Abstract

For radiation-absorbing materials and nonnormal incidence, multilayers with the largest reflectance at a desired wavelength are here demonstrated to require nonsequential optimization when radiation is partially polarized. The thicknesses of multilayers with the largest possible reflectance at a desired wavelength can be calculated sequentially either for s- or p-polarized radiation or for normal incidence and any polarization. The equations that impose the condition of maximum reflectance in the general case of partial polarization do not allow for the simplification that is performed when radiation is s or p polarized, which implies that multilayer optimization for partially polarized radiation is not sequential. An example is given in which the importance of nonsequential optimization is displayed. In contrast to the case of s- or p-polarized radiation, when radiation is partially polarized extra maxima may be found within an optical path difference of much less than half a wavelength.

© 2006 Optical Society of America

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References

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2002

1992

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1980

Apfel, J. H.

Aznárez, J. A.

Calvo-Angós, J.

Carniglia, C. K.

Larruquert, J. I.

Méndez, J. A.

Namioka, T.

Spann, J. F.

Torr, D. G.

Torr, M. R.

Yamamoto, M.

Zukic, M.

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Tables (2)

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Table 1 Layer Thicknesses and Reflectance of ( MgF 2 Yb ) n Multilayers on an Opaque MgF 2 Substrate Optimized for the Largest Possible Reflectance at 132.0 nm at 45 ° for p = 0.4 Incoming Radiation a

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Table 2 Layer Thicknesses, Reflectance, and Total Optical Path over Wavelength for Three Maxima of ( MgF 2 Yb ) 5 Multilayers on an Opaque MgF 2 Substrate Optimized for the Largest Possible Reflectance a

Equations (12)

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r i 1 = f i 1 + r i exp β i 1 + f i 1 r i exp β i ,
β i = 4 π j N i cos θ i x i λ .
p = I p I s I p + I s ,
R = 1 + p 2 R p + 1 p 2 R s ,
R x i max = 0 , i = 1 , , m .
R x 1 = 2 Re ( r 0 * r 0 x 1 ) = 0 ,
R x 2 = 2 Re ( r 0 * r 0 x 1 ) Re ( r 0 x 2 r 0 x 1 ) 2 Im ( r 0 * r 0 x 1 ) Im ( r 0 x 2 r 0 x 1 ) = 2 Im ( r 0 * r 0 x 1 ) Im ( r 0 x 2 r 0 x 1 ) = 0 ,
R x i = 2 Re ( r 0 * r 0 x i ) = 2 Im ( r 0 * r 0 x i ) Im ( r 0 x i r 0 x i 1 ) j = 2 i 1 Re ( r 0 x j r 0 x j 1 ) = 0 , i = 2 , m ,
Im ( r 0 x i r 0 x i 1 ) = Im [ N i cos θ i N i 1 cos θ i 1 ( 1 f i 1 2 ) r i exp β i ( f 1 + r i exp β i ) ( 1 + f i 1 r i exp β i ) ] = 0 .
R x 1 = ( 1 + p ) Re ( r 0 p * r 0 p x 1 ) + ( 1 p ) Re ( r 0 s * r 0 s x 1 ) = 0 ,
R x 2 = ( 1 + p ) A p + ( 1 p ) A s = 0 ,
A q = Re ( r 0 q * r 0 q x 1 ) Re ( r 0 q x 2 r 0 q x 1 ) Im ( r 0 q * r 0 q x 1 ) Im ( r 0 q x 2 r 0 q x 1 ) ,

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