Abstract

The main purpose of this research is to study the anisotropic behavior of dielectric material in thin-film form. First we present a theory based on a 4 × 4 transfer matrix linking tangential components of the electromagnetic field on one interface to the tangential components of the electromagnetic field on the other interface of an anisotropic thin film. A biaxial model is associated with the columnar structure of the layer. The comparison between measurements of the transmission in normal incidence in cross-polarized light and of guided-mode propagation constants with the calculations allows us to study the biaxial behavior of TiO2 films. The excellent consistency between measurements and computations demonstrates the validity of the model based on the columnar structure.

© 1993 Optical Society of America

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References

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  1. D. O. Smith, “Magneto-optical scattering from multilayer magnetic and dielectric films. I: General theory,” Opt. Acta 12, 13–19 (1965).
    [CrossRef]
  2. S. Teitler, B. W. Henvis, “Refraction in anisotropic media,” J. Opt. Soc. Am. 60, 830–834 (1970).
    [CrossRef]
  3. D. Berreman, “Optics in stratified and anisotropic media: 4 × 4 matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  4. M. O. Vassel, “Structure of guided modes in planar multilayers of optically anisotropic materials,” J. Opt. Soc. Am. 64, 166–173 (1974).
    [CrossRef]
  5. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  6. F. Horowitz, “Structure induced optical anisotropy in thin film,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1983);F. Horowitz, H. A. Macleod, “From birefringence in thin films,” in Los Alamos Conference on Optics '83, R. S. McDowell, S. C. Stotlar, eds., Proc. Soc. Photo-Opt. Instrum. Eng.380, 83–87 (1983).
    [CrossRef]
  7. F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. (Paris) 3, 505–520 (1948).
  8. I. Hodgkinson, D. Endelema, “Bound modes in anisotropic multilayer thin film waveguides,” Appl. Opt. 29, 4424–4426 (1990).
    [CrossRef] [PubMed]
  9. A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapor deposited thin films,” Thin Solid Films 47, 219–222 (1977).
    [CrossRef]
  10. H. A. Macleod, “The microstructure of optical thin films,” in Optical Thin Films, R. I. Seddon, ed., Proc. Soc. Photo-Opt. Instrum. Eng.325, 21–28 (1982).
  11. K. H. Guenther, H. K. Pulker, “Electron microscopical investigations of cross sections of optical thin films,” Appl. Opt. 15, 2992–2997 (1976).
    [CrossRef] [PubMed]
  12. E. Pelletier, F. Flory, Y. Hu, “Optical characterization of thin films by guided waves,” Appl. Opt. 28, 2918–2924 (1989).
    [CrossRef] [PubMed]

1990 (1)

1989 (1)

1979 (1)

1977 (1)

A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapor deposited thin films,” Thin Solid Films 47, 219–222 (1977).
[CrossRef]

1976 (1)

1974 (1)

1972 (1)

1970 (1)

1965 (1)

D. O. Smith, “Magneto-optical scattering from multilayer magnetic and dielectric films. I: General theory,” Opt. Acta 12, 13–19 (1965).
[CrossRef]

1948 (1)

F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. (Paris) 3, 505–520 (1948).

Abelès, F.

F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. (Paris) 3, 505–520 (1948).

Berreman, D.

Dirks, A. G.

A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapor deposited thin films,” Thin Solid Films 47, 219–222 (1977).
[CrossRef]

Endelema, D.

Flory, F.

Guenther, K. H.

Henvis, B. W.

Hodgkinson, I.

Horowitz, F.

F. Horowitz, “Structure induced optical anisotropy in thin film,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1983);F. Horowitz, H. A. Macleod, “From birefringence in thin films,” in Los Alamos Conference on Optics '83, R. S. McDowell, S. C. Stotlar, eds., Proc. Soc. Photo-Opt. Instrum. Eng.380, 83–87 (1983).
[CrossRef]

Hu, Y.

Leamy, H. J.

A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapor deposited thin films,” Thin Solid Films 47, 219–222 (1977).
[CrossRef]

Macleod, H. A.

H. A. Macleod, “The microstructure of optical thin films,” in Optical Thin Films, R. I. Seddon, ed., Proc. Soc. Photo-Opt. Instrum. Eng.325, 21–28 (1982).

Pelletier, E.

Pulker, H. K.

Smith, D. O.

D. O. Smith, “Magneto-optical scattering from multilayer magnetic and dielectric films. I: General theory,” Opt. Acta 12, 13–19 (1965).
[CrossRef]

Teitler, S.

Vassel, M. O.

Yeh, P.

Ann. Phys. (Paris) (1)

F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. (Paris) 3, 505–520 (1948).

Appl. Opt. (3)

J. Opt. Soc. Am. (4)

Opt. Acta (1)

D. O. Smith, “Magneto-optical scattering from multilayer magnetic and dielectric films. I: General theory,” Opt. Acta 12, 13–19 (1965).
[CrossRef]

Thin Solid Films (1)

A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapor deposited thin films,” Thin Solid Films 47, 219–222 (1977).
[CrossRef]

Other (2)

H. A. Macleod, “The microstructure of optical thin films,” in Optical Thin Films, R. I. Seddon, ed., Proc. Soc. Photo-Opt. Instrum. Eng.325, 21–28 (1982).

F. Horowitz, “Structure induced optical anisotropy in thin film,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1983);F. Horowitz, H. A. Macleod, “From birefringence in thin films,” in Los Alamos Conference on Optics '83, R. S. McDowell, S. C. Stotlar, eds., Proc. Soc. Photo-Opt. Instrum. Eng.380, 83–87 (1983).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Schematic representation of a thin anisotropic dielectric medium.

Fig. 2
Fig. 2

Model of an anisotropic layer. ϕ is the angle between the plane of propagation and the plane holding the axes of a row of columns and α is the angle between the columnar axis and the normal to the layer surface.

Fig. 3
Fig. 3

Variation versus ϕ of reflectance and transmittance of a TiO2 layer illuminated in normal incidence. Main indices: n1 = 2.236, n2 = 2.283, n3 = 2.234; deposition angle γ = 31.8°; layer thickness, 348 nm. The curves corresponding to incident p polarization are (a) Rpp, (b) Rsp, (c) Tsp, (d) Tpp; the curves corresponding to incident s polarization are (e) Rss, (f) Rps, (g) Tps, (h)Tss.

Fig. 4
Fig. 4

Variation of Tss (θ = 0°, ϕ = 90°) versus geometric thickness for a layer of refractive indices: n1 = 2.236, n2 = 2.283, n3 = 2.234; the angle of incidence of the material during deposition is assumed to be 31.8°.

Fig. 5
Fig. 5

Variation of Tsp (θ = 0°, ϕ = 45°) versus geometric thickness for a layer of refractive indices: n1 = 2.236, n2 = 2.283, n3 = 2.234; the angle of incidence of the material during deposition is assumed to be 31.8°.

Fig. 6
Fig. 6

Variation versus ϕ of reflectance and transmittance of a TiO2 layer illuminated with an angle of 30° to the normal. Main indices: n1 = 2.236; n2 = 2.283; n3 = 2.234; deposition angle γ = 31.8°; layer thickness, 348 nm. The curves corresponding to incident p polarization are (a) Rpp, (b) Rsp, (c) Tsp, (d) Tpp; the curves corresponding to incident s polarization are (e) Rss, (f) Rps, (g) Tps, (h)Tss.

Fig. 7
Fig. 7

Calculated guided-mode propagation constant versus angle ϕ in a single layer. Main indices: n1 = 2.236, n2 = 2.283, n3 = 2.234, deposition angle γ = 31.8°, thickness 291.5 nm. (a) Mode TE0 when ϕ = 0, (b) mode TE1 when ϕ = 0, (c) mode TM0 when ϕ = 0, (d) mode TM1 when ϕ = 0.

Fig. 8
Fig. 8

Relative position of the samples and the crucible in the evaporating chamber.

Fig. 9
Fig. 9

Principle of the m-line technique.

Fig. 10
Fig. 10

Schematic diagram of the apparatus used to measure the change of polarization in normal incidence.

Fig. 11
Fig. 11

Example of agreement between measured and calculated modulation of Tsp versus angle ϕ (defined in Fig. 2).

Fig. 12
Fig. 12

Example of agreement between calculated and measured guided-mode propagation constant versus angle ϕ when the refined values of refractive indices and thicknesses are used (layer 2): (a) mode TE0 when ϕ = 0, (b) mode TE1 when ϕ = 0; (c) mode TM0 when ϕ = 0, (d) mode TM1 when ϕ = 0.

Tables (3)

Tables Icon

Table 1 Refractive Indices and Thicknesses a of Different TiO2 Layers versus the Incidence of the Material during Deposition (γ)

Tables Icon

Table 2 Refined Values of Refractive Indices and Thicknesses of Different TiO2 Layers versus the Incidence of the Material during Deposition (γ)

Tables Icon

Table 3 Measured and Calculated Maximum Values of Tsp (ϕ = 45°)

Equations (34)

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rot E = B t ,
rot H = D t .
( E H ) = Γ ( z ) exp [ i ω ( t sx / c ) ] , ( D B ) = Λ ( z ) exp [ i ω ( t sx / c ) ] ,
Γ ( z ) = ( E x ( z ) E y ( z ) E z ( z ) H x ( z ) H y ( z ) H z ( z ) ) , Λ ( z ) = ( D y ( z ) D y ( z ) D z ( z ) B x ( z ) B y ( z ) B z ( z ) ) .
z ( E x E y H x H y ) = i ω c [ C 11 C 12 0 C 14 0 0 C 23 0 C 31 C 32 0 C 34 C 41 C 42 0 C 44 ] ( E x E y H x H y ) ,
z φ ( z ) = i ω c C φ ( z ) .
φ j ( z ) = φ j ( 0 ) exp ( i ω q j z / c ) .
det ( C + q I ) = 0 ,
C φ j ( 0 ) = q j φ j ( 0 ) .
φ ( d ) = L ( d ) φ ( 0 ) ,
Ψ ( d ) = L ( d ) Ψ ( 0 ) ,
Ψ ( d ) = Ψ ( 0 ) K ( d ) = L ( d ) Ψ ( 0 ) ,
K ( d ) = [ exp ( i ω q 1 d c ) 0 0 0 0 exp ( i ω q 2 d c ) 0 0 0 0 exp ( i ω q 3 d c ) 0 0 0 0 exp ( i ω q 4 d c ) ] .
L ( d ) = Ψ ( 0 ) K ( d ) Ψ ( 0 ) 1 .
( ) N = [ 11 0 0 0 22 0 0 0 33 ] , D N = ( ) N E N .
( E x ( d ) E y ( d ) H x ( d ) H y ( d ) ) = [ L 11 L 22 L 13 L 14 L 21 L 22 L 23 L 24 L 31 L 32 L 33 L 34 L 41 L 42 L 43 L 44 ] ( E x ( 0 ) E y ( 0 ) H x ( 0 ) H y ( 0 ) ) .
( E x ( 0 ) E y ( 0 ) H x ( 0 ) H y ( 0 ) ) = F 0 ( E p + ( 0 ) E s + ( 0 ) E s ( 0 ) E p ( 0 ) ) ,
( E p + ( d ) E s + ( d ) E s ( d ) E p ( d ) ) = F s 1 ( E x ( d ) E y ( d ) H x ( d ) H y ( d ) ) ,
F 0 = [ cos ϴ 0 0 0 cos ϴ 0 0 1 1 0 0 n y y 0 cos ϴ 0 n y y 0 cos ϴ 0 0 n 0 y 0 0 0 n y y 0 ] , F s 1 = [ 1 / cos ϴ s 0 0 1 / n s y 0 0 1 1 / n s y 0 cos ϴ s 0 0 1 1 / n s y 0 cos ϴ s 0 1 / cos ϴ s 0 0 1 / n s y 0 ] ,
[ E p + ( d ) E s + ( d ) E s ( d ) E p ( d ) ] = F s 1 L ( d ) F 0 [ E p + ( 0 ) E s + ( 0 ) E s ( 0 ) E p ( 0 ) ] .
[ E p ( 0 ) E s ( 0 ) ] = [ r pp r ps r sp r ss ] [ E p + ( 0 ) E s + ( 0 ) ] ,
[ E p + ( d ) E s + ( d ) ] = [ t pp t ps t sp t ss ] [ E p + ( 0 ) E s + ( 0 ) ] .
R pp = r pp r pp * , R sp = r sp r sp * , R ps = r ps r ps * , R ss = r ss r ss * , T pp = N t pp t pp * , T sp = N t sp t sp * , T ps = N t ps t ps * , T ss = N t ss t ss * ,
N = n s cos ϴ s n 0 cos ϴ 0 .
[ E x ( 0 ) E y ( 0 ) H x ( 0 ) H y ( 0 ) ] = [ z a H y ( 0 ) E y ( 0 ) y a E y ( 0 ) H y ( 0 ) ] = E y ( 0 ) ( z a ρ 0 1 y a ρ 0 ) ,
y a = i y 0 ( s 2 n 0 2 ) 1 / 2 , z a = i ( s 2 n 0 2 ) 1 / 2 y 0 n 0 2 , ρ 0 = H y ( 0 ) E y ( 0 ) .
[ E x ( d ) E y ( d ) H x ( d ) H y ( d ) ] = [ z s H y ( d ) E y ( d ) y s E y ( d ) H y ( d ) ] ,
y s = i y 0 ( s 2 n s 2 ) 1 / 2 , z s = i ( s 2 n s 2 ) 1 / 2 y 0 n s 2 ,
z s H y ( d ) = ( L 11 z a ρ 0 + L 12 + L 13 y a + L 14 ρ 0 ) E y ( 0 ) , E y ( d ) = ( L 21 z a ρ 0 + L 22 + L 23 y a + L 24 ρ 0 ) E y ( 0 ) , y s E y ( d ) = ( L 31 z a ρ 0 + L 32 + L 33 y a + L 34 ρ 0 ) E y ( 0 ) , H y ( d ) = ( L 41 z a ρ 0 + L 42 + L 43 y a + L 44 ρ 0 ) E y ( 0 ) .
( L 41 z a z s + L 44 z s + L 11 z a L 14 ) ρ 0 = L 12 + L 13 y a L 42 z s L 43 y a z s , ( L 21 z a y s L 24 y s + L 31 z a L 34 ) ρ 0 = L 32 + L 33 y a + L 22 y s + L 23 y a z s .
X 1 = L 32 + L 33 y a + L 22 y s + L 23 y a y s , X 2 = L 12 + L 13 y a L 42 z s L 43 y a z s , X 3 = L 21 z a y s L 24 y s + L 31 z a L 34 , X 4 = L 41 z a z s + L 44 y s + L 11 z a L 14 ,
tan α = tan γ 2 .
φ = T an φ 0 T 0 1 R 0 R pp ( T sp + T pp R sp R 0 1 R 0 R pp ) T an φ 0 T 0 T sp 1 R 0 R pp ,
φ φ ν = ( φ 0 T 0 T sp 1 R 0 R pp ) ( 1 R 0 2 φ 0 T 0 2 ) .

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