1. Introduction

Recently, dynamic Brillouin gratings (DBGs) [1], generated through stimulated Brillouin scattering (SBS), have been envisaged as an efficient architecture to realize wide and fast dynamic tunable microwave photonics filters [2, 3], optical filters [4], optical delay lines [5], all-optical signal processing [6] (e.g. all-optical differentiation, integration and true time reversal), radio signal processing [7] and high resolution sensing [8, 9].

The idea behind DBG is that the Brillouin acoustic wave is driven by the product of two counter-propagating pump fields. Therefore, when both pumps are modulated (either in amplitude, phase, or both) by a signal A(t), the term driving the acoustic wave is (in first approximation) proportional to A(t)A*(t − T − τ), where T is a fixed delay due to the fact that the two pumps enter the fiber from the opposite faces and τ is a further delay due to the specific optical circuit. Loosely speaking, this means that the acoustic wave is proportional to ${\int }_0^{t}A(t^{\prime })A*(t^{\prime }-\tau )\mathrm{d}{t}^{\prime }$, which is nothing but the correlation of the modulating signal A(t). The correlation peak corresponds to the DBG, and its position within the fiber is controlled by τ; in general, however, the correlation may also have side lobes that, in this framework, act as sources of noise.

In polarization maintaining fibers (PMFs) DBGs are generated by the SBS between two counter-propagating pump waves, polarized along one of the principal axes of the fiber and with a frequency separation set at the Brillouin shift Ω_B. A probe signal, polarized along the orthogonal axis, and satisfying a phase-matching condition with the pumps, is reflected by the isotropic acoustic wave generated during the SBS process. Such reflected signal is co-polarized with the probe, propagates in the opposite direction, and is downshifted by Ω_B.

Similarly to fiber Bragg gratings (FBGs) [10], DBGs properties can be designed by properly imposing specific refractive index variations along the fiber through the electrostriction process. The main difference between FBGs and DBGs is that, once realized, FBGs can be weakly and slowly tuned only mechanically while the DBGs can be dynamical reconfigured by a proper design of the acoustic wave which essentially depends on the correlation between the pump waves [6]. In particular, the acoustic wave, and hence the DBG, can be spatially localized (i.e. apodized, borrowing from the FBG terminology), to a short region (in the order of few centimeters) of the fiber by pulsing the pumps [5] or by properly modulating their phase [11, 12]. The amplitude of localized DBGs created by optical pulses oscillates, due to the acoustic wave decay [5]; on the contrary, phase modulation can create both localized and stationary DBGs [11, 12]. In particular with this last approach, variable optical delay lines and narrowband microwave-photonics filters have been realized [2, 5, 11].

Localized and stationary DBGs through phase modulation have been experimentally demonstrated by Antman et al. [11] using the pseudo-random bit sequence (PRBS) modulation format. The slowly varying amplitude of the pump waves at the two ends of the fiber can be described by $A(t)=A_0{\displaystyle {\sum }_n\text{rect}\left[\left(t-n{T}_B\right)/T_B\right]{\mathrm{e}}^{\mathrm{i}\pi {\varphi }_n}}$, where T_B is the PRBS bit duration, ϕ_n ∈ {0,1} is the n-th PRBS bit and both pumps are assumed to have the same constant amplitude A₀. It can be shown [11] that the two waves have maximum correlation at a particular point z₀ where the acoustic wave is given by [11] Q(t, z = z₀) ∝ |A₀|². Outside the correlation point instead, the acoustic wave randomly assumes the instantaneous values ±|A₀|² due to the randomly varying phase given by the PRBS bits, yielding a zero expectation value [11] and forbidding the grow of the acoustic wave. In these conditions the acoustic wave is localized at z₀ and stationary. However, there exist weak fluctuations due to the non zero standard deviation [11], which act as spurious off-peak gratings producing noise. In fact, PRBS has been shown not to be optimal due to the generation of spurious off-peak correlation points along the fiber [13]. Lately, the so-called perfect codes, in particular Golomb codes, have been shown to provide better correlation characteristics with reduced off-peak noise [13, 14]. Nonetheless the optimal case of null off-peak reflections has never been obtained experimentally. The impact of off-peak correlation noise has been analyzed numerically and experimentally only in terms of optical signal-to-noise ratio (OSNR) for few particular cases [11–14]. Eventually, the distributed characterization of DBGs, obtained through pulsed or frequency swept probes, has been limited to uniform [9, 15] or non stationary gratings [8, 16–18], and mainly focused on the determination of the amplitude of the spectral response [19], or of the birefringence for sensing applications [16, 18, 20].

Here, we experimentally generate localized and stationary DBGs in a PMF. DBGs localization is achieved by phase modulation of the pump waves with a PRBSs [11], and different configurations of DBGs lengths and reflectivities are obtained. Heterodyne swept-wavelength interferometry (SWI) is used to measure the complex transfer function of the DBGs thus obtaining the distributed characterization in both modulus and phase. In previous publications similar techniques based on heterodyne detection have been exploited, but they enabled the amplitude characterization only. From the complex transfer function many so far not characterized features of the DBG are assessed: the apodization, the length, the bandwidth, the group delay ripple (GDR), the peak to sidelobe ratio (PSR) and the OSNR. Moreover, by applying post-processing filters, the transfer functions under optimal conditions are estimated and further quantitative insight on the correlation noise is obtained.

2. Experimental setup

The experimental setup is shown in Fig. 1. The fiber used in the experiments is a PMF of length L_f = 5m, with a measured average refractive index n_g = 1.4606 and an average birefringence Δn = 3.77 × 10⁻⁴. In the fast axis the measured Brillouin shift is ${\mathrm{\Omega }}_{\mathrm{B}}^f\approx 10.864\text{GHz}$. Localized DBGs are generated by two counter-propagating pump waves P₁ and P2 with frequencies ν₁ and ${\nu }_2={\nu }_1-{\mathrm{\Omega }}_B^f$, respectively.

 

Fig. 1 Experimental setup. ECL: external cavity laser; PM: phase modulator; FPC: fiber polarization controller; EDFA: erbium-doped fiber amplifier; SC-SSB: suppressed-carrier single sideband modulator; PBS: polarization beam splitter; PMF: polarization maintaining fiber; FRL: fiber-ring laser; PD: photodiode.

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The pumps are generated by an external cavity laser (ECL), whose wavelength is set at 1550 nm, and P₂ is downshifted from P₁ by means of a suppressed-carrier single sideband (SC-SSB1) modulator. Their powers are controlled by erbium-doped fiber amplifiers (EDFAs) and fiber polarization controllers (FPCs) are used to align them to the fast axis of the PMF. The polarization beam splitters (PBSs) are used to couple all waves to the PMF. A phase modulator (PM) is used to modulate both pumps with the same PRBS signal, which actually controls the length and the profile (apodization) of the DBGs.

The DBGs are interrogated by injecting a probe wave S aligned along the slow axis, at frequency ν_s = ν₁ − Δν, where the shift Δν ≈ ν₁Δn/n ≈ 45GHz depends on the birefringence of the PMF [1]. Under this phase-matching condition, S is partially reflected by the DBGs at the frequency ${\nu }_r={\nu }_s-{\mathrm{\Omega }}_B^f$ The DBGs characterization is performed by SWI. A local oscillator (LO) is realized through a high-coherence fiber-ring laser (FRL) (linewidth less than 2 kHz), whose frequency ν_lo (t) is linearly swept over ≈ 4 GHz at rate γ ≈ 350 GHz/s. The signal S is obtained from the LO by a frequency upshift of ${\mathrm{\Omega }}_{\mathrm{B}}^f-f_0$, with f₀ = 5 MHz through SC-SSB2. At the photodiode (PD), the beat between the reflected signal R, at frequency ${\nu }_r={\nu }_s-{\mathrm{\Omega }}_B^f={\nu }_{\text{lo}}(t)+f_0$, and the LO, is detected as a photocurrent given by

Eventually, from Eq. (1), the DBG impulse response h(t) and its spatial profile (actually the filter apodization) h(z) ≜ h(t/(2v_g)) (v_g being the group velocity) can be obtained by applying a discrete Fourier transform (DFT) to measured data. The achieved spatial resolution is about 4 cm, and a single sweep is performed in about 10 ms.

3. Results and discussion

The black solid curve shown in Fig. 2 refers to one of the experimental data obtained when the PRBS frequency is set to f_B = 300MHz. The amplitude |H| of the transfer function and the group delay (GD), defined by

 

Fig. 2 The spectral distribution of the DBG for a PRBS frequency of f_B = 300MHz.

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The impulse response h(t) of the DBG, whose amplitude |h| is shown in Fig. 3 (black solid curve), is obtained by applying the DFT to H(ν). As it can be seen, the correlation peak is at about 3 m, as set by the delay τ₁, and the shape closely matches the expected triangle. The noise outside the correlation peak is due to spurious reflections, which, as explained in the introduction, are caused by the fact that the acoustic wave has zero mean but non zero standard deviation outside the point of perfect correlation [11–14]. The red dashed curve $|\tilde{h}|$ is obtained by filtering h with a Gaussian filter centered at $z_0^{\prime }$ and of length L′ which are the parameters of the triangle fitting the data. Filtering enables to estimate the response of a DBG with ideal correlation characteristics, i.e. with no off-peak correlation noise. In fact, by applying the DFT to $\tilde{h}(t)$, the response of the “ideal” DBG $\tilde{H}$ with no off-peak reflections is calculated and $|\tilde{H}|$ and τ_g are shown in Fig. 2 (red solid curves). By analyzing both filtered and unfiltered spectral responses the characterization of DBGs realized through ideal and PRBS phase-modulation formats can be performed.

 

Fig. 3 The impulse response and the spatial distribution of the DBG for a PRBS frequency of f_B = 300 MHz.

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The GD ${\tilde{\tau }}_g$ (red solid curve) is almost constant in the pass-band, and presents peaks where $|\tilde{H}|$ approaches zero. This is coherent with the fact that DBGs are refractive index gratings like weakly reflective FBGs [22]. Additional proof is given by the dashed curves, which are obtained by numerically calculating the reflectivity of a FBG [10] using the measured data, h(z) (black) and $\tilde{h}(z)$ (red), and an ideal triangle (blue) for the spatial index profile. As it can be seen, in the pass-band the match is very good.

The results that follow present four different configurations (A, B, C and D) of pump waves and probe powers as given in Table 1. Each measure is the result of an average over 10 consecutive sweeps of the interferometer.

Table 1. Pumps and probe powers used in the experiments.

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Figure 4(a) shows the DBG measured length L and bandwidth B as a function of the PRBS frequency f_B. The solid black curves are the theoretical values given by

 

Fig. 4 (a) Measured length L and bandwidth B of the DBGs for different PRBS modulation frequencies f_B. (b) Peak reflectivity R_MAX as a function of the DBGs length L for the pumps and probe power levels given in Table 1.

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The measured reflectivity of the DBGs is reported in Fig. 4(b). The solid curves are the peak reflectivity values as given by Eq. 11 of Ref. [23]

The PSR, which is defined as the ratio between the peak value of the main and of the first side lobes of the amplitude of the transfer function, is measured on the filtered data $|\tilde{H}|$ and is reported in Fig. 5(a) as a function of the PRBS frequency f_B. The theoretical value of the PSR for the ideal sinc² function (solid blue curve of Fig. 2) is about 26.5 dB (red dashed line of Fig. 5(a)). For f_B ≥ 200 MHz the PSR ranges from 19 to 25 dB. The reduced PSR can be attributed to residual measurement noise and to the fact that bit pulses have not an exact rectangular shape. Moreover, the sinc² function only approximates the ideal transfer function of the system, since the acoustic wave, which depends on the correlation between the pumps, slightly depends also on the probe and on the reflected signals.

 

Fig. 5 PSR (a) and GDR (b) for different PRBS frequencies f_B.

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The GDR is the deviation of the GD with respect to its average. In Fig. 5(b) it is reported the peak-to-peak value of the GDR, Δτ_g,pp, calculated on τ_g in the pass-band of H (dashed curves). Similarly, the solid curves of Fig. 5(b) refer to the GDR calculated on ${\tilde{\tau }}_g$, i.e. from the filtered responses $\tilde{H}$ representing gratings with ideal correlation characteristics. For the PRBS, the average ripple is about 10 ns, but it can be as high as 22.5 ns. Moreover, at high f_B, the GDR increases for decreasing powers. The solid curves of Fig. 5(b) instead refer to the GDR calculated on ${\tilde{\tau }}_g$, i.e. from the filtered responses $\tilde{H}$ representing ideal gratings. As it can be seen, for ideal gratings the GDR, except for f_B = 100 MHz, could be lower at least by one order of magnitude with respect to those generated by the PRBS.

The different behavior for f_B = 100MHz (i.e. for a long grating — about 2 m) in both Figs. 5(a) and 5(b) can be attributed to birefringence variations along the fiber, whose effects can be qualitatively observed in Fig. 6, that shows the spectrograms of the grating response for a PRBS frequency of f_B = 100MHz (left) and f_B = 200MHz (right). As it can be seen, for the longer grating (left) the symmetry is broken, and the frequency of the peak along z varies more than for the shorter grating (right), and this can be related to the phase-matching condition ν₁Δn/n given above. In fact, by measuring the frequency shift of the peak along z, we estimated spatial variations of Δn/n in the order of 10⁻⁷. The cumulative effect of these spatial variations is higher for longer gratings, whereas for shorter gratings local birefringence fluctuations are negligible with respect to off-peak correlation noise.

 

Fig. 6 Spectrograms of the grating response for a PRBS frequency f_B = 100MHz (left) and f_B = 200MHz (right).

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The noise is quantified in Fig. 7, which shows the OSNR for different f_B. The OSNR for the PRBS case (dashed curves) is calculated according to

 

Fig. 7 Estimated OSNR for different PRBS frequencies f_B.

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4. Conclusions

We experimentally generated and characterized localized and stationary dynamic Brillouin gratings in a polarization maintaining fiber. Localization of dynamic Brillouin gratings has been achieved by phase modulation of the pumps with pseudo-random bit sequences. A comprehensive characterization of dynamic Brillouin gratings as optical filters has been performed through swept-wavelength interferometry. From the measurement of the complex transfer function of the dynamic Brillouin gratings several features have been determined: the apodization, the length, the bandwidth, the group delay ripple, the peak to sidelobe ratio and the optical signal-to-noise ratio have been measured. All features depend on the pseudo-random bit sequence frequency. The group delay ripple and optical signal-to-noise ratio have been shown to depend also on the pumps and probe powers while the reflectivity depends mainly on the pumps powers. By applying filtering, the performance of gratings generated under ideal conditions (no off-peak reflections) have been compared to those generated by pseudo-random bit sequence that are affected by off-peak correlation noise. The main advantages of Brillouin gratings, over Bragg like, are their fast and broad reconfigurability and their extremely narrow bandwidth, the latter property owed to the long length (even meters) of the gratings. The full characterization of the grating complex transfer function enabled the evaluation of the penalties associated to their use as optical filters, the main drawbacks being the low reflectivity and the noise due to the off peak reflections.