adaptive mirror effects on the performance of annular resonators

Adaptive mirror effects on the performance of annularresonators

J. B. Shellan, D. A. Holmes, M. L. Bernabe, and A. M. Simonoff

Computer studies have shown that azimuthal static mirror misfigures as small as X/20 can lead to severe deg-radation in the far field on axis intensity for resonators with an annular gain region. This poor far-field pat-tern can, however, be dramatically improved by the use of adaptive elements within the resonator. A rearcone in the resonator had a static misfigure applied to it, and an adaptive ring axicon located in the front ofthe resonator was deformed in such a way as to compensate for the misfigure of the rear element of the reso-nator.

1. Introduction

The unstable resonator first described by Siegmanhas been studied extensively both theoreticallyl-4 andexperimentally.56 Some of the useful properties ofunstable resonators include large mode volume and edgecoupling. Because the radiation is not confined to asmall central core, mirror edge effects are important andconventional Laguerre-Gaussian and Hermite-Gaussianmodes are not appropriate. Resonator modes andproperties have thus been studied largely throughcomputer7 and asymptotic techniques.8 If radiationis excluded from the central region by an opaque ob-stacle such as a gain generator, the device becomes anannular unstable resonator.9 -15 The annular designsare especially important for lasers employing a radiallyflowing gain media or a radial excitation source such asa flashlamp.

Figure 1 is a schematic diagram of an annular reso-nator that has been studied at Rocketdyne. Except forthe scraper mirror, the device is rotationally symmetric.The gain generator produces an annular gain region overa large area. The radiation, after traversing both legsof the gain region, is compressed by the inner axicon,and a scraper mirror is used as an output coupler, withthe radiation in the center of the beam being reflectedback into the cavity by the feedback mirror. The cy-lindrical geometry is compact and provides a naturalpressure relief for the outflowing gas.

A beam traversing the gain region of the resonatoralong the top leg from the outer axicon to the rear cone

The authors are with Rockwell International, Rocketdyne Division,Canoga Park, California 91304.

Received 29 September 1979.0003-6935/80/040610-06$00.50/0.© 1980 Optical Society of America.

will, upon reflection from the rear cone, travel the bot-tom leg. This common pass feature of the resonatormakes it relatively insensitive to certain types of mis-alignments and misfigures.'5 For example, if the rearcone has a static misfigure (normal surface displace-ment) of the form (A/N) coso, where 0 is the azimuthalcoordinate, a ray will travel the path shown in Fig. 2.The dotted surfaces represent an ideal mirror surface,while the solid line represents the distorted surface (A/Ncosk). It can be seen, that, although a ray now travelsa shorter distance along the top leg of the resonator, itmust now travel a greater distance along the bottom leg,and the round-trip phase distortion is equal to zero. Theuse of a rear cone or corner cube is an effective passivemethod of reducing the resonator sensitivity to mis-alignments. Because of polarization effects, however,the use of a standard rear cone produces a lowest-ordermode with a null on axis in the far field.' 6 This can bealleviated through the use of a special dielectric coatingthat produces a 90° relative phase shift for the S and Ppolarizations of the resonator.' 7

If a distortion on the rear cone of the form (A/N cos2kis introduced, as shown in Fig. 3, the output of the res-onator is degraded. A beam traveling the = 0 to 'k =1800 path travels a different path length than does abeam traveling the 0 = 90° to 0 = 270° path. Thephase front of the output beam is no longer a plane orspherical surface but is distorted, and the far-field in-tensity is reduced. This problem can be overcome, asshown in Fig. 4, if another optical element, such as theouter axicon, is deformed [in the form (/N) cos2,01.This paper will present computer results indicating thatcompensatory misfigures can in fact greatly increase thefar-field power in the bucket.

II. Field Propagation and Field Modifiers

Resonator solutions are found in a method describedby Murphy and Bernabel 8 or Siegman et al.3"19 A wave

610 APPLIED OPTICS / Vol. 19, No. 4 / 15 February 1980

INNER AXICON(TORIC PARABOLA) rGAIN

FEEDBACK D CONEMIRROR 7 D,$i9'

FS OPTICAL AXIS G

L•DR / OPTICAL AXIS 7

)SCRAPER P /zMIRROR OUTEB

AX ICON(TOR ICPARABOLA)

Fig. 1. Common pass annular resonator. The beam compactorconsists of two toric parabolas with a common ring focus. The focal

length ratio is m.

X/N

._

Fig. 2. Rear cone with a static misfigure of the form (N) coso.Dotted surface and ray represent an ideal mirror surface and ray path.Solid surface and ray represent the distorted surface and ray path.Ray path lengths are the same for the two cases. (Diagram applies

for k = 0 plane.)

/N

- I_

is launched inside the resonator and propagated fromone reference surface to another by the Huygens-Fres-nel integral. After many round trips through the res-onator, it is expected that the resultant field will ap-proximate a resonator solution PyE(r,o) = HE(ro),where H is an integral operator that propagates the fieldthrough the resonator, E(r, 0) is the function describingthe scalar field, and y is the eigenvalue for this mode.The computer runs were made for a bare cavity (nogain), and thus y < 1.3 The Gardner transform is usedto speed up the evaluation of the Fresnel integrals thatare done in polar coordinates.

The Prony method,'8" 9 extended to handle non-symmetric problems, is used to accelerate convergenceof the iterative resonator solution. The fundamentaleigenstate is then analyzed for its Strehl ratio. Fieldmodifiers are used in the computer program for anyperturbation to the aligned axisymmetric resonator suchas misalignments, misfigures, or struts. A phasemodifier simply results from the scalar field propagatinga slightly different distance to a distorted mirror. Thescalar field now becomes

E'(r,o) = E(r,o) exp [4(r,)].

The Strehl ratio is directly affected by the phase per-turbation X?, but, because of the multiple passes a beamtakes through the resonator, a small mirror surfacedistortion (and thus small 4b) can significantly degradethe Strehl ratio. It will be shown that this effect can becompensated by using a deformable mirror.

111. Calculation of Compensatory Mirror Misfigure

We will consider distortions on the rear cone of theform CRn cosp, where R is measured from the centerline of the resonator. The quantity C is a constant, andp is an integer. A geometric optics theory is used topredict the compensatory misfigure needed on the outeraxicon to correct for uncontrolled rear cone errors. Theouter axicon is deformed by a series of piezoelectricactuators, and all distortions are measured normal to

X/N

/017 A

IN

2I

Fig. 3. Rear cone with a static misfigure of the form (/N) cos2.Distorted and ideal ray paths are no longer the same length. (Di-

agram applied for = 0 plane.)

I'

*Fig. 4. Resonator with a rear cone with a static misfigure of the form(X/N) cos2k but with a compensating outer axicon. (Diagram applies

for 1k = 0 plane.)

15 February 1980 / Vol. 19, No. 4 / APPLIED OPTICS 611

C .�x ::.: :t

* .rifiN

li

Z/.

a#XIr r45'

t @ 1 X -I -I T soA:-2. -~~~~~~~~~tCENTERLINEOF

REOO.NATOR

e 1 > a + x~~~~~~~~~~3r

Cos a f.(..I a+Xa + 14r

COO 4 (+4) - +

the surface of an undistorted mirror. In the geometricoptics theory, a virtual source is located a distance Lbehind the feedback mirror, and a ray diverging fromthis point a distance r above the resonator axis willtraverse the resonator and return to Mr, where M is theresonator magnification (see Fig. 5). Upon reflectionfrom the feedback mirror, the ray at Mr will also appearto have originated from the virtual source a distance Lbehind the feedback mirror. The rayleaving thefeedback mirror at r traverses the resonator by first Fig.striking the outer axicon ad radial coordinant a + Xjrand azimuthal coordinant 0, then the rear cone at a +X2r,k, the rear cone a second time at a + X3r,k + r, theouter axicon at a + X4ro + wr, and finally the feedbackmirror at Mr. The parameters M, L, X1, X2, X3, and X4 whereare all functions of the resonator dimensions and feed-back mirror radius of curvature and are easily calcu-lated. The normal surface distortion on the rear coneis given by fr(R, 0), while the unknown compensatory This solutmisfigure on the outer axicon is given by f(R,). int4 Eq. (fo(R,) must be found in terms Offr(R,5). We postu-late that, when the ideal fo is used, an arbitrary ray IV. ModEleaving the feedback mirror at r and returning to Mr As mentravels exactly the same path length as in a perfect un- is taken distorted cavity. Thus, if the rear cone sticks out too misfigurefar, the outer axicon must be pulled in. cone is gin

Quantitatively for small distortions this can be ex- the distortpressed as of actuat

(2)1/2 outer axiccos(O/2)fo(a + Xir,o) + 2 fr(a + ks2r,1k) pattern wj

22)l/2 twenty-fo+ 2 r (a + k\3r,o + r) + cos(O/2)fo(a + k\4r,k + r) = 0. (1) placemen

2 but the stThe cos0/2 and [(2)1/2/2] factors are simply geometric arbitrary factors and account for the rays striking the outer axicon determiniand rear cone at 0/2 and 450 from their normals, re- placemenspectively. For the cases considered, fr(R,) = CRn splines.2 0

cospo (p = even number, since p = odd number re- and a freequires no compensation to first order). The solution outer radiifo(R,k) is given by to set the

-C(1/2 n n positionefo(R,1) = 2 / cosp 0E (n Jan -q(R - a)q given byl2 cosO/2 lq = 0 q2 the mirroiks + (1)Pksll (2) Eq. (2) RI~ + (-1)Pks li enough ac

Fig. 5. Figure used to predict ideal compensatorymisfigure for a distorted resonator [see Eq. (1)].

S. Placement of actuators on the outer axicon.

an n!

Iq nq)!q!,ion can, of course, be verified by substitution1).

eling the Outer Axicon Misfigures

tioned earlier, the distortion on the rear coneis CRn cospo, and the ideal compensatory(at least for geometric optics) on the outer

7en by Eq. (2). It must be remembered that;ion on the outer axicon is produced by a seriesors. Figure 6 is a schematic diagram of the'on with seventy-two actuators arranged in aith three actuators in the radial direction andur in the azimuthal direction. The dis-Ls of the actuators can be controlled precisely,irface between the actuators cannot assumeshape. A simple yet very accurate method forng the surface height once the actuator dis-ts are specified involves the use of bicubicWe used bicubic splines in polar coordinates

e-edge boundary condition for the inner andius of the outer axicon. Equation (2) was usedactuator heights. Each of the actuators wasI so that the mirror height at the actuator wasEq. (2). At locations between the actuators,r surface would not be described precisely byut would differ by only a small amount if

!tuators were used to deform the mirror.


Table 1. Computer Results for Distortions of the Form (X/N) cos24kApplied to the Rear Cone

STREHL RATIO STREHL RATIOMISFIGURE WITHOUT COMPENSATION WITH COMPENSATION

X/120 COS 2 0.9010 0.9996

X/40 COS 2 0.6016 0.9916

4/20 COS 2 0.4012 0.9916

X/10 COS 2 0.1056 0.9911

Table 11. Computer Results for Distortions of the Form (X/40) cospoApplied to the Rear Cone

STREHL RATI O STREHL RATIOMISFIGURE WITHOUT COMPENSATION WITH COMPENSATION

4/40 COS 2 0.6016 0.9916

/40 CO 4 0.5451 0.9916

X/40 COS 6 0.3417 0.9914

/40 COS 8 0O5304 0.9899

X/40 COS 104 --- 0.9697

Table Ill. Computer Results for a Distortion of (c0 + cr + c2 r2) cos6oApplied to the Outer Axicon

Fig. 7. Far-field intensity patternfor a perfect resonator or a reso-

nator with ideal compensation.

V. Computer ResultsLight wavelength X was propagated through a reso-

nator with an equivalent Fresnel number of 8.5 andmagnification of 2.0. The results are summarized inTables 1, 11, and III. A distortion of the form (X/N)cos2o (N = 10,20,40,120) was applied to the rear cone.The distortion is normal to the surface, and X is thewavelength of the light. The heights of the seventy-twoactuators were set according to the method outlined inan earlier section, and their positions on the outer axiconare shown in Fig. 6. Operation of the uncompensatedresonator results in a Strehl ratio ranging from 0.9010to 0.1056, while Table I also indicates that the com-pensated resonator produces a Strehl ratio of 0.9996-

0.9911. The beam clean up resulting from proper ac-tuator setting is excellent.

Table II summarizes misfigures of the form X/40cospo (p = 2,4,6,8,10). The resonator solution did notconverge from the X/40 coslO4 run, and it can be seenthat the beam clean up for this run was not as effectiveas for the other cases. This is to be expected since thecorrective element only has twenty-four actuators in theazimuthal direction, making it difficult to correct forhigher order azimuthal misfigures.

Finally, a case was tested with a radially dependentmisfigure placed on the outer axicon, and the adaptiveelement was moved to the rear cone. The shift in ac-tuator location and misfigured surface, as well as the


. MISFIGURE ON THE OUTER AXICON

6(r,4) = (C + Cr + C2r2) COS 64

. 72 ACTUATORS ON THE REAR CONE

Co = -7.753 x 10 4

C1 = 5.659 x O 5

C2 -1.024 x 06

r IN CM

STREHL RATIO

W ITHOUT W ITHCOMPENSATION COMPENSATION

0.6010 0.9946

Fig. 8. Far-field intensity patternfor a (X/40) cos4 distortion on the

rear cone-uncompensated.

0.

Fig. 9. Far-field intensity patternfor a (X/20) cos2o distortion on the

rear cone-uncompensated.

radially dependent misfigure, make this case differentfrom the last two. The distortion on the outer axiconis of the form (co + cr + 2r 2) cos6k, with c = -7.753X 10-4, C, = 5.659 X 10-5, c2 = 1.024 X 10-6, and r, theradius measured from the center line of the resonator,is given in centimeters. Table III indicates that, aftercompensation, the Strehl ratio is perfect up to the thirddecimal place, despite a Strehl ratio of 0.600 for an un-compensated resonator.

The far-field intensity patterns for four cases is shownin Figs. 7-10. The field from the perfect resonator (ordeformed resonator with ideal compensation), whichhas a high peak intensity, is shown in Fig. 7. If misfi-

gures are applied to the rear cone, the peak field in-tensity decreases and the far-field power spreads overa wider area, if no effort is made to compensate with theadaptive element, as shown in plots 8-10.

VI. Conclusion

Computer studies have shown that mirror misfiguresas small as a fraction of a wavelength can lead to verypoor Strehl ratios for annular resonators. One methodto compensate for these effects is to replace one mirrorof the resonator with an adaptive element, which, oncedeformed, will compensate for misfigures on othermirror surfaces. Geometric optics was used to deter-


Fig. 10. Far-field pattern for a(X/10) cos2o distortion on the rear

cone-uncompensated.

*0

mine the ideal actuator settings for a specific misfigure,and physical optics was used to find the resonatormodes. Results to date indicate that, for typical cy-lindrical lasers, an adaptive element placed at one endof the gain region can be used to compensate extremelywell for misfigures on other mirrors of the resonator.The results show that seventy-two actuators provideda high enough resolution to correct the distortionsconsidered and that geometric optics was sufficient fordetermining the annular corrections, that is, diffractioneffects do not smear out the geometric compensatoryimprovements. A possible method of implementingthis system would be to sample a part of the outputbeam with a grating or Hartman plate and focus itthrough a pinhole. A feedback loop and multidithersystem could then control the adaptive element.

The authors are grateful to W. H. Southwell forhelpful technical discussions and for the use of his polarbicubic spline computer program.

This work was supported by the Air Force WeaponsLaboratory, Kirtland Air Force Base, New Mexico87117, under contract F29601-77-C-0006.

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adaptive mirror effects on the performance of annular resonators

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