Abstract

The differential reflection phase shift, Δ = δpδs, associated with the external reflection of monochromatic light at the surface of an absorbing medium is a monotonically decreasing function of the angle of incidence ϕ which is determined by the complex dielectric function . A new special angle of incidence, denoted by ϕΔ′max, is defined at which the slope Δ′ = ∂Δ/∂ϕ of the Δ–ϕ curve is maximum negative, Δmax, and a transcendental equation is derived that determines this angle. ϕΔ′max differs from the principal angle ϕp at which Δ = 90°. As an example, ϕΔ′ max is calculated by numerical iteration for light reflection at the air–Si interface for photon energies from 1.7 to 5.6 eV in steps of 0.1 eV, and is plotted, along with the associated maximum slope Δmax, vs wavelength λ. It is noted that ϕΔ′ max > ϕp at every λ, a result that may hold in general. Also, for 4.5 ≤ ≤ 5.6 eV, ϕΔ′ max = 90°, so that a maximum negative slope occurs at grazing incidence in this spectral range. Another interesting observation is that, when || ≫ 1 (e.g., for metals in the IR), Δ′(90°) is a direct measure of the extinction coefficient k = Im1/2.

© 1989 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. R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surf. Sci. 16, 14 (1969).
    [CrossRef]
  3. R. M. A. Azzam, “AIDER: Angle-of Incidence Derivative Ellipsometry and Reflectometry,” Opt. Commun. 16, 153 (1976).
    [CrossRef]
  4. V. M. Bermudez, “AIDER (Angle-of Incidence Derivative Ellipsometry and Reflectometry): Implementation and Application,” Surf. Sci. 94, 29 (1980).
    [CrossRef]
  5. R. M. A. Azzam, “Stationary Property of Normal-Incidence Reflection from Isotropic Surfaces,” J. Opt. Soc. Am. 72, 1187 (1982).
    [CrossRef]
  6. D. E. Aspnes, A. A. Studna, “Dielectric Functions and Optical Parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985 (1983).
    [CrossRef]
  7. See, for example, J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, Eds. (McGraw-Hill, New York, 1978).

1983 (1)

D. E. Aspnes, A. A. Studna, “Dielectric Functions and Optical Parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985 (1983).
[CrossRef]

1982 (1)

1980 (1)

V. M. Bermudez, “AIDER (Angle-of Incidence Derivative Ellipsometry and Reflectometry): Implementation and Application,” Surf. Sci. 94, 29 (1980).
[CrossRef]

1976 (1)

R. M. A. Azzam, “AIDER: Angle-of Incidence Derivative Ellipsometry and Reflectometry,” Opt. Commun. 16, 153 (1976).
[CrossRef]

1969 (1)

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

Aspnes, D. E.

D. E. Aspnes, A. A. Studna, “Dielectric Functions and Optical Parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985 (1983).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, “Stationary Property of Normal-Incidence Reflection from Isotropic Surfaces,” J. Opt. Soc. Am. 72, 1187 (1982).
[CrossRef]

R. M. A. Azzam, “AIDER: Angle-of Incidence Derivative Ellipsometry and Reflectometry,” Opt. Commun. 16, 153 (1976).
[CrossRef]

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).

Bennett, H. E.

See, for example, J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, Eds. (McGraw-Hill, New York, 1978).

Bennett, J. M.

See, for example, J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, Eds. (McGraw-Hill, New York, 1978).

Bermudez, V. M.

V. M. Bermudez, “AIDER (Angle-of Incidence Derivative Ellipsometry and Reflectometry): Implementation and Application,” Surf. Sci. 94, 29 (1980).
[CrossRef]

Muller, R. H.

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

Studna, A. A.

D. E. Aspnes, A. A. Studna, “Dielectric Functions and Optical Parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985 (1983).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

R. M. A. Azzam, “AIDER: Angle-of Incidence Derivative Ellipsometry and Reflectometry,” Opt. Commun. 16, 153 (1976).
[CrossRef]

Phys. Rev. B (1)

D. E. Aspnes, A. A. Studna, “Dielectric Functions and Optical Parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985 (1983).
[CrossRef]

Surf. Sci. (2)

V. M. Bermudez, “AIDER (Angle-of Incidence Derivative Ellipsometry and Reflectometry): Implementation and Application,” Surf. Sci. 94, 29 (1980).
[CrossRef]

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

Other (2)

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

See, for example, J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, Eds. (McGraw-Hill, New York, 1978).

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Figures (6)

Fig. 1
Fig. 1

Differential reflection phase shift Δ vs angle of incidence ϕ for a dielectric–dielectric interface (curve a) and dielectric–absorbing medium interface (curve b).

Fig. 2
Fig. 2

(a) Δ vs ϕ curves for the air–Si interface at photon energies = 2.0,2.5, and 3 eV as marked by each curve. (b) Derivative Δ′ = ∂Δ/∂ϕ of the curves in (a) plotted vs ϕ.

Fig. 3
Fig. 3

Same as Fig. 2 except that here = 3.5 and 4 eV.

Fig. 4
Fig. 4

Same as Fig. 2 except that here = 4.5, 5.0, and 5.5.eV.

Fig. 5
Fig. 5

Angle of incidence, ϕΔ′max , of the maximum negative rate of change of Δ with respect to ϕ plotted vs wavelength λ for light reflection at the air–Si interface.

Fig. 6
Fig. 6

Maximum negative rate of change of Δ with respect to ϕ, Δ′max plotted vs wavelength λ for the air–Si interface.

Tables (1)

Tables Icon

Table I Maximum Slope of the Δ-vs-ϕ Curve, Δ max , and the Angle of Incidence at Which it Occurs, ϕΔ′max, for Si at Different Photon Energiesa

Equations (26)

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ρ = ( A B ) / ( A + B ) ,
A = sin 2 ϕ , B = cos ϕ ( sin 2 ϕ ) 1 / 2 ,
= 1 / 0 = r j i
ρ = | ρ |  exp ( j Δ ) .
Δ = Δ / ϕ ,
Δ = 2 Δ / ϕ 2 = 0 .
ρ / ρ = ( | ρ | / | ρ | ) + j Δ ,
Δ = Im F ,
F = ρ / ρ .
Δ =  Im G ,
G = F = ( ρ ρ ρ 2 ) / ρ 2 .
F = ( A B A B ) / ( A 2 B 2 ) ,
G = ( A 2 B 2 ) ( A B A B ) 2 ( A B A B ) ( A A B B ) ( A 2 B 2 ) 2 .
A = sin 2 ϕ , B =  sin ϕ ( sin 2 ϕ ) 1 / 2 ( +  cos 2 ϕ ) , A = 2  cos 2 ϕ , B = cos ϕ ( sin 2 ϕ ) 3 / 2 [ 2 + ( 1 6 sin 2 ϕ ) + 4 sin 2 ϕ ] .
Im [ G ( ϕ ) ] = 0.
F ( 0 ) = 0 ,
Δ ( 0 ) = 0
G ( 0 ) = 2 1 / 2 ,
Δ ( 0 ) = 2 Im 1 / 2
F ( 90 ° ) = 2 ( 1 ) 1 / 2 ,
Δ ( 90 ° ) = 2  Im ( 1 ) 1 / 2 ,
G ( 90 ° ) = 0 ,
Δ ( 90 ° ) = 0.
Δ max  | ϕ   =   90 ° = 2  Im ( 1 ) 1 / 2 .
Δ max | ϕ   =   90 ° = 2  Im 1 / 2 = 2 k ,
N = 1 / 2 = n j k .

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