Ultrashort bragg soliton in a fiber bragg grating
Sien Chi
*, Boren Luo, Hong-Yih Tseng
Institute of Electro-Optical Engineering, National Chiao-Tung University, Hsinchu, 300 Taiwan, ROC Received 31 January 2002; received in revised form 31 January 2002; accepted 1 February 2002
Abstract
The propagation of a nonlinear ultrashort pulse in a photonic bandgap structure is investigated by using the finite-difference time-domain method. The simulation results show that an ultrashort pulse near the bandgap edge can propagate through a nonlinear fiber Bragg grating, even if the broadband spectrum of this ultrashort pulse overlaps the whole forbidden band of the grating. It is also shown that the time delay of such an ultrashort solitary wave is pro-portional to its detuning wavelength from the exact Bragg resonance. Ó 2002 Published by Elsevier Science B.V.
Keywords: Finite-difference time-domain method; Photonic bandgap; Bragg soliton; Gap soliton; Fiber Bragg grating
1. Introduction
Gap solitons are solitary waves propagating in a nonlinear photonic bandgap (PBG) structure [1]. The exact analytic solution to describe such a nonlinear pulse has been obtained from the non-linear coupled-mode equations (NLCMEs). By using the multiple scale method [2], the NLCMEs can be reduced to the nonlinear Schr€oodinger equation (NLSE). Soliton solutions to this ap-proximated NLSE are called Bragg solitons. Bragg solitons exist near the PBG edge and have been widely discussed both in theory [3–7] and experi-ment [8,9]. It has been demonstrated that a Bragg soliton can propagate through a fiber Bragg grat-ing (FBG) [8]. The experimental results are in very good agreement with the NLSE model.
One of the attractive characteristics of a Bragg soliton is the reduction of its group velocity. The experiments have shown that such a soliton-like pulse with 80-ps width can travel with the velocity as low as 70% of the light speed in an unprocessed fiber. Thus all optical buffer based on the slow propagation of a Bragg soliton is an ongoing challenge. Moreover, nonlinear compression for optical pulses by using FBGs is also an interested subject associated with the Bragg soliton propa-gation. Such research may result in applying solitary propagation in FBGs to the practical all-optical communication system. However, to in-vestigate the dynamics of a Bragg soliton, the models of the NLCMEs and the NLSE have the drawback: The NLSE is derived from the NLC-MEs under the low-intensity limit. This limitation restricts a Bragg soliton to a broad pulse, but for a high-speed lightwave system an ultrashort pulse is more practical and necessary.
Optics Communications 206 (2002) 115–121
www.elsevier.com/locate/optcom
*
Corresponding author. Fax: +86-10-62283728. E-mail address: [email protected] (S. Chi).
0030-4018/02/$ - see front matterÓ 2002 Published by Elsevier Science B.V. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 1 7 4 - 4
In this paper, we use the finite-difference time-domain (FD–TD) method to study the nonlinear ultrashort pulse in a PBG structure. The FD–TD method can directly simulate Maxwell’s equations. Hence it provides a robust simulation theory to investigate the characteristics of a Bragg soliton without any approximation. It is shown that a nonlinear ultrashort pulse near the bandgap edge still can propagate through a FBG, even if the broad spectrum of this ultrashort pulse overlaps the whole forbidden band of the FBG. The prop-agating dynamics of such an ultrashort solitary wave is numerically studied and presented.
2. Simulation theory
We consider an electromagnetic field with the electric component Ez polarized along the x-axis
and the magnetic component Hy polarized along
the y-axis. Such an electromagnetic field propa-gates along the x direction in a medium, which is assumed to be isotropic and non-dispersive. Maxwell’s curl equations for this problem are written as: oHy ot ¼ 1 l0 oEz ox ; ð1Þ oDz ot ¼ oHy ot ; ð2Þ Dz¼ e0erðxÞEzþ PzNL; ð3Þ
where l0 is vacuum permeability, er is vacuum
permittivity, e1ðxÞ is the relative material
permit-tivity, Dz is the electric induced polarization
in-cluding the linear and nonlinear contributions of the medium, and PNL
z is the nonlinear polarization
regarding the Kerr nonlinearity. On the basis of the FD–TD method, the finite difference equations for Eqs. (1) and (2) are:
Hyj nþ1=2 iþ1=2 ¼ Hyj n1=2 iþ1=2 þ Dt l0Dx Ezj n iþ1 Ezj n i ; ð4Þ Dzjnþ1i ¼ Dzjni þ Dt Dx Hyj nþ1=2 iþ1=2 Hyjnþ1=2i1=2 ; ð5Þ where Dt and Dx are the finite difference intervals in the temporal and spatial domain, respectively. The procedures of the FD–TD approach are
de-scribed in the following. First Eq. (4) is used to determine Hyjnþ1=2iþ1=2 from the previous values of
Hyjn1=2iþ1=2, Ezj n
iþ1 and Ezj n
i. Second Dzjnþ1i determined
by using Eqs. (5) from the previous values of Dzj n i,
Hyjnþ1=2iþ1=2 and Hyjnþ1=2i1=2. Finally the resulting Dzjnþ1i
are substituted into Eq. (3) to determine Ezjnþ1i
under the Newton iterative procedure: Ehpþ1iz ¼ Dzj
nþ1
e0½erðxÞ þ vð3ÞjEhpiz j2
; ð6Þ
where vð3Þis the third-order susceptibility, p is zero
or positive integral, and Ep z ¼ E
n
z for p¼ 0.
To investigate Bragg solitons in a one-dimen-sional PBG medium, we consider a uniform FBG with the relative material permittivity e1ðxÞ ¼
nðxÞ2, where nðxÞ ¼ n0þ Dn cos 2px K : ð7Þ
Here n0is the linear refractive index at the central
wavelength of the electric field, Dn is the magni-tude of the periodic index variations, and K is the grating period with respect to the Bragg wave-length kB via K¼ kB=2n0. It is noticed that during
the FD–TD process, there is no constraint on the quantity of Dn and the apodized profile of the grating. Thus the FD–TD method is more suitable than the NLCMEs and the NLSE model to in-vestigate the dynamics of a nonlinear pulse in a
Fig. 1. The reflectivity (solid curve) of the uniform FBG and the broadband spectrum (dotted curve) of the incident pulse as functions of the wavelength detuning Dk¼ k kB from the exact Bragg resonance.
realistic rectangular waveguide grating or an ap-odized FBG. Such nonuniform gratings have been widely discussed for pulse compression and all-optical delay line based on the mechanisms of the Bragg soliton. Nevertheless, in the present paper, we focus our attention on how an ultrashort pulse evolves in a uniform FBG with Kerr non-linearity.
3. Numerical results and discussions
The solid curve in Fig. 1 shows the reflectivity RðDkÞ of the uniform FBG in our simulation. The linear refractive index of this FBG is n0¼ 1:5 and
the index variation is Dn¼ 9 104. The central
wavelength of this reflectivity is kB ¼ 1:55 lm. The
dotted curve in Fig. 1 shows the spectrum of the
Fig. 2. Monitoring the propagation of a low-amplitude pulse in the fiber grating. The FD–TD method gives the snapshot of the propagating pulse at t¼ 15 ps, (2) t ¼ 60 ps, (3) t ¼ 105 ps, and (4) t ¼ 150 ps.
adopted incident pulse with a hyperbolic-secant pulse shape initially. The full width at half maxi-mum (FWHM) of this pulse is assumed to be Tf ¼ 5:28 ps; likewise the central wavelength of
this incident pulse is located at k0¼ 1:5492 lm.
Both of the FBG reflectivity and the pulse spec-trum are shown as functions of the Bragg wave-length detuning Dk¼ k kB. Furthermore, the
range with RðDkÞ ¼ 1 exhibits the forbidden band of such a PBG structure. Obviously, the initial pulse spectrum exceeds the PBG edge and even overlaps the whole forbidden band. We emphasize that because of the low-intensity limit for Bragg solitons, the previously demonstrated experiments and simulations have not yet clarified the propa-gation of a nonlinear pulse with such a broadband spectrum. We use FD–TD method to examine the dynamics of this nonlinear ultrashort pulse beyond the low-intensity limit. By choosing a uniform FD–TD space resolution Dx¼ 50 nm, the nu-merical phase error is limited to about 3:6 105,
which is much smaller than the dispersion due to the PBG structure. Fig. 2 shows the evolution of the incident pulse with low peak power P ¼ 1:4 102 W propagating through the FBG.
The Kerr coefficient of this FBG is vð3Þ¼
1:97 109 W1, in which absorbing the effective
core area in three dimensions and the grating length is L¼ 38 mm. After the initial pulse is put into this FBG, Figs. 2(a)–(d) show the pulse shapes at t¼ 15 ps, t ¼ 60 ps, and t ¼ 150 ps, re-spectively. One can see that the shapes of this pulse are asymmetrically broadened as a consequence of its broadband spectrum. The evolution of the peak power and the spatial width of the pulse versus the propagating distance are shown in Fig. 3. During the propagation, the pulse undergoes the large quadratic grating dispersion. Such a large disper-sion is produced by the interference among the multi-layers of the grating. To balance this qua-dratic grating dispersion, we have to increase the peak power of the initial pulse.
Fig. 4 shows the evolution of the nonlinear ul-trashort pulse with peak power P ¼ 1:4 105 W.
Figs. 4(a)–(d) represent the pulse shapes at t¼ 15 ps, t ¼ 60 ps, t ¼ 105 ps, t ¼ 150 ps, re-spectively. It is shown that the peak power and the pulse width are changed very little during the
propagation. Hence the balance between the nonlinearity and the quadratic grating dispersion leads to a soliton-like pulse. Fig. 5 explicitly shows the evolution of the peak power and the spatial width versus the propagating distance. The nu-merical results show that the incident hyperbolic-secant pulse becomes quasi-stable. The pulse adjusts its amplitude and duration periodically because of the interaction between the nonlinearity and the quadratic grating dispersion. The soliton periodic Lsfor such a solitary wave can be defined
[10] by the nonlinear length Lnl via Ls¼
pLnl=2¼ p=ð2c P Þ. Therefore the soliton-like
wave propagates about 16.7 soliton periods. An-other notable characteristic of this solitary wave is its propagating delay with respect to the
propa-Fig. 3. (a) Peak power and (b) spatial pulse width of the low-amplitude pulse versus the propagating distance.
gating time of the light in an unprocessed fiber. For the above hyperbolic-secant pulse with carrier frequency k0¼ 1:5492 lm, the delay after
propa-gating through the grating with length L¼ 38 mm is 42 ps. This delay corresponds to the soliton’s group velocity as low as 72% of the light speed in an unprocessed fiber. The group velocity of our adopted ultrashort pulse is very close to that of the
Bragg solitons demonstrated previously in the ex-periment [8]. The spatial width of the pulse in grating is smaller than the one in the unprocessed fiber. It results from the incidence from the normal group-velocity medium into slow group-velocity medium. However, the spatial length of an optical cycle is unchanged because of the constant average refractive index. It is found that the pulse in
Fig. 4. Monitoring the propagation of the soliton-like pulse in the FBG. The FD–TD method gives the snapshot of the propagating pulse at (1) t¼ 15 ps, (2) t ¼ 60 ps, (3) t ¼ 105 ps, and (4) t ¼ 150 ps.
grating contains less optical cycles than the pulse in the unprocessed fiber. It could be clarified that the pulse is slowing down by periodic medium not by the linear refractive index. Fig. 6 further dis-plays the time delay versus the carrier wavelength of the ultrashort Bragg soliton. One can see that the delay is linearly proportional to the Bragg detuning wavelength. Note that both of the NLCMEs and the NLSE model cannot predict such a relation between the time delay and the carrier wavelength detuning of an ultrashort Bragg soliton. Consequently, it would be useful to apply
the FD–TD method to estimate the group velocity of an ultrashort Bragg soliton, especially for de-signing an all-optical buffer in practical high-speed communication systems.
4. Conclusion
We have applied the FD–TD method to inves-tigate the nonlinear ultrashort pulse in a fiber Bragg grating. The FD–TD method can directly simulate Maxwell’s equation and inherently com-putes the bi-directional electromagnetic field without using any approximation. As a result, our study numerically confirms that an ultrashort solitary wave near the bandgap edge still could propagate through a nonlinear PBG structure, even if its broadband spectrum overlaps the for-bidden gap of the PBG medium. The propagating dynamics that has not yet been clarified by the NLCMEs and the NLSE model is explicitly shown on the basis of the FD–TD method. The present simulations demonstrate that the low-intensity pulse does not yield the NLSE soliton-like prop-agation in Kerr nonlinear Bragg gratings. Such propagation is only ensured by high-power pulses (105 W). This imposes severe limitations on the use
of NLSE Bragg solitons for optical communica-tions. By contrast, solitons can propagate via
near-Fig. 5. (a) Peak power and (b) spatial pulse width of the soli-ton-like pulse versus the propagating distance. It is shown that the pulse adjusts its amplitude and duration periodically be-cause of the interaction between the nonlinearity and the qua-dratic grating dispersion.
Fig. 6. Time delay versus different carried wavelength of the ultrashort Bragg soliton.
resonant self-induced transparency in resonantly absorbing Bragg reflectors with arbitrarily low intensities and are therefore much more suitable for telecommunications [11,12]. Furthermore, the FD–TD method shows that the time delay of an ultrashort Bragg soliton is linearly proportional to the Bragg detuning wavelength. It would be useful to apply the FD–TD method to design an all-op-tical delay line in a realistic high-speed telecom-munication system.
Acknowledgements
This work was supported in part by the Na-tional Center for High-Performance Computing, the National Science Council, Taiwan, ROC under Contract NSC 89-2215-E-009-112, and the Aca-demic excellence program of R.O.C. Ministry of Education under Contract 90-E-FA06-1-4-90X023.
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