Abstract

Angles of incidence for s- and p-polarized light have been computed and confirmed experimentally for which monochromatic interference in transparent thin films on absorbing substrates results in optimum interference fringe contrast (visibility = 1). Under these angles of incidence and with polarized light, film thickness determinations which are not possible at normal incidence or with unpolarized light can be carried out by use of thin-film interference.

© 1980 Optical Society of America

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References

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  1. R. H. Muller, “Thickness measurement of transparent thin films on metal surfaces by light interference,” J. Opt. Soc. Am. 54, 419–420 (1964).
    [Crossref]
  2. A. Vasicek, Optics of Thin Films (North-Holland, Amsterdam, 1960), p. 312.
  3. R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
    [Crossref]
  4. W. Koenig, “Elektromagnetische Lichttheorie,” in Handbuch der Physik, edited by H. Geiger and K. Scheel (Springer, Berlin, 1928), Vol. XX, p. 242.
  5. R. H. Muller, “Principles of Ellipsometry,” in Optical Techniques in Electrochemistry, edited by R. H. Muller, Vol. 9 of Advances in Electrochemistry and Electrochemical Engineering, edited by P. Delahay and C. W. Tobias (Wiley-Interscience, New York, 1973), p. 179.

1969 (1)

R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
[Crossref]

1964 (1)

Koenig, W.

W. Koenig, “Elektromagnetische Lichttheorie,” in Handbuch der Physik, edited by H. Geiger and K. Scheel (Springer, Berlin, 1928), Vol. XX, p. 242.

Muller, R. H.

R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
[Crossref]

R. H. Muller, “Thickness measurement of transparent thin films on metal surfaces by light interference,” J. Opt. Soc. Am. 54, 419–420 (1964).
[Crossref]

R. H. Muller, “Principles of Ellipsometry,” in Optical Techniques in Electrochemistry, edited by R. H. Muller, Vol. 9 of Advances in Electrochemistry and Electrochemical Engineering, edited by P. Delahay and C. W. Tobias (Wiley-Interscience, New York, 1973), p. 179.

Vasicek, A.

A. Vasicek, Optics of Thin Films (North-Holland, Amsterdam, 1960), p. 312.

J. Opt. Soc. Am. (1)

Surf. Sci. (1)

R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
[Crossref]

Other (3)

W. Koenig, “Elektromagnetische Lichttheorie,” in Handbuch der Physik, edited by H. Geiger and K. Scheel (Springer, Berlin, 1928), Vol. XX, p. 242.

R. H. Muller, “Principles of Ellipsometry,” in Optical Techniques in Electrochemistry, edited by R. H. Muller, Vol. 9 of Advances in Electrochemistry and Electrochemical Engineering, edited by P. Delahay and C. W. Tobias (Wiley-Interscience, New York, 1973), p. 179.

A. Vasicek, Optics of Thin Films (North-Holland, Amsterdam, 1960), p. 312.

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Figures (8)

FIG. 1
FIG. 1

Interference in transparent film on absorbing substrate. Definition of terms: Refractive indices of incident medium, n0; of film, nf; and of substrate, n ˆ. Film thickness, d; angles of incidence ϕ and refraction ϕ; reflection coefficients for top and bottom film interfaces, r ˆ 1 and r ˆ 2.

FIG. 2
FIG. 2

Reflection of monochromatic light (wavelength 570 nm) from a film-covered surface. Locus of interference minima (dotted lines) and maxima (dashed lines) as a function of optical path length (2nfdcos ϕ) in the film and phase change (δ1δ2) due to reflection. The reflectance (solid curves) is shown for 0° and 180° phase change and different amplitude reflection coefficients (rr1 = r2).

FIG. 3
FIG. 3

Optimum angle of incidence for monochromatic thin-film interference. Dielectric film of low refractive index (nf = 1.35) on absorbing substrates of complex refractive index n ˆ = n i k, s-polarized light.

FIG. 4
FIG. 4

Optimum angle of incidence for monochromatic thin-film interference. Dielectric film of low refractive index (nf = 1.35) on absorbing substrates of complex refractive index n ˆ = n i k, p-polarized light, Brewster’s angle ϕp = 53.5°.

FIG. 5
FIG. 5

Optimum angle of incidence for monochromatic thin-film interference. Dielectric film of high refractive index (nf = 2.0) on absorbing substrates of complex refractive index n ˆ = n i k, s-polarized light.

FIG. 6
FIG. 6

Optimum angle of incidence for monochromatic thin-film interference. Dielectric film of high refractive index (nf = 2.0) on absorbing substrates of complex refractive index n ˆ = n i k, p-polarized light, Brewster’s angle ϕp = 63.5°.

FIG. 7
FIG. 7

Interference in a tapered cryolite film (nf = 1.34) on a silicon substrate ( n ˆ = 4.14 0.03 i ). Photometrically determined (circles) and computed (curve) fringe visibility as a function of angle of incidence, λ = 546 nm, s polarization.

FIG. 8
FIG. 8

Interference in a tapered cryolite film (nf = 1.34) on a silicon substrate ( n ˆ = 4.14 0.03 i ). Photometrically determined (circles) and computed (curve) fringe visibility as a function of angle of incidence, λ = 546 nm, p-polarization.

Equations (24)

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r ˆ 1 = r 1 e i δ 1 ,
r ˆ 2 = r 2 e i δ 2 ,
δ = ( 4 π / λ ) n f d cos ϕ ,
R = r 1 2 + r 2 2 + 2 r 1 r 2 cos ( δ 1 δ 2 + δ ) 1 + r 1 2 r 2 2 + 2 r 1 r 2 cos ( δ 1 δ 2 + δ ) .
δ 1 δ 2 + δ = m π ,
m = ± 1 , ± 3 , ± 5 etc . minima ,
m = 0 , ± 2 , ± 4 etc . maxima .
R min = ( r 1 r 2 1 r 1 r 2 ) 2 ,
R max = ( r 1 + r 2 1 + r 1 r 2 ) 2 .
r 1 s = | n 0 cos ϕ n 1 cos ϕ n 0 cos ϕ + n 1 cos ϕ | ,
r 1 p = | n 1 cos ϕ n 0 cos ϕ n 1 cos ϕ + n 0 cos ϕ | .
ϕ = sin 1 [ ( n 0 / n f ) sin ϕ ] .
δ 1 s = π .
δ 1 p = 2 π for ϕ < ϕ p ,
δ 1 p = π for ϕ > ϕ p .
r 2 s = ( A 2 + B 2 2 A cos ϕ + cos 2 ϕ A 2 B 2 + 2 A cos ϕ + cos 2 ϕ ) 1 / 2 ,
r 2 p = r 2 s ( A 2 + B 2 2 A sin ϕ tan ϕ + sin 2 ϕ tan 2 ϕ A 2 + B 2 + 2 A sin ϕ tan ϕ + sin 2 ϕ tan 2 ϕ ) 1 / 2 ,
A = ( 1 2 n f 2 { [ ( n 2 k 2 n f 2 sin 2 ϕ ) 2 + 4 n 2 k 2 ] 1 / 2 + ( n 2 k 2 n f 2 sin 2 ϕ ) } ) 1 / 2 ,
B = ( 1 2 n f 2 { [ ( n 2 k 2 n f 2 sin 2 ϕ ) 2 + 4 n 2 k 2 ] 1 / 2 ( n 2 k 2 n f 2 sin 2 ϕ ) } ) 1 / 2 .
δ 2 s = tan 1 ( 2 B cos ϕ A 2 + B 2 cos 2 ϕ ) 0 δ s π ,
δ 2 p = tan 1 ( 2 B cos ϕ ( A 2 + B 2 sin 2 ϕ ) A 2 + B 2 ( 1 / n f 4 ) ( n 2 + k 2 ) 2 cos 2 ϕ ) .
V = R max R min R max + R min .
R min = ( r 1 r 2 1 r 1 r 2 ) 2 = 0.
for ϕ = ϕ m , r 1 = r 2 .