Abstract

The pseudo-Brewster angle of minimum reflectance for the p polarization, the corresponding angle for the s polarization, and the second-Brewster angle of minimum ratio of the p and s reflectances are all determined as functions of the thickness of a transparent film coating an absorbing substrate by numerical solution of the exact equations that govern such angles of the form Re(Z′/Z) = 0, where Z = Rp, Rs, or ρ represent the complex amplitude-reflection coefficients for the p and s polarizations and their ratio (ρ = Rp/Rs), respectively, and Z′ is the angle-of-incidence derivative of Z. Results that show these angles and their associated reflectance and reflectance-ratio minima are presented for the SiO2–Si film–substrate system at wavelength λ = 0.6328 μm and film thickness of up to four periods (≃1.2 μm). Applications of these results are proposed in film-thickness measurement and control.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. M. A. Azzam, M. E. R. Khan, Appl. Opt. 22, 253 (1983).
    [CrossRef] [PubMed]
  2. W. A. Pliskin, in Progress in Analytic Chemistry, Vol. 2, E. M. Murt, W. G. Guldner, Eds. (Plenum, New York, 1969), pp. 1–34.
  3. A direct search method is used. An initial iteration step Δϕin(1°) is selected, and ϕ is scanned in one direction (from 90° to 0). A root of Eq. (2) exists between two successive ϕ values whenever its left-hand side changes sign. The root is refined by successive reduction of Δϕ (0.1,0.01,…) until the absolute value of the left-hand side of Eq. (2) falls below 10−6. The value of the function |Z| is evaluated at the root, and the search is resumed for another root.
  4. R. M. A. Azzam, J. Opt. Soc. Am. 72, 1187 (1982).
    [CrossRef]
  5. R. M. A. Azzam, Opt. Acta 30, 1113 (1983).
    [CrossRef]
  6. See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 316–317.

1983 (2)

1982 (1)

Azzam, R. M. A.

R. M. A. Azzam, M. E. R. Khan, Appl. Opt. 22, 253 (1983).
[CrossRef] [PubMed]

R. M. A. Azzam, Opt. Acta 30, 1113 (1983).
[CrossRef]

R. M. A. Azzam, J. Opt. Soc. Am. 72, 1187 (1982).
[CrossRef]

See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 316–317.

Bashara, N. M.

See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 316–317.

Khan, M. E. R.

Pliskin, W. A.

W. A. Pliskin, in Progress in Analytic Chemistry, Vol. 2, E. M. Murt, W. G. Guldner, Eds. (Plenum, New York, 1969), pp. 1–34.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Opt. Acta (1)

R. M. A. Azzam, Opt. Acta 30, 1113 (1983).
[CrossRef]

Other (3)

See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 316–317.

W. A. Pliskin, in Progress in Analytic Chemistry, Vol. 2, E. M. Murt, W. G. Guldner, Eds. (Plenum, New York, 1969), pp. 1–34.

A direct search method is used. An initial iteration step Δϕin(1°) is selected, and ϕ is scanned in one direction (from 90° to 0). A root of Eq. (2) exists between two successive ϕ values whenever its left-hand side changes sign. The root is refined by successive reduction of Δϕ (0.1,0.01,…) until the absolute value of the left-hand side of Eq. (2) falls below 10−6. The value of the function |Z| is evaluated at the root, and the search is resumed for another root.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Amplitude reflectance |Rp| for p-polarized light vs angle of incidence ϕ in degrees for the air–SiO2–Si system at λ = 632.8 nm and SiO2 film thickness of d = 3D90 = 892.3 nm.

Fig. 2
Fig. 2

(a) Pseudo-Brewster angle of absolute minimum p reflectance ϕ p B p in degrees vs film thickness d in microns for the air–SiO2–Si system at λ = 632.8 nm. (b) Associated minimum p reflectance |Rp min| vs d in microns. (c) Plot of |Rp min| vs ϕ p B p in degrees obtained by eliminating d between Figs. 2(a) and (b).

Fig. 3
Fig. 3

(a) Angular position ϕ 1 min p in degrees of the first minimum of |Rp| found under 90° vs film thickness d in microns for the air–SiO2–Si system at λ = 632.8 nm. (b) Associated minimum p reflectance | R 1 min p | vs d in microns. (c) Plot of | R 1 min p | vs ϕ 1 min p in degrees obtained by eliminating d between Figs. 3(a) and (b).

Fig. 4
Fig. 4

(a) Same as in Fig. 3(a) but for the first maximum found under 90°. (b) Associated maximum p reflectance vs | R 1 max p | d in microns.

Fig. 5
Fig. 5

(a) Pseudo-Brewster angle of absolute minimum s reflectance ϕ p B s in degrees vs film thickness d in microns for the air–SiO2–Si system at λ = 632.8 nm. (b) Associated minimum s reflectance |Rs min| vs d in microns. (c) Plot of |Rs min| vs ϕ p B s in degrees obtained by eliminating d between Figs. 5(a) and (b).

Fig. 6
Fig. 6

(a) Angular position in degrees of the first minimum of |Rs| found under 90° vs film thickness d in microns for the air–SiO2–Si system at λ = 632.8 nm. (b) Associated minimum s reflectance | R 1 min s | vs d in microns. (c) Plot of | R 1 min s | vs ϕ 1 min s in degrees obtained by eliminating d between Figs. 6(a) and (b).

Fig. 7
Fig. 7

(a) Second-Brewster angle ϕ2B in degrees of absolute minimum reflectance ratio |ρmin| vs film thickness d in microns for the air–SiO2–Si system at λ = 632.8 nm. (b) |ρmin| vs d in microns. (c) |ρmin| vs ϕ2B in degrees obtained by eliminating d between Figs. 7(a) and (b).

Fig. 8
Fig. 8

(a) Angular position ϕ1 ext in degrees of the first extremum (maximum or minimum) of the reflectance ratio |ρ| found under 90° vs film thickness in microns for the air–SiO2–Si system at λ = 632.8 nm. (b) Associated |ρ1 ext| vs d in μm. (c) Expansion of Fig. 8(b) over the first thickness period only. dp and ds are the p and s polarizing film thicknesses.

Fig. 9
Fig. 9

Same as in Fig. 1 but for d = 0.27D90 = 80.3 nm.

Fig. 10
Fig. 10

Same as in Fig. 1 but for the s polarization and d = 0.61D90 = 181.4 nm.

Tables (1)

Tables Icon

Table I Critical Points of the |Rp| vs ϕ Curve of Fig. 1 (d = 892.3 nm)

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

D ϕ = ½ λ ( N 1 2 N 0 2 sin 2 ϕ ) 1 / 2 ,
Re ( 1 Z Z ϕ ) = 0 ,
0 d 4 D 90 1.2 μ m .

Metrics