Received December 22, 1993; revised manuscript received May 2, 1994
We have investigated stimulated Brillouin scattering (SBS) in carbon disulfide, using a frequency-narrowedCr,Tm,Ho:YAG laser operating at 2.12 um. Threshold reduction with a novel loop geometry is also demon-strated. A model for the loop-SBS scheme is developed and compared with experimental results. Backscat-ter power, energy, and conjugate-fidelity data are presented.
INTRODUCTIONThere is currently a renewal of interest in the 2-AmHo:YAG laser' because of its inherent eye safety for appli-cations in medicine and laser radar. The laser mediumcan provide high power with high efficiency and has goodatmospheric transmission. However, its low-gain crosssection means that strong pumping is required, whichleads to strong thermal lensing at high repetition ratesand problems with beam quality. Phase conjugation canhelp solve these problems, but the lower peak power avail-able at 2 gum means that stimulated-Brillouin-scattering(SBS) threshold reduction techniques are required.
The SBS threshold can be reduced by use of long in-teraction lengths,2' 3 by repeated refocusing of a beamthrough a series of Brillouin cells,4 or by use of tech-niques similar to those used in photorefractive phaseconjugation.5 These include self-pumped four-wavemixing,6 a loop scheme in which the input beam passesthrough a Brillouin cell twice with the second path cross-ing the first,7 and a ring scheme in which the Stokesoutput of the Brillouin cell is fed back to reseed the input,with the Brillouin cell acting as an amplifier in a ringresonator. 8
In this paper we report for what is the first time to ourknowledge a study of SBS at 2 Am and the use of a newcompact loop geometry to achieve SBS stabilization andthreshold reduction. We first discuss qualitatively thefeatures of the loop scheme and the mechanism respon-sible for its behavior. Then we review the publishedliterature on the loop scheme, discuss the theoretical be-havior and describe experimental studies.
REVIEW OF THE LOOP SCHEMEA schematic layout of a loop is shown in Fig. 1, with in-put beam Al tracing its path around the loop and a Stokesbeam emerging as A2 . In the overlap region grating uswill be driven by interference between Al and A4 and be-tween A2 and A3. Conventional Brillouin amplificationmay also take place, but one can understand the dynamicsof the loop by concentrating on the role of the four-wavemixing grating u alone.
Incident field Al is partly scattered by a weak acous-tic wave u (initially noise) to form A4, and the rest ofAl is transmitted. The two fields then pass around theloop in opposite directions and, after a loop transit time,form beams A3 and A2. The interference fringes formedby these two beams will amplify the acoustic wave us andscatter more radiation from A3 into A2 . If the amplifi-cation rate exceeds the natural damping rate, the acous-tic wave will grow until pump depletion limits it. Oncethe acoustic wave is present, the scattering of Al on thefirst transit will ensure that there are interference fringesformed by A2 and A3 at a later time over the whole ofthe interaction volume, and these will reinforce us. Thusthere is no need to rely on spontaneous noise to act asa continuous input signal, as is the case with conven-tional SBS. The process can operate with SBS or ther-mal gratings. 9
The acoustic wave grows only when the interferencepattern formed by the pump and the Stokes beams onthe second transit precisely replicates the interferencepattern of the first transit, i.e., the fringes formed byAlA4 * match those formed by A2*A3. A phase-conjugateStokes beam ensures that this requirement is met andis one possible output wave front. If Gaussian beamsare used, it is possible to obtain solutions in which theStokes beam is not conjugate.10"1 If there is an aber-rator between the first and the final passes through theBrillouin medium, then a conjugate wave front is still apossible solution, but it is now difficult to find a wavefront that is not conjugate and still meets the require-ment AIA 4*(x, Y)l x IA2*A 3(x, y).1 ll
In addition to four-wave mixing, conventional Brillouinamplification occurs in the loop, and this tends to favorphase conjugation because the conjugate experiences en-hanced Brillouin gain. The Brillouin shift of the four-wave mixing grating depends on the intersection anglebetween the signal and the pump, so the degree to whichconventional Brillouin amplification contributes will de-pend on this angle.
The first investigation of a loop scheme was carriedout by Odintsov and Rogacheva in 1982.7 They used aloop in which the overlapping beams were confined in a
Scott et al.2080 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994
A2
A, v~
A4
A3
US
Fig. 1. Simple loop, showing input beam Al tracing its patharound the loop and emerging as A3 after a loop transit time.The input beam is partly scattered by an initially noise-generatedacoustic grating us to form A4 and, after a loop transit time,A2. In the overlap region the grating us is driven by four-wavemixing interference between Al and A4 and between A2 andA3. Conventional Brillouin amplification may also take placeas indicated by gratings ul and U3.
lightguide; they observed high-fidelity phase conjugationwith the SBS threshold reduced by approximately a factorof 10. They also described a simple theory that identifiedthe mechanism as having an absolute instability arisingfrom the role of feedback in the four-wave mixing process;i.e., once the acoustic wave has started to develop, no
further noise is required, and the final output level isindependent of the initial noise.
Bel'dyugin et al.12 investigated various loop schemeswith, for example, amplifiers inside the loop, and othernonlinearities such as thermal gratings' being responsiblefor the coupling. These and other studies used nearlycollimated beams13 '16 rather than the focused beams wehave used in our study, and beam quality has been anissue. A detailed review of much of the recent Russiananalytical research, particularly the two-dimensionalanalysis, was recently published by Tikhonchuk andZozulya."7
In the West the first study of the Brillouin loop wasmade by Wong and Damzen, 8 who developed a steady-
state theory and described a series of experiments com-paring the loop scheme with a ring geometry. Morerecently workers at TRW'8 reported experiments withboth the loop and the ring geometries.
THEORY OF THE LOOP-SBS INTERACTION
We analyze the behavior of a loop-SBS mirror, payingparticular attention to the parameters that can be influ-enced by experiment. Experimentally we investigate thecase of partially overlapping focused beams that cross ata small angle. In our analysis we use a uniform one-dimensional plane-wave approximation and introduce fac-tors to account for the partial overlap and the effect offocusing.
The experimental scheme that we studied is shown inFig. 2. The input beam is focused through a Brillouincell three times, with the third focus overlapping thefirst, Brillouin four-wave mixing will take place wherethe beams overlap, and additionally Brillouin amplifica-tion may take place elsewhere in the Brillouin cell.
We use the notation of Ref. 19 and assume plane wavesoverlapping as shown in Fig. 1:
In the above relations A co is the frequency differencebetween the input and the conjugate beams (which willbe approximately the Brillouin shift) and 0 is the half-angle between beam 1 and beam 3.
The interference fringes drive acoustic waves accordingto
(1 + ixl)ul + 1 au =-pElE2,1c at
(1 + ix8)u8 + , = -a3(ElE4 + E2E3),
(1 + ixl)u3 + 1 aU3 =-,E3E 4 *,&o0 at
(2)(1 + iXl)u5 + 5 = - BE5E64 .
wo at
The small phase-mismatch term is neglected. u are theacoustic waves, which have resonant frequencies t£oi andlinewidths 6(0 given by
(0sl = Ws3 = s5 2k,Ps,
3o(0 = 1/2
TB,
&)Ss = 2k, P, cos 0,
(3)
where vs is the speed of sound in the medium and TB isthe phonon lifetime.
The first, third, and fourth of Eqs. (2) correspond toconventional Brillouin amplification, and the second cor-responds to Brillouin four-wave mixing. We use detun-ing parameters xi and xs to relate the Stokes frequencyIW2 = (1W - Ae to the Brillouin shift for Brillouin amplifi-cation and four-wave mixing, respectively:
When xl = 0, we are at the center of the Brillouin gaincurve. Substitution for typical values for CS2 scaled to2 ,um (Ref. 20) gives to,, = 1.06 x 1010 s-1 and 6to =2.5 x 107 s-'. Thus, when 0 = 70 mrad, the frequencydifference between co,, and eo8l will equal the Brillouinlinewidth (i.e., when xs = 0, then xl = 1). This detun-ing increases in proportion to 02, and for this and larger
Lens SBS cell Lens
Fig. 2. Experimental geometry showing a compact loop withoverlap at the foci.
Vol. 11, No. 10/October 1994/J. Opt. Soc. Am. B 2081
angles the four-wave mixing process may generate aStokes beam that is not efficiently Brillouin amplified out-side the overlap region.
The acoustic waves scatter the six electric fields:
aE1 = g (hiE 2u1 + hE 4 u,),ar 2/3
aE2 = g (hlE2 ul* + hsE3 Us *)ar 2/+
ar3 2/ (hE 2U + h3E 4 U3 ),
aE4 =ar
aE5 _
g (hsElus* + h3E3u3*),2/3
- h5E6u5 , aE6 - h5E ar = -2/3 5U
(5)
In Eqs. (6) g is the Brillouin gain coefficient, and we haveintroduced hl, h 3, h5, and h5 as empirical enhancementfactors, which will be unity in the case of perfect planewaves and may tend to 2 if speckle interference effectsenhance the gain of the conjugate wave.2 1 It is also pos-sible to introduce complex values for h, and h2 , which thenbecomes h1 (1 + ix1)/(1 + ix,), to include the effect of thedifferent x parameters for the four-wave mixing and theBrillouin amplification.
Overlapping Gaussian beamsWe consider Gaussian beams intersecting at angle sep
20 in the y-z plane. The two beams have axes y ±z sin 0 and intensities given by2 3
2Pi xp 2(x 2 + yi2)1Ii(r) = ex - j (6)
where Ii = I, and I2 for the pump and the Stokes beams,respectively, yi = y ± z sin 0, and the beams have ra-dius w given by co 2 = wo02(1 + Z
2/ZO
2). For a diffraction-
limited pump beam, zo = rW02/A, and the divergencebecomes Odiv = 2(o/zo = 2A/vrco. The growth-rate equa-tion aI1/az = h1gBIlI2 can be integrated first over the x-yplane and then over z = ±0, to give
P2(-°) = P 2(+o)exp ( h1gPz fo
= P2(+cc)exp(h1 g9IiLtfo),
where we have defined the mean pump intensity II, theinteraction length Lt, and the overlap parameter fo as
I = Pi/VWo , Lt = TZO,
fo = -J exp + 2 /(1 + u2 )du,71T J-c \ +
A = 2zo sin 0 20sep
coo Odiv°div = 2oo/Zo.
The parameter fo describes the degree of overlap of two foci. When 0 = 0, the two beams fully overlap, fo = 1. For significant angles of separation °sep betwEthe two beams, fo and the effective interaction length become
div2j/7r Osep
Leff = Ltfo.
(8)
theand*en
In Fig. 3 we plot the overlap parameter obtained fromEqs. (8) and compare it with the approximation in rela-tions (9). Experimentally we would not wish to let thebeams obstruct each other in the far field. This meansthat in practice we tend to operate with sep/Odiv 2 1.5,i.e., with fo s 0.2.
Theory of the Loop with an Internal Brillouin AmplifierWe model the experiment by considering a system of twocoupled nonlinear regions, an overlap region in whichboth four-wave mixing and Brillouin amplification takeplace and one Brillouin amplifier region. To avoid anexcess of minor variables obscuring the physics of theinteraction, we have lumped together all the regions inwhich Brillouin amplification occurs.
We take the interaction length for the overlap zoneto be L = Leff = foLt and for the amplifier to be LA =
(2 - fo)Lt; i.e., the amplifier region consists of two focalregions less an overlap region. We neglect the differencebetween the Stokes shift in the Brillouin amplifier andthe four-wave mixing region, although we discuss it againbelow. The boundary conditions become
El(r = 0, t) = Ei,
E5(rA = LA, t) = yiEl(r = L, t -ti)X
E3 (r = 0, t) = 72E5(rA = 0, t - t)exp(i0A),
E4(r = L, t) = 0,
E6(rA = 0, t) = y2E 4(r = 0, t - t2 )exp(io,),
E2(r = L, t) = yE6(rA = LA, t - t). (10)
In Eqs. (10) Eil is the input field, and 2 are theamplitude reflection coefficients for the mirrors on eitherside of the loop, t and t2 are the relevant transit timesfrom the overlap point on the first transit to the position ofthe second focus, where simple amplification takes place(t1 + t2 = t is the round-loop transit time), and fA =k1Ll00 p and fs = k2Lo0 p are the phase shifts around theloop for frequencies co, and (02, respectively. The opticalpath length of the loop is measured from the overlap pointof the first transit to the overlap point on the third transit.
Le)f/(7 i
Overlap Parameter f.
L 0 1 2 3 4 5
9eepIdiv = Xo/2wO
Fig. 3. Comparison of overlap parameter as function of Osep/Odiv(9) calculated from Eqs. (8) and (9): x separation of axes of beams
1 and 2 at the lens in Fig. 2; wo, beam radius at the lens.
Scott et al.
2082 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994
Growth Rate (1/tj)
-1i
6 8 10 Pump ntetIsity
Fig. 4. Growth rate p versus pump intensity M1 for threedifferent values of cavity transit time t. Curves correspondto fo = 0.2 and t, equal to 0.6, 6, and 60 ns, respectively, for thetop, middle, and bottom curves on the right-hand side.
Transient Behavior of the SBS loopFirst we calculate the transient behavior of the SBS-loop system by assuming undepleted pump beams, i.e.,El, E3 , and E5 are all constant when t > 0 and zerowhen t < 0. Equations (2), (5), and (10) can be solved byapplying Laplace transforms, decoupling, and combiningwith the boundary conditions to give
In our experiment AO changes by 2v when the loop pathlength changes by 11 cm.
The only free variables are s and x, which are bothreal. Applying the inverse transform, we can show thats is an exponential growth or decay rate for A2. Theintensity growth or decay rate for 12 is 2s, and we define adimensionless growth rate p and intensity parameter Ml:
p = 2srB; Ml = gIE12Lt. (14)
Figure 4 shows a plot of growth rate p versus intensityMl for three different values of cavity transit time t. Atlow intensity p tends to -1, corresponding to the normaldecay of any Stokes radiation in the loop. There is aninstability threshold above which p becomes positive. Inthis regime Stokes radiation grows exponentially in timeuntil effects such as pump depletion limit it. The fig-ure also illustrates that the instability threshold itself isindependent of the cavity round-trip time, although thegrowth or decay rate does depend on cavity transit time.Calculations also show that the four-wave mixing is es-sential for introducing an absolute instability, but whenfo = 0.2 the extra Brillouin amplification component re-duces the instability threshold by a factor of 3 comparedwith the threshold for four-wave mixing alone.
This absolute instability requires some initial noise to
be present so that radiation in the loop can build up, butunlike in conventional SBS the noise is no longer neededonce the process is under way, and the threshold is notdependent on the noise intensity. Figure 4 shows aninstability threshold of M = 2.3, a factor of 10 less thanthe conventional SBS threshold.7
The graph also shows that the time constant can tendto infinity near threshold, and the effect of the instabilitywill not be observed in practice unless spontaneous noisehas built up to become comparable in intensity with theincident pump within the duration of the laser pulse.By analogy with the conventional SBS threshold, we candefine an onset threshold that requires that the noisegrow by a factor of exp(25) within the duration of thelaser pulse.
For our experiment we have a loop transit time of6 ns, a phonon lifetime of 20 ns, and a pulse length of210 ns, corresponding to a required growth rate of p =84 ns = 2 .4/TB. In Fig. 4 this onset threshold is M = 5.In comparison the onset threshold for transient SBS willbe M = 25. It is possible to develop this further andconclude that there will be a constant onset energy thatis required at the start of a pulse to create the acousticwave, and this will characterize the transient behavior ofthe loop.
Figure 5 shows the instability threshold and the onsetthreshold as a function of round-trip transmission coeffi-cient T = y' 2y2 for fo = 0.1, 0.2. (The analysis impliesthat these are the principal parameters that character-ize the loop.) Not surprisingly, the threshold increasesas the overlap or transmission coefficient decreases. Al-though the instability threshold remains modest at thelowest values of fo and T, the onset threshold becomeslarge.
STEADY-STATE ANALYSIS
Wong and Damzen8 analyzed the steady-state perfor-mance of the loop scheme. It is impractical to analyzethe behavior of the full system described here, and in-stead we have reduced Eqs. (2), (5), and (10) to a steady-state form and integrated them numerically for a fewspecial cases. This predicts a Stokes output that is zerobelow the instability threshold, has a sharp onset, risesrapidly with increasing pump power, then asymptoticallyapproaches a straight line, and has a slope efficiency of100% and an apparent threshold slightly below the insta-
Intensity gLInstability and Onset Threshold, fe = 0.1, 0.2
14
12
10
8
6
4
2
I . Loop Transmission0 0.2 0.4 0.6 0.8 1
Fig. 5. Instability threshold (lower two curves) and onset inten-sity for a 210-ns pulse (upper two curves) versus loop transmis-sion coefficient for fo = 0.1 and fo = 0.2.
Scott et al.
31
Vol. 11, No. 10/October 1994/J. Opt. Soc. Am. B 2083
bility threshold. This in turn indicates that the instabil-ity threshold provides a good estimate of the steady-stateintensity threshold.
EXPERIMENT
The experiments were carried out at 2.12 Ium with aQ-switched Ho:YAG laser and the apparatus shown inFig. 6. The flash-lamp-pumped Ho laser had a 90-cmoptical-path cavity with a -2.5-mm diameter transverse-mode-selecting aperture. The 2.1207-,um line was se-lected by a birefringent filter and was narrowed to-0.6 GHz by two intracavity uncoated 6talons. The cav-ity was Q switched with an acousto-optic modulator toproduce pulses of as much as 25 mJ with a duration of-210 ns at 1 Hz. The output pulse showed strong axial-mode beating with typically 50% modulation and corre-sponded to approximately six independent modes. Thismode beating was highly uniform throughout the pulse,varied from shot to shot, and was averaged out digitally.The mode beating was neglected in our modeling studies.
The output of the laser was directed through a telescopeto a half-wave plate followed by a polarizer consistingof two ZnSe Brewster plates, permitting the beam to beattenuated by rotation of the half-wave plate. A quarter-wave plate produced circulary polarized light and acted asan isolator to reduce feedback into the oscillator with anextinction coefficient of approximately 10:1.
The beam was directed through a CaF2 wedge for diag-nostics to the phase-conjugate mirror. The diagnosticsconsisted of pyroelectric calorimeters and Au-doped Gedetectors, which monitored the energy and intensity of theinput and backscattered beams and the radiation trans-mitted by the Brillouin medium. The absolute powersensitivity of the fast detectors could be obtained by in-tegrating the voltage signal and calibrating this with theenergy monitors.
As shown in Fig. 6, the laser beam was directedpast mirror Ml to the first lens and focused into theBrillouin cell. The cell was sealed with CaF2 win-dows and filled with CS2 (Aldrich high-performanceliquid-chromatography grade). The length of the ac-tive medium was 10.7 cm. The transmitted beam wasrecollimated by the second lens and directed to dielec-tric mirror M2 having 96% reflectivity. The mirror wastilted a few degrees from normal incidence and returnedthe beam to form a second focus a few millimeters to theside of the first. The transmitted beam was recollimatedby the first lens and directed onto mirror Ml, a Au-coatedGe 6talon with a reflectivity of 98%. The reflection fromthis mirror was refocused by the first lens and aligned toensure overlap of the first and the third focus.
Alignment was straightforward. Mirror Ml will bealigned when the beam passing through the lens on thethird pass is parallel to and has the same collimation asthe input beam. When mirror Ml is partially transmit-ting and intercepts the input beam as well as the beampassing around the loop, then one can align Ml by ensur-ing that the beam passing around the loop is collimatedand parallel to the part of the input beam reflected byMl. This can be done by ensuring that, if the beams arefocused by the same lens, their foci will lie in the sameplace. This is the principle behind our methodology.
Mirror Ml was a Au-coated Ge talon on a kinematicmount and was used only because the alignment tech-nique relied on the fact that the two surfaces of themirror were parallel. Initially the etalon-mirror wasnot present, and the input laser beam passed around theloop and returned to the CaF2 beam splitter, where it wasreflected and refocused by a third lens onto a graphitetarget, where weak breakdown could be observed at thefocus. Next mirror Ml was placed in front of the inputbeam so that the input beam reflected back to the beamsplitter without going around the loop. The spacingbetween the two lenses around the SBS cell was thenadjusted to ensure that when the beam passed around theloop its focus was in the same plane as when the beamwas reflected by the mirror and did not go around the loop.Mirror Ml was then tilted until the focal spot was alignedon the point where the focus was when the mirror wasabsent. The mirror was then translated across so that itdid not block the input beam but reflected the beam afterit had passed around the loop, and the mirror returnedthe beam to the Brillouin cell. Since the front andthe back of the mirror are parallel, the procedure leadsto the beams' overlapping in the Brillouin cell. Finaladjustment was made of mirror Ml while the energy ofthe backscattered beam was monitored. This procedurewas found to be quick, reproducible, and convenient.
ENERGY MEASUREMENTS
The Stokes output energy was measured as a functionof input energy and cross calibrated with a 94% reflec-tivity Cu mirror placed immediately in front of the loop.Measurements were made for the loop and also with thepump beam blocked after it had passed through the cellonce or twice, corresponding to one- or two-focus SBS.The results are given in Fig. 7, with the Stokes outputcharacterized by a threshold and a slope efficiency. Theslope efficiency was 91% for the loop, 87% for two foci, and72% for the single-focus case. The threshold for conven-tional SBS was 10.5 mJ, and this dropped to 8.2 mJ for 2-focus SBS. The threshold for the loop was 4.8 mJ. The
2084 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994
x Total reflector
Loop-e-2-focus
0° -'-1 0 _ .. -f oc s
5 .I..........0
0 5 10
Input energy (mJ)Fig. 7. Plot of reflected energy versus input energy.
80
§~ 60
a)~: 400cL
cm0C' 20
0 JI0 100 200 300 400 500 600 700 800
Time (ns)
Fig. 8. Typical temporal profiles of beams, showing input,transmitted, and backscattered powers. The traces wereelectronically smoothed (averaging the effect of mode beating) todisplay essential behavior.
maximum reflectivity was 64% for the loop, 36% for2-focus SBS, and 21% for single-focus SBS.
Using the single-focus SBS threshold as a bench-mark, we find that the two-focus SBS scheme reducesthe threshold by 1.3 and the loop reduces the thresholdby a factor of 2.2. This includes effects that are due toboth transient and steady-state operation and cannot becompared directly with theory.
Power MeasurementsA typical set of traces for the loop geometry is shownin Fig. 8. The transmitted beam has an initial tran-sient spike followed by a region in which the power is al-most constant. This corresponds to the instability powerthreshold calculated where. The backscattered powerswitches on suddenly and follows the pump power. Thetransmitted pulse is highly reproducible, and it plus theconjugate trace sum together to match the input traceto within a few percent over the whole pulse, indicatingthat all the power is accounted for. The temporal data,as in Fig. 8, can be used to calculate the reflectivity as afunction of time, which reached 90% at the highest pumppowers.
In contrast the set of traces for a misaligned loop, asin Fig. 9, shows that the transmitted and backscatteredbeams are modulated and irregular. This irregular, ran-domly varying pulse shape was also characteristic of all
experiments in which conventional one- and two-focusSBS was observed and was attributed to the effect of ran-dom phase fluctuations in the noise that seeds conven-tional SBS.23
Another way of presenting the data is to plot the outputintensity as a function of input intensity as in Fig. 10.This approach, which converts the two traces into aLissajous figure, was first used by Zel'dovich et al.21 Thecurve is traced in two passes: first on the rise of the in-
_ put and again on the fall, with the transient behaviordisplayed during the initial part of the rise and appar-ent steady-state behavior apparent during the rest of thepulse. Figure 10 shows traces for single-, two-focus, and
_ loop SBS; care has been taken to select traces with a1 5 minimum of modulation for the single- and the two-focus
cases. The initial transient behavior corresponds to aregion near the origin in which the output is zero whilethe input is increasing. After the Stokes beam is estab-lished, the curves all lie close to a straight line with aslope efficiency of near 100%. The single- and two-focustraces were highly variable from shot to shot, but the
- trace for the loop was highly reproducible.Three sets of data for the loop taken at substantially
ed different input energies were overlaid, and after the tran-sient regime they were found to fit the same curve to a re-markable degree of precision. Close to threshold, wherethe signals are small, the curves were remarkably simi-lar and provided an accurate means of measuring the in-stability threshold. At higher intensities there is more
C3
00
0
a.
Time (200 ns/div)
Fig. 9. Temporal profiles, showing slow modulation in reflectedand transmitted beams for a misaligned loop. The traces wereelectronically smoothed. The modulation varied irregularlyfrom shot to shot and was typical of all cases in whichconventional SBS was observed.
50
g 40U,
LI30
0a-0 10co
00 10 20 30
Input Power at focus (kW)
Fig. 10. Plot of reflected power versus inputdifferent geometries.
40 50
power for three
Scott et al.
Vol. 11, No. 10/October 1994/J. Opt. Soc. Am. B 2085
Fig. 11. Plot of experimental onset energy for SBS for a one- anda two-focus and a loop geometry as a function of input energy.The data were obtained by integration of the input pulse overtime until the SBS signal reached 50% of its peak value.
spread in the data. The reasons for this are not certainbut may be associated with self-focusing in the CS2.
This experimental curve of the loop steady-state regimeis qualitatively similar in character to the modelingresults mentioned above except close to the instabilitythreshold, where the model predicts a small region witha slope of greater than 100%. This minor discrepancycorresponds to the regime in which the time constanttends to infinity, so the acoustic wave and the reflectivitywill change slowly and not reach full equilibrium. Thismay account for this slight discrepancy.
These curves can be used to measure the steady-statepower threshold for the various forms of backscattering.The power thresholds are 7, 15, and 22 kW for the loopand the two- and single-focus SBS, respectively.
The simple SBS threshold may be calculated withEq. (8), assuming a diffraction-limited beam and a gainof 65 cm/GW (Ref. 24). This leads to a predicted powerthreshold of 24 kW, close to that observed.
The threshold-reduction factor for two-focus SBS is 1.5,which is less than the value of 1.8 that would be expectedfor two cells with 20% transmission loss. The laser op-erated on several axial modes, so it was expected that theoptical path between the two foci should be set to the lasercavity optical path length to within a small fraction of theoverall coherence length (L,0h = c/Av 50 cm). We metthis condition in most experiments and investigated thedependence in a separate series of studies but found nosignificant effect when the focal separation was varied.
The theory and results of Fig. 5 predict a steady-statethreshold for the loop. In the case of a round-trip loss of64% and an overlap parameter fo of 0.1-0.2 this shouldbe M = 2.4 to M = 3, corresponding to a threshold reduc-tion of 8-10. In practice we see a reduction by a factorof only 3 in the steady-state threshold.
We commented in the Theory section that the transientgrowth rate implied an onset energy at the start of thepulse before the appearance of any backscattered beam.We can measure the onset energy by numerically integrat-ing the input intensity up to the time when the transientspike is seen to end. In Fig. 11 we plot onset energy ver-sus input energy. It can be seen that the onset energyis indeed roughly constant. This appears to be the mostreliable way of characterizing the transient behavior ofthe loop.
Studies of the onset energy for conventional SBSshowed that it is also characterized by a constant on-set threshold approximately independent of pump energy(Fig. 11). The ratios of onset energies for the loop andthe two-focus SBS to that of single-focus SBS are 2.3and 1.3, respectively. We note that Fig. 5 implies a theo-retical onset threshold for the loop of M = 9 to M = 11for the 210-ns pulse duration, compared with M = 26 forconventional SBS; i.e., the loop should exhibit a reductionin the onset threshold of 2.4-2.9. This does agree withthe amount of threshold reduction seen here.
Experimentally we also investigated varying the cross-ing angle between the beams. There was only weakvariation in the power threshold with angle, and thisagreed with the theoretical model. There are some ob-servable differences in the traces. At large angles thetransmitted and the backscattered beams are smooth, butfor small angles a characteristic modulation was observed.This has been suggested2 5 26 as being due to forward Bril-louin scattering, and the modulation frequency is of thecorrect order of magnitude.
Earlier we discussed the longitudinal mode separationgiven by AkAL = 2, where Ak = k2 - k and AL isthe change in loop length between longitudinal modes.For our experiment the calculated change in loop lengthfor a change of one mode was 2.8 cm. We varied thelength to be 164, 167, and 169.5 cm. When the loop was164 and 169.5 cm long, the SBS and throughput powerswere as shown as in Fig. 8. When the loop was variedto be 167 cm, the transmitted power was modulated asshown in Fig. 12, and the SBS was also modulated. Thebeat frequency was observed to be 14 MHz. This modu-lation was previously observed by Anikeev et al.2 7 andwas attributed to the excitation of higher-order transversemodes.
BEAM QUALITY
We characterized our laser beams by making measure-ments in the near field and in the far field, measuringthe latter at the focal plane (not necessarily the beamwaist) of a 4.01-m focal-length mirror and using the rela-tion Df = f OD, where Df is the le 2 diameter at the focalplane. We used a pyroelectric array to produce an imageof the beam and also measured the transmission througha series of calibrated apertures. The latter (energy-in-a-
0.9
0.8
X 0.7
, 0.6
S3 0.5
8 0.4
c 0.3
0.2
0.1
0100 200 300 400 500 600
lime (ns)700 800 900 1000
Fig. 12. Input and throughput power when the loop length isadjusted to make kL = (2n + 1)XT. The modulation frequencyis approximately 14 MHz.
Scott et al.
2086 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994
o 0.8
U) ,V.U 0.6
c: 0.4 _____AetX- Calculated (0.97,3.5)
tion-imitd inut. easued tansmssio thMesruhae ageo
fo 0.2 ide-l Fidelitya)
0.0
0 1 2 3 4 5 6Aperture diameter (mm)
Fig. 13. Near-field output from the oop in the case of diffrac-tion-limited input. Measured transmission through a range ofcalibrated apertures is shown and compared with that predictedfor an ideal Gaussian beam. The fidelity is defined as the ratioof the measured transmission for the backscattered beam to thetransmission of the input beam. The far field is similar indistribution.
bucket) technique was more sensitive in detecting devia-tions from a Gaussian profile at the edge of a beam.
The power transmission of a Gaussian beam with ra-dius wo through a centered circular aperture of radius rois given by
T(ro) = 2 | 2 (rrjexp 2)dr
= I1- expt v (15)
We fitted data to this formula to determine the beamwidths from a series of transmission measurements.These were compared with the theoretical diffractionlimit given by ODD = 4A/7r, or DfD = 10.8 mm 2 .
According to this technique the input beam in the nearfield had 97% of its energy in a Gaussian distributionwith a value of 2 wo = 3.5 mm. In the far field 98% ofthe energy was in a Gaussian profile with 2w) = 3.7 mm,corresponding to 1.2 times the diffraction limit.
When the loop was operated, 97% of the backscat-tered energy in the near field was in a Gaussian beamwith 2cw0 = 3.6 mm (Fig. 13). In the far field 97% of theenergy was in a Gaussian profile with 2w) = 3.4 mm, cor-responding to 1.1 times the diffraction limit. These mea-surements were reproducible both close to threshold andwell above threshold.
For comparison similar measurements were made withthe single-focus SBS. The near field was Gaussian with2w0 = 4.3 mm, and in the far field 97% of the energy wasin a Gaussian profile with 2 = 4.0 mm, correspondingto 1.6 times the diffraction limit.
Next an aberrator plate (a NaCl window that had beenwet with water and roughly repolished) was placed imme-diately in front of mirror Ml. The aberrator was charac-terized by double passing it with a Cu mirror, and in thefar field a highly structured speckle pattern was observedwith an overall divergence angle of 4.2 mrad, correspond-ing to five times the diffraction limit. When the loop-SBSsystem was used to double pass the aberrator, -78% ofthe energy was observed to be in a Gaussian profile witha divergence angle of 1.12 mrad, or 1.3 times the diver-gence limit. The fidelity of the conjugate beam, taken
as the ratio of the transmission through a far-field aper-ture of 1.7 mrad divided by the transmission of the inputthrough the same aperture, was 65% (Fig. 14).
When the aberrrator was used, the input power wasinsufficient to reach threshold for one- or two-focus SBS.
CONCLUSION
We have obtained SBS in CS2, using a frequency-narrowed Cr,Tm,Ho:YAG laser operating at 2.12 mm.We demonstrated single-focus SBS and threshold reduc-tion, using two-focus SBS and a novel loop-SBS scheme.A theory of the threshold-reduction schemes was devel-oped that included the effect of overlapping Gaussianbeams and described their transient and steady-statebehavior.
The analysis for the loop implies that the system ischaracterized by an initial exponential growth regimethat is well characterized by an onset energy followed bya steady-state regime in which the conjugate power de-pends only on the pump power. This was qualitativelyconsistent with our experimental results.
Experimentally the transient behavior in all cases waswell characterized by an onset energy that remained con-stant over a large range of input energies. For con-ventional SBS the backscatter power during the pulsevaried rapidly with time, but in the loop scheme the out-put was highly stable, and there was a fixed power thresh-old. The threshold for conventional SBS was close to thetheoretically predicted value based on the known gain forCS2, and the degree of threshold reduction for two-focusSBS was 1.5, -15% less than expected. The loop schemewas predicted to reduce the onset energy by a factor of 3and to reduce the power threshold by a factor of 8-10.In practice the onset threshold was reduced by the pre-dicted amount, but the power threshold was reduced byonly a factor of 3.
The reason for this outcome is not clear, but it is asystematic discrepancy between theory and experiment.We note that it is consistent with the reduction fac-tor observed by other workers.27 In particular our di-agnostics have separated the transient and the steady-state effects, so we can identify the discrepancy as beingin the steady-state regime. The steady-state theory ofWong and Damzen8 and our own numerical studies in-dicate that the steady-state power threshold should, ifanything, be less than the instability threshold. Exper-imentally our conditions are well defined and appear to
1.0._
,0 0.8
g 0.6.e
E 0.2
0a, 0.2
0.< 0.00 1 2 3 4
Aperture diameter (mm)5 6
Fig. 14. Far field in the case of an input beam aberrated by afrosted salt plate to six times the diffraction limit.
Scott et al.
Vol. 11, No. 10/October 1994/J. Opt. Soc. Am. B 2087
be well matched to the analysis. We are using nearlydiffraction-limited Gaussian beams, and all losses in theloop are measured. We are using multiple-frequency in-put beams (with approximately six modes present), butwe have taken care to match the path length around thecavity to the path length around the resonator, so thatlight that has passed around the loop once is coherentwith the light that it may interfere with. The thresholdfor self-focusing in CS2 is above the power levels that wepredict for the loop threshold, and this threshold powerwould vary, depending on the mode beating in a givenpulse. The detuning between the Brillouin shift for theconventional Brillouin amplifier and the four-wave mix-ing process is less than a Brillouin linewidth. However,we noted above the size of the fluctuations, which werelarger than is typical for conventional SBS, and it is pos-sible that thermal Brillouin scattering somehow modi-fied the threshold.28 (Absorption by the CS2 was suffi-cient to produce thermal lensing after each pulse.) Inthis case a direct comparison of the loop and conventionalSBS will not be an accurate measure of the degree ofthreshold reduction resulting from the loop in isolationof other factors. One possible weakness is the approxi-mation used for the overlap factor at the lens focus. Wehave also carried out experiments with a ring scheme.We get the same discrepancy between theory and experi-ment in that case even though there is no partial overlapof beams.
One noticeable difference between the loop and conven-tional SBS was that the conventional SBS was stronglymodulated because of fluctuations in the initiating noise.In the loop scheme noise is required for starting theprocess but is not required once the process starts.
The phase shift round the loop AkL needed to be a mul-tiple of 27r to produce a conjugate with a low threshold andsmooth temporal profile. The beam quality of the conju-gate was carefully analyzed and shown to be close to thediffraction limit for the loop but 1.6 times the diffractionlimit for conventional SBS. This is consistent with theaccepted model of SBS phase conjugation, in which thebeam inhomogeneities are needed to produce speckle gainenhancement. A simple calculation following Eq. (6) inthis paper assuming input noise over a range of angleswould imply that a diffraction-limited input beam wouldproduce an SBS output with a divergence of 1.4 times thediffraction limit.
Our loop scheme is somewhat different from manypreviously described loop schemes in that we use relayimaging, so the intensity distribution formed on the firsttransit through the SBS cell is replicated on the finaltransit. There are relatively few aberrations in the loop(and the principal one may be the thermally inducedaberration that is due to absorbtive heating of the CS2).The mechanism that produces a conjugate wave frontin preference to a nonconjugate one is not immediatelyobvious, but the experiments show that the fidelity isvery good.
One key factor is that the Brillouin gain of a conjugatebeam is enhanced 2 l when the pump beam is aberrated.This leads to a somewhat lower threshold for a conjugatebeam than for any other in the loop scheme. Modelingsuggests that the degree of reduction in threshold ismodest; if the gain coefficient is enhanced by a factor
of 2 in the amplifier, then the threshold is reducedby approximately 20%.
Gain enhancement occurs in conventional SBS and re-sults in noise that is conjugate to the pump, experiencinghigher gain than the nonconjugate noise. However, thediscrimination is imperfect, and the output is not perfectlyconjugate. In the case of the loop the system is analogousto a resonator. In this case the spatial mode that is con-jugate to the input has a slightly lower threshold thanany other mode, and this will lead to the lower thresh-old mode's oscillating and preventing any other modefrom resonating, even though its threshold is only slightlyhigher. This leads to better discrimination and hence tobetter fidelity.
ACKNOWLEDGMENTS
We thank Barry Feldman and Steven Bowman for helpfuldiscussions and use of equipment. This work was fundedby the Strategic Defense Initiative Office.
*Present address, Defence Research Agency, Malvern,Worcestershire WR14 3PS, UK.
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