Perfect thermal focusing will occur in homogeneouslypumped laser rods, even in the presence of large tem-perature gradients, if the thermal coefficients (thermalconductivity and thermal expansion) are independentof temperature and if birefringence-induced bifocusinghas been compensated for.1 Elimination of bifocusinghas been achieved at high pump powers with radiallypolarized beams.2 The thermal coefficients’ tempera-ture dependence then becomes the dominant source ofhigh-power aberrations.3–5 Inhomogeneous pumpingmay also cause the temperature distribution to have ahigher than quadratic dependence on radial positionand to also be nonradially symmetric. The optical pathdifference (OPD) in the rod is linearly dependent ontemperature �dn�dT � �T�, and thus a nonquadratictemperature distribution will result in higher-orderoptical aberrations, the first of which is sphericalaberration (SA).
For the research described here, Cutting-EdgeOptronics RE60 pump chambers were used. These
chambers housed 6.35 mm diameter by 146 mm longNd�0.6% at�:YAG rods that were side pumped by fivediode arrays. We recorded the pump-distributioncross sections in the rods by photographing the fluo-rescence distributions and found them to be non-uniform (see Fig. 1). First, the pump density was1.7� greater at the center than at the periphery.Second, near the periphery the pump density had afivefold azimuthal deviation of �10% peak to valley.This kind of pump distribution appears frequently inside-pumped rods. The first nonuniformity was dom-inant and was the main source of SA. The secondnonuniformity introduced nonradially symmetric ab-errations, which were weak.
In Section 2 we present wavefront (WF) measure-ments and SA analysis, the effects of SA on beam-quality degradation, and SA’s contribution to theoverall WF distortion. In Section 3 we describe cor-rection based on a relay telescope with sphericallenses and its contribution to SA reduction. In Sec-tion 4 a SA-compensated radially polarized oscillatorwith significantly improved performance is described.Finally, in Section 5 we demonstrate SA compensa-tion together with partial correction of nonradiallysymmetric aberrations in a two-rod amplifier chain.
2. Spherical-Aberration Measurements
WF distortion measurements were performed witha Shack–Hartmann wavefront sensor. A TEM00
The authors are with the Nonlinear Optics Group, Soreq NuclearResearch Center, Yavne 81800, Israel. I. Moshe’s e-mail address [email protected].
�M2 � 1.1� linearly polarized probe beam with a3.5 mm diameter was used. Nd:YAG laser rods sufferthermally induced bifocusing. The following stepswere taken to filter out bifocusing from the OPD: (1)The probe-beam polarization was transformed intocircular polarization because linearly polarizedbeams exhibit astigmatism in thermally induced bi-refringent rods.6 (2) Measurements were performedat the image plane of the rod’s principal plane toprevent size mismatch in the radial and tangentialparts of the beam. (3) While these steps were per-formed, bipolar lensing appeared at the Shack–Hartmann sensor as blurry focal points with centermasses that indicate the average focusing. Removalof the bifocusing was then accomplished by subtrac-tion of focusing (with a quadratic dependence on theradius) from the OPD. The end product was thena focusing-free OPD function, which includespolarization-independent aberrations only.
The probe-beam divergence was controlled by anegative lens to facilitate symmetric passage throughthe pumped rod (the same absolute value for the WFcurvature at the rod’s entrance and exit but oppositesign). This arrangement ensures maximum overlapbetween the rod and laser-beam volumes. To mea-sure the output WF, we relay imaged the beam ontoa WF sensor (HASO-32 Shack–Hartmann sensorfrom Imagine Optics). The measured WF (after tip
and tilt were subtracted) was fitted to a radially sym-metric fourth-order polynomial:
OPD�r� � C0 � C2r2 � C4r
4. (1)
The C4 OPD coefficient was then extracted (Seidelcoefficient). C4 and the calculated beam-quality deg-radation were investigated as functions of pumppower. At the maximum pump power �3.4 kW� andwith lasing, C4 equaled 1.4 � 10�13 �m�3. Usingthe method described in Ref. 7, we may describethe beam-quality degradation as M2 � ��Min
2�2 ��MSA
2�2�1�2, where Min2 is the incident beam quality
and �MSA2 indicates the beam-quality degradation
induced by spherical aberration. �MSA2 is a function of
C4. Calculating the beam-quality degradation for theresultant C4 value yields �MSA
2 � 3.2 and an output-beam quality M2 � 3.4. This C4 measurement is atthe same level as obtained by calculation of the OPDby use of the temperature distribution derived fromthe measured pump distribution. The OPD is also atthe same level as the theoretical result, calculatedaccording to Ref. 3 for 500 W power and a pumpdistribution of � � 0.42 (� gives a pump distributionclose to our measurements).
3. Spherical-Aberration Compensation
There are several single-pass options to compensatefor SA in rod-based pump chambers:
(1) SA induced by temperature-dependent ther-mal conductivity, which is positive in YAG, may bereduced by more homogeneous pumping or even elim-inated with a pump distribution that has a slightcentral dip that introduces negative SA.3 Implemen-tation of this method in 2 kW output pump chambersis described in Ref. 8.
(2) Direct compensation can be accomplished byuse of an aspheric element designed to introduce thesame absolute value but opposite sign as the aberra-tion’s phase shift.9 For strong SA correction, suchaspheric elements were found to be expensive.
(3) The positive thermal SA imposed by thetemperature-dependent thermal conductivity and bythe center-peaked pump-distribution can be compen-sated for by the negative SA generated in positivespherical lenses. In one setup a relay telescope thatintroduces SA without additional focusing (Fig. 2) isadded. This method is inexpensive and is available offthe shelf.
Fig. 2. Positive spherical aberration introduced by thermal ef-fects in the strongly pumped laser rod is eliminated by addition ofa relay telescope with spherical lenses that introduce negativespherical aberration. A tight focal spot may then be obtained.
Fig. 3. SA compensation by use of a relay-imaging telescope basedon plano–convex lenses oriented in a (plano–convex)(convex–plano)configuration.
Fig. 1. Transverse pump distribution in the laser rod. The pumpdistribution was captured by photographing the 1 �m fluorescenceemitted during pumping.
The third method was chosen and a relay-imagingtelescope based on spherical lenses was designed. Goodcompensation could be achieved by use of relay imag-ing based on two-plano–convex lenses with 12.9 mmradii of curvature in a (plano–convex)(convex–plano)arrangement (Fig. 3). Based on Ref. 10, C4 was cal-culated for this scheme to have a value of ���1.4� 10�13 �m�3, as required to correct the rod’s ther-mal SA.
SA correction was tested with a SA compensator(SAC) coupled to a single laser rod. Measurementresults appear in Fig. 4. The SA was minimized to anegligible value near the desired working point of3 kW electrical pump power (during lasing, 3.4 kWof pump power would produce an equivalent amountof heat). The dependence of C4 on pump power waslinear over the range tested but yielded a sufficientlywide operation span of low-SA values near the work-ing point.
In the past, power from our single-rod radially polar-ized oscillator was limited by blurring of the foci fromthe two orthogonal polarizations.11 SA correctionshould allow higher output powers to be achieved. TheSAC was added to the symmetric plano–plano resona-tor (Fig. 5). This resonator was designed to producea radially polarized beam as described in Ref. 2.The strong SA in the laser rod limited the electricalpump power to 2.8 kW, at which point 150 W of out-put power was achieved in an M2 � 3 beam. Intro-ducing SA compensation facilitated pumping atmaximum power and enlarged the beam size inside
the rod while it increased the overlap efficiency. Theoutput power from the resonator could then be in-creased by 60% to 250 W at an M2 value of 3.8 in aradially polarized beam.
5. High-Order Aberration Compensation in MultirodAmplifier Systems
Figure 6 depicts the measured peak-to-valley WF de-formation as quantified by the Zernike third-order SAdivided by the peak-to-valley WF deformation intro-duced by the other high-order aberrations. It is clearthat the SA was completely eliminated at one pumppower and that the remaining contribution to WFdeformation then came from other higher-order ab-errations.
Important higher-order aberrations occur becauseof pump-diode fivefold symmetry about the laser rod.These aberrations, which are not radially symmetric,may be compensated for in two- (or more) rod systemswhen the orientation of the side-pumping diodes inone pump chamber is rotated with respect to those ofthe other chambers to produce a more nearly homo-geneous pump distribution from the composite. Wetested this concept in a two-rod system, in which twoSACs were added after each rod (Fig. 7). These SACspermitted compensation for spherical aberration and36° rotation of one pump chamber’s fivefold symme-try with respect to the other pump chamber. Thisrotation was caused by inverse imaging of the SACthat had been placed between the pump chambers.
Measured wavefront and far-field intensity distri-butions are depicted in Fig. 8. Aberration compensa-tion was provided by reduction of the C4 SA coefficientby �2 orders of magnitude and by reduction of thepeak-to-valley wave distortion (azimuthal aberra-tions) by a factor of 6. The residual-low-intensity
Fig. 4. Measured spherical-aberration coefficient with and with-out compensation as functions of electrical pump power. Note that,with compensation, spherical aberration is totally eliminated at apump power of 2.95 kW.
Fig. 5. Symmetric plano–plano resonator with a SAC. Resonatorlength and aperture size were designed to produce a radially po-larized beam at the maximum pump power.
Fig. 6. Measured peak-to-valley (P to V) WF deformation intro-duced by Zernike third-order SA divided by the peak-to-valley WFdeformation introduced by the other high-order aberrations.
Fig. 7. Schematic of the two-rod amplifier with SAC.
wing in the far-field image of the compensatedscheme is due to birefringence-induced bifocusing,5which may be compensated for by use of radiallypolarized oscillators.
Improvement in the high-order aberration value byuse of rotated pump chambers is also demonstratedin Fig. 9. This figure presents the Zernike analysis forthe measured wavefront in the SA-compensated two-rod system, with the image of the pump-chamberdiodes rotated or in line (double the aberration valueof a single rod). The X and Y axes indicate the Zernikecoefficient index and value, respectively. One cansee that SA on the X axis was eliminated in bothschemes; however, the scheme with rotated pumpchambers showed lower higher-order aberrationvalues.
Table 1 summarizes the measured beam-qualitydegradation generated in a two-pump-chamber mod-ule with various levels of aberration correction.
��M2�2 is the difference between output and input�M2�2 according to �M2�2 � �Min
2�2 � ��M2�2.
6. Conclusions
Thermally induced spherical-aberration was analyzedand measured in high-power Nd:YAG laser rods. SAcompensators based on a relay-imaging telescope werepositioned adjacent to the pump chambers. Our exper-iments showed excellent SA elimination without SAC-induced focusing or additional aberrations (owing toSAC lens misalignment). The SAC was implementedin a radially polarized oscillator and facilitated a 60%increase in output power without significant beam-quality degradation. Radial polarization and SACstogether eliminated the main aberrations in uni-formly pumped rod-based amplifiers. Scale-up ofwave front maintenance was demonstrated in a two-rod–two-SAC amplifier module �6.3 kW electricalpump power). Higher-order aberrations were par-tially corrected in these modules by rotation of thepump chambers.
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Fig. 8. Measured output beam wavefront and focal-plane inten-sity distribution after the two-rod system, with and without SACs.Wavefronts are presented after subtraction of undistorted focusingand include SA and other higher-order aberrations.
Fig. 9. Values of Zernike coefficients in two-rod systems, with thediode-arrays oriented at 0° and 36°.
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