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U-M EECS 410 - Stimulated Brillouin scattering parasitics in large optical windows

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410 OPTICS LETTERS / Vol. 12, No. 6 / June 1987Stimulated Brillouin scattering parasitics in large opticalwindowsJ. M. Eggleston and M. J. KushnerSpectra Technology Inc., 2755 Northup Way, Bellevue, Washington 98004Received May 1, 1986; accepted March 18,1987The growth of optical radiation, scattered transverse to the pump axis by stimulated Brillouin scattering (SBS) inoptical windows, is considered. Basic equations are presented, and an analytic expression that determines theparasitic buildup time is derived for a transverse SBS geometry. Losses suffered by the scattered optical radiationare included in a bulk-loss term. Calculations are performed for fused-silica windows and compared with anumerical model. This parasitic process may affect the design of laser systems that will generate multinanosecond,multikilojoule, narrow-band pulses in the ultraviolet region.Stimulated Brillouin scattering" 2(SBS) is an allowednonlinear process in all media and has previously3been associated with damage processes in mirrors andwindows. Scaling of UV excimer lasers for kilojoule,short-pulse (10 nsec), narrow-band (<0.5 cm-') opera-tion creates the potential for transverse SBS process-es. The SBS-generated optical wave would be a para-sitic wave that removes energy from the transmittedwave and simultaneously generates an intense acous-tic wave, which could fracture large windows.The SBS process is a three-wave mixing processinvolving two optical waves and an acoustic wave inthe SBS medium. A compressional acoustic wave cre-ates an index grating that scatters energy from oneoptical field to the other. The two optical fields forma traveling standing-wave pattern with high and lowfield intensity regions, which, in turn, drives the com-pression and rarefaction of the acoustic wave. As withall nonlinear mixing processes, energy and momentummust be conserved.Conservation of momentum requires that the wavevectors for the three waves be coplanar; thus thephase-matching diagram of Fig. 1 can be used. Sincethe speed of light is many orders of magnitude largerthan typical sound-wave velocities, then to a high de-gree of accuracycoa 2cop nasin(0/2), (la)c4s= Wp- 4a, (lb)Oa (0/2) - 900, (1c)where cop, wS, and Wa are the radial frequency of thepump, Stokes, and acoustical waves, respectively; 0 isthe angle between the pump and Stokes wave vectors;Oa is the angle between the pump and acoustical wavevectors; Va is the speed of sound in the SBS medium; nis the optical index of refraction in the SBS medium;and c is speed of light.The pump and Stokes electric fields are defined bythe slowly varying envelopes P and S through therelationsEs = (7r/c)l12S exp[i(wt -Kr)] + c.c.,Ep = (7r/c)"'2P exp[i(wpt - Kpr)] + c.c.*(2a)(2b)The intensities of the Stokes and pump fields are 1512/2 and IP12/2, respectively. The equations of motionfor SBS can be derived from Maxwell's equations andfrom the equations for continuous media, which in-clude electrostrictive terms.2Invoking the slowlyvarying envelope approximation, the SBS equationsare given by\6t n Ax(2rbn Ax( t) Sp a 2P,-)S = PP -_ 2S.1) p = 9oC SP*,(3a)(3b)(3c)where xp and x8are coordinates in the direction ofpropagation of the pump and Stokes waves, respec-tively; Tb is the intensity decay time of the acousticwave; go is the steady-state gain coefficient for theStokes wave; ap (as) is the background loss for thepump (Stokes) intensity per unit time; and p is propor-tional to the complex amplitude of the acoustic wave.These equations are identical to those for liquids andsufficiently dense gases.'KKpFig. 1. Phase-matching diagram for SBS showing thepump, Stokes, and acoustic wave vectors._} °a0146-9592/87/060410-03$2.00/0 © 1987, Optical Society of AmericaJune 1987 / Vol. 12, No. 6 / OPTICS LETTERS 411INCIDENT PUMPXs TES; RXp ACOUSTICWAVE ' TRANSMITTEDPUMPTHICKtFig. 2. Transverse SBS geometry. A uniform pump fieldin time and space is normally incident upon a window. AStokes field propagating normal to the pump field builds up,reflecting off the sides of the windows with reflection coeffi-cient R, and eventually depletes the pump field.The angularly dependent gain coefficient in an iso-tropic or cubic solid for a monochromatic pump isgiven by5'627rn7P1221-CpOXs2vaAb sin 0/2 (4)where P12is the elasto-optical coefficient, po is thedensity of the solid, XA is the Stokes wavelength invacuum, and Avb is the FWHM linewidth of backscat-tered Stokes fluorescence (0 = 180°). For backwaveSBS, sin 0/2 is 1, and the gain expression reduces to theform given by Ippen.6The FWHM linewidth of the transverse Stokeswave (Av) is a function of the acoustic frequency givenbyV (-) AVbO, (5)where Wao is the frequency of the acoustic wave wherethe backscattered linewidth Avbo was measured. Thepump wavelength and the scattering-angle depen-dence of the acoustic frequency (Wa) are given in Eq.(la). The decay time of the acoustic intensity is relat-ed to the line width by1/rb = 27riAv. (6)Since the acoustic frequency is inversely proportionalto the optical wavelength, the frequency dependenceof the gain is determined solely by the wavelengthvariation of the optical index of refraction. The re-sponse time, however, varies as the square of the opti-cal wavelength.The gain for backwave SBS in fused-silica fibers hasbeen experimentally measured6at 535.5 nm to be 4.3cm/GW. This is within experimental error of the the-oretical value6of 5.8 cm/GW. The index of refractionat 535 nm is 1.46, and at 248 nm it is 1.51. Includingthe effect of the index of refraction on linewidth, thetheoretical gain for transverse SBS at 248 nm is 9.7cm/GW. The frequency shift and response time7'8(frb) for transverse SBS at 248 nm should be 4.9 GHzand 0.52 nsec (Av = 0.3 GHz), respectively.Consider the geometry in Fig. 2. The pump beamhas a temporally flat top, uniform in the transversedimensions, with an intensity Ipo. The leading edge ofthe pump beam is normally incident upon a window attime t = 0. Inside the window, a uniform Stokes wavewith an initial intensity of Iso is propagating at rightangles to the pump wave. If the loss seen by theStokes wave at the window edges is distributed into anequivalent continuous loss coefficient (as), then theinitial conditions, the boundary conditions, and theequations of motion become invariant

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