Numerical approximation for Brillouin fiber ring resonator

Cristina Elena Preda; Andrei A. Fotiadi; Patrice Mégret

doi:10.1364/OE.20.005783

1. Introduction

The Stimulated Brillouin Scattering (SBS) is one of the most dominant nonlinear effects in optical fibers. This is a process initiated by spontaneous scattering of a pump wave on occasional hypersound waves caused by thermal noise. In the last years, the interest for Brillouin fiber ring lasers has significantly increased due to the large spectrum of their applications. The large scale of their peak power, the ultra-narrow linewidth and the low threshold power are the most exploited features. In spite of their high-power delivery, their application is limited because of the pulse-to-pulse peak-power fluctuations [1, 2]. Therefore, it is of great interest to develop new models and methods for controlling such amplitude fluctuations which are mainly due to the stochastic nature of the spontaneous Brillouin scattering that initiates the SBS. For instance, the general form of the non-linear rate equations system of the SBS is difficult to solve. In [3,4] are provided analytical and numerical solutions for steady state stimulated Brillouin scattering in single mode optical fibers and in continuous-wave pumping. The SBS, used for compressing lasers pulse duration, is studied also in dynamic regime, but in linear cavity in [5,6]. Simulation and approximation tools are proposed. For a fiber ring resonator, theoretical investigations of the transient SBS instability which follow a switch-on of the pump signal are addressed in [7].

In this paper we present an iterative method for solving the SBS rate equations system. This technique allows to understand and to explain the behavior of the fiber ring resonator when it serves as generator for the SBS and thus, it has the role of a mirror for self-Q-switching laser operation. More precisely, we propose an iterative method for estimating the outputs: transmitted pump and Stokes waves power from the ring resonator, as well as the enhancement of pump power inside the resonator when an optical pulse is incident into it through a fiber directional coupler. Furthermore, we present experimental results that match the theoretical results obtained with our iterative method.

2. Theory and computational method

SBS is a nonlinear interaction between a laser pump wave and first-order Stokes wave through the intermediary of the sound wave. The coupling of the optical waves can be written as [8, 9]:

\begin{array}{l} \frac{\partial E_{p}}{\partial z} + \frac{n}{c} \frac{\partial E_{p}}{\partial t} = - \frac{α}{2} E_{p} + i \frac{γ ω_{p}}{4 ρ_{0} n c} ρ E_{s}, \\ \frac{\partial E_{s}}{\partial z} - \frac{n}{c} \frac{\partial E_{s}}{\partial t} = \frac{α}{2} E_{s} - i \frac{γ ω_{s}}{4 ρ_{0} n c} ρ^{*} E_{p}, \\ \frac{\partial ρ}{\partial t} + \frac{Γ}{2} ρ = i \frac{γ q^{2}}{16 π Ω} E_{p} E_{s}^{*} + f, \end{array}

where E_p(z, t) represents the slowly-varying complex field amplitude of the pump wave with the frequency ω_p and the wave number κ_p. E_s(z, t) represents the slowly-varying complex field amplitude of Stokes wave with the frequency ω_s and the wave number κ_s. ρ(z, t) is the complex amplitude of the variation of the material density from the mean value ρ₀. We have introduced the acoustic frequency Ω = ω_p − ω_s and the acoustic wave number q = κ_p + κ_s. The various constants used are: α the linear fiber loss coefficient, n the refraction index, c the speed of the light in a vacuum, γ the electrostrictive constant and Γ = 1/τ_p the phonon decay rate with τ_p the photon life time. Here, f(z, t) is a Langevin noise source that describes the thermal fluctuations in the density of the fiber that leads to spontaneous Brillouin scattering. This noise source is modeled as a centered Gaussian noise having the auto-correlation function of type 〈f(z, t), f^*(z′, t′)〉 = Yδ(z − z′)δ(t − t′). As in [8], the noise intensity is given by:

Y = 2 κ_{B} T ρ_{0} / (τ_{p} v_{A}^{2} S_{eff})

, where κ_B is Boltzmann’s constant, T is the temperature, S_eff is the effective core area and v_A is the acoustic velocity in this material.

The methodology that we use is similar as that employed in [10]. We simplify the spatio-temporal system of the coupled rate equations by neglecting the time derivatives of the complex field amplitudes and thus by considering, for the acoustic wave, an expression no depending on time [11]:

ρ (z) = \frac{2}{Γ} (\frac{i γ q^{2}}{16 π Ω} E_{p} (z) E_{s} {(z)}^{*} + f (z)) .

Thus, the classical system (1) which couples, through the electrostriction process, the evolution of two optical fields E_p and E_s with acoustic field ρ, is reduced from three to two coupled equations for the complex field amplitudes of the pump and the Stokes wave:

\begin{array}{l} \frac{d E_{p} (z)}{d z} = - \frac{α}{2} E_{p} (z) - g_{E} {| E_{s} (z) |}^{2} E_{p} (z) + i b E_{s} (z) f (z), \\ \frac{d E_{s} (z)}{d z} = \frac{α}{2} E_{s} (z) - g_{E} {| E_{p} (z) |}^{2} E_{s} (z) - i b E_{p} (z) f^{*} (z), \end{array}

where the gain coefficient is given by g_E = (nc/16π)g_B and the constant b by

b = \sqrt{g_{B} {cv}_{A} / ρ n Γ} / g_{E}

, with g_B the standard Brillouin gain coefficient of the medium.

The obtained approximation is generally valid for a continuous-wave pumping but it works as well for a regime where the incident pulse duration is much longer than the round-trip time (t_r) in the ring cavity and superior to the phonon life time in this environment. Such a model, usually used to obtain the solutions in the steady-state regime, can be also used in the transient or pulsed regime when the time scales are longer than the round-trip time.

We adapt this system of two non linear rate equations to a ring resonator which plays the role of a Brillouin laser cavity (see Fig. 1(a)). This cavity is formed by a single-mode fiber loop of length L which is direct-coupled connected to the coupler which can be characterized by an intensity coupling coefficient κ. In our model, we consider that the time evolution of the incident signal is periodic and has a secant-hyperbolic shape with a FWHM ∼ 500 t_r. With these considerations, the role of the pulse shape is taken into account by using for each new m round-trip a different value E_in(mt_r) of the incident field (Fig. 1(a)). After several recirculation pass of the pump wave in the cavity, we obtain a concentration of energy inside the ring due to the incident power accumulated inside. Thus, using a weak incident power, one can obtain in this way high values of the power generating non-linear effects in the ring as the SBS.

Fig. 1 (a) Schematic representation of pulsed pumping Brillouin fiber ring laser with a zoom on the time evolution of the incident field E_in(t). (b) Schematic representation of the integration method with Δz the integration step.

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For every m^th round-trip of pump and Stokes waves in the resonator, the rate equations system (3) describing the spatial distribution of the pump wave amplitude $E_{p}^{(m)} (z)$ and of the Stokes wave amplitude $E_{s}^{(m)} (z)$ along the fiber ring is solved, as is shown in Fig. 1(b) in four steps: step 1 and step 2 followed by the steps 1′ and 2′, respectively. In the following, we describe these four steps. Assuming that the pump and the Stokes waves propagate with the same velocity, we can consider that during the forward-propagation of the pump on the first half of the resonator (step 1), the Stokes wave is backward-propagated on the second half of the resonator (step 1′). These two steps are followed by the forward-propagation (backward-propagated) of the pump (Stokes) on the second half (step 2 (2′)) by using the values of the Stokes (pump) field already found in the first part of the integration (step 1′ (1)). In this way, the system (3) of the coupled equations is decoupled: the first equation is used to numerically propagate the pump, whereas the second one, to propagate the Stokes wave. These ordinary differential equations can be now independently integrated using a numerical approximation method such as the fourth-order Runge-Kutta. Let denote by ℜE_p, ℑE_p, ℜE_s and ℑE_s the real and imaginary part of the pump and Stokes fields. Then, the pump propagation is described by the following two equations, coupled through the values of Stokes field $ℜ E_{s}^{(m)} (z)$ and $ℑ E_{s}^{(m)} (z)$ :

\begin{array}{l} \frac{d ℜ E_{p}^{(m)} (z)}{d z} = - \frac{α}{2} ℜ E_{p}^{(m)} (z) - g_{E} {| E_{s}^{(m)} (z) |}^{2} ℜ E_{p}^{(m)} (z) - b ℑ E_{s}^{(m)} (z) f (z), \\ \frac{d ℑ E_{p}^{(m)} (z)}{d z} = - \frac{α}{2} ℑ E_{p}^{(m)} (z) - g_{E} {| E_{s}^{(m)} (z) |}^{2} ℑ E_{p}^{(m)} (z) + b ℜ E_{s}^{(m)} (z) f (z), \end{array}

with the boundary conditions for each new m round-trip calculated as a superposition [12, 13] between the value of the incident field at the moment m × t_r and the circulating field of the precedent round-trip which are transmitted through the coupler (Fig. 1(a)):

E_{p}^{(m)} (0) = i \sqrt{1 - κ} E_{in} (m t_{r}) + \sqrt{κ} E_{p}^{(m - 1)} (L) e^{- i Δ ϕ_{p}},

where Δϕ_p is detuning from resonance which replaces the linear phase shift for the pump wave ϕ_p and is defined by: Δϕ_p = 2πnL/λ_p − 2pπ, p being an integer. We may observe that, for the first pass, the boundary conditions depend only on the value of the incident field of the pump.

In order to obtain the spatial distribution of the Stokes field, we use the second equation of the system (3) in which the pump and Stokes fields are counterpropagating. As these two propagations are decoupled, we impose a change of the sign in this equation. Put z′ = L − z with z′ ∈ [0,L] and one obtains the following two equations coupled through the values of pump $ℜ E_{p}^{(m)} (z)$ and $ℑ E_{p}^{(m)} (z)$ :

\begin{array}{l} \frac{d ℜ E_{s}^{(m)} (z^{'})}{d z^{'}} = - \frac{α}{2} ℜ E_{s}^{(m)} (z^{'}) + g_{E} {| E_{p}^{(m)} (z^{'}) |}^{2} ℜ E_{s}^{(m)} (z^{'}) - b ℑ E_{p}^{(m)} (z^{'}) f^{*} (z^{'}), \\ \frac{d ℑ E_{s}^{(m)} (z^{'})}{d z^{'}} = - \frac{α}{2} ℑ E_{s}^{(m)} (z^{'}) + g_{E} {| E_{p}^{(m)} (z^{'}) |}^{2} ℑ E_{s}^{(m)} (z^{'}) + b ℜ E_{p}^{(m)} (z^{'}) f^{*} (z^{'}), \end{array}

with the boundary condition:

E_{s}^{(m)} (0) = \sqrt{κ} E_{s}^{(m - 1)} (L) e^{- i Δ ϕ_{s}},

where Δϕ_s is the linear phase shift per round-trip of the Stokes wave. The linear phase shift of the Stokes wave is different of the pump linear phase shift Δϕ_p and it is defined according to the following expression [7]: Δϕ_s = Δϕ_p − 2πν_BnL/c, where ν_B is the acoustic frequency.

Based on these observations, for each m round trip in the ring, the values of the Stokes (pump) field appearing in the previous Eqs. (4) and (6), respectively, are calculated as fallows: for the propagation on the first half of the round trip they are given according to the values obtained and stored in the previous round trip (using Eqs. (6) and (4), respectively) : $E_{s}^{(m)} (z) = E_{s}^{(m - 1)} (z^{'})$ , $E_{p}^{(m)} (z^{'}) = E_{p}^{(m - 1)} (z)$ and, for the second part of the resonator they are given by those found in the first step integration of the Stokes (pump) field: $E_{s}^{(m)} (z) = E_{s}^{(m)} (z^{'})$ , $E_{p}^{(m)} (z^{'}) = E_{p}^{(m)} (z)$ . From these last relations and from the boundary conditions (7) let observe that for the first round trip, the Stokes field will never appear without the contribution of the noise f(z). That justifies the introduction of the noise source term in the initial system (1).

After each complete tour of the resonator of t_r duration, we obtain the spatial distribution of fields and the new initial conditions computed using Eqs. (5) and (7). Simultaneously we obtain the expressions for the output fields after m^th round-trip:

E_{p}^{out} (m t_{r}) = \sqrt{κ} E_{in} (m t_{r}) + i \sqrt{1 - κ} E_{p} (L, m t_{r}) e^{- i Δ ϕ_{p}},

E_{s}^{out} (m t_{r}) = i \sqrt{1 - κ} E_{s} (0, m t_{r}) e^{- i Δ ϕ_{s}} .

By Eq. (8) one computes the time evolution (at each mt_r) of the resonator transmission as :

{T (mt}_{r}) = {| E_{p}^{out} ({mt}_{r}) / E_{0} |}^{2}

and by Eq. (5) one calculates the time evolution of the enhancement coefficient Q inside the resonator :

{Q (mt}_{r}) = {| E_{p}^{(m)} (0, {mt}_{r}) / E_{0} |}^{2}

. Finally, the relation (9) allows to obtain the Stokes power in the output of the laser :

P_{S} ({mt}_{r}) = (n / 2) ε_{0} {cS}_{eff} {| E_{S}^{out} ({mt}_{r}) |}^{2}

. The round-trip number is fixed at the beginning of the simulation such that it covers several periods of the incident signal.

3. Experimental setup and results

The theoretical results were checked experimentally in the optical fiber ring resonator shown in Fig. 2. As the single-mode fiber (SMF) used is 2 m long, the free spectral range (SFR) of this ring cavity is approximately 100 MHZ. This value is about 7 times higher than the FWHM Δν_B (≈ 14.5 MHz at 1550 nm) of the Brillouin gain curve. This condition restricts the Stokes oscillation to a single longitudinal mode [14]. The Brillouin fiber ring laser is pumped by a periodical signal delivered by a laser diode Tunics with an external diffraction grating and emitting at 1550 nm. The line bandwidth of pump laser is less than 10 MHZ, that it 10 times smaller than the SFR of the cavity. Thus, the pump is narrowband source. For the experiments, the laser wavelength was continuously tuned in an available range in order to maximize the Brillouin scattering effect. When the effect is maximal it ensures us that the ring is resonant for both: the pump and the Brillouin emission [15]. The experimental setup is totally automated through a C + + application that drives simultaneously a high-speed multifunction data acquisition card (National Instruments PCI-6259) and a fast sampling oscilloscope (Tektronix TDS7104). The signals used to drive the pump laser and the trigger of the oscilloscope are generated by the digital-analog converter of the data acquisition card with 2.86 MS/s sample rate. In the experiments, the power of the pump laser is directly modulated by the current driving the pump diode. This periodic modulation signal, with an average power of about 20 mW, is applied on the ring cavity through the coupler (10/90). Each period of the incident signal (the black signals on Fig. 2) is accompanied by one pulse emitted by SBS inside the ring resonator (the red signals on Fig. 2) with an average power of about 0.4 mW. The Brillouin laser emission obtained inside the ring and transmitted by the coupler, the pump transmitted by the ring coupler and the 1% of the pump modulation passed through the first coupler (1/99) are monitored by high speed detectors (D) with 150 MHz bandwidth and digitized by the oscilloscope. To realize a relevant comparison between the experience and the results given by the theoretical model, the 1% of the pump detected after the first coupler was used, after recalculations, as incident signal in simulations. In Fig. 3 we show that experimental (Fig. 3(a)) and simulation results (Fig. 3(b)) for the transmitted pump and Stokes signal are concordant for the same conditions in dynamical regime with a secant-hyperbolic periodical signal as input. The parameters used for the numerical calculations are those of the conventional fiber, with the following values: the acoustic frequency ν_B = 11 GHz, the effective mode area S_eff = 85 μm², the linear refractive index n = 1.454, the density ρ₀ = 2210 kg/m³, the acoustic velocity v_A = 5960 m/s, the photon live time τ_p = 22 ns, the linear fiber loss coefficient α = 2.3 × 10⁻⁵ m⁻¹ and the pump wavelength λ = 1550 nm. For both situations, experiment and simulation, we observe a phenomena of depletion in the transmitted pump that corresponds to the SBS signal emission. The shape of the pulses is qualitatively the same with a pulse duration of about 350 ns (FWHM) and its repetition period is of about 140μs which was imposed by the repetition period of the pumping signal (notice that it can be modified).

Fig. 2 Experimental setup of the Brillouin fiber ring laser with all the fiber components with polarization maintaining (PM).

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Fig. 3 (a) Experimental results. (b) Simulation results obtained on the resonance conditions Δϕ_p = 0. Transmitted signals (black) and SBS signals (red).

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4. Conclusion

We have proposed a new methodology for approximating the time evolution of the power for Stokes wave, of the pump enhancement and of the pump transmission in dynamic regime. This method, accurate and easily implementable, can be used for modeling the dynamically pumped Brillouin lasers with a wide variety of the pump pulse shape or phase modulation. This work provides a starting point for further investigation in modeling and understanding Brillouin fiber lasers and similar systems. Our iterative method allows to obtain a good agreement between the experimental results and the predictions given by the SBS model.

Acknowledgments

This research was supported by the European Regional Development Fund and the Wallonia (Mediatic projet), the Interuniversity Attraction Pole program VI/10 of the Belgian Science Policy, the FP7 IRSES projects and the programs of Russian Federal Agency on Science and Innovation. The authors thank Dr. G. Ravet for helpful discussions.

References and links

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Numerical approximation for Brillouin fiber ring resonator

Abstract

1. Introduction

2. Theory and computational method

3. Experimental setup and results

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Equations (9)

Optics Express