transverse-mode selection in apertured super-gaussian resonators: an experimental and numerical...

Transverse-mode selection in aperturedsuper-Gaussian resonators: an experimental andnumerical investigation for a pulsed CO2 Dopplerlidar transmitter

E. Galletti, E. Stucchi, D. V. Willetts, and M. R. Harris

A representative prototype of a high-energy, long-pulse, and narrow-bandwidth pulsed CO2 laser suitablefor a spaceborne Doppler wind lidar application has been developed. We obtained 10 J of output energyat greater than 8% efficiency in long, narrow-bandwidth, single-longitudinal, and transverse-modepulses. We used a positive branch unstable resonator with a fourth-order super-Gaussian mirror as theoutput coupler. Experiments were carried out to assess the effect of intracavity hard apertures ofdifferent diameters that induce diffractive perturbation of the theoretical field and reduce the transverse-mode selectivity of the cavity. An upper limit to the choice of the mirror soft radius has been found,which allows optimization of the trade-off between laser efficiency and beam quality. We determinedexperimentally that a value of 0.75–0.8 for the ratio between the exp~21! diameter of the beam intensityand the laser clear aperture gave a single-transverse-mode operation without significant loss of efficiency.© 1997 Optical Society of America

Key words: Laser resonators, super-Gaussian mirrors, mode discrimination, CO2 lasers.

1. Introduction

The generation of beams of high energy or powerwithout loss of significant optical quality is still anissue for the designers of both solid-state and gaslasers. Currently, the solution that is widely consid-ered the most suitable is the use of a positive branchunstable resonator ~PBUR! with a radially smoothoutput coupling. Single-transverse mode ~STM! op-eration can be obtained in a conventional PBUR witha hard-edge output coupler, but only for a restrictedset of cavity parameters. The use of a variable re-flectivity mirror ~VRM! as an output coupler givesmore design flexibility and theoretically allows aSTM to be produced over a wider range of conditions.VRM properties have been studied for a long time,1,2but extensive application of this kind of cavity was

E. Galletti and E. Stucchi are with Centro Informazioni Studi eEsperienze Tecnologie Innovative, P.O. Box 12081, I-20134 Milan,Italy. D. V. Willetts and M. R. Harris are with the Defence Re-search Agency, St. Andrews Road, Great Malvern, WorcestershireWR14 3PS, U.K.Received 8 March 1996; revised manuscript received 24 June

1996.0003-6935y97y061269-09$10.00y0© 1997 Optical Society of America

delayed until a straightforward mirror manufactur-ing technology became available.3 Mirrors with aGaussian profile were addressed first because of theirrather easier analytical treatment4,5; later the moregeneral super-Gaussian design, which includes theGaussian one as a special case, was introduced6 withthe aim of optimizing the energy extraction from theactive medium, which is of great importance for awide variety of applications. Examples include ma-terial working, in which high energy or power beamswith good focusing properties are needed; pump la-sers for harmonic or parametric frequency conver-sion, in which high intensity, low divergence, and asmooth near-field profile are required; and lasersources for remote sensing systems, in which highenergy and low divergence are demanded. Becausethe analytical theory applies to empty resonatorswith an infinite radial dimension, a common problemis how tightly the mode can be fitted in the actualaperture of the active medium without one introduc-ing significant diffraction effects on the output beam.In addition, even for the case of a uniformly pumpedactive medium, the theoretical mode can be alterednoticeably by gain saturation compared with the coldcavity case.7 We addressed these topics while work-ing on the design and subsequent experimental eval-uation of a pulsed CO2 laser demonstrator for a

20 February 1997 y Vol. 36, No. 6 y APPLIED OPTICS 1269

spaceborne coherent Doppler wind lidar, which hasbeen promoted and supported by the European SpaceAgency. We obtained good results8,9 using a super-Gaussian cavity, but concern arose about thetransverse-mode discrimination because of the pres-ence of secondary power spikes on a fraction of theoutput pulses. Substantial work exists on the de-sign rules and the performance obtainable with dif-ferent super-Gaussian mirrors for solid-statelasers,10,11 but a comprehensive study applied to gaslasers, and specifically to pulsed CO2, is availableonly for the basic Gaussian case.12 The aim of ourresearch is to fill the gap at least for a configurationthat, though specific, has great practical interest, andadds an experimental contribution to the already-available trade-off studies on Doppler lidar emit-ters.13,14

2. Super-Gaussian Theory Review

The generalized super-Gaussian mirror ~SGM! hasan intensity reflectivity profile given by6

R~r! 5 R0 exp@22~rywm!n#, (1)

where R0 is the intensity peak reflectivity, r is theradial coordinate, wm is the exp~22! mirror spot sizeor soft radius, and n is the super-Gaussian order thatgives the usual Gaussian profile when equal to 2 anda conventional hard-edge when set to infinity. Inthe consideration of an empty resonator, the cavity-mode profile is super-Gaussian of the same order asthat of the mirror, so that the field amplitude u0 canbe expressed as

u0~r! } exp@2~rywi!n#, (2)

where the beam-field spot size wi is related to that ofthe mirror by

wi 5 wm ÎnMn 2 1, (3)

where M is the resonator magnification. One canobtain the output field by simply multiplying u0 bythe complementary super-Gaussian transmission:

uout~r! 5 u0~r!Î1 2 R~r!. (4)

From the above equations it is apparent that a super-Gaussian field has a higher amplitude at low r thana Gaussian field having the same spot size, and cor-respondingly it has shorter tails at large r; in princi-ple the super-Gaussian profile thus can fit better inthe active medium. The theoretical results outlinedabove are no longer valid if diffraction becomes im-portant, i.e., for large values of n that give a hard-edgelike cavity ~n . 10! as well as for spot sizes thatare too large. It is usually assumed that the resultsof the SGM theory could be applied to a finite-sizeresonator when the beam power outside the actuallaser radius has a negligible value1 or, similarly,when the theoretical intensity profile is cut belowsome small percentage of its peak value.10,11 A com-monly used parameter12,14 to distinguish between dif-ferent designs is the ratio wbya, where wb is the

1270 APPLIED OPTICS y Vol. 36, No. 6 y 20 February 1997

exp~21! radius of the beam intensity given by

wb 5 wiy În 2 5 wm În ~Mn 2 1!y2, (5)

and a is the laser clear aperture radius.

3. Doppler Lidar Transmitter Demonstrator

The laser performance demonstrator built for the Eu-ropean Space Agency Doppler wind lidar program isbased on an e-beam-sustained gain module with a5 3 5 3 70 cm3 discharge volume. It operates at apressure of 57 kPa with a CO2:N2:He gas mixture ina ratio of 1:2:3 and is pumped by an 8-ms-long currentpulse delivering approximately 125 J. The main la-ser specifications, which drove the design phase, are10 Jypulse of output energy, 8% efficiency, 5-ms pulselength, 200-kHz bandwidth, and diffraction-limiteddivergence. The laser has to operate in the single-longitudinal mode and the STM on a single lasertransition at 9.25 mm ~9R24!. Given the active vol-ume needed to obtain the specified energy, the choiceof a large cross section is required to avoid intrapulsefrequency chirp that is due to laser-induced mediumperturbation,15 which could degrade severely thebeam spectral purity.Most of the specifications have been demonstrated

successfully with a super-Gaussian resonator withn5 4 and a spot size large enough to fill efficiently thecross section of the medium.8,9 We performed a sys-tematic study of the aperturing effects by selecting amirror whose soft radius was small compared withthe dimensions of the gain medium. Hard circularapertures of adjustable size were introduced into thecavity, and these then defined the limiting aperture.For practical convenience, the geometric parametersof the cavity were otherwise held constant for boththe mirrors used. All the relevant parameters arelisted in Table 1; the cavity configuration is shown inFig. 1. To avoid the use of a concave diffraction grat-ing, we obtained laser tuning using a plane gratingcoupled with a positive lens to obtain a wavelength-selective equivalent concavemirror. The laser couldbe readily injection seeded with a cw beam incidentonto the zeroth order of the diffraction grating.A simulation code has been developed to obtain

Table 1. Resonator Configurations

Parameter Mirror 1 Mirror 2

Operating wavelength 9.25 mm ~9R24!SGM order n 5 4Peak reflectivity 80%Mirror soft radius @at exp~22!# 20.1 mm 14.6 mmRadius of curvature 25.86 m 26 mLens focal length 7.93 mLens–grating distance 7 cmResonator length 1 mResonator magnification 1.41 1.33Beam soft radius @at exp~22!# 26.5 mm 17.7 mmBeam soft radius @at exp~21!# 22.3 mm 14.9 mmBeam wbya ratio @at exp~21!# 0.89 0.60Theoretical output coupling 60% 55%

more insight into laser operation and to evaluate thebest performance obtainable from the necessarilylimited experimental results. The code computesthe propagation of a wave front in the resonator,which is considered as a periodic lens waveguide ofinfinite length but limited transverse aperture. Thegain is collapsed to a small number of gain sheetsdistributed inside the cavity.16,17 To reduce memoryand speed requirements, the code applies to the caseof an axially symmetric laser with a zeroth-order az-imuthal symmetry, so that all the variables are sim-ply one-dimensional arrays, and the propagation canbe computed from theHuygens–Fresnel integral witha quasi-fast Hankel transform algorithm.18–20 Eachgain sheet is described by its own population densi-ties, whose dynamics, as well as the photon numberincrement, can be computed using rate equationsthat are solved by discrete steps each time a gainsheet is crossed. We checked the active mediummodel21–23 by measuring the small signal gain avail-able in the actual laser, with the encouraging resultshown in Fig. 2. Nevertheless the code significantlyoverestimates the output energy with respect to ourexperimental results. However, good performancehas been obtained as far as relative changes of theoutput energy between different configurations areconcerned, as well as good predictions of the actualnear-field beam profile.

Fig. 2. Comparison between experimental small signal gain ~solidcurves! and the result of the simulation program ~dash–dot curve!.The experimental data come from two subsequent measurementsand differ slightly because of electrical noise and interferences.

Fig. 1. Schematic diagram of the optical cavity showing the po-sition of the gain sheet used by the computer simulation code.

4. Experimental Results

Wemonitored the near-field beam profiles using boththermal paper, to get an immediate appreciation ofthe overall shape, and a cruciform array of pyroelec-tric detectors to obtain a quantitative measurementof the energy distribution in the horizontal and ver-tical directions. Figure 3~a! shows the spot we ob-tained on thermal paper using the large spot-sizemirror 1, where an interference pattern of horizontalfringes is clearly visible on the beam image. Its or-igin can be guessed when one considers that the dis-charge cross section is a square whose sides aredefined vertically by ceramic discharge limiters andhorizontally by the electrodes. The bottom is actu-ally a grid that allows e-beam injection, whereas thetop is a solid block of gold-coated stainless steel. Be-cause the fringes are parallel to the discharge elec-trodes, the effect is likely to be related to the fieldreflected by the metallic surface of the upper sidealthough it was sandblasted to obtain a fine rough-ness. The corresponding image produced by thesmall spot-size mirror 2 @Fig. 4~a!# is much smoother.The same behavior can be seen on the array detectordata in Figs. 3~b! and 4~b!, where the beam profilepredicted by the simulation code as well as the the-oretical profile are also shown. The actual beam sizeis well estimated by the code and is significantlywider than that computed from the cold cavity the-ory, as expected because of the presence of a nonuni-form radial gain that arises from active mediumsaturation. The energy and efficiency data obtainedduring these tests are listed in Table 2.Operation in a STM can be checked effectively by

looking for the presence of low-frequency beats in thetemporal output pulse shape measured by direct de-tection; when many longitudinal modes are present,the signal has to be low-pass filtered, for example, at20 MHz, to suppress the high-frequency longitudinalbeats at approximately 150 MHz. Satisfactorypulses have been obtained @Fig. 5~a!#, but on a frac-tion the presence of extra spikes and beatings is anindication of the onset of a higher-order transversemode @Fig. 5~b!#. The occurrence of these extrapulses is greater whenmirror 1 is used, whereas theirtime position is rather constant at approximately8–10 ms after the start of the pump-current pulse.Their presence is even more puzzling if we considerthat they also appear sometimes after the end of themain pulse, with a delay of more than 14 ms. Noevidence of the presence of more than one emissionline was found when a diffractive spectrometer wasused; the heterodyne detection setup that we used tomeasure the intrapulse frequency chirp allowed us tocheck that the extra pulses were at the same wave-length as the main one, i.e., there was not a differentline coming in, because they beat with the local os-cillator. The presence of extra spikes has been re-ported24 for the case of misaligned PBUR withconventional hard-edge coupling, although to ourknowledge no examples are present in the availableliterature on VRM resonators.


Fig. 3. Near-field ~NF! measurement of the output beam obtained with the SGM with wm 5 20.1 mm ~mirror 1!. ~a! Beam spot onthermal paper. ~b! Pyroelectric array data and simulated beam profile.

To assess the limit of the parameter wbya corre-sponding to a true multitransverse-mode operation,we took measurements using the smaller spot-sizemirror, but with a variable iris diaphragm placed inthe cavity between the lens and the grating and care-fully aligned onto the optical axis. Figure 6 showsthe measured values of output energy as a function ofthe intracavity aperture diameter; the ratio wbyaranges from 0.6 for the full discharge to 1.24 when a24-mm limiting aperture is used. It can be seen thatthe output energy is more or less constant for aper-tures larger than 38 mm, although some effect starts

at 36 mm, which is reasonable behavior when oneconsiders that the theoretical exp~22! intensity di-ameter is 35.4 mm. For smaller apertures the en-ergy scales linearly with the diameter and not as itssquare, which is likely to be due to the changes inbeam shape induced by diffraction that balances thereduction in cross section. The same plot also showsthe trend computed by the simulation code, whichhas been normalized to the mean experimental en-ergy obtained for the larger diameters so as to com-pensate for code errors when one evaluates theoutput energy. The agreement with experimental

Fig. 4. Near-field ~NF! measurement of the output beam obtained with the SGM with wm 5 14.6 mm ~mirror 2!. ~a! Beam spot onthermal paper. ~b! Pyroelectric array data and simulated beam profile.


data is fairly good, and it predicts the linear change inenergy for small diameters. The line representingsimulated data has been plotted as a dotted curvebelow 34mm ~wbya. 0.88!, where the code simulatesa cavity-beam profile that changes round trip byround trip and oscillates around a mean distribution;this can indicate that a multitransverse-mode oper-ation is to be expected. Actually, the experimentaltemporal shape of output pulses shows that STMoperation is sustained for clear apertures to 36 mm~wbya 5 0.83!, although the occurrence of extra

Fig. 5. Examples of output pulses for mirror 2 cavity with a full50-mm-diameter aperture: ~a! normal STM output, ~b! presenceof extra spikes. The detection chain has a bandwidth of 20 MHzand provides a negative voltage signal proportional to the receivedintensity.

Table 2. Summary of Main Performances Obtained from the LaserDemonstrator

Parameter Mirror 1 Mirror 2

Output pulse energy 8.3 J 5.9 JPumping efficiency 9.0% 6.2%Pulse length .5 ms FWHMOverall bandwidth ,200 kHz FWHM

spikes is more frequent, whereas a multitransverse-mode behavior appears for diameters below 32 mm~wbya 5 0.93!. The experimental efficiency datahave been also scaled taking into consideration theactive volume defined by the intracavity iris dia-phragm to highlight the efficiency performance of dif-ferent wbya designs. As shown in Fig. 7, a broadmaximum exists around wbya 5 0.9, but it is evidentthat the efficiency optimization leads to a critical re-gion for the transverse-mode selection. The validityof a STM limit of the order of 0.9 is also supported bythe performance obtained with mirror 1 ~wbya 50.89!, which supports the STM only if it is carefullyaligned and shows extra spikes more frequently.The near-field measurements for the apertures with36- and 32-mm diameter are shown in Figs. 8 and 9.Although they show a much less smooth profile thanthat obtained at full aperture, no reliable indicationcan be obtained from these results about thetransverse-mode structure.

Fig. 6. Comparison between measured and simulated output ef-ficiencies with different intracavity-limiting apertures.

Fig. 7. Experimental efficiency versus wbya ratio scaled to ac-count for the active volume defined by the cavity aperture.


Fig. 8. Near-field ~NF! measurement of the output beam with a 36-mm diameter intracavity aperture and mirror 2. ~a! Beam spot onthermal paper. ~b! Pyroelectric array data and simulated beam profile.

We took far-field measurements by focusing an at-tenuated sample of the output beam and measuringthe fraction of energy transmitted through a pinholeplaced in the focal plane as a function of the holediameter. In spite of the smooth experimental nearfield, even in the case of the small spot-size mirrorand full laser aperture, the theoretical and simulatedpredictions in the far field are significantly betterthan the experimental data ~Fig. 10!. However, theresults are practically identical for both mirrors used~Fig. 11!, notwithstanding the noticeable differences

in near-field smoothness; in our opinion this indicatesthat our data lay on the limit of experimental accu-racy, which is probably set by pointing jitter and bybeam degradation that are due to steering and atten-uation. Approximately 60% of the total beam en-ergy is encircled within the Airy radius of a planewave whose diameter is equal to the discharge width,a result that is similar to other available data.25The far-field pattern dependence on aperture diame-ter has also been addressed, both with simulations~Fig. 12! and measurements ~Fig. 13!. As for full

Fig. 9. Near-field ~NF! measurement of the output beam with a 32-mm diameter intracavity aperture and mirror 2. ~a! Beam spot onthermal paper. ~b! Pyroelectric array data and simulated beam profile.


aperture measurements, although simulations showa significant change in the energy distributions be-tween the different cases, the measured data are sim-ilar for all the diameters and thus strengthen thehypothesis that the results obtained are at the limitof experimental accuracy.From all the measurements we carried out, it is

clear that the observations of beats, and possibly ofextra spikes, on the temporal shape of the outputpulse is themost useful and sensitive tool to check theonset of higher transverse modes, whereas little or noinformation can be gained from near- and far-fieldprofiles.

5. Conclusions

The feasibility of a high-energy, long-pulse, andnarrow-bandwidth CO2 pulsed transmitter suitablefor a spaceborne Doppler wind lidar application hasbeen demonstrated. However, some concern still ex-

Fig. 10. Experimental ~crosses! and simulated ~solid curve! far-field pattern for mirror 2; the distribution for the theoretical SGMprofile is also shown ~dash-dot curve!.

Fig. 11. Comparison between the far-field results for the twoSGM mirrors in the case of full laser aperture.

ists about the trade-off between output efficiency andbeam quality, especially because of the evidence ofhigher-order transverse-mode onset even in the caseof a resonator design that fits quite loosely within thefinite dimension of the active medium. On the otherhand, our experiments have demonstrated that atrue multitransverse-mode operation is associatedwith a rather strong perturbation of the theoreticalbeam, i.e., wbya of approximately 0.9 or greater.The sporadic presence of higher-order modes, whichappear as extra spikes even with a wbya ratio as lowas 0.6, supports the existence of mode perturbationscaused by the active medium, for example, producedby shock waves propagating inside the gas because ofthe pumping process26,27 or because of refractive-index changes that are due to the laser action itself.28

Fig. 12. Simulation of the encircled energy integral in the far-field versus the diffraction angle withmirror 2 and in considerationof different intracavity-limiting apertures ~solid curves!. The dis-tribution computed for the theoretical SGM output beam is alsoshown ~dash-dot curve!.

Fig. 13. Experimental encircled energy integral in the far-fieldversus the diffraction angle with mirror 2 and different limitingapertures.


The possible responsibility of shock waves can beevaluated roughly by one defining an unperturbedradius r0 for whichwbyr0 is equal to the above-quotedwbya limit and then computing the time needed forthe perturbation to reach it coming from the dis-charge edge. The wave propagation velocity is of theorder of 0.5 mm ms21, so that in our case the resultingtime is approximately 16 ms, which is quite large,although of the same order, with respect to the typical10-ms delay that we observed, but is instead in accor-dance with the presence of long-delayed extra pulses.The matter of why a higher-order transverse modeappears several microseconds after the buildup of theoptical pulse, which is initially both the STM and thesingle-longitudinal mode if injected, is thus still anunsolved problem whose solution would require fur-ther experiments, more validation of the numericalpredictions, and investigation of the active mediumphenomena, which are likely to underlie this prob-lem. Nevertheless, apart from this topic, the limitthat has been found experimentally for the onset ofmultimode operation states an upper boundary to thechoice of the soft radius that has a more generalvalidity and is useful for the design trade-off of thefourth-order SGM resonators. The laser efficiencyshows a broad maximum around wbya 5 0.9; consid-ering the results on mode discrimination, a designvalue ofwbya between 0.75 and 0.8 should be taken soas to obtain STM operation without much loss ofefficiency.

This research has been funded by the EuropeanSpace Research and Technology Centre contract, De-velopment of a CO2 laser for spaceborne Dopplerwind lidar applications @European Space Agency con-tract 9504y91yNLyPB~SC!#. We thank Errico Ar-mandillo and Callum Norrie of European SpaceResearch and Technology Centre for useful discus-sions during the entire program. The laser bread-board was built with the essential contribution ofother team members, General Electric Company(GEC) Marconi ~Borehamwood, U.K.! and Dornier~Friedrichshafen, Germany!, and we particularlythank Paul Schwarzenberger and Steven Wallace ofGEC Marconi for their contribution to earlier exper-iments.

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transverse-mode selection in apertured super-gaussian resonators: an experimental and numerical...

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