Abstract

A new magnitude is unveiled from the fact that a multilayer can be optimized for the largest reflectance in a layer-by-layer sequence: This magnitude has been named inreflectance. Inreflectance equals the modulus of the complex reflectance once elevated to the inverse of the complex refractive index of the next outer layer, including the inclination term. The maximization of inreflectance at every internal layer, proceeding sequentially starting with the innermost layer, results in a multilayer with the largest reflectance. At a given layer in the multilayer, inreflectance is different from reflectance when the next outer material absorbs radiation, whereas the two magnitudes are coincident when the next outer layer is transparent. The outermost layer of the multilayer is intrinsically different from the internal layers, and its optimization is performed through reflectance and not through inreflectance. Every maximum of inreflectance is found at a layer thickness that is larger than that for the reflectance maximum. The new magnitude is illustrated with some examples in the extreme ultraviolet, a spectral range in which all materials absorb radiation. Inreflectance can also be used to design multilayers in which some layers have fixed thicknesses and the rest of the layers have to be optimized taking into account the fixed layers.

© 2005 Optical Society of America

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References

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  1. F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. (Paris) 3, 504–520 (1948).
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2004 (1)

2002 (2)

2001 (2)

1992 (1)

1987 (1)

1986 (1)

1980 (1)

1977 (1)

1950 (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

1948 (1)

F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. (Paris) 3, 504–520 (1948).

Abelès, F.

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. (Paris) 3, 504–520 (1948).

Apfel, J. H.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975).

Carniglia, C. K.

Cruddace, R. G.

Gursky, H.

Keski-Kuha, R. A. M.

Larruquert, J. I.

Mayer, H.

H. Mayer, Physik dünner Schichten (Wissentschaftliche Verlagsgesellschaft, 1950).

Meekins, J. F.

Namioka, T.

Spiller, E.

E. Spiller, Soft X-Ray Optics (SPIE, 1994), p. 143.

Vinogradov, A. V.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975).

Yamamoto, M.

Zeldovich, B. Y.

Ann. Phys. (Paris) (2)

F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. (Paris) 3, 504–520 (1948).

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Other (3)

H. Mayer, Physik dünner Schichten (Wissentschaftliche Verlagsgesellschaft, 1950).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975).

E. Spiller, Soft X-Ray Optics (SPIE, 1994), p. 143.

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Figures (4)

Fig. 1
Fig. 1

Comparison of reflectance and inreflectance as a function of thickness for a Si 5 x ( Rh Mo Si ) multilayer optimized for the largest reflectance at 13.4 nm at normal incidence. A dot highlights the switch from the outermost Mo to Si layer.

Fig. 2
Fig. 2

Representation in the complex plane of (a) reflectance and (b) the complex expression of inreflectance for a growing Si 5 x ( Rh Mo Si ) multilayer optimized for the largest reflectance at 13.4 nm at normal incidence. The dots in reflectance display the maximum reflectance at each layer. The dot in inreflectance displays the switch from the outermost Mo to Si layer.

Fig. 3
Fig. 3

Comparison of reflectance and inreflectance as a function of thickness for a Si C 1 x ( Mg F 2 Al 2 O 3 C B 4 C Si C ) multilayer optimized for the largest reflectance at 83.4 nm at normal incidence.

Fig. 4
Fig. 4

Representation in the complex plane of (a) reflectance and (b) complex expression of inreflectance for a growing Si C 1 x ( Mg F 2 Al 2 O 3 C B 4 C Si C ) multilayer optimized for the largest reflectance at 83.4 nm at normal incidence.

Equations (40)

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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 .
R x m max = 2 Re ( r m + 1 * r m + 1 x m ) = 0 , i = m ,
R x i max = 2 Re ( r m + 1 * r m + 1 x i ) = 2 Im ( r m + 1 * r m + 1 x m ) Im ( r i + 2 x i r i + 2 x i + 1 ) k = i + 1 m 1 Re ( r k + 2 x k r k + 2 x k + 1 ) = 0 , i = 1 to m 1 .
Re ( r m + 1 * r m + 1 x m ) = R Re ( r m + 1 x m r m + 1 ) = 0 , i = m ,
Im ( r i + 1 x i r i + 2 x i + 1 r i + 2 r i + 1 ) = λ 4 π Re ( r i + 1 x i N i + 1 cos θ i + 1 r i + 1 ) = 0 , i = 1 to m 1 .
Im ( u i ) = 0 , i = 1 to m ,
u m = ( N m cos θ 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 ,
u i = N i cos θ i N i + 1 cos θ i + 1 ( 1 f m + 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 .
Re ( r m + 1 x m r m + 1 ) = Re [ ln ( r m + 1 ) x m ] = 0 , i = m ,
Re ( r i + 1 x i N i + 1 cos θ i + 1 r i + 1 ) = Re [ 1 N i + 1 cos θ i + 1 ln ( r i + 1 ) x i ] = 0 , i = 1 to m 1 .
ln ( z ) = ln ( m ) + i θ .
Re [ ln ( r m + 1 ) x m ] = x m Re [ ln ( r m + 1 ) ] = 0 , i = m ,
Re [ 1 N i + 1 cos θ i + 1 ln ( r i + 1 ) x i ] = x i Re { ln [ r i + 1 ( N i + 1 cos θ i + 1 ) 1 ] } = 0 , i = 1 to m 1 .
x m ( ln r m + 1 ) = 0 , i = m ,
x i [ ln r i + 1 ( N i + 1 cos θ i + 1 ) 1 ] = 0 , i = 1 to m 1 .
ln r m + 1 = extreme r m + 1 = extreme , i = m ,
ln r i + 1 ( N i + 1 cos θ i + 1 ) 1 = extreme r i + 1 ( N i + 1 cos θ i + 1 ) 1 = extreme , i = 1 to m 1 .
r i + 1 ( N i + 1 cos θ i + 1 ) 1 x i = 4 π λ r i + 1 ( N i + 1 cos θ i + 1 ) 1 Im ( u i ) , i = 1 to m 1 ,
r m + 1 x m = 4 π λ r m + 1 Im ( u m ) , i = m .
( 1 ) m i + 1 det i , m ( 2 R x k 1 x k 2 ) ext > 0 , i = 1 to m .
Im ( u i x i ) k = i + 1 m Re ( u k ) > 0 , i = 1 to m .
Im ( u i ) x i > 0 , i = 1 to m .
2 r m + 1 x m 2 < 0 , i = m ,
2 r i + 1 ( N i + 1 cos θ i + 1 ) 1 x i 2 < 0 , i = 1 to m 1 .
r i + 1 ( N i + 1 cos θ i + 1 ) 1 = r i + 1 Re ( N i + 1 cos θ i + 1 ) ( N i + 1 cos θ i + 1 ) 2 exp [ φ r i + 1 Im ( N i + 1 cos θ i + 1 ) N i + 1 cos θ i + 1 2 ] ,
φ r i + 1 ( N i + 1 cos θ i + 1 ) 1 = Re ( N i + 1 cos θ i + 1 ) N i + 1 cos θ i + 1 2 φ r i + 1 Im ( N i + 1 cos θ i + 1 ) N i + 1 cos θ i + 1 2 ln r i + 1 .
r i + 1 N i + 1 cos θ i + 1 2 Re ( N i + 1 cos θ i + 1 ) N i + 1 cos θ i + 1 = r i + 1 exp ( φ r i + 1 tan φ N i + 1 cos θ i + 1 ) ,
φ r i + 1 N i + 1 cos θ i + 1 2 Re ( N i + 1 cos θ i + 1 ) N i + 1 cos θ i + 1 = φ r i + 1 tan ( φ N i + 1 cos θ i + 1 ) ln r i + 1 ,
r i + 1 x i = r i + 1 tan ( φ N i + 1 cos θ i + 1 ) φ r i + 1 x i , i = 1 to m 1 ,
r m + 1 x m = 0 , i = m .
Re ( u i ) = λ 4 π N i + 1 cos θ i + 1 2 [ Re ( N i + 1 cos θ i + 1 ) φ r i + 1 x i Im ( N i + 1 cos θ i + 1 ) r i + 1 x i r i + 1 ] , i = 1 to m 1 ,
Re ( u m ) = λ 4 π φ r m + 1 x m , i = m .
r i + 1 x i = 4 π λ r i + 1 Im ( N i + 1 cos θ i + 1 ) Re ( u i ) , i = 1 to m 1 ,
φ r i + 1 x i = 4 π λ Re ( N i + 1 cos θ i + 1 ) Re ( u i ) , i = 1 to m 1 ,
φ r m + 1 x m = 4 π λ Re ( u m ) , i = m .
R x h ( m ) max = 2 Re [ r l + 1 * r l + 1 x h ( m ) ] = 0 , i = m ,
R x h ( i ) max = 2 Re [ r l + 1 * r l + 1 x h ( i ) ] = 2 Im [ r l + 1 * r l + 1 x h ( m ) ] Im [ r h ( i + 1 ) + 1 x h ( i ) r h ( i + 1 ) + 1 x h ( i + 1 ) ] k = i + 1 m 1 Re [ r h ( k + 1 ) + 1 x h ( k ) r h ( k + 1 ) + 1 x h ( k + 1 ) ] = 0 , i = 1 to m 1 .
Re [ r l + 1 x h ( m ) r l + 1 ] = Re [ ln ( r l + 1 ) x h ( m ) ] = 0 r l + 1 = extreme , i = m ,
Re [ r h ( i + 1 ) x h ( i ) N h ( i + 1 ) cos θ h ( i + 1 ) r h ( i + 1 ) ] = Re { 1 N h ( i + 1 ) cos θ h ( i + 1 ) ln [ r h ( i + 1 ) ] x h ( i ) } = 0 r h ( i + 1 ) ( N i + 1 cos θ i + 1 ) 1 = extreme , i = 1 to m 1 .

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