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/
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
As seen in the figure n
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(Bt+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
where
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
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
which describes the amplitude of the driven damped resonant oscillator, and from Eq.(3) the partial differential equation for E
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
where n
The frequency shift of the Stokes depends on the angle, φ, between the momentum vectors of the pump and scattered radiation through the relation
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
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
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 g0 is the value of the SBS gain coefficient at the exact Brillouin resonance and I
Results of calculations for the case when F(
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(
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, =
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
In contrast to the spectral features of the Stokes emission, those of the medium’s response,
Examples of the spectra,
It follows from Eq.(22) that when the spectral width,
From Eq. (22) the SBS induced dispersion of refractive index, n() (/)|
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
which shows that the SBS-induced n
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
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
Here I
Suppose that the input Stokes signal is an optical pulse, the time dependent intensity of which is given by
where I
Figs 7 and 8 show the calculated relative output Stokes pulse powers, PS(t) = |ES(t)|2S (S is the effective area of fiber–mode cross section), shapes, amplitudes and delays for four input pulse durations, t
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 PS0eGef, where Gef 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
For a long input Stokes pulse, t
For t
For the short pulses, t
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
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() ’()/2n0, of the medium, where n0 is the refractive index of a medium without SBS. In the limit of small gain, G < 1, the Bessel function I0(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 I0(x) = 1, the integral in Eq. (28) can be taken for ES0(t) given by Eq. (30). It results in the following analytic expression for (t),
where b = t
The temporal dynamics of the induced acoustic wave amplitudes and their spectra for values of t
In the first case of a long Stokes pulse, that is t
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
The imaginary part of () described by Eq. (13) gives the refractive index of the medium modified by the SBS interaction, n() = n0 + n() n0 + (/)()/2n0 and its corresponding group index, ng() = n0 + [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
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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