Abstract

A method is proposed to determine the optical constants of uniaxial crystals by ellipsometry. The same scheme works for absorbing and nonabsorbing crystals. The quantities measured are ratios of the four reflection amplitudes rss, rsp, rps, and rpp, and angles. The common zeros of rsp and rps determine the symmetry direction (optic axis in the plane of incidence) at which rpp/rss is measured to obtain one equation linking the ordinary and extraordinary dielectric constants o and e, and the inclination χ of the optic axis to the normal to the reflecting plane. Measurement of rpp/rss at right angles to the symmetry direction, and of rsp/rps away from the symmetry direction gives two more equations for the unknowns. The method can be used on microscopic crystal faces.

© 1997 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  2. J. Lekner, “Ellipsometry of anisotropic media,” J. Opt. Soc. Am. A 10, 1579–1581 (1993).
    [CrossRef]
  3. J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys., Condens. Matter 3, 6121–6133 (1991).
    [CrossRef]
  4. J. Lekner, “Brewster angles in reflection by uniaxial crystals,” J. Opt. Soc. Am. A 10, 2059–2064 (1993).
    [CrossRef]
  5. J. Lekner, “Bounds and zeros in reflection by uniaxial crystals,” J. Phys., Condens. Matter 4, 9459–9468 (1992).
    [CrossRef]
  6. See, for example, Sec. 9-1 of J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987).
  7. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw–Hill, New York, 1950), Sec. 24.7.
  8. See, for example, the articles and references in E. D. Palik ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).
  9. N. H. Hartshorne, A. Stuart, Crystals and the Polarising Microscope, 4th ed. (Edward Arnold, London, 1970).
  10. E. E. Wahlstrom, Optical Crystallography, 4th ed. (Wiley, New York, 1969).
  11. F. Yang, G. W. Bradberry, J. R. Sambles, “A method for the optical characterization of thin uniaxial samples,” J. Mod. Opt. 42, 763–774 , 1241–1252, 1447–1458 (1995).
    [CrossRef]

1995 (1)

F. Yang, G. W. Bradberry, J. R. Sambles, “A method for the optical characterization of thin uniaxial samples,” J. Mod. Opt. 42, 763–774 , 1241–1252, 1447–1458 (1995).
[CrossRef]

1993 (2)

1992 (1)

J. Lekner, “Bounds and zeros in reflection by uniaxial crystals,” J. Phys., Condens. Matter 4, 9459–9468 (1992).
[CrossRef]

1991 (1)

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys., Condens. Matter 3, 6121–6133 (1991).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bradberry, G. W.

F. Yang, G. W. Bradberry, J. R. Sambles, “A method for the optical characterization of thin uniaxial samples,” J. Mod. Opt. 42, 763–774 , 1241–1252, 1447–1458 (1995).
[CrossRef]

Hartshorne, N. H.

N. H. Hartshorne, A. Stuart, Crystals and the Polarising Microscope, 4th ed. (Edward Arnold, London, 1970).

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw–Hill, New York, 1950), Sec. 24.7.

Lekner, J.

J. Lekner, “Ellipsometry of anisotropic media,” J. Opt. Soc. Am. A 10, 1579–1581 (1993).
[CrossRef]

J. Lekner, “Brewster angles in reflection by uniaxial crystals,” J. Opt. Soc. Am. A 10, 2059–2064 (1993).
[CrossRef]

J. Lekner, “Bounds and zeros in reflection by uniaxial crystals,” J. Phys., Condens. Matter 4, 9459–9468 (1992).
[CrossRef]

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys., Condens. Matter 3, 6121–6133 (1991).
[CrossRef]

See, for example, Sec. 9-1 of J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987).

Sambles, J. R.

F. Yang, G. W. Bradberry, J. R. Sambles, “A method for the optical characterization of thin uniaxial samples,” J. Mod. Opt. 42, 763–774 , 1241–1252, 1447–1458 (1995).
[CrossRef]

Stuart, A.

N. H. Hartshorne, A. Stuart, Crystals and the Polarising Microscope, 4th ed. (Edward Arnold, London, 1970).

Wahlstrom, E. E.

E. E. Wahlstrom, Optical Crystallography, 4th ed. (Wiley, New York, 1969).

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw–Hill, New York, 1950), Sec. 24.7.

Yang, F.

F. Yang, G. W. Bradberry, J. R. Sambles, “A method for the optical characterization of thin uniaxial samples,” J. Mod. Opt. 42, 763–774 , 1241–1252, 1447–1458 (1995).
[CrossRef]

J. Mod. Opt. (1)

F. Yang, G. W. Bradberry, J. R. Sambles, “A method for the optical characterization of thin uniaxial samples,” J. Mod. Opt. 42, 763–774 , 1241–1252, 1447–1458 (1995).
[CrossRef]

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

J. Phys., Condens. Matter (2)

J. Lekner, “Bounds and zeros in reflection by uniaxial crystals,” J. Phys., Condens. Matter 4, 9459–9468 (1992).
[CrossRef]

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys., Condens. Matter 3, 6121–6133 (1991).
[CrossRef]

Other (6)

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

See, for example, Sec. 9-1 of J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987).

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw–Hill, New York, 1950), Sec. 24.7.

See, for example, the articles and references in E. D. Palik ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).

N. H. Hartshorne, A. Stuart, Crystals and the Polarising Microscope, 4th ed. (Edward Arnold, London, 1970).

E. E. Wahlstrom, Optical Crystallography, 4th ed. (Wiley, New York, 1969).

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Figures (4)

Fig. 1
Fig. 1

Reflection by a uniaxial crystal: xy is the reflecting face and zx is the plane of incidence, with the inward normal along the z-axis. The optic axis c (long-dashed line) is at angle χ to the normal, and the plane containing c and the z-axis cuts the xy plane at angle ϕ to the x axis.

Fig. 2
Fig. 2

Theoretical reflection amplitudes for the cleavage faces of calcite, as a function of the azimuthal angle ϕ, at 45° angle of incidence. The azimuthal angle goes from 0 to 2π as the crystal is rotated a full revolution. The angle between the normal to a cleavage plane and the optic axis is 44.61°. The refractive indices at 633 nm are no=1.655, ne=1.485. Note the common zeros of rsp and rps at ϕ=0 and π and also that the largest amplitude rss has been brought into the figure by adding 0.28.

Fig. 3
Fig. 3

Deduced o values for calcite and selenium at ϕ=45° and θ=45°, assuming 10% errors in the measured values of r1, r2, and r3. The plotted values are obtained by varying r1, r2, and r3 by 10% in magnitude away from their exact values, with eight different phases equally spaced around the unit circle, so that |r-rexact| =|rexact|/10. The true values of o are (1.655)22.739 for calcite at 633 nm and (3.38+0.65i)2 11.0 +4.4i for selenium at 620 nm.

Fig. 4
Fig. 4

Detail of the calcite error ovals, at ϕ=45° and θ =45° and 60°. The deduced o values are obtained by taking 10% magnitude errors in r1, r2, and r3, as in Fig. 3. The curves are ellipses fitted to the two points on the real axis, one focus being at the exact value o=(1.655)22.739. The ellipses all have small ellipticities, ranging from 0.037 for the r2 (60°) ellipse to 0.137 for the r3 (60°) ellipse. The r3 errors orbit about the right focus, the others about the left focus.

Equations (21)

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c=(α,β,γ)α2+β2+γ2=1,
c=(sin χ cos ϕ, sin χ sin ϕ, cos χ).
ρP=rpp+rsp tan Prps+rss tan P,ρA=rpp+rps tan Arsp+rss tan A,
rsp=β(αqo+γK)Frps=β(αqo-γK)F
rss(β=0)=q1-qoq1+qorpp(β=0)=Q-Q1Q+Q1
qγ˙2=γ(ω/c)2-K2,γ=o+γ2(e-o).
r1rss-rpprss+rppβ=0=1qoQ-q12q1(qo-1Q).
r2rss-rpprss+rppα=0=[(1-γ2)o+γ21 sin2 θ][1qoqe-oq12]q1[(1-γ2)o(oqe-1qo)+γ21(oqo-1qe)sin2 θ]
 
qe2=oγ [e(ω/c)2-K2].
Q2=e(ω/c)2-K2oe.
r3rsp+rpsrsp-rps=αqoγK=tan χ cos ϕ qoK.
γ2=(qo cos ϕ)2(qo cos ϕ)2+K2r32,
tan2 θpp(β=0)=oe-1γ1(γ-1),
γ=cos χ=13 tan(A/2),
α=N cos ϕ,β=N sin ϕ,N=1+2 cos A3 cos(A/2)
rps(-α, -β, γ)=rsp(α, β, γ).
ρP=rpp+rsp tan Prps+rss tan P.
rsp=ρrssrps=ρ1-ρρ0-ρ1 rss
rpp=ρ0(ρ1-ρ)ρ0-ρ1 rss.
ρ-1=ρ0ρ1-2ρ0ρ+ρ1ρ2ρ1-ρ0-ρ.

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