Abstract

A multilayer coating alters the amplitude and phase of a reflected wave front. The amplitude effects are multiplicative and well understood. We present a mathematical formalism that can be used to describe the phase effects of coating in a general case. On the basis of this formalism we have developed an analytical method of estimating the wave-front aberrations introduced by the multilayer coating. For the case of field-independent aberrations, we studied both uniform and graded multilayer coatings. For the case of field-dependent aberrations, we studied only the effects of a uniform multilayer coating. Our analysis is based on a coated plane mirror tilted with respect to an incident converging beam. Altogether we have found, up to the second order, the following aberrations: a field-dependent piston, a field-squared-dependent piston, defocus, field-independent tilt, field-independent astigmatism, and anamorphic magnification. To obtain numerical results we apply our analysis to the specific case of a plane mirror tilted 8.2 deg with respect to an incident converging beam with a numerical aperture of 0.1. We find that the magnitudes of the field-independent aberration coefficients for the graded coating are approximately ten times smaller than those for the uniform coating. We show that a coating can introduce anamorphic magnification.

© 2001 Optical Society of America

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References

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  1. D. J. Reiley, R. A. Chipman, “Coating-induced wave-front aberrations: on-axis astigmatism and chromatic aberration in all-reflecting systems,” Appl. Opt. 33, 2002–2012 (1994).
    [CrossRef] [PubMed]
  2. T. E. Jewell, “Effect of amplitude and phase dispersion on images in multilayer-coated soft-x-ray projection systems,” in Soft-X-Ray Projection Lithography, J. Bokor, ed., Vol. 12 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1991), pp. 113–188.
  3. N. J. Duddles, “Effects of Mo/Si multilayer coatings on the imaging characteristics of an extreme-ultraviolet lithography system,” Appl. Opt. 37, 3533–3538 (1998).
    [CrossRef]
  4. Code V is a product of Optical Research Associates, Pasadena, Calif. (1999).
  5. Y. Watanabe, M. Suzuki, N. Mochizuki, M. Niibe, Y. Fukuda, “Optical design for soft x-ray projection lithography,” Jpn. J. Appl. Phys. 30, 3053–3057 (1991).
    [CrossRef]
  6. Center for X-Ray Optics (CXRO), http://cindy.lbl.gov/optical_constants (1998).
  7. J. E. Bjorkholm, “Reduction imaging at 14 nm using multilayer-coated optics: printing of features smaller than 0.1 um,” J. Vac. Sci. Technol. B 8, 1509–1513 (1990).
    [CrossRef]
  8. A. H. Macleod, Thin-Film Optical Filters, 2nd ed. (Hilger, London, 1986).
    [CrossRef]
  9. zemax is a product of Focus Software, Inc., Tucson, Ariz. (1999).
  10. mathematica is a product of Wolfram Research, Inc., Champaign, Ill. (1999).

1998 (1)

1994 (1)

1991 (1)

Y. Watanabe, M. Suzuki, N. Mochizuki, M. Niibe, Y. Fukuda, “Optical design for soft x-ray projection lithography,” Jpn. J. Appl. Phys. 30, 3053–3057 (1991).
[CrossRef]

1990 (1)

J. E. Bjorkholm, “Reduction imaging at 14 nm using multilayer-coated optics: printing of features smaller than 0.1 um,” J. Vac. Sci. Technol. B 8, 1509–1513 (1990).
[CrossRef]

Bjorkholm, J. E.

J. E. Bjorkholm, “Reduction imaging at 14 nm using multilayer-coated optics: printing of features smaller than 0.1 um,” J. Vac. Sci. Technol. B 8, 1509–1513 (1990).
[CrossRef]

Chipman, R. A.

Duddles, N. J.

Fukuda, Y.

Y. Watanabe, M. Suzuki, N. Mochizuki, M. Niibe, Y. Fukuda, “Optical design for soft x-ray projection lithography,” Jpn. J. Appl. Phys. 30, 3053–3057 (1991).
[CrossRef]

Jewell, T. E.

T. E. Jewell, “Effect of amplitude and phase dispersion on images in multilayer-coated soft-x-ray projection systems,” in Soft-X-Ray Projection Lithography, J. Bokor, ed., Vol. 12 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1991), pp. 113–188.

Macleod, A. H.

A. H. Macleod, Thin-Film Optical Filters, 2nd ed. (Hilger, London, 1986).
[CrossRef]

Mochizuki, N.

Y. Watanabe, M. Suzuki, N. Mochizuki, M. Niibe, Y. Fukuda, “Optical design for soft x-ray projection lithography,” Jpn. J. Appl. Phys. 30, 3053–3057 (1991).
[CrossRef]

Niibe, M.

Y. Watanabe, M. Suzuki, N. Mochizuki, M. Niibe, Y. Fukuda, “Optical design for soft x-ray projection lithography,” Jpn. J. Appl. Phys. 30, 3053–3057 (1991).
[CrossRef]

Reiley, D. J.

Suzuki, M.

Y. Watanabe, M. Suzuki, N. Mochizuki, M. Niibe, Y. Fukuda, “Optical design for soft x-ray projection lithography,” Jpn. J. Appl. Phys. 30, 3053–3057 (1991).
[CrossRef]

Watanabe, Y.

Y. Watanabe, M. Suzuki, N. Mochizuki, M. Niibe, Y. Fukuda, “Optical design for soft x-ray projection lithography,” Jpn. J. Appl. Phys. 30, 3053–3057 (1991).
[CrossRef]

Appl. Opt. (2)

J. Vac. Sci. Technol. B (1)

J. E. Bjorkholm, “Reduction imaging at 14 nm using multilayer-coated optics: printing of features smaller than 0.1 um,” J. Vac. Sci. Technol. B 8, 1509–1513 (1990).
[CrossRef]

Jpn. J. Appl. Phys. (1)

Y. Watanabe, M. Suzuki, N. Mochizuki, M. Niibe, Y. Fukuda, “Optical design for soft x-ray projection lithography,” Jpn. J. Appl. Phys. 30, 3053–3057 (1991).
[CrossRef]

Other (6)

Center for X-Ray Optics (CXRO), http://cindy.lbl.gov/optical_constants (1998).

A. H. Macleod, Thin-Film Optical Filters, 2nd ed. (Hilger, London, 1986).
[CrossRef]

zemax is a product of Focus Software, Inc., Tucson, Ariz. (1999).

mathematica is a product of Wolfram Research, Inc., Champaign, Ill. (1999).

T. E. Jewell, “Effect of amplitude and phase dispersion on images in multilayer-coated soft-x-ray projection systems,” in Soft-X-Ray Projection Lithography, J. Bokor, ed., Vol. 12 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1991), pp. 113–188.

Code V is a product of Optical Research Associates, Pasadena, Calif. (1999).

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Figures (9)

Fig. 1
Fig. 1

Schematic comparison of a uniform coating versus a graded coating. Only one layer pair is drawn here and the diagrams are not to scale.

Fig. 2
Fig. 2

Multilayer-coated plane mirror configurations considered in this paper. (a) Multilayer-coated mirror tilted 8.2° with respect to a converging wave for the field-independent aberrations case. (b) Multilayer-coated mirror tilted at 8.2° with respect to a converging wave and a chief ray angle of 1.43 deg. The NA equals 0.1 in (a) and (b).

Fig. 3
Fig. 3

Geometry associated with the angle of incidence for the field-independent aberrations case. The tilted plane mirror is not shown.

Fig. 4
Fig. 4

Geometry associated with the angle of incidence for the field-dependent aberrations case. The tilted plane mirror is not shown.

Fig. 5
Fig. 5

Surface plots of W 011, W 020, and W 022. (a) The axis labeled W is measured in units of W 011, the wave-front departure from a sphere at the edge of the pupil. A similar interpretation is applied to (b) and (c). The x and y axes represent normalized pupil coordinates.

Fig. 6
Fig. 6

Phase and reflectance introduced by a uniform multilayer coating at a mirror tilt angle of 8.2 deg. s-polarized light is shown as a solid curve, and p-polarized light is shown as a dashed curve.

Fig. 7
Fig. 7

Phase and reflectance introduced by a graded multilayer coating. The graded multilayer coating’s effect on s-polarized light is shown as a solid curve, and the effect on p-polarized light is shown as a dashed curve.

Fig. 8
Fig. 8

Tangential wave-front error versus pupil position. The zemax result is shown as a dashed curve, and the analytical result is shown as a solid curve.

Fig. 9
Fig. 9

Sagittal wave-front error versus pupil position. The zemax result is shown as a dashed curve, and the analytical result is shown as a solid curve.

Tables (5)

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Table 1 Taylor-Series Expansion Coefficients of the Phase-Shift Dependence on Angle of Incidence for a Uniform Coating at a Mirror Tilt Angle of 8.2-deg

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Table 2 Field-Independent Aberration Coefficients for a Uniform Coating at an 8.2-deg Tilta

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Table 3 Taylor-Series Expansion Coefficients of the Phase-Shift Dependence on Angle of Incidence for a Graded Coating at a Mirror Tilt Angle of γ = 8.2°

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Table 4 Field-Independent Aberration Coefficients for the Graded Coating on a Plane Mirror and γ = 8.2°

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Table 5 Field-Dependent Aberration Coefficients for a Uniform Coating on a Plane Mirror Tilted at γ = 8.2°a

Equations (15)

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ySi=0.999934-j1.8211×10-3, yMo=0.92275-j6.2189×10-3, ysubstrate=0.978713-j1.05687×10-2.
Λimean=Λ0cos imean.
iρˆ, θ, γ, NA=cos-1ρˆNA sinθsinγ+1-NA2 cosγNA2ρˆ2-1+11/2,
iρˆ, θ, h, β, NA, ū, γ=cos-1ρˆ NA sinθsinγ-hū sinβsinγ1-NA21-ū2+cosγ1-NA2ρˆ2NA2+h2ū21-NA21-ū2+1-NA2-2hūρˆNA1-NA21-ū2cosθ-β1/2,
ϕiρˆ, θ, NA, γϕγ-NAρˆ sinθϕγ+12NA2ρˆ2ϕγ+12NA2ρˆ2 cos2 θ×cotγϕγ-ϕγ,
ϕiρˆ, θ, h, β, NA, ū, γϕγ-NAρˆ sinθϕγ+12NA2ρˆ2ϕγ+12NA2ρˆ2 cos2θcotγϕγ-ϕγ+hū sinβϕγ+12 h2ū2cotγcos2βϕγ+sin2βϕγ-NAhūρˆcotγcosβcosθϕγ+sinβsinθϕγ,
Wρˆ, θ=-λ2πNAϕγρˆ sin θ=W011ρˆ sin θ,
Wρˆ, θ=λ4πNA2ϕγρˆ2=W020ρˆ2,
Wρˆ, θ=λ4πNA2cotγϕγ-ϕγρˆ2 cos2θ=W022ρˆ2 cos2θ.
Wρˆ, θ, h, β=λ2π ūϕγh sinβ=W100h sinβ,
Wρˆ, θ, h, β=λ4π ū2ϕγcotγcos2β+ϕγsin2βh2=W200h2,
Wρˆ, θ, h, β=-λ2πNAūϕγcotγhρˆ cosβcosθ=W111Chρˆ cosβcosθ,
Wρˆ, θ, h, β=-λ2πNAūϕγhρˆ sinβsinθ=W111Shρˆ sinβsinθ.
Δx=1NA W111Ch,
Δy=1NA W111Sh.

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