1. INTRODUCTIONThe goal of this investigation is to provide the dataneeded for the design of a solid-state laser system that re-lies on phase locking and phase conjugation to generate ahigh-brightness beam. As shown by previous work,1–4 apromising scheme for such a laser system consists of alow-energy master laser and several optical amplifiers op-erating in a parallel arrangement. The beam of a masterlaser is divided to get a bundle of subbeams. Each ofthese beams is amplified in a first pass through an opticalamplifier. After reflection on a phase-conjugating andphase-locking element, the subbeams retrace their opticalpath and recombine to form a high-energy, nearlydiffraction-limited beam. With phase conjugation, phasedistortions introduced by the amplifiers are corrected,while phase locking ensures that the recombined sub-beams exhibit no relative phase shift, i.e., the pistonphase error of the subbeams vanishes. Depolarizationcorrection can be incorporated with the scheme presentedby Basov et al.5
In the visible and near-infrared, stimulated Brillouinscattering (SBS) is an effective means to obtain phase-conjugated light. By overlapping the subbeams in a com-mon SBS medium, phase locking is achieved, addition-ally.
Phase locking as a function of various properties of thebeams involved was investigated by Carrol et al.,6 Falket al.,7 Sternklar et al.,8 and Moyer et al.9 This investi-gation extends the phase-locking database by providingfurther details of the relationship between beam energies,energy ratios of the beams, beam overlap geometry, andphase locking.
2. EXPERIMENTAL LAYOUT ANDDIAGNOSTICSA scheme of the experimental setup used in this investi-gation is shown in Fig. 1. The master laser is a flash-lamp pumped Nd: YAG laser operating at 1064 nm. In-jection seeding with a monolithic Nd: YAG ring laserprovides single-longitudinal-mode operation. Themaster-laser radiation is linearly polarized and has apulse width (FWHM) of 17.3 6 0.5 ns, a pulse energy asgreat as 50 mJ, a pulse repetition rate of 10 Hz, a coher-ence length of more than 3 m, and a beam quality of M2
less than 1.1. Broadband operation is not investigated.An optical isolator consisting of two polarizers and a
Faraday rotator is used to avoid damage to the master la-ser by pulsed light that is returning from the SBS cell andis transmitted by the thin-film polarizers (TFP). With abeam splitter the master-laser beam is divided into twoidentical subbeams. Beam a propagates through amp a.A half-wave plate is used to attenuate beam b. Thereaf-ter, the beams are transmitted by the two TFP’s that onthe return pass separate the amplified beams from the in-put beams. By passing twice through the quarter-waveplates, the amplified beams are polarized perpendicularto the input beams.
The two input beams are focused into a high-pressuregas cell that contains nitrogen at 75 bars, which results inan SBS-threshold energy of Wth ' 17.2 mJ. The SBScell consists of a stainless steel tube (36-mm inside diam-eter, 2-mm wall thickness, 0.9-m length) with BK7-glasswindows (9.5-mm thickness, both sides AR coated) at-tached to each end and sealed by grommets. This SBScell was operated in sealed off mode for at least 5 3 106
pulses without any observable changes in performance.This is in contrast to preliminary experiments with pres-surized SF6 in which the windows of the cell were con-taminated with fluoride and subsequently destroyed afterless than 106 pulses. With pressurized nitrogen it wasobserved that, if optically induced dielectric breakdownshave occurred inside the SBS medium, the performance ofthe SBS cell is reduced as well as the threshold energy foran optically induced dielectric breakdown. But the ini-tial performance is recovered after some hours without la-ser operation.
The interference between the output beams is recordedby a CCD line camera.
Carroll et al.6 and Moyer et al.9 show that phase lock-ing is nearly independent of the f-number of the focusinglens. Therefore this investigation is limited to a singlef-number that was chosen to be large enough to com-pletely avoid optical breakdown of the SBS medium.
Inside the SBS cell both beams have a beam-waist di-ameter of ds0 5 260 mm, a full divergence angle of Us
5 5.7 mrad, and a distance of z0 5 0.82 m between thelens plane and the beam waist. The beam diameter inthe plane of the focusing lenses is ds(0) 5 4.7 mm. TheRayleigh range is zR ' 46 mm. The beam waist is lo-cated approximately in the center of the SBS cell. Mea-surements of the beam parameters were taken accordingto the draft of the standard ISO/DIS-11146:1995.10 Thetemporal behavior of the pulses is recorded by a digital-storage oscilloscope.
Inside the Brillouin cell the interaction volume of thetwo beams is characterized by the following parameters(see Fig. 1): the separation of the beam centers in theplane of the lenses (NFS), the separation of the beam cen-ters at the beam waists measured parallel to the tabletop(FFS), and the separation perpendicular to FFS (FFS').In some experiments the focusing lens of beam a was
Fig. 1. Experimental setup used in this investigation. Circledinset, beam overlap region viewed from above; quadratic inset,cross section of the beam waist. BS, beam splitter with 50% re-flectivity; TFP’s, thin-film polarizers.
translated along the beam propagation path, thus intro-ducing a separation of the beam waists in the beam-propagation direction (BWS).
FFS is negative if the beams cross behind the beamwaists, and BWS is negative if the beam waist of beam bis behind that of beam a. Here W3 /W1 is the energy ra-tio of beam b to beam a, measured in front of the SBS cell.
Changes in the relative phase of the two output beamswere measured interferometrically. The interferencepattern of each pulse arising from the two slightly tiltedoutput beams is recorded with a CCD line camera.Thereafter, a computer program is used to calculate theperiod and the phase of this interference pattern. Bysubtraction of the phase of the interference pattern of twoconsecutive pulses the absolute value of the relativephase difference d is obtained. This method ensures thatreflectivity fluctuations do not affect the results. Here dis identical to the change of the relative phase differenceof the two optical beams and ranges from zero to p.
When we measure d for more than 1000 pulses a prob-ability distribution p(d) is found (see Fig. 2). As given by
E0
e
p~d!dd 5 Ee
p
p~d!dd,
the value e of this distribution is a measure of the qualityof the phase locking of the two beams. Here e is normal-ized by introduction of the phase-locking coefficient
K 5 1 2 2e/p,
which ranges from 0 to 1 where K 5 1 denotes a per-fectly constant phase relation between the two beams andK 5 0 denotes no phase locking at all.
The injection seeding of the master laser was not per-fectly stable, and although the diagnostics rejected mostof the multimode pulses, this instability has to be recog-nized as the main source of experimental error. Residualnoise in the diagnostics (fluctuations of the output-beamoptical path length, noise of the CCD line camera, uncer-tainty of the diagnostic software) lends to e 5 l/67 (i.e.,K 5 0.94), even for perfectly phase-locked beams.
Fig. 2. Typical result of ;1000 interferometrical measurementsof the change of the relative phase difference d of the two outputbeams from pulse to pulse. For 50% of all pulses, d is smallerthan e. Here e is a measure of the quality of the phase locking(here e 5 l/31 corresponding to K 5 0.87). This measurementcorresponds to the marked data point in Fig. 7.
3. PHASE LOCKING AS A SEQUENTIALPROCESSStimulated Brillouin scattering is a process in which aweak signal wave is amplified by many orders of magni-tude by stimulated scattering in an electrostrictive me-dium. The signal wave is phase conjugated and slightlyfrequency shifted with respect to the pump beam. Nor-mally, the signal wave originates from scattering of thepump beam at a density fluctuation of the medium. As aconsequence, an arbitrary phase shift occurs betweenpump and signal wave.
The phase difference d between two signal beams gen-erated by two mutually coherent pump beams vanishes ifthe four-wave mixing process described below is success-fully implemented.
The electrical fields of the pump and signal beams maybe given by
E1~r, t ! 5 ePE1 expH iFvt 2v
cPr 2 fP~r!G J , (1)
E2~r, t ! 5 ePE2 expH iF ~v 2 vB!t 1v 2 vB
cPr
1 fP~r! 1 f1G J , (2)
E3~r, t ! 5 eSE1 expH iFvt 2v
cSr 2 fS~r!G J , (3)
E4~r, t ! 5 eSE2 expH iF ~v 2 vB!t 1v 2 vB
cSr
1 fS~r! 1 f2G J , (4)
where E1 ,E3 are the pump waves and E2 ,E4 are thephase-conjugated and Brillouin-shifted signal waves (seeFig. 3). Here eP ,eS are unit vectors that describe the po-larization, P is a unit vector in the propagation directionof the first pump wave E1 , similar to S for the secondpump wave E3 . Here fP(r),fS(r) describe phase distor-tions of the two pump beams and are assumed to be onlyfunctions of (r 3 P) and (r 3 S), respectively. Heref1 ,f2 describe arbitrary piston phase shifts.
The interference of the second pump beam E3 and thefirst signal beam E2 results in the intensity distribution
Fig. 3. The pump beams E1 , E3 , the phase conjugated beamsE2 , E4,4WM , E4 , the unit vectors S, G, P, and the angles betweenthem.
I2,3~r, t ! 5 1/2e0cn~v!uE2 1 E3u2
} E22 1 E3
2 1 2~eP • eS!E2E3
3 cosFvBt 2v
c~Pr 1 Sr! 1
vB
cPr
2 fP~r! 2 fS~r! 2 f1G , (5)
where P 1 S 5 2 cos (a) G, with the unit vector G in thedirection of the bisectrix of the angle 2a between P and S.Here I2,3 describes an intensity grating that has a spacingof
lgrating, 2, 3 ' l~v!/2 cos a, (6)
and that is traveling with a velocity of
vgrating, 2, 3 ' cvB/2v cos a (7)
along the direction of G. The grating is normal to G.For small angles 2a the wave (E4,4WM) generated by re-
flection of the first pump wave E1 on this grating is de-scribed by Eq. (4) for f2 5 f1 . This means that reflec-tion of E1 on the acoustic grating driven by theinterference of E2 and E3 generates a fourth waveE4,4WM . E4,4WM is in phase with E2 and can be a seed forthe stimulated scattering of E3 .
The process that leads to phase locking of two beams ofa pulsed laser can be described as a sequential process:
1. The first pump beam E1 is reflected by SBS. Thisresults in the first signal beam E2 .
2. E2 interferes with the second pump beam E3 [seeEq. (5)]. This interference generates an acoustic grating.
3. E1 is reflected by this grating. This results inE4,4WM , which is in phase with E2 .
4. If E3 still has not been reflected by SBS, the secondsignal beam E4 will originate from stimulated scatteringof E3 seeded by E4,4WM .
4. COMPARISON OF THEORY ANDEXPERIMENTSPhase locking was measured as a function of the param-eters W1 , W3 /W1 , NFS, FFS, FFS' , and BWS as de-scribed in the corresponding figure captions. Here FFS'
is always zero (except in the experiments shown in Figs.10 and 11), and BWS is always zero (except in the experi-ments shown in Figs. 9, 12, and 13).
Simultaneously, measurements of the transmission ofthe SBS cell and the temporal evolution of the SBS pulseswere taken.
A. Energy and Energy Ratio of the Two BeamsAs can be seen by observing the SBS pulses on an oscillo-scope, the lower the pump energies, the more jitter theleading edges of the SBS pulses will show with respect tothe pump pulses. In a similar way, the reflectivity of theSBS cell shows jitter (see Dane et al.11). If the time pe-riod between a pump pulse and its SBS pulse is measuredfor a large number of pulses, a broad distribution isfound. The higher the pump beam energy, the more thisdistribution will be narrowed.
As a consequence, if two beams of identical pulse shapeand timing are reflected by SBS, a time period betweenthe leading edges of the two SBS pulses is expected. Thistime period becomes larger if the pump-beam energies arereduced. It will also become larger if only one of thepump beams is attenuated, i.e., a pump-beam energy im-balance is introduced. Basically, during this time period,the four-wave mixing process leading to phase locking cantake place. It can be expected that the longer this timeperiod, the more effective the phase locking.
As a result, for pump beams of equal energy reducedphase-locking efficiency will be observed at increasingpump energy. The measurements of phase locking as afunction of the pump energy show that phase locking is atoptimum for energies close to SBS threshold and deterio-rates to half the optimum value at ;10Wth , as can beseen in Fig. 4. It should be noted that for optimized FFSa much weaker influence of the pump energy on phaselocking can be expected.
Concerning the energy ratio W3 /W1 , it is to be ex-pected that, for equal energy of the two pump beams,phase locking will be the worst. The larger the energyimbalance, the better the phase locking should be. Thisis shown by measurements of phase locking as a functionof W3 /W1 summarized in Fig. 5 (also see Fig. 7 comparedwith Fig. 8). The measurements shown in Fig. 5 weretaken with an energy of W1 5 7Wth for the strongerpump beam. Operating at 40Wth with the stronger pumpbeam, Carroll et al.6 did not observe a correlation betweenthe energy ratio and phase locking. This has to be attrib-uted to the fact that the timing of the SBS pulses of apump beam at 40Wth and one at 20Wth is nearly identical.Whereas, at about 7Wth for the stronger pump beam,even small changes of the energy ratio have a detectableinfluence on the phase locking (see Fig. 5).
B. Effective Interaction Length of the BeamsThe four-wave mixing process leading to phase locking islimited because of the geometrical overlap volume of thetwo pump beams as a function of the angle between them.An overlap length lo can be defined as the length overwhich the two pump beam centers approach to within onebeam diameter. Then NFS 5 2ds(0) corresponds to lo5 45 mm and NFS 5 4ds(0) corresponds to lo5 23 mm.
Therefore it is expected that phase locking is stronglydependent on the angle between the two pump beams.Phase locking as a function of NFS was measured atequal energies of the pump beams (see Fig. 6). ForNFS 5 4.1ds(0), phase locking is reduced to half themaximum value. This is in agreement with the resultspresented by Carroll et al.6 For W3 /W1 , 1 it was ob-served that phase locking is far less dependent on NFS,e.g., NFS 5 6ds(0) still has no negative effect on phaselocking at W3 /W1 5 0.6.
C. Location of the Four-Wave Mixing Process Along thePropagation PathThe location of the beam overlap volume along the propa-gation path has a strong influence on phase locking. It isexpected that optimum phase locking is achieved by gen-eration of the phase-correlated seed E4,4WM close to the
point along the propagation path where SBS of the secondpump beam would originate in absence of the first pumpbeam.12 Locating the overlap volume behind the ‘start-
Fig. 4. Phase locking as a function of the energy of the beams atan energy ratio of W3 /W1 5 1.0 and FFS 5 0. This graphshows the average of a number of data sets with NFS’s rangingfrom 2ds(0) to 4ds(0). Each individual data set shows a simi-lar function.
Fig. 5. Phase locking as a function of the energy ratio of thebeams at FFS 5 0, NFS 5 2.2ds(0), and W1 5 7Wth . Similardependencies were obtained for other FFS’s, too.
Fig. 6. Phase locking as a function of NFS at an energy ratio ofW3 /W1 5 1.0 and FFS 5 0. NFS 5 ds(0) corresponds to anangle of 2a 5 5.7 mrad between the pump beams. This graphshows the average of a number of data sets with W1 ranging from1.2Wth to 9.2Wth . Each individual data set shows a similarfunction.
ing point’ of SBS of the first pump beam is expected toprevent phase locking. This is because E2 will not beavailable inside the overlap volume, and therefore thefour-wave mixing process will not be possible. Phaselocking as a function of FFS was measured for an energyratio of W3 /W1 5 1.0 (see Fig. 7) and W3 /W1 5 0.3 (seeFig. 8).13
At an energy ratio of W3 /W1 5 1.0, phase lockingshows a well-defined maximum (FWHM 5 5.6ds,0) atFFS 5 21.6ds,0 . A phase-locking coefficient of K
5 0.77 was achieved (i.e., e 5 l/17). This is in agree-ment with the results of Sternklar et al.8 and Carrollet al.6
At an energy ratio of W3 /W1 5 0.3, phase locking wasfar less sensitive to FFS, and a phase-locking coefficient ofK 5 0.85 (i.e., e 5 l/27) was obtained. Operating atabout 40Wth , Carroll et al.6 did not observe relaxed align-ment requirements when introducing an energy imbal-ance in the pump beams (see Subsection 4.A).
D. Beam-Waist Separation Along the Beam PropagationPathWhen we translate the focusing lens of beam a, the beamwaist of beam a is translated relative to that of beam b.The beams were aligned to keep the overlap volume at thebeam waist of beam b. For negative BWS the beam waist
Fig. 7. Phase locking as a function of FFS at an energy ratio ofW3 /W1 5 1.0, NFS 5 2.2ds(0), and W1 5 7.8Wth . Filled andopen circles represent two individual data sets.
Fig. 8. Phase locking as a function of FFS at an energy ratio ofW3 /W1 5 0.3, NFS 5 2.2ds(0), and W1 5 7.8Wth . An FFS of22ds,0 results in the overlap volume being centered at 20.93zRbehind the beam waists. Filled and open circles represent twoindividual data sets. The measurement of p(d) that corre-sponds to the marked data point is shown in Fig. 2.
of beam a was in front of the overlap volume and, for posi-tive BWS, was behind it. Figure 9 shows the results of ameasurement of phase locking versus BWS at an energyratio of W3 /W1 5 0.3. As expected from the results pre-sented in the Subsection 3.C, phase locking vanishes forlarger negative values of BWS.
E. Misalignment of the Beams Perpendicular to FFS(FFS')Changing FFS' will not affect the location of the overlapvolume along the beam propagation path but changes theoverlap volume’s size. For optimum phase locking,FFS' 5 0 has to be maintained. Figure 10 shows the re-sults of a measurement of phase locking as a function ofFFS' at an energy ratio of W3 /W1 5 0.3. A total align-ment tolerance of DFFS' 5 2.6ds,0 is indicated. At anenergy ratio of W3 /W1 5 1.0 a total alignment toleranceof DFFS' ' ds,0 was obtained. This is in agreementwith the results of Sternklar et al.8
F. Transmission of the SBS Cell as a Function of PhaseLockingBy use of an energy ratio of W3 /W1 5 0.3 the energy ofthe second pump beam was only ;2.3Wth . That means
Fig. 9. Phase locking as a function of the separation of the beamwaists in the beam propagation direction (BWS) at an energy ra-tio of W3 /W1 5 0.3. The overlap volume is centered at thebeam waist of beam b, NFS 5 2.2ds(0), and W1 5 7.8Wth . Forpositive BWS the beam waist of beam a is behind the beam cross-ing. Filled and open circles represent two individual data sets.
Fig. 10. Phase locking as a function of FFS' at an energy ratioof W3 /W1 5 0.3 at FFS 5 0, NFS 5 2.2ds(0), and W15 7.8Wth . Filled and open circles represent two individualdata sets.
that for this beam the reflectivity of the SBS cell was notsaturated. While no saturation occurs, the SBS reflectiv-ity improves if the noise level that starts the process rises.Basically, the four-wave mixing process leading to phaselocking provides a wave E4,4WM that replaces the noise asthe seed of the stimulated scattering process. As a con-sequence, it is expected that for the second pump beamthe reflectivity of the SBS cell will significantly improve if
Fig. 11. Transmission of the SBS cell as a function of FFS' atan energy ratio of W3 /W1 5 0.3, FFS 5 0, NFS 5 2.2ds(0), andW1 5 7.8Wth . The filled circles represent the data of beam a,the open circles that of beam b.
Fig. 12. Transmission of the SBS cell as a function of BWS forthe conditions of the measurements presented in Fig. 9. Thefilled circles represent the data of beam a, the open circles that ofbeam b.
Fig. 13. Delay of the pump pulse and the signal pulse of beam bagainst BWS for the conditions of the measurements presentedin Fig. 9. The origin of the time scale is chosen arbitrarily.
phase locking is achieved. Additionally, the time periodbetween pump pulse and SBS pulse is expected to becomeshorter.
These improvements shown by measurements of thetransmission of the SBS cell and the delay of the risingedge of the pump pulse entering the SBS cell and the SBSpulse emerging from the SBS cell (see Figs. 11, 12 and13). These measurements were taken simultaneouslywith the acquisition of the phase-locking data, i.e., Fig. 10corresponds to Fig. 11, and Fig. 9 corresponds to Figs. 12and 13.
The delay of the rising edges of the pump pulse and theSBS pulse was determined by 50% of the peak intensityvalues.
The transmission data were obtained by dividing theaveraged beam energies in front of and behind the SBScell. The transmission measurements can be comparedwith the results of Mangir and Rockwell,14 who used pres-surized nitrogen as a SBS medium, too. In preliminaryexperiments the reflectivity of the SBS cell was measuredwith a setup in which the pump beam propagates througha thin-film polarizer, a quarter-wave plate, and is focusedinto the SBS cell. By comparing the energy of the pumpbeam to the backward traveling beam that is reflectedfrom the polarizer, the reflectivity of the SBS cell is deter-mined. For single-longitudinal-mode pulses from the os-cillator described in the previous sections and nitrogen at75 bars as a SBS medium, the reflectivities shown in thefollowing table were determined.
For pulse width (FWHM) from 13 ns to as great as 26 ns,no dependency of the reflectivity on the pulse width wasobserved.
5. SUMMARYPhase locking of two beams by mutual reflection in anSBS cell has been investigated. The SBS-cell was filledwith nitrogen at 75 bars. The SBS-threshold energy fora nearly diffraction-limited single-longitudinal-modebeam of a pulsed Nd:YAG laser with tP 5 17.3 ns was17.2 mJ. The reflectivity of the SBS cell was nearly 80%at eight times SBS-threshold energy. I determinedphase locking by measuring the change of the relativephase difference of the beams from pulse to pulse for morethan 1000 pulses. The sensitivity of phase locking on thealignment of the beams was established. The influenceof energies and energy ratios of the beams on phase lock-ing was determined. Phase locking of K > 0.85 wasachieved. This corresponds to a pulse-to-pulse changefor the phase difference of the output beams of e < l/27for 50% of all pulses. A model of the phase-locking pro-cess for pulsed laser beams that emphasizes the sequen-tial nature of the process has been presented. The influ-ence of a variety of beam characteristics on phase lockingwas deduced and compared with the experimental re-
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13. In these experiments FFS522ds,0 resulted in the overlapvolume being centered at 20.93zR behind the beam waist.
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