Optimizing coherent lidar performance with graded-reflectance laser resonator optics

Optimizing coherent lidar performance withgraded-reflectance laser resonator optics

David M. Tratt

We demonstrate how the design of graded-reflectance output coupler unstable laser cavities may betailored to significantly enhance the overall power transmission efficiency of a given laser system relativeto that of a conventional diffractively coupled unstable resonator. The importance of these findings incoherent lidar applications is explained with particular emphasis on projected space-based systems.

Key words: Lidar, graded optics.

1. Introduction

With plans for the orbital deployment of a LaserAtmospheric Wind Sounder now well advanced, 2 theissue of optimization of system efficiency in coherentlidar applications has come under increasing scru-tiny. Since laser devices are generally of relatively lowefficiency, their operation aboard space-based plat-forms with -the attendant limited power resourcesposes certain engineering challenges. Thus the pri-mary impetus of this investigation is to assess howthe efficiency of a coherent lidar wind sensor may beimproved by the incorporation of graded-reflectivityoptics into the transmitter laser cavity.

The limiting measurement range of a given coher-ent lidar system is dependent on several factors. Onesuch consideration is the efficiency of coherent mix-ing in the propagation process, which depends interalia on the transmitted beam profile and thus on thenature of the laser cavity optics employed. Thisparticular issue has received much attention in recentyears as increasingly sophisticated optical coatingtechniques have become accessible to the opticalengineering community.3-5 In particular, the applica-tion of the graded-lamina output optic mirror(GLOOM) concept to unstable resonator structuresoffers significant improvements in output quality(and therefore propagation efficiency) over the moreconventional unstable cavity scheme that uses thediffractive output optic mirror (DOOM) arrange-ment.6 7 In the coherent lidar idiom, the propagation

efficiency refers to that proportion of the transmittedradiation that falls within the first Airy zone in thefar field, since it is this component that plays thecentral r6le in the coherent mixing process. The termGLOOM may be regarded as a generic description fora variety of graded-reflector concepts, including thosethat employ variable-phase or graded-index coatings.However, here we will restrict our considerations tothe case of a GLOOM with radially variable reflec-tance.

A separate but closely related issue in this respectinvolves consideration of the extraction efficiency of agiven resonator configuration, as inferred from itsfilling factor Q. 8 In Ref. 8 it was pointed out that theoverall suitability of a given resonator configurationin a coherent lidar context should be assessed withreference not only to the far-field beam brightness(expressed in terms of the resonator antenna effi-ciency q), but also the cavity-filling factor. To assist incarrying out such a study, we have previously intro-duced a figure of merit, Q, which combines both ofthese attributes to yield a measure of the overallpower transmission efficiency of a given resonatorstructure8 :

Q = qQU- (1)

Using this procedure, we are able to compare theoverall performance of a variety of GLOOM unstable-cavity configurations to gain greater insight into thebenefits of this approach in relation to coherent lidarapplications.

The author is with the Jet Propulsion Laboratory, Mail Stop169-214, California Institute of Technology, 4800 Oak GroveDrive, Pasadena, California 91109.

Received 21 August 1991.0003-6935/92/214233-07$05.00/0.c 1992 Optical Society of America.

I1. DOOM and GLOOM: Dependence of ResonatorPerformance on Output Coupling

The particular GLOOM concept adopted for thisstudy is the so-called super-Gaussian profiled re-flector, whose radial variation of reflectivity R(r) is

20 July 1992 / Vol. 31, No. 21 / APPLIED OPTICS 4233

expressed by9

R(r) = Ro exp[-2(r/wm)n], (2)

where Ro is the on-axis reflectance, r is the radialcoordinate, and wm is the 1/e2 radius of the reflec-tance profile. The parameter n is the super-Gaussianorder and has a lower bound of 2, which correspondsto the Gaussian profiled reflector. The antenna prop-erties of the latter class of resonator were comprehen-sively investigated in Ref. 8.

For a confocal Cassegrain unstable resonator witha coupler of the form described by Eq. (2) the l/e2

intensity radius w0 of the internal mode is9

W = W.(Mn -1)1/n, (3)

where M is the cavity magnification. The internalmode Im(r) is described by 9

Im(r) = exp[-2(r/wo)N. (4)

Thus the output transverse beam profile may berepresented by a compound super-Gaussian formal-ism:

1(r) = Im(r)[1 - R(r)] = exp[-2(r/wo)n]- Ro exp[-2(Mr/wo)n].

(5)

In both Eqs. (4) and (5), constants of proportionalityhave been neglected.

In this fashion, a super-Gaussian GLOOM gener-ates a smooth, well-defined output beam profile withhigh propagation efficiency,10 although as the super-Gaussian order increases, there will come a point atwhich this determinism breaks down and diffractioneffects once more become important.11 By contrast, aconventional DOOM-coupled unstable resonator emitsa heavily structured transverse output profile owingto diffraction effects imparted by the hard-edgedcharacter of the coupler itself. Thus although theextraction efficiency in this case is close to maximum,the propagation efficiency suffers severely. In addi-tion, the fast spatial modulations of the cavity modecan perturb the gain medium to such an extent(through differential heating processes) that beambreakup becomes a real possibility within the dura-tion of the laser pulse.12

The exact point at which the super-GaussianGLOOM begins to evince significant diffraction fea-tures is in general dependent on the magnitude ofdR(r)/dr relative to the laser wavelength, so it ispossible to conceive of situations in which the onset ofdiffraction effects might not occur until n exceeds 10to a substantial degree. Indeed, experiments with aGLOOM-coupled Nd:YAG laser appear to indicatethat performance advantages over the DOOM arrange-ment accrue even for n = 35, although the modeprofile in this case was definitely influenced by diffrac-tion effects.10 This question can be addressed moredefinitively in the future, but the analysis here has

been extended to n = 20 in order to cover a majority ofsituations.

The energy extraction efficiency of a given cavityarrangement is proportional to the overlap integral ofthe three-dimensional mode descriptor with the avail-able gain volume, which is often termed the cavity-filling factor. Because a confocal unstable resonatoremits a nominally collimated output beam, we mayneglect the z dependence in the overlap integral andconsider only the mode area (i.e., the integral of theoutput irradiance with respect to area). Thus Q, forour case may be described as the ratio of the modearea of the subject resonator to that of a plane wave,'3for which I 1 (r < a, where a is the radius of thetransceiver pupil):

f In(r)dr

Q = raJ 0d'

We are assuming here that the limiting aperturewithin the subject laser system is exactly matched tothe transceiver pupil (a justifiable assumption for aproperly optimized lidar).

11. Coherent Receiver Model

The coherent receiver model used here follows thedescription given previously.8 We assume a coaxialmatched-pupil transceiver geometry and invoke thebackpropagated local oscillator treatment (antennatheorem) to model the coherent mixing process.'4 Theantenna efficiency for a diffuse (i.e., aerosol) target isthen given by'5

TT X JO' ~ IT(p)IL(p)d'p

[f IT(r)d2r I IL(r)dr]

(7)

in which z denotes the range, is the operatingwavelength, and I. is the transverse irradiance distri-bution function. r and p are the radial ordinates at thetransceiver pupil and the target plane, respective, Tand A are the pupil transmission function and area(=-7ra2), respectively, and the subscripts refer to thetransmitter beam (T) and the backpropagated localoscillator (L).

We assume radially symmetric transverse irradi-ance profiles concentric with the transceiver pupil, sothat

JO IT(r)d2r

f IT(r)d'r(8)

Combining Eqs. (7) and (8), we obtain

9 = (Xz/a)'Jf IT(p)IL(p)2pdp

(9)

[f: IT(r)2lTrdr] fa IL(r)2Trdr]

(6)

4234 APPLIED OPTICS / Vol. 31, No. 21 / 20 July 1992

1.00 1.25 1.50 1.75

a/W 0

Fig. 1. Coherent lidar antenna efficiency as a function of ttransmitter truncation parameter for selected super-GaussiGLOOM cavity structures with M = 2 and Ro = 0.8: curve a, n2; curve b, n = 6; curve c, n = 10. The upper and lower horizonidashed lines represent the uniform and the Gaussian transmitllimits, respectively, which were obtained earlier.8,15

in which IT(r) is defined by Eq. (5). IL(r) is assumedbe Gaussian at the transceiver pupil with a trunction parameter a/wL = 0.84 (WL being the 1/intensity radius), since previous studies indicate ththis configuration is optimum for both Gaussiarand compound-Gaussian8 transmitted-beam profilEThe far-field beam irradiances Ix(p) are determined 1

applying the Fraunhofer treatment in the diffractionlimit (i.e., neglecting atmospheric propagation ef-fects8"16). The alternative form of Eq. (9) that appearsin the previous paper8 results when the numerator istransformed from planar to solid-angle coordinatesby using the substitution given by Eq. (2) of Ref. 8.(This is essentially a computational device that ren-ders the integration limits finite.)

IV. Parametric Behavior of Antenna Efficiency andFigure of Merit

Figure 1 shows sample output from the lidar trans-ceiver model for three selected values of the GLOOMsuper-Gaussian order n and for a cavity magnificationM = 2. We may immediately note that relatively little

he power transmission efficiency is lost for moderate n.The reduced o (the optimum antenna efficiency for a

tal;ter

toa-/e2

athL5

es.

by

-3.45

-3.50

a

-3.55

-3.2

> n=2

m

F~

(a)

io. n=6

aa

n = 10

(a) (b)

Fig. 2. Transmitter irradiance patterns (a) immediately after thetransceiver pupil (field width 3wo) and (b) in the far field (fieldwidth: three Airy zones) for the resonators described in Fig. 1,from which the corresponding optimum truncation parametershave been inferred.

0.3 0.4 0.5 0.6Ro

0.7 0.8 0.9

Fig. 3. Dependence of optimum antenna efficiency on M andRo: (a) n = 2, (b) n = 4, and (c) n = 6. The upper and lowerhorizontal dashed lines in each figure represent the uniform andthe Gaussian transmitter limits, respectively [the uniform transmit-ter limit is off scale in (a)].


-3.0

-3.5

m -4.01

I I I I I II I I I I I

UNIFORM LIMIT

GAUSSIAN LIMIT

I I I I I I I 1 -

-4.51

gi n ).75

_ ,,~ "

_~~~~~ M=2.5- 2.0

1.5

_-, _. _. .. . . . . . . . . . ... ... . . .

-3.3

-3.4

-3.5

-3 .6

given value of n) evident at n = 10 most likelyrepresents a best-case scenario, since output-irradi-ance relation Eq. (5) probably loses validity beforethis point is reached, as explained in Section II. Thecorresponding transmitted-beam profiles and the far-field irradiance distributions for these three examplesare depicted in Fig. 2, from which similar conclusionsmay be inferred by observing the relative growth ofthe second Airy zone in the far field as n increases.

The relationship between o and Ro for selectedvalues of n and M is recorded in Fig. 3. The valueschosen for M are similar to those most frequentlyencountered in actuality, while Ro is permitted tovary over its entire meaningful range. Most notice-able of all from this figure is the change in characterof the featured dependence when n exceeds 2. For thecase of a super-Gaussian GLOOM with n = 2 thereexists an optimum value of R0, which is itself depen-dent on M. However, for n > 2 the relationshipbecomes monotonic with -no inversely correlating toRo. The implication to be drawn from Fig. 3 is that thesuper-Gaussian GLOOM resonator designer shouldstrive (laser performance considerations permitting)toward Ro 0.6 or less. This conclusion is alsosupported by Figs. (4)-(6), which collectively providea comprehensive picture of GLOOM resonator effi-ciency. Panel (a) in these figures depicts the increas-ingly severe degradation of the coherent mixing effi-ciency with respect to the super-Gaussian order as Rois permitted to increase, with the appearance of an

-3.5

-4.0 4

0.9

0.8 g (c) A=00.9

optimum n for each value of Ro. In particular, note forn = 2 that there is an almost total lack of sensitivityto these parameters, as was first observed in the priorstudy.8

The data contained in Figs. 4(b), 5(b), and 5(c)imply the following trend:

lim[a/wo]pt = 1,n en

(10)

which follows as a natural consequence of the ten-dency

i dI=T(r)

dr(11)

The family of curves represented in Figs. 4(b), 5(b),and 6(b) reside in close proximity to one another, sowe have chosen to reproduce only those shown for thesake of clarity. This is even more the case for theextraction efficiency plots [Figs. 4(c), 5(c), and 6(c)],which evince the least dependence of all and for whichonly the end-member data sets are reproduced here.Of course, Eq. (6) implies that Q, should be indepen-dent of Ro. In point of fact, however, an indirectdependence is passed through from the optimizationprocedure for the truncation parameter [Figs. 4(b),5(b), and 6(b)], which is interpreted as a weak modu-lation of a, the upper integration limit in Eq. (6).

Finally, Figs. 4(d), 5(d), and 6(d) show the corre-

1.4 I I I I I I I I 1

wI-w

0

0zIt

F-

0

a

0 5 10 15 20 0 5 10 15SUPER-GAUSSIAN ORDER, n SUPER-GAUSSIAN ORDER, n

Fig. 4. Dependence of (a) the optimum antenna efficiency, (b) the optimum truncation parameter, (c) cavity energy extraction, and (d) theoverall figure of merit on the super-Gaussian order and the on-axis reflectance of the GLOOM output coupler. The cavity magnification is1.5. The upper and lower horizontal dashed lines in (a) represent the uniform and the Gaussian transmitter limits, respectively.


1

1

Pm

F

30

a:wus

"IH

(ra-z0

0za:

0

0.5 R= 0.2- 0 03 .

0.4

0.4 02-0.50.2~~ I I I I I I I i 01L2 0.70.3 0.80.2 0.1I

0 5 10 15 20 0 5 10 15 20

SUPER-GAUSSIAN ORDER, n SUPER-GAUSSIAN ORDER, n


a'09'

-4.5

0.9

0.8

0.7

3 0.6a0.5

0.4

0.3

0.2200 5 10 15

SUPER-GAUSSIAN ORDER, n

a:w

a:

a-

z00C.a:H

n

0

0.5

0.4

a 0.3

0.2

0.1



1.0

0.8

0.6

0cc

0.4

0.2

0.01.0 1.5 2.0 2.5 3.0

M

Fig. 7. Dependence of output coupling y on M and Ro for aGLOOM with radially variable reflectance. Each curve is labeledwith its respective y.

sponding variation of Q, which is the figure of meritdefined by Eq. (1). Inspection of these curves tells usthat significant degradation of Q for a given n doesnot occur until Ro exceeds 0.6, but that the systemsensitivity to this parameter becomes weaker as Mincreases (as do all the dependences illustrated).However, the output coupling y for a variable-reflectance GLOOM is reliant on both Ro and M9"10"13:

y = 1 - Ro/M 2. (12)

This function is described by the family of curves inFig. 7, from which it is evident that in order tosimultaneously achieve R < 0.6 and M > 1.2(magnifications less than this value being subject toincreased misalignment sensitivity'7), it is necessaryto accept the condition y > 0.6. This result wouldappear incompatible with low-gain laser media, but itretains some utility for large multijoule transverselyexcited atmospheric CO2 devices (such as the pro-posed Laser Atmospheric Wind Sounder transmit-ter), for which optimum output coupling y is usuallyfound to be in the range 0.4-0.6. Setting Ro = 0.6 andBy = 0.6 so as to maximize M, we see that Eq. (12)returns M = 1.23, so that from Fig. 7 there appearslittle scope for departure from y = 0.6. This some-what restrictive situation is ameliorable if greateroutput coupling is an option. Clearly, the dictates ofthe application under consideration will determinewhich of these features is the less tolerable.

In summary, all of the dependences representedhere become weaker as the cavity magnification in-creases, with o asymptotically approaching the ideallimit of a uniform (plane-wave) transmitter profile asn, M and Ro - 0. However, the parametricbehavior of 90 is of scarcely more than academicinterest here, since the overall system efficiency ismore effectively codified by the figure of merit Q,which is defined in section I.

V. Conclusions

We have investigated the power transmission proper-ties of super-Gaussian GLOOM unstable resonators.The examination of numerous data sets suggests thata super-Gaussian order of between 6 and 10 offers anoptimal trade-off between the mode profile determin-ism of the GLOOM coupler and the onset of nontriv-ial diffraction effects likely to be prevalent for n > 10.Furthermore, it appears that on-axis reflectivitiesRo > 0.6 result in output mode patterns with signifi-cantly inferior propagation characteristics. However,this condition is mitigated by the interdependence ofR0, M, and y, with the result that very-high-gainsystems (which require correspondingly less feed-back) would seem to be the most amenable to theGLOOM approach.

Perhaps the principal drawback to the use ofGLOOM elements within a high-energy laser systemis their relative susceptibility to optical damage, aswas briefly alluded to previously.8 However, it shouldbe feasible to fabricate more robust coating struc-tures by implementing more advanced depositiontechniques, such as the graded-refractive-index con-cept.8 Such an entity would incorporate more thanone graded parameter, so we have introduced theterm GLOOM as a generic description that covers allsuch combinations.

It is certainly possible to specify GLOOM cavityparameters that yield little or no advantage over thecorresponding DOOM configuration (as previouslynoted in Ref. 8). However, it has been the purposehere to provide a formalism by which cavity designersmay assess how best to optimize a given set ofresonator parameters. Thus, while each of the above-mentioned criteria must obviously be applied withdue regard to required laser performance, it neverthe-less seems clear that super-Gaussian reflectance pro-filed optics offer some potential gains in the area oflaser efficiency enhancement in general, and long-range coherent lidar in particular, as was concludedin a recent preliminary report. 9

This research was carried out at the Jet PropulsionLaboratory, California Institute of Technology, un-der contract with the National Aeronautics and SpaceAdministration. Computing resources were providedby the Jet Propulsion Laboratory/California Insti-tute of Technology Supercomputing Project.

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Optimizing coherent lidar performance with graded-reflectance laser resonator optics