Abstract

Generalized Abelès relations for one anisotropic thin film [E. Cojocaru, Appl. Opt. 36, 2825–2829 (1997)] are developed for light propagation from an isotropic medium of incidence (with refractive index n 0) within a multilayer anisotropic thin film coated onto an anisotropic substrate. An immersion model is used for which it is assumed that each layer is imaginatively embedded between isotropic gaps of zero thickness and refractive index n 0. This model leads to simple expressions for the resultant transmitted and reflected electric field amplitudes at interfaces. They parallel the Abelès recurrence relations for layered isotropic media. These matrix relations include multiple reflections while they deal with total fields. They can be applied directly to complex stacks of isotropic and anisotropic thin films.

© 2000 Optical Society of America

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References

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  1. J. Mentel, E. Schmidt, T. Mavrudis, “Birefringent filter with arbitrary orientation of the optic axis: an analysis of improved accuracy,” Appl. Opt. 31, 5022–5029 (1992).
    [CrossRef] [PubMed]
  2. P. J. Valle, F. Moreno, “Theoretical study of birefringent filters as intracavity wavelength selectors,” Appl. Opt. 31, 528–535 (1992).
    [CrossRef] [PubMed]
  3. J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum. Electron. QE-18, 1253–1258 (1982).
    [CrossRef]
  4. Z. M. Li, B. T. Sullivan, R. R. Parsons, “Use of the 4 × 4 matrix method in the optics of multilayer magnetooptic recording media,” Appl. Opt. 27, 1334–1338 (1988).
    [CrossRef] [PubMed]
  5. D. A. Holmes, D. L. Feucht, “Electromagnetic wave propagation in birefringent multilayers,” J. Opt. Soc. Am. 56, 1763–1769 (1966).
    [CrossRef]
  6. S. Teitler, B. W. Henvis, “Refraction in stratified, anisotropic media,” J. Opt. Soc. Am. 60, 830–834 (1970).
    [CrossRef]
  7. J. Schesser, G. Eichmann, “Propagation of plane waves in biaxially anisotropic media,” J. Opt. Soc. Am. 62, 786–791 (1972).
    [CrossRef]
  8. G. J. Sprokel, “Reflectivity, rotation, and ellipticity of magnetooptic film structures,” Appl. Opt. 23, 3983–3989 (1984).
    [CrossRef] [PubMed]
  9. M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
    [CrossRef]
  10. G. D. Landry, T. A. Maldonado, “Complete method to determine transmission and reflection characteristics at a planar interface between arbitrarily oriented biaxial media,” J. Opt. Soc. Am. A 12, 2048–2063 (1995).
    [CrossRef]
  11. E. Cojocaru, “Generalized Abelès relations for an anisotropic thin film of an arbitrary dielectric tensor,” Appl. Opt. 36, 2825–2829 (1997).
    [CrossRef] [PubMed]
  12. P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72, 507–513 (1982).
    [CrossRef]
  13. C. Gu, P. Yeh, “Extended Jones matrix method. II,” J. Opt. Soc. Am. A 10, 966–973 (1993).
    [CrossRef]
  14. F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. 3, 504–520 (1948).
  15. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Chap. 4, pp. 59–62.
  16. F. Horowitz, “Generalized Abelès relations for an anisotropic thin film with an arbitrary dielectric tensor: comments,” Appl. Opt. 37, 4268–4269 (1998).
    [CrossRef]
  17. D. M. Spink, C. B. Thomas, “Optical constant determination of thin films: an analytical solution,” Appl. Opt. 27, 4362–4362 (1988).
    [CrossRef] [PubMed]
  18. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  19. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  20. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–718.
  21. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1957), Chap. 4, pp. 107–109.
  22. T. Motohiro, Y. Taga, “Thin film retardation plate by oblique deposition,” Appl. Opt. 28, 2466–2482 (1989).
    [CrossRef] [PubMed]
  23. H. Fu, S. Sugaya, M. Mansuripur, “Measuring distribution of the ellipsoid of birefringence through the thickness of optical disk substrates,” Appl. Opt. 33, 5994–5999 (1994).
    [CrossRef] [PubMed]
  24. Y. Tomita, T. Yoshino, “Optimum design of multilayer-medium structures in a magneto-optical readout system,” J. Opt. Soc. Am. A 1, 809–817 (1984).
    [CrossRef]
  25. G. J. Sprokel, “Reflectivity, rotation, and ellipticity of magnetooptic film structures,” Appl. Opt. 23, 3983–3989 (1984).
    [CrossRef] [PubMed]
  26. S. Sugaya, M. Mansuripur, “Effect of tilted ellipsoid of birefringence on readout signal in magneto-optical disk data storage,” Appl. Opt. 33, 5999–6008 (1994).
    [CrossRef] [PubMed]

1998 (1)

1997 (1)

1995 (1)

1994 (2)

1993 (1)

1992 (2)

1990 (1)

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[CrossRef]

1989 (1)

1988 (2)

1984 (3)

1982 (2)

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum. Electron. QE-18, 1253–1258 (1982).
[CrossRef]

P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72, 507–513 (1982).
[CrossRef]

1979 (1)

1972 (2)

1970 (1)

1966 (1)

1948 (1)

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

Abelès, F.

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

Berreman, D. W.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–718.

Cojocaru, E.

Eichmann, G.

Feucht, D. L.

Fu, H.

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1957), Chap. 4, pp. 107–109.

Gu, C.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Chap. 4, pp. 59–62.

Henderson, D. M.

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum. Electron. QE-18, 1253–1258 (1982).
[CrossRef]

Henvis, B. W.

Holmes, D. A.

Horowitz, F.

Landry, G. D.

Li, Z. M.

Lotspeich, J. F.

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum. Electron. QE-18, 1253–1258 (1982).
[CrossRef]

Maldonado, T. A.

Mansuripur, M.

Mavrudis, T.

Mentel, J.

Moreno, F.

Motohiro, T.

Parsons, R. R.

Schesser, J.

Schmidt, E.

Spink, D. M.

Sprokel, G. J.

Stephens, R. R.

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum. Electron. QE-18, 1253–1258 (1982).
[CrossRef]

Sugaya, S.

Sullivan, B. T.

Taga, Y.

Teitler, S.

Thomas, C. B.

Tomita, Y.

Valle, P. J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–718.

Yeh, P.

Yoshino, T.

Ann. Phys. (1)

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

Appl. Opt. (11)

E. Cojocaru, “Generalized Abelès relations for an anisotropic thin film of an arbitrary dielectric tensor,” Appl. Opt. 36, 2825–2829 (1997).
[CrossRef] [PubMed]

F. Horowitz, “Generalized Abelès relations for an anisotropic thin film with an arbitrary dielectric tensor: comments,” Appl. Opt. 37, 4268–4269 (1998).
[CrossRef]

D. M. Spink, C. B. Thomas, “Optical constant determination of thin films: an analytical solution,” Appl. Opt. 27, 4362–4362 (1988).
[CrossRef] [PubMed]

Z. M. Li, B. T. Sullivan, R. R. Parsons, “Use of the 4 × 4 matrix method in the optics of multilayer magnetooptic recording media,” Appl. Opt. 27, 1334–1338 (1988).
[CrossRef] [PubMed]

J. Mentel, E. Schmidt, T. Mavrudis, “Birefringent filter with arbitrary orientation of the optic axis: an analysis of improved accuracy,” Appl. Opt. 31, 5022–5029 (1992).
[CrossRef] [PubMed]

P. J. Valle, F. Moreno, “Theoretical study of birefringent filters as intracavity wavelength selectors,” Appl. Opt. 31, 528–535 (1992).
[CrossRef] [PubMed]

G. J. Sprokel, “Reflectivity, rotation, and ellipticity of magnetooptic film structures,” Appl. Opt. 23, 3983–3989 (1984).
[CrossRef] [PubMed]

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

H. Fu, S. Sugaya, M. Mansuripur, “Measuring distribution of the ellipsoid of birefringence through the thickness of optical disk substrates,” Appl. Opt. 33, 5994–5999 (1994).
[CrossRef] [PubMed]

G. J. Sprokel, “Reflectivity, rotation, and ellipticity of magnetooptic film structures,” Appl. Opt. 23, 3983–3989 (1984).
[CrossRef] [PubMed]

S. Sugaya, M. Mansuripur, “Effect of tilted ellipsoid of birefringence on readout signal in magneto-optical disk data storage,” Appl. Opt. 33, 5999–6008 (1994).
[CrossRef] [PubMed]

IEEE J. Quantum. Electron. (1)

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum. Electron. QE-18, 1253–1258 (1982).
[CrossRef]

J. Appl. Phys. (1)

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (3)

Other (3)

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Chap. 4, pp. 59–62.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–718.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1957), Chap. 4, pp. 107–109.

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Figures (4)

Fig. 1
Fig. 1

(a) Multilayer structure containing N anisotropic thin films of thicknesses d m , m = 1… N, that are coated onto the anisotropic substrate numbered N + 1. Layer interfaces are parallel to the xy plane. Wave vectors that lie in the xz plane are denoted k0 ± in the isotropic medium of incidence of refractive index n 0 and by k vi ± in the anisotropic regions of refractive indices n vi ±, where v = α, β and i = 1… N + 1. (b) Illustration of the immersion model. Each anisotropic layer is imaginatively embedded between two isotropic gaps of refractive index n 0 and thickness d 0 that is set equal to zero. The isotropic (dotted) regions are numbered 0i, with i = 1… N + 1. Wave vectors and refractive indices are specified for all the isotropic and anisotropic regions.

Fig. 2
Fig. 2

Relationship between the (x pi , y pi , z pi ) principal-axis system of the ith biaxial layer and the (x, y, z) laboratory-coordinates system. The transformation between these coordinate systems is specified by the Euler angles (x convention) ϕ i , θ i , and ψ i .

Fig. 3
Fig. 3

(a) Intensity of transmitted light and (b) phase retardation Δ t versus Euler angle ϕ1 for normally incident waves in air (n 0 = 1) upon the system containing two biaxial thin layers that are coated onto a glass substrate (n g = 1.5). The biaxial layers have the same principal refractive indices (n px = 1.60, n py = 1.65, n pz = 2.25) and thickness (d/λ = 0.25) but different orientations of the principal-axis systems: ψ1 = 15°, θ1 = 45°, ψ2 = π - ψ1, θ2 = π - θ1, and ϕ2 = π - ϕ1.

Fig. 4
Fig. 4

Variations of polar Kerr rotation angle θ K (a) against Euler angle θ when ϕ = ψ = 45° and (b) against Euler angle ϕ when θ = 1° and ψ = 45°.

Equations (74)

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k0σ=ω/cξxˆ+σζ0zˆ,
kviσ=ω/cξxˆ+ζviσzˆ,
E0iσ=Esiσsˆ+Epiσpˆσexpjωt-k0σr,
H0iσ=k0σ×E0iσ/ω.
H0iσ=n0/cEsiσpˆσ-Epiσsˆexpjωt-k0σr.
Eiσ=v=α,β Eviσeˆviσ exp-jkviσrexpjωt,
kviσ×eˆviσ=hˆviσnviσω/ccos ηviσ,
Hiσ=v=α,βnviσ/cEviσhˆviσ cos ηviσ exp-jkviσr×expjωt.
E¯iσ=Eαiσ, EβiσT,  E¯0iσ=Esiσ, EpiσT,
τ˜1+E¯01+=E¯1++ρ˜1+E¯1-,
τ˜1-E¯01-=ρ˜1-E¯1++E¯1-,
X˜1+|z1E¯1++ρ˜1+X˜1-|z1E¯1-=τ˜1+X0+|z1E¯02+,
X˜1-|z1E¯1-+ρ˜1-X˜1+|z1E¯1+=τ˜1-X0-|z1E¯02-,
τ˜2+X0+|z1E¯02+=X˜2+|z1E¯2++ρ˜2+X˜2-|z1E¯2-,
τ˜2-X0-|z1E¯02-=ρ˜2-X˜2+|z1E¯2++X˜2-|z1E¯2-,
X˜2+|z2E¯2++ρ˜2+X˜2-|z2E¯2-=τ˜2+X0+|z2E¯03+,
X˜2-|z2E¯2-+ρ˜2-X˜2+|z2E¯2+=τ˜2-X0-|z2E¯03-,
τ˜N+X0+|zN-1E¯0N+=X˜N+|zN-1E¯N++ρ˜N+X˜N-|zN-1E¯N-,
τ˜N-X0-|zN-1E¯0N-=ρ˜N-X˜N+|zN-1E¯N++X˜N-|zN-1E¯N-,
X˜N+|zNE¯N++ρ˜N+X˜N-|zNE¯N-=τ˜N+X0+|zNE¯0N+1+,
X˜N-|zNE¯N-+ρ˜N-X˜N+|zNE¯N+=τ˜N-X0-|zNE¯0N+1-,
τ˜N+1+X0+|zNE¯0N+1+=X˜N+1+|zNE¯N+1+,
τ˜N+1-X0-|zNE¯0N+1-=ρ˜N+1-X˜N+1+|zNE¯N+1+,
zl=m=1l dm, X˜i±|zl=exp-jω/czlζαi±00exp-jω/czlζβi±.
X0±|zl=expjω/czlζ0.
χ˜i±=X˜i±|di, χ0i±=X0±|di.
X˜i±|zi=χ˜i±X˜i±|zi-1,  X0±|zi=χ0i±X0±|zi-1.
¯i±=X˜i±|zi-1E¯i±,  ¯0i±=X0±|zi-1E¯0i±
τ˜i+¯0i+=¯i++ρ˜i+¯i-,
τ˜i-¯0i-=ρ˜i-¯i++¯i-,
τ˜i+¯0i+1+=χ˜i+¯i++ρ˜i+χ˜i-¯i-,
τ˜i-¯0i+1-=χ˜i-¯i-+ρ˜i-χ˜i+¯i+.
τ˜˜i¯0i+¯0i-=ρ˜˜i¯i-˜i+,
τ˜˜i¯0i+1+¯0i+1-=ρ˜˜iχ˜˜i¯i-¯i+,
τ˜˜i=τ˜i+Õ2Õ2τ˜i-,  ρ˜˜i=ρ˜i+Ĩ2Ĩ2ρ˜i-,  χ˜˜i=χ˜i-Õ2Õ2χ˜i+.
¯0i+¯0i-=M˜˜i¯0i+1+¯0i+1-,
M˜˜i=τ˜˜i-1ρ˜˜iχ˜˜i-1ρ˜˜i-1τ˜˜i.
E¯01+E¯01-=M˜˜1M˜˜2M˜˜Nτ˜˜N+1-1I˜2ρ˜N+1-¯N+1+.
t˜g=τ˜N+1+
r˜g=τ˜N+1--1ρ˜N+1-τ˜N+1+.
A˜B˜C˜D˜=M˜˜1M˜˜2M˜˜N,
E¯01+E¯01-=A˜B˜C˜D˜I˜2r˜gt˜g-1˜g+,
t˜f=t˜gA˜+B˜r˜g-1,
r˜f=C˜+D˜r˜gA˜+B˜r˜g-1.
χ˜i±=expjω/cdiζiI˜2,  τ˜i+=τ˜i-=tsi00tpi,  ρ˜i+=ρ˜i-=rsi00rpi,
xpi, ypi, zpiT=R˘ix, y, zT,
kˆpviσ=R˘ikˆviσ=Ri13, Ri23, Ri33T,
c2invi4+c1invi2+c0=0,
c2i=npx2Ri132+npy2Ri232+npz2Ri332,  c1i=-npx2npy21-Ri332-npx2npz21-Ri232-npy2npz21-Ri132,  c0=npxnpynpz2.
J˘pvi=ε˘p-nvi2I˘3
Vpviσ=J˘pvi-1kpviσ.
eˆpviσ=signVpviσ·kˆpviσVpviσ/N1vi,
dˆpviσ=ε˘peˆpviσ/N2vi,
hˆpviσ=kˆpviσ×dˆpviσ,
ηviσ=-signkˆpviσ·eˆpviσcos-1dˆpviσ·eˆpviσ.
tfp=tf21+tf22,  tfs=tf11+tf12,
nα2+=ε+jε121/2,  nβ2+=ε-jε121/2,  nα2-=nβ2+,  nβ2-=nα2+,
eˆα2±=±1, j, 0T/21/2,  eˆβ2±=±1, -j, 0T/21/2,  hˆα2±=j, 1, 0T/21/2,  hˆβ2±=±j, 1, 0T/21/2.
eˆviσ=evxiσ, evyiσ, evziσT,  hˆviσ=hvxiσ, hvyiσ, hvziσT,
avi±σ=n0evxiσ±μviσhvyiσ cos φ0,
bvi±σ=ζ0evyiσ±μviσhvxiσ,
δi+=aβi++bαi-+-aαi++bβi-+,
δi-=aβi--bαi+--aαi--bβi+-,
τ˜i±=2ζ0/δi±aβi±±±bβi±-aαi±±bαi±.
τi12-=-2ζ0bβi+-/δi-,  τi22+=-2ζ0bαi-+/δi+.]
ρi11±=aβi±±bαi-aαi±bβi±/δi±,
ρi12±=aβi±±bβi-aβi±bβi±/δi±,
ρi21±=aαi±bαi±-aαi±±bαi/δi±,
ρi22±=aβi±bαi±-aαi±±bβi/δi±.
ρi11+=aβi++bαi---aαi+-bβi-+/δi+,  ρi11-=aβi--bαi++-aαi-+bβi+-/δi-].
rg11=aβ+bα+-aα+bβ+/δ,
rg12=bα+bβ--bα-bβ+/δ,
rg21=aα-aβ+-aα+aβ-/δ,
rg22=aα-bβ--aβ-bα-/δ.

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