Abstract

The extended Jones matrix method is applied for determination of the transmission and reflection matrices for a normally incident plane wave upon an homogeneous and lossless biaxial thin layer. The elements of these matrices are expressed by simple analytical relations. By using these relations one can express analytically the polarization-dependent optical parameters to be determined by generalized ellipsometry.

© 1999 Optical Society of America

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References

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  1. D. A. Holmes, D. L. Feucht, “Electromagnetic wave propagation in birefringent multilayers,” J. Opt. Soc. Am. 56, 1763–1769 (1966).
    [CrossRef]
  2. S. Teitler, B. W. Henvis, “Refraction in stratified, anisotropic media,” J. Opt. Soc. Am. 60, 830–834 (1970).
    [CrossRef]
  3. J. Schesser, G. Eichmann, “Propagation of plane waves in biaxially anisotropic layered media,” J. Opt. Soc. Am. 62, 786–791 (1972).
    [CrossRef]
  4. D. J. De Smet, “Reflection from an oriented biaxial surface,” Appl. Opt. 26, 995–998 (1987).
    [CrossRef] [PubMed]
  5. M. A. Dreger, J. H. Erkkila, “Improved method for calculating phase-matching criteria in biaxial nonlinear materials,” Opt. Lett. 17, 787–788 (1992).
    [CrossRef] [PubMed]
  6. 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]
  7. P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72, 507–513 (1982).
    [CrossRef]
  8. E. Cojocaru, “Generalized Abeles relations for an anisotropic thin film of an arbitrary dielectric tensor,” Appl. Opt. 36, 2825–2829 (1997).
    [CrossRef] [PubMed]
  9. R. M. A. Azzam, N. M. Bashara, “Generalized ellipsometry for surfaces with directional preference: application to diffraction gratings,” J. Opt. Soc. Am. 62, 1521–1523 (1972).
    [CrossRef]
  10. M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
    [CrossRef]
  11. M. Schubert, B. Rheinlander, C. Cramer, H. Schmiedel, J. A. Woollam, C. M. Herzinger, B. Johs, “Generalized transmission ellipsometry for twisted biaxial dielectric media: application to chiral liquid crystals,” J. Opt. Soc. Am. A 13, 1930–1940 (1996).
    [CrossRef]
  12. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1957), Chap. 4, pp. 106–109.
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–718.

1997 (1)

1996 (2)

1995 (1)

1992 (1)

1987 (1)

1982 (1)

1972 (2)

1970 (1)

1966 (1)

Azzam, R. M. A.

Bashara, N. M.

Born, M.

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

Cojocaru, E.

Cramer, C.

De Smet, D. J.

Dreger, M. A.

Eichmann, G.

Erkkila, J. H.

Feucht, D. L.

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1957), Chap. 4, pp. 106–109.

Henvis, B. W.

Herzinger, C. M.

Holmes, D. A.

Johs, B.

Landry, G. D.

Maldonado, T. A.

Rheinlander, B.

Schesser, J.

Schmiedel, H.

Schubert, M.

Teitler, S.

Wolf, E.

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

Woollam, J. A.

Yeh, P.

Appl. Opt. (2)

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (2)

Opt. Lett. (1)

Phys. Rev. B (1)

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
[CrossRef]

Other (2)

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1957), Chap. 4, pp. 106–109.

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

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Figures (4)

Fig. 1
Fig. 1

Relationship between the (x p , y p , z p ) principal axes and the (x, y, z) laboratory coordinate systems. The transformation between these coordinate systems is specified by three Euler angles (x convention): ϕ, θ, and ψ.

Fig. 2
Fig. 2

Relationship between the (E i σ, H i σ, S i σ) orthogonal triplets and the (x, y, z) laboratory coordinate system. The relationship is specified by angles γ i σ, α i σ, and η i σ.

Fig. 3
Fig. 3

Specific values of Euler alignment angle ϕ, which are defined by Eqs. (11), plotted versus θ and ψ: (a) ϕ0, to yield uncoupled waves, and (b) ϕ eq , to yield equal transmission and reflection coefficients of the p and s modes. At certain pairs of θ and ψ values, ϕ0 jumps abruptly from -π/4 to π/4 [as shown in (a)], and ϕ eq is zero. These pairs of θ and ψ values are plotted on the curve in (c). On both sides of this curve, ϕ0 and ϕ eq change signs: On the right-hand side, ϕ0 < 0 and ϕ eq > 0; on the other side, ϕ0 > 0 and ϕ eq < 0.

Fig. 4
Fig. 4

Variations of |t pp | against θ and ψ at three values of ϕ: (a) ϕ = 0, (b) ϕ = π/4, and (c) ϕ = ϕ eq . Similarly, at these values of ϕ, variations of |t sp | (which is equal to |t ps |) versus θ and ψ are shown: (d) ϕ = 0, (e) ϕ = π/4, and (f) ϕ = ϕ eq .

Equations (24)

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M¯=R¯zψR¯xθR¯zϕ.
R¯za=cos asin a0-sin acos a0001,
R¯ya=cos a0-sin a010sin a0cos a.
Eˆipσ=signVipσ·kˆpσVipσ/Nipσ,
Eˆiσ, Hˆiσ, SˆiσT=R¯yηiσR¯zαiσR¯yγiσxˆ, yˆ, zˆT.
tpp=F1t tan2 α1+F2t/1+tan2 α1,
tss=F1t+F2t tan2 α1/1+tan2 α1,
tsp=F1t-F2ttan α1/1+tan2 α1,  tps=tsp;
rpp=-F1r tan2 α1+F2r/1+tan2 α1,
rss=F1r+F2r tan2 α1/1+tan2 α1,
rsp=-F1r-F2rtan α1/1+tan2 α1,  rps=-rsp.
rpp=-r01 tan2 α1+r02/1+tan2 α1,
rss=r01+r02 tan2 α1/1+tan2 α1,
rsp=-r01-r02tan α1/1+tan2 α1, rps=-rsp.
tan 2αi=2m11m12-m31m32cot2 Ωp/m112-m122+cot2 Ωpm322-m312,
tan 2αi=tan 2ϕ+β/1-β tan 2ϕ,
β=cos θ sin 2ψ/cos2 ψ-cos2 θ sin2 ψ+sin2 θ cot2 Ωp.
tan 2ϕ0=-β,  tan 2ϕeq=1/β,
tan α1=m12+gm32 cot Ωp/m11+gm31 cot Ωp,
|rp|2+nb/n0|tp|2=1+ζG-2/χ2,
|rs|2+nb/n0|ts|2=1-ζG-2,
ζ=tan α1χ-tan α11+χ tan α1/1+tan2 α12,
G=F1rF2r*+F1r*F2r+nb/n0F1tF2t*+F1t*F2t,
|rs|2+nb/n0|ts|2+χ2|rp|2+nb/n0|tp|2=1+χ2.

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