Stimulated Brillouin scattering

Stimulated Brillouin scattering Chapter 0.1 STIMULATED BRILLOUIN SCATTERING Robert G Harrison and Dejin Yu NONLINEAR LIGHT SCATTERING Department of Physics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, United Kingdom Email: R.G.Harrison@hw.ac.uk Contents 0.1.1 Stimulated Brillouin scattering 0.1.2 Characteristics of SBS 0.1.3 SBS equations 0.1.4 Steady-state solutions 0.1.5 SBS in optical bers 0.1.6 Nonlinear dynamics in SBS 0.1.7 Concluding remarks Theory and experiment nomena have been investigated under ideal conditions of continuous plane wave excitation 10 Stimulated Brillouin scattering, gain, threshold, acoustic wave, Stokes wave, electrostriction, optical absorption, Bulk media, optical bers, nonlinear dynamics Stimulated Brillouin scattering (SBS) has a long history, stemming some four decades from the early 1960's when the foundations of this phenomenon were rst established, through theory and experiment As with nonlinear optical interaction in general, the need of high power laser excitation ned most, if not all, earlier experimental work and applications to the pulsed operating regime This work has been covered in many reviews on the subject and are also well documented in texts on nonlinear optics1; 2; 3; The purpose of this review is not therefore to treat this same ground but rather to concentrate on more recent developments in the eld A physical picture and theoretical background of this nonlinear process is provided to describe its familiar steady-state behavior From this, analysis is extended to include the roles of nonlinear refraction, spontaneous scattering and external feedback on the SBS behavior Their contributions, separate and collective, are shown to render SBS dynamically unstable, exhibiting many of the features of nonlinear dynamical systems, from stochastic-type amplied spontaneous emission to periodic, quasiperiodic and even chaotic behavior This is discussed in the context of the ever-growing body of literature on the generation of SBS in optical ber in which these phe- 0.1.1 Stimulated Brillouin scattering Brillouin scattering is a spontaneous light scattering process from acoustic waves in materials, so named following the discovery5 of this phenomenon by Brillouin in 19226 The process is said to be spontaneous since the light scattering is caused by thermal uctuations in media and the incident light eld is sufciently weak that its presence does not change the dielectric property of the material By de nition, the scattered light waves which are shifted to lower frequencies with respect to the frequency of incident light eld are called Stokes waves while those shifted to higher frequencies are called anti-Stokes waves By contrast, when the incident light eld is strong the Stokes wave experiences gain and stimulated Brillouin scattering occurs, characterized by exponential light ampli cation Such optical gain can amplify the scattered light by a factor of e30 1013; thus up to 100% of incident power can be converted into scattered light under appropriate conditions Physically, the two optical waves, incident and scattered, interact in the material and excite an acoustic wave through electrostriction The generated acoustic wave traveling in the medium, in turn, causes a periodic modulation of refractive index as a volume grating, which scatters the pump light through Bragg di raction The scattered Stokes eld is downshifted in frequency because of the Doppler shift associated with a grating moving at the acoustic velocity The incident and Stokes waves interfere to generate traveling intensity fringes When the velocity and spacing of these fringes match the velocity and the wavelength of the acoustic wave, the acoustic wave is ampli ed through electrostriction causing gain and thus stimulated Brillouin scattering Such a positive feedback process leads to exponential ampli cation of the Stokes wave Note that the anti-Stokes component in SBS has much lower gain while the Stokes emission Stimulated Brillouin scattering is more commonly observed experimentally In this dent laser; the transmitted laser being depleted until review, we therefore restrict ourselves to the Stokes the incident laser power in the back edge drops and component SBS ceases at time t2 During SBS emission, the transmitted laser power remains nearly unchanged Figure Oscilloscope traces of power of (a) incident laser, (b) and the peaked-value conversion e ciency is as high scattered Stokes wave, and (c) transmitted laser in ethyl ether (Re- as 80% produced from Maier8 ) (a) Power [MW] (b) (c) t1 25 Time [nsec] t2 50 0.1.2 Characteristics of SBS There are two basic types of experiment gurations: i) SBS generator, and ii) SBS ampli er The former, normal SBS, is a genuine scattering process involving the interaction of incident and scattered light waves with an acoustic wave In this guration, only one incident light eld is supplied externally and both scattered light and acoustic waves are generated during the scattering process in the medium The latter process is best described as a parametric interaction, in which an intense pump wave at !1 together with a weak input signal wave at !2 are incident on the Brillouin medium and the ampli cation of the signal wave is accompanied by a generation of an \idler" acoustic wave at = !1 ?!2 For backward scattering, the two SBS gurations are schematically shown in Fig Below, we mainly deal with the SBS generator and will not distinguish the incident from pump waves and the scattered from signal waves The phenomenon of SBS was rst discovered by Figure (a) SBS generator and (b) SBS ampli er gurations Chiao et al in 19647 They found that when a E1 , E2 are the forward and backward electric elds, and represents the acoustic wave in the medium k1 , k2 and q are wave vectors of su ciently intense Q-switched ruby laser beam of the three wave, de ned below frequency !1 passed through quartz or sapphire, a coherent acoustic wave at a frequency was excited within the crystal while a coherent optical E1 k1 ρ beam at a frequency !2 was simultaneously generq k2 E2 ated and ampli ed, whose frequency was downshifted by the acoustic frequency from the incident wave, (a) i.e., !2 = !1 ? Both the acoustic and scattered optical waves were emitted in speci c directions, in ρ q E1 k1 which the scattered light occurred predominantly in E2 k2 E k2 the backward direction Subsequently, a considerable number of experimental and theoretical investi(b) gations appeared (See Maier8 for a historic review) Typical are those of Maier8, in which a ruby pulsed laser with a maximum output power of MW over a diameter of mm and a beam divergence of mrad Stimulated Brillouin scattering has the following was used to generate SBS in a liquid cell containing distinct characteristics: either ethyl ether or n-hexane The time traces of the incident laser power, the stimulated Brillouin power Its generation requires high incident light and the transmitted laser power, recorded with a intensity, but not ultra-short light pulses time resolution of 0.3 ns, are shown in Fig As Scattered light in SBS has a much seen, the Stokes wave occurs once the incident laser narrower spectrum than Raman and power reaches a certain level at time t1 ; after which Rayleigh-wing scattering the Stokes emission follows the waveform of the inci- Stimulated Brillouin scattering 3 Scattered Stokes light occurs in the backward direction with respect to the incident wave The rst and second of these are directly linked to their spontaneous scattering properties Table shows typical values of frequency shift B and linewidth B of the backward scattered Stokes component and relaxation time of the material Table excitation9 , for the three commonly encountered spontaneous light scattering processes, Raman, Brillouin and Rayleigh The steady-state gain factors GB of the corresponding stimulated scattering processes in bulk solid media are also illustrated Of these, spontaneous Brillouin scattering has the narrowest linewidth and highest gain and, compared with Raman scattering, a much smaller frequency shift and longer relaxation time Typical parameters for spontaneous Raman, Brillouin and Rayleigh-wing light scattering processes9 Scattering process Raman Brillouin Rayleigh-wing Frequency shift B (cm?1 ) 1000 Linewidth B (cm?1) 5 10?3 It has been established from experiments that there exists a well-de ned threshold value of incident light power for a given material, beyond which an exponential light ampli cation process happens, leading to stimulated Brillouin scattering Below this conventional spontaneous Brillouin scattering occurs In bulk solid media7 , the typical Brillouin threshold power density is of order 102 ? 103 MW/cm2 for an interaction length cm We will see below that such a requirement for high power density can be dramatically decreased using waveguide media such as optical bers1 In bulk media, stimulated Brillouin scattering can be generated using a giant pump pulse providing its pulse duration is longer than the response time of the acoustic or phonon lifetime, typically a few nanoseconds (Table 1), otherwise only stimulated Raman scattering is possible In the quantum-mechanical description of SBS, when an incident photon is annihilated a Stokes photon and an acoustic phonon are created simultaneously The energy and momentum are conserved during each scattering event and, as a result, the frequencies and the wave vectors of the three waves satisfy the following conditions !1 = !2 + ; k1 = k2 + q; Relaxation time (sec) 10?12 10?9 10?12 (1) where the frequencies f!1; !2 ; g and wave vectors fk1 ; k2 ; qg correspond to the incident laser, scattered Stokes and generated acoustic waves, respectively The scattering geometry is shown Fig 3, where is the scattering angle between the incident and scattered waves The dispersion relation of the acoustic wave gives B = qv, where B = B is the Bril- Gain factor GB (cw/MW) 10?3 10?2 10?3 louin frequency and v is the velocity of the acoustic wave The wave number q = jqj and frequency B of the acoustic wave exhibit a strong dependence on the scattering angle , given as q = jk1 ? k2 j ' 2k sin ; B = qv ' 2kv sin : (2) (3) Here we have used the fact that the magnitudes of the wave vectors of the two light waves are nearly the same, i.e., k = jk1 j ' jk2 j since the frequency shift B is much smaller than the frequencies of the two light waves From Eq (3), we see that the maximum frequency shift occurs in the backward scattered wave, max = 2kv at = , and forward scattering ( = 0) leads to a frequency shift = Geometry of stimulated Brillouin scattering (a) Diagram of the two light wave vectors and (b) relation of the three wave vectors Figure k1 k1 θ k2 (a) k2 θ q (b) In principle, stimulated Brillouin scattering can occur in any directions provided that energy and momentum conservation (1) is satis ed However, SBS experiments reveal that a strong Stokes beam appears mainly in the backward direction with a small angular divergence This follows since in the back- Stimulated Brillouin scattering ward direction there is the longest interaction dis- where e1 and e2 are the unit vectors corresponding tance and consequently, the Stokes eld is most ef- the incident and scattered elds and the c:c: stands for complex conjugate, and the density change is fectively ampli ed along this direction written as 0.1.3 SBS equations (z; t) = 21 A3 (z; t)ei(qz? t) + c:c: (8) The three-wave interaction process of SBS involves The evolution equations of the two electric eld two light waves and an acoustic wave, which are amplitudes A1 and A2 of the forward pump and coupled through the process of electrostriction The backward Stokes waves under the slowly-varying light wave obeys Maxwell's wave equation subject to approximation2 are derived as nonlinear polarization 2 @ P(nl) r2 E ? nc2 @@tE2 ? cn @E @t = @t2 ; (4) where E is the total electric eld of light wave and P(nl) is the nonlinear polarization is the linear power absorption coe cient, n is the linear refractive index of the medium, c and are the light speed and permeability in the free space The acoustic wave is described in terms of density change around its equilibrium value due to the process of electrostriction in the presence of light elds The damped longitudinal acoustic wave equation is derived from the linearized Navier-Stokes equation2; 3; 10 , given as @ + ? @ ? v r2 @t2 B @t = ? 21 "0 e r2 hE Ei; !1 c P 1; ! c P: 2n 2n (9) (10) The relation between the nonlinear polarization P(nl) and the density change can be established using the fact that the density uctuation of the medium gives rise to a change in dielectric constant and a nonlinear polarization is, in turn, generated We express the dielectric constant as a function of density ( = + ) and the change in the dielectric constant is = ( ) ? ( ) ' @@ = e ; (11) where e = (@ =@ ) is the electrostrictive constant and the derivative is taken at = From this, we obtain (5) where hE Ei stands for a time average taken over an oscillation period of the light wave ?B is the acoustic damping coe cient related to the spontaneous Brillouin linewidth B by ?B = B v is the acoustic velocity and e is the electrostrictive constant, both are determined by the materials Suppose that all elds are plane waves in which the forward incident and backward scattered waves propagate in the +z and ?z directions in the medium, respectively, and the generated density wave propagates in the +z direction, as illustrated in Fig Noting that the amplitudes are slowlyvarying in time and space, the total electric eld E(nl=) E1 +(nlE) and nonlinear polarization P(nl) = P1 + P2 are expressed in terms of two frequencies at !1 and !2 as i(k1 z?!1 t) e1 A1 (z; t)e + e2 A2 (z; t)ei(?k2 z?!2 t) + c:c:; ( nl ) P (z; t) = e1 P1 (z; t)ei(k1 z?!1 t) + e2 P2 (z; t)ei(?k2 z?!2 t) + c:c:; n @A1 + @A1 + A = i c @t @z n @A2 ? @A2 + A = i c @t @z 2 E(z; t) = (6) (7) P(nl) = "0 E = "0 e E: (12) Care should be taken in using Eq (11) The density uctuation , hence the dielectric constant change , may be responsible for Brillouin and Rayleigh scattering processes The former originates from the pressure uctuation at constant entropy whereas the latter is due to the entropy uctuation at constant pressure11 Thus, we must select the components related to SBS in Eq (12) Substituting (6) into (5), we nd that hE Ei contains a term 21 (e1 e2 )A1 A2 ei (k1 +k2 )z?(!1 ?!2 )t] + c:c:, which can resonantly drive an acoustic wave at frequency = !1 ? !2 and wavenumber q = k1 + k2 in the stimulated Brillouin scattering process So the generated acoustic wave will, in turn, resonantly drive optical elds through parametric coupling From Eqs (6)-(8) and (12), the slowly-varying amplitudes of nonlinear polarization P(nl) (z; t) at !1 and !2 are expressed as P1 = "40 e A2 A3 ; P2 = "40 e A1 A3 : 0 (13) On introducing the expression (13) into Eqs (9) and (10) and assuming the acoustic wave to be highly Stimulated Brillouin scattering damped in Eq (5), the complete coupling wave equaAs seen from Eq (20), the exponential growth of tions describing SBS are then obtained as the Stokes signal is characterized by the gain factor GB , the peak value of Brillouin gain spectrum resulting from the process of electrostriction, which has a n @A1 + @A1 + A = i e !1 A A ; (14) Lorentzian spectral pro le3; 9; 13; 14 with a HMFW c @t @z 4nc n @A2 ? @A2 + A = i e !2 A A ; (15) being ?B The gain factor GB is determined by propc @t @z 2 4nc erties of the medium and the incident pump light @A3 + ?B A = i "0 e q2 A A : (16) wave Table lists some typical values of the fre1 @t quency shift B = B =2 , linewidth B = ?B =2 In deriving Eq (16), the space evolution of the sound and Brillouin gain factor GB for a number of liquid wave has been ignored since it can only travel a and solid materials at wavelength 6940 A of ruby short distance from its excitation location3 , typically laser It follows that for most of the liquid media, the frequency shift ranges from 4000 to 6000 MHz < 10?5 m and the Brillouin linewidth is about a few hundreds of MHz 0.1.4 Steady-state solutions Under steady-state conditions, applicable for a cw or quasi-cw pump, the time derivatives in Eqs (14){ (16) can be dropped and a set of two coupled intensity equations are obtained dI1 = ? I ? G I I ; B 12 dz dI ? dz = ? I2 + GB I1 I2 ; (17) (18) where I1 = " cnjA1 j and I2 = " cnjA2 j are the intensities of forward traveling and backward scattered light waves GB is called the steady-state SBS gain factor or Brillouin gain, given by 2 2 GB = 2nc3!1 e Brillouin gain and SBS threshold Bv : (19) Frequency shift B , linewidth B and steady-state gain factor GB of stimulated Brillouin scattering for a number of liquid and solid materials2; 3; Table Substance CS2 Acetone n-Hexane Toluene CCl4 Methanol Ethanol Benzene H2 O Cyclohexane Optical glass SiO2 B (MHz) 5850 4600 5910 4390 4250 4550 6740 5690 5550 11000-16000 17000 B (MHz) 52.3 224 222 579 520 250 353 289 317 774 10-106 78 GB (cm/MW) 0.130 0.020 0.026 0.013 0.006 0.013 0.012 0.018 0.0048 0.0068 0.004-0.025 0.0045 measurable exponential ampli caLet us rst examine the Brillouin gain and SBS tionExperimentally occurs for an incident light intensity I1 (0) such threshold To so, we take the small signal ap- that1; proximation and neglect pump depletion In this case, the Stokes eld grows exponentially with space GB I1 (0)Le ? L ' 20 ? 30: (21) in the backward direction and its intensity output is given by Such an incident intensity is de ned as the SBS threshold For example, in optical bers, this factor has been selected as 211 This value accounts G I (0)Le ? L B I2 (0) = I2 (L)e ; (20) for I (0)=I (0) 1% For the steady-state gain fac2 where L is the interaction length in the medium, tors given in Table 2, we can estimate the threshLe = ? e? L = is the e ective interaction old pump intensities Ith through the equation (21) GB = 0:02 cm/MW, L = 10 cm and = length and I2 (L) is the intensity of the Stokes eld at Take the end of z = L When ! 0, Le ! L while when cm?1 as 2an example, the SBS threshold is Ith ' 150 L 1, Le ! 1= In practice, no Brillouin signal MW/cm exists at z = L in a SBS generator and the Stokes wave grows from noise or spontaneous Brillouin scat- Pump depletion and gain saturation tering in the medium However, it has been shown that the steady state system can be treated as an Once the SBS threshold is reached, a large part of inequivalent SBS ampli er through injecting a weak cident light power is transferred into the Stokes wave signal I2 (L) at z = L at the Stokes frequency12 and the pump wave becomes depleted This phe- Stimulated Brillouin scattering havior, we examine the dependence of the transmitted pump and scattered light intensities on the incident pump strength For simplicity, we solve I1 (z = L) and I2 (z = 0) form Eqs (22) and (23) for = and plot them as a function of the incident pump strength in Fig It is obvious that signi cant exponential ampli cation occurs when the value GB I1 (0)L ranges from 20 to 25, which is the SBS threshold, beyond which the scattered Stokes inten(22) sity increases linearly with the pump strength while (23) the transmitted pump remains unchanged Moreover, since the net gain is proportional to the intensity I1 of the forward traveling pump wave in the medium, gain saturation therefore happens as the pump depletion occurs (24) nomenon is best shown in a SBS ampli er guration In this case, it is necessary to solve the steady state coupled-intensity equations (17) and (18) It is noted however that, for 6= 0, an analytical solution to the equations (17) and (18) has not so far been obtained For = 0, an analytical solution to Eqs (17) and (18) can be readily obtained as follows15 I1 (z) = (1Q?(zb)0?)Qb(z) I1 (0); b (1 ? b ) 0 I2 (z) = Q(z) ? b I1 (0); where the function Q(z ) is Q(z) = e(1?b0 )G0 z ; and the two parameters, b0 and G0 , are ; G0 = GB I1 (0): b0 = II2 (0) (0) Figure (a) Transmitted pump and (b) scattered Stokes intensities as a function of the incident pump strength GB I1 (0)L (25) The parameter b0 is a measure of the Brillouin efciency and G0 is sometimes called the small-signal gain associated with the SBS process In the general case, 6= 0, numerical solutions must be sought; this is a two-point boundary problem Figure shows pump depletion and growth of back-scattered SBS signal with propagation distance for two di erent pump levels Plotted are the relative intensities, normalized to the incident pump intensity I1 (z = 0) One can see that most of the power transfer occurs within rst 20 ? 30% of interaction distance, and 55% and > 70% of the incident power are scattered back to the incident end for the Steady-state SBS with feedback two pump levels We next consider SBS in the presence of external Figure Steady state solutions to the SBS equations (17) and (18) feedback, in which the Brillouin medium is enclosed Solid and dotted lines correspond to forward pump and backward in a Fabry-Perot cavity consisting of two mirrors with Stokes intensities, respectively (a) and (b) G0 L = 10:0, (c) and power re ectivities R1 and R2 , respectively The sys(d) G0 L = 20:0 Absorption of L = 0:01 is used and the injection tem as such is a Brillouin laser In this case, apart signal at z = L is assumed to be I2 (L)=I1 (0) = 0:001 from Eqs (17) and (18) responsible for the forward pump and backward scattered elds, the steady-state coupled intensity equations should include two additional equations describing the backward pump and forward Stokes elds due to back-re ections They are Further, in order to clearly see how the backscattered Stokes intensity exhibits the threshold be- ? dIdz3 = ? I3 ? GB I3 I4 ; (26) dI4 = ? I + G I I ; (27) B 34 dz where I3 and I4 are intensities of the re ected pump and Stokes signals The boundary conditions couple the two groups of elds and are formulated to be I1 (0) = R1 I3 (0) + Iin; I2 (L) = R2 I4 (L) + ; (28) Stimulated Brillouin scattering I3 (L) = R2 I1 (L); I4 (0) = R1 I2 (0) + ; (29) where Iin is the incident intensity within the cavity, and (' ) are the initial Stokes intensities at the two ends of the medium Dammig et al.16 gave an SBS threshold condition by extending Smith's de nition12 The result for the SBS threshold is expressed in terms of a transcendental equation, which is not convenient to use Following the concept of Dammig et al., we can derive an explicit expression for the SBS threshold in the presence of feedback Near the SBS threshold, the Stokes wave is relatively weak and, as a result, both the interaction between the re ected pump and Stokes waves and the pump depletion can be neglected In this case, the steady-state solutions of the simpli ed equations together with their boundary conditions lead to ? L ? L+GB I1 (0)Le ; I2 (0) = ( + R2 e ?2)eL+GB I1 (0)L e ? R1 R2 e I1 (0) = ? R IRin e?2 L : (30) (31) Threshold for SBS Oscillation: The oscillation for the Stokes emission takes place when the denominator of Eq (30) vanishes Thus, we obtain the steady-state oscillation threshold for SBS to be GB I1 (0)Le = L ? ln(R1 R2 ); (32) where I1 (0) is given by (31) Eq (32) implies that SBS oscillation happens when the gain of the Stokes wave is equal to the natural round-trip loss in the medium, a condition similar to that in a normal laser4 We can alternatively de ne the SBS threshold condition according to the SBS threshold without feedback To this end, we express I2 (0) = rth I1 (0), where rth is given by rth = exp(21) and the factor of 21 corresponds to the SBS threshold value without feedback Under these conditions, the SBS threshold is given as 0.1.5 SBS in optical bers The study of stimulated Brillouin scattering phenomenon in optical bers is of special importance since their long interaction length and the high power densities they support through their small core area contribute to low threshold SBS and allow cw excitations It was shown that SBS can occur for pump power as low as a few milliwatts in a 13-km low loss single-mode ber17 Stimulated Brillouin scattering in optical bers was nevertheless rst observed by Ippen and Stolen in 1972 under pulsed excitation18 Two glass bers with a core radius a = 1:9 m and linear loss = 1300 dB/km were investigated, one 5.76 m and the other 18.5 m A pulsed xenon laser operating at 5335 A with a pulse duration 600 ns was used as pump source Oscilloscope traces of incident, transmitted pump and back-scattered Stokes signals are shown in Fig for the shorter ber The threshold pump power was estimated as 2.3 W for L = 5:76 m and < W for the longer ber (L = 18:5 m) The oscillatory feature evident in the Stokes and transmitted pump elds was explained by Ippen and Stolen in terms of an oscillation due to nite ber length18 In the following section, we will see that such a temporal behavior comes from intrinsic dynamical instability in SBS Experimental observation for bre length 5.76 m (a) Incident pump and transmitted signals and (b) backward scattered Stokes signal (200 ns/div) (Reproduced from Ippen and Stolen18 ) Figure (a) i h GB I1 (0)Le = 21 ? ln R1 R2 e21 + + R2 e? L : (33) If R1 R2 e21 (1+ 42 R2 e? L ), then Eq.(33) is reduced to the threshold condition for SBS oscillation given by Eq.(32) Assuming R1 = R2 = R, this implies (b) approximately R > e?10:5 10?5 For example, R = 4% and = leads to an SBS threshold The theoretical description of SBS in optical bers GB I1 (0)Le ' 6:4 is similar to that in section 0.1.4 The following points should however be considered 8 Stimulated Brillouin scattering Only two propagation directions are involved in optical bers { the pump wave travels in the forward direction while the scattered Stokes wave has to propagate backward Three interacting signals are no longer plane waves It can be shown however that in single-mode optical ber, the three elds are equivalently viewed as plane waves after integrating the transverse distribution functions over the cross-section in bers1 Polarization of electric elds has an effect on the gain factor with GB redened by KGB , where K = (e1 e2 )2 is called the depolarization factor The value of K ranges from 1/2 to 1, the two limiting cases correspond to polarization-scrambled and polarizationpreserving bers, respectively Spontaneous Brillouin linewidth B in optical bers is enlarged by a factor of e = ? 2:5 due to the guided-wave nature and inhomogeneity of the dopant compared with that in the corresponding bulk silica materials1; 19 Nonlinear refraction should be considered in nonlinear wave coupling Self- and cross-phase modulation may lead to new type of dynamics 0.1.6 With inclusion of the two e ects from the depolarization and additional spectrum-broadening, the SBS gain factor (19) is now modi ed as 2 GB = 2nc3!1 e K v : B (34) Note that B in Eq (refeqn34) is the Brillouin linewidth in optical bers which includes the enhancement factor e with respect to the linewidth in bulk media Furthermore, the Brillouin gain is assumed to have a Lorentzian spectral pro le as in bulk media1; 2; We note however that recent experiment provides evidence of the inhomogeneous spectral broadening of SBS in an optical ber20 The more interesting aspect of SBS in optical bers lies in instabilities and a variety of dynamical behavior that comes from interplay between gain and nonlinear refraction It is well known that the latter is responsible for self- and cross-phase modulation of the interacting electric elds in optical ber1 We consider this topic in the next section Nonlinear dynamics in SBS Apart from the dynamic response of SBS observed when the pulse width of the incident laser is comparable with the phonon lifetime, the Stokes emission displays relaxation oscillations with a period of 2Tr when a long pulse or cw laser is employed as a pump source21, where Tr is the transit time in the medium In the following, we will see that external feedback22 and/or strong nonlinear refraction23 can turn these relaxation oscillations into stable periodic oscillations, quasiperiodic motion, and even chaotic emission While pulsed laser excitation of SBS in bulk media tends to mask such features by complex transient type behavior they are clearly manifested under cw pumping using single-mode optical ber as the SBS gain medium Investigation of dynamical behavior in SBS provides an insight into a fundamental issue in nonlinear optical systems, that is, whether in general external feedback is a necessary condition for the onset of dynamical instabilities and chaos Silberberg and Bar-Joseph were perhaps the rst to point out that self-oscillations and chaos can indeed occur in an optical system through counter-propagating wave interaction, without external feedback24 This phenomenon was predicted theoretically25 and observed experimentally26; 27 in the SBS ampli er guration The gain-feedback mechanism results in dynamical instability, in which the gain is experienced by side-modes to the light elds and the (internal) feedback originates from inherent nonlinear interaction due to scattering from the grating formed by the interference between the two counter-propagating elds rather than through an external cavity In normal SBS (SBS generator), only the pump beam is supplied externally and the counterpropagating Stokes eld is generated through the nonlinear interaction, and as such this case deserves special attention Earlier theory provided analytical evidence of unstable behavior in SBS involving a single beam pump and Stokes signal in a semi-in nite medium28 In a nite-length medium, theoretical analysis established a quasiperiodic route to chaos for the Stokes emission on increasing the strength of nonlinear refraction23 The numerical simulations were based on the coupled amplitude equations (14)-(16) with inclusion of the e ects of self- and cross-phase modulation29 , distributed spontaneous scattering30, and detuning of acoustic wave @A + @A + A = ?g BC + iu jAj2 + 2jBj2 A; (35) @ @ @B ? @B + B = g AC + iu jBj2 + 2jAj2 B; (36) @ @ Stimulated Brillouin scattering @C + (1 + i )C = AB + f ( ; ); n @ B (37) where = ct=nL and = z=L are the normalized time and space coordinates, u = U0g0 = (!L=c)n2jAP j2 is the strength of nonlinear refraction fn ( ; ) represents a Langevin noise (spontaneous scattering) source30 The eld amplitudes of the three waves are normalized to that of the incident light wave AP = A1 (0) at the input end of = see how external feedback, even weak, suppress noise and determines the form of dynamical behavior Time evolution of scattered Stokes signals for nonlinear refraction strength u0 = 0:4 and pump strength g0 = 30:0; (a) without noise and (b) with noise Figure A = AA1 ; B = AA2 ; C = i" q2?jBA j2 A3 : (38) e P P P Parameters in Eqs (35){(37) are de ned as g0 = U0 = n? L B = 2Bc ; = ?B ?=2 ; B "0 e2 K !L jA j2 ; P 0v P B c 0v P B n ; "0 e2 (39) where P is the wavelength of the incident light wave and n2 is the nonlinear refraction1; For the case without noise and detuning ( = 0), Figure 7(a) demonstrates that nonlinear refraction plays a key role in generating sustained motion and interplay between the gain and nonlinear refraction is responsible for the chaotic dynamics Detailed numerical calculations showed a quasiperiodic route to chaos on increasing pump strength g0 23; 31 and within the chaotic regime a Type-III intermittency for g0 = 40:0 exists31 However, with inclusion of the spontaneous scattering noise term in the model description, the dynamical behavior of SBS emission becomes largely noise-dominated, as shown in Fig 7(b) Existing experimental observations show good qualitative agreement with these theoretical results16; 32; 33 Further investigation by Chow and Bers predicted quasiperiodic motion and chaotic Stokes emission in stimulated Brillouin scattering34 They considered the important role of detuning ( 6= 0) but ignored the e ect of nonlinear refraction (u = 0) To date, no experiment has rmed this nding Figure Stable and unstable regions of SBS with external feedback and in the presence of nonlinear refraction, shown in the g0 ? R plane The dashed lines, (a) and (b), correspond to the case of u0 = and A = The two solid lines, (c) and (d), give the the critical values determined by using the full model31 in which the acoustic damping and loss parameters used are A = 148, = 0:0414 and u0 = 0:4 (Reproduced from Yu et al.35 ) 20 Pump strength g0 = L; 15 (d) 10 (b) Unstable The presence of external feedback not only reduces the SBS threshold as mentioned above, but may also turn relaxation oscillations into sustained dynamics in stimulated Brillouin scattering process In experiments using optical bers, unless specially treated, a natural re ectivity of 4% exists between the glass-air interface, which is su cient to provide nonnegligible optical feedback In this section, we shall (c) (a) Stable SBS dynamics with feedback Stable Reflectivity R(%) 10 The dynamical aspects of stimulated Brillouin scattering in optical ber media with weak feedback were rst investigated theoretically by Bar-Joseph et al.22 Instabilities were predicted using a truncated SBS model and sustained periodic SBS oscillations were experimentally observed In the general case where the full model is used, we have numeri- 10 Stimulated Brillouin scattering Figure (a) Time-averaged Stokes signal and (b) transmitted pump as a function of input pump power The regions I{V mark parameter windows of di erent dynamical behavior (Reproduced from Harrison et al.36 ) 800 SBS, Transmitted Pump (mW) V IV 600 (a) periodic motion occurs On increasing the external pump power, the SBS emission follows a quasiperiodic route to chaos These time traces together with their phase portraits are depicted in Fig 10 Detailed theoretical analysis and numerical calculations show a remarkable agreement with these experimental measurements35; 37 Time series (left-hand column) shows periodic (a), quasiperiodic (c), chaotic (e), and periodic (g), corresponding to four regions I-IV in Fig Corresponding phase portraits are shown in the right-hand column (b), (d), (f) and (h) (Reproduced from Harrison et al.36 ) Figure 10 SBS Intensity (Arbitrary units) cally investigated the stability of SBS35 For a set of typical parameters, the stability diagram is illustrated in Fig The two dashed lines, (a) and (b), correspond to the case predicted by Bar-Joseph et al.22 Firstly, as seen, the instability threshold at low pump intensity is close to the threshold for SBS given by Eq (32) in section 0.1.4 Such an instability threshold is not sensitive to u0 and is only slightly upward shifted on increasing u0 Thus, the presence of nonlinear refraction will slightly increase the instability threshold (also SBS threshold) Next, at high pump, a large u0 can however lead to a notable extension of the unstable region through upshift of the upper boundary35 For example, the amount of shift of the boundary is g0 ' 3:7 for an increase of u0 from u0 = 0:0 to 0.4 for R = 3:3% The e ect of nonlinear refraction is however not only to increase the unstable region but also to considerably enhance the complexity of dynamical behavior that exists in the unstable region III 400 II (b) 200 I 0 0.5 1.0 1.5 Pump Power (W) A number of experimental studies have been made of the dynamical behavior of SBS in optical bers with external feedback16; 33; 36 , revealing that both the SBS and transmitted pump signals may exhibit a rich variety of reoccurring and classi able dynamical features: limit cycle, two-frequency oscillation, period-two and period-three bifurcation and perhaps fully developed chaos For example, a typical experiment by Harrison et al.36 classi ed the SBS dynamics in ve regions, as shown in Fig presented by time-averaged SBS and transmitted pump intensities Beyond the SBS threshold, the stochastic emission shown in Fig 7(b) is suppressed and stable 12 16 Time t (units of Tr) 0.1.7 Concluding remarks In this chapter we have addressed the optical phenomenon of stimulated Brillouin scattering A physical picture and theoretical background of this nonlinear process has been provided to describe both its steady state and dynamical behavior Traditional generation of SBS in bulk media using high power pulsed pump radiation was brie y reviewed Particular attention was paid to recent work on the generation of SBS under cw pumping in optical ber and the new insight this has provided into the unstable, chaotic and stochastic nature of this phenomenon Stimulated Brillouin scattering 11 It is hoped that this review will provide the reader of Brillouin-gain spectra for single-mode optical bers; 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