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Self-interference patterns and their application toinertial-fusion target characterization

Mark D. Wittman and R. Stephen Craxton

The sphericity and wall-thickness uniformity requirements of direct-drive inertial-fusion targets are ofthe order of less than 1%. These shells display self-interference patterns ~SIP’s! when irradiated witha spatially incoherent, narrow-bandwidth light source and viewed with a compound microscope. Thesepatterns are distinct concentric fringes when the target is uniform, whereas faint, distorted, or discon-tinuous fringes indicate a nonuniform target. We determined the wall thickness to within 60.5 mm bycounting the number of fringes in the SIP, independent of the outside diameter. Thickness uniformityis verified to an accuracy better than 0.05 mm. The wall thickness of gas-filled targets can be determinedto this accuracy without knowledge of the type of gas or its pressure. The SIP fringe technique is usedto select polymer shells typically of 800- to 1000-mm diameter and 5- to 12-mm wall thickness. The fringelocations have been modeled by use of ray tracing and agree well with actual measurements of well-characterized shells. Details of the formation of the SIP fringes, a theoretical model, and the methodused for quantitative measurement of the shell-wall thickness with the SIP are presented with validationexamples. © 1999 Optical Society of America

OCIS codes: 220.4840, 120.3180, 120.3940, 180.3170, 310.6860, 350.2660.

1. Introduction

In direct-drive fusion, a high degree of uniformity andsymmetry of laser irradiation are required for high-density compression, which is a prerequisite for suc-cessful nuclear fusion. Equally important is thetarget uniformity: both the sphericity and the wallthickness should be uniform to less than ;1%. Inthe past, dual-beam interference microscopy1 hasbeen used at the University of Rochester’s Laboratoryfor Laser Energetics to characterize the wall thick-ness and uniformity of transparent targets to be im-ploded on the OMEGA laser system.2 With thisechnique, an interference pattern is formed betweenne beam that passes through the target and a secondeam, split off from the first, that passes around thearget. By comparison with computer-generatedemplates, these interference patterns can yield botharget wall thickness and its uniformity to a highegree of accuracy.3We describe here an alternative interferometric

The authors are with the Laboratory for Laser Energetics, Uni-versity of Rochester, 250 East River Road, Rochester, New York14623-1299.

Received 15 March 1999; revised manuscript received 21 May1999.

0003-6935y99y255365-07$15.00y0© 1999 Optical Society of America

technique that is simpler to use and that provides arapid characterization of both the wall thickness andthe uniformity of single-shell targets. These are typ-ically polystyrene ~CH! shells, which are selected priorto being coated with layers of various materials orbeing filled with D2, DT, or some other desired gas.These shells have a remarkable property: when theyare irradiated with a spatially incoherent, narrow-bandwidth light source and viewed with only a com-pound microscope, they display self-interferencepatterns ~SIP’s! such as the one shown in Fig. 1.These patterns are distinct concentric fringes whenthe target is uniform, but faint, distorted, or discontin-uous fringes when the target is nonuniform. Previ-ously, SIP’s have not been used because they areclearly observed only in targets of very high quality,with a uniformity typically better than 1%.

This technique is currently being used for the pre-liminary selection of CH shells typically of 800- to1000-mm diameter and 5- to 12-mm wall thickness.The fringe locations have been modeled with ray trac-ing and agree well with actual measurements of well-characterized shells. Shells can be selected with thewall thickness known to 60.5 mm and with a unifor-mity better than 0.05 mm.

2. Origin of the Self-Interference Pattern

SIP formation results from multiple reflections ofrays within the shell walls. The three relevant

1 September 1999 y Vol. 38, No. 25 y APPLIED OPTICS 5365

7

a

5

beam paths are illustrated in Fig. 2. Beam 1 passesstraight through the shell, beam 2 undergoes tworeflections on the input side, and beam 3 undergoestwo reflections on the output side. For a perfectshell, the emerging wave fronts of beams 2 and 3 arevirtually identical, so they combine coherently andinterfere with the wave front of beam 1 to form theSIP. For an imperfect shell in which the input andthe output thicknesses are different, not one, but twoseparate SIP’s are formed, one corresponding to theinput side ~beams 1 and 2 interfering! and the otherto the output side ~beams 1 and 3!.

In Fig. 2, all rays are shown backprojected ~withdashed lines! to the point on the object plane ~z 5 0!from which they appear to come. Exact ray-tracingcalculations show that a ray incident at a height riappears to come from a height ra in the object plane,where the difference between ri and ra is negligibly

366 APPLIED OPTICS y Vol. 38, No. 25 y 1 September 1999

small ~typically less than 0.1 mm except very close tothe edge of the target! for each of the three beampaths. This is true only for the object plane z 5 0.Since the source is spatially incoherent, two rays caninterfere only if they originate from the same incidentray; thus the only object plane that permits SIP for-mation is the midplane z 5 0. In this sense thefringes can be described as being localized in thisplane. In contrast, if the illumination was spatiallycoherent as in the dual-beam interferometry tech-nique described by Gram et al.,3 interference fringescould be obtained for any object plane.

For a spherically symmetric target, one can cal-culate the locations of the interference fringes byplotting the optical path difference OPD2 2 OPD1between beams 2 and 1 as a function of apparentradius ra in the object plane ~Fig. 3!. @OPDi isdefined as the optical path difference ~in centime-

Fig. 1. Compound-microscope image of a SIP produced by a sym-metric target when the target is illuminated with narrow-bandwidth light. The CH capsule has an 850-mm diameter and a-mm thickness.

Fig. 2. Ray paths through the target of ~a! beam 1, ~b! beam 2, and ~c! beam 3. In each case the rays enter from the left, the emerging wavefront is drawn on the right, and the emerging rays are backprojected ~dashed lines! to their apparent origin in the object plane ~z 5 0!. For

perfect target, wave fronts 2 and 3 are virtually identical, add coherently, and form the SIP through combination with wave front 1.

Fig. 3. Optical path difference ~OPD2 2 OPD1! between beams 2and 1 for a representative CH target with an 850-mm outer diam-eter, a 5-mm thickness, and a refractive index at 546 nm of 1.59.The abscissa is the apparent radius ra in the object plane ~Fig. 2!,which is almost identical to the incident radius ri. The filledcircles correspond to integer values of optical path difference andthus give the radii of the centers of the bright fringes.

rifisii~sgafn

scfte

cp

Fn~

v

acf

ters! between a ray of beam i ~i 5 1 to 3! and aeference ray passing through vacuum, but in Fig. 3t is plotted in waves.# In this example, six brightringes are seen with optical path differences rang-ng from 29 to 24 waves, and one can simply con-truct the loci of greatest intensity in annterferogram by drawing circles at the correspond-ng radii. For targets with nonuniformities in thex, y! plane, i.e., targets that are not rotationallyymmetric about the z axis, one can form interfero-rams by tracing a grid of rays through the targetnd drawing a contour plot of the optical path dif-erence with the contour levels chosen to be integerumbers of waves.4In place of Fig. 3, the universal curves of Fig. 4

can be used to predict the behavior of all perfectlyuniform targets of interest. In this figure, the op-tical path differences OPD1 and OPD2 along withthe difference OPD2 2 OPD1 are all plotted againstthe normalized radius riyRshell, where Rshell is theaverage of the inner and the outer shell radii.~Similar results are obtained by normalizing to ei-ther the inner or the outer radius.! Graphs aresuperposed for CH targets with outer diameter d 5250 to 1500 mm and thickness t 5 2 to 20 mm;pecifically, the four extreme combinations are in-luded. All curves are virtually identical exceptor the low-aspect-ratio combination ~d 5 250 mm,5 20 mm!, which is not of current interest for

xperiments on OMEGA.The values of OPD for rays passing through the

1

enter of a perfect target with respect to parallel raysassing externally to the target are given by

OPD1 5 2t~n 2 1!, (1)

OPD2 5 OPD3 5 4tn 2 2t, (2)

OPD2 2 OPD1 5 2tn. (3)

or example, for n 5 1.59 and a wavelength l 5 546m, as used throughout this paper, the difference inOPD2 2 OPD1!yt between the center and the edge is

D~OPD2 2 OPD1!yt 5 0.62 from Fig. 4. ~The edge isunderstood to correspond to 95% of the innershellradius, i.e., it approaches the last ray that is trans-mitted through the target.! Thus, if N bright fringesare counted, D~OPD2 2 OPD1! ' Nl and

t < Nly0.62 5 0.88N mm. (4)

The accuracy of Eq. ~4! is limited by the accuracywith which D~OPD2 2 OPD1! can be estimated bycounting fringes. The method works as long as thetime difference between interfering rays @~OPD2 2OPD1!yc, where c is the speed of light# is less than thecoherence time of the source. For the low-pressuremercury-vapor source5 used in the study reportedhere, this criterion is satisfied for wall thicknesses t #15 mm.

Ray trajectories have been calculated both withexact ray tracing and with a paraxial approximationthat includes third-order spherical aberration. Theparaxial approximation does not accurately predictthe wave front near the edge of the target, wherehigher-order spherical aberration is present. Theerror incurred, however, is approximately equal ineach of wave fronts 1, 2, and 3; thus D~OPD2 2 OPD1!and D~OPD3 2 OPD1! are nearly identical for theparaxial and the exact ray-tracing treatments, andresults from the two methods agree closely.

One notable property of the SIP is its sensitivity tot. For n 5 1.59, the quantity detected is OPD2 2OPD1 5 3.2t @from Eq. ~3!# compared with OPD1 51.2t @from Eq. ~1!#, as would apply to conventionaltwo-beam interferometry. This method is thusroughly three times more sensitive to changes in tar-get thickness.

Another property of the SIP is that for very smalldifferences between tL and tR—the thicknesses on theleft and the right of the target in Fig. 2,respectively—a single interferogram is not formed.For a half-wave difference in the OPD along the axis,enough to destroy the SIP, the necessary thicknessdifference derived from

OPD2 2 OPD3 5 ~OPD2 2 OPD1! 2 ~OPD3 2 OPD1!

5 2tL n 2 2tR n 5 0.5l,

which gives

tL 2 tR 5 0.086 mm, (5)

Fig. 4. Universal curves governing the formation of the SIP.With plotting the OPD divided by the shell thickness t on theertical axis and the normalized radius ~riyRshell! on the horizontal

axis, the three quantities OPD1yt, OPD2yt, and ~OPD2 2 OPD1!ytre virtually independent of shell diameter and thickness. Theurves shown here are for four shells with outer diameters rangingrom 250 to 1500 mm and thicknesses ranging from 2 to 20 mm.

The dashed curves correspond to a 250-mm diameter and a 20-mmthickness.

September 1999 y Vol. 38, No. 25 y APPLIED OPTICS 5367

ttfSccgss

fdtItwwdtloc

fco

5

corresponds to a 1.2% peak-to-valley thickness vari-ation for a typical 7-mm shell. Targets displaying adistinct SIP have a much better uniformity than this.

Some examples of calculated SIP’s are given inFigs. 5–7. Figure 5 shows interferograms of threetargets with various wall thicknesses from which Eq.~4! can be verified. Figure 5~b! matches very wellhe experimental interferogram of Fig. 1. In Fig. 6,he interferograms of three targets with slightly dif-erent wall thicknesses t show the sensitivity of theIP to t, as alluded to above. The OPD through theenter changes by 0.29 waves for each 0.05-mmhange in thickness. Each change is clearly distin-uishable, especially if the location of the first or theecond clear fringe is measured. Finally, Fig. 7hows a combination of two SIP’s with a 1% nonuni-

Fig. 5. Calculated SIP’s for three perfectly symmetric CH targetsrom 5 to 9 mm. The outer circles indicate the edge of the targetorresponds to Fig. 3 and that for t 5 7 mm corresponds to Fig. 1.f the number of bright fringes by 0.88.

Fig. 6. Calculated SIP’s for three CH targets with slightly differedetected if attention is paid to the location of the inner fringes.

368 APPLIED OPTICS y Vol. 38, No. 25 y 1 September 1999

ormity. The nonuniformity is directed along the zirection @Fig. 7~a!#, at 45° to the y and the z direc-ions @Fig. 7~b!#, and along the y direction @Fig. 7~c!#.n each case the heavy and the light lines indicate thewo SIP’s; in both Figs. 7~a! and 7~b! a distinct SIPould not be seen in practice, so the nonconcentricityould be easily detected. In Fig. 7~c!, where theefect is aligned perpendicular to the viewing direc-ion, an up ~down! shift can be observed in the calcu-ated fringe pattern but would probably not be readilybserved in practice. In Fig. 7~c! the two SIP’s addoherently as the two thicknesses tL and tR are equal.

Actual target imperfections rarely match the sim-plified imperfections shown in Fig. 7. An example ofan imperfect target is shown in Fig. 8. The fringeson the bottom are not too different from those of Fig.

with an outer diameter of 850 mm, but with thicknesses t rangingthe other circles are interference fringes. The SIP for t 5 5 mmarget thickness in micrometers can be estimated by multiplication

ll thicknesses. Thickness differences as small as 0.05 mm can be

, all, andThe t

nt wa

lAsttertrcaCriceViwafttbbm

ip

1, but extra fringes are observed near the top where,clearly, the target contains a region of excessivethickness. There also appears to be some moirebeating in this area between the two SIP’s, one ofwhich is stronger because the microscope is focusedcloser to its plane of localization.

Fig. 7. Calculated SIP’s formed by interference between beams 1target with a 1% thickness nonuniformity for three different oriensurface is spherical but shifted 0.05 mm in the direction of D.! ~a!interference pattern would be seen in practice; thus the existenceuniformity.

Fig. 8. Example of an interference pattern formed from a poor-quality shell. Two SIP’s are produced, as in Fig. 7, but they arenot concentric.

1

It does not require a large deviation from sphericalsymmetry for the two SIP’s to lack coherent combi-nation. When the apparent positions ra in the objectplane of the two rays associated with beams 2 and 3~Fig. 2! differ by the spatial coherence length of theight source imaged onto this plane, coherence is lost.n alternative and more general approach is to con-ider every incident ray, including rays other thanhe parallel set shown in Fig. 2, as independent owingo the spatially incoherent nature of the source. Forach incident ray there are three emerging rays, cor-esponding to beam paths 1 through 3. Projectingheir trajectories back, there is both a position whereays 1 and 2 cross for an ideal target and a point oflosest approach for a nonideal target. There will beseparate point of closest approach for rays 1 and 3.onstructive interference occurs if the backprojectedays pass sufficiently close to each other. For andeal target, rays with different angles of incidenceross in rotated midplanes but may still be closenough to each other in the object plane to interfere.iewed alternatively, each angle of incidence results

n an interference pattern that appears to be formedithin the target, with a certain localization depthlong the propagation direction; the patterns for dif-erent angles then add in intensity. This property ofhe system makes it unnecessary to illuminate theargets with collimated light and enhances therightness of the images. It suffices to use a narrow-andwidth extended source such as a low-pressureercury-vapor lamp.From Fig. 2, it is evident that the SIP is largely

ndependent of the gas inside the target, whatever itsressure. This is because the interfering rays have

2 ~light lines! and between beams 1 and 3 ~heavy lines! for a CHns of the nonuniformity given by the unit vectors D. ~The innerSIP’s are out of phase by a half-wave in the center, so no distinctistinct SIP indicates a target with better than 0.05-mm thickness

andtatioTwo

of a d

September 1999 y Vol. 38, No. 25 y APPLIED OPTICS 5369

stwt

teT

Ff

wi~

5

tfmut

Table 1. Comparison Between the Calculated and the Measured SIP

5nfwtgSt

5

a small angle and lateral displacement relative toeach other; thus they traverse almost the same opti-cal path through the gas. This allows the target’swall thickness to be measured, after it is pressurizedwith fusion fuel, without accurate knowledge of thefill pressure. This is not possible with dual-beaminterferometry since the optical path through the tar-get is relative to an equivalent path in air. In thiscase, the refractive index difference between the fillgas and the surrounding air must be accounted forand subtracted from the total optical path lengththrough the target so that the target’s wall thicknesscan be determined.

3. Fringe Visibility

Uniform shells possess a relatively high fringe visi-bility with respect to nonuniform shells since the re-flected wave fronts ~beams 2 and 3! superpose both inpace and in phase, thereby interfering construc-ively to modify the amplitude of the transmittedave front ~beam 1!. Assume electric field ampli-

ude transmission coefficients T1 and T2 and a reflec-tion coefficient R at each interface. ~T1 and T2 applyto rays passing from air to shell and from shell to air,respectively. Similar coefficients R1 and R2 could bedefined, but they are equal in magnitude.! In prac-ice, T1, T2, and R depend on the angle of incidence atach interface, but the assumption of single values for1, T2, and R is good near the target center. For n 5

1.59, R 5 ~n 2 1!y~n 1 1! 5 0.228. Then the am-plitudes of transmitted waves 1–3 are given respec-tively by

A1 5 A0 T12T2

2,

A2 5 A0 T12T2

2R2,

A3 5 A0 T12T2

2R2. (6)

or a perfectly symmetric shell, waves 2 and 3 inter-ere constructively to produce a wave with amplitude

A2 1 A3 5 2A0 T12T2

2R2. (7)

Interference between these two waves and the purelytransmitted wave 1 gives amplitudes

A1 6 ~A2 1 A3! 5 A0 T12T2

2~1 6 2R2!, (8)

with 6 indicating constructive ~destructive! interfer-ence. The fringe visibility is then

V ;Imax 2 Imin

Imax 1 Imin5

~1 1 2R2!2 2 ~1 2 2R2!2

~1 1 2R2!2 1 ~1 2 2R2!2 (9)

54R2

1 1 4R4 , (10)

here Imax and Imin denote maximum and minimumntensities. Without waves 2 and 3 combining, Eq.8! would be replaced by

A1 6 A2 5 A0 T12T2

2~1 6 R2! (11)

370 APPLIED OPTICS y Vol. 38, No. 25 y 1 September 1999

for interference between waves 1 and 2, with

V9 52R2

1 1 R4 . (12)

Thus the visibility is greater by approximately afactor of 2 when waves 2 and 3 constructively inter-fere. For example, for n 5 1.59, R 5 ~n 2 1!y~n 1 1!

0.228, V 5 0.206, and V9 5 0.104.

4. Experimental Verification

The wall thicknesses of several glass shells weremeasured with dual-beam interferometry with anuncertainty of 60.05 mm. Their outside diameterswere also determined with a calibrated compoundmicroscope to within 63 mm. The refractive index ofhe glass shells was measured by fracturing shellsrom the same glass batch, immersing them in index-atching fluid, and varying the fluid temperaturentil the glass shards could not be differentiated fromhe fluid.6 This method utilizes the temperature de-

pendence of the refractive index of the index-matching fluid and results in a very sensitiverefractive-index measurement with an uncertaintyas low as 60.0002. The shells were then imagedwith the same compound microscope used to measuretheir outside diameter, but with a 10-nm-bandwidthinterference filter centered on a 546-nm wavelengthplaced between the diffuser and the condenser of themicroscope; this time the diameters of the SIP fringeswere measured.

A comparison between the measured SIP fringediameters and the calculated ones for a specific shellof thickness 2.89 6 0.05 mm is given in Table 1.Calculated fringe diameters are given for 2.89 and2.93 mm. The latter thickness, well within the un-certainty of the thickness measurement, gives thebetter agreement between the measured and the cal-culated SIP fringe diameters.

The SIP fringe diameters ~when normalized to theshell diameter! are much less sensitive to small er-

Fringe Diameters for a Glass ~n 5 1.4648 6 0.0003! Shella

Calculated Fringe Diameters~mm!

Measured FringeDiameters ~mm!t 5 2.89 mm t 5 2.93 mm

94 110 112 6 5160 168 167 6 3202 210 206 6 3238 242

aGlass shell has a 255 6 3-mm outside diameter illuminated with46-nm light with a 10-nm bandwidth. The measured wall thick-ess of 2.89 6 0.05 mm was obtained with a Mach–Zehnder inter-erence microscope. The SIP fringe diameters were measuredith a calibrated eyepiece reticule while the shell was viewed

hrough a compound microscope. A wall thickness of 2.93 mmave the best agreement between the calculated and the measuredIP fringe diameters and is within the uncertainty of the wall-hickness measurement. ~The outermost predicted fringe was not

observed in the measured SIP.!

StIfflo

rors in the outside diameter than in the wall thick-ness because the universal curves of Fig. 4 dependprimarily on the ratio riyRshell. As noted above, the

IP is more sensitive to wall-thickness variationshan the dual-beam interferogram of the same target.n particular, the positions of the SIP innermostringes provide information not so readily availablerom dual-beam interferometry because of the prob-em of establishing the piston, i.e., the absolute valuef the optical path through the target center.

5. Conclusions

The SIP fringe technique is now routinely used topreselect targets based on their wall thickness andnonconcentricity prior to high-precision interferomet-ric characterization. This technique requires only acompound microscope with a narrow-bandwidth in-terference filter or a stereo microscope and a diffusemercury-vapor illumination source. We determinedthe wall thickness to within 60.5 mm by counting thenumber of fringes in the SIP, independent of theoutside diameter, and verified the thickness unifor-mity to an accuracy better than 0.05 mm. In addi-tion, the wall thickness of gas-filled targets can bedetermined to the same accuracy without knowledgeof the gas type or pressure.

This research was supported by the U.S. Depart-ment of Energy ~DOE! Office of Inertial Confine-

1

ment Fusion under Cooperative Agreement DE-FC03-92SF19460, the University of Rochester, andthe New York State Energy Research and Develop-ment Authority. The support of DOE does not con-stitute an endorsement by DOE of the viewsexpressed in this paper.

References1. G. M. Halpern, J. Varon, D. C. Leiner, and D. T. Moore, “Laser

fusion microballoon wall-thickness measurements: a compar-ative study,” J. Appl. Phys. 48, 1223–1228 ~1977!.

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3. R. Q. Gram, M. D. Wittman, C. Immesoete, H. Kim, R. S. Crax-ton, N. Sampat, S. Swales, G. Pien, J. M. Soures, and H. Kong,“Uniform liquid-fuel layer produced in a cryogenic inertial fu-sion target by a time-dependent thermal gradient,” J. Vac. Sci.Technol. A 8, 3319–3323 ~1990!.

4. M. K. Prasad, K. G. Estabrook, J. A. Harte, R. S. Craxton, R. A.Bosch, Gar. E. Busch, and J. S. Kollin, “Holographic interfero-grams from laser fusion code simulations,” Phys. Fluids B 4,1569–1575 ~1992!.

5. Green Monochromatic Lamp, Edmund Scientific Company, 101East Gloucester Pike, Barrington, N.J. 08007-1380.

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