## 1. Introduction

It is well known that the velocity of a light pulse in a medium, referred to as the group velocity, is smaller than the phase velocity of light, c/* n*, where c is the speed of light in vacuum, and

*is the refractive index of the medium. The difference between phase and group velocity of light is a result of two circumstances: a pulse is generically composed of a range of frequencies, and the refractive index,*n

*, of a material is not constant but depends on the frequency,*n

*, of the radiation,*ω

*=*n

*(*n

*). A group index*ω

n

_{ g }(

*) =*ω

*+*n

*(d*ω

*/d*n

*) is used to quantify the delay (or advancement),*ω

t

_{ g }, of an optical pulse,

t

_{ g }=

n

_{ g }L/c, which propagates in a medium of length L, where c/

n

_{ g }is called the group velocity (Brillouin, 1960).

For about a century studies of this phenomenon, now topically referred to as slow light (SL), were mostly of a scholastic nature. In general the effect is very small for propagation of light pulses through transparent media. However when the light resonantly interacts with transitions in atoms or molecules, as for gain and absorption, the effect is greatly enhanced. Fig. 1 shows the gain (inverted absorption) spectral profile around a resonance together with its refractive index dispersion profile, the gradient of which results i_{
g
}(* ω*).

As seen in the figure n_{
g
}(ω) peaks at line centre and it is here that the group delay is a maximum. However in reality for a meaningful delay the gain required must be high and this leads to competing nonlinear effects, which overshadow the slowing down (Basov et al., 1966). On the other hand in the vicinity of an absorbing resonance the corresponding absorption is much too high to render the group effect useful. An exciting breakthrough happened in the early nineties when it was shown that group velocities of few tens of meters per second were possible with nonlinear resonance interactions (Hau et al., 1999). Two important features of nonlinear resonances make this possible: substantially reduced absorption, or even amplification, of radiation at a resonance, and sharpness of such resonances; the sharper a resonance, the higher dn/dω and so the stronger the enhancement of group index, and hence the greater the pulse is delayed.

Widely ranging applications for slow light have been proposed, of which those for telecommunication systems and devices (optical delay lines, optical buffers, optical equalizers and signal processors) are currently of most interest (Gauthier, 2005). The essential demand of such devices is compatibility with existing telecommunication systems, that is they must be of wide enough bandwidth (10 GHz) and able to be integrated seamlessly into such systems.

Of the various nonlinear resonance mechanisms and media, which allow sufficiently long induced delays, stimulated Brillouin and Raman scattering (SBS and SRS) in optical fiber are deemed to be among the best candidates. Currently SBS is the most actively investigated and many experimental and theoretical papers on pulse delaying via SBS in optical fiber have been published in the last few years, see the review paper (Thevenaz, 2008) and references therein. In this process the pulse to be delayed is a frequency down-shifted (Stokes) pulse. This is transmitted through an optical fiber through which continuous wave (CW) pump radiation is sent in the opposite direction to prime the delay process. It is supposed that the Stokes pulse is amplified by parametric coupling with the pump wave and a material (acoustic) wave in the medium (Kroll, 1965), and the amplification is characterised by a resonant-type gain profile. The dispersion of refractive index associated with this profile (which is similar to that in Fig.1) can then be used to increase the group index for optical pulses at the Stokes frequency (Zeldovich, 1972).

Along with obvious device compatibility, there are several other advantages of the SL via SBS approach for optical communications systems: slow-light resonance can be created at any wavelength by changing the pump wavelength; use of optical fibre allows for long interaction lengths and thus low powers for the pump radiation, the process runs at room temperature, it uses off the shelf telecom equipment, and SBS works in the entire transparency range of fibers and in all types of fiber. Currently a main obstacle to applications of this approach is the narrow SBS gain spectral bandwidth, (Thevenaz, 2008), which is typically 120-200 MHz in silica fiber in the spectral range of telecom optical radiation (~1.3-1.6 m) (Agrawal, 2006).

This chapter reviews our ongoing work on the physical mechanisms that give rise to pulse delay in SBS. In section 2 the theoretical background of the SBS phenomenon is given and the main working equations describing this nonlinear interaction are presented. In section 3 ways by which the SBS spectral bandwidth may be increased are addressed. Waveguide induced spectral broadening of SBS in optical fibre is considered as a means of increasing the bandwidth to the multi-GHz range. An alternative way widely discussed in the literature, (Thevenaz, 2008), is based on spectral broadening of the pump radiation. However it is shown through analytic analysis of the SBS equations converted to the frequency domain that pump radiation broadening by any reasonable amount has only a negligible effect on increasing the SBS bandwidth. Importantly in this section we show that, irrespective of the nature of the broadening considered, the SBS gain bandwidth remains centred at the Brillion frequency which is far removed from the centre frequency of the Stokes pulse. Consequently the associated group index, which is enhanced at and around the SBS gain centre, cannot lead to group index induced delay of a Stokes pulse as claimed in the literature (Thevenaz, 2008). In section 4 the actual physical mechanisms by which a Stokes pulse is delayed through SBS are examined. Analytical analysis of the equations in the time domain shows that the SBS amplification process does not amplify an external the Stokes pulse and so again cannot induce group delay of this pulse. Rather the delay is shown to be predominantly a consequence of SBS gain build-up determined by inertia of the acoustic wave excitation. Finally in section 5 conclusions are drawn from this work in regard to current understanding of SL in SBS.

## 2. Theory of stimulated Brillioun scattering

In SBS, the resonance in a medium’s response occurs at the Brillouin frequency, _{B}, which is the central frequency of the variation of density in a medium, (z,t) = 1/2{(z,t)exp[-i(_{B}t+qz)] + c.c.}. This density variation is resonantly induced by an electrostrictive force resulting from interference of two plane counter-propagating waves, the forward-going (+z direction) Stokes and backward-going (-z direction) pump optical fields, E_{
S
}(z,t) = 1/2{E_{
S
}(z,t)exp[-i(_{
S
}t-k_{
S
}z)] + c.c.} and E_{
p
}(z,t) = 1/2{E_{
p
}(z,t)exp[-i(_{
p
}t+k_{
p
}z)] + c.c.}, respectively, where (z,t), E_{
S
}(z,t) and E_{
p
}(z,t) are the amplitudes of the acoustic wave and of Stokes and pump fields, with = _{
p
} - _{
S
} and q = k_{
p
} + k_{
S
}, _{
S
} and k_{
S
}, and _{
p
} and k_{
p
} being their radian frequencies and wavevectors and c.c. is the abbreviation for complex conjugate. In an isotropic medium (z,t) is described by the equation, (Zeldovich et al., 1985),

where _{
s
} is the speed of a free acoustic wave, A is its damping parameter, ^{2}
^{2}/z^{2} in the chosen plane wave model, and _{0} are the dielectric function and equilibrium density of the medium, and E(z,t) = E_{
p
}(z,t) + E_{
S
}(z,t). Since the amplitude (z,t) is supposed to be slowly varying in space, then ^{2}(z,t) -q^{2}(z,t) and Eq.(1) is usually reduced to

This is then the equation for the induced acoustic wave. It is a typical equation for an externally driven damped resonant oscillator, in which the right-hand side is the driving force, _{B} = q_{
s
} = 2_{
s
}
_{
p
}/(c/n+_{
s
}) 2n_{
s
}
_{
p
}/c is the resonant frequency of the oscillator, known as the Brillouin frequency, _{B} is the FWHM spectral width of the resonant profile with 2/_{B} being the decay time of the acoustic wave.

The pump field reflected by the induced acoustic wave is a new Stokes field, which in turn interacts with the pump field to further electrostrictively enhance the acoustic wave and so the Stokes field and so forth. Increase of the Stokes field in SBS is therefore a direct consequence of increase of reflectivity of the acoustic wave for the pump field. As such, so called “SBS gain” characteristics are determined by the reflectivity, spectral characteristics and dynamics of the acoustic wave. In the approximation that the CW pump radiation is not depleted over the interaction length, L, the spatial/temporal evolution of the Stokes signal is described by the nonlinear wave equation,

Eqs (2) and (3) are the basic equations, which describes the SBS phenomenon in an optically lossless medium in the small signal plane wave approximation. Since the density and Stokes field amplitudes, (z,t) and E_{
S
}(z,t), vary slowly in both space and time and the acoustic wave in SBS attenuates strongly, their evolution is usually reduced to two well known first order equations: from Eq.(2) the relaxation equation for (z,t),

which describes the amplitude of the driven damped resonant oscillator, and from Eq.(3) the partial differential equation for E_{
S
}(z,t),

Here = - _{B} is the difference between the acoustic drive frequency, , and the resonant Brillouin frequency and asterisk, *, marks complex conjugate. The right-hand side of Eq. (5) is a source of the Stokes emission.

## 3. Spectral broadening of SBS

In the literature on group index induced slow light it is argued that rate at which optical pulses may be delayed is ultimately determined by the spectral bandwidth of the resonance responsible for slow light generation in the material (Boyd & Gauthier, 2002). So, the narrower the bandwidth the larger is the delay. On the other hand, to minimize pulse distortion the bandwidth must exceed substantially that of the optical pulse to be delayed and consequently determines a lower limit for the duration of the optical pulse. This argument is correct for systems in which a resonance in the material is in resonance with the optical pulse to be delayed, such as those based on electromagnetically induced transparency and coherent population oscillation (Boyd & Gauthier, 2002). However as shown below this does not apply to SBS since the resonance occurs around the Brillouin frequency, _{B}, which is far from the frequency of the Stokes pulse to be delayed. This point has been overlooked in the literature on SL via SBS and as a consequence has led to misinterpretation of experimental findings of Stokes pulse delay in SBS. This issue is considered in some detail in section 4 where it is shown the Stokes delay arises from the inertial build up time of SBS and not group index delay as has been claimed throughout the literature. Nevertheless it is still of academic interest to consider ways in which the spectral bandwidth of SBS may be increased and this is considered below.

The physical mechanism responsible for _{B} is attenuation of the Brillioun acoustic wave, in liquids and solid optical media this is predominantly due to viscosity (Zeldovich et al., 1985). Such spectral broadening is homogeneous in nature. For bulk silica, _{B}, scales with pump radiation wavelength, , as _{B} 240/^{2} MHz (Heiman et al. 1979), where is in m. It is evident from this expression that the shorter the radiation wavelength the wider the spectrum, so for radiation in the short wavelength transmission window of silica, 0.2 m, _{B} is expected to be ~2 GHz compared to ~ 20 MHz at telecom wavelengths, 1.3-1.6 m. The SBS gain bandwidth in fibers may also be broadened through varying fiber design, doping concentration, strain and/or temperature (Tkach et al., 1986, Shibata et al., 1987, Azuma et al., 1988, Shibata et al., 1989, Yoshizawa et al., 1991, Tsun et al., 1992, Yoshizawa & Imai, 1993, Shiraki et al., 1995, LeFloch & Cambon, 2003). However the highest achieved line-width enhancement factor, compared to _{B} is ~5, (Yoshizawa et al., 1991). A potentially attractive solution to increasing _{B} is by waveguide induced spectral broadening (Kovalev & Harrison, 2000), which is discussed in some detail below (Sect. 3.1). Spectral broadening of the pump radiation has also been proposed (Stenner et al., 2005, Herraez et al., 2006) as a means for broadening _{B} and is currently a subject of considerable activity (Thevenaz, 2008). However, as shown below (see Sect. 3.2) the effect is in fact negligible.

### 3.1. Waveguide induced spectral broadening of SBS

Due to the waveguiding nature of beam propagation in optical fiber and its effect on the SBS interaction, such propagation has been shown to render the Stokes spectrum inhomogeneous (Kovalev & Harrison, 2000), the bandwidth of which is massive in fibers of high numerical aperture, NA (Kovalev & Harrison, 2002). The nature of the broadening arises from the ability of optical fiber to support a fan of beam directions within an angle 2θ_{
c
} (Fig. 2), where θ_{
c
} is the acceptance angle of the fiber, defined as

where n_{
cl,co
} are the refractive indices of the fiber cladding and core, respectively.

The frequency shift of the Stokes depends on the angle, φ, between the momentum vectors of the pump and scattered radiation through the relation _{
B
}(φ)=4n_{
s
}sin(φ/2)/λ. So the range of _{
B
}(φ) in a fiber will be from _{
B
}(π) = 4n_{
s
} /λ to _{
B
}(π-2θ_{
c
}) = 4n_{
s
}cosθ_{
c
} /λ. For every _{
B
}(φ) there corresponds a homogeneously broadened line of the form

The Stokes spectrum, broadened by guiding, is then the convolution of frequency-shifted homogeneously broadened components, each generated from a different angular component of the pump and Stokes signal (such broadening is inhomogeneous by definition). The shape of the broadened Brillouin linewidth is described by the equation (Kovalev & Harrison, 2002),

where θ_{
c
} is linked to NA through Eq.(6). Fig. 3 shows _{i}(,θ_{
c
}) for five values of NA.

Intuitively, the dispersion and group index profiles, which are associated with the convolutionally broadened SBS gain spectrum (dashed lines in Fig.4a), are expected to also be convolutionally broadened (solid lines in Fig.4a and
Fig.4b). As seen (Fig.4b) the maximum n_{
g
} is expected to be more or less constant and so it’s value for each and all the homogeneous spectral components that contribute to the group index profile is the same.

The shape of the group index spectrum is determined more precisely by numerical simulation (Kovalev et al., 2008). The group index in the case of the SBS resonance in optical fiber can be expressed as (Okawachi, 2005, Kovalev & Harrison, 2005),

where g_{0} is the value of the SBS gain coefficient at the exact Brillouin resonance and I_{
p
} is the pump radiation intensity. When several resonant frequencies, _{
B
}, exist in the medium, and the distribution of their relative amplitudes over the range from _{
B
}(π) to _{
B
}(π-2θ_{
c
}) is some function F(_{
B
}), the spectrum of the “broadened” group index, n_{
gb
}(), is described by the convolution integral,

Results of calculations for the case when F(_{
B
}) = 1 in the range from _{
B
}(π) to _{
B
}(π-2θ_{c}) are presented in Fig.5. For the sake of illustration the results are centred by shifting the limits of integration in such a way that the lower limit is _{1} = _{0} - m_{B}/2 and _{2} = _{0} + m_{B}/2, where m, which is called the rate of broadening, varies from 2 to 100. As seen the width of the profile increases continuously with increasing m. The original shape of the profile (individual components in Fig 4b) is retained for a broadening of m 2. Beyond this, the profile becomes top-hat, at m 4, and for m > 4 it exhibits a dip, the depth of which increases with increasing m. For m > 20 the dip tends to becomes flat-bottomed. It can therefore be seen that the value of group index can stay constant over a broad range of frequencies, especially when m > 60. However the price for this is a reduced magnitude of the group index. However, as seen in Eq.(9), this may be compensated for by increase of the pump intensity.

Earlier work has shown that waveguide induced broadening is dependant on the numerical aperture of fiber through the equation (Kovalev & Harrison, 2002),

It follows from Eq.(11) that in the calculations above, m = 2 corresponds to NA = 0.12, which is standard for single-mode telecom fiber. However, it is now readily possible to realise single-mode fiber with much higher NA, ~0.8 (Knight et al., 2000). For such fiber the broadening is ~15 GHz, which is comparable with the needs of telecom devises. As noted above this analysis assumes that the homogeneously broadened Brillouin gain contributions to the inhomogeneous profile are uniformly distributed, F(_{
B
}) = 1. It is relatively straight forward to account for alternative distributions by introducing their appropriate shape function F(_{
B
}) into Eq.(10).

These considerations therefore show that waveguide induced spectral broadening of the gain bandwidth of SBS in optical fiber is potentially massive (> 10 Gb/s), and readily achievable.

### 3.2. On the effect of the pump spectral width on spectral broadening of SBS

It has been proposed in (Stenner et al., 2005) that spectral broadening of the pump radiation may lead to comparable spectral broadening of the Stokes pulse in SBS. This approach has since been the focus of many publications, (Minardo et al., 2006, Shumakher et al., 2006, Zhu & Gauthier, 2006, Zadok et al., 2006, Chin et al., 2006, Schneider et al., 2006, Kalosha et al., 2006, Zhu et al., 2007, Song & Hotate, 2007, Lu et al., 2007, Zhang et al., 2007-1& -2, Yi et al., 2007, Shi et al., 2007, Pant et al., 2008, Ren & Tomita, 2008, Sakamoto et al., 2008, Wang et al., 2008, Schneider et al., 2008, Cheng et al., 2008), aimed at high data rate applications of SL. In this section the validity of this assertion is examined and it is shown that the effect is in fact negligible.

This may be seen from examining the spectral features of the medium’s response and the Stokes emission through Fourier transformation of Eqs (4) and (5) using the following basic properties of Fourier transforms, F() S[f(t)], (Korn & Korn, 1967),

where f(t) is a function of time. Here and are Fourier transform frequencies, which are the difference frequencies of the acoustic, Stokes and pump signals from their respective line centers. Eqs (4) and (5) then give

Note that here we have primed the Stokes and pump fields within (13), which are responsible for inducing the acoustic wave, to distinguish them from the generated Stokes field, (LHS of (14)), and from the pump field, which generates the new Stokes field, (the field

Consider the case of a typical SBS slow light experiment in which the spectrum of the Stokes signal corresponds to that of a temporally smooth pulse and the spectrum of the pump radiation is the Fourier-transform of a continuous wave field the amplitude of which is randomly fluctuating in time. As seen, the right-hand sides of these equations are proportional to the convolution integrals of spectra

Eq. (13) is an algebraic equation, the solution of which gives the spectrum of the medium’s response

where

is the function which determines the spectrum of the driving force for the medium’s response. The spectrum of the medium’s response is then given by the modulus of

The spectrum of the Stokes field is described by the first order differential equation, Eq.(14), the solution of which is

where

is the function which determines the spectrum of the source of the generated Stokes field. Equations (15)-(18) describe the spectral features of the SBS-induced material response and Stokes field when both optical fields, pump and Stokes, are non-monochromatic. Note that solution Eq.(17) in the spectral domain is entirely consistent with the analytical solution of Eqs. (4) and (5), previously obtained in the temporal domain for stimulated scattering induced by non-monochromatic pump and monochromatic Stokes fields in (Kroll, 1965, Charman et al., 1970, Akhmanov et al., 1971, Akhmanov et al., 1988), and for monochromatic pump and non-monochromatic Stokes fields in (Kovalev et al., 2009).

It is easily seen that the solution (17) for the Stokes field differs substantially from that usually deduced in textbooks from (4) and (5) in the steady state approximation (that is when both pump and Stokes fields are considered monochromatic),

which results from the equation for the Stokes field of the form, (Zeldovich et al., 1985),

Here it is again important to remember that is a detuning parameter, the value of which is the difference between the frequencies of the monochromatic pump and Stokes fields as chosen, = _{
p
} - _{
S
}, and the resonant Brillouin frequency, _{B}, |E_{
p
}E_{
p
}*| is the pump radiation intensity, and g_{0} is the standard steady state SBS gain coefficient. Equation (20) means that the Stokes field is amplified in the medium with gain proportional to the pump radiation intensity. To appreciate the difference between this case and our case let us consider both pump and Stokes fields in Eqs.(13)-(18) to be monochromatic. The spectra of the amplitudes of driving forces and of the medium’s response then reduce to -functions at = 0 and these depend on z only. The equation for the Stokes field is then,

The physical meaning of Eq.(21) is substantially different from that of Eq.(20), though their mathematical forms may look similar. Equation (21) describes the spatial evolution of the amplitude of the Stokes field, which results from reflection of the pump field by the induced acoustic wave. Since the Stokes field on the RHS of Eq.(21) is responsible for creating the acoustic wave, it is not the same as the reflected Stokes field on the LHS of this equation and therefore it is distinguished by its prime. Though for this monochromatic case the Stokes fields have the same frequency, their roles still remain physically distinct as in our general treatment above. Such distinction is not made in the text-book treatment that leads to Eq.(20) and to its familiar exponential solution, Eq.(19), which displays “gain” and a “modified propagation constant” for the Stokes field (see (Zhu et al., 2005)). Evidently the solution for Eq.(21) cannot be the same. As such, though the RHS of this equation has both real and imaginary parts (in the case of non-zero detuning, ), this does not modify the propagation constant for the reflected Stokes field and therefore it can have no bearing on changing the refractive and group index for this field.

Returning now to the solutions (15)-(18) of Eqs. (13) and (14) for the general case in which either one of the fields or both have nonzero bandwidth, they display three important features of the SBS interaction: i) the external input Stokes signal, as seen in Eq.(17), propagates through a non-absorbing medium without gain or measurable loss (its energy loss for creating the acoustic wave is usually negligible), ii) the SBS-generated Stokes signal is a result of reflection of the pump radiation by the acoustic wave, which is created by the pump and the original Stokes fields (see Eqs. (4) and (5)), and iii) each spectral component of the generated Stokes signal arises from a range of spectral components of the non-monochromatic pump and Stokes fields, see Eqs. (17) and (18).

To see the consequences of this, consider the case when the growth of the Stokes field along z is small. As such, the z dependence of _{E}() (see Eq. (18)). The latter is the SBS-induced contribution to the Stokes signal, and is determined by which of

In contrast to the spectral features of the Stokes emission, those of the medium’s response, _{B} appears in addition to the convolution integral, F_{
}(). From Eq. (15),

Examples of the spectra,

It follows from Eq.(22) that when the spectral width, _{
}, of the driving force, |F_{}()| (shown by dashed curves in Fig. 6(a)), is narrower than _{B}, the width, , and centre frequency of _{
}()|, (the convolution of _{
} increases, grows and for _{
}> _{B} it saturates at ~1.7_{B} (solid curves 3 and 4 in Fig. 6(a)), and the effect of the detuning on the features of the medium’s spectrum becomes negligible (compare solid curves 3 and 4 in Figs. 6(a) and 6(b)). In essence, this means that irrespective of how broad the bandwidth of the broadband pump and/or Stokes emission is/are, the bandwidth of the material response spectrum is predominantly determined by the features of the material and can never be much greater than that of the medium’s resonant response (~_{B}). This is exactly the features expected from an externally driven damped resonant oscillator (see Eq.(2)).

From Eq. (22) the SBS induced dispersion of refractive index, n() (/)|_{0}, and its corresponding induced group index at the frequency of the Stokes radiation, _{
}(’)| to be Gaussian in shape centred at frequency _{B}, F_{
}(’) = F_{
a}exp[-(’-_{B})^{2}/_{
}
^{2}], where _{
S
} is the Stokes signal intensity. The induced group index at absolute frequency ’ is then

where

The dispersion of the SBS-induced group index, which is described by Eq. (23), has a maximum at ’ = _{B} - and a minimum at ’ = _{B} + , where is functionally dependent on _{
} and is < _{B}/2. These maximum and minimum, in principle, may be of very high amplitude, however it is critically important to note that they are located at ’ _{B}. At the Stokes frequency, ’ = _{S}, since _{S}>> _{B}> _{B} (_{S} 1.710^{4}
_{B} in silica), Eq.(23) reduces to

which shows that the SBS-induced n_{
g
}(_{S}) is independent of both _{B} and _{B}, and is negligibly small for any reasonable bandwidth, _{
}, of

This may also be seen using the following less rigorous simple argument. The resonant Brillouin frequency, _{B}, that is the frequency of the acoustic wave, for Stokes radiation excited by a monochromatic pump radiation, is _{B} = 2n_{
s
}
_{p}/c. Broadening of the pump radiation spectrum by _{p} then results in pump induced broadening of the acoustic wave spectrum, _{B}, = 2n_{
s
}
_{p}/c. It follows that for _{p} of ~2(12-25) GHz (Zu et al., 2007, Song & Hotate, 2007), which is of order of _{B} for SBS excited by 1.55 m pump radiation in silica fiber (n 1.45, _{
s
} 6 km/s), _{B} 2(0.6-1.5) MHz << _{B} 216 MHz. Clearly then pump induced broadening of the acoustic wave spectrum is negligible compared to its homogeneous spectral width, _{B}, for any reasonable value of the pump spectrum width up to few tens of GHz.

## 4. Effect of acoustic wave inertia on Stokes pulse delay in SBS

In this section the underlying physical processes that give rise to Stokes pulse delay in SBS are addressed.

In typical SBS-based slow light experiments the CW pump power is kept below the value at which the SBS interaction experiences pump depletion. The pump power is therefore constant throughout the interaction length (in lossless media). This is also an underlying reason why the contribution of spontaneous scattering to the SBS interaction is considered sufficiently small to be ignored in theoretical treatments of this problem (Song et al., 2005, Okawachi et al., 2005, Zhu et al., 2005). Equations (4) and (5) with appropriate boundary conditions are therefore sufficient for describing the evolution of a Stokes pulse in a medium.

It is convenient to introduce the new temporal coordinate t’ = t – zn/c and suppose that the centre frequency of the Stokes pulse spectrum coincides with the resonant Brillouin Stokes frequency, that is = 0. In terms of the new variables Eqs (4) and (5) can be rewritten as

This set of equations has an analytic solution, which can be obtained using Reimann’s method (Bronshtein & Semendyaev, 1973). Consider the case addressed in typical SL experiments, in which the duration of the Stokes pulse is much less than its transit time in the medium and the pump is CW monochromatic radiation. Assuming that there are no acoustic waves in the medium before a Stokes pulse enters, and E_{
S
}(t’ 0) = 0, E_{
S
}(z=0,t’) = E_{
S0
}(t’) and E_{
p
}(z,t’) E_{
p
} = const, the solutions for the Stokes field and the density variation are then

Here I_{
p
} = |E_{
p
}|^{2} is the pump radiation intensity in [W/cm^{
2
}], I_{0,1}(x) are the Bessel functions of imaginary argument x, and g is the SBS gain coefficient,

Suppose that the input Stokes signal is an optical pulse, the time dependent intensity of which is given by

where I_{
S0} is the intensity at the peak of the pulse, t_{
p
} is the FWHM pulse duration. The shape of the pulse is shown in Figs. 7 and 8 (curves for G = gI_{p}L =0) and it is a good approximation for pulses actually used in experiments (Pohl & Kaiser, 1970). Since the SBS exponential gain, G, in the fiber is supposed to be below the SBS threshold, this, according to (Kovalev & Harrison, 2007), limits G to ≤12 for standard silica fibers of > 0.1 km length.

Figs 7 and 8 show the calculated relative output Stokes pulse powers, P_{S}(t) = |E_{S}(t)|^{2}S (S is the effective area of fiber–mode cross section), shapes, amplitudes and delays for four input pulse durations, t_{
p
}, and different G. Here the decay time, , of the hyper-sound wave in silica is taken to be = 18 ns at a pump radiation wavelength ~1.55 m. It should be emphasized that the decay time of the acoustic wave, which is determined by the viscosity of fused silica (Pinnow, 1970), is used in these calculations rather than the SBS spectral width. This is because the latter depends on intrinsic characteristics of the fiber (core/cladding design, doping type and concentration, numerical aperture, etc. and the ambient environment (mechanical, thermal and electromagnetic fields (see Sec. 3), all of which can result in substantial variation of the resonant Brillouin frequency, _{B}. However they do not appreciably effect the viscosity of the medium.

It follows from Figs 7 and 8 that the induced delay of the output Stokes pulse, its duration and peak power, which is estimated to be P_{S0}e^{Gef}, where G_{ef} is the effective SBS exponential gain, in all cases increase with increase of G. Rates of these growths depend substantially on the ratio of pulse duration to acoustic wave decay time, t_{
p
}/, as shown in Fig. 9.

For a long input Stokes pulse, t_{
p
} = 200 ns that is t_{
p
}/ = 200/18 >> 1, the output pulse, on increase of G, remains similar in form (Fig. 7(a)), and is increasingly delayed following T_{
d
}/G 11 ns as shown by the thick dashed line in Fig. 9(a). Additional calculations have shown that it may coincide with T_{
d
} G/2 9G ns (shown by the thin solid line in Fig. 9(a)), but this happens only when t_{
p
}/ 2.50.5 and >100. For t_{
p
}/ < 2 the growth of T_{
d
} falls below the value T_{
d
} 9G ns. The duration of the output pulse compared to that of the input is slightly broadened with increase of G (see dashed line in Fig. 9(b)), and G_{ef} decreases very slightly compared to G (dashed line in Fig. 9(c)).

For t_{
p
} = 18 ns (Fig. 7(b)), that is t_{
p
}/ = 1, T_{
d
} for the output pulse again increases with G though not linearly; at lower G (0 to ~2) with slope T_{
d
} /G 3 ns, and at higher G (> 2) with slope T_{
d
} /G 9 ns (dotted line in Fig. 9(a)). The pulse broadening in this case increases substantially with G, by a factor 5 at G > 10 (dotted line in Fig. 9(b)) and G_{ef} decreases notably, by a factor of ~3 at lower G to ~1.5 at G 10 (dotted line in Fig. 9(c)).

For the short pulses, t_{
p
} 4 ns (Fig. 8), that is for t_{
p
}/ < 1, significant new features appear. For G < 4 the output pulses approximately retain their shape with only a slight increase of T_{
d
} with G (T_{
d
} /G 0.1-0.15 ns for t_{
p
} = 4 ns and T_{
d
} /G 0.03 ns for t_{
p
} = 0.5 ns, as is shown in inset of Fig. 9(a), solid and dashed lines respectively). In both cases G_{ef} for the leading peak decreases substantially, by a factor ~10. For G between 4 and 6 there is substantial growth of the power in the tail of the pulses and for G > 8 the maximum of the pulse shifts to the tail (see Figs. 8(a) and 8(b)). This is because of the long decay of the acoustic wave excited by the short Stokes pulse interacting with the CW pump. The dependence of T_{
d
} on G for G > 8 then follows the linear relation T_{
d
} (9G – 25) ns (thick solid line in Fig. 9(a)). The pulse broadening factor is then ~20 for t_{
p
} = 4 ns (solid line in Fig. 9(b) at G > 6) and ~200 for t_{
p
} = 0.5 ns.

It is interesting to note that analytical results presented in Figs. 7 and 8, which are obtained in the small signal limit, give dependencies of pulse delay and broadening on G quite similar to these obtained numerically both for long pulses, t_{
p
}/ 15 and 5 in (Zhu et al, 2005), and for short pulses, 0.1 < t_{
p
}/ < 2 in (Kalosha et al., 2006). There are however some quantitative differences, which are important to highlight. The numerical modelling in (Zhu et al., 2005) gives T_{
d
} = G/2 dependence for t_{
p
}/ 15, while calculations here predict this only for t_{
p
}/ in the region ~2.50.5, while for t_{
p
}/ > 5 the slope T_{
d
} /G is a factor 1.2-1.3 steeper. Also the shape of the T_{
d
} upon G dependence for t_{
p
}/ 5 in ((Zhu et al, 2005)) is similar to that in these calculations but for t_{
p
}/ 1. As to the numerical results presented in (Kalosha et al., 2006), the difference with findings here is most probably because all their results were obtained for an input Stokes power well above that for onset of pump depletion (input Stokes power is of 1 mW compared with pump power of 20 mW).

To understand the underlying nature of the behaviour described above consider through Eq. (28) the temporal and spectral characteristics of the complex dielectric function variation in the medium, = (/) = ’ + i’’, induced by the interaction of the pump and Stokes signals; () results in the SBS gain and ’() is responsible for modification of the refractive index, n() ’()/2n_{0}, of the medium, where n_{0} is the refractive index of a medium without SBS. In the limit of small gain, G < 1, the Bessel function I_{0}(x) in Eq. (28) may be set to unity, and analysis is greatly simplified. While this approximation does not allow us to describe gain narrowing of the SBS spectrum typical for higher G, it still captures reasonably well the trends in the temporal and spectral features of , which determine those of the SBS gain and modified refractive index.

When I_{0}(x) = 1, the integral in Eq. (28) can be taken for E_{S0}(t) given by Eq. (30). It results in the following analytic expression for (t),

where b = t_{
p
}/3.5, and

The temporal dynamics of the induced acoustic wave amplitudes and their spectra for values of t_{
p
}/ considered above are shown in Fig. 10 (the dynamics for t_{
p
} = 0.5 ns is not shown on the timescale of Fig. 10(a)). The curves in Fig. 10(a) represent different characteristic types of SBS interaction: curves 1 give an example of quasi-steady state interaction when t_{
p
}/ >> 1, curves 3 demonstrate transient type interaction when t_{
p
}/ << 1, and curves 2 are for an intermediate case when t_{
p
}/ 1.

In the first case of a long Stokes pulse, that is t_{
p
}/ >> 1, the shape of the acoustic wave pulse shown in Fig. 10(a) as the solid curve 1 almost reproduces the shape of the input Stokes pulse (dotted curve 1). The spectrum of the excited acoustic wave in this case, Fig. 10(b), reproduces the spectrum of the input Stokes pulse, shown by the solid and dashed curves 1 respectively. These are both narrower than the Lorentzian-shaped spectrum corresponding to = 18 ns (dotted curve 5).

This is to be expected since Eq. (2) is in essence the equation for the amplitude of a driven damped oscillator; for such a system the spectrum of the induced oscillations is fully determined by the spectrum of the driving force when it is narrower than the reciprocal decay time of the oscillator.

In the case of shorter Stokes pulses, t_{
p
}/ 1, (dashed curves 2 and 3 in Fig. 10(a)) the spectrum of the driving force, that is that of the input Stokes pulse, is broad band, (dashed curves 2 and 3 and also curve 4 for t_{
p
} = 0.5 ns pulse in Fig. 10(b)). As seen the dynamics of the medium’s response (solid curves 2 and 3 in Fig. 10(a)) and its spectra (solid curves 2, 3 and 4 in Fig. 10(b)) differ substantially from those of the Stokes pulses and their spectra. In the temporal domain the maximum amplitude of the induced (t) decreases with decrease of pulse duration and a long tail appears after a Stokes pulse, which decays exponentially with a decay time of . The spectra of (t) in these cases are narrower, increasingly so for shorter pulses, than the spectra of the driving force (input Stokes pulse). Their width is then determined predominantly by the reciprocal decay time of the oscillator (dotted curve 5 in Fig. 10(b)) as is to be expected for a damped oscillator, which is driven by a broad-band force. Reduction of amplitude of the induced (t) with decrease of pulse duration results in reduced amplitude of the output Stokes pulse and its effective G as shown in Fig. 9(c).

The imaginary part of () described by Eq. (13) gives the refractive index of the medium modified by the SBS interaction, n() = n_{0} + n() n_{0} + (/)()/2n_{0} and its corresponding group index, n_{g}() = n_{0} + [dn()/d] (Okawachi et al., 2005, Zhu et al., 2005). Spectra of the Stokes pulses and the group indices induced by these pulses are shown in Fig. 11 for the four different t_{
p
}/. It is important to remember (see section 3.2, Eqs (23) and (24)) that the group index profile is frequency shifted from the centre frequency of the Stokes pulse by ~1.710^{4}
_{B} and therefore has negligible effect on the Stokes pulse delay. In the literature on SL via SBS, it is assumed that their centre frequencies coincide. However even if this were so it is clear from Fig. 11 that the spectral width of a Stokes pulse is bigger than that of the SBS induced group index regardless of t_{
p
}/. As such, in this hypothetical case, while some, central, part of the input spectrum may experience a group delay, other parts of the spectrum will experience group advancement or neither delay or advancement (see Figs. 11(c) and 11(d)).

It therefore follows that regardless of the pulse length of the input Stokes pulse the pulse delays associated with SBS amplification of a Stokes pulse, as described above, cannot be attributed to SBS induced group delay. They are predominantly a consequence of the phenomenon of SBS build-up.

## 5. Conclusions

The results presented in this chapter raise the question of whether slow light, as first discussed in (Zeldovich, 1972), can be realised. To answer this recall the nature of the group delay effect. It is a linear phenomenon exhibited by a pulse propagating through a medium with normal dispersion of refractive index (Brillouin, 1960). The effect is greatly enhanced in the vicinity of a medium’s resonance, which for a gain medium is normally dispersive. For SBS the maximum of this resonantly enhanced dispersion is centred around the Brillioun frequency, _{B}, and it is all but negligible at the Stokes frequency. As such the acoustic resonance can have next to no effect in enhancing or modifying the natural group index in the medium for the Stokes signal. Consequently pulse delay associated with Stokes pulse induced SBS cannot be attributed to SBS induced group delay. It is predominantly a consequence of the phenomenon of SBS build-up, which arises from the inertia of the medium in responding to the optical fields. Also, spectral broadening of the pump radiation by any reasonable amount has next to no effect on the SBS spectral bandwidth of the excited acoustic wave in the medium, which is commonly believed to determine the SBS gain bandwidth (Thevenaz, 2008).

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