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Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Editor: James C. Wyant
  • Vol. 45, Iss. 2 — Jan. 10, 2006
  • pp: 235–264

Unified analysis and mathematical representation of film-thickness behavior of film–substrate systems

A. Rahman M. Zaghloul and Mohamed S. A. Yousef  »View Author Affiliations


Applied Optics, Vol. 45, Issue 2, pp. 235-264 (2006)
http://dx.doi.org/10.1364/AO.45.000235


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Abstract

The ellipsometric function ρ of a film–substrate system is studied as the film thickness d is kept constant and the angle of incidence ϕ is changed. The generated constant-thickness contours (CTCs) are characterized by an introduced mathematical behavior indicator that represents a group of CTCs. The behavior of each group is developed and studied in the four planes ϕ d , X , Z ,   and   ρ , where X is the film-thickness exponential function and Z is a previously introduced intermediate plane. In the ϕ d plane the film-thickness domain is identified and divided into a sequence of disconnected thickness subdomains (DTSs), depending on only N 0   and   N 1 , and their number depending on the range in which N 0 / N 1 lies. The behavior of the CTCs in the successive planes X, Z, and ρ is then studied in each DTS, and the CTC's space is divided into disconnected subfamilies according to the behavior indicator. Equivalence classes that reduce the infinite number of subfamilies into a finite number are then introduced. The transformation from each plane to the next is studied with the origin of the Z plane mapped onto the point at of the ρ plane, forming a singularity. A multiple-film-thickness inequality is derived to determine the unique solution of the film thickness. The type of reflection being internal or external at both ambient–film and film–substrate interfaces affects the analysis and is also considered. To conclude we introduce the design of polarization-preserving devices and a novel oscillating single-element ellipsometer to fully characterize zero film–substrate systems as examples of applying the knowledge developed here.

© 2006 Optical Society of America

OCIS Codes
(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry
(120.4530) Instrumentation, measurement, and metrology : Optical constants
(310.3840) Thin films : Materials and process characterization
(310.6860) Thin films : Thin films, optical properties

ToC Category:
Ellipsometry and Polarimetry

Citation
A. Rahman M. Zaghloul and Mohamed S. A. Yousef, "Unified analysis and mathematical representation of film-thickness behavior of film-substrate systems," Appl. Opt. 45, 235-264 (2006)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-45-2-235


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References

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  28. phivT is the angle of total reflection at the ambient-film interface. phivT = 43.23° when N0 = 1.46 and N1 = 1.
  29. The arc length of the exponential function of Eq. (3) as the angle of incidence changes from 0° to 90° is obtained by the definite integral ℒ=(2pid/lambda)∫phiv=0phiv=90((-2N02 sin2 phiv cos phiv)/(N12−N02 sin2 phiv)1/2) dphiv.
  30. For the case of external reflection at the ambient-film interface (N0 < N1) the ratio of any term to the one before in the sequence of Eq. (39) is less than unity. Hence it is a convergent sequence.
  31. In general, a domain is called m simply connected if the boundary of the same consists of m distinct boundaries.
  32. A singular point of a function is isolated if the function is analytic at each point in some deleted neighborhood of that point.
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  34. The cases of internal and total reflection at any or both of ambient-film and film-substrate interfaces are, however, beyond the scope of this paper and are considered elsewhere.
  35. A binary relation ℜ on a set is called an equivalence relation on tau provided the following three properties hold: (1) For all a ϵtau, (a, a) ϵ ℜ. (2) For all a and b in tau, if (a, b) ϵ ℜ, then (b, a) ϵ ℜ. (3) For all a, b, and c in tau, if (a, b) ϵ ℜ and (b, c) ϵ ℜ , then (a, c) ϵ ℜ. A relation that satisfies (1) is called reflexive. A relation that satisfies (2) is called symmetric. A relation that satisfies (3) is called transitive.
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  37. One of the possible experimental techniques for scanning the Delta behavior of the rho-CTC is to use the polarizer-surface-analyzer null ellipsometry described in detail in 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).
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