## Abstract

We show that in optical fiber the threshold exponential gain, *G _{th}*, for stimulated Brillouin scattering initiated by spontaneous Brillouin scattering is functionally and strongly dependent on the material, length and numerical aperture of the fiber and the pump wavelength. For silica fiber we show that the value of

*G*at

_{th}*λ*≅1 µm ranges from as low as ~5 in long fiber (≥ few kms) to ~10–12 in fibers of ~100 m length and ~20–23 for very short fibers (<10 cm).

©2007 Optical Society of America

Stimulated Brillouin scattering (SBS) [1] continues to stimulate interest for its practical importance especially in optical fiber-based systems such as optical communication lines [2], fiber lasers and amplifiers [3], CW phase conjugation [4], and most recently controllable optical delay [5]. Of the parameters that characterise SBS, perhaps the most widely used is its threshold. In reality, however, there is no physical threshold for SBS, since the Stokes emission is in essence amplification of the spontaneous Brillouin scattering (SpBS) that initiates it; (we are not concerned here with SBS systems with external feedback for which there is a threshold). For practical convenience, the idea of a threshold is nevertheless introduced to describe the pump power, P_{th}, at which the output Stokes power is some measurable fraction, *r*, of P_{th}; *r* is then used as a threshold criterion [6]. However, since P_{th} is a characteristic of the SBS sample rather than the phenomenon itself, it is more meaningful to characterise the SBS threshold by the exponential gain, *G _{th}*=

*g*

_{0}P

_{th}

*L*

_{eff}/

*A*[7], where

*g*

_{0}is the SBS gain coefficient of the medium,

*L*

_{eff}is its effective length and

*A*is the effective cross-sectional area of the interaction region. Physically

*G*describes the amplification of the seeded Stokes signal to its output power level of rP

_{th}_{th}. There are however problems with existing treatments of the SBS threshold in fiber. Foremost,

*G*is commonly taken to be constant, of value ~21, independent of the fiber and pump radiation characteristics. Further, this value, which was originally obtained for

_{th}*r*=1 [2], is used in experiments in spite of substantial variation of the actual experimental value of

*r*; we have found it to vary from ~10

^{-4}to ~10

^{-1}[8–11], while in many other cases it has been simply left undetermined, e.g. [12]. This aside, it is speculated, though surprisingly there is no evidence of its actual use in the literature, that the criterion for the SBS threshold is

*r*≅10

^{-2}, the so called 1% criterion; the physical justification for this criterion is sensible since it marks the transition from the small signal to pump depletion regime of the SBS interaction. Also, in theory it is widely assumed that the Stokes signal is seeded by a single Stokes photon per spatio-temporal mode [2,6], instead of spontaneous Brillouin scattering (SpBS) as classically described.

In this work we provide a physically consistent theoretical treatment of *G _{th}* which considers the Stokes emission to be initiated by SpBS and uses the 1% criterion for the threshold. We show from our analysis that, in fiber

*G*depends explicitly and strongly on the length,

_{th}*L*, and the numerical aperture (NA) of the fiber as well as the fiber material and pump radiation wavelength,

*λ*. These dependencies are a consequence of the fact that the power/intensity of the SpBS seed scales with SBS interaction length, solid angle of incident pump radiation and SpBS coefficient respectively. For silica fiber we show that the value of

*G*at

_{th}*λ*≅1 µm, for a fiber with NA=0.2 ranges from as low as ~5 in long fiber (≥few kms) to ~10-12 in fiber of ~100 m length and ~20–23 in very short fiber (<10 cm).

In treating SBS we follow the approach originally developed by Tang [13]. Since the SpBS seed is temporally incoherent with a spectrally distributed intensity, the total Stokes intensity integrated over its spectrum at the output (z=0) is given by

where I_{S}(0,*ω*) is the spectral density of the Stokes intensity. Its spatial evolution (along the direction -z) in the field of the pump wave with constant intensity, *I _{p}*(z)=

*I*(0)=const, (no pump depletion and

_{p}*L*≪1/α, where α is the linear optical attenuation coefficient) is described by the equation

with *g*(*ω*)=*g*
_{0}/[1+4(*ω*-*ω*
_{B})^{2}/Γ^{2}], where g0 is the line-centre SBS gain coefficient, *ω*
_{B} is the resonant frequency and Γ is the acoustic phonon decay rate, which varies as the square of the length of the phonons wave vector. The second term, **I**
^{sp}
* _{S}*(0,

*ω*), on the right-hand side of Eq.(2) represents the spectral density of the intensity of the backward-propagating SpBS signal per unit length of the interaction region, which is:

with Θ being the solid angle subtended by the pump radiation and K the SpBS coefficient [14],

where *ρ* is the density, and *γ* is the electrostrictive constant of the medium, *v* is the acoustic wave velocity.

Since an externally applied Stokes input is absent, that is I_{S}(*L*,*ω*)=0, Eq. (2) can be explicitly integrated, giving the spectral density of the output Stokes intensity,

By integrating Eq. (5) over the spectrum one obtains the expression for *I*
_{S}(0),

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

*L*. This expression shows that the main contribution to the Stokes seed signal in the case of distributed SpBS initiation of SBS is made by the part of the interaction region with an effective length

*L*/

*G*

^{3/2}. We note that this length, and therefore the intensity of the SpBS seed signal, depends on

*G*, decreasing with increase of

*G*. If we then use, as discussed in the introduction, the criterion

*r*=10

^{-2}for the threshold of SBS, the equation for

*G*will be

_{th}The product ΘK*L* is the ratio of the intensity of SpBS signal to the pump intensity at z=0, which can be measured at z=0 in the absence of SBS amplification. This equation demonstrates the explicit dependence of *G _{th}* on the length of the interaction region,

*L*, the angular spread of the scattered radiation, Θ, and the scatterability of the medium described by the coefficient K (see Eq. (4)). Fig. 1 shows the dependency of

*G*on ΘK

_{th}*L*. It follows from this graph that an uncertainty in the value of ΘK

*L*of factor 2 (dotted lines in Fig. 1) leads to an uncertainty in

*G*of less than 5%.

_{th}Let us consider how *G _{th}* depends on the manner of SBS excitation in two commonly used experimental configurations: those of focusing a diffraction limited pump beam into a bulk medium and into an optical fiber.

In the first case, the effective length of the SBS interaction is defined by the Fresnel length, *L*=*l _{c}*=4nλF

^{2}/πD

^{2}, where F is the focal length of the lens, D is the diameter of laser beam on the lens [15], and Θ, the solid angle, in which the pump and Stokes beams are propagating [13], can be estimated as Θ=πD

^{2}/4F

^{2}. Accounting for this in Eq. (7), the term ΘK

*L*≈nλK. This shows that in the case of focusing a pump beam into a bulk SBS medium ΘK

*L*is independent of the focal length of the lens and the interaction length, and is fully determined, for a given radiation wavelength, only by the scattering coefficient, K, of the medium. Calculated values of K and bulk

*G*in different media for

_{th}*λ*=1.06 µm are given in Table 1. The parameters of the materials are taken from [14–16] and from standard handbooks.

These results show that in liquids at a given radiation wavelength (1.06 µm) *G _{th}*, is approximately constant spanning the range from ~22 to ~24, which is in good agreement with values widely used in the literature (we note that

*G*for CS

_{th}_{2}(here 22.8) is lower than that calculated (23.66) in [6] using the same criterion for the threshold which is simply due to the difference in nature of the seed signal as we discussed in the introduction). In silica,

*G*is ~28, which is appreciably higher than in the other materials because K is two orders of magnitude lower. In Fig. 2 we show the dependence of

_{th}*G*on

_{th}*λ*for bulk silica. In the last line of Table 1 we give the estimate for

*G*for one of the currently popular materials for highly nonlinear optical fibers [16].

_{th}In the case of SBS in waveguide media, such as optical fibers, the situation is very different. Firstly, the interaction length is determined by the physical length of the waveguide, *L*, (or, in a waveguide with optical losses, by the effective length *L*
_{eff}=(1-e^{-αL})/α)), and secondly, Θ≈π(NA)^{2}/n^{2} [17], where NA is the numerical aperture of the waveguide. So for optical fiber, *G _{th}* depends explicitly on the length and numerical aperture of the fiber in addition to its dependence on the pump radiation wavelength. Calculated results for lossless silica fiber are shown in Fig. 3(a–c); (our calculations have shown that in silica fiber the quantum-per-mode contribution to the Stokes seed power, as discussed in the introduction, [2,6,18,19], is of ~6 times smaller than that from SpBS). Evidently

*G*is not constant as commonly taken in the literature, but varies substantially, being as low as ≤5 in long fiber (~10 km length) with high NA (~0.2) and as high as ~20–23 in short fiber (<10 cm), the latter tending to its value in bulk silica (see Table 1) in the shortest fibers with lowest NA. These estimates are based on Eq. (7) which assumes negligible losses. However for long fiber (>1 km), such losses must be considered since it is well known that they reduce the efficiency of the SBS interaction [19]. This will certainly modify the trend, predicted by Eq. (7), of

_{th}*G*steadily decreasing with increasing

_{th}*L*, limiting the minimum of

*G*to a level of perhaps ~5–6.

_{th}We have verified our theoretical predictions by comparing them with experimentally measured values, both obtained in our laboratory and taken from the literature. 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 [20]. A single mode CW Nd:YAG laser with variable output power of 0–2 W was used as the pump source. Special care was paid to measuring the pump power, P_{p}(0), at which the Stokes signal power was equal to ~1% of P_{p}(0); the 1% SBS threshold criterion. A multi-mode (50 µm core diameter) fiber with NA=0.2, loss coefficient α≅0.23 km^{-1} and of length 3.6 km was investigated. The numerical value of the SBS gain coefficient for this fiber, g_{0}≈1.1·10^{-11} m/W, was estimated, taking into account waveguide-induced spectral broadening of the SBS gain profile [17,20]. We note that the fiber was of telecom grade and wound on a spool without tension so spectral broadening due to fiber imperfections was negligible compared to waveguide-induced broadening [20]. The entrance face of the fiber was cleaved perpendicular to the fiber axis, and the fiber was tilted to avoid reflection of pump radiation to the detector. The exit face of the fiber was cleaved with the surface tilted at angle ~10° from perpendicular to the fiber axis to prevent feedback. Measurements gave the threshold pump power, P_{th}, to be 350±30 mW, and *G _{th}*=g

_{0}P

_{th}

*L*/

*A*to be 6.9±0.6. This matches well the calculated value of

*G*=7.3 for this tested sample (see Fig. 3(a)).

_{th}We also compared results of our calculations of *G _{th}* in fibers with those in Ref [21] which we believe to be correctly measured. The procedure in Ref [21] involved measurements of the SBS amplification of an externally seeded Stokes signal vs pump power and also the amplified SpBS signal emerging from the fiber at a particular pump power, the latter being straightforwardly converted to

*G*using the results of the former. In our calculations here we supposed that standard telecom fibers with NA=0.12 were used in [21]. Along with our own measurements these results are summarised in Table 2. As seen there is reasonably good correspondence between measured and calculated

*G*. For commonly used fiber length ≥100 m the value of

_{th}*G*is ≤10–12 and for fiber of length ≥km

_{th}*G*. drops to ≤7.

_{th}In conclusion, we have developed a physically consistent theory for describing the threshold value of exponential gain, *G _{th}*, for SBS initiated by SpBS in both bulk media and optical fiber. For bulk media we show

*G*to be approximately constant whereas for fiber it is explicitly and strongly dependent on the fiber material, length and numerical aperture and the pump radiation wavelength.

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

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