Numerit[WIN32][1700][1703]Pz qffffff)@`"Aj@ s@?@ffffff9@?@ffffff9@ffffff)@ffffff)@       Times New RomanArialSymbol Courier New G Zd;O-c@ ףp=jR@y1zy2zy3zy4zy5z?@?@?@?@ ?@ >@>@ >@ >@ >@>@>@>@>@>@ >@  ???>>??Wave amplitudeDistance [arb]-1.21.20.40.010.01-6620.010.01E #0@H@(w1k1)!yz(y}y}z(yydbdbd E `"Qb@+@w2=w1-WB!z(yxz(yx{(zdbdbdbd E `"Qb@+@qB=2k1!z(zxyxz(ydbdbd E `"Qb@ffffff-@!WB=fqBvs!{(zx"z(zxz(z!!dbd bdbd!!!!!!E DDDDD @H@e vs!z(z  dbd      X fffffv1@ffffff@tE 3@wwwwwW@(w2-k2)!yz(y}yx}z(yydbd bd E """""1@wwwwwW@(WBqB)!y{(z}yz(zydbdbd vvvpvvvpvvvpvvvpvvvpvvvpvvvp)vvv4Stimulated Brillouin Scattering :vvvUri Levy October 15, 2001vvvvvvA. The scattering processvvvavvvStimulated Brillouin scattering (SBS) is a non-linear optical process where three waves interact:*vvv1. A forward propagating light wave - ! ,vvv2. A backwards "scattered" light wave - ) .vvv3. A forward propagating sound wave * vvv=vvvIn simple words, the SBS process can be described as follows:vvvA small portion of a forward propagating light wave is backwards scattered by a sound (or acoustic) wave. The backwards scattered light wave is frequency shifted such thatvvv! " vvv{See, for example, Robert W. Boyd, "Nonlinear Optics" pp. 332}. The two light waves interact to generate a "standing" wave (moving forward "slowly" at the velocity of sound).vvvThe standing light wave generates a sound wave through a process known as electrostriction (material compression in the presence of electric field).uvvvThe wave-vector of the INTENSITY of the standing wave (and hence of the sound wave) is, to a very good approximation,vvv" # vvvor in terms of wavelength - the wavelength of the sound wave is half the wavelength of the light wave. For example, if the wavelength of light (in matter) is 1 micron, the wavelength of the sound wave is determined as 0.5 micron.TvvvNow, the dispersion relations of sound in matter determine the sound wave frequency:vvv# $ vvvvvvwhere % is the sound velocity. From 2 and 3 we can see how the frequency of the back-reflected wave (given in equation 1) is determined by the wavelength of the "exciting" light wave (equation 2) and the velocity of sound (equation 3).vvvIf the intensity of the input light is increased, both the acoustic wave intensity and the back-reflected light intensity will also increase. This process is important in optical fiber communication. It sets a limit to the power transmission capability through the fiber. Beyond a certain input power level, the Brillouin scattering becomes so strong that all added optical input power is back-reflected. vvv&vvvWe can now turn to see the simulation.vvvvvvB. The simulation'vvvAll five waves discussed above are shown in figure 1. For simplicity, only the forward propagating wave is first shown. You can switch to the program pane and de-comment the waves progressively. Note that the (apparently standing) purple and two green waves are actually moving forward slowly.vvvvvv0t =' secondsvvv J{Av\! . Simulation of the stimulated Brillouin scattering process. The red and blue optical waves generate a "standing" (slowly moving forward) wave shown in purple. The intensity of the standing wave (dark green) promotes a sound wave (light green) through electrostriction. The blue light wave is back-reflected by the sound wave.{AvuThe wavelength of the sound wave is half the wavelength of light. The frequency down-shift of the back-reflected light wave equals the frequency of the sound wave. vvv q8ffffff)@j@fffffvq@?@ffffff9@?@ffffff9@ffffff)@ffffff)@        Times New RomanArialSymbol Courier New , `Stimulated Brillouin Scattering`October 15, 2001 `Parameters4delay_seconds = 0.02 `sets the delay duration f = 10 `1/secT = 1/f `sec h = -0.05 lamda = 3omega1 = 2*pi*flamda1 = lamdak1 = 2*pi/lamda1omega2 = omega1*(1+h)lamda2 = lamda/(1+h)k2 = 2*pi/lamda2t = 0 dt = T/100`Viewing window"z = -2.0*lamda..2.0*lamda len 61report`Run the waveswhile t < 10.2*T;`The five waves. Comment out / de-comment as you see fit., `Forward propagating light, w1, k1y1 = cos(k1*z - omega1*t)- `Backwards propagating light, w2 < w1y2 = cos(-k2*z - omega2*t)*0.75 `Standing wave (y1 + y2)3y3 = (cos(k1*z - omega1*t)+cos(-k2*z - omega2*t))/3/ `Intensity of the standing wave (y3*y3)7y4 = ((cos(k1*z - omega1*t)+cos(-k2*z - omega2*t))/3)^2 `Acoustic wave)y5 = cos(-(k2+k1)*z + (omega1-omega2)*t)t += dtwait delay_secondsrefreshd:\program files\numerit\ delay_secondsfThlamdaomega1lamda1k1omega2lamda2k2t dtz y1 y2 y3 y4 y5      C G  B B  B C   @B   @C  B C     C  G B  B f  BY7  B  BA G B  BA B!  B  BA G B  BA@ C#  B  BA G B  BA@ C E% @G B  A B@' N ( )6N 0.02{Gz?10$@1?0.05?3@2@0100Y@2.0@61N@10.2ffffff$@0.75?