High-power solid-state lasers based on rod geometrygain elements can have large amounts of sphericalaberration in addition to overall lensing and strainbirefringence.1,2 Side pumping with laser-diode ar-rays is particularly subject to this kind of aberration,since it is difficult in this configuration to pump therod uniformly. Even if the rod were uniformlypumped, nonlinear thermo-optic properties of mate-rials such as Nd:YAG result in spherical aberration.3It has been well documented that the use of theseintensely pumped rods in laser oscillators results inseverely compromised beam quality through a com-bination of wave front distortion and strain inducedbilensing, even in unpolarized lasers.1,2,4 The com-pensation of aberrations with fixed elements such ascorrector plates and diffractive optical elements isfrustrated by the fact that the lensing and the aber-rations are dependent on the amount and the distri-bution of stimulated emission5 in the laser medium.
Various approaches to an assessment of the impactof aberrations on the performance of laser oscillatorshave been reported. Numerical beam propagationalgorithms1 can be used to find an eigenmode in thepresence of a radial quartic phase aberration by re-petitive Fox–Li-type propagations of an initial beamprofile in the resonator. This eigenmode is thenevaluated for its overall loss and saturation of the
The author is with Cutting Edge Optronics, Inc., 20 Point WestBoulevard, St. Charles, Missouri 63301. His email address [email protected].
Received 8 May 2002; revised manuscript received 16 September2002.
gain medium. The behavior of the resonator maythen be studied as a function of its parameters.Whereas valid results are readily obtained for a sin-gle mode, multimode operation2 is less clear, becausehigher-order modes are not all mutually coherent,and thus a particular superposition does not neces-sarily return in the same amplitude and phase aftera round trip. The results of such iterative calcula-tions, if they do converge on a beam structure thatreplicates the phase and amplitude on an iteration ofthe computation, must represent a superposition ofcoherent higher-order modes �a set for which theGouy phase is constant� for the resonator. This is amore restrictive condition than that of the actual res-onator, which might support other modes with differ-ing Gouy phases.
A more recent approach uses moments of the beamderived from the Wigner distribution function of thefield.7 Beams consisting of resonator eigenmodesmust reproduce their second moments on making around trip of the resonator, as represented by its raytransfer or ABCD matrix. There are three such mo-ments �treating only a single transverse dimension�,and propagation laws for these moments establishthree equations in these three moments with the ma-trix elements as parameters. An introduction of aquartic phase aberration4 does not increase the num-ber of equations but does introduce three additionalfourth and sixth moments of the beam as variables.Unique solutions for all of these variables are there-fore not possible from the eigenfunction relationsalone, but useful relationships between the secondand higher moments are obtained.
It would be more useful and instructive now tohave a truly analytic approach to multimode beamformation, which breaks the beam down into elemen-tary components that behave simply and indepen-
dently during propagation and whose interaction inan oscillator is limited to competition for availablegain. This kind of approach was taken in a previousstudy7 to determine the dependence of beam qualityon power extraction in rod geometry lasers. We re-port in this paper an extension of that work to includethe effects of spherical aberration.
2. Helicoid Modes
In the aforementioned paper we showed how a modalanalysis of a multimode oscillator can be accom-plished with helicoid modes with intensity profiles
Im�r� �2
��2m! �2r2
�2�m
exp�� 2r2
�2� . (1)
These profiles represent a nested set of ring modeswith peak radii �rm�2 � �2m�2. Each mode in theset has a fundamental scale size � that is common toall members of the set. Each mode has a beam qual-ity M2 � m 1, and the modes are normalized so thatthe area integral of their intensity equals unity totalpower. Thus the overall beam quality of an incoher-ent superposition of such modes is the power-weighted mean of the M2 terms of the modes. Anumerical algorithm was derived to calculate a self-consistent set of mode powers that takes into accountthe cross-saturation of the gain medium owing to theoverlapping intensity profiles of the modes.
Now, because the helicoid mode profiles exist in alimited range of radius, they would seem to be suitedto approximating eigenmodes of resonators that havepurely radially aberrated intracavity lenses. To testthe appropriateness of using helicoid modes as ap-proximate eigenmodes in the presence of sphericalaberration, we performed some numerical experi-ments. Selecting an ABCD matrix for a stable res-onator �see Section 4 for the resonator model�, weimposed a quartic phase screen at each iteration rep-resenting a round trip of the resonator and startedwith a helicoid of index m and scale �m. This stablemode scale size is determined by the values of theround-trip ABCD matrix. Then the mode was cy-cled through a numerical propagator �the Fresnelmodule of Paraxia™� that used the mth Hankeltransform to propagate the mode. After five cyclesof applying the phase screen and the propagator tothe output wave of the previous cycle, the result wascompared with the original input wave. Examplesare shown in Figs. 1 and 2. The input modes sufferlittle distortion after propagation, confirming the ap-propriateness of their use as eigenmodes in this case.
Now, these results would seem to be in contrast tothose of Stein,1 particularly illustrated in Fig. 6 ofthat paper showing radial intensity plots of the seedprofile and that of the perturbed mode. The per-turbed mode may not in fact be the lowest-order modeof the aberrated resonator. The plots suggest that itmay be a superposition of two Laguerre–Gaussianmodes where 2p m � 2, with p as the radial indexand m as the tangential. These modes have thesame Gouy phase and are therefore coherent. The
mode p, m� � 1, 0� has a central lobe and a sur-rounding ring, while the 0, 2� mode is an annuluswhose radius corresponds to the outer ring of the1, 0� mode. The scale size of this mode is muchsmaller than that of the unperturbed mode and, aswe shall see below, this is consistent with what isexpected for this resonator in the case of negativespherical aberration coefficients. In the analysisabove, care was taken to make the seed mode scalesize correspond to the axial resonator focal power,which is stronger than the average focal power overthe aperture.
3. Quartic Focal Power Aberration of Laser Rods
The focal power D of a lens with an r4 phase aberra-tion behaves as
Dm�r� � ��1 � ��rm�
b �2� , (2)
where � is the axial focal power, b is the apertureradius, is the aberration coefficient, and �rm� is theradius of the peak intensity of the helicoid mode of
Fig. 1. Input m � 5 helicoid after five round trips of resonatorwith a quartic phase screen. Curve fit to m � 5 helicoid shownabove shows mode is stable to within 1 part in 1000. Curve fitfunction, y � 1.000669 2�x�0.8821024�2�5 exp�2�x�0.8821024�2�;R � 0.9999959. Input scale size was 0.8844 mm.
Fig. 2. Input m � 10 helicoid after five round trips of resonatorwith a quartic phase screen. Curve fit to m � 10 helicoid shownabove shows mode is stable to within 1 part in 1000. Curve fitfunction, y � 0.99033062�x�0.8836908�2�5 exp�2�x�0.8836908�2�; R � 0.9995867. Input scale size was 0.8844 mm.
index m. In terms of the Zernike coefficients, thecoefficient is
�1
1 �Z21
3Z42
� brz�2
, (3)
and this is related to the Seidel coefficient a by
a ��
b2 . (4)
It is instructive to determine the magnitude of thephase aberration commonly encountered, particu-larly in high-power diode-pumped rod amplifier mod-ules. A commercial amplifier module �model RE50manufactured by Cutting Edge Optronics, Inc.�, ca-pable of producing over 500-W cw multimode powerin an oscillator, was characterized with a Shack–Hartmann wave-front sensor to determine the wave-front distortion up to the fourth order in Zernikepolynomials. The module was driven at a current of21 A, producing a total pump power from the diodearray of 1200 W. The shape of the wave-front dis-tortion, neglecting spherical terms, is shown in Fig. 3,along with the Zernike coefficients. The degree ofspherical aberration is due to a combination oftemperature-dependent thermal conductivity and re-fractive index in Nd:YAG, and nonuniform pumping.The presence of a beam that is extracting significant
amounts of power also influences spherical aberra-tion by locally altering the branching of energy flowthrough Nd levels and changing the ratio ofradiative-to-nonradiative relaxation.5 This resultsin heat rate reductions that depend on the local in-tensity of the extracting beam. This not only re-duces the power of the overall rod thermal lens butalso can reduce dramatically the amount of sphericalaberration, particularly when the beam is most in-tense on axis. In an oscillator this presents the dy-namically messy situation of a slow feedback loopbetween the formed beam and the optics of the reso-nator that formed it, which we decline to address here.
4. Effect on Resonator Optics
The fact that helicoid modes are shaped like annuliwhose radii increase with increasing mode indexcauses them to experience only the effective power ofthe laser rod at the radius of their peak. The aber-ration then may cause the scale size of the helicoidmodes to vary with mode index, because the effectiverod lens power differs for each mode. The possibilitythat � in Eq. �1� can vary with m in our analysisdemands that hereafter we specify the mode index inthe scale size with a subscript, i.e., �m.
To illustrate this behavior, consider the simple res-onator model shown in Fig. 4. It consists of: 1� ahighly reflective �HR� mirror with convex curvature Cto compensate the thermal lensing of the rod, 2� alaser medium with a focal power D modeled as asimple lens, 3� a propagation distance L and, 4� a flatoutput coupler mirror. The propagation distancebetween the HR mirror and the rod is neglected inthis analysis. The ray-transfer matrix for this res-onator, starting from the position just to the right ofthe rod �the analytic plane� is
� 1 2L2�C � Dm� 1 � 4LDm � 4LC� . (5)
Since helicoid modes are a subset of Laguerre–Gaussian modes, which are resonator eigenmodes,the resulting mode scale size is
�m � (�L� � 1
1 � �Dm � C� L��Dm � C� L�1�2)1�2
(6)
Fig. 3. Measured wave front distortion in a 450-W pump chamberwith a 5-mm diameter rod and the spherical component removed.Vertical scale is in microns. Zernike radius � 2.0 mm. Zernikecoefficients in microns: Z20 � �0.08, Z21 � �9.433 �not plotted�,Z22 � 0.419, Z30 � 0.088, Z31 � �0.187, Z32 � �0.06, Z33 � 0.012,Z40 � 0.054, Z41 � �0.006, Z42 � 0.449 � � 0.195�, Z43 � 0.116,and Z44 � 0.067.
Fig. 4. Optical resonator model used in simulations. The ana-lytic plane, unless otherwise specified, lies immediately to the rightof the laser rod. Mode scale sizes and wave front curvatures aredetermined at this plane.
Figure 5 graphs the scale size of the modes in the rodas a function of rod focal power D. The fundamentalmode size � in the gain medium has a U-shapedgraph �“U graph” hereafter� of the sharply delimitedextent from C to C 1�L, which represents the re-gion of geometric resonator stability. On the rightside of the U-graph, d��dD � 0; on the left side,d��dD � 0. Now Eq. �6� relates �m to Dm, whilesubstitution of �rm� into Eq. �2� and isolating �m givesthe further relation
�m � ��� � Dm�2b2
m ��1�2
, (7)
also between Dm and �m, which we chart in Fig. 5 asa family of curves �“m-index curves” hereafter� forvarious m radiating from the point �0, �� to illustrateits behavior. Where these curves cross the U graphare solutions for the scale size and the effective lens-ing for a particular mode. Modes whose peak radiilie within the rod aperture require � � Dm � ��1 � �, which region is outlined in dashes in Fig. 5. Theconvex curvature of the back mirror can be used totranslate the U graph laterally in the diagram so thatthe m-index curves can be made to cross it anywhere.What matters for now is that the m-index curves cancross the U graph in either the regime where the Ugraph has a positive slope or a negative slope. In theformer case, the mode scale decreases with the index;in the latter the scale increases with the index. Twopeculiarities are noted.
First, � may lie beyond the stability region of the Ugraph, and since the m-index curves sweep to the left
some of them will intersect the U curve in regionsthat represent stable modes within the rod aperture.But m � 0 �whose graph is a vertical line at �� andother low-order modes will not intersect the U graphwhere oscillation is possible. The aberrated oscilla-tor therefore experiences a virtual obscuration cen-tered on the rod axis.
Second, when C is close to D, some m-index curvescross the U graph twice, indicating two solutions forscale size. Furthermore, there is obviously a maxi-mum mode index that may lie within the accessablerange of rod focal powers. In this case, the resonatorexperiences a virtual aperture that is smaller thanthe rod aperture. The maximum mode index, m�,falls where the effective lens power is
Dm� �18 �2� �
3L
� 6C � �4�� � C�2
� �9L
� 4�� � C�� 1L�
1�2� . (8)
If Dm� � ��1 � � then there exists a virtual aperture.This allows the possibility that the modal superposi-tion may not completely fill the gain medium, result-ing in a reduced extraction efficiency.
The dual solutions for some values of mode indexpresent the scenario that a pair of modes can existwith two different scales. Near the maximum modeindex, the scales are very close, leading to large over-lap and cross-saturation between the differentbranches. Modes with large cross-saturation typi-cally do not both oscillate; the dominant �lowest-loss,highest-gain� mode quenches the oscillation of theless-favored mode. Lower scale size is often enoughto guarantee winning the competition, hence theouter mode seldom occurs. Where the scale size isgreatly different, reducing mode competition, theouter mode is so large that it suffers some truncationloss to the rod aperture and is unfavored in compar-ison with other modes of higher index, but smallersize. We have assumed in the numerical modelingof this study that only the lower scale intersections ofm-index curves represent viable modes.
Symmetric and periodic resonators are so com-monly encountered in high-power lasers that theydeserve special consideration here. The U-graphfunction for a symmetric resonator is
�m � (�L� � 1
4 � �Dm � C� L��Dm � C� L�1�2)1�2
, (9)
where L is now the total resonator length and C is thepower of negative lenses or curves on rod ends used tocompensate overall rod lensing. The strategy forachieving brightness in these oscillators is to adjust Cand L to have the m-index curves of Eq. �7� intersectEq. �6� for low-order modes only. Symmetric reso-nators have inherently smaller mode scales for agiven L so that it is necessary to make L large to limitmode content for a given aperture. This makes theU graph so narrow that both d��dD regimes are
Fig. 5. Solutions to mode scale size satisfying both Eq. �6� and Eq.�7�. U-shaped curve is a graph of Eq. �6� having a width of theinverse of the cavity length. It is displaced to the right by thevalue of the end mirror curvature C. Equation �7�, represented bya family of curves with helicoid mode index as the parameter,radiating from ��, 0�, modes of index 10, 30, 80, and 160, areillustrated. The dashed rectangle of width � shows the range ofpossible values of D that fall within the radius of the rod withaberration coefficient . Where the radiating curves cross theU-shaped curve are solutions for effective mode lens power andscale size. From zero to two solutions are possible numerically,but these are often eliminated by falling outside the range � .Clearly, a critical value of m exists, above which there is no solu-tion. Dual solutions are possible with markedly differing scalesize, but the same index.
present in the same oscillator, and the system mayhave both a virtual obscuration and a virtual aper-ture.
5. Impact of Mode Scale Variation on Beam Quality
A given rod aperture will be filled with all modes thathave low truncation loss out to a maximum modeindex. The larger that mode index, the higher M2
will be. From the above characterization, resona-tors with d��dD � 0 for Eq. �6� mode scales decreasewith increasing mode index, allowing higher-ordermodes to pack into a given radius. The opposite istrue of resonators with d��dD � 0 for Eq. �6�: Theyinvariably contain fewer or lower-order modes. Thisleads to the surprising conclusion that some resona-tor designs will show improved beam quality whenspherical aberration is present, while others will bedegraded.4
It is important also to recognize that without ab-erration all modes have the same radius of curvatureat the initial plane in the resonator. Aberration notonly induces changes of scale size between modes butalso makes their radii of curvature different. As wehave seen with scale size, there is a progressive al-teration of mode curvature with index, and this ef-fectively causes the mean wave front of the beam toassume a nonspherical shape. The curvature foreach mode, Cm, is simply
Cm � C � Dm, (10)
and since Dm for most modes is progressively smalleras mode index increases, we find that the curvature ofthe mode is negative �converging� at low indices andthat its absolute magnitude decreases with increas-ing index. This is a characteristic of negative spher-ical aberration, and it should be no surprise that thebeam in the resonator has had the same sense ofspherical aberration in the rod imprinted on it.Thus there seem to be two components of beam qual-ity degradation in a spherically aberrated resonator:1� increasingly high modal content and 2� an overallnonspherical wavefront. The spatial separation ofthe component modes allows one to correct at leastpart of the degradation with external optics. Thisresult is valid at the analytic plane, but not neces-sarily anywhere else in the resonator. For instance,to the left of the aberrating rod in our model all wavefronts must have the same curvature as the HR mir-ror and Cm � C. Since the mode content has notchanged to the left of the rod but the wave frontaberration is gone, the beam quality must be betteron the left of the rod than at the analytic plane on theright. This surprising result not only is borne out bythe numerical results below but also is predicted byintensity moment theory4 of aberrated resonators,which we shall discuss below.
6. Results of Numerical Modeling
We carried out a detailed simulation of these kinds oflaser oscillators that used an iterative process to findthe steady-state solution for the partitioning of power
among modes that had scale sizes perturbed byspherical aberration. Typical parameters for the la-ser were selected on the basis of a high-power diode-pumped Nd:YAG gain module that was capable ofover 450-W multimode oscillator power. The simu-lation parameters are shown in Table 1.
The method used for numerical simulation was thesame as that described in Ref. 6. The average totalintensity profile is an incoherent superposition ofthese profiles with coefficients Pm, where this valuerepresents the circulating power of the mth mode:
I�r� � �Pm Im�r�. (11)
In the steady state the effective gain for each trans-verse mode is equal to the loss, including truncationloss. The effective gain for a particular mode is themode-intensity-weighted mean value of the gain inthe presence of the saturating intensity profile of allother oscillating modes, hence
gn�Pn� � �0
2�
2� �0
b
In�r�
�g0 rdr
1 �1Is��
m�nPm Im�r� � Pn In�r��
� �n, (12)
where g0 is the round-trip unsaturated gain; Is is thebidirectional saturation intensity, assuming the in-tensity is the same in both resonator directions; and�n is the total loss, including output coupling anddiffraction on resonator apertures, which can differfrom mode to mode and therefore bears a mode indexsubscript. Provided we know the other Pms, we cansolve Eq. �12� numerically by iteration to determinethe value of Pn, which is consistent with them. Thenthe process is repeated with the other modes, updat-ing the set of Pm as we go. The resulting set of modepowers is summed to find the total power. After thecomplete pass through all modes we begin again tosee if the mode powers represent a steady state,which is detected when the fractional change in Pmfor every mode with each pass is below a giventhreshold, say, 10�4. We have used a value of 1500W�cm2 for bidirectional saturation intensity, consis-tent with an effective emission cross-section of 3 �10�19 cm2 and a fluorescence lifetime of 230 ms.
Table 1. Simulation Parameters
Single pass unsaturated gain coefficient 0.65Rod diameter 6.35 mmResonator length 25 cmBack mirror curvature variableSaturation intensity, bidirectional 1500 W�cm2
The gain used in the calculations corresponds to adiode pump power of approximately 1440 W. Theresonator model was for flat-end mirrors and a simplelens representing the focal power of the pumped rodplaced at the HR mirror. Up to 140 modes are usedin the simulations.
In contrast with the method of Ref. 7, it is neces-sary to use modes of the form of Eq. �2� with the scalesizes, �m, determined self-consistently for each modeindex before proceeding with the saturation calcula-tion. This is done by setting Eq. �6� equal to Eq. �7�and solving the resulting quartic equation in Dm.These Dm are then used in Eq. �6� to determine �m.
We first treat the case d��dD � 0. Figure 6 plotsthe radial intensity profiles of various modes wherethe back mirror is flat and the axial lens power isvaried. The plots clearly show the effect of the vir-tual obscuration progressing until only a single high-order mode can oscillate at the periphery of the laserrod. We demonstrated this behavior experimentallyin a resonator by reimaging the beam profile in thelaser rod on a CCD camera. The profiles are shownin Figs. 7�a�–7�c�. The resonator configuration wasas in Fig. 4, using a pump chamber with a 6.35-mmrod and a rod-to-output mirror spacing of 25 cm.The HR mirror used was flat. As the current isturned up to increase the lensing in the rod to acritical point to expand the low-order mode sizes, theyare seen rather to quench, leaving only the modesoscillating in the rod periphery.
In the simulated case where d��dD � 0, the axiallens focal power is held constant, and the back mirrorcurvature is varied. Radial intensity profiles forthese examples are plotted in Fig. 8. In this series ofplots the effect of the virtual aperture is evident, asthe beam profile progressively withdraws from theperiphery of the rod.
Figure 9 shows the results of the simulations onpower and beam quality. The crosses on the dia-gram represent d��dD � 0 resonators. In these
cases the beam quickly became annular and wouldnot reach a beam quality less than M2 � 18. Thepower dropped drastically because of a poor extrac-tion of the thin annular beam. The d��dD � 0 res-onators, represented by circles, achieved better beamquality, but not without a significant sacrifice inpower. The scatter in the data is due to variations inmode filling that are sensitive to a variation in backmirror curvature. The solid curve is the fit to theresults of the calculations without aberration.
Clearly, the virtual aperture effect is the reason forthe poor extraction efficiency with this kind of rodaberration. Since the cause of the virtual apertureis the shape of the U curve, are there other resonatortechniques that can alter this enough to overcome themode index limit? One possibility is the resonatorsthat use a graded reflectivity output coupler �GROC�with a Gaussian reflectivity profile.8 GROC resona-tors have a U curve that has no hard limits to therange of mode scales in D. Where the m-indexcurves sweep away from the U curve in Fig. 6, theGROC resonator can allow the U curve to interceptthem to establish a stable mode that better fills therod aperture. The analysis of higher-order modes inresonators with Gaussian mirrors that use helicoidmodes is simplified by the fact that the resultingcomplex helicoid modes have spherical wave fronts,which is not the case with more general modes.7
7. Comparison with Intensity Moment Theory
Intensity moment theory4 involving higher degreemoments of the beam intensity distribution has beenused to derive relationships that are present betweenthese moments in a resonator that has the transversequartic phase aberration
�� x� � exp��ikax4�, (13)
where the aberration coefficient a is
a ��
b2 (14)
in terms of variables used previously in this paper.In addition to the moments, �x2�, �u2�, and �xu���x2��related to squared beam width, squared divergence,and mean curvature�, the moments �x4�, �x6�, and�x3u� are developed to describe a beam that is quar-tically aberrated in only one dimension. One of themost important products of this theory is a predictionof beam quality in terms of these higher moments.The formula for beam quality in terms of M2 is
M2 �2�
��x2���4K2a2�x2�2 � 2Ka�x2��1
L� 2��
� ��
L� �2��1�2
, (15)
where K is the coefficient of kurtosis, defined as �x4���x2�2. This parameter is a measure of the “sharp-ness” or flatness of a distribution. For instance, aflat distribution �“platykurtic”� has K � 9�5, and a
Fig. 6. Numerical simulation mode profile results of cases of op-eration where U-graph slope is positive. Conditions are as inTable 1. Back mirror curvature, C � 0. Axial lens power � isvaried between profiles: a, � � 0.039; b, � � 0.03999; c, � � 0.042;d, � � 0.044; e, � � 0.046.
Gaussian distribution has K � 3. The formulashows the interesting property that an aberrationwill improve beam quality when 1�L � 2�. This isequivalent to the case when d��dD � 0, and boththeories predict the same qualitative result. Thefact that our mode calculations make it possible tocompute the beam’s second moment and kurtosis sothat M2 may not only be computed directly but alsoinferred from Eq. �15�. We did this for two caseswith positive and negative slopes, and the results areplotted in Fig. 10.
The curves are qualitatively similar for both theo-ries, approaching the same values for a � 0. Oth-erwise, the intensity moment formula gives a greatereffect than helicoid mode analysis. This effect ap-pears to be a constant scaling of the value of a, the a
of helicoid theory being about 2�3 of the a of intensitymoment theory. This difference can be understoodon the basis of the difference in symmetry of theproblems that the two theories treat. The publishedintensity moment theory is strictly one-dimensionalCartesian and is extended to two dimensions if wetreat the aberration function as the product of twoone-dimensional quartic aberrations, or
�� x� � exp �ika� x4 � y4� . (16)
However, helicoid mode theory, while calculating mo-ments �hence, M2� in terms of rectangular �not radial�coordinates, treats the aberration as a true radial
Fig. 7. Experimental demonstration of the virtual obscuration effect in an aberrated oscillator. Resonator as in Fig. 5. L � 25 cm, C �0. Image of rod is projected onto CCD camera through a flat back mirror. Drive current to 80 bar pump chamber is �a� 21.5, �b� 22, and�c� 22.5 amps. Output power is �a� 277, �b� 187, and �c� 101 W.
quartic function, which expanded to rectangular co-ordinates becomes
�� x� � exp �ika� x4 � 2x2y2 � y4� . (17)
This is clearly a different case owing to the crossproduct of the coordinates. To treat the radiallysymmetric aberration, intensity moment theory willneed to include additional higher moments such as�x2y2� and �u2v2�. The effect this will have can beseen to reduce the impact of the aberration in theradial case when compared with the rectangular. Inthe case of rectangular symmetry, the aberration di-rects wavefront normals away from the y axis by thesame magnitude regardless of the y coordinate.With radial symmetry, the wave front normals havex components that are maximal for y � 0 and arereduced away from the x axis. Furthermore, for agiven x, the weighting factor �intensity� falls awayfrom the x axis in the radial case but not in therectangular case, so these components also contributeless to the overall effect on the beam.
A noteworthy conclusion from Fig. 10 is how lowthe quartic coefficient has to be in order not to have asignificant impact on mode quality in these specificresonators. In the case of our experimental laser,the aberration coefficient must be reduced by a factorof over 40. Some resonator designs are less sensitivethan others, but they are also lower in beam quality.
Equation �15� also predicts that mode quality willimprove in some resonators and worsen in others,depending on the sign of 1�L � 2�. If this factor ispositive, that is when the length of the resonator isless than half of the focal length of the lens, the modequality improves. This is the same region in whichd��dD � 0, so both intensity moment theory andhelicoid mode theory predict the same result. Inten-sity moment theory also predicts that beam qualitywill be different on one side of the rod than the other,which we confirmed by separate calculation of beamcharacteristics for helicoid modes on the left side ofthe rod in Fig. 4. Beam quality is in fact slightlybetter there than on the right for the few examples wehave tested, but the amount of improvement to beexpected in all cases is difficult to generalize fromhelicoid mode theory. Intensity moment theorygives a formula for the improvement that depends onthe aberration coefficient a �x4�, �xu�, �x2�, and �x3u�,which must be determined by numerical simulationwith a technique such as ours.
8. Conclusions
We have investigated a technique for the decomposi-tion of a laser oscillator beam into quasi-eigenmodesof the resonator in the presence of axially symmetricspherical aberration. Our previous work, whichuses these kinds of modes to analyze the unaberrated
Fig. 8. Numerical simulation results showing mode profiles ofresonators operated with negative U-graph slopes. Conditionsare as in Table 1. Axial lens power, � � 0.04. Back mirrorcurvature is varied between profiles: a, C � 0.03; b, C � 0.035; c,C � 0.036; d, C � 0.037; e, C � 0.038; f, C � 0.039; and g, C �0.0399.
Fig. 9. Overall graph of all simulations showing power versusmode quality.
Fig. 10. Comparison of results of mode quality versus aberrationcoefficient for two resonator designs with intensity moment theoryand helicoid mode theory. Parameters chosen for the illustratedtwo cases are as in Table 1, with back mirror curvature chosen tomake d��dD � 0 �C � 0� and d��dD � 0 �C � 0.037� when theaxial rod focal power � was � 0.04.
resonator, allowed us to calculate output power,beam quality, and higher moments of the intensitydistribution. These were compared with the one-dimensional Cartesian intensity moment theory andfound to have a quantitative difference that likely isdue to the different symmetries that are treated.The impact of spherical aberration on oscillator beamquality depends on the regime of oscillator operationand can either improve or worsen beam quality.The kind of spherical aberration commonly encoun-tered in Nd:YAG lasers has a detrimental impact onlaser efficiency owing to the virtual aperture�obscu-ration effect peculiar to these resonators. We sug-gest that properly designed GROC resonators can toa degree overcome some of the virtual aperture effect.
In this study, we have chosen to ignore the effectsof rod strain birefringence and bilensing. Unlessthese effects can be compensated by the manipulationof polarization, even unpolarized lasers will experi-ence other sources of mode perturbation that frus-trate the achievement of high power and good beamquality. At the high thermal loading that manysolid-state laser systems are capable of achieving,compensation may be only partial.9 It may be pos-sible to analyze such laser oscillators with modifiedvector eigenmodes based on another subset ofLaguerre–Gaussian modes.
Further studies in this area are appropriate.First, the lemma that helicoid modes are quasi-eigenmodes of a resonator with a radial quartic phaseaberration needs more rigorous mathematical sub-stantiation. Such a treatment would reveal the lim-
its and conditions over which the approximation isvalid, which are not obvious from these numericalexperiments. Second, intensity moment theoryshould be recast in cylindrical coordinates so that theresults of both approaches can be compared.
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