## Abstract

We show that in the generation of stimulated Brillouin scattering (SBS) in long (>1km) fiber, the threshold for observing coherent Stokes emission occurs at notably lower gain than the classical lasing threshold gain for SBS. The effect is due coherent regenerative amplification of strong spontaneous Brillouin scattering, which occurs in the presence of feedback. The magnitude of the effect scales with the length of interaction.

© 2008 Optical Society of America

Stimulated Brillouin scattering (SBS) continues to be of practical importance particularly in regard to optical fiber-based systems such as optical communication lines [1], fiber lasers and amplifiers [2], CW phase conjugation [3], and most recently controllable optical delay [4]. It is well known that in the presence of optical feedback, the Stokes emission may exhibit lasing [5]. The threshold exponential gain, *G _{th(osc)}*, for laser oscillation is commonly described by the standard condition for lasing, in a Fabry-Perot cavity

*G*= 2

_{th(osc)}*αL*- ln(R

_{1}R

_{2}) [6], where

*L*is the length of a medium, α is the intensity loss coefficient and R

_{1,2}are the power reflection coefficients of the cavity mirrors for the Stokes emission. In this paper we show that while this relation is upheld in a fiber of short length, in fiber of moderate to long length (> 1 km) the gain at which coherent Stokes emission first emerges is notably lower than

*G*

_{th(osc)}. We show that the effect is due to the high strength of the single-pass amplified spontaneously scattered Stokes emission in long fiber, which provides a strong seed for coherent regenerative amplification of the Stokes signal in the cavity. Along with the stochastic single-pass amplified Stokes emission, it provides a substantial contribution to the detectable output signal even when the actual SBS exponential gain is lower than that determined by the classical oscillation condition.

Our experimental arrangement and detection system is typical of that used in investigations of SBS in optical fiber, details of which can be found in [7]. A single mode CW Nd:YAG laser with variable output power of 0-2 W was used as the pump source. Multi-mode silica graded index fiber (core/cladding diameter 50/125 μm, numerical aperture NA = 0.2 and α ≈ 0.23 km^{-1}) of sample lengths from 0.2 to ∼4.2 km were investigated. The numerical value of the SBS gain coefficient for this fiber, g ≈ 1.1×10^{-11} m/W, was estimated taking into account waveguide-induced spectral broadening of SBS [8]. The entrance and exit faces of the fiber-samples were cleaved perpendicular to the fiber axis and, to avoid the contribution of reflected pump radiation to the detected signal, the fiber’s axis was tilted with respect to the optical axis of incident pump radiation. Fiber-end reflectivity, R, determined by Fresnel’s law, was estimated as ∼3.5%. The overall accuracy of measurements was ≤ 10%.

Typical examples of the dynamics of the Stokes emission for relatively short, 0.34 km, and long, 4.2 km, fiber at a pump power slightly above (∼1-1.1 times) the minimum pump power for detecting Stokes emission, P_{th,}, are shown in Figs. 1(a) and 1(c).The traces exhibit periodic trains of pulses, of period equal to the cavity round-trip time, *t _{r}* = 2n

*L*/c, and pulse duration

*t*

_{p}= (0.1-0.2)

*t*. Such dynamics are typical of mode-locked laser emission [9] and a manifestation of multimode cavity feedback. The peak power of the pulses in Figs. 1(a) and 1(c) corresponds to an SBS reflectivity, of ∼5-10%. Note that the modulation in trace 1(c) for the longer fiber also exhibits small amplitude noise-like behaviour. At higher pump power, ∼1.2P

_{r}_{th,}the dynamics in the short fiber sample, remains modulated at

*t*while its reflectivity increases to ≥ 50–100%, (see Fig. 1(b)). For progressively higher pump powers this modulation is gradually suppressed while a temporally stable component emerges and grows, toward eventual complete stable emission, which is attributable to spectral phase conjugation [10]. On route a sequence of higher harmonics of

_{r}*t*are observed, similar to that reported in single-mode fibers [6,11]. For the longer fiber sample, on increase of pump power, (see Fig. 1(d) for ∼1.6P

_{r}_{th}), the dynamics also remains periodically modulated at

*t*but the superimposed noise-like structure, occurring on a sub-microsecond time-scale, grows substantially along with the emergence and growth of the temporally stable component. At higher pump power, ≥ 2P

_{r}_{th}, the dynamical component comprises almost totally irregular spiking [7].

In Table 1, second column, we show measured P_{th} for each of the tested fiber samples. Direct conversion of P_{th}, to its corresponding threshold exponential gain, *G*
_{th}, through the relation *G*
_{th} = gP_{th}
*L*
_{eff}/*S*
_{eff}, where *S*
_{eff} is the effective cross sectional area and *L*
_{eff} the effective length (*L*
_{eff} = (1-e^{-αL})/α) of the SBS interaction, is nontrivial because of substantial uncertainty in the actual value of *S*
_{eff} in multimode fiber [12] and in the contribution of depolarisation [13]. To avoid these uncertainties we normalize our data for the fiber samples to that for the shortest fiber, *L* = 0.2 km, for which P_{th} = 0.95 W, giving *G*
_{th} ∼6.7. This is justified since it is expected that the spatial distribution and the rate of depolarisation of radiation in all the fiber samples is the same since they are taken from the same spool. The values of *G*
_{th} for all the samples are shown in the forth column in Table 1. Since for short fiber, *G*
_{th}, within experimental error, is the same as *G _{th(osc)}*, we also take

*G*∼ 6.7 to determine

_{th(osc)}*G*for the other fibers, calculated from the standard laser threshold condition,

_{th(osc)}*G*= 2α

_{th(osc)}*L*- ln (R

_{1}R

_{2}) with α = 0.23 km

^{-1}, and R = R

_{1,2}= 3.5%. These are shown in the last column of Table 1, and as seen are consistently higher than those of

*G*

_{th}. We note that, while in the shorter samples the difference is within our measurement accuracy of ∼10%, for the longer samples the difference is ∼25%. Furthermore, for the longer fibers the calculated values of

*G*are paradoxically higher than their corresponding values of

_{th(osc)}*G*, for zero feedback, which for example is 6.9±0.7 for the sample

_{th}*L*= 3.6 km [14].

To understand the underlying physics of our observations we consider three fundamental physical effects which simultaneously contribute to the Stokes output emission: seeding of the Stokes emission by spontaneous Brillouin scattering, its parametric amplification in the SBS active medium, and its regenerative coherent amplification because of cavity feedback. We take the pump radiation to be monochromatic and of constant intensity, *I _{p}*(z) =

*I*(0) = |

_{p}*E*(0)|

_{p}^{2}, (no pump depletion) propagating in the z direction in a medium from 0 to

*L*.

Evolution of the slowly varying amplitude of the intra-cavity Stokes field in the backward direction -z, from *L* to 0, is governed by the equation,

where *E’ _{S}*(z,t) is the SpBS seed field per unit length taken to be uniformly distributed along z. Here we suppose linear optical attenuation losses to be small, that is

*L*≪ 1/α. Thus in the direction z, from 0 to

*L*, the forward propagating Stokes field remains constant. Reflection of the Stokes field at the entrance (z = 0) and exit (z =

*L*) ends of the fiber,

*r*

_{1,2}, provide the boundary conditions. A rigorous analytical solution of this equation is not straightforward since the seed SpBS signal is incoherent. To account for this we consider the spatial evolution of the Stokes field

*E*(z,t) for each of its monochromatic Fourier components, E

_{S}_{S}(ω,z), which is described as, [14],

Here g(*ω*) = g_{0}/[1+4(*ω*-*ω _{B}*)

^{2}/Γ

^{2}], g

_{0}is the line-centre SBS gain coefficient,

*ω*is the resonant Stokes frequency and Γ is the FWHM width of the gain profile. The second term on the right-hand side of Eq. (2) represents the field of the SpBS radiation per unit length of the interaction region within a solid angle ϴ subtended by the pump radiation, and K is the SpBS coefficient.

_{B}To find E_{S}(*ω*,z) with proper boundary conditions we consider first a single-pass amplification of the Stokes field, E_{SS}(*ω*,z), seeded by SpBS, and second an intra-cavity regenerative amplification of the Stokes field, E_{SC}(*ω*,z), seeded by E_{SS}(*ω,L*). The resulting output field amplitude is then E_{S}(*ω*,0) = (1-*r*
_{1}
^{2})^{1/2}[E_{SS}(*ω*,0) + E_{SC}(*ω*,0)].

For the first stage the process can be treated in a way similar to that in [15]. Since at this stage an externally applied Stokes input is absent, that is E_{SS}(ω,*L*) = 0, Eq. (2) can be explicitly integrated, giving the spectral density of the output Stokes intensity (inside the medium), |E_{SS}(*ω*,z)|^{2}, which is,

where *G*(*ω*) = g(*ω*)*I _{p}*(0)

*L*is the exponential SBS gain. The electric field of this radiation, after reflection at z = 0, propagation from z = 0 to z =

*L*and reflection at z =

*L*, can then be considered as a seed field, E

_{SC}(

*ω,L*) =

*r*

_{1}

*r*

_{2}E

_{SS}(

*ω*,0), for further intra-cavity amplification.

According to standard theory of radiation amplification in a Fabry-Perot cavity with an intra-cavity gain medium [16], the output field, E_{SC}(*ω*,0), when resonant with a longitudinal cavity mode, is

in which it is understood that *r*
_{1}
*r*
_{2}exp[*G*(*ω*)/2] ≤ 1. The important feature of this equation is that its denominator,

can become small, ε ≪ 1, and even zero. In the first case the system operates as a high gain regenerative amplifier for the input field, while the second case corresponds to the standard oscillation condition.

When many cavity modes fall under the gain bandwidth envelope, which is typical for the case under consideration (>20 modes when *L* ≥ 200m), the output signal usually exhibits irregular spiking because the phase of the emission in the different modes are arbitrary and independent. However, as we experimentally observe (Fig. 1), the detected signal exhibit a periodic train of pulses, providing evidence that a number (*N*) of these modes are phase-locked. As such the peak intensity of the pulses is the intensity of the Stokes emission in each mode, given by Eq. (4), times *N*
^{2}. The total output intensity of the Stokes emission, *I _{S}*(0), is then the sum of this and that given by Eq. (3) both integrated over the spectrum,

where *G* = g_{0}
*I _{p}*(0)

*L*,

*δω*is the spectral width of emission in each mode which contributes to the periodic signal. The first (SS) term in the bracket describes the mean intensity of single-pass amplification of Stokes emission seeded by SpBS, which is stochastic. The second (SC) term describes the peak intensity of periodic emission due to regenerative amplification of this Stokes signal in the cavity, which is evidently coherent. Results of calculations are presented in Fig. 2 for the two fiber samples used in the experiments, length

_{c}*L*

_{1}= 0.2 km and

*L*

_{2}= 4.2 km.

We use ϴ (≈ π(NA)^{2}/n^{2} [8]) ≡ 0.06, where NA = 0,2 and n = 1.46 is the refractive index of the fiber, K ≡ 6.5×10^{-4} km^{-1} for silica [14] and (*r*
_{1}
*r*
_{2})_{ef}=exp(-G_{th(osc)}/2), which accounts for the fiber losses through effectively reduced *r*
_{1}
*r*
_{2}. Based on our measurements (see Figs. 1(a)–1(c)) *N* is estimated to be ∼4–6 and δω_{c}/Γ ≈ 2×10^{-3} (δω_{c} from spectra in Refs [11,17], and taking Γ ≡ 2π×35 MHz for silica at λ ≡ 1.06 μm [18]). The SS term of Eq. (6) is plotted as the thin line 1a for fiber *L*
_{1} and 2a for fiber *L*
_{2} in Fig. 2. They increase monotonically with *G* and at *G* ≈ 12 (not shown in Fig. 2) and ∼7.6 respectively, exceed the detection threshold, taken to be 0.01P_{p}, the so called 1% SBS threshold criterion. We note that this our choice of the SBS threshold criterion is actually the quantification of that described in [19]. The SC term of Eq. (6) is plotted as the dashed lines, curve 1b for fiber *L*
_{1} and curve 2b for fiber *L*
_{2}. As seen both grow with *G* up to a peak value, which occurs when the denominator in the SC term of Eq. (6) approaches zero which marks the classical condition for onset of SBS lasing, G_{th(osc)}. This occurs at *G*
_{th(osc)} = 6.7 for fiber *L*
_{1} and 8.6 for fiber *L*
_{2}. The thick solid lines 1and 2 are the total signals for the short, and long fiber, that is the sum of curves (terms) SS and SC.

The SC curves 1b and 2b intersect the SBS threshold line, P_{th}, at *G*’s lower than their
respective SS curves 1a and 2a. For the short fiber (curve 1b) this intersection occurs at *G* ≈ 6.7, which practically coincides with the lasing threshold *9*
_{th(osc)}, since the curve here is all but vertical, that is *G* = *G*
_{th} = *G*
_{th(osc)}. As seen this curve also coincides with line 1 above P_{th}, showing the contribution from SC is much greater than that from SS for shorter fiber. Hence from the onset of detectable Stokes emission the output is coherently modulated, arising overwhelmingly from regenerative amplification of the SpBS (term SC). This agrees with our experimental observations and also other reports of periodically modulated Stokes emission [6,11,20,21]. By contrast for the longer fiber a detectable Stokes emission occurs notably below *G* = *G*
_{th(osc)}, ∼8.6 down to *G* ≡ 6.7. At *G* ≥ 6.7 the total SBS signal, curve 2, comprises significant contributions from both the coherent SC term (curve 2b) and stochastic SS term (curve 2a). This again is what we experimentally observe (see Fig. 1(c)). We also see from Fig. 2 that curves 1 and 2 intersect the threshold line at approximately the same *G*
_{th}. In fact the same occurs for all the fiber samples which is also in pleasing agreement with our experimental findings (see column 4 of Table 1). Thus the principle features of the SBS emission at and close to the threshold of observation, P_{th}, are well captured by our model description. We should emphasise that our treatment strictly applies only at and around this threshold region since pump depletion is not taken into account. It cannot therefore correctly capture our experimental observations at higher *G* as for example trace d in Fig. 1 for P∼1.6 P_{th} for which pump depletion is significant.

In conclusion we have shown that for long optical fiber in the presence of feedback, SBS Stokes emission occurs notably below the classical lasing threshold. The emission is that of stochastic single pass amplified spontaneous Brillouin scattering and its coherent regenerative amplification in the cavity. Both components are detectable below the lasing threshold in long fiber because the strength of spontaneous scattering is high due to the long interaction length. We more generally conclude that conventional understanding of the laser threshold, as a singularity, marking the transition from stochastic to coherent lasing emission, is not strictly correct for fiber based lasers seeded by strong spontaneous scattering (Brillouin, Raman and active-media systems).

This work was supported by the Engineering and Physical Sciences Research Council, UK, Grant No GR/R56105/01.

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