Abstract

Southwell’s analysis of optical multilayers within the limits of very thin films has been extended to include absorption in the multilayer for predicting the effective values of the refractive index n e and extinction coefficient k e of mixed-composition binary homogeneous films over a wide spectral region, including the high-absorption (k > 10-2) region. It has been found that n e in general is a complicated function of the optical parameters (n 1, k 1, n 2, k 2) and volume fractions (f 1, f 2) of the component materials in a homogeneous layer, and the expression for n e becomes the same as that predicted by the Drude model in the spectral region where the layers are transparent. Moreover, according to the present analysis, the volume fractions of the product of the refractive index and the extinction coefficient of the component materials of a binary composite film are additive and the sum equals the product of the effective refractive index and extinction coefficient of the composite film.

© 2001 Optical Society of America

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References

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  1. A. Feldman, E. N. Farabaugh, W. K. Haller, D. M. Sanders, R. A. Stempniak, “Modifying structure and properties of optical films by coevaporation,” J. Vac. Sci. Technol. A 4, 2969–2974 (1986).
    [CrossRef]
  2. J. S. Chen, S. Chao, J. S. Kao, H. Niu, C. H. Chau, “Mixed films of TiO2–SiO2 deposited by double electron beam coevaporation,” Appl. Opt. 35, 90–96 (1996).
    [CrossRef] [PubMed]
  3. N. Gluck, D. B. Taker, J. P. Heuer, R. L. Hall, W. J. Gunning, “Properties of mixed composition Si/ZnSe and ZnSe/LaF3 infrared optical thin film,” Appl. Opt. 31, 6127–6132 (1992).
    [CrossRef] [PubMed]
  4. H. O. Sankur, J. DeNatale, W. J. Gunning, “Stress, scatter and structure dependence on composition in thin films of Si–YF3 and ZnSe–SrF2,” Appl. Opt. 30, 495–499 (1991).
    [CrossRef] [PubMed]
  5. Y. Tsou, F. C. Ho, “Optical properties of hafnia and coevaporated hafnia: magnesium fluoride thin films,” Appl. Opt. 35, 5091–5094 (1996).
    [CrossRef] [PubMed]
  6. F. Cheliebou, A. Boyer, L. Martin, “Studies on MgO-stabilized zirconia thin films in the UV-visible region,” Thin Solid Films 249, 86–90 (1994).
    [CrossRef]
  7. A. Basu, B. S. Verma, T. K. Bhattacharya, M. Kar, R. Bhattachryya, “Optical and mechanical characteristics of mixed composition infrared optical thin films–zinc selenide–cryolite and magnesium fluoride–thorium fluoride systems,” Opt. Optoelectron.: Theory Devices Applications 1, 562–567 (1998).
  8. A. Feldman, “Modeling refractive index in mixed component systems,” in Modeling of Optical Thin Films, M. R. Jackson, ed., Proc. SPIE821, 129–132 (1987).
    [CrossRef]
  9. W. H. Southwell, “Coating design using very thin high- and low-index layers,” Appl. Opt. 24, 457–460 (1985).
    [CrossRef] [PubMed]
  10. M. Kar, B. S. Verma, A. Basu, R. Bhattacharyya, “Optical equivalence between digital and homogeneously mixed films,” presented at the Department of Atomic Energy—Board of Research in Nuclear Sciences Workshop on Thin Film Multilayers, Mumbai, India, 6–8 Oct. 1999.
  11. H. A. McLeod, Thin Film Optical Filters, 2nd ed. (McGraw-Hill, New York, 1986), p. 35.
  12. H. Demiryont, “Optical properties of SiO2–TiO2 composite films,” Appl. Opt. 24, 2647–2650 (1985).
    [CrossRef]
  13. K. Kinosita, M. Nishibori, “Porosity of MgF2-films—evaluation based on changes in refractive index due to adsorption of vapors,” J. Vac. Sci. Technol. 6, 730–733 (1969).
    [CrossRef]
  14. K. L. Chopra, S. K. Sharma, V. N. Yadava, Thin Solid Films 20, 209–215 (1974).
  15. M. Harris, H. A. Macleod, S. Ogure, “The relationship between optical inhomogeneities and film structure,” Thin Solid Films 57, 173–178 (1979).
    [CrossRef]

1998

A. Basu, B. S. Verma, T. K. Bhattacharya, M. Kar, R. Bhattachryya, “Optical and mechanical characteristics of mixed composition infrared optical thin films–zinc selenide–cryolite and magnesium fluoride–thorium fluoride systems,” Opt. Optoelectron.: Theory Devices Applications 1, 562–567 (1998).

1996

1994

F. Cheliebou, A. Boyer, L. Martin, “Studies on MgO-stabilized zirconia thin films in the UV-visible region,” Thin Solid Films 249, 86–90 (1994).
[CrossRef]

1992

1991

1986

A. Feldman, E. N. Farabaugh, W. K. Haller, D. M. Sanders, R. A. Stempniak, “Modifying structure and properties of optical films by coevaporation,” J. Vac. Sci. Technol. A 4, 2969–2974 (1986).
[CrossRef]

1985

1979

M. Harris, H. A. Macleod, S. Ogure, “The relationship between optical inhomogeneities and film structure,” Thin Solid Films 57, 173–178 (1979).
[CrossRef]

1974

K. L. Chopra, S. K. Sharma, V. N. Yadava, Thin Solid Films 20, 209–215 (1974).

1969

K. Kinosita, M. Nishibori, “Porosity of MgF2-films—evaluation based on changes in refractive index due to adsorption of vapors,” J. Vac. Sci. Technol. 6, 730–733 (1969).
[CrossRef]

Basu, A.

A. Basu, B. S. Verma, T. K. Bhattacharya, M. Kar, R. Bhattachryya, “Optical and mechanical characteristics of mixed composition infrared optical thin films–zinc selenide–cryolite and magnesium fluoride–thorium fluoride systems,” Opt. Optoelectron.: Theory Devices Applications 1, 562–567 (1998).

M. Kar, B. S. Verma, A. Basu, R. Bhattacharyya, “Optical equivalence between digital and homogeneously mixed films,” presented at the Department of Atomic Energy—Board of Research in Nuclear Sciences Workshop on Thin Film Multilayers, Mumbai, India, 6–8 Oct. 1999.

Bhattacharya, T. K.

A. Basu, B. S. Verma, T. K. Bhattacharya, M. Kar, R. Bhattachryya, “Optical and mechanical characteristics of mixed composition infrared optical thin films–zinc selenide–cryolite and magnesium fluoride–thorium fluoride systems,” Opt. Optoelectron.: Theory Devices Applications 1, 562–567 (1998).

Bhattacharyya, R.

M. Kar, B. S. Verma, A. Basu, R. Bhattacharyya, “Optical equivalence between digital and homogeneously mixed films,” presented at the Department of Atomic Energy—Board of Research in Nuclear Sciences Workshop on Thin Film Multilayers, Mumbai, India, 6–8 Oct. 1999.

Bhattachryya, R.

A. Basu, B. S. Verma, T. K. Bhattacharya, M. Kar, R. Bhattachryya, “Optical and mechanical characteristics of mixed composition infrared optical thin films–zinc selenide–cryolite and magnesium fluoride–thorium fluoride systems,” Opt. Optoelectron.: Theory Devices Applications 1, 562–567 (1998).

Boyer, A.

F. Cheliebou, A. Boyer, L. Martin, “Studies on MgO-stabilized zirconia thin films in the UV-visible region,” Thin Solid Films 249, 86–90 (1994).
[CrossRef]

Chao, S.

Chau, C. H.

Cheliebou, F.

F. Cheliebou, A. Boyer, L. Martin, “Studies on MgO-stabilized zirconia thin films in the UV-visible region,” Thin Solid Films 249, 86–90 (1994).
[CrossRef]

Chen, J. S.

Chopra, K. L.

K. L. Chopra, S. K. Sharma, V. N. Yadava, Thin Solid Films 20, 209–215 (1974).

Demiryont, H.

DeNatale, J.

Farabaugh, E. N.

A. Feldman, E. N. Farabaugh, W. K. Haller, D. M. Sanders, R. A. Stempniak, “Modifying structure and properties of optical films by coevaporation,” J. Vac. Sci. Technol. A 4, 2969–2974 (1986).
[CrossRef]

Feldman, A.

A. Feldman, E. N. Farabaugh, W. K. Haller, D. M. Sanders, R. A. Stempniak, “Modifying structure and properties of optical films by coevaporation,” J. Vac. Sci. Technol. A 4, 2969–2974 (1986).
[CrossRef]

A. Feldman, “Modeling refractive index in mixed component systems,” in Modeling of Optical Thin Films, M. R. Jackson, ed., Proc. SPIE821, 129–132 (1987).
[CrossRef]

Gluck, N.

Gunning, W. J.

Hall, R. L.

Haller, W. K.

A. Feldman, E. N. Farabaugh, W. K. Haller, D. M. Sanders, R. A. Stempniak, “Modifying structure and properties of optical films by coevaporation,” J. Vac. Sci. Technol. A 4, 2969–2974 (1986).
[CrossRef]

Harris, M.

M. Harris, H. A. Macleod, S. Ogure, “The relationship between optical inhomogeneities and film structure,” Thin Solid Films 57, 173–178 (1979).
[CrossRef]

Heuer, J. P.

Ho, F. C.

Kao, J. S.

Kar, M.

A. Basu, B. S. Verma, T. K. Bhattacharya, M. Kar, R. Bhattachryya, “Optical and mechanical characteristics of mixed composition infrared optical thin films–zinc selenide–cryolite and magnesium fluoride–thorium fluoride systems,” Opt. Optoelectron.: Theory Devices Applications 1, 562–567 (1998).

M. Kar, B. S. Verma, A. Basu, R. Bhattacharyya, “Optical equivalence between digital and homogeneously mixed films,” presented at the Department of Atomic Energy—Board of Research in Nuclear Sciences Workshop on Thin Film Multilayers, Mumbai, India, 6–8 Oct. 1999.

Kinosita, K.

K. Kinosita, M. Nishibori, “Porosity of MgF2-films—evaluation based on changes in refractive index due to adsorption of vapors,” J. Vac. Sci. Technol. 6, 730–733 (1969).
[CrossRef]

Macleod, H. A.

M. Harris, H. A. Macleod, S. Ogure, “The relationship between optical inhomogeneities and film structure,” Thin Solid Films 57, 173–178 (1979).
[CrossRef]

Martin, L.

F. Cheliebou, A. Boyer, L. Martin, “Studies on MgO-stabilized zirconia thin films in the UV-visible region,” Thin Solid Films 249, 86–90 (1994).
[CrossRef]

McLeod, H. A.

H. A. McLeod, Thin Film Optical Filters, 2nd ed. (McGraw-Hill, New York, 1986), p. 35.

Nishibori, M.

K. Kinosita, M. Nishibori, “Porosity of MgF2-films—evaluation based on changes in refractive index due to adsorption of vapors,” J. Vac. Sci. Technol. 6, 730–733 (1969).
[CrossRef]

Niu, H.

Ogure, S.

M. Harris, H. A. Macleod, S. Ogure, “The relationship between optical inhomogeneities and film structure,” Thin Solid Films 57, 173–178 (1979).
[CrossRef]

Sanders, D. M.

A. Feldman, E. N. Farabaugh, W. K. Haller, D. M. Sanders, R. A. Stempniak, “Modifying structure and properties of optical films by coevaporation,” J. Vac. Sci. Technol. A 4, 2969–2974 (1986).
[CrossRef]

Sankur, H. O.

Sharma, S. K.

K. L. Chopra, S. K. Sharma, V. N. Yadava, Thin Solid Films 20, 209–215 (1974).

Southwell, W. H.

Stempniak, R. A.

A. Feldman, E. N. Farabaugh, W. K. Haller, D. M. Sanders, R. A. Stempniak, “Modifying structure and properties of optical films by coevaporation,” J. Vac. Sci. Technol. A 4, 2969–2974 (1986).
[CrossRef]

Taker, D. B.

Tsou, Y.

Verma, B. S.

A. Basu, B. S. Verma, T. K. Bhattacharya, M. Kar, R. Bhattachryya, “Optical and mechanical characteristics of mixed composition infrared optical thin films–zinc selenide–cryolite and magnesium fluoride–thorium fluoride systems,” Opt. Optoelectron.: Theory Devices Applications 1, 562–567 (1998).

M. Kar, B. S. Verma, A. Basu, R. Bhattacharyya, “Optical equivalence between digital and homogeneously mixed films,” presented at the Department of Atomic Energy—Board of Research in Nuclear Sciences Workshop on Thin Film Multilayers, Mumbai, India, 6–8 Oct. 1999.

Yadava, V. N.

K. L. Chopra, S. K. Sharma, V. N. Yadava, Thin Solid Films 20, 209–215 (1974).

Appl. Opt.

J. Vac. Sci. Technol.

K. Kinosita, M. Nishibori, “Porosity of MgF2-films—evaluation based on changes in refractive index due to adsorption of vapors,” J. Vac. Sci. Technol. 6, 730–733 (1969).
[CrossRef]

J. Vac. Sci. Technol. A

A. Feldman, E. N. Farabaugh, W. K. Haller, D. M. Sanders, R. A. Stempniak, “Modifying structure and properties of optical films by coevaporation,” J. Vac. Sci. Technol. A 4, 2969–2974 (1986).
[CrossRef]

Opt. Optoelectron.: Theory Devices Applications

A. Basu, B. S. Verma, T. K. Bhattacharya, M. Kar, R. Bhattachryya, “Optical and mechanical characteristics of mixed composition infrared optical thin films–zinc selenide–cryolite and magnesium fluoride–thorium fluoride systems,” Opt. Optoelectron.: Theory Devices Applications 1, 562–567 (1998).

Thin Solid Films

F. Cheliebou, A. Boyer, L. Martin, “Studies on MgO-stabilized zirconia thin films in the UV-visible region,” Thin Solid Films 249, 86–90 (1994).
[CrossRef]

K. L. Chopra, S. K. Sharma, V. N. Yadava, Thin Solid Films 20, 209–215 (1974).

M. Harris, H. A. Macleod, S. Ogure, “The relationship between optical inhomogeneities and film structure,” Thin Solid Films 57, 173–178 (1979).
[CrossRef]

Other

A. Feldman, “Modeling refractive index in mixed component systems,” in Modeling of Optical Thin Films, M. R. Jackson, ed., Proc. SPIE821, 129–132 (1987).
[CrossRef]

M. Kar, B. S. Verma, A. Basu, R. Bhattacharyya, “Optical equivalence between digital and homogeneously mixed films,” presented at the Department of Atomic Energy—Board of Research in Nuclear Sciences Workshop on Thin Film Multilayers, Mumbai, India, 6–8 Oct. 1999.

H. A. McLeod, Thin Film Optical Filters, 2nd ed. (McGraw-Hill, New York, 1986), p. 35.

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Figures (6)

Fig. 1
Fig. 1

(a) Digital-type film with j identical pairs of thin layers with refractive indices N 1 and N 2 and thicknesses d 1 and d 2, respectively. (b) Equivalent homogeneous mixed-composition layer.

Fig. 2
Fig. 2

Experimental data of a refractive index of 100% TiO2 layer (thickness, 0.2666 µm) fitted to the Cauchy function with a Marquardt regression algorithm: n1=A0+B0λ2+C0λ12;A0=2.35468,B0=1.74498E-2,C0=6.67044 E-6.

Fig. 3
Fig. 3

Experimental data of the extinction coefficient of 100% TiO2 layer (thickness, 0.266 µm), fitted to a logistic equation, with a Marquardt regression algorithm: k1=A11+λC1B1-1, A1=0.32667, B1=12.03521,C1 = 0.37459.

Fig. 4
Fig. 4

Comparison of —, the experimentally observed data of the effective refractive index n e of a TiO2 (52%) + SiO2 (48%) layer with those predicted by ●, the present analysis/model.

Fig. 5
Fig. 5

Comparison of —, the experimentally observed data of the effective extinction coefficient k e of a TiO2 (52%) + SiO2 (48%) layer with those predicted by ●, the present analysis/model.

Fig. 6
Fig. 6

Refractive index n f of a single-component porous film (n s = 1.40, n v = 1.0) as a function of the packing density p (0 ≤ p ≤ 1) according to various models.

Equations (31)

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M=cos ϕi sin ϕNiN sin ϕcos ϕ,
M=12πidλ2πidN2λ1
M12=12πid1+d2λ2πiN12d1+N22d2λ1.
deff=d1+d2,
Ne2=N12d1d1+d2+N22d2d1+d2.
f1=d1d1+d2,  f2=d2d1+d2.
ne2=12n12-k12f1+n22-k22f21+1+2n1k1f1+n2k2f2n12-k12f1+n22-k22f221/2,
ke=1nen1k1f1+n2k2f2.
10-3 < k1, k2<10-2,n12  k12, n1>1,n22  k22, n2>1.
ne2=12n12f1+n22f21+1+2n1k1f1+n2k2f2n12f1+n22f221/2,
ke=1nen1k1f1+n2k2f2,
k1, k2 <10-3,n1  k1,  n1>1,n2  k2,  n2>1,
ne2=n12f1+n22f2,
ke=1nen1k1f1+n2k2f2,
ne2=n12f1+n22f2,ke0.
neexp=a0+b0λ2+c0λ12,
a0=0.20400650 exp+01,b0=0.23814860 exp-01,c0=0.85318690 exp-06,
keexp=a1.0+λcb,
a=0.29178180 exp+00,b=0.12137540 exp+02,c=0.35374530 exp+00.
ne=nen1, k1, n2, k2, f1, f2,
Δne=nen1 Δn12+nek1 Δk12+nen2 Δn22+nek2 Δk22+nef1 Δf12+nef2 Δf221/2.
nenfkekf for the film of single-component optical material,n1nsk1ks for the solid part of the film,n2nvk2kv for voids,
nf2=12ns2-ks2p+nv2-kv21-p×1+1+2nsksp+nvkv1-pns2-ks2p+nv2-kv21-p21/2,
nfkf=nsksp+nvkv1-p,
p=volume of solid part of the filmtotal volume of the film solid+voids.
nf2=ns2p+nv21-p
nf=1-pnv+pns Kinosita and Nishibori13,
nf2=1-pns2+2nv2+pnv2+2ns21-pns2+2+pnv2+2Chopra et al.14,
nf2=1-pnv4+1+pnv2ns21+pnv2+1-pns2Harris et al.15.
n1=A0+B0λ2+C0λ12;A0=2.35468,B0=1.74498E-2,C0=6.67044 E-6.
k1=A11+λC1B1-1, A1=0.32667, B1=12.03521,C1 = 0.37459.

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