Quantum Processes and Metrology Group

"Minimizing spatial-dispersion-induced birefringence in crystals used for precision optics by using mixed crystals of materials with the opposite sign of the birefringence"

John Burnett, Zachary Levine, and Eric Shirley

National Institute of Standards and Technology
Gaithersburg, MD 20899

Abstract

We recently measured and calculated an intrinsic birefringence in CaF₂, SrF₂, and BaF₂ cubic crystals in the ultraviolet (UV). These results present serious problems for use of these crystalline materials for precision optics in the UV, e.g., for UV optical lithography. Since we found that the intrinsic birefringence of CaF₂ has an opposite sign to that of SrF₂ and BaF₂, we propose a correction solution in which a mixed crystal Ca_1-xSr_xF₂ or Ca_1-xBa_xF₂ is made which effectively zeroes out the birefringence of the crystal at a given wavelength, for appropriately chosen values of x. The appropriate x values are determined approximately by the mixtures where the linearly interpolated birefringences go to zero. Optical components such as lenses and beam splitters could be made of these mixed crystals that would be free of intrinsic birefringence aberrations at a chosen wavelength. These could be used as components of precision optical systems at this wavelength, such as photolithography projection systems.

We have recently measured and calculated for the first time an intrinsic, spatial-dispersion-induced birefringence n=n_<-110> - n_<001> near 193 nm and 157 nm in CaF₂ and similar intrinsic birefringences in SrF₂ and BaF₂, but with an opposite sign [1-2]. Our measured values are shown in Table 1 below, along with our calculated values. The stated uncertainties of the measurements are standard uncertainties due to statistical errors. The calculated values have estimated relative uncertainties of 15 %, but absolute standard uncertainties no smaller than 2 × 10^-7. The stated theoretical uncertainty takes into account issues of convergence with respect to several numerical cutoffs. However, it does not take into account the systematic uncertainties of the overall theoretical framework, which remain to be assessed by comparison to measurement as this study continues.

Table 1. Our measurements and calculations of n=n_<-110> - n_<001> for CaF₂, SrF₂, and BaF₂.

Material	n × 107(193 nm)		n × 107(156 nm)
	(measured)	(calculated)	(measured)	(calculated)
CaF2	-3.4 ± 0.3	-1.3	-11.8 ± 0.4	-18
SrF2	+6.6 ± 0.2	+9.8	+5.7 ± 0.3	+7.3
BaF2	+19 ± 2	+27	+33 ± 3	+52

In the references [1-2] we suggested one method to partially correct the effect in optical systems by combining <111> oriented lenses with transverse crystal axes rotated relatively by 60°. Here we propose a method to null out the effect at a given wavelength in each optical element. CaF₂, BaF₂, and SrF₂ all have the same fluorite crystal structure (space group Fm3m). Mixed crystals retaining the cubic symmetry can be made from these, e.g., Ca_1-xSr_xF₂, Ca_1-xBa_xF₂, and Ca_1-x-yBa_xSr_yF₂, using the vacuum Stockbarger technique or gradient freeze technique, by mixing CaF₂, SrF₂, and/or BaF₂ powders in the premelt material. The result, for example for Ca_1-xSr_xF₂, is a crystal with optical properties intermediate between those of CaF₂ and SrF₂, and the variation should be roughly linear with the x value. The intrinsic birefringence for Ca_1-xSr_xF₂ would also be expected to be intermediate between that for CaF₂ and SrF₂. We showed that the spatial-dispersion-induced intrinsic birefringence depends approximately on a single parameter [1]. Since the value of this intrinsic birefringence for SrF₂ has an opposite sign from that for CaF₂ in a wavelength range including 193 nm and 157 nm, and n should be a continuous function of x at a given wavelength, then a value x can be found for which n=0 at a given wavelength in the range. If the intrinsic birefringence varies approximately linearly with x between the two endpoints, then the value of x required to null out the effect should be determined approximately by the mixture where the interpolated by birefringence goes to zero, x  |n(CaF₂)/[n(SrF₂) - n(CaF₂)]|. From the values in the Table, this gives for Ca_1-xSr_xF₂ x  0.34 near 157 nm. It has been demonstrated that single-crystal, solid solutions Ca_1-xSr_xF₂ can be formed for any value of x [3]. Similarly, Ca_1-xBa_xF₂ solid solutions can be formed for at least some values of x. A value of x for the mixture Ca_1-xBa_xF₂ may be found to null out n at a given UV wavelength in the range. Using the quaternary mixture Ca_1-x-yBa_xSr_yF₂ allows a nulling at two wavelengths, or a broadband correction. This is possible because the n for SrF₂ and BaF₂ have different wavelength dependencies. Numerous other mixed crystal fluoride combinations that zero out n are similarly possible.

The mixed crystal materials with intrinsic birefringence nulls targeted for a given wavelength can be ground and polished into optical elements such as lenses and beam splitters, just as with CaF₂. These elements however, will have minimal aberrations associated with the intrinsic birefringence at the target wavelength. Thus they are appropriate for high-precision optics for that wavelength, where such aberrations are unacceptable. For example, for 157 nm lithography optics, values of n for the optical materials higher than ~1 × 10^-7 are considered unacceptable. Our measured value at 156 nm for CaF₂, the principal material being considered for the optics in these systems, is n = -11.8 × 10^-7, making CaF₂ unacceptably birefringent for 157 nm lithography applications. The same is also true for 193 nm lithography. Optics fashioned from the mixed crystals as discussed above, will have intrinsic birefringences and the consequent aberrations within the acceptable range and meet the design requirements of 157 nm lithography systems.

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[1]		John H. Burnett, Zachary H. Levine, and Eric L. Shirley, "Intrinsic birefringence in calcium fluoride and barium fluoride," (PDF 70 kB ), Phys. Rev. B 64, 241102(R) (2002).
[2]		John H. Burnett, Zachary H. Levine, and Eric L. Shirley, "Intrinsic Birefringence in 157 nm Materials," in Proceedings of the Second International Symposium on 157 nm Lithography, ed. by R. Harbison (International SEMATECH, Austin, 2001).
[3]		E.G. Chernevskaya and G.V. Anan'eva, "Structure of Mixed Crystals Based on CaF2, SrF2, and BaF2," Sov. Phys. Solid State 8, 169 (1966).

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