Modeling of the gain distributionfor diode pumping of a solid-state laser rodwith nonimaging optics

R. John Koshel and 1. A. Walmsley

We investigate the absorption distribution in a cylindrical gain medium that is pumped by a source ofdistributed laser diodes by means of a pump cavity developed from the edge-ray principle of nonimagingoptics. The performance of this pumping arrangement is studied by using a nonsequential, numerical,three-dimensional ray-tracing scheme. A figure of merit is defined for the pump cavities that takes intoaccount the coupling efficiency and uniformity of the absorption distribution. It is found that thenonimaging pump cavity maintains a high coupling efficiency with extended two-dimensional diode arraysand obtains a fairly uniform absorption distribution. The nonimaging cavity is compared with two otherdesigns: a close-coupled side-pumped cavity and an imaging design in the form of a ellipticalcavity. The nonimaging cavity has a better figure of merit per diode than these two designs. It alsopermits the use of an extended, sparse, two-dimensional diode array, which reduces thermal loading of thesource and eliminates all cavity optics other than the main reflector.

1. IntroductionDiode lasers are a logical choice for the pump sourcefor highly efficient, high-output-power, solid-stateamplifiers or lasers in the quasi-cw regime. Becauseof the narrow linewidth of the laser emission, diodepumping is more efficient than flash-lamp pumpingand reduces the problems caused by thermal loadingin the solid-state gain medium. Because the radiantexistance of a diode array is quite different from thatof an arc lamp or flash lamp, however, the problem ofcoupling the diode radiation into the laser gain me-dium with a prescribed distribution is significantlydifferent from that of a lamp-pumped system. Herewe address some of the issues associated with cou-pling radiation from an extended, non-Lambertian,nonhomogeneous source into a smooth absorptiondistribution.

Several configurations for the diode pumping of asolid-state gain medium exist, and we categorizethem into two groups: end pumping and side pump-ing. The end-pumping configuration uses diodeswhose output is directed along the mode axis of thelaser. Three geometries that use end pumping are

The authors are with The Institute of Optics, University ofRochester, Rochester, New York 14627.

Received 16 April 1992.0003-6935/93/091517-11$05.00/0.e 1993 Optical Society of America.

close-coupled,' tightly folded resonator,2 3 and fiber-coupled4 5 pump cavities. End-pumping schemes cou-ple a large amount of the pump radiation into thesolid-state medium in such a way that the gaindistribution closely matches a TEMOO mode; there-fore, end pumping provides an efficient method ofpumping for this mode. Unfortunately, end-pump-ing geometries are not easily scalable to outputpowers greater than a few decawatts 6 because of theincreased complexity of the coupling optics or ofalignment. Close-coupled end pumping is limited bythe size of the TEMOO mode for reasonable configura-tions. One solution to this limitation is to pack thediodes more densely and another is to use morepowerful diodes, but both solutions cause a buildup ofheat in the diodes. To compensate for this buildupwe must reduce the duty cycle of the diodes, which inturn reduces the amount of available pump power.Fiber coupling provides an efficient method of usingmore diodes but requires more collimating optics;therefore, the complexity of the system increasesdramatically, and its alignment becomes trouble-some.

These factors make end pumping unfeasible for thehigh-power quasi-cw operation of a solid-state laser,so for the highest power lasers, side pumping isfrequently used. In side-pumped configurations theoutput from the diodes is directed along paths thatare different from those of the mode axis of the laser.

20 March 1993 / Vol. 32, No. 9 / APPLIED OPTICS 1517

There are several possible configurations, such asclose-coupled, imaging, and nonimaging configura-tions. The main advantage of side pumping is that itpermits a larger number of diodes to be used, andtherefore is common in the highest energy pulsed-laser systems. It is not a common configuration inlow-powered systems for several reasons. First, itnecessitates a reduced absorption path length in thegain medium. In end pumping the path length istypically long enough to absorb almost all of theincident pump radiation, but for side pumping thelength of the absorption path is usually less than twoBeer's lengths, except in multiple-pass pump-cavitydesigns. A consequence of this length is that unlessthe solid-state medium is highly absorptive, there is asignificant amount of loss caused by pump radiationpassing through the gain medium and not beingabsorbed. Second, the distribution of gain withinthe laser medium is not typically well matched to themode profile, which leads to smaller efficiencies. 7

Although not as efficient as end-pumping methods,a close-coupled side-pumping geometry that usesdiodes placed in proximity to the surface of the gainmedium8 9 does give a fairly good match of theabsorption distribution to the mode profile. It iscurrently the most efficient side-pumping method,with reported optical slope efficiencies from a Nd:YAGof 47.7% in a multimode operations and 32% in aTEMOO operation. Unfortunately, the total amountof pump power absorbed is limited by the number ofdiodes that can be placed adjacent to the gain medium.Only a few diodes may surround a cylindrical gainmedium of small diameter, and therefore this systemis not scalable to high output powers in the quasi-cwregime and is not suitable for our purpose here.

To obtain a high output power from the solid-statelaser or amplifier we must use more diodes, so opticalcomponents that efficiently transfer the pump radia-tion to the gain medium must be included in thedesign. One way to make this inclusion is to incorpo-rate into diode-pumped designs the reflective, imag-ing pump cavities often used in lamp-pumped lasers.An imaging pump cavity is one for which one point ofthe pump source and one point in the gain mediumare conjugate. An example of this cavity is a cylindri-cal, elliptically shaped reflector in which the diodearray is centered along the line through one focus,perpendicular to the major axis of the ellipse, and theaxis of a cylindrical gain medium is placed at the otherfocus. An imaging pump cavity permits the sourceto be located away from the gain medium so that alarger number of diodes, less densely packed, may beused. The imaging properties of an elliptical cavityprovide efficient coupling when the source and ab-sorber are located at the foci, but if either is movedaway from its respective focus the coupling falls offdramatically. Therefore, there is a limit to the sizeof the pump source and coupling efficiency, whichmakes this pump cavity unsuitable for extremelyhigh-power operation.

The most suitable pump cavity is one that effi-

ciently concentrates the radiation from many diodelasers into the solid-state gain medium when thediodes are not densely packed and are placed over alarge entry aperture. By sparsely packing the diodeswe can reduce thermal loading problems in the array;therefore wavelength-emission variations can be re-duced. This diode source is hereafter called an ex-tended, sparsely packed, two-dimensional diode array(ESDA). A possible candidate for a pump cavity thatcan efficiently transfer the radiation from an ESDA tothe gain medium is a nonimaging concentrator. Theprimary purpose of a nonimaging concentrator is toachieve the highest throughput of light from theinput aperture (i.e., the source) to the exit aperture(i.e., the surface of the gain medium); therefore, thetransfer efficiency of these pump cavities is high.In addition, the nonimaging cavity design offers otherbenefits: scalability and alignment-tolerance insen-sitivity.

To obtain the efficient operation of a solid-statelaser or amplifier, we see that the absorption distribu-tion, which can be assumed to match the gain distribu-tion,1 must be mode matched to the output mode ofthe resonator. 12 For a TEMOO mode as the output,the ideal absorption distribution should be Gaussian.For such an output this ideal precludes the use ofslab-laser geometries that use zig-zag beam propaga-tion. These systems give a highly efficient multi-mode operation but an inefficient TEMOO mode opera-tion.13 For cylindrical gain media the end-pumpingand close-coupled side-pumping geometries approachthe ideal TEMOO mode-matched absorption distribu-tion, but they have the total pump-power limitationsdescribed previously. Although the nonimagingpump cavity provides a large amount of pump power,it does not give a Gaussian absorption profile. Oneof the main purposes of this study is to examine theabsorption profile generated in a cylindrical lasermedium when pumped by an ESDA by means of anonimaging concentrator.

The considerations outlined above lead to a numberof design goals. The system examined in this studyis for an optical amplifier with a TEMOO input andoutput. The amplifier is to be operated at a highaverage power with a low repetition rate; therefore, alarge number of diodes must be used as the pumpsource, but cooling of the laser rod is not deemed to bea problem. The research is easily extended to anoscillator in the quasi-cw regime. To reduce thermalloading problems in the source, we find that the arealpacking density of the diode array must be small;therefore an ESDA is to be used. A simple systeminvolving only the source, the gain medium, and thereflector is desired to alleviate alignment concernsand the overall expense of the system. A solutionthat satisfies most of these design criteria is thecompound parabolic concentrator (CPC) for a circularabsorber, which is designed from the edge-ray princi-ple of nonimaging optics.' 4 The CPC is designed toconvert the radiation from a large-area homogeneoussource emitting uniformly into a small angle to a

1518 APPLIED OPTICS / Vol. 32, No. 9 / 20 March 1993

small-area homogeneous output emitting uniformlyinto a large angle; thus the source tendue is pre-served in this design. This preservation implies thatthe irradiance is larger at the output surface than atthe entrance aperture, leading to a net concentrationof the pump radiation. The ESDA is a large-areasource with a moderate emission angle (i.e.,OFWHM 'r/3), and the CPC pump cavity converts theemitted radiation onto the smaller rod-surface area;therefore, to preserve the 6tendue, the CPC increasesthe maximum output angle. The only conditionunfulfilled by the ESDA is that it is not a homoge-neous source with a uniform angular emission, whichimplies that the laser rod surface is nonuniformlyilluminated. Previous research with CPC's has inves-tigated the energy distribution on the absorber sur-face arising from nonhomogeneous, nonuniformsources,15 but in the case of optical pumping thisinvestigation must be extended to finding the energydistribution within the absorber.

In Section 2 the modeling procedure is discussed.In particular, a figure of merit (FOM) is defined,which facilitates the design of the cavity and permitscomparisons of the nonimaging design with otherdesigns. The FOM takes into account both thecoupling efficiency and uniformity of the gain distribu-tion. In Section 2 we also give results of a six-diodeclose-coupled side-pumping geometry as an example.The results for the nonimaging pump cavity whenpumped by an extended two-dimensional array aregiven in Section 3. A comparison to the close-coupled side-pumping geometry is also made in thissection. In Section 4, the nonimaging pump cavity iscompared with an aspheric imaging pump cavity inthe form of an ellipse. Finally, Section 5 presents adiscussion of issues raised by this study and gives ourconclusions.

2. Theoretical Model Used in Calculating the GainDistributionThe proposed system consists of a pump-cavity geom-etry of similar symmetry to that shown in Fig. 1.The rod is placed along the axis of the cavity (labeledas the z direction), and the extended two-dimensional

2D DiodeArray -

Laser Rod

Fig. 1. Three-dimensional view of the proposed pump cavity.

Power

0.5 .2

Fig. 2. Radiation-emission pattern of individual diode lasers,comprising a Gaussian distribution in the angle in the tangentialdirection (perpendicular direction of the diode) and two equallyoffset Gaussian distributions in the angle in the sagittal direction(parallel direction of the diode).

diode array is placed near the mouth of the reflectorwith the diodes pointing downward parallel to the yaxis.

A. Geometry of the Source

The diode laser is modeled as a point source, with anonuniform elliptical beam cross section as shown inFig. 2.16 The radiance in the perpendicular directionis described by a Gaussian angular profile, and therabbit-ear profile in the parallel direction is modeledby the summation of two offset Gaussian angulardistributions separated by 20,,. Here art is the stan-dard deviation in the angle of the perpendiculardirection and as is the standard deviation in the angleof one of the lobes in the parallel direction.17

The plane containing the small divergence angle ofthe diode output is aligned parallel to the axis ofsymmetry of the pump cavity. The reason for thisalignment is that the size of the pump cavity in-creases dramatically with a smaller entrance angle, sothis choice returns a reasonable entrance-aperturedimension. A three-dimensional ray trace is per-formed. Because of the cylindrical symmetry of thecavity, only ray directions without a component in thez direction (i.e., the tangential direction) must betraced to the surface of the gain medium. Uponstriking the absorber, the entire ray set, includingskew rays, must be traced because of the dependenceon the z component in the refraction equation.

Figure 3 shows the procedure for determining theangle of the ray when Mt tangential rays are tracedfrom each diode. The area under the Gaussiandistribution is divided into Mt regions of equal area,and a ray is traced at the angle that lies at themidpoint of each of these regions.18 Using ou, we usethe same procedure to determine the angles in onelobe in the sagittal direction. The number of raystraced per lobe M,(Ot) is given as a function of thetangential ray angle Ot by

Ms(ot) =Mts(1 - 0t2 /Otm 2 )1/2

°-t

20 March 1993 / Vol. 32, No. 9 / APPLIED OPTICS 1519

/

A3A2 Al A. A2 A3A

4- 1 1 1 1 1 ~~0-33 32 -1 01 02 03

Fig. 3. Gaussian radiation pattern in the angle of an individualdiode laser. The ray-angle set is determined by setting Ai = A fori= -1, 2,. .,±(M - 1)/2;theangles0i,aresetatthemidpointsof each of these regions along the 0 axis.

where Ot is the tangential ray angle that is currentlybeing traced and 0

t,m is the maximum tangential rayangle from the diode. Each ray is arbitrarily given apower rating of one unit, so each diode has a totalpower equal to the number of rays traced from it:

Mt

Pciode = 2 E M3(O 1i). (2)

A standard diode source is used throughout furthersections to facilitate a comparison between the vari-ous types of pump cavities. This source is a pointsource emitting an angular radiation pattern as shownin Fig. 2 with standard deviation angles of Ut = rr/12and a = wr/90 in the tangential and sagittal direc-tions, respectively; the lobes in the sagittal directionare centered at Os = Tr/45. From each diodesource in the tangential direction, 501 rays are traced;therefore, the power level (or number of rays traced)per diode is 63070 units as determined from Eq. (2).Only rays from the linear array of diodes in a crosssection of the pump cavity are traced. The plots,however, show the absorption integrated over thelength of the rod. The effects of the additional lineararrays of the two-dimensional array can be easilydetermined from the rays traced. A linear arrayconsisting of 51 diodes spaced 1 mm apart, placedsymmetrically in the cavity, and with all of the diodespointing down is used throughout this study. Fi-nally, the height of the diode array above the rod isoptimized to provide the best FOM, which is de-scribed in Subsection 2.D.

B. Design of the Nonimaging Pump Cavity

To achieve a high output power, we must concentratea large amount of pump radiation in the absorber; theprimary design goal of nonimaging optics is to achieve

exactly this. The principal performance parameterfor nonimaging optical systems is the concentrationratio C:

C =A/A', (3)

where A is the surface area of the input aperture andA' is the surface area of the output aperture thatpasses all transmitted rays. Suppose that all trans-mitted rays have an input angle less than or equal to0 a, hereafter denoted as the acceptance angle; then,because the 6tendue is preserved, these rays strike theoutput aperture.19 The theoretical maximum concen-tration ratio for a two-dimensional concentrator whenthe entrance and exit spaces have indices of refractionof n' and n, respectively, occurs when the output lightis emitted into an angle of'rr/2 and is

C2D = n'/n sin 0a. (4)

The refractive imaging systems that can achieve thisconcentr,-.ion ratio are severely restricted in size,20

where, 3 reflective imaging and nonimaging cavitiesare easily scalable. The loss of image formation isnot significant in the application discussed here; infact, we want to avoid the formation of an image ofthe diode array in the gain medium.

A nonimaging concentrator that gives the maxi-mum concentration for a two-dimensional cavity isbased on the edge-ray principle.2' It is labeled as theCPC and is designed for a planar absorber.'4 Thistype of concentrator has recently been proposed foruse in the diode side pumping of solid-state slabmedia.22 Fortunately, the edge-ray principle, andtherefore the CPC, can be extended to nnplanarabsorbers. 23 24 The CPC for a circular absorber,hereafter labeled simply CPC, has been investigatedin lamp-pumped systems25 and solar-pumped sys-tems,26 27 but it has only been recently investigated indiode-pumped systems.28 The edge-ray principle fora circular absorber of radius a is a two-region reflec-tor defined by29

p = aO for 101 < a + I/2, (5a)

0 + a + Tr/2 - cos(O - Oa)a 1 + sin(O - Oa)

for Oa + fl/2 < 0 1 < 3/2 - Oa- (5b)

As shown in Fig. 4, for a ray that enters the CPC at anangle of O and strikes the rod tangent to its surface,p = RA is the distance along the ray path from the lastintersection point of the ray with the CPC, A, to thetangent point on the rod, R. The angle measuredfrom the negative y axis to the line segment joiningthe origin 0 and the previously described tangentpoint R is 0 = DOR. The surface of the firstregion, parameterized in Eq. (5a) for 101 < Oa + rr/2,is the involute of the circular absorber (in Fig. 4 it isthe curve B'DB) and ensures that rays entering atangles less than the design angle of 0a will strike theabsorber after a finite number of reflections. Equa-

1520 APPLIED OPTICS / Vol. 32, No. 9 / 20 March 1993

P

y

C'

p e xNB'' D B

Fig. 4. Cross section of the CPC pump cavity. The cavity isdesigned with the edge-ray principle of nonimaging optics; 0a is theacceptance angle of the cavity and a is the radius of the laser rod.

tion (5b) gives the parameterization of the surface ofthe second region (in Fig. 4, it is the two curves B'C'and BC) and ensures that any ray entering theentrance aperture at the design angle of Oi = a strikesthe circular absorber tangent to its surface after onereflection. Notice that a does not have to equal theradius of the solid-state laser rod; it is simply a designparameter.

The entrance aperture size, CC', is equal to2arr/sin(0a); therefore, as the design acceptance angleis decreased, the size of the cavity aperture increases.This phenomenon permits many diodes to be placedon or above CC', but it also creates a rather largeaperture through which rays can be ejected from thecavity after a single pass through the gain medium.Additionally, because the CPC size scales to 1/sin(0a),to get a reasonably sized concentrator we must choosea moderate value of Oa. For this reason the CPC isdesigned with the larger of the two diode-emissionangles (i.e., the divergence angle from the perpendicu-lar direction).

In the calculation presented in Section 3j a is equalto the radius of the solid-state laser rod. The rod isair cooled for this amplifier design, but a gap for acooling jacket can be included by either offsetting therod or by choosing a larger value of a than the truerod radius, which permits room for a cooling jacket.In addition, the reflector wall is assumed to be coatedwith silver, which is 98.8% reflective at. normalincidence at the pump wavelength. Finally, a isequal to twice the angular standard deviation of thediode source in the tangential direction (i.e., Ut).This value ensures that 95% of the pump radiationstrikes the absorber. The effects of changing theacceptance angle have been studied. Optimizing theFOM by ray tracing with varying'Oa values shows thatthe optimum pump cavity is obtained for an accep-tance angle that lies within 5% of the condition of thetwo standard deviations.

Tolerancing of CPC cavity designs for other applica-

tions has been studied by other authors. 0 It hasbeen shown that the shape tolerance of the CPC is notrestrictive (i.e., large-scale deviations in the reflectorprofile are allowed), but the surface must still besmooth (i.e., small-scale deviations in the reflectorprofile must be minimized) so that only specularreflections are obtained. If the surface is not smooth,some of the pump radiation is ejected by scatteringbefore striking the rod. Inconsistencies in the large-scale shape of the CPC average out over all inputangles so that some rays entering at less than 0 a maybe ejected but some rays with input angles greaterthan Oa may strike the rod. One method to minimizethe effects of fabrication errors on the large scale is todesign a cavity that uses a value of Oa that is largerthan the maximum angle from the diode source,which provides a larger tolerance in the large-scaledeviation of the CPC surface. Unfortunately, thismethod of removing large-scale errors does not givegood uniformity in the absorption distribution.

C. Geometry of the AbsorberThe cylindrical solid-state laser gain medium has aradius of ar and can be encased in a cooling jacket ofradius a. To calculate the gain distribution withinthis rod, we divide the absorber into Nr equally spacedradial zones and Na equally spaced angular zones asshown in Fig. 5. The radial zones increase fromzone 1 at the surface of the rod to zone Nr near thecenter, whereas the angular zone 1 is located adjacentto the positive x axis and is then labeled counterclock-wise to zone Na. During ray tracing the path lengthin each intercepted zone is found; with this value andthe effective absorption coefficient of the solid-statematerial, the power absorbed within each region iscalculated.

As we previously stated, cooling jackets have beenomitted in this study. The inclusion of a cooling

Radial Zone N,

ua

Radial Zone 1

Fig. 5. Cross-sectional view of the laser rod. The rod is dividedinto Na angular zones and Nr radial zones. The angular zones aremeasured counterclockwise from the positive x axis, whereas theradial zones are measured from the surface of the rod toward itscenter.

20 March 1993 / Vol. 32, No. 9 / APPLIED OPTICS 1521

jacket requires a change in the CPC design. Threealternatives in the design of the CPC are offsettingthe rod to allow a cooling jacket, designing the CPCfor the radius of the cooling jacket, or employing a V-or W-shaped groove at the bottom of the concentrator.The first design alternative introduces loss becausethere is a gap through which rays can pass withoutstriking the absorber. By designing a cavity for thecooling jacket,3 ' we can remove the loss if the edgerays of the cooling jacket surface also strike the laserrod tangentially (i.e., a, < arnc, where n, is the indexof the cooling fluid). The final alternative involvesremoving the cusp at the bottom of the CPC andinserting V- or W-shaped grooves that reflect radia-tion to the absorber. 32 33

In this study the gain medium is a Nd:YAG cylinderwith a radius of 5 mm. The effective absorptioncoefficient, °aeff = 0.25 mm-', and the effective index ofrefraction, neff = 1.824, for the Nd:YAG at 809 nm areassumed.34 Here Nr and Na are equal to 75 and 72,respectively, which gives a total of 5400 zones.Additionally, it is assumed that the rod is coated totransmit 99% of the incident pump radiation.

D. Definition of the Figure of MeritIf the coupling efficiency of the source radiation intothe absorber is the main consideration, then theconcentration ratio C in Eq. (3) is an adequate FOMdescribing the performance of the system. However,for an amplifier or laser in which the distribution ofthe absorbed power is important, C is not a goodFOM. The concentration ratio is useful in the begin-ning stages of the design of the CPC, but it ismisleading further along in the design process. Itsays nothing about the uniformity of the light thatstrikes the absorber and about the absorption-distribution profile that is obtained within the rod.The major problem is that the concentration ratio isdefined in terms of the output and input surfaceareas. In the diode-pumping application both theuniformity of gain and the coupling efficiency are ofequal importance. A FOM that addresses thesepoints should be established.

The FOM for a given pump geometry is determinedfrom the radial and angular gain distributions andfrom the amount of pump radiation coupled into therod. The desired gain-distribution profile is a two-dimensional Gaussian distribution with its peak lo-cated at the center of the rod. The correct gainprofile has been obtained when the radial gain distri-bution is fit well by a Gaussian distribution in thesense of minimizing the error both radially,

Nr

Erad = z jPr,i - PG(ri) (6)

and in the angle,

Na

Eang = Paj - Pu(Oi)I. (7)j=l

Equations (6) and (7) give the sum of the absoluteerror over the zones of the radial and angular gaindistributions, respectively; Pr,i and Pa j signify thedata determined for each individual zone during raytracing in the radial and angular gain distributions,respectively; and PG(r) is the best restricted Gaussianfit in the radius r to the ray-trace data. The fit isrestricted in the sense that the fitted width of theGaussian is exponentially weighted to values that arenear the half-width of the laser rod. Pu(4,) is thebest uniform fit in angle +. Each of the absoluteerrors is then normalized by its respective averageover the ray-trace gain-distribution data: Prad radi-ally and Pang angularly. Finally, the FOM is calcu-lated by forming the weighted sum of these threeterms,

FOM = wpow (1 Pabs Erad Eang+ Wrad + Wang p

Pin lyrad ang

= FOMpo + FMrad + FMang, (8)

where Pabs is the power absorbed inside the rod, Pin isthe total power from the diode sources, i.e., thesummation over the number of diodes of Eq. (2), andthe w terms represent the weightings. Here FOMPOW,FOMrad, and FOMang represent the component FOM'sfor each of the three terms that determine the overallFOM. For the remainder of this study the weight-ings are set equal to one, so each of the FOM terms isequally important. This setting gives the FOM avalue that is greater than or equal to zero, with valuesapproaching zero indicating pump-cavity designs thatclosely match the desired gain distributions and stillmaintain a high coupling efficiency. It is to be notedthat although the FOM is not a direct measure of anycharacteristics of the output of the solid-state ampli-fier or laser, a lower FOM value means that more ofthe pump radiation is being absorbed to match thedesired mode; therefore, the system has a lowerthreshold. We also define a FOM that is specifiedper diode, FOMdiode, which is the FOM divided by thenumber of diodes being used in the particular pumpgeometry. This value reflects the performance ofthe pump scheme for large numbers of diodes (thusfor high pump powers), because with an increasingnumber of diodes and a constant FOM, FOMdiodedecreases. As an example, the FOM for a six-diodeclose-coupled side-pumping geometry is calculatedhere.

The six diodes have been placed 2.5 mm from thesurface of the solid-state laser rod and are pointedtoward the center of the rod.35 There are no cou-pling optics, nor is there a coolingjacket. The sourceand absorber parameters given previously have beenused. A radial plot of the gain distribution inte-grated over the length of the rod is shown in Fig. 6,whereas an angular plot is shown in Fig. 7. Theradial gain-distribution data are normalized to thearea of the annular regions, i.e., the gain powerdensity is plotted. Because the angular plot sumsthe gain over a wedge that has an area equal to other

1522 APPLIED OPTICS / Vol. 32, No. 9 / 20 March 1993

0.0 0.2 0.4 0.6 0.8 1.0r/a

Fig. 6. Radial gain density distribution for the close-coupledside-pumping design when six diodes are used to pump. Thehorizontal axis is the ratio of the radius of the radial zones to theradius of the rod.

wedges, no normalization is required, so the gainpower is plotted. Figure 8 shows a gray-scale plotthat is a cross-sectional view of the absorption inte-grated over the length of the rod. The shades of graysignify the amount of power absorbed in each zone,black signifies the maximum absorption, and whiterepresents little or no absorption. The gray-scaleplot is normalized to the area of the regions made byintersections of the radial and angular spokes; there-fore, the gain density is plotted. The figures showthat the gain distribution is Gaussian, but only overapproximately half the diameter of the rod. Outsideof this radius the absorption from the diodes does notoverlap to make a uniform angular distribution.The various FOM values are given in Table 1.

8000

7000 F

6000 F

5000 FIDP., 4000 F

3000 F

2000 F

1000F

00 2 4 6

8 (radians)

Fig. 7. Angular gain distribution for the close-coupled side-pumping design when six diodes are used to pump. The horizon-tal axis is the angle in radians measured counterclockwise from thepositive x axis of the laser rod.

Fig. 8. Gray-scale gain density distribution for the close-coupledside-pumping design when six diodes are used to pump. Blacksignifies near or at maximum absorption for this plot, whereaswhite signifies little or no absorption.

3. Performance of the Compound ParabolicConcentrator Pump Cavity and Comparison with theClose-Coupled Side-Pumping Geometry

The radial plot for the CPC pump cavity and extendedtwo-dimensional diode-array pump source is shownin Fig. 9, the angular plot is shown in Fig. 10, and thegray-scale plot is shown in Fig. 11. The radial plotshows that the power density steadily increases untilthe radius corresponding to the critical angle of thelaser rod is reached. The radial power density re-mains approximately constant from this radius to thecenter of the laser rod. This constancy shows thatthe CPC cavity is able to transform the input radia-tion emitted over a full-width angle of 20a into afull-width angle of Tr. The absorption coefficient ofthe gain medium is the primary factor for the radialabsorption profile, because more of the pump radia-tion is absorbed at outer radial zones as Oteff isincreased. The angular plot indicates that there is alack of rotational symmetry in the absorption profile,with the minima occurring near the top and bottom ofthe laser rod and the maxima occurring on the sidesof the rod. The reason for this occurrence is that theray-intercept density on the surface of the rod is amaximum on the sides of the rod and a minimum onthe top and bottom of the rod, so caustics are foundon the sides of the rod. To alleviate this phenome-non we must change the placement of the individualdiodes so that the top and bottom receive more pumpradiation and the sides receive less. This change isaccomplished by placing the diodes closer to the edgeof the cavity wall while reducing the number of diodestoward the center of the cavity trough. The CPCalso directs more of the pump radiation toward the

20 March 1993 / Vol. 32, No. 9 / APPLIED OPTICS 1523

12000

10000F

8000 F

G)116000 F

4000 F

2000

0

- f * , * . . , . . ; , ; , . . . . . ,

. ~ ~ .* . . . . . .

Table 1. Merit Values for the Various Cavity Designs

Cavity Design FOMrad FOMang FOMpow FOM FOMdiode

Close-coupled, six-diode 0.1561 0.3388 0.0861 0.5810 0.0968CPC, ESDA 0.2450 0.2293 0.1284 0.6027 0.0118Truncated elliptical, ESDA 0.3051 0.2503 0.4041 0.9595 0.0188

lower half of the rod than toward the top half;therefore, by adjusting the absorption coefficient wecan obtain a more uniform angular absorption profile.Another method of obtaininga more uniform distribu-tion is the use of different reflector designs that give adesired ray distribution but at the loss of concentra-tion. 3637 Finally, the gray-scale plot shows the re-gions inside the rod where caustic formation is aproblem. As we can see from Fig. 11, there are, aswe expected, caustics on the lower edge of the circledescribed by the critical angle radius.38 The variousFOM values for this design are given in Table 1. Incomparison with the example shown in Section 2, theFOMrad for the CPC is higher because the radial gaindistribution is uniform rather than Gaussian, as it isin the close-coupled case. The FOMang actually de-creases for the CPC in comparison with that of theclose-coupled design, showing that a more uniformangular absorption profile is obtained. The FOMPOWis lower for the close-coupled design because the raypath lengths within the rod are longer on averagethan those for the CPC.

The overall FOM values are nearly equal, butbecause the CPC permits many more diodes to beused, the FOMode is much lower for the CPC design.A small FOMdiode indicates that the CPC design ismore suitable for high-power applications than aclose-coupled geometry is because it permits a largenumber of diodes to be placed across the entranceaperture.

50000Fr ' -1

40000 F

a0

-3,1

I0

30000[

20000 F

10000

0. . . . . . . . . . . . . . . . . . . .

0.0 0.2 0.4 0.6 0.8 1.0r/a

Fig. 9. Radial gain density distribution for the CPC side-pumpingdesign of Section 3 and the truncated elliptical side-pumpingdesign of Section 4.

4. Discussion and Comparison with Imaging Systems

To investigate the utility of a CPC pump cavity in thehigh-output-power regime, we need comparisons withother types of pump cavities, particularly a large-entrance-aperture imaging system. One possible im-aging design is the cylindrical, elliptical pump cavitythat is typically used in lamp side pumping. As wenoted previously, refractive imaging systems cannotbe designed to couple an ESDA efficiently and stillachieve a high concentration (see Ref. 20). Theelliptical pump cavity is a reflective design; therefore,the size of the cavity is easily scaled to accommodatethe size of the diode array. Additionally, refractiveimaging pump cavities cannot illuminate the entiresurface of the rod unless they contain multiple-element lens systems (such as the close-coupled side-pumping geometry of Section 2). As in the CPCpump-cavity design, reflective imaging designs suchas the elliptical cavity have a single surface that canpossibly illuminate the entire surface of the rod, sothat pump-cavity optics are at a minimum. Forthese reasons imaging systems should employ reflec-tors and not refractors. The elliptical pump cavity isone of the closest imaging systems to the nonimagingCPC pump cavity, but its utility for diode side pump-ing should be studied. The performance of an ellipti-cal cavity is predicated on the idea that the twogeometric foci are optical conjugates. Thus, an ellip-tical cavity truncated at the latus rectum has a largeentry aperture in which an ESDA can be located.

60000 r . . . . i

50000 .

40000

i 30000P.,

20000

10000

00 2 4 6

0 (radians)

Fig. 10. Angular gain distribution for the CPC side-pumpingdesign of Section 3 and the truncated elliptical side-pumpingdesign of Section 4.

1524 APPLIED OPTICS / Vol. 32, No. 9 / 20 March 1993

'1

i, x,,.S's 't~~~~~~~~~~~~~~1

,, w,,.,,

- CPC pump cavity-I- Ellipticalpumpcavity - -- CPC pump cavity

--- Elliptical pump cavity

. . . . . . . . . . . .

Fig. 11. Gray-scale gain density distribution for the CPC side-pumping design.

Any ray that intercepts this plane after passingthrough the cavity is considered to be ejected.

The design conditions are that the rod is centeredalong one of the focal lines and that the diode array ispositioned to lie perpendicular to the major axialplane of the elliptical cavity and to contain the otherfocal line. The shape of the elliptical pump cavity isnot determined by the parameters of the source andthe absorber. Initially we chose an elliptical pumpcavity for which the latus rectum length was the sameas the width of the ESDA (i.e., 50 mm), and the fociwere separated so that there was direct illuminationof the rod surface by 30% of the radiation from asingle diode located at the source focus. This config-uration tends to focus right at the bottom surface ofthe rod, leading to an absorption distribution thatlacks uniformity. Absorption uniformity can be rem-edied somewhat by expanding the size of the ellipticalcavity, but this expansion causes the coupling effi-ciency to decrease because the diode is moved awayfrom the rod surface. In a simple optimization, thebest elliptical design had a major axis length that wasequal to the array dimension and a minor axis lengththat was 80% of the array size. Figures 9, 10, and 12show the radial, angular, and gray-scale plots, respec-tively, for such a cavity when the source and laser rodparameters are the same as those used in Section 3.Table 1 gives the various FOM values obtained forthis cavity design. It is to be noted that the ellipticalcavity gives a poor overall FOM value in comparisonwith that of the other two cavities. Whereas theFOMrad and FOMa,1g values are comparable betweenthe elliptical and CPC pump cavities, the FOMPOW hasalmost quadrupled for the elliptical cavity in compari-son with that of the CPC. This increase means thatthere is approximately four times as much pump

Fig. 12. Gray-scale gain density distribution for the truncatedelliptical side-pumping design.

radiation that is never coupled into the gain medium.The reason for this radiation loss lies in the nature ofthe cavity design: the elliptical cavity images onlyfrom one focus to the other. As the individual diodeposition lies further away from the focus, more of itsradiation is ejected from the cavity without everstriking the laser rod. This phenomenon is illus-trated in Fig. 13, where the coupling efficiency isshown for an individual diode as it is moved off axis inboth the truncated elliptical and CPC pump cavities.As we can see, the CPC maintains a uniform coupling

1.0

0.9

-0

0

9

.

0.81

0.71

0.6

0.5

0.41

0.31

0.2

0.1

0.0

0 1 2 3 4 5 6r/a

Fig. 13. Effect on the power absorbed in the laser rod when anindividual diode laser is moved off axis for the CPC and truncatedelliptical side-pumping cavities. The horizontal axis, x/a, signi-fies the position of the diode in units of rod radii.

20 March 1993 / Vol. 32, No. 9 / APPLIED OPTICS 1525

3 -a- -a- ~-3 U-U--U^-

.I.

'44

'4

cPC pump caviy 1-a-- Ellintical Pumn cavity I

a- a- U

. . . . . . . . . . . .

- --rev--r---r ---'-J

efficiency over the width of the entrance apertureuntil the diode is located near the edge of the cavity,but the elliptical cavity coupling efficiency falls offquickly as the diode is moved off axis. Furthermore,as the diode is moved off axis, the radiation that doesstrike the laser rod in the elliptical cavity is incidenton the rod surface, on average, with angles approach-ing the tangent. This circumstance has a twofoldeffect: it gives a shorter path length within the rod(i.e., less of the pump radiation is absorbed) andlocates more gain near the critical angle radius of therod rather than at its center. This effect is evident inthe radial plot of Fig. 9, in which the gain densityincreases until the critical angle radius is reached.Beyond this radius the gain density drops off until thecenter of the rod is reached. At the center of the rodthe imaging qualities of the elliptical cavity dominate,and a sharp peak in the gain density is obtained. Incomparison with the close-coupled geometry of Sec-tion 2, the main asset of the cavity lies in its ability tocouple the output from many diodes into the laser rod(indicated by a small FOMdiode value). It is still not asefficient as the CPC design, however. The CPCachieves better results because the reflector is de-signed with the specifications of the absorber (i.e., thelaser rod radius) and the source (i.e., the diode-arrayemission angle) explicitly included, whereas the ellip-tical cavity is specified by two points, the foci, and itseccentricity and takes no account of the systemparameters.

5. ConclusionsIn high-power systems, the ESDA side-pumping CPCdesign is the logical choice for the pump cavity.First, side pumping permits more diode radiation tobe coupled into the gain medium than in end pumping.Second, the CPC pump cavity has nearly the sameeffectiveness as the close-coupled geometry, as shownby FOM values; however, the CPC permits muchmore pump radiation to be coupled into the gainmedium, as shown by FOMdiode. Third, the use of anESDA reduces thermal loading problems in both thelaser rod, because of the wavelength matching of thesource-emission band to the solid-state absorptionband, and in the diode array, because of the sparsepacking of the individual diodes. Finally, the CPC ismore effective at coupling radiation from an ESDAthan are imaging designs such as the truncatedelliptical pump cavity, because even though suchdesigns can be used to couple a large amount of pumpradiation, they do so inefficiently.

Refinements of the CPC design may permit evenbetter control of the uniformity of the absorptiondistribution, as is discussed in Refs. 36 and 37.However, the edge-ray principle on which the CPCdesign is based appears to be inadequate for dealingwith non-Lambertian, nonhomogeneous sources.The edge-ray-principle design achieves maximum con-centration from one surface to another in the two-dimensional case. In diode pumping, one is inter-ested in the transmission from a surface (the diode

array) to a volume (the gain medium). The FOMdefinitions proposed here are a first attempt to definea quantity that is more suited to this application thanthe concentration ratio.

The authors thank their colleagues, especially D.Caffey, at the U. S. Army Center for Night Vision andElectro-Optics. This research was partially sup-ported by the U.S. Army Research Office-UniversityResearch Initiative Center for Opto-Electronic Sys-tems Research. During this research, R. John Ko-shel was supported by fellowships from the Depart-ment of Education, Xerox, and the New York StateCenter for Advanced Optical Technology.

References and Notes1. L. C. Conant and C. W. Reno, "Laser diode pumped Nd:YAG

laser," Appl. Opt. 13, 2457-2458 (1974).2. T. M. Baer, D. F. Head, and M. Sakamoto, "High efficiency

diode-bar pumped solid state laser using a tightly foldedresonator," in Conference on Lasers and Electro-Optics, Vol.11 of 1989 OSA Technical.Digest Series (Optical Society ofAmerica, Washington, D.C., 1989), p. 416.

3. T. M. Baer, D. F. Head, and P. Gooding. "High peak powerQ-switched Nd:YLF laser using a tightly folded resonator," inConference on Lasers and Electro-Optics, Vol. 7 of 1990 OSATechnical Digest Series (Optical Society of America, Washing-ton, D.C., 1990), p. 24.

4. T. Y. Fan, "Diode-pumped solid state lasers," Linc. Lab. J. 3,413-425 (1990).

5. T. Y. Fan, "Efficient coupling of multiple diode laser arrays toan optical fiber by geometric multiplexing," Appl. Opt. 30, 620(1991).

6. As of February 1992, the maximum reported output powersfor the tightly folded resonator was 10 W when fiber couplingwas used: M. S. Keirstead and T. M. Baer, "10 W, TEMOOoutput from a diode-pumped, solid-state laser," in Conferenceon Lasers and Electro-Optics, Vol. 10 of 1991 OSA TechnicalDigest Series (Optical Society of America, Washington, D.C.,1991), p. 490. In a geometric multiplex end-pump design, a15-W multimode and a 6-W TEMoo have been obtained: S. C.Todwell, J. F. Seamans, C. E. Hamilton, C. H. Muller, andD. D. Lowenthal, "Efficient, 15-W output power, diode-endpumped Nd:YAG laser," Opt. Lett. 16, 584-586 (1991).

7. L. R. Marshall, A. Kaz, and R. L. Burnham, "Highly efficientTEMOO operation of transversely diode-pumped Nd:YAGlasers," Opt. Lett. 17, 186-188 (1992).

8. A. D. Hays and R. Burnham, "Quasi-CW diode-array sidepumped 946-nm neodymium laser," in Conference on Lasersand Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series(Optical Society of America, Washington, D.C., 1990), p. 4.

9. R. Burnham, "High-power transverse diode-pumped solid-state lasers," Opt. Photon. News 1(1), 4-8 (1990).

10. T. H. Allik, W. W. Hovis, D. P. Caffey, and V. King, "Efficientdiode-array-pumped Nd:YAG and Nd:Lu:YAG lasers," Opt.Lett. 14, 116-118 (1989).

11. Throughout this study the absorption distribution is calcu-lated for a number of pump cavities with diode pump sources.The gain distribution is directly related to the absorptiondistribution by multiplication of the quantum defect value(e.g., for diode pumping at 810 nm of Nd:YAG, this value isapproximately 0.8).

12. We assume that matching the gain distribution to the cavityTEMoo mode profile is optimum. Some authors claim thatthe optimum transverse gain profile is a delta function alongthe mode axis of the gain medium, although these authorsaccount for neither the effects of amplified spontaneous emis-

1526 APPLIED OPTICS / Vol. 32, No. 9 / 20 March 1993

sion and optical damage nor diffraction. See, e.g., D. G. Hall,R. J. Smith, and R. R. Rice, "Pump-size effects in Nd:YAGlasers," Appl. Opt. 19, 3041-3043 (1980).

13. M. K. Reed, W. J. Kozlovsky, R. L. Byer, G. L. Harnagel, andP. S. Cross, "Diode-laser-array-pumped neodymium slaboscillators," Opt. Lett. 13, 204-206 (1988).

14. W. T. Welford and R. Winston, High Collection NonimagingOptics (Academic, San Diego, Calif., 1989), Chap. 4.

15. Ref. 14, Chap. 9.16. 0. Svelto, Principles of Lasers, 3rd ed. (Plenum, New York,

1989), p. 354. The parallel junction is typically three to fivetimes larger; therefore, the divergence angle in the perpendicu-lar direction is three to five times larger than the divergenceangle in the parallel direction.

17. The full standard deviation angle is twice the standard devia-tion angle of the Gaussian curve in both the tangential and thesagittal directions. The relation between these angles andthe FWHM divergence angles of the individual diodes isOFWHM = 2.35482 (t,,').

18. Other ray-tracing angle selection algorithms were considered,but because of the small dimensionality of the space and adeterministic result in question, a deterministic ray set waschosen. For example, a Monte Carlo ray-tracing methodwould give results similar to those of these deterministic raysets. It is also noted that, because the cavity is cylindrical innature, the three-dimensional ray trace offers little advantageover the two-dimensional trace; however, the three-dimen-sional trace was performed for completeness. The accuracy ofthe deterministic ray trace in two-dimensional space is propor-tional to one over the square root of the number of rays. Toback up this claim we traced ray sets larger by 1 order ofmagnitude and found that there are no appreciable differencesbetween the determined absorption distributions.

19. I. M. Bassett, W. T. Welford, and R. Winston, "Nonimagingoptics for flux concentrators," in Progress in Optics, E. Wolf,ed. (Elsevier, London, 1989) Chap. III, p. 171.

20. Refractive imaging systems are available with a N.A. of 0.85,which corresponds to an object-space angle of approximately58 ; therefore the maximum possible C2D is not obtained.Further, these systems are microscope objectives with a focallength of approximately 3 mm. This length gives a lensradius of approximately 5 mm, so the lens system is small andcannot be enlarged easily without introducing a large amountof aberration. Therefore refractive imaging systems cannotbe scaled up to permit large-area pump sources.

21. Ref. 14, Chap. 1.22. P. Lacovara, P. Gleckman, R. L. Holman, and R. Winston,

"Nonimaging concentrators for diode-pumped slab lasers," inNonimaging Optics: Maximum Efficiency Light Transfer, R.Winston and R. L. Holman, eds., Proc. Soc. Photo-Opt. In-strum. Eng. 1528, 135-141 (1991).

23. A. Rabl, "Solar concentrators with maximal concentrations forcylindrical absorbers," Appl. Opt. 15, 1871-1873 (1976).

24. Ref. 14, App. H.25. J. D. Kuppenheimer, Jr., "Design of multilamp nonimaging

laser pump cavities," Opt. Eng. 27, 1067-1071 (1988).26. P. Gleckman, "Achievement of ultrahigh solar concentration

with potential for efficient laser pumping," Appl. Opt. 27,4385-4391 (1988).

27. R. M. J. Benmair, J. Kagan, Y. Kalisky, Y. Noter, M. Oron, Y.Shimony, and A. Yogev, "Solar-pumped Er, Tm, Ho:YAGlaser," Opt. Lett. 15, 36-38 (1990).

28. R. J. Koshel and I. A. Walmsley, "Diode pumping of solid-statelasers with nonimaging optics: a theoretical study," in An-nual Meeting, Vol. 15 of 1990 OSA Technical Digest Series(Optical Society of America, Washington, D.C., 1990), p. 126.

29. The parameterization of the CPC for a circular absorberfollows Rabl23; however, in Eqs. (8) and (9) of Sec. II in Rablone reads that = r/2, when this substitution for 8 is notsupposed to be done until Sec. III (specific to a circularabsorber). Welford and Winston complete the theory in thecorrect order, but the notation of Rabl is used throughout thisstudy.

30. Ref. 14, Chap. 9.31. R. Winston, "Ideal flux concentrators with reflector gaps,"

Appl. Opt. 17, 1668-1669 (1978).32. W. R. McIntire, "New reflector design, which avoids losses

through gaps between tubular absorbers and reflectors," Sol.Energy 25, 215-220 (1980).

33. W. R. McIntire and R. Winston, "Design considerations forreducing optical losses due to gaps between absorber andreflector," in Proceedings of the 1981 Annual Meeting of theAmerican Section of the International Solar Energy Society,B. H. Glenn and G. E. Franta, eds. (American Section,International Solar Energy Society, Newark, Del., 1981), pp.211-273.

34. An effective absorption coefficient of the solid-state material isused by comparing the overlap of the absorption-bandwidthprofile of the solid-state medium (Nd:YAG) and the emission-bandwidth profile of the diode array. This comparison alsogives an effective index of refraction, but the index is fairlyconstant over the emission bandwidth of a diode laser (typi-cally 2-4 nm). The diode parameters used in this study weretaken from the literature and were assumed to have a band-width of 2 nm centered at 809 nm.

35. This is not a requirement but provides symmetry of theabsorption distribution. For an example of off-axis diodepump sources see M. Kuzumoto, K. Kuba, and S. Yaga,"Continuous-wave operation of a YAG laser by off-centered LDside-pumping," in Conference on Lasers and Electro-Optics,Vol. 11 of 1989 OSA Technical Digest Series (Optical Society ofAmerica, Washington, D.C., 1989), p. 414.

36. D. G. Burkhard and D. L. Shealy, "Design of reflectors whichwill distribute sunlight in a specified manner," Sol. Energy 17,221-227 (1975).

37. M. J. J. B. Maes and A. J. E. M. Janssen, "A note on cylindricalreflector design," Optik (Stuttgart) 88, 177-181 (1991).

38. The critical angle radius is the radius of the caustic formed byrays that strike the rod surface tangentially. These rays areanalogous to the rays that enter the CPC pump cavity at theacceptance angle of 0a. This radius is the interior circle of thegray-scale plots of the CPC and elliptical pump cavities (seeFigs. 11 and 12).

20 March 1993 / Vol. 32, No. 9 / APPLIED OPTICS 1527