Pulse Switching in Nonlinear Fiber Bragg Gratings Pedro M. Ramos, Jorge R. Costa, and Carlos R. Paiva Instituto de Telecomunicações, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal Phone: +351-218418489 Fax: +351-218418472 e-mail:
[email protected] Abstract The influence of the Kerr-like nonlinearity in the switching characteristics of fiber Bragg gratings is studied. Two types of apodization profiles are considered: uniform and raisedcosine apodized gratings. The influence of the bit-rate and the wavelength detuning of the incident pulses in the nonlinear response of the grating is analyzed. The deterioration in the pulses caused by the nonlinear effects is also observed. I. INTRODUCTION Fiber Bragg Gratings (FBG’s) are a key element in fiber optical communication systems. Due to their fabrication process, they have very low insertion loss, immunity to electromagnetic interference, electrical isolation and light weight. The wide variety of applications has made it a very developed technology. Either in dense wavelength division multiplexing (DWDM) channel equalization [1], demultiplexing [2], noise reduction [3] or in dispersion management [1], FBG’s are becoming one of the most stable and versatile technologies in the optical field. For dispersion management systems, linearly chirped gratings are used to compensate the dispersion caused by group velocity dispersion (GVD) in fiber links. Instead of using long optical fibers ( > 1km ) with different dispersions to counteract the GVD, a small FBG ( < 1m ) can be tailored to compensate the GVD and even higher-order dispersion. In FBG’s, two counter-propagating modes interact. For the design wavelength of the grating the incident electrical field is reflected. The range of wavelengths reflected depends on the length and strength of the grating. This bandwidth is designated as the band-gap zone where no forward propagation can occur. In this paper, the use of FBG’s for all-optical switching is considered. Since the wavelength characteristics of the grating depend on the effective refractive index of the fiber, Kerr-like nonlinearity can alter the wavelength response of the grating. This results in wavelength and power dependent switching – i.e., the output state depends on the input power, wavelength of the incident pulses and also on the shape of the pulses. The nonlinear effect causes the central wavelength of the grating to increase. This increase can be used to switch pulses if the initial wavelength is near a transition from zero to one (or one to zero) in the reflectivity curve. However, the rise of the nonlinearities causes a pulse breakup that indirectly reduces the peak power of the pulses. For this reason, the initial operating point should be as close as possible to the transition which should be as abrupt as possible – in order to minimize the necessary nonlinear effect.
II. GRATING EQUATIONS FBG’s are achieved by exposing optical fiber to a pattern of ultraviolet intensity (varying along the fiber/grating length). The ultraviolet light causes variations in the fiber refractive index that can be modeled by [4] 2π n ( z ) = n0 + ∆n ( z ) cos z (1) Λ where n0 is the refractive index of the fiber outside the grating, ∆n( z ) is the maximum refractive index variation and Λ is the spatial period of the grating – the design wavelength of the grating is λ B = 2n0 Λ . Two apodization profiles are considered: uniform and raised-cosine. In the uniform grating, the maximum refractive index variation is constant ∆n ( z ) = ∆n0 . (2)
For the raised-cosine profile, the amplitude of the modulations slowly rise from the beginning of the grating, they achieve a maximum value in the middle of the grating and then decrease until the end, according to z − L 2 1 1 ∆n ( z ) = ∆n0 + cos 2π (3) L 2 2 where L is the grating length. The refractive index for the two profiles is shown in Fig. 1. n(z)
Uniform Grating
n0+∆n0 n0
z z=0
n(z)
z=L
Gaussian Grating
n0+∆n0 n0
z z=0
z=L
Fig. 1 – Apodization profiles of the gratings. In the real gratings, the number of periods is much higher (105). The electrical field in the FBG is composed of two counter-propagating LP01 modes. a+ is the envelope of the electrical field mode traveling in the longitudinal positive direction, while a− corresponds to the mode propagating in the opposite direction. The coupling between these two modes and the wavelength detuning define the grating characteristics.
The propagation in the grating with Kerr-like nonlinearity is governed by a set of nonlinear-coupled mode equations (NLCME) [5] 2 ∂a± n ∂a 2 = ±i ∆ a± + α a± a± + 2 a∓ + κa∓ ∓ 0 ± (4) c ∂t ∂z where the coupling coefficient κ is π ∆n ( z ) κ= , (5) λB ∆ is the detuning between the carrier wavelength ( λC ) and the design wavelength of the grating 1 1 ∆ = 2πn0 − (6) . λC λ B α represents the nonlinear Kerr-like effect πn α= 2 , (7) λB where n2 is the nonlinear coefficient of the original fiber – in the linear regime α = 0 . The strength of the grating is the maximum of the coupling coefficient, i.e., π ∆n0 . (8) κ0 = λB In the continuous-wave (CW) regime the input pulses have a negligible spectral width and the last term of (4) is omitted. The power reflection coefficient in CW is
a− ( z = 0 )
+∞
−∞ +∞ −∞
Uniform
a+ ( z = 0 )
0.5
0.0 1549.7
1549.9
.
(9)
a− ( z = 0, t ) dt
λ [nm]
1550.1
1550.3
Fig. 2 – Spectrum reflectivity of the gratings with κ 0 = 200 m -1 and spectral profiles of the input pulses ( B = 1Gb/s ) for λ C = 1550 nm and λ C = 1550.1nm . Uniform gratings can have a large reflection bandwidth and maintain a high reflection coefficient in that bandwidth. However, outside the bandwidth, their sidelobes have considerable amplitude and can cause the pulse to breakup which may result in unnecessary bit-rate limits. The sidelobes are caused by the abrupt interfaces at the grating limits. In Fig. 3 the reflected and transmitted pulses are shown for a Gaussian input pulse centered at λ C = 1550 nm and λ C = 1550.1nm for the uniform grating presented in Fig. 2.
2
For pulses, the power reflection coefficient is the ratio between the energies of the reflected and incident pulses at the beginning of the grating
∫ R= ∫
Raised-Cosine
)
(
R=
1.0
λC=1550nm
∆t ≅ 24ps
1.0
|a+(z=0)| |a-(z=0)| 0.5
2
a+ ( z = 0, t ) dt 2
.
(10)
The gratings considered have L = 3.5 cm , n0 = 1.447 and λ B = 1550 nm . The NLCME system is solved numerically using a fourth order modified predictor-corrector method [6]. III. LINEAR REGIME The input pulses considered are Gaussian with t 2 (11) a+ = pIN exp − τ0 where pIN is the input peak power and τ0 is the temporal width of the pulses ( τ0 = 0.85τ FWHM ). In the linear regime, the grating responses do not depend on the input peak power. Therefore, in this section pIN = 1 and α = 0 are used. Fig. 2 shows the spectral reflectivity of the uniform and raised-cosine gratings with κ0 = 200 m −1 – obtained in CW. It also presents the spectrum of a Gaussian input pulse in the reflection region of the grating ( λ C = 1550 nm ) and in the transmission region ( λ C = 1550.1nm ). The input pulse has a temporal width of τ FWHM = 117.6 ps (with a time-slot of 10τ0 , the bit-rate is B = 1Gb/s ).
0.0 -0.25
0
0.25
0.5
t [ns]
λC=1550.1nm
∆t ≅ 202.5ps
1.0
0.75
|a+(z=0)| |a+(z=L)| 0.5
∆t ≅ 408ps
|a-(z=0)| 0.0 -0.25
0
0.25
t [ns]
0.5
0.75
Fig. 3 – Reflected and transmitted pulses at λ C = 1550 nm and λ C = 1550.1nm – uniform grating ( κ 0 = 200 m -1 ). For λ C = 1550 nm , only the reflected pulse is visible – R = 99.99% . The delay caused by the grating ( 24 ps ) is minimal and corresponds to the propagation time along 2.48mm – i.e., the pulse is reflected after traveling approximately only 7% of the grating length.
For λ C = 1550.1nm , the input pulse is mainly transmitted – R = 16.15% . The delay in the transmitted pulse corresponds to the propagating time along the full distance of the grating ( n0 L / c = 169 ps ). The discrepancy in the actual values is caused by the high-dispersion of the grating near the band-gap limit. The reflected field presents two small pulses. These pulses are caused by the reflections at the interfaces of the grating. The first pulse has a negligible delay (interface at z = 0 ) while the second pulse has a delay that corresponds to approximately the propagation along twice the length of the grating (reflection at z = L ). To reduce the sidelobes of the gratings, apodized gratings are used. Since, there are no abrupt interfaces, the sidelobes are drastically reduced – the raised-cosine profile is considered to be one of the best, since the sidelobes are very small. The pulses obtained for the situation of Fig. 3 with the raised-cosine grating are presented in Fig. 4.
1.0
|a+(z=0)| |a-(z=0)| 0.5
∆λ BG = λ B
∆n0 κ = λ 2B 0 . n0 πn0
(12)
Both gratings have a slightly larger spectral width (FWHM) – ∆λ = 0.114 nm . The Gaussian pulse ( B = 1Gb/s ) has a spectral width of ∆λ FWHM = 0.03nm . For λ C = 1550 nm , almost all the power of the pulse is reflected since it is in the spectral region of the grating where R 1 – Fig. 2. For higher bit-rates (i.e., larger spectral width of the pulses), the reflected signal has several small pulses at the end of the main pulse. In fact, part of the spectral components of the signal is in the transmission region but is still reflected by the sidelobes of the grating. This pulse deterioration is higher for uniform gratings since their sidelobes are more intense. In the rest of this paper, only raised-cosine gratings are considered. IV. NONLINEAR REGIME In the nonlinear regime, the response of the gratings depends on the input peak power of the pulses. Introducing the normalizations a e± = ± (13) pIN
λC=1550nm
∆t ≅ 97.8ps
The wavelength band-gap of a grating can be estimated for long gratings by [7]
|a+(z=L)|
and Γ = αpIN , the propagation equations (4) are modified to
0.0 -0.25
0
0.25
0.5
t [ns] 1.0
(14)
λC=1550.1nm
∆t ≅ 183.1ps
|a+(z=0)| |a+(z=L)| 0.5
(
∂e± 2 = ±i ∆ e± + Γ e± e± + 2 e∓ ∂z
2
) + κe ∓ nc ∂∂et . (15) ∓
0
±
Parameter Γ represents the intensity of the nonlinear effects. In the linear regime Γ = 0 since α = 0 and pIN ≠ 0 . Fig. 5 shows the level lines of the reflection coefficient for different values of the nonlinear parameter and the central wavelength of the pulses.
|a-(z=0)| 500 0.0 -0.25
0
0.25
0.5
t [ns]
Fig. 4 – Reflected and transmitted pulses at λ C = 1550 nm and λ C = 1550.1nm – raised-cosine grating ( κ 0 = 200 m -1 ). For λ C = 1550 nm ( R = 99.46% ), the reflected pulse has a considerable delay (97.8ps). This delay is caused by the fact that the profile increases slowly and therefore the pulse has to travel well into the grating until it is reflected – the propagation distance corresponds to approximately 10.13mm ( 29% of the grating length). For λ C = 1550.1nm ( R = 0.2% ), the delay of the transmitted pulse corresponds to the propagation time along the grating length. The actual value is closer to 169 ps because the dispersion of the raised-cosine grating is smaller than the dispersion in the uniform grating [7].
Γ -1 [m ]
0.5
0.1 0.2
0.3 0.4
0.5
0.5
0.6
0.6
0.7
0.5 0.4 0.3
0.2
250 0.8
0.1
R=0.9
0 1549.8
1549.9
1550
λC [nm]
1550.1
1550.2
Fig. 5 – Level lines of the reflection coefficient as a function of the nonlinear parameter Γ and the wavelength of the carrier λC ( κ 0 = 200 m -1 ).
For higher nonlinear effects, the central wavelength of the grating increases while its bandwidth is reduced – the spectral response of the grating is no longer symmetric around λ B . The reflection region is also drastically reduced. In fact, for Γ > 500 m−1 , the reflection coefficient is always less than 62%. The refractive index of the fiber depends on the intensity of the electrical field ( I ) with n0 ( I ) = n0 L + n2 I
(16)
A grating can be used as an intensity dependent switch. The wavelength of the input pulse should match the design wavelength of the grating. In the linear regime the pulse is reflected corresponding to the ON position. Increasing the power of the signal, the pulses are mainly transmitted with a residual reflection – OFF position. The use of a grating as a switch is schematically presented in Fig. 7. e+(z=0)
where n0 L is the refractive index in the linear regime. For higher input pulse peak powers, there is a shift in the spectral response of the grating to higher wavelengths ( n2 > 0 ). Fig. 6 shows the spectrum of the pulses presented in Fig. 4 ( λ C = 1550.1nm ) for the linear ( Γ = 0 ) and nonlinear regimes ( Γ = 100 m −1 ). 1.0
0.5
|e+(z=0,λ)| |e-(z=0,λ)|
0.0 1549.95
1550
1550.05
1550.1
1550.15
e-(z=0) Fig. 7 – Implementation of a grating as an intensity dependent switch.
The raised-cosine grating with κ 0 = 200 m -1 can be used as a switch for a 1Gb/s system at the design wavelength. The ON position corresponds to Fig. 4 ( λ C = 1550 nm ) with a reflection coefficient of RON = 99.46% . The OFF position is presented in Fig. 8 with Γ = 14000 m -1 and ROFF = 2.52% .
1550.2 1.0
λ [nm]
1.0
Λ
Γ=0 |e+(z=L,λ)|
e+(z=L)
|e+(z=L)|
|e+(z=0)|
Γ=100m-1
|e+(z=0,λ)|
0.5
|e-(z=0,λ)| 0.5
|e-(z=0)|
|e+(z=L,λ)| 0.0 -0.25
0.0 1549.95
1550
1550.05
1550.1
1550.15
1550.2
λ [nm]
Fig. 6 – Reflected and transmitted pulses in the spectral domain of the pulses of Fig. 4 ( λ C = 1550.1nm ) with Γ = 0 and Γ = 100 m −1 . In the linear regime, the pulse is in the transmission region of the grating. Therefore, the spectrum of the initial and transmitted pulses is almost identical ( R = 0.2% ). For Γ = 100 m -1 there is an increase in the central wavelength of the grating and part of the pulse is in the reflection region. In fact, there is a small-reflected pulse at the expense of a reduction and deterioration of the transmitted pulse ( R = 21.3% ). The spectrum of the transmitted pulse is wider than the initial spectrum and is shifted to lower wavelengths. The width of the spectrum of the reflected pulse is maintained albeit being shifted to higher wavelengths.
0
t [ns]
0.25
0.5
Fig. 8 – Reflected and transmitted pulses for λ C = 1550 nm , B = 1Gb/s and Γ = 14000 m -1 . For higher bit-rates, the spectral width of pulses is higher and the grating bandwidth must also be higher. This is achieved by using stronger gratings ( > κ0 ). Fig. 9 presents the results obtained with κ0 = 600 m-1 ( ∆λ 0.342 nm ) in the two switching positions for a bit-rate of B = 4 Gb/s ( τ FWHM = 29.4 ps and ∆λ FWHM = 0.12 nm ). The reflection coefficients are RON = 99.78% and ROFF = 8.21% . With the increase of the grating bandwidth, the nonlinear effects have to be more intense to produce a larger shift in the grating response and shift the pulse to the transmission region. However, with the increase of the nonlinear effects, the spectral broadening of the pulses increases and part of it remains in the reflection region – i.e., there is an undesirable increase of ROFF .
1.0
|e+(z=0)| |e-(z=0)| 0.5
|e+(z=L)|
Γ=0 0.0 -0.1
0
0.1
0.2
0.3
0.4
0.5
t [ns] |e+(z=0)|
1.0
|e+(z=L)| 0.5
|e-(z=0)|
Γ=22000m 0.0 -0.1
0
0.1
0.2
0.3
t [ns]
0.4
-1
0.5
Fig. 9 – Reflected and transmitted pulses for λ C = 1550 nm , B = 4 Gb/s and κ0 = 600 m-1 . For low bit-rates, the grating response practically corresponds to the nonlinear CW regime. In this regime, a high constant input power produces a sequence of pulses in the transmission pulse [8]. Fig. 10 presents a situation where the bit-rate is sufficiently low ( B = 0.5Gb/s ) for the nonlinearities ( Γ = 450 m -1 ) to produce a sequence of pulses in the transmitted pulse.
1.0
|e+(z=0)|
|e+(z=L)|
0.5
0.0 -0.5
-0.25
0
t [ns]
0.25
0.5
0.75
Fig. 10 – Incident and transmitted pulses for λ C = 1550 nm , B = 0.5Gb/s , κ0 = 200 m -1 and Γ = 450 m −1 .
V. CONCLUSIONS For uniform gratings, the sidelobes can cause the pulse breakup due to the fact that some spectral components of the pulses are transmitted while others are reflected. The sidelobes are caused by the reflections that occur at the interfaces of the gratings. To reduce the sidelobes of the gratings, apodization profiles are used. By suitably choosing
the correct profile the reflections can be reduced. The raisedcosine profile can reduce the amplitudes of the sidelobes up to 40dB . For this profile the interface reflections are practically non-existent. The reflection coefficients of the two gratings for λ C = 1550.1nm and B = 1Gb/s are R = 16.15% (uniform) and R = 0.2% (raised-cosine). The difference is caused by the sidelobes of the gratings. The nonlinear effect causes the central wavelength of the grating to increase. This variation can be used to switch pulses if the initial wavelength is near one of the transitions in the reflectivity curve. However, to produce large wavelength shifts (necessary for high bit-rates), very high nonlinearities are required. With the increase of the nonlinear effects the pulse deterioration also increases. Based on the linear and nonlinear propagation results, an all-optical switching scheme is proposed. In this scheme, the design wavelength of the grating should match the central wavelength of the incident pulses. The low peak power pulses are reflected and can, through the use of a optical circulator, be separated from the high-intensity pulses – which are transmitted through the grating. VI. REFERENCES [1] M. Ibsen, M. K. Durkin, M. N. Zervas, A. B. Grudinin, and R. I. Laming, “Custom design of long chirped bragg gratings: application to gain-flattening filter with incorporated dispersion compensation,” IEEE Photon. Technol. Lett., vol. 12, pp. 498-500, May 2000. [2] J. Hubner, D. Zauner, and M. Kristensen, “Strong sampled Bragg gratings for WDM applications,” IEEE Photon. Technol. Lett., vol. 10, pp. 552-554, Apr. 1998. [3] S. Y. Set, H. Geiger, R. I. Laming, M. J. Cole, and L. Reekie, "Optimization of DSF- and SOA-based phase conjugators by incorporating noise-suppressing fiber gratings," IEEE J. Quantum Electron., vol. 33, pp. 16941698, Oct. 1997. [4] P. M. Ramos, J. R. Costa, and C. R. Paiva, “A numerical study of pulse switching in nonlinear nonuniform fiber Bragg gratings” Proc. LEOS 2000, Porto Rico, pp. 661662, Nov. 2000. [5] C. de Sterke, and J. Sipe, “Switching dynamics of finite periodic nonlinear media: A numerical study,” Phys. Rev. A, vol. 42, pp. 2858-2869, Sept. 1990. [6] C. de Sterke, K. Jackson and B. Robert, “Nonlinear coupled-mode equations on a finite interval: A numerical procedure,” J. Opt. Soc. Am. B, vol. 8, pp. 403-412, Feb. 1991. [7] T. Erdogan, “Fiber grating spectra,” J. Light. Tecnhol., vol. 15, pp. 1277-1294, Aug. 1997. [8] H. Winful and G. Cooperman, "Self-pulsing and chaos in distributed feedback bistable optical devices," Appl. Phys. Lett., vol. 40, pp. 298-300, Feb. 1982.
