Multichannel optical filters with an ultranarrow bandwidth based on sampled Brillouin dynamic gratings

We first propose a multichannel optical filter with an ultranarrow 3-dB bandwidth based on sampled Brillouin dynamic gratings (SBDGs). The multichannel optical filter is generated when an optical pulse interfaces with an optical pulse train based on an ordinary stimulated Brillouin scattering (SBS) process in a birefringent optical fiber. Multichannel optical filter based on SBDG is generated with a 3-dB bandwidth from 12.5 MHz to 1 GHz. In addition, a linearly chirped SBDG is proposed to generate multichannel dispersion compensator with a 3-dB bandwidth of 300 MHz and an extremely high dispersion value of 432 ns/nm. The proposed multichannel optical filters have important potential applications in the optical filtering, multichannel dispersion compensation and optical signal processing. ©2014 Optical Society of America OCIS codes: (060.2310) Fiber optics; (060.4370) Nonlinear optics, fibers; (290.5900) Scattering, stimulated Brillouin; (190.2055) Dynamic gratings. 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Introduction
Recently, multichannel optical filters have attracted great interests thanks to their important applications in multi-wavelength fiber laser [1,2], multi-channel optical signal processing [3], optical DWDM systems [4,5], multichannel dispersion compensators [6,7].These multichannel optical filters have been successfully generated with dielectric thin-film filters, array waveguides (AWGs), Fabry-Pérot filters, fiber Bragg gratings [8][9][10], unbalanced Mach-Zehnder interferometers [11] and high birefringence fiber [12].However, the channel bandwidth is limited to about dozens of GHz using these techniques.An interesting means of implementing periodic multichannel filters is based on a sampled fiber Bragg grating, the channel bandwidth of the multichannel filter is fixed uniquely by the grating length [13][14][15].By introducing the phase shift and proper adjustment of the phase between the grating samples, the bandwidth can be reduced to several GHz [16].The narrow bandwidth multichannel filters are useful for application in ultra-dense wavelength-division multiplexing systems and high resolution spectrum measurement.
Brillouin dynamic grating (BDG) which is generated in polarization-maintaining fibers (PMFs) has attracted much attention [17][18][19].A Brillouin grating in fact is a moving periodically modulated refractive index associated with an acoustic wave, which results from the interaction between two counter-propagating pump waves through electrostriction effect if the phase matching condition for these waves is satisfied [20,21].Other applications of the dynamic grating induced by stimulated Brillouin scattering (SBS) include controllable delay lines [22], distributed sensing [23][24][25][26], microwave photonic filtering [27] and optical signal processing [28,29].While applications of dynamic gratings have been well explored, the possibility of creating sampled Brillouin dynamic gratings (SBDG), i.e. dynamic gratings which has excellent wavelength selectivity and reconfigurable spectral responses, has recently been presented in Ref [30], a detailed analysis, such as the spectral characteristics of a SBDG by different pump pulse schemes, has not been addressed.
In this paper, we first demonstrate theoretically a multichannel filter with ultra-narrow 3-dB bandwidth based on a SBDG.In this investigation, it is clearly shown that a wide variety of multichannel spectral responses can be achieved in based on Gaussian and Sinc apodized and linearly chirped SBDGs.The introduction of sampling function into the field of dynamic gratings with a 3-dB bandwidth should likewise open up many new applications.The properties of SBDG with and without chirps in the grating period can be utilized in WDM systems for filtering and dispersion compensation, which provides great flexibility in the design of functional WDM devices.

Operating principle and model
When imposing a periodic modulation onto the fiber's refractive index using a rectangular sampling slit.The normalized index change induced by a rectangular sampling slit along the fiber axis z can be described by: Where ( ) is the "dc" refractive index modulation, ( ) Λ is the grating period, ( ) represents the phase change of refractive index.
a(z) is an apodization profile over the total length L of the grating and P is the sampling period, δ() is a Dirac delta-function, and ⊗ is a convolution symbol.
The transfer function of a grating is the Fourier transform of the refractive index changes in the grating The refractive index variation of the uniform "seed" grating is modulated further by a periodic amplitude sampling function.The refractive index modulation results in a reflection spectrum comprising a discrete and periodic series of nearly-identical reflection bands or bandgaps (i.e., periodic multichannel filter).In the work, we apply the sampling technique incorporating with the SBS effect in an optical fiber to implement an ultra-narrow multichannel filter.
The BDG is first demonstrated in a polarization-maintaining fiber (PMF) [19], it is possible to generate a BDG in a PMF along one fixed polarization state and observe the scattering from this grating in the other orthogonal polarization state at a shifted frequency.Figure 1 represents the principle to generate SBDG in a PMF.When we use one pulse and one counter-propagating periodic pulse trains at distinct frequencies v1 and v2 (v2>v1) as Brillouin pumps with a frequency difference equal to the Brillouin shift (v B ), at their crossing position in the fiber, an acoustic wave is generated as a result of the standard SBS process [20].The SBDG will be obtained, and the reflected signal is detected.The Brillouin grating can be generated at a specific location controlled by the delay of the two pumping pulses, and the numbers of the BDG inside the PMF can be varied accordingly, which is mainly decided by the repetition rate of the pulse trains adjusted shorter than the acoustic lifetime [31], i.e., ~10ns.By changing the repetition rate, position of the grating inside the fiber and the numbers of the BDG inside the PMF can be varied accordingly.Figure 2 shows the schematic diagram of refractive index modulation of the proposed SBDG.Refractive index change associated with induced acoustic wave amplitude can be represented as: To estimate refractive index change, according to the parameters in the following Table 1.So the refractive index of ~8 × 10 −8 .It worth noting that the index change given in Eq. ( 3) is an estimation under the condition of the steady state, i.e., the pumping laser works either in CW (or Semi-CW with high repetition rate pulse) or in the pulse status but with a pulse width considerably larger than the photon lifetime.The dynamic spectral response of the SBDG along the time will be analyzed in Section 4. The sampling period, sampling duty cycle and sample number can be expressed as , , , where f is the repetition rate, c is the optical velocity, L grating is usually referred to the sampling length, which is decided by pumping pulse width.The interaction length of the two pulses is (t p1 + t p2 )c/2n, where t p1 and t p2 are the pulse widths of pump 1 and pump 2, respectively.However, due to the phonon excitation time, the effective length of Brillouin grating is only half of the interaction length, L grating = (t p1 + t p2 )c/4n.The wavelength spacing Δλ is determined by sample period P, can be given by 2 , 2 For a weak dynamic grating, its 3-dB bandwidth is only determined by the length of the grating L grating , can be simply given by 2 , 2 grating nL λ δλ = (6) So we can tune the repetition rate and pulse widths of the pump pulses to vary the wavelength spacing and 3-dB bandwidth.

Simulation results
We now give several examples that demonstrate the effects of apodization and chirp on the optical properties of SBDGs.The main calculation steps that used for the reflection spectrum calculation are as follows.1) The amplitude distribution of density, and refractive index variations, are calculated along the fiber over one period.2) The reflection spectrums of SBDGs are calculated as a function of wavelength by the transmission matrix method.

Uniform SBDG
In the first simulation, a uniform SBDG is demonstrated.Pulse width of pump1 and pump2 are 0.5 ns and the pumping pulse powers are 200 mW, respectively.The effective length of Brillouin grating L grating is 0.0517 m correspondingly.The pulse temporal spacing of pumping pulse is 2.5 ns and the length of PMF is 5 m.The "ac" index change in the nonzero regions is 6e-8, and there are 487 sections with an "on-off" duty cycle of 10%.The reflection spectrum of a SBDG exhibiting periodic comb spectral response is shown in Fig. 3(a) with a channel spacing of 200 MHz.When sampling the rectangular function the obtained reflection spectrum envelope is the sinc function, so in the Fig. 3(a) the reflectance peaks are inevitable of different amplitudes.The reflectivity of the band at the center is the highest and reflectivity of other bands is reduced by a factor determined by a sinc-function.Figure 3(b) shows a zoom-in-view of one-channel spectral response with a central wavelength of 1549.9993nm, the 3-dB bandwidth is 12.5 MHz.The maximum peak reflectivity of −1.65 dB, limited by the pump pulse powers used.
When the Pulse width and repetition rate of pump1 and pump2 is tuned to 0.002 ns and 100 GHz, respectively, the effective length of Brillouin grating L grating is 0.00020548 m correspondingly.The length of PMF is 1m.The reflection spectrum of a SBDG is shown in Fig. 4(a) with a channel spacing of 50 GHz which fits to the ITU grid. Figure 4(b) shows a zoom-in-view of one-channel spectral response with a central wavelength of 1549.9667nm, the 3-dB bandwidth is 1 GHz.The maximum peak reflectivity is −22.5 dB.It can be concluded that by changing the pulse width and repetition rate, the wavelength spacing can be adjusted from 200 MHz to 50 GHz, which can meet the requirements in different applications.The spectra at the wavelengths of 1549.9667nm with the maximum peak reflectivity of −22.5dB.

Gaussian and Sinc Apodized SBDG
When we use the Gaussian or Sinc-shaped pulse as the pump pulses, it will generate a Gaussian or Sinc apodized modulation of refractive index in the PMF.When the input is Gaussian pulse, and its Fourier transform is also a Gaussian profile.The Gaussian laser pulse is described by the following function To obtain a flat-top spectral response, a Sinc-shaped pulse employed and is described as ) where τ FWHM is the full width at half-maximum (FWHM) pulse duration.To demonstrate the effect of Gaussian and Sinc apodization, Fig. 5 and Fig. 6 show the reflection spectrum versus wavelength for SBDG.In the Gaussian pulses input, the Gaussian spatial profile is used within the samples, so the reflectance peaks are inevitable of different amplitudes.The reflectivity of the band at the center is the highest and reflectivity of other bands is reduced by a factor determined by a Gauss-function.The comparison of Gaussian and Sinc pump pulses input, shows that the same wavelength spacing 50 GHz and reflected spectral reflection line with suppressed sidelobes 13.05 dB.The maximum peak reflectivity of Gaussian apodized SBDG in center wavelength is 13.23 dB, which is higher than Sinc apodized SBDG.The spatial profile of the individual grating samples in the Sinc apodized SBDG follows a Sinc form, ensuring that the reflectivity envelope is square, as required.

Linearly chirped SBDG
The linearly chirped SBDG can be obtained using two counter-propagating linearly chirped laser pulses.For the linear chirp effect, in the index modulation function (1), the local grating period can be expressed as ( ) ( ) where C and Λ 0 refer to the chirp coefficient and the fundamental grating period, respectively.For Gaussian-shaped chirped pulse, the amplitudes of pump pulses can be represented as following: where C 1 , C 2 are the chirp coefficients, τ 1 , τ 2 are the pulse widths, A 10 , A 20 are the amplitudes of pump pulses.Ω B is the Brillouin angular frequency shift.The instantaneous frequencies of the pulses are: The two chirped pump pulses will induce period varies with distance, which include a nonzero z-dependent phase term 0.5dφ/dz in the self-coupling coefficient.In terms of more readily understandable parameters, the phase term for a linear chirp is In contrast to an SBDG, in an SCBDG, the period of the seed grating is linearly chirped.We have demonstrated that if specific conditions between the linear chirp and the sampling period are satisfied, then a suitable interference between the individual spectrally-overlapped wavelength channels can to a discretization of the otherwise quasi-continuous reflection spectrum, resulting again a periodic comb spectrum.
In order to achieve 50 GHz wavelength spacing, the repetition rate and pulse width of pump pulse laser must be less than 50 GHz and 0.01 ns.However the chirped dynamic grating couldn't be generated for the 0.01 ns pulse width of chirp pulse laser.So in the simulation, Pulse width of pump1 and pump2 are 0.5 ns and the pump pulse powers are 200 mW, respectively.The effective length of Brillouin grating L grating is 0.05137 m correspondingly.The repetition rate of pump pulse is 400 MHz and the length of PMF is 5 m.A SCBDG appropriate for 100-channel filtering with 1 GHz wavelength spacing is shown in Fig. 7.The averaged power is −3 dBm and the ripple is less than 1 dB.The wavelength spacing in SCBDG the spacing between neighboring passbands can be precisely controlled because it varies with the repetition rate, which is in the order of millimeters.The maximum index change values are the same as the index change and the length of the SBDG in Fig. 5 and Fig. 6, respectively.The frequency chirp parameters in the pulse pump 1 and pump 2 is dλ D /dz = -1 nm/cm, which contributes to the marked improvement of the filter performance: flat "top" with very small ripple in peak transmittance, steep edge and nearly linear phase response.A one-channel zoom-in-view of Fig. 7 is shown in Fig. 8.The 3-dB bandwidth is 300 MHz, and the reflection response has a flat passband, steep edge and high reflectivity.The phase response of SCBDG is also investigated, wherein the group delay of the filter is illustrated in Fig. 8.The dispersion value in each channel is 342 ns/nm which is at least 2-3 order higher than that achieved in a general linearly chirped fiber Bragg grating.
The filter performance is near-identical among channels.The maximum reflectivity in each channel decreases when the chirp coefficients are increased due to the increasing of the grating's spatial frequencies (i.e., the grating spans a larger bandwidth).In the case of the chirped seed gratings, the inter-band wavelength spacing is clearly the same as that for the SBDG.These features can be used to compensate dispersion in optical communication system with variable dispersion and in a microwave photonics subsystem which requires flexible and large optical delay line.
In the SCBDG, when the pulse width is smaller than 0.1 ns, and the BDG length accordingly is smaller than 0.0102 m, the chirp effect becomes poor, as the spectra and group delay of a SCBDG shown in Fig. 9.The dispersion value in each channel is close to zero.

Discussions
It worth noting that in the simulations, we assume that there is no decay of the BDG, but in fact the exponential decay of the acoustic phonons imprints an amplitude distortion on the signal.In an optical fiber, time decay constant is around 10-12ns and it is difficult to find another suitable material with substantially longer phonon lifetimes.The acoustic lifetime is also a limiting factor for the maximum sampling numbers, since it requires periodic regeneration of the grating within the time it completely decays, and need continual pulses.In the SBDG, the scattered light exponentially decays with the time constant of 2δ, since the acoustic wave decays with time, which is given by Thus, we can see that acoustic wave intensity decays twice faster that acoustic wave amplitude.So we introduce the refractive index modulation attenuation with time and simulate the optical spectrum in the same simulation conditions.Figure 10(a) shows the optical spectra with the time, the black, blue and red lines represent the different time 15 ns, 24 ns and 45 ns, and the Fig. 10 (b) shows an enlarged portion of the middle of a channel.From the figure, we can see that the exponential decay of dynamic grating affect the shape of each peak, but don't change the overall spectral shape, wavelength numbers and wavelength spacing.To increase the number of pulses and pulse trains at regular time intervals, the reflectivity will increase.It is worth noting that to avoid the degradation of the spectral response of the SBDG along the time, the pumping 1 and 2 should both works at a high rate repetition rate.

Conclusion
We proposed a multichannel optical filter with an ultra-narrow 3-dB bandwidth based on SBDGs for the first time to our knowledge.The multichannel optical filter was generated when an optical pulse interfaces with an optical pulse train based on an ordinary SBS process in a birefringent optical fiber.Multichannel optical filter based on SBDG was generated with a 3-dB bandwidth from 12.5 MHz to 1 GHz.addition, a linearly chirped SBDG was proposed to generate multichannel dispersion compensator with a 3-dB bandwidth of 300 MHz and an extremely dispersion value of 432 ns/nm.The proposed multichannel optical filters have important potential applications in the optical filtering, multichannel dispersion compensation and optical signal processing.

Fig. 1 .Fig. 2 .
Fig. 1. (a).Principle to generate SBDG in a PMF using the ordinary SBS process.One pulse 1 and one counter-propagating periodic pulse trains 2 at distinct frequencies v1 and v2 (v2>v1) are used as Brillouin pumps with a frequency difference equal to the Brillouin shift.(b).Pump1 and pump2 in a PMF along one fixed polarization axis, and the scattering from the grating in the other orthogonal polarization axis at a shifted frequency Δv.

Fig. 7 .Fig. 8 .
Fig.7.Reflection spectra versus wavelength for a SCBDG with the wavelength spacing of the SBDG is 0.008nm (1 GHz), respectively, when Sinc-shaped chirped pump pulses as input pulse and the chirp coefficient of pulse pump 1 and pump 2 are −1 nm/cm.

Fig. 9 .
Fig. 9. (a) Reflection spectra versus wavelength for SCBDG in one channel optical spectra from Fig. 7. (b) Group delay versus wavelength and its value is close to zero.

Fig. 10 .
Fig. 10.(a) Reflection spectra with the time, and the black, blue and red lines represent the different time 15ns, 24 ns and 45 ns.(b) Reflection spectra in the enlarged portion of the middle of a channel.