Gadolinium scandium gallium garnet crystals~GSGG! codoped with neodymium and chromiumhave demonstrated two to three times the lasing ef-ficiency of the more commonplace Nd-doped yttriumaluminum garnet ~YAG!.1 This increased efficiencyis due to the absorption by Cr31 over broad absorp-tion bands ~420–520 and 560–700 nm! and a rela-tively efficient and fast radiationless energy transferto Nd31.2 If high energy in a short pulse is also
esired, as from a Q-switched oscillator or from anamplifier, then Nd:Cr:GSGG has the added advan-tage of a stimulated-emission cross section that ishalf that of Nd:YAG, so twice the energy can be storedin a Nd:Cr:GSGG laser element than in Nd:YAG be-fore the onset of energy-draining parasitic oscilla-tions.3
The absence of GSGG-based lasers is due to the poorthermo-optical properties of GSGG.4 Figures 1 and 2show thermal focusing and birefringence measure-ments performed by us on laser rods and heads used inthe oscillators to be described. KK-1 filter materialcontains a samarium dopant and is optimized for usewith Nd:YAG with transmission starting to drop at500 nm, whereas KF-2 filter material has high trans-
The authors are with the Nonlinear Optics Group, Soreq NuclearResearch Center; 81800 Yavne, Israel.
Received 8 January 1998; revised manuscript received 14 April1998.
mission to 350 nm. Both filters are used in the Kigrelaser head flow tubes that house the rod and the flashlamps. The measurements used continuous wave,TEM00 probe beams expanded to fill the rods ~thermalensing, green He–Ne; birefringence, Nd:YAG! andmployed standard measurement techniques.6 Re-ults are in general agreement with those of otheresearchers.7 Clearly, the six to nine times greater
thermal lensing of GSGG is a significant factor in thedesign of a low-divergence laser system, whereas thethree times greater birefringence must be dealt with ifanything other than low average power in a polarizedbeam is required. Note, too, that as the dependenceof thermo-optical parameters on pump power becomesgreater, the sensitivity to average pump-powerchanges becomes more critical and the operating rangeof a static resonator becomes narrower.
We demonstrate the use of adaptive optics in aGSGG oscillator to increase the operating range sub-stantially. Birefringence losses were substantiallyreduced through incorporation of a previously devel-oped reentrant resonator design.8
2. Overview of the Oscillator
We describe application of adaptive optics to hemi-spherical resonators. The same techniques were ap-plied by us to other resonator designs with otherlasents, as we describe in a separate publication.9Hemispherical resonators were investigated for sev-eral reasons. The first, historical, is that the effi-ciency of an existing TEM00 oscillator in anoscillatoryphase-conjugated multiple-pass amplifiersystem could be increased by a factor of 5 while only
0 September 1998 y Vol. 37, No. 27 y APPLIED OPTICS 6415
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minimum changes were made to the system, bymeans of alteration of the resonator from one opti-mized to maximize thermal lensing stability to oneoptimized for maximum TEM00 output.10 The sec-ond reason is that the hemispherical resonator rep-resents a means of maximizing intracavity fluencenear the output coupler where optical parametric os-cillators and other intracavity frequency-conversiondevices are optimally placed.11 The third is that thehemispherical resonator operates at the limit of sta-bility @g1~R1! 5 1, g2~R2! 5 0# with performance vari-ations easily monitored through changes in outputenergy and mode structure, thus providing an idealtest-bed for demonstration of dynamic stabilizationtechniques.12
Figure 3 shows the complete oscillator layout. Thelaser rod was obtained from Litton-Airtron and haddimensions of 0.635 cm 3 7.62 cm. Faces were flat–
Fig. 2. Thermally induced birefringence from GSGG and YAGmeasured with the same-diameter rod and the same type of laserhead.
416 APPLIED OPTICS y Vol. 37, No. 27 y 20 September 1998
flat, at 90° to the barrel axis, and antireflection coated.The barrel surface was ground to 1.4 3 1024-cm rough-ness. Cr and Nd doping concentrations were 2 3 1020
cm23. The laser head was from Kigre and used aSm-doped KK1 filter to house the rod and a Xe-filledflash lamp while providing water-cooling channels.Surrounding the filter was a barium sulfate diffusereflector. The output coupler was a two-etalon reso-nant reflector that comprised 1.0- and 1.5-cm uncoatedfused-silica etalons separated by 5.6 cm, which pro-vided nearly optimum output coupling of 40% andwhich, together with a 0.02-cm-thick 20% coatedfused-silica etalon, narrowed the bandwidth to 270 60 MHz. The KD*P Pockels cell was obtained fromast Pulse Technology.We formed the reentrant resonator by placing a 45°
erbium gallium garnet in a permanent magnet Fara-ay rotator ~Litton-Airtron! between the laser rodnd the backmirror. On double passage through theirefringence-inducing laser rod, the depolarizationas in large part corrected and the beam was in largeart reflected by the polarizer toward the reentrantirror, where it was retroreflected. The lengths of
he output and the reentrant arms were approxi-ately 94 cm, with the exact length of the reentrant
rm adjusted to place the beam waist on the reen-rant mirror whence optimum performance ensued.he importance of using the reentrant resonator de-ign is shown in Fig. 4, where output and birefrin-ence loss are plotted for two-mirror linear and thehree-mirror reentrant resonator resonators.
The backmirror was set at 500 cm to produce aarge-fill-factor, flattopped beam ~multimode! insen-itive to thermal lensing variations. Output mirroreflectivities were 60% and 40% for the linear and theeentrant resonators, respectively.
3. Concept of the Variable-Radius Mirror
Figure 5 depicts the variable-radius mirror ~VRM!esign and its application as the hemispherical res-nator backmirror.13 The VRM consisted of a sta-
tionary negative lens and a concave mirror mountedupon a translation stage aligned parallel to the opti-cal axis. The concave mirror was moved to controlthe beam’s effective radius of curvature. This typeof device has one degree of freedom. The equationgoverning the effective radius of curvature of the
Fig. 3. Schematic of the hemispherical resonator, showing all thecomponents.
Fig. 1. Thermal lensing in the rods and laser heads used in thisstudy.
b
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VRM, Reff ~and the light phase front in the regionetween the rod and the negative lens! is given by
Reff 5Flens~Rmirror 1 D!
Flens 2 Rmirror 2 D, (1)
where Flens is the focal length of the VRM negativelens, Rmirror is the radius of curvature of the VRMconcave mirror, and D is the spacing between the twoVRM elements. The sign convention is such thatnegative lenses and concave mirrors have negativefocal lengths and radii of curvature. If Reff is fixedby an external consideration, i.e., resonator require-ments, and if Flens and Rmirror have been selected,then we can use Eq. ~1! to determine the required
Fig. 4. Performance comparison of a standard two-mirror linearcavity and a three-mirror reentrant resonator. EPFN 5 9 J. Thetwo-mirror cavity birefringence loss is the light rejected from theQ-switched polarizer after double passage through the rod.
Fig. 5. Schematic of the VRM comprising a negative lens withfocal length Flens separated by a distance D from a concave mirror
ith radius of curvature Rmirror. Together with a laser rod thathas a thermal focal length Ft, the output plane is reimaged backnto itself in the hemispherical resonator.
20
element spacing. Similarly, Eq. ~1! can be used asthe basis of a sensitivity analysis to fix VRM opticsparameters.
The combined effect of the rod thermal focus andthe VRM is to focus the laser beam onto the outputcoupler. Adjustment of the VRM minimizes ther-mally induced variations in the beam’s radius of cur-vature and diameter at points between the outputcoupler and the rod. Were the VRM to remain staticwhile average pump power was varied, the laser per-formance would degrade: Reduced pump powerwould move the focus beyond the output coupler andmultiple transverse modes would be generated, andincreased pump power would move the focus inwardof the output coupler and the resonator would bedriven into the unstable region where output is se-verely reduced. By varying the VRM effective ra-dius of curvature, we can compensate for the effect ofchanging rod thermal focusing.
The hemispherical resonator equation that governsproper focus of the VRM in the presence of thermallensing is
Reff 5Ft~L1 1 L2! 2 L1 L2
L1 2 Ft, (2)
where Ft is the thermal lensing focal length, L1 is thedistance between the output coupler and the rod, andL2 is the distance between the rod and the VRM’snegative lens ~Fig. 5!. Equations ~1! and ~2! can beused together with known resonator, optics, and ther-mal focusing parameters to calculate the requiredReff and D as a function of average pump power. Theconditions of our hemispherical resonator were L1 5130 cm, L2 5 20 cm, Flens 5 ~2!10 cm, Rmirror 5 ~2!20cm, and Ft
21 5 ~4 DyKW!PPFN ~Fig. 1 for Nd:Cr:GSGG with a KK1 filter!. PPFN is the electricalpower obtained from the pulse-forming network~PFN!. The results are plotted in Fig. 6. Two sin-gularities appear, one for each of Eqs. ~1! and ~2!.The lowest pump-power singularity derives from Eq.~2! and occurs when Reff36` and the VRM behavesas a flat mirror. Although they are not treated bythis model, higher-order aberrations will dominateresonator dynamics here, and a single-degree-of-freedom VRM will not function effectively. The sec-ond singularity appears in Eq. ~1! and occurs as D30. In this case the VRM component spacing becomesextremely sensitive to variations in pump power, andstable VRM operation cannot be expected. In prac-tice, the region of effective VRM operation extendsonly to the first singularity. Proper VRMyresonatordesign can move the first singularity beyond the laseroperating regime. With the present VRM it wasfound necessary to control D to an accuracy of 625mm, well within the capabilities of our Oriel opticallyencoded stepper-motor-driven translation stage.
Reff can be controlled through use of a feedback loopor with the aid of look-up tables. Use of a feedbackloop implies that, following a change in pump param-eters, the laser rod reaches a new equilibrium statefor which resonator optimization is required.
September 1998 y Vol. 37, No. 27 y APPLIED OPTICS 6417
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Look-up tables are used when the laser is run undernonsteady-state conditions in which each shot re-quires a different VRM setting. Use of look-up ta-bles requires that the concave mirror’s translationstage contain an optical encoder. Look-up tablesmust be built up during a learning mode. Closed-and open-loop modes of operation have both beenvalidated.
Feedback requires measurement of a laser param-eter. The appropriate parameter here is far-fieldbeam brightness. We measured far-field brightnessby extracting a small portion of the output beam,splitting it into two, and focusing the first partthrough a far-field aperture and onto a TEM00 trans-
ission photodiode and the second part onto a wholeeam photodiode. The two signals were transferredo a computer, where far-field brightness was calcu-ated and a dither-correction signal was sent to theRM translation stage.By adding additional components, one can correct
ther aberrations. In this oscillator it was necessaryo correct static astigmatism introduced by the laserod. Correction was accomplished by addition of aylindrical zoom lens that comprised ~1! and ~2! 1-m
focal-length cylindrical lenses aligned with parallelcylindrical axes ~Fig. 7!. We controlled the strengthof the zoom lens by adjusting the lens separation andcontrolled the cylindrical axis by adjusting the rota-tion of the two lenses. Far-field images show thatastigmatism correction was complete ~Fig. 7!.
Addition of the cylindrical zoom lens to the basicRM leads to the concept of a distributed adaptiveptics system in which one can achieve extra degreesf freedom for aberration correction by adding addi-ional discrete components rather than by adding ad-itional piezoelectric transducers to a flexibleirror.14 Depending on the application, this ap-
proach allows larger amounts of aberration to be cor-rected in a more cost effective and rugged package,although the package will be bulkier.
418 APPLIED OPTICS y Vol. 37, No. 27 y 20 September 1998
A type of bipolar focusing occurs in the reentrantresonator when rod and Faraday rotator double-passed light acquires a focusing term that depends oneach ray’s radial and azimuthal position with respectto the polarizer axes.10 The mode-distorting effect ofthis bipolar focusing, which is a function of averagepump power, was not observed following correction ofthe static astigmatism with the zoom telescopic lens.
4. Oscillator Performance with Dynamic Correction ofThermal Focusing
Figure 8 shows the oscillator output energy with theVRM in closed-loop feedback and in static modes.The pulse-repetition frequency ~PRF! of the oscillatorwas varied in 1-Hz increments, and the oscillator was
Fig. 8. Output energy of the hemispherical resonator with theVRM in closed-loop feedback mode and with the VRM in staticmode focused for low-repetition-rate operation. With feedback,the output was TEM00 at every point.
Fig. 6. Theoretical VRM effective radius of curvature and com-ponent spacing based on the experimentally measured thermalfocusing of Nd:Cr:GSGG in a KK1-filtered laser head.
Fig. 7. Schematic of the zoom cylindrical lens added to the one-
degree-of-freedom VRM and its efficacy in correcting astigmatism;f 5 1 m in the current oscillator.
fr
allowed to come to dynamic equilibrium. LargerPRF increments could also be implemented. Ifpump conditions varied by a large amount sufficientto terminate lasing, then the VRM was rough focusedby reference to the look-up tables before feedbackcontrolled the fine focusing. With feedback, con-stant output energy in a pure TEM00 beam wasachieved. As much as 150 mJ of energy in a TEM00beam at 20 Hz could be obtained from the oscillatorwithout adjustment of the PFN voltage. HigherTEM00 power would require use of a modified VRM.With the VRM in static mode but aligned for low PRFoperation, the effect of thermal lensing was to quicklycause a reduction in output energy. Static backmir-ror radii were also selected for hot operating condi-tions, but the beam then became multiple transversemode under reduced thermal loading conditions.
Beam quality as a function of shot number duringa zero warm-up time sequence is shown in Fig. 9.The number above each far-field image is the shotnumber within the sequence. At a PFN energy of6.2 J and at a PRF of 10 Hz, it took 1.2 s for the laserstabilize with the VRM held static at the steady-statefocus. VRM control by reference to the look-up ta-bles resulted in optimal beam quality for each shot inthe sequence. Thus the VRM controlled by look-uptables permitted operation with zero warm-up time.
5. Conclusions
Dynamic stabilization of thermal focusing in an os-cillator operating at the edge of stability was demon-
Fig. 9. Far-field beam profiles from the instant the oscillator wasturned on. The numbers are the shot numbers within the se-quence. EPFN 5 6.2 J. a, The VRM is maintained static at its hotocus. b, The VRM focus is adjusted based on look-up tables de-ived earlier during a learning mode.
20
strated with a simple one-degree-of-freedom adaptiveoptic. Microprocessor control was achieved inclosed-loop feedback or open-loop look-up tablemodes. This variable-radius mirror, consisting of anegative lens and a translator-mounted concave mir-ror, proved effective in compensating for GSGG ther-mal focusing throughout the tested range of 0–1.5 D.Compensation of higher levels of thermal focusing ispossible. Additional discrete optical elements wereinserted to form a distributed adaptive-optics system.In particular, a cylindrical zoom lens was added toeliminate beam astigmatism. Bipolar focusing wasnot observed over the pump-power range tested.
Development of the VRM concept is proceeding inseveral directions. Testing of VRMycylindricalzoom lens distributed adaptive-optics systems de-signed to correct higher levels of thermal focusingand astigmatism are under way, the goal being toproduce TEM00 beams from GSGG stable resonatorsat the tens-of-watts level. VRM’s are being success-fully applied to unstable resonators with Gaussianoutput couplers to produce low-divergence beamsover the range of 0 to .30 W in flash-lamp-pumpedNd:YAG with the goal of achieving 0–50 W of powerfrom GSGG. Application to diode-pumped YAG os-cillators operating at .100 W is also planned.VRM’s are also being successfully used in GSGG andglass phase-conjugated multiple-pass amplifiers tomaintain near-field beam collimation. Finally, pre-liminary engineering design of compact VRM’s con-trolled by dedicated microprocessors has begun.
At some point it will be necessary to correct forhigher-order aberrations. Then flexible mirrors con-trolled by multiple piezoelectric transducer actuatorsmay be required. Even here, the VRM’s as describedmay find use in correcting for the high levels of lower-order aberrations that are beyond the range of piezo-electric transducer–controlled adaptive optics.
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