development of a low-cost and miniaturized fiber Bragg grating strain sensor system, Jpn. J. Appl. Phys. 56 (2017) 052502. [6] F. Urban, J. Kadlec, R. Vlach, R. Kuchta, Design of a pressure sensor based on optical fiber Bragg grating lateral deformation, Sensors 10 (2010) 11212. [7] V. Kothari, A. Bedi, S. Kumar, and L. Singh, Fiber Bragg Grating based Temperature Sensor for Biomedical applications, in Frontiers in Optics 2017, OSA Technical Digest (online) (Optical Society of America, 2017), paper JTu2A.59 (2017). [8] D.C. Betz, G. Thursby, B. Culshaw, W.J. Staszewski, Acousto-ultrasonic sensing using fiber Bragg grating, Smart Mater. Struct. 12 (2003). [9] D. Barrera, J. Madrigal, S. Sales, Tilted fiber Bragg gratings in multicore optical fibers for optical sensing, Opt. lett. 42 (2017) 1460–1463. [10] G. Alvarez-Botero, F.E. Baron, C.C. Cano, O. Sosa, M. Varon, Optical sensing using fiber Bragg gratings: fundamentals and applications, IEEE Instrum. Measure. Mag. 20 (2017) 33–38. [11] T. Manjo Divagar, S. James Raja, S. Ilayaraja, Optical pulse compression and its effects using fiber Bragg grating, Int. J. Adv. Res. Comput. Eng. & Technol. (IJARCET) 3 (2014) 1497. [12] W.S. Al-Younis, M.H. Bataineh, Pulse compression using Fiber Bragg gratings, 2017 4th International Conference on Electrical and Electronic Engineering (ICEEE), Ankara, 2017, pp. 240–245. [13] Julijanas Želudevičius, Rokas Danilevičius, Kęstutis Regelskis, Optimization of pulse compression in a fiber chirped pulse amplification system by adjusting dispersion parameters of a temperature-tuned chirped fiber Bragg grating stretcher, J. Opt. Soc. Am. B 32 (2015) 812–817. [14] Z. Yu, W. Marguilis, O. Tarasenko, H. Knape, P.Y. Fonjallaz, Nanosecond switching of fiber Bragg grating, Opt. Express 15 (2007) 14948. [15] Ľ. Scholtz, L. Ladányi, J. Müllerová, Numerical modeling of all optical self switching in chalcogenide fiber Bragg grating, 2015 16th IEEE International Symposium on Computational Intelligence and Informatics (CINTI), Budapest, 2015, pp. 123–127. [16] H. Zoweil, J.W.Y. Lit, Bragg grating with periodic non-linearity as optical switch, Opt. Commun. 212 (2002) 57. [17] E. Yousefi, M. Hatami, S. Dehghani, Optimization of Bistability in nonlinear chalcogenide fiber Bragg grating for all optical switch and memory applications, Int. J. Opt. Photon. (IJOP) 11 (2017) 49–55. [18] E. Yousefi, M. Hatami, A. Torabi Jahromi, All-optical ternary signal processing using uniform nonlinear chalcogenide fiber Bragg gratings, J. Opt. Soc. Am. B 32 (2015) 1471. [19] S. Pawar, S. Kumbhaj, P. Sen, P.K. Sen, Optical multistability in nonlinear fiber Bragg grating, Optoelectron. Adv. Mater. 6 (2012) 25. [20] A. Melloni, M. Chinello, M. Martinelli, All optical switching in phase-shifted fiber Bragg grating, IEEE Photonics Technol. Lett. 12 (2000) 42. [21] M. Karimi, M. Lafouti, A.A. Amidiyan, J. Sabbaghzadeh, All-optical flip-flop based on nonlinear effects in fiber Bragg gratings, Appl. Opt. 51 (2012) 21.
Fig. 12. The flowchart of our proposed method to investigate the pulse propagation through the multistability FBGs.
5
Optical Fiber Technology 54 (2020) 102075
E. Yousefi and M. Hatami
(2019) 88–94. [32] A. Biswas, J. Vega-Guzman, M.F. Mahmood, S. Khan, Q. Zhou, S.P. Moshokoa, M. Belic, Solitons in optical fiber Bragg gratings with dispersive reflectivity, Optik 182 (2019) 119–123. [33] A. Biswas, M. Ekici, A. Sonmezoglu, M.R. Belic, Optical solitons in fiber Bragg gratings with dispersive reflectivity for parabolic law nonlinearity by extended trial function nonlinearity, Optik 183 (2019) 595–601. [34] A. Biswas, J. Vega-Guzman, M.F. Mahmood, M. Ekici, Q. Zhou, S.P. Moshokoa, M.R. Belic, Optical solitons in fiber Bragg graings with dispersive reflectivity for parabolic law nonlinearity using undetermined coefficients, Optik 185 (2019) 39–44. [35] A. Biswas, M. Ekici, A. Sonmezoglu, M.R. Belic, Optical solitons in fiber Bragg gratings with dispersive reflectivity for quadratic cubic nonlinearity, Optik 185 (2019) 50–56. [36] F. Emami, M. Hatami, A. Keshavarz, A.H. Jafari, A heuristic method to simulate the pulse propagation in nonlinear fiber Bragg gratings, Opt. Quant. Electron. 41 (2009) 429–439. [37] E. Yousefi, M. Hatami, Unique solution of short pulse propagation in nonlinear fiber Bragg grating, Int. J. Opt. Photonics (IJOP) 7 (2013) 85–90. [38] G.P. Agrawal, Applications of Nonlinear Fiber Optics, Academic Press, 2001. [39] G.P. Agrawal, Nonlinear Fiber Optics, Academic, 2013. [40] J.M. Harbold, F.O. Ilday, F.W. Wise, Highly nonlinear As-S-Se glasses for all-optical switching, Opt, Lett. 27 (2002) 119.
[22] A.F.G.F. Filho, J.R.R. de Sousa, A.F. de Morais Neto, J.W.M. Menezes, A.S.B. Sombra, Periodic modulation of nonlinearity in a fiber Bragg grating: a numerical investigation, J. Electromagn. Anal. Appl. 4 (2012) 53. [23] E. Yousefi, M. Hatami, All optical self signal processing using chalcogenide nonlinear fiber Bragg grating, Optik 125 (2014) 6637. [24] L. Scholtz, J. Müllerová, Numerical studies on wavelength-selective all-optical switching using optical bistability in nonlinear chalcogenide FBGs, 2015 17th International Conference on Transparent Optical Networks (ICTON), Budapest, 2015, pp. 1–4. [25] H. Lee, G.P. Agrawal, Nonlinear switching of optical pulses in fiber Bragg gratings, IEEE J. Quantum Electron. 39 (2003) 508–515. [26] Ľ. Scholtz, L. Ladányi, J. Müllerová, Cross phase modulation switching in chalcogenide fiber Bragg grating, 2016 ELEKTRO, Strbske Pleso, 2016, pp. 632–636. [27] Y.L. Kim, J.H. Kim, S. Lee, D.H. Woo, S.H. Kim, T.H. Yoon, Broad-band all optical flip-flop based on optical bistability in an integrated SOA/DFB-SOA, IEEE Photonics Technol. Lett. 16 (2004) 398. [28] Y. Yosia, P. Shum, Optical bistability in periodic media with third fifth- and seventh- order nonlinearities, J. Lightwave technol. 25 (2007) 875. [29] A. Zakery, S.R. Elliott, Optical Nonlinearities in Chalcogenide Glasses and Their Applications, Springer, 2007. [30] Y. Yosia, S. Ping, Double optical bistability and its application in nonlinear chalcogenide-fiber Bragg gratings, Phys. B 394 (2007) 293. [31] A. Biswas, M. Ekici, A. Sonmezoglu, M.R. Belic, Solitons in optical fiber Bragg gratings with dispersive reflectivity by extended trial function method, Optik 182
6