annular converging wave cavity

Annular converging wave cavity

R. A. Chodzko, S. B. Mason, and E. F. Cross

A new type of resonator that generates an annular geometric mode by use of spherical mirrors has been de-

veloped. The four-element cavity consists of an external confocal unstable resonator with a double-sided

450 coupling mirror and a flat feedback mirror. The flat feedback mirror is placed on the plane wave side

of the confocal cavity to form an annular mode between the feedback mirror and the coupling mirror. A

plane annular wave (matched to an annular gain medium) is fed back into the unstable resonator that alter-

nately converges and diverges to produce a diverging annular output beam. Experiments were performed

on a cw HF laser. Observations of mode patterns on the flat feedback mirror and the convex mirror and of

far-field beam quality were made. Far-field measurements indicated near-diffraction-limited beam quality

for a peak on-axis intensity mode on the convex mirror. A nearly uniform annular mode was observed on

the flat feedback mirror.

1. Introduction

The unstable resonator first described by Siegman1

has been used to generate a single transverse modeoutput at large-cavity Fresnel numbers. The activemedium typically is placed inside the resonator toproduce an annular beam for edge coupling 2 3 or a ho-mogeneous beam for continuous coupling. 2 For thepresent work, positioning the unstable resonator ex-ternal to the active medium for transverse mode controlhas been proposed. By means of an external edge-,coupled, confocal unstable resonator (with a double-sided 450 coupling mirror), an annular geometric modecan be formed, which can be matched to an arbitraryannular gain medium. Recently, 4 mode control in anannular medium has been obtained with toroidal mir-rors, while the present method employs spherical mir-rors. The output then can be obtained in the form ofa diverging or parallel annular beam. Feedback be-tween the active medium and the external unstablecavity can be obtained by generating a series of alter-nately converging and diverging spherical waves.

A geometric theory for this converging wave resonatorwas developed and applied to determine the effectivereflectivity. The presence and suppression of parasiticstable cavity modes is also considered. Preliminaryexperiments with a converging wave resonator wereperformed on a cw HF linear flow device. Measure-ments of the intensity distribution of internal cavitymodes and beam quality were made. The work re-ported herein was completed in September 1974.

The authors are with Aerospace Corporation, P.O. Box 95085, Los

Angeles, California 90045.Received 7 February 1976.

11. Cavity Description and Theory

Figure 1 is a schematic diagram of a conventionalpositive branch confocal unstable resonator. Figure1(a) shows the standard single-ended output tech-nique,1 where the 450 output coupling mirror is placedclose to the convex mirror. 3 Most (but not all) of theoutput radiation is extracted from the coupling mirrorsurface facing the concave mirror since most of thegeometric expansion occurs on this side. It is possible,however, to obtain a double-ended output by the use ofa double-sided, 45° output coupling mirror. For aconfocal unstable resonator, the two output beamsconsist of a parallel beam and a diverging spherical wavewith their obscuration ratios dependent on the axialposition of the coupling mirror. In Fig. 1, the activeregion is assumed to be inside the resonator, and theoutput beams are formed as the result of the geometricexpansion of diverging spherical waves.

The converging wave cavity that applies externalfeedback to the confocal unstable resonator fortransverse mode control is shown in Fig. 2. The four-element cavity consists of a confocal unstable resonatorwith a double-sided output coupling mirror in con-junction with a flat, totally reflecting feedback mirror.The feedback mirror is placed on the plane wave outputside of the double-ended unstable resonator, thusforming an annular mode within the geometric opticsapproximation. If a gain medium is placed between theoutput coupling mirror and the flat feedback mirror,this resonator will extract power from an annular por-tion of the active region. If the active region is annularin shape, this resonator should efficiently extract powerfrom the medium. It should be noted that, if the shapeof the hole in the coupling mirror is varied, one cangenerate a variety of annular cylindrical modes, whichinclude a circular or a square annulus.

September 1976 / Vol. 15, No. 9 / APPLIED OPTICS 2137

a) SINGLE-ENDED OUTPUT

intensity distribution. Also, the gain seen by the cavitywill be the gain averaged over an annular region, i.e.,spatial-gain variations increase the threshold for os-cillation.

The positive branch confocal unstable cavity shownin Fig. 1 will have a round trip magnification M givenby3

M = -R1 /R 2 , (1)

and the radii of curvature of the mirrors are relatedby

""- DOUBLE-SIDEDCOUPLING MIRROR

b) DOUBLE-ENDED OUTPUT

Fig. 1. Confocal unstable cavity configurations.

R1 + R 2 = 2L. (2)

It can be shown from Fig. 4 that the diverging waveobscuration Ml is given by

CONFOCAL -/UNSTABLECAVITY

[i

-z DIVERGING SPHiERICALWAVE OUTPUT BEAM

- DOUBLE-SIDED COUPLING MIRROR

ANNULAR GAIN REGION

- ANNULAR GEOMETRIC MODE

- - FLAT TOTAL REFLECTORFEEDBACK MIRROR

Fig. 2. Confocal converging wave cavity with external feedback.

Figure 3 shows the various geometric parameters tobe considered in the design of the confocal convergingwave cavity. These parameters include the mirror radiiR1 and R2, the mirror separation L, the distance be-tween the concave mirror and the coupling mirror L,the diverging wave magnification of the output beamM1, the plane wave magnification M2, the diameter ofthe hole in the output coupling mirror Do, the distancebetween the feedback mirror and the coupling mirrorLF, and the thickness of the feedback annulus T2.Another parameter, not shown in Fig. 3, is the confocalunstable cavity magnification M.

Figure 4 shows an equivalent lens sequence for theconfocal converging wave cavity, which illustrates theprinciple of operation within the geometric optics ap-proximation. Thus, a plane annular wave is launchedinto the sequence from the active medium. The con-verging wave is reduced in size down to the diffrac-tion-limited spot size of the order of V/I and subse-quently expanded such that a fraction of the incidentwave (of diameter M2Do) is reflected back into the gainregion to provide feedback and a fraction (of diameterM1 Do) is output-coupled in the form of a divergingspherical wave. Since the converging wave is limitedby a diffraction in this device, any nonuniformities inintensity caused by spatial variations in the gain dis-tribution will be smoothed out to provide an outputbeam in the form of an annulus with a more uniform

M = M - a(M - 1)

within the geometric optics limit. The plane wave ob-scuration M 2 is given by

M1 M 2 = M. (4)

Figure 5 shows a plot of M1 and M 2 vs a for variousvalues of M from Eqs. (3) and (4). Thus, if the outputcoupling mirror is positioned at various locations, therelative values of Ml and M 2 can be varied over a widerange, from Ml = M and M 2 = 1 to Ml = 1 and M 2 = M.These limiting values will not be attainable, of course(a = 0 and a = 1), because of the finite size of the cou-pling mirror. In the configuration shown in Fig. 3, theplane wave obscuration M2 corresponds to the feedbackthrough the active medium. For a given device, theannular gain region will be specified and, hence, M 2 willbe specified. Thus, for a given value of M, M1 and a willthen be determined. Equations (3) and (4) also showthat Ml is linearly proportional to M for a fixed valueof a, while M 2 approaches a finite limit of

MIRRORRADIUS R

Fig. 3. Confocal converging wave cavity geometric parameters.

(M2- 1DO/2

{Mr-1Do/2

.L

Fig. 4. Equivalent lens sequence for confocal converging wavecavity.

2138 APPLIED OPTICS / Vol. 15, No. 9 / September 1976

, . _ Ia_,

i I-. _all

(3)

f

a

Fig. 5. Variation of plane wave M 2 and diverging wave M1 magni-

fication with position of coupling mirror a.

M 2 = 1/(1-a) (5)

as M - A, which is a result of the assumed confocalunstable'cavity geometry.

Another parameter of interest for the confocal con-verging wave resonator in Fig. 2 is the reflectivity r and

the transmissivity t of the device. It is clear that theseproperties will differ from a conventional unstableresonator since what is normally output-coupled is now

being fed back into the resonator. In Fig. 4, it can beseen that an annular input from the active region willfirst converge and then expand from the diffraction-limited spot size. A fraction

n=l ( M22 n=l()

(1 - IM22 )

(1- /M2 ) (10)

and

nil (M2 2 M2) n=l M2)

(1/M 22 - /M2 )

(1 - 1/M2 ) (11)

where it is assumed that M > 1. Note from Eqs. (10)and (11) that r + t = 1. In Fig. 6, the reflectivity r isplotted as a function of magnification M for variousvalues of a. Thus, as M varies from 1 to o for a fixedvalue of a, r varies from a to 1 - (1 - a) 2 . The value ofr = a in the limit of M = 1 is not valid, however, sincethe series indicated in Eqs. (10) and (11) do not con-verge.

It can be concluded from Fig. 6 that the reflectivityr increases with increasing M in contrast to the con-ventional unstable resonator. The variation of r withM is not large, however, with a maximum variation of50% for a = 0.5. It is also seen that, if the 450 couplingmirror is suitably positioned, a wide range of values forthe reflectivity can be obtained to satisfy the thresholdcondition for the given active medium. The primaryparameter that controls the reflectivity r is seen to bea as shown in Fig. 7. Thus, for small values of M (nearunity), the reflectivity is nearly linearly dependent on

a.The confocal converging wave resonator (see Figs. 2

and 4) differs from the conventional unstable resonatorin that there occur many passes within the empty res-onator before the return (reflected) wave traverses theactive medium. Within the geometric optics approxi-mation, the number of round-trip passes Np is given by(assuming DF = DiMN)

ln(DF/Di)- ln(M)

(12)

r = 1 - (1/M 22)

will then be fed back, and a fraction

M 12

-1 1 1

M 2 M22 M2

(6)1.00

(7)

will be output-coupled in the form of a divergingspherical wave within the geometric optics approxi-mation. Subsequently, more plane waves will be fed

back into the active region and more spherical waves will

be output-coupled until all the energy from the incidentannular plane wave is either reflected or transmitted.Thus, for the nth wave, a fraction

r

0.50 -

(8)

0.10

rn = rlt and arciis reflected, and a fraction

0

a = 0.90

c = 0. 75

c = 0. 25

a .1

I L I I I I I I I1.0

tn = (M2 -M2) (M2)(9

is transmitted. The total reflectivity r and transmis-sivity t are given by

5.0M

Fig. 6. Variation of reflectivity r with confocal unstable cavitymagnification M for various coupling mirror locations a.


cc = 0. 50

IIn U.\1.0

1.00

r

K

0.50 F-

0

M = 5.0

M = 3.0-

= 1.50

= 1.0

I I I I

U 0.5a

1.0

Fig. 7. Variation of reflectivity r with coupling mirror position a forvarious values of magnification M.

where DF is the maximum beam diameter in the cavityand Di is the initial beam diameter (Di VXE, basedon diffraction). Thus, 2Np round trips are required toget in and out of the sequence shown in Fig. 4 for thefirst geometric wave (r1 and t) and one more for eachsubsequent wave (r 2 ,r3 ,.. . ,r,,). This implies morereflection losses for the converging wave cavity andmore sensitivity to cavity mirror aberrations than forthe conventional edge-coupled unstable cavity. FromEqs. (8), (9), and (12), it can be seen that these addi-tional losses can be reduced by increasing the magnifi-cation M. Thus, if the magnification M is increased-3.0, most of the energy will be contained in the firstgeometric wave (r1 + t1 0.9). If DF L 2.5 cm and Di

1.25 mm, then NP 3 round trips (assuming L = 1m and = 3 X 10-4 cm).

The above analysis is valid within the geometric op-tics approximation. It is of interest to determine theconditions when diffraction effects will become im-portant for the converging wave unstable cavity shownin Fig. 3. One condition is that the annular mode in thefeedback region (M2) must remain a nearly geometricannulus in one round trip from the coupling mirror andback over the distance 2LF. The two characteristiclengths defining the feedback mode are the hole diam-eter in the coupling mirror D and the thickness of theannulus T2. Normally, D > T 2, so that the geometricoptics approximation will be valid if

LD - (T 22/A) > 2LF, (13)

where LD is a characteristic distance over which thethickness of the annulus will not spread appreciablyfrom its geometric value. Thus, it can be concludedfrom Eq. (13) that

LD = T 22

NF

2LF 2XLF 2 (14)

where NF = Do2/4XLF is the Fresnel number based onthe feedback mirror distance.

A plot of Eq. (14) is shown in Fig. 8. The ratio LD/2LF is plotted vs NF for various values of the plane wavemagnification M 2 . The horizontal line (LD/2LF = 1)indicates where the geometric theory applies and wherediffraction losses are expected. Thus, for the conditionsLD/2LF < 1, it would be expected that the reflectioncoefficient (see Fig. 6) calculated from the geometricanalysis be too large. It can be concluded from Fig. 8that the diffraction losses obtained from a large NFnumber device and small feedback magnification (M2small) could also be obtained from a small NF numberdevice and large feedback magnification.

A further theoretical consideration is the formationof parasitic stable cavity modes in the converging waveresonator.5 The region between the concave mirror andthe flat feedback mirror in Fig. 3 can form a stable res-onator such that bouncing ball modes will exist in anannular region. These parasitic modes will have poorphase coherence, and high radiation flux levels will buildin this region similar to a closed cavity6 condition.These parasitic modes have been successfully sup-pressed experimentally when the distance LF was in-creased and the mirror radius R was decreased suchthat an unstable cavity was formed. A negative branchnonconfocal unstable cavity is formed if the condi-tion

(1 (A+ D) = 1 2(LF + L) <-1(15)

is satisfied, where A and D are the elements of the raymatrix for the negative branch unstable cavity.

A final theoretical consideration for the confocalconverging wave cavity is the existence of a peak in-tensity region on the convex mirror in Fig. 3. The ex-istence of a peak (on-axis) intensity region is predictedfrom the quasigeometric model since the convergingwave collapses down to a spot with a diameter of theorder of \/XE (first Fresnel zone) before expandingagain to provide feedback. This hot spot could ther-mally load the cavity mirrors seriously at high powerlevels. The quasigeometric model is too simple topredict correctly the intensity distribution of the in-ternal cavity modes, however, and a diffraction analysisis required to determine the Fresnel number depen-dence. A solution to the converging wave unstable

0.0 2.0 .00

L0/LfI r

7-A,

I . . .I V I I I I I I I I I I I I , I I - I I I10 100 1000 0,000

N

CEOIETRIC THEORY APPLIES

LARGE DltFRACTION LOSSS

Fig. 8. Variation of diffraction losses LD/2LF with feedback mirrorFresnel number (NF) for various values of plane wave magnification

M 2.


I I

ca w w7 ,e o S

I I I I~~~~~~~~~~~

100,

,.,

. . I. ,

Table I. Confocal Converging Wave Cavity Design Parameters

Cavity M M, M2 L LF R, R, a

1 4.8 2.88 1.66 121 cm 91 cm 305 cm -64 cm 0.51

2 2.3 1.42 1.62 41 153 146 -64 0.68Do N = D0

2/(4XL) LDI( 2LF) NF r 1/2 (A + D)1 1.25 cm 11.6 3.43 17.8 0.67 0.002 1.25 cm 34 1.75 9.1 0.75 -1.48

cavity problem within the Fresnel-Kirchhoff scalarwave diffraction approximation has not yet been ob-tained. Experimental measurements have provided theinformation available thus far on the intensity distri-bution of the internal cavity modes.

111. Experimental Results

A list of design parameters for the confocal converg-ing wave cavities that were applied to a cw HF chemicallaser are presented in Table I. These cavities were

applied to an arc-driven supersonic diffusion cw HF(X2.8 X 10-4 cm) linear flow (as opposed to a radial flow)

device.7 A modified three-dimensional, modular,54-slit, 1.25 cm X 18 cm nozzle array was used at flowconditions corresponding to -2 kW closed-cavity power.Two cavity designs were evaluated with magnificationsof M = 4.8 and 2.3 and reflectivities of r = 0.67 and 0.75,respectively, calculated from the geometric theory. Theprimary difference between the two designs is the factor1/2(A + D) of 0.00 and -1.48, respectively, indicatingthe presence or elimination of parasitic stable cavitymodes. The first experiments were conducted withcavity 1 until it became clear from the data that par-asitic modes were oscillating. Cavity 2 was then usedand it was verified that no parasitic modes existed, thusyielding the true modes of the confocal converging waveresonator.

The experimental measurements included the outputpower, near-field intensity distribution, intensity dis-tribution of the internal cavity modes on the convexmirror, internal cavity mode on the flat feedback mirror,

and the far-field beam quality. The experimental ap-paratus used for measuring the intensity distributionof the internal cavity modes and far-field beam qualityis shown in Fig. 9.

An ir vidicon camera recorded the diffuse infraredimage of the HF wavelengths (filter with a 2.6-3.0-gmbandpass) on both the convex mirror and the flatfeedback mirror. The gold-coated BeCu mirrors pro-vided a diffuse image that closely approximated that of

an ideal (Lambertian) diffuse reflector. These ir imageshad a small amount of geometric distortion caused byviewing off-axis approximately 150. While the diffuseimage on the convex mirror was viewed, beam qualitymeasurements were made simultaneously in terms ofthe far-field power distribution. The far-field powerdistribution was measured with two InSb detectorsviewing the total power (reference detector) andtransmitted power (signal detector) through an apertureat the focal plane by means of diffuse reflectors.Variable aperture diameters were obtained with a ro-tating disk.8 All measurements were made with a

low-duty cycle internal cavity shutter to reduce thermaldistortion effects on the cavity mirrors (20-msec pulseswith 1 sec between pulses). The spatial resolution ofthe ir vidicon measurements was -1 mm.

Cavity 1 (Table I) was first applied to the cw HF ac-tive medium. The cavity resonated with an outputpower of -100 W, and an approximately uniform an-nular output beam was obtained. It was found, how-ever, that very high internal cavity flux levels were oc-curring in an annular region between the feedbackmirror and the 450 coupling mirror. This could beobserved from the light emitted as a result of the heatingof small particles (from the arc flow) impinging on theCaF2 Brewster windows. From past experience, thislight was emitted only when internal cavity flux levelsexceeded several kilowatts per square centimeter.These high internal cavity flux levels were inconsistentwith the low output power obtained from the resonator.Furthermore, beam-quality measurements of the an-nular output beam showed that less than 5% of thetheoretical power was within the diffraction limitedmain lobe. From the above data, it was concluded thatthe cavity was not performing as predicted by the geo-metric theory and that parasitic stable cavity modeswere removing power from the active medium withinan annular region. Thus, the region between thefeedback mirror and the concave mirror formed a stablecavity of high reflectivity such that high internal cavityflux levels could be obtained similar to the closed-cavitytechnique reported by Spencer et al. 6 The multimodestable cavity would have poor phase coherence and,hence, the poor beam quality of the annular output.

Subsequently, cavity 2 (Table I) was applied to thecw HF active medium with a factor 1/2(A + D) = -1.48(as opposed to 0.00 for cavity 1), forming a negativebranch unstable resonator in the feedback region. The

/ LCa F2 WEDGE BEAM SPLITTER

In Sb REFERENCEDETECTOI

Fig. 9. Experimental apparatus for measuring internal cavity modes

and beam quality of converging wave cavity.


I-

Z 0. 5

-jw0!

SINGLE TV FRAMEPHOTOGRAPH

negative branch unstable cavity was achieved by in-creasing the distance LF [Eq. (15)], decreasing theconcave mirror radius R, and decreasing the externalcavity length L. With cavity 2, the high internal cavityradiation flux levels obtained with cavity 1 were notobserved, and good output beam quality was obtainedas discussed below. Furthermore, when the convexmirror was blocked, no lasing was observed with the irvidicon on the flat feedback mirror. It was concludedthat cavity 2 would yield the true modes of the confocalconverging-wave resonator, and the data given belowapply to this resonator.

Cavity 2 was prealigned with a He-Ne laser with boththe technique discussed by Krupke and Sooy3 and apellicle method. After prealignment, the cavity reso-nated at an output power of -100 W, and fluctuationsin the output indicated near-threshold operation. Thelow output power was attributed to both diffractionlosses and a poor matching of active medium to theannular mode. The converging wave cavity differs fromthe conventional unstable cavity in that it sees gainaveraged over an annular region. The highly nonuni-form gain distribution in the linear flow device couldreduce this averaged gain to near-threshold values.Thus, if one assumes an effective reflectivity r = 0.5(assuming additional mirror absorption losses) for theconverging wave cavity, a threshold gain of -2%/cm iscalculated. This gain is reasonable if for the cw HFlaser one averages over an annular region with a 2-cmi.d. and a 1.25 cm o.d. For a uniform annular gain dis-tribution this effect would not be present.

It was found that two dominant modes exist in theconverging wave cavity: a peak, on-axis, intensity mode-and an annular mode. The selection of either of the twomodes was found to be a very sensitive function of cavitymirror tilt; an approximately 25-girad tilt of the concavemirror was sufficient to change from one mode to the

T (averagedover couplingmirror hole)

Fig. 10. Intensity distribution onconvex mirror in peak intensity

mode (ir vidicon data).

-5 0 5r, mm

SINGLE SCAN ACROSS CENTERLINE

other. Figure 10 shows the intensity distribution on theconvex mirror from the ir vidicon data recorded duringa single on-pulse corresponding to the peak intensitymode. The pattern at the left is a photograph of asingle ir TV frame, and the plot at the right is the rela-tive intensity vs radial position on the convex mirrorscanned across the peak intensity region. The solid linecorresponds to a Gaussian curve fit. If an axisymmetricsolution is assumed, the Gaussian fit yields a peak in-tensity about four times the intensity averaged over the1.25-cm-diam hole in the 45° coupling mirror. Thedotted line shows the first Fresnel zone region. Thus,the effect of diffraction for this particular resonator withan inner Fresnel number of N = 34 is to reduce con-siderably the maximum intensity of the mode.

Figure 11 shows the intensity distribution on theconvex mirror from the ir vidicon data recorded duringa single on-pulse corresponding to the annular mode.The solid line in the single centerline scan is an ap-proximate fit of the data. This annular first-order ra-dial mode cannot be predicted from the simple geo-metric model, and a diffraction analysis is required topredict it. It was observed that the annular mode wasquite stable and would remain indefinitely once thecavity mirrors were set. The near-field intensity dis-tribution of the diverging output beam was observed,and the intensity pattern was an approximately uniformannulus for both the peak intensity and annularmodes.

Beam quality measurements were made while ob-serving the mode patterns on the convex mirror. It wasfound that the beam quality was good only for the peakintensity mode. Thus, Thermofax burn patterns takenat the Siegman focal plane showed a maximum center-line intensity for the peak intensity mode and an an-nular pattern for the annular mode. Figure 12 showsthe far-field power distribution measured during a


1.

-U

zwz~0.-

IJJc:

SINGLE TV FRAMEPHOTOGRAPH

Fig. 11. Intensity distribution onconvex mirror in annular mode (ir

vidicon data).

x, mmSINGLE SCAN ACROSS CENTERLINE

1.0

P/ Po

InSb DETECTOR OUTPUT5 msec/cm div

UPPER TRACE: TRANSMITTED POWER

LOWER TRACE: TOTAL POWER

Fig. 12. Far-field power distribu-tion with peak intensity mode.

0.5 1.0d, mm

1.5

15cm FROM COUPLER POLAROID PHOTOOF IR VIDICON

DATA

Fig. 13. Internal cavity mode onflat mirror.

130 cm FROM COUPLER

He-Ne PELLICLE BEAM


00

1 1 1 1 1 1 -

0.5

20-msec on-pulse with the peak intensity internal cavitymode. The data indicate more than half of the theo-retical power within the diffraction-limited main lobe.The deviation from theory could be due in part to slightnonuniformities observed in the near-field intensity.

Figure 13 shows the intensity distribution on the flatfeedback mirror from the ir vidicon data recordedduring a single on-pulse. This pattern did not appearto depend on whether the peak intensity or annularmodes were oscillating, although no correlation exper-iments were conducted (this would require two ir vid-icon cameras). The pattern on the flat mirror wasfound to be very sensitive to flat mirror tilt angle witha 25-grad change sufficient to deviate from a uniformannulus. Notice the annulus is rather uniform and doesnot show significant intensity variation that one wouldexpect from the nonuniform gain profile. This resultindicates that diffraction effects are smoothing the in-tensity distribution, and gain averaging is probablyoccurring. Notice the halo around the annulus from their vidicon data. This halo is believed to be due to dif-fraction losses over the distance 2LF. A He-Ne pelliclebeam was generated inside the unstable cavity and theplane wave (M2) output beam was propagated a dis-tance of 15 cm and 130 cm, as shown in Fig. 13. It isseen that significant diffraction spreading occurs at theHe-Ne wavelength ( = 0.63 gm); this effect will be evenworse at the HF/DF wavelengths (X 2.8 m or X _ 3.8AM). It can be concluded that the approximate crite-rion LD/2LF = 1 (see Fig. 8) is insufficient to determinewhether the geometric theory applies; further diffrac-tion calculations should be made.

IV. Conclusions

A new type of resonator has been developed for cou-pling coherent power from an annular gain region. Theresonator, which uses conventional spherical mirrors,is termed a converging wave cavity. The device differsfrom a conventional confocal unstable cavity in that themode is formed by a standard confocal unstable cavityplaced outside of the active region. Feedback throughthe annular active region is provided by a double-sided450 coupling mirror and a flat feedback mirror.

Preliminary experiments have been performed on acw HF laser with this device at a cavity Fresnel numberN = 34. Measurements of the output power, near-fieldintensity distribution, far-field beam quality, and in-tensity distribution of the internal cavity modes weremade. The data show an approximately uniform an-nular mode in the feedback region, an internal cavitymode with peak on-axis intensity distribution on theconvex mirror of the confocal cavity, and near-diffrac-tion-limited beam quality. Further experiments atlarge Fresnel numbers (N and NF) as well as a diffrac-tion analysis of the modes should be made.

This paper reflects research supported by the AirForce Weapons Laboratory (AFWL) and Space andMissile Systems Organization (SAMSO) under ContractF04701-74-C-0075.

References1. A. E. Siegman, Proc. IEEE 53, 227 (1965).2. R. A. Chodzko, H. Mirels, F. S. Roehrs, and R. J. Pedersen, IEEE

J. Quantum Electron. QE-9, 523 (1973).3. W. F. Krupke and W. R. Sooy, IEEE J. Quantum Electron. QE-5,

575 (1969).4. L. W. Casperson and M. S. Shekhani, Appl. Opt. 14, 2653

(1975).5. It should be noted that an alternate configuration to Figs. 2 and

3 that apply feedback on the plane wave (M2 ) side of the double-sided coupling mirror (producing a diverging output beam) is toapply feedback on the diverging wave (MI) side, thus producinga parallel output beam. Feedback on the diverging wave side canbe accomplished by means of a concave totally reflecting feedbackmirror or a lens in conjunction with a flat feedback mirror (whichpermits coupling to a cylindrical annular gain medium). Equa-tions (3) and (4) are slightly modified for the case of feedback onthe diverging wave side.

6. D. J. Spencer, D. A. Durran, and H. A. Bixler, Appl. Phys. Lett. 20,164 (1972).

7. D. J. Spencer, H. Mirels, and D. A. Durran, J. Appl. Phys. 43,1151(1972).

8. R. Wey and G. R. Wisner, "An Integrated Irradiance Analyzer forReal Time Laser Beam Diagnostics," Report UARL-148, UnitedAircraft Research Laboratories, East Hartford, Conn. (Oct.1972).

E. N. Bellows (left) and Robert Poirier of Galileo Electro-Optics Corporation


annular converging wave cavity

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