Abstract

A model is derived for the reflectance optimization of an inhomogeneous coating made of absorbing materials. The model is applicable mainly for spectral regions where no transparent materials are available, such as in the extreme ultraviolet. The complex refractive index is assumed to take values within a given continuous domain and in a given sequence. The coating design is generated through a series of layer elements with a small refractive-index contrast across interfaces; the thickness of the element is calculated in terms of the refractive-index increment at the interface. The coating is optimized element by element starting from the substrate. When the refractive index varies both continuously and smoothly, the thickness element is of first order in the refractive-index increment. Suggestions are given on how to optimize a more general coating that alternates continuous and smooth refractive-index domains along with discrete indices, which results in a succession of inhomogeneous coatings and finite layers. An example is given to illustrate the model. A new material selection rule is obtained to discriminate whether the addition of a material on top of a partly grown coating will increase or decrease the reflectance of the coating. As a consequence, the model, which is highlighted toward the maximization of reflectance, can be used analogously for reflectance minimization such as for antireflection coatings.

© 2006 Optical Society of America

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References

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  1. M. Schlick, "Über die Reflexion des Lichtes in einer inhomogenen Schicht," Thesis dissertation (Friedrich-Wilhelms-Universität zu Berlin, 1904).
  2. W. Geffcken, "Reflexion elektromagnetischer Wellen an einer inhomogenen Schicht," Ann. Phys. (Paris) 40, 385-392 (1941).
  3. F. Abeles, "Recherches sur la propagation des ondes électromagnétiques sinuoïdales dans les milieux stratifiés. Application aux couches minces," Ann. Phys. 5, 596-640 (1950).
  4. R. Jacobsson and J. O. Mårtenson, "Evaporated inhomogeneous thin films," Appl. Opt. 5, 29-34 (1966).
    [CrossRef] [PubMed]
  5. A. Dobrowolski and D. Lowe, "Optical thin film synthesis program based on the use of Fourier transforms," Appl. Opt. 17, 3039-3050 (1978).
    [CrossRef] [PubMed]
  6. W. H. Southwell, "Spectral response calculations of rugate filters using coupled wave theory," J. Opt. Soc. Am. A 5, 1558-1564 (1988).
    [CrossRef]
  7. R. Jacobsson, "A review of the optical properties of inhomogeneous thin films," in Inhomogeneous and Quasi-inhomogeneous Optical Coatings, J.A.Dobrowolski and P.G.Verly, eds., Proc. SPIE 2046, 2-8 (1993).
  8. M. Yamamoto and T. Namioka, "Layer-by-layer design method for soft-x-ray multilayers," Appl. Opt. 31, 1622-1630 (1992).
    [CrossRef] [PubMed]
  9. J. I. Larruquert, "New layer-by-layer multilayer design method," J. Opt. Soc. Am. A 19, 385-390 (2002).
    [CrossRef]
  10. J. I. Larruquert, "Layer-by-layer design method for multilayers with barrier layers: application to Si/Mo multilayers for extreme-ultraviolet lithography," J. Opt. Soc. Am. A 21, 1750-1760 (2004).
    [CrossRef]
  11. J. I. Larruquert, "Inreflectance: a new function for the optimization of multilayers with absorbing materials," J. Opt. Soc. Am. A 22, 1607-1614 (2005).
    [CrossRef]
  12. J. I. Larruquert, "Reflectance enhancement with sub-quarterwave multilayers of highly absorbing materials," J. Opt. Soc. Am. A 18, 1406-1414 (2001).
    [CrossRef]
  13. J. I. Larruquert, "General theory of sub-quarterwave multilayers with highly absorbing materials," J. Opt. Soc. Am. A 18, 2617-2627 (2001).
    [CrossRef]
  14. J. I. Larruquert, "Reflectance enhancement in the extreme ultraviolet and soft x rays by means of multilayers with more than two materials," J. Opt. Soc. Am. A 19, 391-397 (2002).
    [CrossRef]
  15. J. I. Larruquert, "Sub-quarterwave multilayers with enhanced reflectance at 13.4 and 11.3 nm," Opt. Commun. 206, 259-273 (2002).
    [CrossRef]
  16. J. Gautier, F. Delmotte, M. Roulliay, F. Bridou, M. F. Ravet, and A. Jéreome, "Study of normal incidence of three-component multilayer mirrors in the range 20-40 nm," Appl. Opt. 44, 384-390 (2005).
    [CrossRef] [PubMed]
  17. http://www-cxro.lbl.gov/opticallowbarconstants.
  18. B. L. Henke, E. M. Gullikson, and J. C. Davis, "X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV,Z=1-92," At. Data Nucl. Data Tables 54, 181-342 (1993).
    [CrossRef]
  19. D. A. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation (Cambridge U. Press, 1999), p. 91.
  20. For instance, see E. Spiller, Soft X-ray Optics (SPIE, 1994), Eqs. 2.18a and 2.18b.
    [CrossRef]

2005

2004

2002

2001

1993

B. L. Henke, E. M. Gullikson, and J. C. Davis, "X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV,Z=1-92," At. Data Nucl. Data Tables 54, 181-342 (1993).
[CrossRef]

1992

1988

1978

1966

1950

F. Abeles, "Recherches sur la propagation des ondes électromagnétiques sinuoïdales dans les milieux stratifiés. Application aux couches minces," Ann. Phys. 5, 596-640 (1950).

1941

W. Geffcken, "Reflexion elektromagnetischer Wellen an einer inhomogenen Schicht," Ann. Phys. (Paris) 40, 385-392 (1941).

Abeles, F.

F. Abeles, "Recherches sur la propagation des ondes électromagnétiques sinuoïdales dans les milieux stratifiés. Application aux couches minces," Ann. Phys. 5, 596-640 (1950).

Attwood, D. A.

D. A. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation (Cambridge U. Press, 1999), p. 91.

Bridou, F.

Davis, J. C.

B. L. Henke, E. M. Gullikson, and J. C. Davis, "X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV,Z=1-92," At. Data Nucl. Data Tables 54, 181-342 (1993).
[CrossRef]

Delmotte, F.

Dobrowolski, A.

Gautier, J.

Geffcken, W.

W. Geffcken, "Reflexion elektromagnetischer Wellen an einer inhomogenen Schicht," Ann. Phys. (Paris) 40, 385-392 (1941).

Gullikson, E. M.

B. L. Henke, E. M. Gullikson, and J. C. Davis, "X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV,Z=1-92," At. Data Nucl. Data Tables 54, 181-342 (1993).
[CrossRef]

Henke, B. L.

B. L. Henke, E. M. Gullikson, and J. C. Davis, "X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV,Z=1-92," At. Data Nucl. Data Tables 54, 181-342 (1993).
[CrossRef]

Jacobsson, R.

R. Jacobsson and J. O. Mårtenson, "Evaporated inhomogeneous thin films," Appl. Opt. 5, 29-34 (1966).
[CrossRef] [PubMed]

R. Jacobsson, "A review of the optical properties of inhomogeneous thin films," in Inhomogeneous and Quasi-inhomogeneous Optical Coatings, J.A.Dobrowolski and P.G.Verly, eds., Proc. SPIE 2046, 2-8 (1993).

Jéreome, A.

Larruquert, J. I.

Lowe, D.

Mårtenson, J. O.

Namioka, T.

Ravet, M. F.

Roulliay, M.

Schlick, M.

M. Schlick, "Über die Reflexion des Lichtes in einer inhomogenen Schicht," Thesis dissertation (Friedrich-Wilhelms-Universität zu Berlin, 1904).

Southwell, W. H.

Spiller, E.

For instance, see E. Spiller, Soft X-ray Optics (SPIE, 1994), Eqs. 2.18a and 2.18b.
[CrossRef]

Yamamoto, M.

Ann. Phys.

F. Abeles, "Recherches sur la propagation des ondes électromagnétiques sinuoïdales dans les milieux stratifiés. Application aux couches minces," Ann. Phys. 5, 596-640 (1950).

Ann. Phys. (Paris)

W. Geffcken, "Reflexion elektromagnetischer Wellen an einer inhomogenen Schicht," Ann. Phys. (Paris) 40, 385-392 (1941).

Appl. Opt.

At. Data Nucl. Data Tables

B. L. Henke, E. M. Gullikson, and J. C. Davis, "X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV,Z=1-92," At. Data Nucl. Data Tables 54, 181-342 (1993).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

J. I. Larruquert, "Sub-quarterwave multilayers with enhanced reflectance at 13.4 and 11.3 nm," Opt. Commun. 206, 259-273 (2002).
[CrossRef]

Other

R. Jacobsson, "A review of the optical properties of inhomogeneous thin films," in Inhomogeneous and Quasi-inhomogeneous Optical Coatings, J.A.Dobrowolski and P.G.Verly, eds., Proc. SPIE 2046, 2-8 (1993).

http://www-cxro.lbl.gov/opticallowbarconstants.

M. Schlick, "Über die Reflexion des Lichtes in einer inhomogenen Schicht," Thesis dissertation (Friedrich-Wilhelms-Universität zu Berlin, 1904).

D. A. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation (Cambridge U. Press, 1999), p. 91.

For instance, see E. Spiller, Soft X-ray Optics (SPIE, 1994), Eqs. 2.18a and 2.18b.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Circular refractive-index distribution defined in Eq. (19) compared with the refractive indices of materials with the most extreme optical constants at 13.4 nm.

Fig. 2
Fig. 2

Ratio of element thickness to parameter increment calculated with Eq. (13) as a function of the parameter. The refractive-index function of t is defined in Eq. (19). The inset shows the innermost elements represented versus the logarithm of t. The symbols display the element thickness obtained by exact calculation.

Fig. 3
Fig. 3

Complex representation of the reflectance of the growing coating for the refractive-index function defined in Eq. (19). The first element corresponds to the center of the plot. Reflectance starts spiraling around the center with the coating growth.

Fig. 4
Fig. 4

Log plot of the normal reflectance of the inhomogeneous coating as a function of wavelength. Solid curve, the refractive index is set constant through the spectral domain. Short dashed curve, both k and δ = 1 n vary with the square of wavelength. Long dashed curve, multilayer alternating two materials. The refractive-index function is defined in Eq. (19).

Fig. 5
Fig. 5

Log–log plot as a function of t of the term in inequality (17), whose positive value indicates that the reflectance is increased by the addition of each layer element. The refractive-index function is defined in Eq. (19).

Equations (23)

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r i + 1 = f i + 1 + r i exp β i 1 + f i + 1 r i exp β i ,
R = r m + 1 * r m + 1 .
Im ( u i ) = 0 ,
u m = N m ( 1 f m + 1 2 ) r m exp β m ( f m + 1 + r m exp β m ) ( 1 + f m + 1 r m exp β m ) , i = m ( outermost layer ) ,
u i = N i N i + 1 ( 1 f i + 1 2 ) r i exp β i ( f i + 1 + r i exp β i ) ( 1 + f i + 1 r i exp β i ) , i = 1 to m 1 ( inner layers ) .
u i = 1 Δ N i 2 N i z ( r i exp β i ) + ( Δ N i 2 N i ) 2 z ( r i exp β i ) [ z ( r i exp β i ) 1 ] ,
z ( r ) = 1 r + 2 + r .
Im [ Δ N i 2 N i z ( r i exp β i ) ] Im { ( Δ N i 2 N i ) 2 z ( r i exp β i ) [ z ( r i exp β i ) 1 ] } = 0 + O ( Δ N i 3 ) .
Im [ Δ N i 1 2 N i z ( r i ) ] Im { ( Δ N i 1 2 N i ) 2 z ( r i ) } = 0 + O ( Δ N i 3 ) .
Δ 2 N i 1 = Δ N i Δ N i 1 O ( Δ N i 2 )
Im { Δ N i 2 N i ( r i 1 r i exp β i ) ( exp β i 1 ) } = Im { ( Δ N i 2 N i ) 2 [ z 2 ( r i exp β i ) z ( r i exp β i ) z ( r i ) ] } Im [ Δ 2 N i 2 N i z ( r i ) ] + O ( Δ N i 3 ) .
Δ x i = λ 4 π Im { Δ N i 2 2 N i 2 [ 2 z ( r i ) z 2 ( r i ) ] } + Im [ Δ 2 N i N i z ( r i ) ] Re [ Δ N i ( 1 r i r i ) ] .
Δ x ( t ) = λ 4 π Im { N ̇ 2 2 N 2 [ 2 z ( r ) z 2 ( r ) ] + N ̈ N z ( r ) } Re [ N ̇ ( 1 r r ) ] Δ t ,
Δ x i = λ 4 π Im [ Δ N i N i z ( r i ) ] + Im { Δ N i 2 2 N i 2 [ z ( r i ) z 2 ( r i ) ] } Re [ Δ N i ( 1 r i r i ) ] .
Im ( u i x i ) k = i + 1 m Re ( u k ) > 0 , i = 1 to m .
Re ( u k ) = 1 Re [ Δ N k 2 N k z ( r k exp β k ) ] + O ( Δ N 2 ) > 0 .
Im ( u i x i ) k = i + 1 m Re ( u k ) = 2 π λ Re [ Δ N i ( 1 r i exp β i r i exp β i ) ] + O ( Δ N 2 ) > 0 .
Re [ Δ N i ( 1 r i r i ) ] > 0 .
Im [ Δ N i Δ N i 1 ] > 0 ,
N ( t ) = N c + ρ exp ( j t ) ,
Δ N ( t ) = j ρ exp ( j t ) Δ t ,
Δ 2 N ( t ) = ρ exp ( j t ) Δ t 2 ,
Re { [ N ( t ) N ( t ) ] [ 1 r ( t ) exp β ( t t ) r ( t ) exp β ( t t ) ] } > 0 ,

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