Abstract

We consider the application of the Bragg–Pippard (BP) equations for form birefringence to a tilted-columnar biaxial thin film with columns of index n c and voids of known index n v. In such a situation the three forward BP equations that express the principal refractive indices n 1, n 2, and n 3 as functions of n c, n v, the packing fraction p c, and the depolarization factors L 1, L 2, and L 3 can be inverted. The procedure described for adding dispersion to the principal indices involves entry to the BP model via the inverted equations, modification of n c to allow for dispersion, and then exit from the model via the forward BP equations. We discuss the introduction of composite columns to the model to allow for angular dependence of n c and the selection of suitable dispersion functions for bulk tantalum oxide, titanium oxide, and zirconium oxide. Theory and experiment both show that the dispersion of the normal-incidence birefringence Δn of the thin films is several times larger than the dispersion of the individual principal refractive indices.

© 2001 Optical Society of America

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References

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  1. G. W. Mbise, D. Le Bellac, G. A. Niklasson, C. G. Granqvist, “Angular selective window coatings: theory and experiments,” J. Phys. D 30, 2103–2122 (1997).
    [CrossRef]
  2. I. J. Hodgkinson, Q. H. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1998).
    [CrossRef]
  3. W. L. Bragg, A. B. Pippard, “The form birefringence of macromolecules,” Acta Crystallogr. 6, 865–867 (1953).
    [CrossRef]
  4. I. J. Hodgkinson, Q. H. Wu, J. C. Hazel, “Empirical equations for the principal refractive indices and column angle of obliquely deposited films of tantalum oxide, titanium oxide, and zirconium oxide,” Appl. Opt. 37, 2653–2659 (1998).
    [CrossRef]
  5. I. J. Hodgkinson, Q. H. Wu, “Vacuum deposited biaxial thin films with all principal axes inclined to the substrate,” J. Vac. Sci. Technol. A 17, 2928–2932 (1999).
    [CrossRef]
  6. I. J. Hodgkinson, Q. H. Wu, B. Knight, A. Lakhtakia, K. Robbie, “Vacuum deposition of chiral sculptured thin films with high optical activity,” Appl. Opt. 39, 642–649 (2000).
    [CrossRef]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).
  8. H. K. Pulker, G. Paesold, E. Ritter, “Refractive indices of TiO2 films produced by reactive evaporation of various titanium–oxygen phases,” Appl. Opt. 15, 2986–2991 (1976).
    [CrossRef] [PubMed]
  9. Technical information from Patinal Newsletter No. 3, E. Merc, Patinal-Centre, 64578 Gernsheim, Germany, June1994.
  10. H. Kuster, J. Ebert, “Activated reactive evaporation of TiO2 layers and their absorption indices,” Thin Solid Films 70, 43–47 (1980).
    [CrossRef]
  11. P. J. Martin, “Ion-based methods for optical thin film deposition,” J. Mater. Sci. 21, 1–25 (1986).
    [CrossRef]
  12. D. Smith, P. Baumeister, “Refractive index of some oxide and fluoride coating materials,” Appl. Opt. 18, 111–115 (1979).
    [CrossRef] [PubMed]
  13. T. Motohiro, Y. Taga, “Thin film retardation plate by oblique deposition,” Appl. Opt. 28, 2466–2482 (1989).
    [CrossRef] [PubMed]
  14. I. J. Hodgkinson, Q. H. Wu, “Serial bideposition of anisotropic thin films with enhanced linear birefringence,” Appl. Opt. 38, 3621–3625 (1999).
    [CrossRef]
  15. A. Zuber, H. Jänchen, N. Kaiser, “Perpendicular-incidence photometric ellipsometry of biaxial anisotropic thin films,” Appl. Opt. 35, 5553–5556 (1996).
    [CrossRef] [PubMed]
  16. T. Motohiro, Y. Takeda, T. Hioki, S. Noda, “Simultaneous oblique deposition from opposite azimuthal directions for fabrication of thin film retardation plates,” in International Symposium on Polarization Analysis and Applications to Device Technology, H. Yokota, T. Yoshizawa, eds., Proc. SPIE2873, 214–217 (1996).
    [CrossRef]
  17. I. J. Hodgkinson, Q. H. Wu, K. E. Thorn, M. D. Arnold, “Vacuum-deposited thin film polarizing elements for use with linearly and circularly polarized light at visible and near infrared wavelengths,” in Complex Mediums, A. Lakhtakia, W. S. Weiglhofer, R. Messier, eds., Proc. SPIE4097, 266–279 (2000).
    [CrossRef]

2000 (1)

1999 (2)

I. J. Hodgkinson, Q. H. Wu, “Vacuum deposited biaxial thin films with all principal axes inclined to the substrate,” J. Vac. Sci. Technol. A 17, 2928–2932 (1999).
[CrossRef]

I. J. Hodgkinson, Q. H. Wu, “Serial bideposition of anisotropic thin films with enhanced linear birefringence,” Appl. Opt. 38, 3621–3625 (1999).
[CrossRef]

1998 (1)

1997 (1)

G. W. Mbise, D. Le Bellac, G. A. Niklasson, C. G. Granqvist, “Angular selective window coatings: theory and experiments,” J. Phys. D 30, 2103–2122 (1997).
[CrossRef]

1996 (1)

1989 (1)

1986 (1)

P. J. Martin, “Ion-based methods for optical thin film deposition,” J. Mater. Sci. 21, 1–25 (1986).
[CrossRef]

1980 (1)

H. Kuster, J. Ebert, “Activated reactive evaporation of TiO2 layers and their absorption indices,” Thin Solid Films 70, 43–47 (1980).
[CrossRef]

1979 (1)

1976 (1)

1953 (1)

W. L. Bragg, A. B. Pippard, “The form birefringence of macromolecules,” Acta Crystallogr. 6, 865–867 (1953).
[CrossRef]

Arnold, M. D.

I. J. Hodgkinson, Q. H. Wu, K. E. Thorn, M. D. Arnold, “Vacuum-deposited thin film polarizing elements for use with linearly and circularly polarized light at visible and near infrared wavelengths,” in Complex Mediums, A. Lakhtakia, W. S. Weiglhofer, R. Messier, eds., Proc. SPIE4097, 266–279 (2000).
[CrossRef]

Baumeister, P.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Bragg, W. L.

W. L. Bragg, A. B. Pippard, “The form birefringence of macromolecules,” Acta Crystallogr. 6, 865–867 (1953).
[CrossRef]

Ebert, J.

H. Kuster, J. Ebert, “Activated reactive evaporation of TiO2 layers and their absorption indices,” Thin Solid Films 70, 43–47 (1980).
[CrossRef]

Granqvist, C. G.

G. W. Mbise, D. Le Bellac, G. A. Niklasson, C. G. Granqvist, “Angular selective window coatings: theory and experiments,” J. Phys. D 30, 2103–2122 (1997).
[CrossRef]

Hazel, J. C.

Hioki, T.

T. Motohiro, Y. Takeda, T. Hioki, S. Noda, “Simultaneous oblique deposition from opposite azimuthal directions for fabrication of thin film retardation plates,” in International Symposium on Polarization Analysis and Applications to Device Technology, H. Yokota, T. Yoshizawa, eds., Proc. SPIE2873, 214–217 (1996).
[CrossRef]

Hodgkinson, I. J.

I. J. Hodgkinson, Q. H. Wu, B. Knight, A. Lakhtakia, K. Robbie, “Vacuum deposition of chiral sculptured thin films with high optical activity,” Appl. Opt. 39, 642–649 (2000).
[CrossRef]

I. J. Hodgkinson, Q. H. Wu, “Serial bideposition of anisotropic thin films with enhanced linear birefringence,” Appl. Opt. 38, 3621–3625 (1999).
[CrossRef]

I. J. Hodgkinson, Q. H. Wu, “Vacuum deposited biaxial thin films with all principal axes inclined to the substrate,” J. Vac. Sci. Technol. A 17, 2928–2932 (1999).
[CrossRef]

I. J. Hodgkinson, Q. H. Wu, J. C. Hazel, “Empirical equations for the principal refractive indices and column angle of obliquely deposited films of tantalum oxide, titanium oxide, and zirconium oxide,” Appl. Opt. 37, 2653–2659 (1998).
[CrossRef]

I. J. Hodgkinson, Q. H. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1998).
[CrossRef]

I. J. Hodgkinson, Q. H. Wu, K. E. Thorn, M. D. Arnold, “Vacuum-deposited thin film polarizing elements for use with linearly and circularly polarized light at visible and near infrared wavelengths,” in Complex Mediums, A. Lakhtakia, W. S. Weiglhofer, R. Messier, eds., Proc. SPIE4097, 266–279 (2000).
[CrossRef]

Jänchen, H.

Kaiser, N.

Knight, B.

Kuster, H.

H. Kuster, J. Ebert, “Activated reactive evaporation of TiO2 layers and their absorption indices,” Thin Solid Films 70, 43–47 (1980).
[CrossRef]

Lakhtakia, A.

Le Bellac, D.

G. W. Mbise, D. Le Bellac, G. A. Niklasson, C. G. Granqvist, “Angular selective window coatings: theory and experiments,” J. Phys. D 30, 2103–2122 (1997).
[CrossRef]

Martin, P. J.

P. J. Martin, “Ion-based methods for optical thin film deposition,” J. Mater. Sci. 21, 1–25 (1986).
[CrossRef]

Mbise, G. W.

G. W. Mbise, D. Le Bellac, G. A. Niklasson, C. G. Granqvist, “Angular selective window coatings: theory and experiments,” J. Phys. D 30, 2103–2122 (1997).
[CrossRef]

Merc, E.

Technical information from Patinal Newsletter No. 3, E. Merc, Patinal-Centre, 64578 Gernsheim, Germany, June1994.

Motohiro, T.

T. Motohiro, Y. Taga, “Thin film retardation plate by oblique deposition,” Appl. Opt. 28, 2466–2482 (1989).
[CrossRef] [PubMed]

T. Motohiro, Y. Takeda, T. Hioki, S. Noda, “Simultaneous oblique deposition from opposite azimuthal directions for fabrication of thin film retardation plates,” in International Symposium on Polarization Analysis and Applications to Device Technology, H. Yokota, T. Yoshizawa, eds., Proc. SPIE2873, 214–217 (1996).
[CrossRef]

Niklasson, G. A.

G. W. Mbise, D. Le Bellac, G. A. Niklasson, C. G. Granqvist, “Angular selective window coatings: theory and experiments,” J. Phys. D 30, 2103–2122 (1997).
[CrossRef]

Noda, S.

T. Motohiro, Y. Takeda, T. Hioki, S. Noda, “Simultaneous oblique deposition from opposite azimuthal directions for fabrication of thin film retardation plates,” in International Symposium on Polarization Analysis and Applications to Device Technology, H. Yokota, T. Yoshizawa, eds., Proc. SPIE2873, 214–217 (1996).
[CrossRef]

Paesold, G.

Pippard, A. B.

W. L. Bragg, A. B. Pippard, “The form birefringence of macromolecules,” Acta Crystallogr. 6, 865–867 (1953).
[CrossRef]

Pulker, H. K.

Ritter, E.

Robbie, K.

Smith, D.

Taga, Y.

Takeda, Y.

T. Motohiro, Y. Takeda, T. Hioki, S. Noda, “Simultaneous oblique deposition from opposite azimuthal directions for fabrication of thin film retardation plates,” in International Symposium on Polarization Analysis and Applications to Device Technology, H. Yokota, T. Yoshizawa, eds., Proc. SPIE2873, 214–217 (1996).
[CrossRef]

Thorn, K. E.

I. J. Hodgkinson, Q. H. Wu, K. E. Thorn, M. D. Arnold, “Vacuum-deposited thin film polarizing elements for use with linearly and circularly polarized light at visible and near infrared wavelengths,” in Complex Mediums, A. Lakhtakia, W. S. Weiglhofer, R. Messier, eds., Proc. SPIE4097, 266–279 (2000).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Wu, Q. H.

I. J. Hodgkinson, Q. H. Wu, B. Knight, A. Lakhtakia, K. Robbie, “Vacuum deposition of chiral sculptured thin films with high optical activity,” Appl. Opt. 39, 642–649 (2000).
[CrossRef]

I. J. Hodgkinson, Q. H. Wu, “Vacuum deposited biaxial thin films with all principal axes inclined to the substrate,” J. Vac. Sci. Technol. A 17, 2928–2932 (1999).
[CrossRef]

I. J. Hodgkinson, Q. H. Wu, “Serial bideposition of anisotropic thin films with enhanced linear birefringence,” Appl. Opt. 38, 3621–3625 (1999).
[CrossRef]

I. J. Hodgkinson, Q. H. Wu, J. C. Hazel, “Empirical equations for the principal refractive indices and column angle of obliquely deposited films of tantalum oxide, titanium oxide, and zirconium oxide,” Appl. Opt. 37, 2653–2659 (1998).
[CrossRef]

I. J. Hodgkinson, Q. H. Wu, K. E. Thorn, M. D. Arnold, “Vacuum-deposited thin film polarizing elements for use with linearly and circularly polarized light at visible and near infrared wavelengths,” in Complex Mediums, A. Lakhtakia, W. S. Weiglhofer, R. Messier, eds., Proc. SPIE4097, 266–279 (2000).
[CrossRef]

I. J. Hodgkinson, Q. H. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1998).
[CrossRef]

Zuber, A.

Acta Crystallogr. (1)

W. L. Bragg, A. B. Pippard, “The form birefringence of macromolecules,” Acta Crystallogr. 6, 865–867 (1953).
[CrossRef]

Appl. Opt. (7)

J. Mater. Sci. (1)

P. J. Martin, “Ion-based methods for optical thin film deposition,” J. Mater. Sci. 21, 1–25 (1986).
[CrossRef]

J. Phys. D (1)

G. W. Mbise, D. Le Bellac, G. A. Niklasson, C. G. Granqvist, “Angular selective window coatings: theory and experiments,” J. Phys. D 30, 2103–2122 (1997).
[CrossRef]

J. Vac. Sci. Technol. A (1)

I. J. Hodgkinson, Q. H. Wu, “Vacuum deposited biaxial thin films with all principal axes inclined to the substrate,” J. Vac. Sci. Technol. A 17, 2928–2932 (1999).
[CrossRef]

Thin Solid Films (1)

H. Kuster, J. Ebert, “Activated reactive evaporation of TiO2 layers and their absorption indices,” Thin Solid Films 70, 43–47 (1980).
[CrossRef]

Other (5)

T. Motohiro, Y. Takeda, T. Hioki, S. Noda, “Simultaneous oblique deposition from opposite azimuthal directions for fabrication of thin film retardation plates,” in International Symposium on Polarization Analysis and Applications to Device Technology, H. Yokota, T. Yoshizawa, eds., Proc. SPIE2873, 214–217 (1996).
[CrossRef]

I. J. Hodgkinson, Q. H. Wu, K. E. Thorn, M. D. Arnold, “Vacuum-deposited thin film polarizing elements for use with linearly and circularly polarized light at visible and near infrared wavelengths,” in Complex Mediums, A. Lakhtakia, W. S. Weiglhofer, R. Messier, eds., Proc. SPIE4097, 266–279 (2000).
[CrossRef]

I. J. Hodgkinson, Q. H. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1998).
[CrossRef]

Technical information from Patinal Newsletter No. 3, E. Merc, Patinal-Centre, 64578 Gernsheim, Germany, June1994.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

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Figures (8)

Fig. 1
Fig. 1

Basic model of a tilted-columnar thin film: θ v , deposition angle; ψ, column angle; 1–3, principal axes.

Fig. 2
Fig. 2

Dependence of principal refractive indices and column index of titanium oxide films on deposition angle; n 1, n 2, and n 3 were calculated with the empirical equations, Eqs. (9)–(11), and n c was calculated from Eq. (4).

Fig. 3
Fig. 3

Packing fraction p c and B-P depolarizing parameters L 1, L 2, and L 3 of titanium oxide films, calculated with Eqs. (5)–(8).

Fig. 4
Fig. 4

Model of a tilted-columnar thin film adapted to allow for dependence of n c on deposition angle.

Fig. 5
Fig. 5

Dispersion of principal refractive indices and birefringences simulated for a titanium oxide film deposited at an angle of 57°.

Fig. 6
Fig. 6

Normal-incidence birefringence as a function of deposition angle.

Fig. 7
Fig. 7

Ratio of dispersion constants for normal-incidence birefringence and for the bulk material.

Fig. 8
Fig. 8

Experimental values of normal-incidence birefringence of titanium oxide coatings.

Tables (1)

Tables Icon

Table 1 Constants for Refractive Index and Column Angle Functions

Equations (24)

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L1+L2+L3=1.
nj2=nv2+pcnc2-nv21+1-pcnc2-nv2Lj/nv2, j=1, 2, 3
nv=1,
nc2=n12+nv2n12-n22n22-nv2+n12-n32n32-nv2,
pc=n12-nv2nc2-nv2,
L1=0,
L2=nv2n12-n22nc2-n12n22-nv2,
L3=1-L1-L2.
n1,633θv=A0+A2θv2,
n2,633θv=B0+B2θv2,
n3,633θv=C0+C2θv2.
ψθv=tan-1E1 tan θv.
np=sin2 ψ/n12+cos2 ψ/n22-1/2,
Δn=n3-np,
n633nc,6330=A02+A02-B02B02-1+A02-C02C02-11/2;
nc,633θv=n1,6332θv+n1,6332θv-n2,6332θvn2,6332θv-1+n1,6332θv-n3,6332θvn3,6332θv-11/2,
nc,6332θv=pθvn6332+1-pθvnv2,
pθv=nc,6332θv-1n6332-1.
nλ=n6331+D1λ2-16332,
nc,λθv=pθvnλ2+1-pθvnv21/2,
nj,λ=nj,6331+Dj1λ2-16332,  j=1, 2, 3.
Δnjk,λ=Δnjk,6331+Djk1λ2-16332, jk=12, 13, 32,
Δnλ=Δn6331+DΔn1λ2-16332.
Δnλ=Δn6331+9.2×1041λ2-16332.

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