Abstract

A general theory of multilayers with enhanced reflectance has been developed based on the superposition of sub-quarterwave layers of various highly radiation-absorbing materials. The theory has been developed by second-order expansion of the multilayer reflectance with respect to the optical-constant differences between the materials in the multilayer. The current paper completes and improves the theory that was developed in a previous paper [J. Opt. Soc. Am. A 18, 1406 (2001)] by including the case of nonnormal incidence and general radiation polarization and by providing more-accurate film thickness values of the optimized multilayer than with the previous theory. The theory provides an accurate approach to the design of a new concept of multilayer coatings with more than two materials. The new multilayers are adequate to enhance the reflectance of the materials particularly in the far and the extreme ultraviolet.

© 2001 Optical Society of America

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References

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  1. J. I. Larruquert, “Reflectance enhancement with sub-quarterwave multilayers of highly absorbing materials,” J. Opt. Soc. Am. A 18, 1406–1414 (2001).
    [CrossRef]
  2. J. I. Larruquert, R. A. M. Keski-Kuha, “Reflectance measurements and optical constants in the extreme ultraviolet for thin films of ion-beam-deposited SiC, Mo, Mg2Si, and InSb, and evaporated Cr,” Appl. Opt. 39, 2772–2781 (2000).
    [CrossRef]
  3. G. M. Blumenstock, R. A. M. Keski-Kuha, M. L. Ginter, “Extreme ultraviolet optical properties of ion-beam-deposited boron carbide thin films,”in X-Ray and Extreme Ultraviolet Optics, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2515, 558–564 (1995).
    [CrossRef]
  4. J. I. Larruquert, R. A. M. Keski-Kuha, “Reflectance measurements and optical constants in the extreme ultraviolet of thin films of ion-beam-deposited carbon,” Opt. Commun. 183, 437–443 (2000).
    [CrossRef]
  5. E. D. Palik, Handbook of Optical Constants of Solids II (Academic, San Diego, Calif., 1998).
  6. J. I. Larruquert, R. A. M. Keski-Kuha (unpublished data).
  7. E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1998).
  8. J. A. Méndez, J. I. Larruquert, J. A. Aznárez, “Preservation of FUV aluminum reflectance by overcoating with C60 films,” Appl. Opt. 39, 149–156 (2000).
    [CrossRef]

2001 (1)

2000 (3)

Aznárez, J. A.

Blumenstock, G. M.

G. M. Blumenstock, R. A. M. Keski-Kuha, M. L. Ginter, “Extreme ultraviolet optical properties of ion-beam-deposited boron carbide thin films,”in X-Ray and Extreme Ultraviolet Optics, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2515, 558–564 (1995).
[CrossRef]

Ginter, M. L.

G. M. Blumenstock, R. A. M. Keski-Kuha, M. L. Ginter, “Extreme ultraviolet optical properties of ion-beam-deposited boron carbide thin films,”in X-Ray and Extreme Ultraviolet Optics, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2515, 558–564 (1995).
[CrossRef]

Keski-Kuha, R. A. M.

J. I. Larruquert, R. A. M. Keski-Kuha, “Reflectance measurements and optical constants in the extreme ultraviolet for thin films of ion-beam-deposited SiC, Mo, Mg2Si, and InSb, and evaporated Cr,” Appl. Opt. 39, 2772–2781 (2000).
[CrossRef]

J. I. Larruquert, R. A. M. Keski-Kuha, “Reflectance measurements and optical constants in the extreme ultraviolet of thin films of ion-beam-deposited carbon,” Opt. Commun. 183, 437–443 (2000).
[CrossRef]

J. I. Larruquert, R. A. M. Keski-Kuha (unpublished data).

G. M. Blumenstock, R. A. M. Keski-Kuha, M. L. Ginter, “Extreme ultraviolet optical properties of ion-beam-deposited boron carbide thin films,”in X-Ray and Extreme Ultraviolet Optics, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2515, 558–564 (1995).
[CrossRef]

Larruquert, J. I.

Méndez, J. A.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1998).

E. D. Palik, Handbook of Optical Constants of Solids II (Academic, San Diego, Calif., 1998).

Appl. Opt. (2)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

J. I. Larruquert, R. A. M. Keski-Kuha, “Reflectance measurements and optical constants in the extreme ultraviolet of thin films of ion-beam-deposited carbon,” Opt. Commun. 183, 437–443 (2000).
[CrossRef]

Other (4)

E. D. Palik, Handbook of Optical Constants of Solids II (Academic, San Diego, Calif., 1998).

J. I. Larruquert, R. A. M. Keski-Kuha (unpublished data).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1998).

G. M. Blumenstock, R. A. M. Keski-Kuha, M. L. Ginter, “Extreme ultraviolet optical properties of ion-beam-deposited boron carbide thin films,”in X-Ray and Extreme Ultraviolet Optics, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2515, 558–564 (1995).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Representation in the complex plane of the refractive indices of the materials used in the examples of Section 3. A dot marks the outermost material used in the multilayer. The arrows point the next material in the film underneath. A circumference is also plotted to illustrate the case described in Subsection 2.B. The first refractive-index increment and angle subtended from Nc are plotted. In the example, five materials with refractive indices N1 to N5 located in the circumference are assumed.

Fig. 2
Fig. 2

Calculated reflectance versus the wavelength for single opaque layers of B4C and Ir and for the following multilayer that was optimized at 53.6 nm, 45° incidence and s polarization: 7.44-nm B4C/ opaque Ir.

Fig. 3
Fig. 3

Calculated reflectance versus the angle of incidence with respect to the normal for single opaque layers of Mo, B4C, and Ir and for the following multilayer that was optimized at 53.6 nm, 45° incidence and s polarization (starting with the outermost layer): 4.01-nm B4C/4.48-nm Mo/14.66-nm Ir/ opaque C60.

Tables (3)

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Table 1 Film Thickness of Different Sub-Quarterwave Multilayers Calculated by Four Different Methodsa

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Table 2 Reflectance at 83.4 nm at Normal Incidence of Different Sub-Quarterwave Multilayers Calculated by Three Different Methodsa

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Table 3 Reflectance of Sub-Quarterwave Multilayers Optimized at 53.6 nm, 45° Incidence Angle, and s -polarized (p=-1) and Nonpolarized (p=0) Radiation

Equations (52)

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gi-1=fi-1+gi exp(βi)1+fi-1gi exp(βi),
gm=fm,
βi=4πiNi cos θixiλ,
Ni cos θi=(Ni2-N02 sin2 θ0)1/2.
Ninc sin θinc=N0 sin θ0.
r=g0=f0+g1 exp(β1)1+f0gi exp(βi),
gi-1fi-1+gi exp(βi)+O(ΔN3),i=2tom.
g1=i=1mfi expj=2iβj+O(ΔN3).
r=rinc,1+(1-rinc,12)t¯1-rinc,1(1-rinc,12)(t¯1)2+O(ΔN3),
t¯i=j=imtj,tj=fj exp(β¯j),β¯j=k=1jβk
R=Rinc,1+2 Re[st¯1]-2 Re[rinc,1s(t¯1)2]+|1-rinc,12|2t¯1*t¯1+O(ΔN3),
srinc,1*(1-rinc,12).
Rxi=2 Res t¯1xi-4 Rerinc,1st¯1 t¯1xi+2|1-rinc,12|2×Ret¯1* t¯1xi=0,i=1tom.
(-1)m-i deti,m2Rxjxk<0,i=1tom,
Rxi=2 4πλ{-Im[sNi cos θit¯i]+2 Im[rinc,1sNi cos θit¯1t¯i]-|1-rinc,12|2 Im[Ni cos θit¯1*t¯i]}=0,i=1tom,
Rxi=-2 4πλ Im[sNi cos θit¯i]=0,i=1tom.
Im[su1 exp β10]=0,
Imuiui-1 exp βi0=0,i=2tom,
ui=Ni cos θifi,i=1tom.
tan ϕmax,10=-Im[su1]Re[su1],
tan ϕmax,i0=-Im[ui/ui-1]Re[ui/ui-1],i=2tom,
ϕmax,i0=4πnixmax,i0λ,i=1tom.
Im[su1]<0,
Im[ui/ui-1]<0,i=2tom,
ImsΔN1cos θ1>0,
ImΔNiΔNi-1<0,i=2tom;
ImsΔN1 cos θ11-N02 sin2 θ0N12 cos2 θ1>0,
ImΔNiΔNi-1<0,i=2tom.
R=Rinc,1+2|s|j=1m |fj|Re[Nj cos θj]|Nj cos θj|×exp-4πλ k=1j Im[Nk cos θk]xk0+O(ΔN2),
p=Ip-IsIp+Is.
ImΔN11-p2 sscos θ1+1+p2sp cos θ1×1-N02 sin2 θ0N12 cos2 θ1>0,
ImΔNiΔNi-1<0,i=2tom.
tan ϕmax,10=-Im1-p2ssu1s+1+p2spu1pRe1-p2ssu1s+1+p2spu1p,
tan ϕmax,i0=-Im1-p2ssuis+1+p2spuip1-p2ssui-1s+1+p2spui-1pRe1-p2ssuis+1+p2spuip1-p2ssui-1s+1+p2spui-1p,
i=2tom,
R=1-p2Rs+1+p2Rp.
R=Rinc,1+|s|ncnc2+kc2 j=1m|ΔNj|exp-4πλkck=1jxk0.
su1=-sΔN12,
uiui-1=ΔNiΔNi-1,i=2tom.
R=Rinc,1+|s|ncnc2+kc2 j=1m|ΔNj|exp-kcncϕ¯j,
ϕ¯j=k=1jϕmax,k0.
ϕi=12(αi-1+αi).
R-R=|s|ncnc2+kc2 |ΔNi-1|exp-kcncϕ¯i-1+|ΔNi|exp-kcncϕ¯i-|ΔNi-1+ΔNi|exp-kcncϕ¯i,
R-R>|s|ncnc2+kc2[|ΔNi-1|+|ΔNi|-|ΔNi-1+ΔNi|]×exp-kcncϕ¯i>0.
R({xmax,l0+Δxmax,l})=R({xmax,l0})+j=1m Rxj({xmax,l0})Δxj+12 j=1mk=1m 2Rxjxk ({xmax,l0})ΔxjΔxk+O(Δx3).
0=Rxi ({xmax,l0+Δxmax,l})
=Rxi ({xmax,l0})+j=1m 2Rxixj ({xmax,l0})Δxj+O(Δx2),i=1tom.
j=1m 2Rxixj ({xmax,l0})Δxj=-Rxi ({xmax,l0})+O(Δx2), i=1tom.
2Rxixj ({xmax,l0})=24πλ2{-Re[sNi cos θiNj cos θjt¯i0]+2 Re[rinc,1sNi cos θiNj cos θj(t¯j0+t¯10)t¯i0]-|1-rinc,12|2Re[Ni cos θit¯i(Nj cos θjt¯10*-Nj* cos θj*t¯j0*)]}, i=1tom, ji.
R({xmax,l0+Δxmax,l})=R({xmax,l0})-12 j=1mk=1m 2Rxjxk({xmax,l0})ΔxjΔxk+O(Δx3).
Rxi=1-p2 Rsxi+1+p2 Rpxi,
2Rxixj=1-p2 2Rsxixj+1+p2 2Rpxixj.

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