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

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 *X* is the film-thickness exponential function
and *Z* is a previously introduced intermediate plane. In the *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

- See, for example, D. Goldstein,Polarized Light (Marcel Dekker, 2003).
- H. Zhu, L. Liu, Y. Wen, Z. Lu, and B. Zhang, "High-precision system for automatic null ellipsometric measurement," Appl. Opt. 41, 4536-4540 (2002). [CrossRef] [PubMed]
- 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). [CrossRef]
- A. R. M. Zaghloul, "Ellipsometric function of a film-substrate system: applications to the design of reflection-type optical devices and to ellipsometry," Ph.D dissertation (University of Nebraska-Lincoln, 1975).
- R. M. A. Azzam, "Ellipsometry of transparent films on transparent substrate," Surf. Sci. 96, 67-80 (1980). [CrossRef]
- R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977), Sec. 4.3.
- W. H. Weedon, S. W. McKnight, and A. J. Devaney, "Selection of optimal angles for inversion of multiple-angle ellipsometry and reflectometry equations," J. Opt. Soc. Am. A 8, 1881-1891 (1991). [CrossRef]
- S. F. Nee and H. E. Bennett, "Accurate null polarimetry for measuring the refractive index of transparent materials," J. Opt. Soc. Am. A 10, 2076-2083 (1993). [CrossRef]
- S. Li, "Jones-matrix analysis with Pauli matrices: application to ellipsometry," J. Opt. Soc. Am. A 17, 920-926 (2000). [CrossRef]
- F. Sagnard, F. Bentabet, and C. Vignat, "Theoretical study of method based on ellipsometry for measurement of complex permittivity of materials," Electron. Lett. 36, 1843-1845 (2000). [CrossRef]
- M. S. A. Yousef and A.-R. M. Zaghloul, "Ellipsometric function of a film-substrate system: characterization and detailed study," J. Opt. Soc. Am. A 6, 355-366 (1989). [CrossRef]
- M. Ghezzo, "Thickness calculations for a transparent film from ellipsometric measurements," J. Opt. Soc. Am. 58, 362-372 (1968). [CrossRef]
- R. M. A. Azzam and A.-R. M. Zaghloul, "Determination of the refractive index and thickness of a transparent film on a transparent substrate system from the angles of incidence of zero reflection-induced ellipticity," Opt. Commun. 24, 351-353 (1978). [CrossRef]
- A.-R. M. Zaghloul, R. M. A. Azzam, and N. M. Bashara, "Inversion of the nonlinear equations of reflection ellipsometry on film-substrate systems," Surf. Sci. 5, 87-96 (1976). [CrossRef]
- Y. Yoriume, "Method of numerical inversion of the ellipsometry equation for transparent film," J. Opt. Soc. Am. 73, 888-891 (1983). [CrossRef]
- K. Vedam, R. Rai, F. Lukes, and R. Srinivason, "Simultaneous and independent determination of the refractive index and the thickness of thin films by ellipsometry," J. Opt. Soc. Am. 58, 526-532 (1968). [CrossRef]
- D. A. Tonova, "Inverse profiling by ellipsometry: a Newton-Kantorovitch algorithm," Opt. Commun. 105, 104-112 (1994). [CrossRef]
- T. Easwarakhanthan, S. Ravelet, and P. Renard, "An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates," Appl. Surf. Sci. 90, 251-259 (1995). [CrossRef]
- S. Bosch, J. Perez, and A. Canillas, "Numerical algorithm for spectroscopic ellipsometry of thick transparent films," Appl. Opt. 37, 1177-1179 (1998). [CrossRef]
- A.-R. M. Zaghloul and M. S. A. Yousef, "Ellipsometric function of a film-substrate system: detailed analysis and closed-form inversion," J. Opt. Soc. Am. A 16, 2029-2044 (1999). [CrossRef]
- M. Elshazly-Zaghloul and A.-R. M. Zaghloul, "Closed-form inversion of the ellipsometric function of a film-substrate system: absorbing-substrate optical constant," J. Opt. Soc. Am. A 22, 1630-1636 (2005). [CrossRef]
- S. C. Warnick and M. A. Dahleh, "Ellipsometry as a sensor technology for the control of deposition processes," in Proceedings of the 37th IEEE Control System Society (Institute of Electrical and Electronics Engineers, 1998), Vol. 3, pp. 3162-3167.
- I. G. Rosen, T. Parent, B. Fidan, C. Wang, and A. Madhukar, "Design, development, and testing of real-time feedback controllers for semiconductor etching processes using in situ spectroscopic ellipsometry sensing," IEEE Trans. Control Syst. Technol. 10, 64-75 (2002). [CrossRef]
- W. G. Chinn and N. E. Steenrod, First Concepts of Topology: the Geometry of Mappings of Segments, Curves, Circles, and Disks (Random House, 1966).
- M. Holtz, K. Steffens, and E. Weitz, Introduction to Cardinal Arithmetic (Birkhaeuser, 1999).
- [x] is defined as the greatest integer that is ≤x (the step function).
- A domain [a, b] is divided into a sequence D(i) of disconnected subdomains if ∪D(i) = [a,b] and D(i) ∩ D(j) = { } ∀i ≠ j. Rearrangement may be ascending or descending.
- phivT is the angle of total reflection at the ambient-film interface. phivT = 43.23° when N0 = 1.46 and N1 = 1.
- 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.
- 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.
- In general, a domain is called m simply connected if the boundary of the same consists of m distinct boundaries.
- A singular point of a function is isolated if the function is analytic at each point in some deleted neighborhood of that point.
- A limit of a function f(x) at a point x = xi exists if and only if limx-->xi−0f(x)=limx-->xi+0f(x)=f(xi).
- 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.
- 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.
- The determination of drphiv is described in detail in Refs. 3, 4, and 14. See also, D. A. Holmes, "On the calculation of thin-film refractive index and thickness by ellipsometry," Appl. Opt. 6, 168-169 (1967).
- 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).
- A. R. M. Zaghloul, D. A. Keeling, W. A. Berzett, and J. S. Mason, "Design of reflection retarders by use of nonnegative film-substrate systems," J. Opt. Soc. Am. A 22, 1637-1645, (2005). [CrossRef]
- A. R. M. Zaghloul, M. Elshazly-Zaghloul, W. A. Berzett, and D. A. Keeling, "Thin-film coatings: an ellipsometric function approach I. Nonnegative transmission systems, polarization devices, coatings, and closed-form design formulas," submitted to J. Opt. Soc. Am. A .
- R. M. A. Azzam, "Simultaneous reflection and refraction of light without change of polarization by a single-layer-coated dielectric surface," Opt. Lett. 10, 107-109 (1985). [CrossRef] [PubMed]

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