Abstract

The ratio ρt = Tp/Ts of the complex amplitude transmission coefficients for the p and s polarizations of a transparent unbacked or embedded thin film is examined as a function of the film thickness-to-wavelength ratio d/λ and the angle of incidence ϕ for a given film refractive index N. The maximum value of the differential transmission phase shift (or retardance), Δt = argρt, is determined, for given N and ϕ, by a simple geometrical construction that involves the iso–ϕ circle locus of ρt in the complex plane. The upper bound on this maximum equals arctan{[N − (1/N)]/2} and is attained in the limit of grazing incidence. An analytical noniterative method is developed for determining N and d of the film from ρt measured by transmission ellipsometry (TELL) at ϕ = 45°. An explicit expression for Δt of an ultrathin film, d/λ ≪ 1, is derived in product form that shows the dependence of Δt on N, ϕ, and d/λ separately. The angular dependence is given by an obliquity factor, fo(ϕ) = 21/2 sinϕ tanϕ, which is verified experimentally by TELL measurements on a stable planar soap film in air at λ = 633 nm. The singularity of fo at ϕ = 90° is resolved; Δt is shown to have a maximum just short of grazing incidence and drops to 0 at ϕ = 90°. Because N and d/λ are inseparable for an ultrathin film, N is determined by a Brewster angle measurement and d/λ is subsequently obtained from Δt. Finally, the ellipsometric function in reflection ρr is related to that in transmission ρt.

© 1991 Optical Society of America

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References

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  1. R. M. A. Azzam, “Simple and Direct Determination of Complex Refractive Index and Thickness of Unsupported or Embedded Thin Films by Combined Reflection and Transmission Ellipsometry at 45° Angle of Incidence,” J. Opt. Soc. Am 73, 1080–1082 (1983); “Ellipsometry of Unsupported or Embedded Thin Films,” J. Phys. (Paris) Colloq. 44, C10-67–C10-70 (1983).
    [CrossRef]
  2. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  3. See, for example, W. R. LePage, Complex Variables and the Laplace Transform for Engineers (Dover, New York, 1961), Sec. 3.10.
  4. R. M. A. Azzam, “Polar Curves for Transmission Ellipsometry,” Opt. Commun. 14, 145–147 (1975).
    [CrossRef]
  5. See, for example, A. E. Taylor, Calculus with Analytic Geometry (Prentice Hall, Englewood Cliffs, NJ, 1959), Ch. 7.
  6. D. A. Holmes, “Wave Optics Theory of Rotary Compensators,” J. Opt. Soc. Am. 54, 1340–1347 (1964).
    [CrossRef]
  7. R. M. A. Azzam, A.-R. M. Zaghloul, “Determination of the Refractive Index and Thickness of a Transparent Film on a Transparent Substrate from the Angles of Incidence of Zero Reflection-Induced Ellipticity,” Opt. Commun. 24, 351–354 (1978).
    [CrossRef]
  8. A. R. Reinberg, “Ellipsometer Data Analysis with a Small Programmable Desk Calculator,” Appl. Opt. 11, 1273–1274 (1972).
    [CrossRef] [PubMed]
  9. F. Abelès, “Un Théoreme relatif à la réflexion metallique,” C. R. Acad. Sci. 220, 1942–1943 (1950).
  10. Wonder Bubbles, Chemtoy Corp., Chicago, IL 60624. At normal room temperature and pressure, the stable film can stay intact, without rupture, for 15 min or longer.
  11. R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surf. Sci. 16, 14–33 (1969).
    [CrossRef]
  12. This is not the case for a partially transmitting thin film with k ≠ 0. Even with a numerically small k value (e.g., 0.01), the difference Δr − Δt assumes anomalously large values in the immediate neighborhood of the pseudo-Brewster angle.

1983

R. M. A. Azzam, “Simple and Direct Determination of Complex Refractive Index and Thickness of Unsupported or Embedded Thin Films by Combined Reflection and Transmission Ellipsometry at 45° Angle of Incidence,” J. Opt. Soc. Am 73, 1080–1082 (1983); “Ellipsometry of Unsupported or Embedded Thin Films,” J. Phys. (Paris) Colloq. 44, C10-67–C10-70 (1983).
[CrossRef]

1978

R. M. A. Azzam, A.-R. M. Zaghloul, “Determination of the Refractive Index and Thickness of a Transparent Film on a Transparent Substrate from the Angles of Incidence of Zero Reflection-Induced Ellipticity,” Opt. Commun. 24, 351–354 (1978).
[CrossRef]

1975

R. M. A. Azzam, “Polar Curves for Transmission Ellipsometry,” Opt. Commun. 14, 145–147 (1975).
[CrossRef]

1972

1969

R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surf. Sci. 16, 14–33 (1969).
[CrossRef]

1964

1950

F. Abelès, “Un Théoreme relatif à la réflexion metallique,” C. R. Acad. Sci. 220, 1942–1943 (1950).

Abelès, F.

F. Abelès, “Un Théoreme relatif à la réflexion metallique,” C. R. Acad. Sci. 220, 1942–1943 (1950).

Azzam, R. M. A.

R. M. A. Azzam, “Simple and Direct Determination of Complex Refractive Index and Thickness of Unsupported or Embedded Thin Films by Combined Reflection and Transmission Ellipsometry at 45° Angle of Incidence,” J. Opt. Soc. Am 73, 1080–1082 (1983); “Ellipsometry of Unsupported or Embedded Thin Films,” J. Phys. (Paris) Colloq. 44, C10-67–C10-70 (1983).
[CrossRef]

R. M. A. Azzam, A.-R. M. Zaghloul, “Determination of the Refractive Index and Thickness of a Transparent Film on a Transparent Substrate from the Angles of Incidence of Zero Reflection-Induced Ellipticity,” Opt. Commun. 24, 351–354 (1978).
[CrossRef]

R. M. A. Azzam, “Polar Curves for Transmission Ellipsometry,” Opt. Commun. 14, 145–147 (1975).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Holmes, D. A.

LePage, W. R.

See, for example, W. R. LePage, Complex Variables and the Laplace Transform for Engineers (Dover, New York, 1961), Sec. 3.10.

Muller, R. H.

R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surf. Sci. 16, 14–33 (1969).
[CrossRef]

Reinberg, A. R.

Taylor, A. E.

See, for example, A. E. Taylor, Calculus with Analytic Geometry (Prentice Hall, Englewood Cliffs, NJ, 1959), Ch. 7.

Zaghloul, A.-R. M.

R. M. A. Azzam, A.-R. M. Zaghloul, “Determination of the Refractive Index and Thickness of a Transparent Film on a Transparent Substrate from the Angles of Incidence of Zero Reflection-Induced Ellipticity,” Opt. Commun. 24, 351–354 (1978).
[CrossRef]

Appl. Opt.

C. R. Acad. Sci.

F. Abelès, “Un Théoreme relatif à la réflexion metallique,” C. R. Acad. Sci. 220, 1942–1943 (1950).

J. Opt. Soc. Am

R. M. A. Azzam, “Simple and Direct Determination of Complex Refractive Index and Thickness of Unsupported or Embedded Thin Films by Combined Reflection and Transmission Ellipsometry at 45° Angle of Incidence,” J. Opt. Soc. Am 73, 1080–1082 (1983); “Ellipsometry of Unsupported or Embedded Thin Films,” J. Phys. (Paris) Colloq. 44, C10-67–C10-70 (1983).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

R. M. A. Azzam, A.-R. M. Zaghloul, “Determination of the Refractive Index and Thickness of a Transparent Film on a Transparent Substrate from the Angles of Incidence of Zero Reflection-Induced Ellipticity,” Opt. Commun. 24, 351–354 (1978).
[CrossRef]

R. M. A. Azzam, “Polar Curves for Transmission Ellipsometry,” Opt. Commun. 14, 145–147 (1975).
[CrossRef]

Surf. Sci.

R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surf. Sci. 16, 14–33 (1969).
[CrossRef]

Other

This is not the case for a partially transmitting thin film with k ≠ 0. Even with a numerically small k value (e.g., 0.01), the difference Δr − Δt assumes anomalously large values in the immediate neighborhood of the pseudo-Brewster angle.

See, for example, A. E. Taylor, Calculus with Analytic Geometry (Prentice Hall, Englewood Cliffs, NJ, 1959), Ch. 7.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

See, for example, W. R. LePage, Complex Variables and the Laplace Transform for Engineers (Dover, New York, 1961), Sec. 3.10.

Wonder Bubbles, Chemtoy Corp., Chicago, IL 60624. At normal room temperature and pressure, the stable film can stay intact, without rupture, for 15 min or longer.

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Figures (10)

Fig. 1
Fig. 1

Reflection and transmission of light by an unbacked or embedded thin film at oblique incidence.

Fig. 2
Fig. 2

Constant-angle-of-incidence contours (CAICs) of the transmission ellipsometric function ρt for a glass film (N = 1.5) in air as a coaxial family of circles in the complex plane. The angle of incidence ϕ takes values from 10 to 80° in steps of 10°.

Fig. 3
Fig. 3

Geometrical construction for finding the maximum differential transmission phase retardance, Δtmax, associated with the passage of light through a transparent film (or dielectric slab) of a given refractive index and at a given angle of incidence. See text.

Fig. 4
Fig. 4

Maximum transmission phase retardance, Δ ^ t max, of a transparent film or dielectric slab as a function of the angle of incidence ϕ for three values of the film refractive index: 1.5, 2.5, and 4.

Fig. 5
Fig. 5

Upper bound on the transmission phase retardance, Δ ^ t max, in the limit of grazing incidence, for a transparent film or dielectric slab as a function of the film refractive index N.

Fig. 6
Fig. 6

Least normalized film thickness ζ, which is required to produce the maximum phase retardance values indicated in Fig. 4, plotted vs the angle of incidence ϕ.

Fig. 7
Fig. 7

Family of constant thickness contours (CTCs) of the transmission ellipsometric function ρt in the complex plane for a glass film (N = 1.5) in air. The film thickness as a fraction of λ/2 takes values from 0.1 to 1 in steps of 0.1.

Fig. 8
Fig. 8

Obliquity factor fo(ϕ) as a function of the angle of incidence ϕ. The continuous line is calculated from Eq. (28), whereas the points marked by X are obtained from ellipsometric measurements of the transmission phase retardance of a stable planar soap film in air at λ = 633 nm.

Fig. 9
Fig. 9

Transmission phase retardance, Δt, at large angles of incidence, 80 ≤ ϕ ≤ 90°, for a glass film (N = 1.5) with d/λ = 0.01 in air. Instead of the singularity predicted by the obliquity factor, Eq. (28), Δt reaches a maximum at an angle ~1° away from grazing incidence and drops to 0 at exactly ϕ = 90°.

Fig. 10
Fig. 10

Index factor, Eq. (30).

Equations (36)

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ρ t = [ ( 1 - r p 2 ) / ( 1 - r s 2 ) ] [ ( 1 - r s 2 X ) / ( 1 - r p 2 X ) ] ,
X = exp [ - j 4 π ( d / λ ) S 1 ] ,
S 1 = ( N 1 2 - N 0 2 sin 2 ϕ ) 1 / 2 .
N 1 > N 0 sin ϕ ,
X = exp ( - j 2 π ζ ) .
ζ = d / D ϕ ,
D ϕ = λ / 2 S 1
N = N 1 / N 0 ,
ρ t + = 1 ,
ρ t - = [ ( 1 - r p 2 ) / ( 1 + r p 2 ) ] / [ ( 1 - r s 2 ) / ( 1 + r s 2 ) ] .
tan Δ t max = C T / O T = C T / ( O A O B ) 1 / 2 = C T / O B 1 / 2 ,
Δ t max = arctan [ ( r s 2 - r p 2 ) / ( 1 - r s 4 ) 1 / 2 ( 1 - r p 4 ) 1 / 2 ] .
Δ ^ t max = arctan { [ N - ( 1 / N ) ] / 2 } .
ζ = ( 1 / 2 π ) arc cos [ ( r p 2 + r s 2 ) / ( 1 + r p 2 r s 2 ) ]
X = [ ρ t ( 1 - r s 2 ) - ( 1 - r p 2 ) ] / [ ρ t r p 2 ( 1 - r s 2 ) - r s 2 ( 1 - r p 2 ) ]
X = 1 ,
ρ t - ( 1 + R s ) = R s 2 ρ t - R s ( 1 + R s ) ,
R s = r s 2 ,
ρ t = a + j b ,
= tan ψ t ( cos Δ t + j sin Δ t ) ,
R s 2 - 2 μ R s + 1 = 0 ,
μ = ( a - 1 ) / [ ( a - 1 ) 2 + b 2 ] .
R s = μ ± ( μ 2 - 1 ) 1 / 2 ,
N = ( 1 + R s ) 1 / 2 / ( 1 + R s 1 / 2 ) .
d = { [ - arg X / ( 2 π ) ] + m } D ϕ ,
Δ t = 2 π ζ { [ r s 2 / ( 1 - r s 2 ) ] - [ r p 2 / ( 1 - r p 2 ) ] } ,
Δ t / ( 2 π ) = [ ( N 1 2 - N 0 2 ) 2 / 2 N 0 N 1 2 ] ( sin ϕ tan ϕ ) ( d / λ ) .
f o ( ϕ ) = Δ t ( ϕ ) / Δ t ( 45 ° ) ,
= 2 1 / 2 sin ϕ tan ϕ .
f n ( N ) = [ Δ t ( 45 ° ) / 2 π ] / ( d / λ ) ,
= ( 1 / 2 1.5 ) [ N - ( 1 / N ) ] 2 ,
ρ r = G ( N , ϕ ) ρ t ,
G ( N , ϕ ) = ( r p / r s ) [ ( 1 - r s 2 ) / ( 1 - r p 2 ) ]
Δ r = Δ t ± π ,             ϕ < ϕ B ,
Δ r = Δ t ,             ϕ > ϕ B .
tan ψ r = G tan ψ t ,

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