Principal angle, principal azimuth, and principal-angle ellipsometry of film-substrate systems
JOSA, Vol. 67, Issue 8, pp. 1058-1065 (1977)
http://dx.doi.org/10.1364/JOSA.67.001058
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Abstract
When the film thickness is considered as a parameter, a system composed of a transparent film on an absorbing substrate (in a transparent ambient) is characterized by a range of principal angle ø¯_{min} ≤ ø¯ ≤ ø¯_{max} over which the associated principal azimuth ψ¯ varies between 0° and 90° (i.e., 0° ≤ ψ¯ ≤ 90°) and the reflection phase difference Δ assumes either one of the two values: +π/2 or −π/2. We determine the principal angle ø¯(d) and principal azimuth ψ¯(d) as functions of film thickness d for the vacuum-SiO_{2}-Si system at several wavelengths as a concrete example. When the film thickness exceeds a certain minimum value, more than one principal angle becomes possible, as can be predicted by a simple graphical construction. We apply the results to principal-angle ellipsometry. (PAE) of film-substrate systems; the relationship between ø¯ and ψ¯ during film growth is particularly interesting.
© 1977 Optical Society of America
Citation
R. M. A. Azzam and A.-R. M. Zaghloul, "Principal angle, principal azimuth, and principal-angle ellipsometry of film-substrate systems," J. Opt. Soc. Am. 67, 1058-1065 (1977)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-67-8-1058
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References
- These definitions of the principal angle and the principal azimuth can be extended to include the case of an arbitrary transparent ambient. We should also mention that the value of Δ = + π/2, instead of − π/2, is based on the choice of conventions discussed in the paper by R. H. Muller, "Definitions and Conventions in Ellipsometry," Surface Sci. 16, 14–33 (1969) [also in Proceedings of the Symposium on Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North-Holland, Amsterdam, (1969)].
- See, for example, K. Kinosita and M. Yamamoto, "Principal-Angle-of-Incidence Ellipsometry," Surface Sci. 56, 64–75 (1976) [also in Proceedings of the Third International Conference on Ellipsometry, edited by N. M. Bashara and R. M. A. Azzam (North-Holland, Amsterdam, 1976)].
- P is measured from the plane of incidence, positive in a counterclockwise sense when looking into the beam.
- C. V. Kent and J. Lawson, "A photoelectric method for the determination of the parameters of elliptically polarized light," J. Opt. Soc. Am. 27, 117–144 (1937).
- H. M. O'Bryan, "The Optical Constants of Several Metals in Vacuum," J. Opt. Soc. Am. 26, 122–127 (1936).
- At this uv spectral line of mercury, the refractive indices of SiO_{2} and Si are assumed to be 1.5 and (1.67-j3. 59), respectively. [Ellipsometric Tables of the Si-SiO_{2} System for Mercury and He-Ne Laser Spectral Lines, edited by G. Gergely (Akademiai Kiado, Budapest, 1971).]
- R. M. A. Azzam, A. -R. M. Zaghloul, and N. M. Bashara, "Ellipsometric function of a film-substrate system: Applications to the design of reflection-type optical devices and to ellipsometry," J. Opt. Soc. Am. 65, 252–260 (1975).
- See Eq. (13) and Fig. 3 of Ref. 7 and also Fig. 3 (b) of this paper.
- All CAIC's between 66.3° and 84.7° that appear in Fig. 2 intersect the imaginary axis of the complex ρ plane at four points. It is clear, however, that two and three points of intersection (and tangency) will occur, e.g., at angles of incidence between 66.3° and 75°.
- A. -R. M. Zaghloul, R. M. A. Azzam, and N. M. Bashara, "Design of film-substrate single-reflection retarders," J. Opt. Soc. Am. 65, 1043–1049 (1975).
- R. M. A. Azzam, A. -R. M. Zaghloul, and N. M. Bashara, "Design of film-substrate single-reflection linear partial polarizers," J. Opt. Soc. Am. 65, 1472–1474 (1975).
- R. M. A. Azzam, A. -R. M. Zaghloul, and N. M. Bashara, "Polarizer-surface-analyzer null ellipsometry for film-substrate systems," J. Opt. Soc. Am. 65, 1464–1471 (1975).
- At these five wavelengths, the refractive indices of SiO_{2} are 1.48, 1.475, 1.47, 1.46, and 1.46, and those of Si are (5. 06-j3. 04), (6. 63-j2. 74), (5. 63-j0. 29), (4. 83-j0. 116), and (3. 85-j0. 02), respectively (after Gergely, Ref. 6).
- Exact symmetry of ø¯(d) and ψ¯(d) around the line d=d_{s}, occurs in the limit of zero absorption in the substrate.
- The reduced-thickness curve (RTC) is obtained by subtracting from the ordinate of each point on the line d = const the proper multiple of the thickness period D_{ø} that is required to bring that point vertically down below the D_{ø} boundary curve of the reduced-thickness zone (RTZ) (see the discussion in Ref. 12).
- The equations that give d when m_{1} = m_{2} and m_{1} = m_{2} + 1 are the same as Eqs. (18a) and (18b), respectively, in Ref. 12.
- See, for example, the method discussed in Sec. IV of Ref. 7.
- See, for example, M. M. Ibrahim and N. M. Bashara, "Parameter correlation and computational considerations in multiple-angle ellipsometry," J. Opt. Soc. Am. 61, 1622–1629 (1971).
- This contour is readily derived from Fig. 4 by plotting ø¯(d) vs ψ¯(d) for different values of d.
- It is interesting to observe that when the film-substrate system acts as a p or s reflection polarizer, such a condition is detected experimentally by the extinction of the reflected beam (the sample under measurement and the polarizer of the ellipsometer now operate as a pair of crossed polarizers). The null in both O'Bryan and Kent and Lawson's ellipsometers becomes due to the reflected beam being extinguished, and not because it is circularly polarized.
- Furthermore, the coincident branches also become exactly symmetrical around the ψ¯ =45° line.
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