1. Introduction

The control of the group velocity of light has recently received attention of many research groups. It has potential applications in optical communications where it can be used as buffers in all-optical router. Slow light in optical fibers using Brillouin interaction as physical mechanism has been evidenced [1, 2]. Since the 80’s, Brillouin scattering is regarded as a penalty in optical communications limiting the maximum input power into the fiber. At the contrary, Brillouin scattering is an important effect used in optical fiber sensors. Distributed Brillouin sensors are used to measure strain and temperature over 51 km in a single span without additional amplification [3].

Distributed Brillouin sensor based on a Brillouin Optical Time Domain Analysis (BOTDA) is a device where a Stokes and a pump beam counter-propagates, where frequency and time dependent variations of Stokes (or pump) intensity is detected. These sensors have shown the best performances in terms of sensing length, spatial resolution, temperature, and, strain accuracies [3–5]. The interaction is the maximum when the optical frequency difference between pump (v_p) and Stokes (v_s) corresponds to the Brillouin frequency (v_B). At the Brillouin frequency, the Stokes beam is amplified at the expense of the pump. v_B is the longitudinal acoustic phonon frequency and is a local material signature. An increase (decrease) in temperature or strain induces a proportional increase (decrease) of v_B. The sensor detects the Brillouin frequency variation as a function of position. The distributed nature of the sensor is achieved by field modulation of the probe beam. Due to the resonant nature of the interaction, the pulses propagating in the fiber experience a group velocity change. That has an impact on the location accuracy of the Brillouin sensor. Reference [6] has theoretically discussed four Brillouin sensor configurations using undepleted solutions of the steady-state coupled amplitude equations [7]. It was found that for the BOTDA set-up introduced in [4], slow light brings a location error larger than twice of the spatial resolution. This means that the definition of the spatial resolution of the distributed sensors (associated with the pulse width) should be re-considered, as the slowing light may bring the location error of 2-3 time of the pulse width.

The purpose of this work is to re-examine the validation of the pulse width as spatial resolution including Brillouin slow light for different pump and probe power over long and short sensing length for uniform and non-uniform strain/temperature profile. Our results show that pulse delay is significantly mitigated due to pump depletion over the sensing length. As pump depletion effect was neglected in Ref. [6], we account for this effect in our approach and compute spatial resolution versus location accuracy when Brillouin slow light is taken into account. Pump depletion is an important feature of the BOTDA. In fact, higher Stokes power increases the contrast of the interference fringes due to the interaction between the pump and probe beam, strengthening electrostriction, and then enhancing the local interaction. It improves the sensitivity to Brillouin frequency variations, and, as a consequence, results in better strain and temperature resolution. However, when depletion is too high as it happens for strong pump or strong Stokes pulse, the interaction between pump and probe tends to be reduced due to power imbalance and gain saturation [3, 8]. Hence strain and temperature measurement accuracies first increase when pump or probe power are increasing due to stronger interaction, which helps to detect small strain/temperature sections along the fiber length. Then this accuracy decreases due to gain saturation and pump power depletion, especially for long sensing length where moderate depletion gives better result [8] leading to better location accuracy. Moreover it is known that under large pump depletion Brillouin frequency shift at a given location influences the gain shape at the next positions [3, 8, 9] due to the power dependence of the phonon relaxation time [8]. To avoid spectrum distortion, it is then required to select a sensor operation mode that generates moderate pump depletion, which is close to a small gain dynamic range over the whole sensing fiber to maintain uniform strain/temperature resolution along the sensing length. Hence there is a trade-off between depletion and uniform gain over the sensing length [8].

In order to account for pump depletion, we solved analytically the coupled amplitude equations for stimulated Brillouin scattering (SBS) at steady-state condition. We obtained solutions for Stokes and pump phases that are expressed as integrals of pump and Stokes intensities respectively. These intensities are calculated from a set of two implicit equations [10] that are solutions of the coupled intensity equations for SBS at steady state. Eventually, the group delay is calculated from the differentiation of the phase respective to frequency. We have used these solutions to calculate the group delay for various fiber length cases and close to those of references 3, 4, 5 and 6. Through these calculations, we found that the pulse delay is increased by low Stokes power and small depletion, while distributed Brillouin sensors require stronger pump and Stokes interaction which contributes to pump depletion, smaller delay and better sensor performances. By taking pump depletion into account, we have shown that delay is negligible for several kilometers under typical BOTDA operational configuration as specified in Ref. [8].

2. Theoretical model

In the present work, we use the steady state approximation which is valid for pulses larger than the phonon lifetime (Δτ≈10 ns), which is equivalent to a spatial resolution w>1 m. We consider the steady state coupled amplitude equations that describe interaction between two counter-propagating laser beams in a single-mode optical fiber [7]:

where z is the position along the fiber, α is the fiber attenuation coefficient, A_s and A_p are the Stokes and the pump amplitudes respectively. The Stokes pulse originates from z=L and the pump from z=0. We introduced the complex gain coefficient κ that can be written as

Here g_B is the line center Brillouin gain coefficient, Δv=v_p-v_s-v_B is the detuning frequency, Δv_B is the Brillouin gain natural linewidth. The first term is the well-known Brillouin gain coefficient. The second term is related to the refractive index of the gain medium. It is possible that κ depends on position when the v_B is not uniform over the whole fiber length. We assume that the solutions have the form

where G_s (G_p) is the Stokes (pump) amplitude coefficients, which is gain in the case of the Stokes beam (loss in the case of the pump beam) and Φ_s (Ψ_p) is the Stokes (pump) phases. Substituting this ansatz in Eq. (1) and replacing κ by Eq. (2), we obtain the solutions for pump and Stokes phases as follows:

I_s(z,Δv) = |A_s|² and I_p(z,Δv) = |A_p|² are Stokes and pump intensities, respectively. φ can be moved out of the integral when the Brillouin frequency distribution is constant over the whole fiber length.

The intensities, I_s(z,Δv) and I_p(z,Δv), can be calculated by solving analytically the coupled intensity equations for SBS [10]. We use the same notation as in reference [10] by defining ∑(z,Δv)=I_p(z,Δv)+I_s(z,Δv) and Δ(z,Δv)=I_p(z,Δv)-I_s(z,Δv). Then the solutions for pump and Stokes intensities can be written in the implicit form by the following:

where ∑₀=∑(0, Δv) and Δ₀=Δ(0,Δv). The initial conditions of the coupled intensity equations are I_p(0,Δv)=I_p0 and I_s(L,Δv)=I_sL, where I_p0 is the input pump intensity, and, I_sL is the input Stokes intensity. G_s(G_s) at z is given by the logarithm of the ratio of I_s(z, Δv) (I_p(z, Δv)) to I_sL (I_p0).

Once the phase of the Stokes wave is known, it is straightforward to calculate the group delay affecting Stokes pulses propagating in an optical fiber. Similarly, we can calculate the delay on a pump pulse. Stokes and pump group delay, noted as τ_s(z,Δv) and τ_p(z,Δv), respectively, are then expressed as

Complex gain coefficient can be moved out of the integral when the Brillouin frequency distribution is uniform as shown in the right hand side of Eqs. (6).

In the undepleted pump approximation, Eq. (6a) reduces to the first term. The integrals yields I_p0L_eff, where L_eff=(1-e ^-αL)/α is the fiber effective length, and the group delay at Δv=0 becomes [6]

The comparison of Eq. (6a) and Eq. (7) shows that the pump frequency and position dependence of the pump intensity impacts significantly the delay. It is important to take pump depletion into consideration to have a complete picture of Brillouin slow light in optical fibers.

3. Numerical simulations results and discussion

In our simulations, we first compute the pump and Stokes intensities from Eqs. (5) for various positions along the fiber and frequency detuning ranging from -500 MHz to +500 MHz. We have implemented the Newton-Raphson method to solve these two equations [11]. We then integrate Eq. (4) for each of these frequencies and positions by using the extended Simpson rule [11]. Eventually, we calculated the group delay with the help of Eq. (6). All simulation parameters are: Δv_B = 35 MHz, A_eff = 80 μm², g_B = 2.3×10^-11 m/W, α= 0.35 dB/km. These computations were performed for fiber lengths of 100 m, 1 km and 10 km. Note that our model is based on steady states solutions. It is valid for pulses as small as 10ns, meaning that 1 m is the smallest spatial resolution for which the whole discussion is valid.

 

Fig. 1. (a). Group delay of Stokes beam with P_sL= 0.1mW (blue), P_sL= 1 mW (red), P_sL= 10 mW (green) and undepleted (violet) for L = 10 km and P_p0= 6 mW; (b) Group delay of three Stokes beam with P_sL= 0.1 mW (blue), P_sL= 1 mW (red), P_sL= 10 mW (green) and undepleted (violet) for L = 100 m and P_p0= 6 mW.

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Two extreme cases distinct by the fiber length are presented in Fig. 1. Figure 1(a) shows the delay variation as a function of position in a 10 km long fiber. The delay of the Stokes pulse at the fiber output (z=0) is 20 ns for P_sL = 0.1 mW, 11 ns for P_sL = 1m W and 3 ns for P_sL = 10 mW while the undepleted [Eq. (7)] case yields 55 ns when the pulse exits the fiber. Both delay variations with position remain far below the undepleted approximation. Hence lower Stokes power induces more delay. Because of the depletion effect, the real Brillouin gain is much smaller than the undepleted case. In contrast to Ref. [6], the location error due to slow light effect is well below the spatial resolution defined by the pulse width. Figure 1(b) depicts the delay variation with position in a 100 m long fiber. In 100 m applications, the targeted spatial resolution can be as low as 20 cm, which corresponds to a pulse width of 2 ns.

 

Fig. 2. (a). Gain of Stokes beam with P_sL= 0.1 mW (blue), P_sL= 1 mW (red), P_sL= 10 mW (green) for L = 10 km and P_p0= 6 mW; (b) Stokes and pump power distribution along a 10km long fiber for P_sL= 0.1 mW (blue), P_sL= 1 mW (red), P_sL = 10 mW (green) and P_p0= 6 mW; Solid lines refer to pump power and dashed lines refer to Stokes power.

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The comparison of Figs. 1(a) and 1(b) shows that the discrepancy between depleted and undepleted approaches reduces as the fiber length decreases. That can be understood when we consider Fig. 2(a) which represents the Stokes pulse gain distribution along fibers of 10 km. When the input Stokes power is small (P_sL = 0.1 mW), the Stokes gain increases monotonically as does the Stokes pulse power. At the Stokes input, the pump power is large enough to contribute to the amplification of the pulse everywhere in the fiber [Fig. 2(b)]. In the case of an input Stokes of 1 mW, the gain first is constant as the pulse moves towards the other fiber end. In that region, the pump is strong enough to balance the effect of the natural loss of the fiber. As the pulse gets closer to the pump origin, the amplification rises as the pump is stronger. When the input Stokes power is 10 mW, the pump is strongly depleted when z=L. The energy transfer from the pulse cannot compensate the natural fiber loss. The amplification increases only in the first 2 km of the fiber.

 

Fig. 3. (a). Brillouin loss spectrum of the stressed section for P_sL= 10 mW (blue), P_sL= 1 mW (red), P_sL= 20 μW (green); (b) Peak frequency detected on the Brillouin loss spectrum of the stressed section as a function of input Stokes power. Simulation parameters are L = 10 km, P_p0= 6 mW, Δl = 1m and v_Bs-v_B = 35 MHz.

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Figure 2 clearly emphasize the role of gain build-up and pump depletion. The overall high gain for Stokes power of 0.1 mW indicates high delay as shown in Fig. 1. In the 10 mW case, small delay is associated with small overall gain, while Stokes power and gain distributions vary significantly over the last 2 km of the pulse propagation in the fiber. This behavior is the result of a stronger interaction of pump with Stokes, which is a condition for better strain and temperature accuracy as electrostriction is enhanced.

If pump depletion limits the pulse delay in the optical fiber, it is necessary to consider its impact on Brillouin frequency shift detection. The results shown in Fig. 3 come from the modeling of a 10 km long fiber whose first section (near the output of the Stokes signal) is stressed such that the Brillouin frequency is shifted by 35 MHz. The stressed section length Δl is 1 m. Figure 3(a) shows the normalized Brillouin loss spectrum at the stressed section. It is clear that the spectrum peak frequency is close to the 35 MHz shift at power in the milliwatt range although the frequency shift is underestimated at lower power. That behavior is summarized in Fig. 3(b) where the detected peak frequency of the Brillouin loss spectrum is presented as a function of the input Stokes power for Δl = 1 m. Similar trends are observed for longer sections.

If depletion is required, it should remain moderate because too high depletion brings in three kinds of problems. First, a large depletion of the pump would make it impossible to sense temperature and strain over the whole fiber length. Only the initial sections of the fiber would be measurable while no pump power would be left along the rest of the fiber to achieve interaction between pump and Stokes beams [3]. Second, there is always the risk to have the measured spectrum contaminated with non-local effects [3, 8, 9]. Third, large power contributes to the increase of the phonon relaxation time [8], which in turn degrades spatial resolution and SNR. After experimental results presented in [8], we know that SNR improves as a function of power until it reaches a maximum around 5 mW. SNR then drops degrading the frequency resolution. These effects should be avoided for distributed sensor systems. Therefore moderate depletion is recommended for the pump and probe power of 4-5 mW [3, 8]. A trade-off between large gain and pump depletion over the entire sensing length must be achieved: on the one hand, the sensor needs a large dynamic range, which is obtained with low Stokes power but with large delays; on the other hand, power depletion brings stronger interaction and then lower gain influences strain and temperature resolution, but significantly reducing delay. Those two issues must be accounted to provide the best spatial resolution combined with optimum temperature and strain resolution.

 

Fig. 4. (a) Gain of three Stokes beam with P_sL= 0.1 mW (blue), P_sL= 1 mW (red), P_sL= 10 mW (green) for L = 100 m and P_p0= 6 mW; (b) Stokes and pump power distribution along a 100 m long fiber for P_sL= 0.1 mW (blue), P_sL= 1 mW (red), P_sL= 10 mW (green) and P_p0= 6 mW; Solid lines refer to pump power and dashed lines refer to Stokes power.

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For much shorter fiber (L = 100 m), neither gain nor powers suffer strong variation over the length. At the contrary, they vary monotonically as a function of position as shown in Fig. 4. Although depletion is weak, it is obvious that it cannot be neglected in the case of 10 mW Stokes and 6 mW pump case. Moreover, these power settings contribute to lower delay, and, also better operation for the distributed sensor system. The depletion process enhanced distributed sensor performance and reduced the pulse delay.

For a given fiber length with an appropriate selection of Stokes and pump powers, pulse delay can be mitigated and the location error is minimized. For example, the location error in the 100 m can be lowered further as it is reported in Fig. 5 where a Stokes power larger than 10 mW is used. Figure 5 also reveals that the undepleted approximation is the asymptotic case of our model. When the Stokes power diminishes, the delay increases until it reaches a plateau of undepleted delay.

 

Fig. 5. Group delay of output Stokes beam (z=0 m,) as a function of input Stokes power for three fiber lengths (L = 100 m, blue, L = 1 km, green, L= 10 km, red) and P_p0 = 6mW.

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The purpose of the Brillouin sensor is to detect temperature and/or strain variations that are linearly related to Brillouin frequency changes. Moreover, in practical conditions, the Brillouin frequency is never uniform due mainly to environmental stresses on the fiber. These variations shift local resonances and then reduce pump depletion along the fiber. Ultimately, they affect group index and hence decrease group delay. In order to estimate qualitatively the impact of a non-uniform Brillouin frequency distribution, we study two profile cases for a 1 km long fiber [12]. These profiles are made of a short section (stressed section) of different length (Δl = 10m and Δl = 100 m) whose Brillouin frequency is up-shifted by 50 MHz and located in the middle of the test fiber. The Brillouin frequency is unshifted elsewhere along the fiber. A non-uniform Brillouin frequency profile limits the delay as shown in Fig. 6(a) where those two cases are compared with a uniform fiber. It is clear that the delay decreases when the fiber strain or temperature is non-uniform because during propagation pulse encounters sections that are off-resonance. The amplitude of the delay reduction increases if the section is larger. That behavior can be associated to the largest gain drops that occurs for the case of the fiber whose “stressed” section is longer (Δl = 100 m) as presented in Fig. 6(b). The short section case (Δl = 10 m) does not differ excessively from the uniform. The same analysis was conducted for stressed section located elsewhere in the fiber and for longer fibers (10 km), both leading to similar results.

 

Fig. 6. (a). Group delay of a Stokes beam at the Brillouin frequency of a fiber as a function of position for uniform Brillouin frequency profile (blue), Δl = 10 m (red), Δl = 100 m (green); (b) Gain of a Stokes beam at the Brillouin frequency of a fiber as a function of position for uniform Brillouin frequency profile (blue), Δl = 10 m (red), Δl = 100 m (green); parameters are P_p0= 6 mW, P_s0= 1 mW. The non-uniform cases are up-shifted by 50 MHz.

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The pulse delay depends on the pump power, the Stokes power, fiber length as well as Brillouin frequency distribution. These parameters strongly influence the actual gain experienced by the pulse and hence the Stokes group delay. Uniform Brillouin frequency leads systematically to higher delay. It corresponds to the highest location error for distributed sensor, and it is less than the pulse width.

4. Conclusions

In order to study the effect of Brillouin slow light on BOTDA based sensor, we have solved the steady state coupled amplitude equations for SBS in optical fibers. We obtained integral forms for Stokes and pump phases as well as Stokes and pump group delays. Our solutions take into account pump depletion. We found that moderate pump depletion mitigates significantly the Stokes pulse delay: the maximum delay induced at resonance is only a fraction of BOTDA spatial resolution, which means that the use of pulse width to define the spatial resolution is valid when Brillouin slow light is considered. Non-uniform strain and temperature distribution in a fiber reduces furthermore delay related location inaccuracy in BOTDA.