The drive of the semiconductor industry to deviceswith smaller dimensions necessitates the need forlithographic tools with shorter-wavelength sources.The next generation of lithographic tools will utilizeexcimer laser irradiation at 193 nm. Over the pro-jected ten-year lifetime of these systems, the opticalcomponents will be subjected to a total of 40 3 109
pulses with incident fluences ranging from 0.1 to afew millijoules per square centimeter per pulse.
Under such large irradiation doses even the bestcommercial fused silica, a key lens material in a pro-jection optics system, undergoes subtle changes,namely, color center formation and laser-induceddensification; see, for example, Ref. 1 and referencestherein. The latter phenomenon refers to a struc-tural rearrangement, causing a reduction in volumealong the irradiated area. This change leads to anincrease in local refractive index and a decrease inpath length, both of which can cause undesirablewave-front distortions. Recent long-term irradia-tion experiments,2 aimed at verifying the scaling ofdensification phenomenon with irradiation dose, re-vealed the formation of filamentary damage tracksunder certain irradiation conditions. Specifically,
E. M. Wright and M. Mansuripur are with the Optical SciencesCenter, University of Arizona, Tucson, Arizona 85721. Thee-mail address for E. M. Wright is [email protected]. V. Liberman and K. Bates are with MIT Lincoln Laboratory,Lexington, Massachusetts 12420-9108. K. Bates is also withIBMySEMATECH, Austin, Texas 78741.
for small spot size irradiation geometries ~millimeter-cale aperture! microchannel tracks originated at theack of the sample and propagated toward the frontlong the beam direction. The tracks first appearedround the edges of the aperture before they spreadnto the interior. The onset of microchannel forma-ion appeared to be a function of dose. Thus, forxample, at incident fluences of 10 mJycm2, 108
pulses were required to observe the damage.Longer samples developed microchannels at smallerirradiation doses than shorter ones.
In contrast, a crystalline material, such as calciumfluoride, does not undergo densification and does notappear to show microchannel formation. Althoughit is not a priori clear that compaction always pre-cedes microchannel formation or that they are evencausally related, this raises the question of whetherminimizing the effects of compaction will necessarilyeliminate microchannel formation. Our goal here isto show through numerical simulations that the spa-tial damage patterns observed in microchannel for-mation experiments are correlated with the priordevelopment of light hot spots that are due tocompaction-induced self-focusing effects. In thisway we demonstrate that compaction is a necessaryprecursor to microchannel formation. Furthermore,our results contribute to the physical understandingof the development of compaction and microchannelformation, and we speculate on the mechanism bywhich the light hot spots produce the microchannels.
2. Basic Model
Our model for compaction-induced effects on lightpropagation is described by the following paraxialwave equation for the field envelope %, scaled suchthat I 5 u%u2 is the energy per unit area per pulse in
0 September 1999 y Vol. 38, No. 27 y APPLIED OPTICS 5785
k
st
wrE
w
0
sH
5
millijoules per square centimeter, including therefractive-index change that is due to compaction3
Dn:
]%
]z5
i2k S ]2
]x2 1]2
]y2D% 1 ik0Dn%, (1)
where we adopt a longitudinal reference frame thatmoves at the group velocity of the pulse, z is thepropagation direction of the field, the first term on theright-hand side describes transverse beam diffrac-tion, and k 5 n0k0 5 n0wyc is the light wave vector,with n0 as the background refractive index. We ob-tained the compaction-induced change in refractiveindex from the corresponding change in density Dr byusing Dnyn0 5 aDryr0, with r0 as the backgrounddensity, where for a fixed energy per unit area perpulse incident on the material1,4,5
Dr
r05 kSNI2
t Dc
, (2)
with c as a constant dependent on the specific silica,is in parts per million, N is the number of pulses in
millions incident on the medium, and t is the pulselength in nanoseconds. We next generalize thismodel to allow the field intensity to change in re-sponse to the compaction. To do this we introduce adimensionless time variable T 5 Nc, which is a mea-ure of the number of pulses, and differentiate Eq. ~2!o yield
ddT
Dn~r, T! 5 gI2c~r, T!, (3)
where I is now allowed to vary with the time variable,and g 5 akn0ytc. By using the formal solution forthe refractive-index change, one can write the parax-ial wave equation as
]%
]z5
i2k
¹T2% 1 ik0g *
T
dT9u%~r, T9!u4c%~r, T!. (4)
To perform numerical simulations based on thismodel we make use of the fact that the light propa-gation and compaction occur on vastly different timescales. In particular, we assume that the materialremains fixed over a duration Tc, work out the fieldpropagation, and then in the next step update thematerial state, and so on. This iterative procedureis described by the coupled equations
]%l
]z5
i2k
¹T2%l 1 ik0Dnl~r!%l (5)
for the field envelope in step l of the procedure, andfor the material
Dnl~r! 5 Dnl21~r! 1 gTcu%l21~r!u4c, (6)
hich is the first-order approximation for theefractive-index change we obtained by integratingq. ~3! over a duration Tc. The first equation is the
786 APPLIED OPTICS y Vol. 38, No. 27 y 20 September 1999
paraxial wave equation for the field envelope %l in-cluding the refractive-index change that is due tocompaction Dnl. One can solve this paraxial waveequation numerically by using the beam propagationmethod3 over the medium thickness L for a fixedinitial condition
%l~x, y, z 5 0! 5 %in~x, y!, (7)
Equation ~6! describes the evolution of the refractive-index change that is due to compaction Dnl at step lfrom the results of the previous step l 2 1. To bemore specific, for the first step of l 5 0 we set Dn0 50 and solve for the field throughout the medium %0according to the paraxial wave equation. In the nextstep we solve for the induced refractive-index stepDn1 in terms of u%0u2 according to the material model.Next we work out %1 by using the paraxial waveequation including Dn1, and from this we work outDn2, and so on. This scheme allows for feedbackbetween the changing material properties that aredue to the field, and the changing field profile that isdue to the material.
3. Numerical Simulations
We have performed a series of numerical simulationsof Eqs. ~5! and ~6! by using DIFFRACT.6 For the sim-ulations discussed here we considered a sample oflength L 5 25 mm, which is uniformly illuminated
ith I 5 50 mJycm2 over a 2a 5 0.35-mm-diameteraperture by t 5 20-ns pulses. The range of param-eter values found for silica in the literature were c 5.43–0.7 and k 5 1027 2 0.8 3 1026, depending on
the particular sample. As representative of silicaparameters we used n0 5 1.5, c 5 0.5, a 5 0.3,1 andk 5 0.6 3 1026, yielding g 5 6 3 1028 cm2ym. Inaddition, we checked that the conclusions drawn fromour numerical simulations were not sensitive to smallchanges in our choice of parameters. For our adia-batic approximations to apply we require gI2cTc ,, 1,o that the refractive-index change per step is small.ere we set gI2cTc 5 3.7 3 1027 so that Tc 5 0.12.Figure 1 shows the evolution of the field intensity
profile at the sample output z 5 25 mm for a variablenumber of iterations Nit of our numerical procedure.The corresponding number of pulses to which themedium has been exposed is N 5 ~NitTc!
2 in millions.Before the onset of compaction the output shows cir-cular diffraction rings characteristic of linear Fresneldiffraction with Fresnel number ^ 5 n0a2ylL 5 9.5;see Fig. 1~a!.7 However, after 20 iterations, or 5.8million pulses, some azimuthal modulation devel-oped on the outer ring at the aperture edge; see Fig.1~b!. After further iterations this developed intowell-formed hot spots where the intensity is en-hanced; see Fig. 1~c! ~13 million pulses!. After addi-tional iterations the initial hot spots on the outermostring can become asymmetric, being more intense onone side rather than the other as in Fig. 1~d! ~52million pulses!, and the inner rings can also start todevelop azimuthal modulations followed by hot spots.We note that for these simulations we do not account
stodp
Tseb
p
for microchannel formation that can intervene beforethe later stages of hot spot formation shown in Fig.1~d!.
Physically, the field hot spots arise fromcompaction-induced self-focusing, and similar hotspot patterns were predicted and observed in single-pulse experiments for media with large self-focusingnonlinearities.8,9 In contrast, the compaction-induced effects are small per pulse and hot spot for-mation must therefore accumulate over many pulses.
To the best of our knowledge there are no publishedmeasurements of the formation of hot spots in outputbeam profiles. However, what we have at our dis-posal are experimental measurements of the patternsof microchannel formation in fused-silica samples af-ter many millions of pulses. It seems reasonable tospeculate that compaction-induced hot spots arewhat ultimately leads to microchannel formation, in-asmuch as the material areas exposed to the hot spotsare subject to higher intensities over long periods oftime. If this speculation is correct then we shouldexpect a correlation between the pattern of hot spotsand those of microchannel formation. Figure 2 is aphotograph of microchannel formation on the backsurface of a fused-silica sample. The sample hadbeen irradiated from the front side with 5 3 107
pulses of a 193-nm excimer laser at a laser energy of50 mJycm2ypulse. The beam was incident on theample through a hard aperture, 0.35 mm in diame-er. By side illumination of the sample with a fiber-ptic source the microchannels can be seen as brightots around the periphery of the irradiated area. Inarticular, it can be seen that the microchannels form
Fig. 1. Output beam profile I~x, y, z 5 L, T! for ~a! T 5 0 showingFresnel diffraction rings before the onset of compaction, ~b! T 520Tc or N 5 5.8 million pulses, ~c! T 5 30Tc or N 5 13 million
ulses, ~d! T 5 60Tc or N 5 52.2 million pulses.
20
an array around the incident aperture similar to theintensity hot spots observed in the numerical simu-lations, so that correlation exists between the spatialpatterns formed in the experiment and use of ourmodel. We note, however, that there are nearlytwice as many microchannels in Fig. 1 as there areintensity hot spots but we attribute this to uncer-tainty in the material constants for the model. Inparticular, by varying the model parameters we canchange the number of hot spots, but our main pointhere is not one-to-one comparison but rather that ourmodel displays spatial patterns of hot spots of thesame nature as the patterns of microchannels ob-served, hence validating the basic physics underlyingour model.
Finally, we speculate on how the appearance ofcompaction-induced hot spots could lead to micro-channel formation and the volcano effect seen in theexperiments whereby the material is evacuated fromsmall channels starting near the output face. So farwe have discussed the hot spots appearing at theoutput face but these form throughout the sampleand lead to large longitudinal ~z! intensity gradients.
hese intensity gradients are centered on the hotpots and produce a large volume force by way oflectrostriction with a longitudinal component giveny fz~r, T! 5 ge]u%~r, T!u2y]z, where ge is a constant
proportional to the electrostrictive constant for theglass.10 Clearly, the volume force is largest at theoutput face where the hot spots are most intense andis directed toward the output face. Furthermore,although the electrostrictive effect arising from a sin-gle pulse might not be sufficient to cause microchan-nel formation, the accumulative action of manypulses could weaken the material and ultimately leadto the volcano effect.
4. Summary and Conclusions
In summary, we have presented numerical simula-tions of spatial pattern formation of hot spots that are
Fig. 2. Photograph of microchannel formation on the back surfaceof a fused-silica sample irradiated from the front side with 5 3 107
pulses of a 193-nm excimer laser at a laser energy of 50 mJycm2ypulse incident through a hard aperture of 0.35-mm diameter.
September 1999 y Vol. 38, No. 27 y APPLIED OPTICS 5787
silica: laser-fluence and material-grade effects on the scaling
5
due to compaction-induced self-focusing over manyultraviolet pulses and have shown that these pat-terns correlate with the spatial patterns of micro-channel formation observed in experiments. Thiscorrelation provides confidence that compaction is aprecursor to microchannel formation, and we havespeculated that electrostrictive effects associatedwith the large intensity gradients that are due to thehot spots could provide the physical mechanism ofmicrochannel formation.
The Lincoln Laboratory portion of this research wasperformed under Cooperative Research and Develop-ment Agreements with SEMATECH. Opinions, in-terpretations, conclusions, and recommendations arethose of the authors and are not necessarily endorsedby the United States Air Force or our employers.
References and Notes1. R. E. Schenker and W. G. Oldham, “Ultraviolet-induced den-
sification in fused silica,” J. Appl. Phys. 82, 1065–1071 ~1997!.2. V. Liberman, M. Rothschild, J. H. C. Sedlacek, R. S. Uttaro,
and A. Grenville, “Excimer-laser-induced densification of fused
788 APPLIED OPTICS y Vol. 38, No. 27 y 20 September 1999
law,” J. NonCryst. Solids 244, 159–171 ~1999!.3. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent
propagation of high energy laser beams through the atmo-sphere,” Appl. Opt. 10, 129–160 ~1976!.
4. D. C. Allan, C. Smith, N. F. Borrelli, and T. P. Seward III,“193-nm excimer-laser-induced densification of fused silica,”Opt. Lett. 21, 1960–1962 ~1996!.
5. N. F. Borrelli, C. Smith, D. C. Allan, and T. P. Seward III,“Densification of fused silica under 193-nm excitation,” J. Opt.Soc. Am. B 14, 1606–1615 ~1997!.
6. The simulations reported in this article were performed byDIFFRACT, a software product of MM Research, Inc., Tucson,Ariz.
7. A. J. Campillo, J. E. Pearson, S. L. Shapiro, and N. J. Terrell,Jr., “Fresnel diffraction effects in the design of high-powerlaser systems,” Appl. Phys. Lett. 23, 85–87 ~1973!.
8. A. J. Campillo, S. L. Shapiro, and B. R. Suydam, “Periodicbreakup of optical beams due to self-focusing,” Appl. Phys.Lett. 23, 628–631 ~1973!.
9. A. J. Campillo, S. L. Shapiro, and B. R. Suydam, “Relationshipof self-focusing to spatial instability modes,” Appl. Phys. Lett.24, 178–181 ~1974!.
10. R. W. Boyd, Nonlinear Optics ~Academic, Boston, 1992!, Chap. 8.
