Polarization-dependent phase lockingin stimulated Brillouin scattering systems
X. Hua and Joel Falk
Measurements of the mutual coherence of the output beams from a seeded, two-pump-beam, stimulatedBrillouin scattering system are reported. Mutual coherence depends on the relative polarizations of thepump beams and the seed beam. A seed beam can phase-lock the Stokes outputs even if the pump beamsare orthogonally polarized. Four-wave mixing is responsible for this phase locking.
Stimulated Brillouin scattering (SBS) has been inves-tigated during the past decade as a means of lockingtogether the optical phases of multiple laser beams.1-7Phase locking is necessary if multiple, small-aper-ture, low-power laser beams are to be combined into alarge-aperture, high-power laser beam. Valley andco-workers'3 and Sternklar et al.7 have shown that iftwo pump laser beams are well overlapped in theirfocal volumes then they produce Brillouin scatteredoutputs that have essentially the same optical phases.Loree et al. 4 examined phase locking in a seeded SBSsystem, i.e., a system where a low-level, Stokes inputis introduced to initiate the scattering process. Hefound good locking of the Stokes outputs if the twopump beams were overlapped by the seed beam.Hua et al.8 and Chu et al.9 suggested that themechanism for phase locking in both the seeded andunseeded SBS systems is four-wave mixing of thepump and Stokes beams. This paper is designed toelucidate locking mechanisms in seeded SBS systems.
Two strong pumping beams, E1 and E3, and a seedbeam counterpropagating to E1 are incident upon aBrillouin active medium (Fig. 1). The pumpingbeams are assumed to be undepleted by the SBSinteraction that produces two backscattered Stokesoutputs, E2 and E4. The pumping beams are par-tially overlapped so that both input beams interact
X. Hua is with the Department of Physics and Astronomy,University of Pittsburgh, Pittsburgh, Pennsylvania 15260. J.Falk is with the Department of Electrical Engineering, Universityof Pittsburgh, Pittsburgh, Pennsylvania 15261.
with both Stokes beams. The angle between thepumping beams is small but nonzero.
In this paper we examine the degree of phaselocking between the Stokes outputs. In particular,we examine the dependence of phase locking on thepolarizations of seed, pump, and output Stokes beams.We define a mutual coherence function S, whichmeasures the degree of phase locking between E2 andE4. The function S quantifies the degree of spatialand temporal coherence between the two outputs andis defined by
S E2iI E 4 Icos 4)d2rdtSi = (1)(5 E 2i 12d2 rdt I E4i 12d2rdt)
The amplitudes of the Stokes outputs E2 and E 4 aremeasured in a plane where the two outputs areoverlapped. The angle 4) is the temporal phasebetween E2 and E4. The integrals are over spatialcross section (d2r) and time (dt). 8 9 In some experi-ments the coherence function is different for theStokes polarizations parallel and perpendicular to aseed beam. To indicate this polarization dependencewe have written Si, E2i, and E4i, where i = x, yindicates the polarization of the Stokes fields recorded.This coherence function, except for this polarizationdependence, is identical to one discussed earlier in theliterature. 8
We examine the probability density function (PDF)of S for seeded SBS systems for various polarizationsof pump and seed beams. The seed beam is polarizedparallel to one pump beam, and the pump beams areeither polarized parallel or perpendicular to each
Fig. 1. Pump beams El and E3 produce Stokes beams E2 and E4 ina Brillouin active material. In some experimental situations, alow-level Stokes seed beam is backinjected along E 2.
other. The seed beam sets the temporal phase foreach Stokes beam that it drives. For example, lock-ing is observed when both pump beams and the seedare polarized in the same direction. For orthogo-nally polarized pump beams, a different behavior isobserved for each of two perpendicular Stokes polar-izations; locking is observed for one output polariza-tion but not the other. The experiments reportedhere help to elucidate the physics of phase locking andthe role of four-wave mixing in multibeam SBS.
Experiments
SBS was produced in ethyl alcohol by a frequency-doubled, Nd:YAG (532-nm), 7-ns, 10-pulse/s laser.The laser ran single-longitudinal mode on a signifi-cant fraction of its output pulses, and our data-collection system was gated so that data were recordedonly when the laser produced the narrow-bandwidth,single-mode output. As shown schematically in Fig.2, we used a phase grating to split the incident laserbeam into many orders.5' 8 Two grating orders werefocused into a cell 2 cm in diameter and 25 cm inlength that contained ethanol. A half-wave plate(X/2) was sometimes used to rotate the nominallyhorizontal (x) polarization of pump beam E3 by 90°.A horizontally (x) polarized seed beam was generatedin an auxiliary SBS cell. In some experiments the
polarization of the seed beam was passed through asecond half-wave plate and the Stokes beam E2 wasseeded with a vertically polarized (y) input.
The two SBS Stokes return signals were recom-bined in the grating. The energy in the vertical andhorizontal polarizations for each of three outputorders from the grating was measured on a largenumber of pulses. The detection system used silicondetectors, gated amplifiers, analog-to-digital convert-ers, and a microcomputer. These measurements,along with measured grating parameters, permittedthe determination of Si [Eq. (1)] on each laser pulse.5'8'9To form the probability distribution of S, we calcu-lated its value for each of 103-104 laser pulses andsorted these measured values into 200 bins equallydistributed between + 1 and -1. A histogram of thenumber of pulses in each bin is proportional to thePDF of S. Consequently, the plots of fractionalnumber of counts versus bin number are in effectplots of the PDF of S.
Experiments with an unseeded system and withpump beams polarized parallel or orthogonal to eachother have been reported in Ref. 8. For the seededconjugator, with both pump beams and the seed beampolarized horizontally (x) and parallel to one another,the measured distribution of S is shown in Fig. 3.The narrow distribution of S indicates strong phaselocking. The distribution is not centered near bin200 (S = +1) because of the poor spatial overlapbetween the seeding Stokes beam and the pumpbeams. Although the foci of the pump and seedbeams had a common center and occurred in roughlythe same plane, the pump beam's radius (50 tim) wassubstantially smaller than that of the seed beam (500Pm). Poor overlap causes poor conjugate fidelity,which results in poor spatial coherence between E2
0.10
SBS cell
zz'1
(n.4-C
0L3
Fig. 2. Schematic of the experimental apparatus. The phasegrating is used to create two 532-nm, 0.4-mJ, 7-ns pump beamsfrom the laser's output. The prism equalizes the path lengths forseed and pump beams. The thin-film polarizer (TFP) separatesthe Stokes outputs into two polarizations. The vertically (y)polarized Stokes output is detected on channels 1-3 (CH1-CH3)and the horizontally polarized output is recorded on channels 4-6(CH4-CH6). P, P2, polarizers; L1-L5, lenses; BS-BS 3 , beamsplitters; R1-R4, highly reflecting mirrors; and /2, half-waveplates.
0.05[
00 200
Bin NumberFig. 3. Distribution of S for the dual-pump-beam, seeded SBSsystem. The pump and seed beams are all polarized in the same(horizontal) direction.
and E4, which moves the recorded distribution awayfrom S = + 1. However, good phase locking (asdenoted by the narrow width of the distribution) ismaintained. Seeded SBS systems behave as seedamplifiers rather than as phase conjugators, and thusmultiple SBS outputs are not nearly spatially identi-cal. 10
For pumping beams polarized perpendicular toeach other, i.e., El polarized in thex direction (horizon-tal) and E3 polarized vertically (y) and a seed B2polarized in the y direction, the measured distribu-tion depends on the Stokes polarization observed.The equations governing the process, given in Ref. 8,can be simplified by taking a Laplace transform of theequations and writing the resulting equations in the sor Laplace domain. The resulting equations can beseparated into expressions for components E2x, E2y,E4x, and E4y, and the x and9 components of the Stokessignals incident upon the silicon detectors can becalculated. The equations become
(k2 V)E2X = s + [E1*E2 .(S)]E1-,
(j. V)E2y9 = KK 2(k2 ''~y + f [Ei*E4.(S)]E3 exp i9, (3)
(k4 V)E 4 ye =
(k4* V)E4x =
S + [E3*E4y(s)]E3 :e,
Kl + [E3*E2y(s)]El exp i9.
Note that the polarization of Stokes beam 4 isrotated by the half-wave plate and the polarizationdirections x and 9 written in Eqs. (2)-(5) refer topolarizations at the detectors.
All the x polarized components, Eqs. (2) and (4), aredescribed by and arise from normal SBS. Equations(2) and (4) contain no common terms and thus E2X andE4y are not coupled together. The measured distribu-tion of S,, i.e., the distribution for the x polarizedStokes outputs, is shown in Fig. 4. These Stokescomponents have no means of exchanging phaseinformation, and neither phase locking nor any influ-ence of the seed beam on Sx is observed. S,, is nearlyuniformly distributed over the range -1 to + 1.
The 9 components are due to four-wave mixing.Equations (3) and (5) are coupled linear differentialequations. The 9 polarized outputs, E2y and E4x, areboth driven by the input seed and thus both outputscarry its optical phase and are locked together.Figure 4 also shows the distribution Sy, i.e., thedistribution for the Stokes y polarization. Theseoutputs have optical phases that are set by the seedbeam and a narrow distribution of S results. Poorspatial overlap between the seed and pump beams
0.10
z1-z
cU4-C
0C-)
0.051
C200
Bin NumberFig. 4. Distribution of S for a seeded SBS system. The pumpbeams are polarized perpendicular to one another and the seedbeam is polarized parallel to E3. The open-square data show S.,the distribution found from the horizontally (x) polarized light onthe detectors. The open-circle data indicate Sy, the distributionfound from the (y) vertically polarized light on the detectors.
results in a peak well removed from S = + 1 (bin 200).However, the sharp peak and small distributionwidth indicate good phase locking.
Conclusions
We have examined phase locking in SBS systems,especially in seeded systems. If the polarizations ofseed and pump beams are the same, then the Stokesoutput phase can be locked to that of the seed beam.If the polarizations of the pump beams are orthogonaland the seed is polarized along the polarizationdirection of one pump beam, then there are substan-tial differences in the degree of phase locking ob-served in the two output Stokes polarizations. Theoutputs in one polarization are coupled together byfour-wave mixing and are thus phase locked. TheStokes outputs with the orthogonal polarization buildup as normal SBS beams, are uncoupled, and are notphase locked.
The work supported in this paper was sponsored bythe U.S. Air Force, Air Force System Command,Phillips Laboratory, Kirtland Air Force Base, NewMexico.
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