Abstract

Kramers-Kronig analysis of spectra of reflectance has usually been done only with radiation incident as nearly normal to the sample surface as possible. Any effects of obliquity have been assumed to be negligible. However, it is not much more difficult or time consuming to do Kramers-Kronig analysis of spectra taken with almost any angle of incidence, provided that polarized radiation is used to obtain the data. The method for such analysis for radiation incident at almost any angle is described in this paper. The method fails with π polarized radiation if the angle of incidence lies between the two somewhat different Brewster angles at the high and low frequency ends of the spectrum. The rather small error in a typical analysis caused by neglecting a 15° angle of incidence is illustrated.

© 1967 Optical Society of America

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References

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  1. T. S. Robinson, Proc. Phys. Soc. (London) B65, 910 (1952).
  2. For a recent review and improvements, see G. Andermann, A. Caron, D. A. Dows, J. Opt. Soc. Am. 55, 1210 (1965).
    [CrossRef]
  3. M. Born, E. Wolf, Principles of Optics (Macmillan Co., New York, 1964), p. 48, 2nd ed.
  4. W. Kaiser, W. G. Spitzer, R. H. Kaiser, L. E. Howarth, Phys. Rev. 127, 1950 (1962).
    [CrossRef]
  5. G. Bird, M. Parrish, J. Opt. Soc. Am. 50, 886 (1960).
    [CrossRef]
  6. M. Hass, M. O’Hara, Appl. Opt. 4, 1027 (1965); similar polarizers, developed independently by H. Marshall of Perkin-Elmer Corp., Norwalk, Connecticut, are now commercially available.
    [CrossRef]

1965

1962

W. Kaiser, W. G. Spitzer, R. H. Kaiser, L. E. Howarth, Phys. Rev. 127, 1950 (1962).
[CrossRef]

1960

1952

T. S. Robinson, Proc. Phys. Soc. (London) B65, 910 (1952).

Andermann, G.

Bird, G.

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan Co., New York, 1964), p. 48, 2nd ed.

Caron, A.

Dows, D. A.

Hass, M.

Howarth, L. E.

W. Kaiser, W. G. Spitzer, R. H. Kaiser, L. E. Howarth, Phys. Rev. 127, 1950 (1962).
[CrossRef]

Kaiser, R. H.

W. Kaiser, W. G. Spitzer, R. H. Kaiser, L. E. Howarth, Phys. Rev. 127, 1950 (1962).
[CrossRef]

Kaiser, W.

W. Kaiser, W. G. Spitzer, R. H. Kaiser, L. E. Howarth, Phys. Rev. 127, 1950 (1962).
[CrossRef]

O’Hara, M.

Parrish, M.

Robinson, T. S.

T. S. Robinson, Proc. Phys. Soc. (London) B65, 910 (1952).

Spitzer, W. G.

W. Kaiser, W. G. Spitzer, R. H. Kaiser, L. E. Howarth, Phys. Rev. 127, 1950 (1962).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan Co., New York, 1964), p. 48, 2nd ed.

Appl. Opt.

J. Opt. Soc. Am.

Phys. Rev.

W. Kaiser, W. G. Spitzer, R. H. Kaiser, L. E. Howarth, Phys. Rev. 127, 1950 (1962).
[CrossRef]

Proc. Phys. Soc. (London)

T. S. Robinson, Proc. Phys. Soc. (London) B65, 910 (1952).

Other

M. Born, E. Wolf, Principles of Optics (Macmillan Co., New York, 1964), p. 48, 2nd ed.

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Figures (1)

Fig. 1
Fig. 1

Real and imaginary parts of for CaF2 from Kramers-Kronig analysis of reflectance of π polarized and of σ polarized radiation when a 15° angle of incidence is ignored.

Equations (9)

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G ( ω ) [ Z ( ω ) - 1 ] / [ Z ( ω ) + 1 ] 2 .
ρ ( ω ) e i φ ( ω ) [ Z ( ω ) - 1 ] / [ Z ( ω ) + 1 ] .
ρ ( ω ) = ± [ G ( ω ) ] / 2 1 .
φ ( ω 0 ) = ( ω 0 / π ) 0 ln G ( ω ) - ln G ( ω 0 ) d ω ω 0 2 - ω 2 .
Z σ ( ω ) = ± [ n 2 ( ω ) - sin 2 θ ] ½ cos θ
Z π ( ω ) = ± n 2 ( ω ) cos θ [ n 2 ( ω ) - sin 2 θ ] ½
Z ( ω ) = 1 + ρ ( ω ) e i φ ( ω ) 1 - ρ ( ω ) e i φ ( ω ) = 1 ± [ G ( ω ) ] ½ e i φ ( ω ) 1 [ G ( ω ) ] ½ e i φ ( ω ) .
( ω ) n 2 ( ω ) = Z σ 2 ( ω ) cos 2 θ + sin 2 θ .
( ω ) = n 2 ( ω ) = [ Z π 2 ( ω ) / ( 2 cos 2 θ ) ] × [ 1 ± ( 1 - 4 cos 2 θ sin 2 θ ) Z π 2 ( ω ) ) ½ ] .

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