Robert W. Collins, Center for Photovoltaics Innovation and Commercialization and Department of Physics and Astronomy, University of Toledo, Toledo OH 43606
Corresponding author: [email protected]

Introduction
Thin film and multilayered materials having thicknesses on the nano to microscale (0.5 nm  5 μm) are critical components for numerous technologies. Substrates on which these materials are fabricated can range in size from a few mm^{2} for specialized single crystals  designed for lattice matching and exceptional single crystalline quality in thin film form, to m^{2} for glass plates  designed for low cost production of amorphous, nanocrystalline, or polycrystalline thin films.
Increases in the functionality of films and multilayers have led to a growing demand for probes of their thicknesses and other key properties, applicable at all technology development stages [1]. Polarized light spectroscopies, encompassing ellipsometry and polarimetry performed in reflection, serve as noninvasive optical probes of films and multilayers (see Fig. 1) [26]. Through the additional measurement of phase shift difference, polarized light spectroscopies achieve significant advantage over reflectometry.

Figure 1. Optical configurations are shown for (a) ellipsometry, and (b,c) polarimetry. Instruments include: (a) rotatingpolarizer spectroscopic ellipsometer; (b) single rotatingcompensator polarimeter for Stokes vector spectroscopy, and (c) dual rotatingcompensator polarimeter for Mueller matrix spectroscopy.
All configurations incorporate a detection system consisting of a spectrograph and photodiode array for high speed spectroscopy. Ref.: R.W. Collins, I. An, J. Li, and J.A. Zapien, “Multichannel ellipsometry”, in: H.G. Tompkins and E.A. Irene, (eds.), Handbook of Ellipsometry, (William Andrew, Norwich NY, 2005), p. 481.

Reflectometry exploits the change in real irradiance I upon reflection; however, the technique of ellipsometry exploits the change in a complex polarization state parameter ξ, which includes both a relative amplitude ξ and a phase difference δ: ξ = ξexp(iδ) [2]. Thus, the irradiance change upon reflection is characterized by the reflectance R = I_{r}/I_{i }, the ratio of the reflected (r) to incident (i) beam irradiances, whereas the polarization state change is described analogously, but as a complex number ρ = ξ_{r}/ξ_{i} = (ξ_{r}/ξ_{i})exp{i(δ_{r } δ_{i})}.
Thus, ellipsometry provides two angles, a ratio of the relative amplitudes defined (historically) by ψ = tan^{1} (ξ_{r}/ξ_{i}), and a shift (upon reflection) in the phase differences.
Δ = δ_{r } δ_{i} [2]. Ellipsometry is less sensitive to the surface defects and macroscopic roughness that may scatter power out of the optical beam because, ultimately, it is based on the measurement of the shape (not the size) of the polarization ellipse, characterized by tilt and ellipticity angles (Q, χ). The ellipsometry angle Δ also exhibits extraordinary sensitivity to film thickness. High precision ellipsometers can detect the formation of hundredths of a monolayer, while characterizing the complex dielectric function of thin films at the monolayer level [7].
Ellipsometry Measurement
The simplest ellipsometry measurement is that of a bare isotropic substrate. If the dielectric function ε_{a} of the ambient medium (typically air or vacuum) and the oblique angle of incidence θ_{i} are known, then (ψ, Δ) provide directly the real and imaginary parts (ε_{1s}, ε_{2s}) of the complex dielectric function of the substrate for the given optical wavelength λ of measurement [2]. This capability provides the unique opportunity for spectroscopic ellipsometry to measure directly the wavelength or photon energy (ħω) dependence of the complex dielectric function ε_{s}. Thus, spectroscopic ellipsometry of substrates or opaque thin films enables development of a database of material dielectric functions [8] that can then be applied to assist in the analysis of ellipsometry data on single thin films and multilayers.
The next step in the progression of ellipsometric measurement involves determination of the thickness d and the real and imaginary parts of the complex dielectric function (ε_{1f}, ε_{2f}) of a single unknown ideal, isotropic thin film on a known isotropic substrate. Because there are three unknowns in a single photon energy problem and only two data values (ψ, Δ), multiple measurements are required [2]. Because spectroscopic ellipsometry is powerful in its own right, this same multiple measurement approach is also most desirable for solving the single film data analysis problem. Although there is always one more unknown parameter, d, than measured (ψ, Δ) values, it is still possible to solve the single film problem using a single pair of (ψ, Δ) spectra [9]; however, even greater success results from analyzing multiple spectra at different θ_{i} [10].
Features in the dielectric function (ε_{1}, ε_{2}) of a substrate or thin film can be described by lineshape parameters, including amplitudes A_{n}, energies E_{n}, and broadenings Γ_{n },which provide information on physical characteristics [11,12]. For example, A_{n} reflects material density; E_{n} reflects composition, strain, and material temperature; and Γ_{n} reflects defect density or grain size, and ordering. In many cases, such information cannot be determined directly and must be established through correlations with direct measurements. If ε_{f} can be expressed accurately as an analytical function of lineshape parameters, then the single film analysis problem can be reduced to leastsquares regression  determination of only ħωindependent parameters. The validity of the bestfit model for (ε_{1f}, ε_{2f}) is evaluated using the mean square error [13].
A single thin film on a substrate can rarely be modeled adequately assuming a single layer between the semiinfinite ambient and substrate. Film surfaces and interfaces are microscopically rough to some degree with possible film/substrate interdiffusion, as well. In fact, microscopic roughness, i.e., roughness having an inplane scale much smaller than the light wavelength, can be incorporated into the optical model as one or more layers. The complex dielectric functions of these layers are modeled as effective media of the underlying and overlying materials. The Bruggeman approximation has been applied most widely for determining ε for microscopic mixtures based on the dielectric functions and volume fraction of the components [14,15]. Consequently, roughness regions can be incorporated into least squares regression through the addition of ħωindependent thickness parameters [13].
Polarimetry Measurement
The presence of a variety of nonidealities in thin films motivates spectroscopic polarimetry, the measurement of the fourcomponent Stokes vector of the optical beam before (S_{i}) and after (S_{r}) oblique reflection [16]. This vector describes not only the polarization ellipse shape (Q, χ), but also the irradiance I and degree of polarization p. I and p can provide information on macroscopic roughness, whose inplane scale is on the order of the wavelength and scatters irradiance out of the beam, as well as on nonuniformities that lead to a distribution of properties over the probed area  typically 0.1  1 cm^{2}. By expanding polarimetry to the measurement of the sample's 4x4 Mueller matrix M_{S }, anisotropic systems can be characterized  including thicknesses and principal axis complex dielectric functions of substrates and films [13,16].
Outlook
From research to manufacturing, there exists a critical need to perform polarization spectroscopies at high speed  from milliseconds to seconds (see Fig. 1). For example in research, such measurements performed in situ and in real time during thin film growth at a single spot on the substrate surface can provide critical information on the development of optimum processes (see Fig. 2). For evaluating the uniformity of optimum processes and transitioning to full scale manufacturing, ex situ mapping spectroscopies are critical, in which case measurements are performed on a grid over the full surface area of the film/substrate (see Fig. 3). Finally, online single point or mapping polarization spectroscopies can be applied to monitor production output.High speed requires integration of detector arrays into the instruments, and these challenges are compounded by the need to extract spectra in the Stokes vector of the beam or the Mueller matrix of the sample in many applications [1719]. In fact, these challenges have defined the current directions in instrumentation development [20].

Figure 2. Results are shown for (a) the effective layer thickness and (b) the real and imaginary parts of the dielectric function from a leastsquares regression analysis of spectroscopic ellipsometry data collected during the growth of a ptype amorphous silicon (aSi:H) layer component of an aSi:H nip solar cell on a polymer substrate in a rolltoroll plasma deposition system. The effective thickness is given by d_{eff} = d_{b} + f_{s}d_{s} + f_{i}d_{i} , where d_{b}, d_{s}, and d_{i} are the bulk, surface roughness, and i/pinterface roughness layer thicknesses, respectively, which are all determined in the analysis, and f_{s} and f_{i} are the player volume fractions in the surface and interface roughness layers. The dielectric function is deduced as an analytical formula (CodyLorentz expression) which provides the band gap of the player material at the deposition temperature (108°C). Ref.: L.R. Dahal, Z. Huang, D. Attygalle, M.N. Sestak, C. Salupo, S. Marsillac, and R.W. Collins, "Application of real time spectroscopic ellipsometry for analysis of rolltoroll fabrication of Si:H solar cells on polymer substrates", 35th IEEE Photovoltaics Specialists Conference, June 2025, 2010, Honolulu, HI, (IEEE, Piscataway NJ, 2010) p. 631.


Figure 3. (a) A map of the bulk layer thickness determined by ex situ spectroscopic ellipsometry at high speed is shown for a CdS thin film deposited on top of a NSGPilkington TEC15 SnO_{2}:F coated glass panel. The mapped area is 35 cm x 80 cm. The two plots (top and right) show the CdS thickness variations along the X and Y axes. The experimental (y, D) spectra and best fit results for the two indicated points are shown in (b). Experimental and best fit spectra at positions (X = 2.0, Y = 0) and (X = 21.8, Y = 0) show the effect of differences in the CdS thickness; confidence limits and mean square errors are also given.
Ref.: Z. Huang, J. Chen, M. N. Sestak, D. Attygalle, L.R. Dahal, M.R. Mapes, D.A. Strickler, K.R. Kormanyos, C. Salupo, and R. W. Collins, "Optical mapping of large area thin film solar cells", 35th IEEE Photovoltaics Specialists Conference, June 2025, 2010, Honolulu, HI, (IEEE, Piscataway NJ, 2010) p. 1678.

References
[1] I. P. Herman, Optical Diagnostics for Thin Film Processing, (Academic, New York, 1995).
[2] R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, (NorthHolland, Amsterdam, 1977).
[3] H.G. Tompkins, A User’s Guide to Ellipsometry, (Academic, New York, 1992).
[4] H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications, (John Wiley & Sons, West Sussex UK, 2007).
[5] H.G. Tompkins and E.A. Irene, (eds.), Handbook of Ellipsometry, (William Andrew, Norwich NY, 2005).
[6] Five conferences on spectroscopic ellipsometry and polarimetry have been held. The proceedings of these conferences provide snapshops of this rapidly evolving field:
(a) A. C. Boccara, C. Pickering, and J. Rivory, (eds.), Proceedings of the First International Conference on Spectroscopic Ellipsometry, (Elsevier, Amsterdam, 1993); also published as Thin Solid Films 233234 (1993);
(b) R. W. Collins, D. E. Aspnes, and E. A. Irene, (eds.), Proceedings of the Second International Conference on Spectroscopic Ellipsometry, (Elsevier, Amsterdam, 1998); also published as Thin Solid Films 313314 (1998);
(c) M. Fried, J. Humlicek, and K. Hingerl, (eds.), Proceedings of the Third International Conference on Spectroscopic Ellipsometry, (Elsevier, Amsterdam, 2003);
(d) H. Arwin, U. Beck, and M. Schubert, (eds.), Proceedings of the Fourth International Conference on Spectroscopic Ellipsometry, (WileyVCH, Weinheim Germany, 2007);
(e) H.G. Tompkins et al., (eds.), Proceedings of the Fifth International Conference on Spectroscopic Ellipsometry, (Elsevier, Amsterdam, 2011, in press).
[7] D.E. Aspnes and A.A. Studna, “High precision scanning ellipsometer”, Appl. Opt. 14, 220 (1975).
[8] Handbooks tabulate the ħωdependence of the optical properties of solids. Data are presented in terms of the real and imaginary parts of the complex dielectric function (ε_{1}, ε_{2}), or the index of refraction and extinction coefficient (n, k). The two pairs are related according to ε_{1} = n^{2}  k^{2} and ε_{2} = 2nk. See, for example:
(a) E.D. Palik, Handbook of Optical Constants of Solids, (Academic, New York 1985); (b) E.D. Palik, Handbook of Optical Constants of Solids II, (Academic, New York, 1991).
[9] D.E. Aspnes, “Spectroscopic ellipsometry”, in: B.O. Seraphin (ed.), Optical Properties of Solids: New Developments, (NorthHolland, Amsterdam, 1976) p. 799.
[10] J.A. Woollam, B. Johs, C.M. Herzinger, J.N. Hilfiker, R. Synowicki, and C. Bungay, “Overview of variable angle spectroscopic ellipsometry (VASE), Parts I and II”, Proc. Soc. PhotoOpt. Instrum. Eng. Crit. Rev. 72, 3 (1999).
[11] F. Wooten, Optical Properties of Solids (Academic, New York, 1972).
[12] R.W. Collins and A.S. Ferlauto, “Optical physics of materials”, in: H.G. Tompkins and E.A. Irene, (eds.), Handbook of Ellipsometry, (William Andrew, Norwich NY, 2005), p. 93.
[13] G.E. Jellison, Jr., “Data analysis for spectroscopic ellipsometry”, in: H.G. Tompkins and E.A. Irene, (eds.), Handbook of Ellipsometry, (William Andrew, Norwich NY, 2005), p. 237.
[14] D.E. Aspnes, “Optical properties of thin films”, Thin Solid Films 89, 249 (1982).
[15] H. Fujiwara, J. Koh, P.I. Rovira, and R.W. Collins, “Assessment of effectivemedium theories in the analysis of nucleation and microscopic surface roughness evolution for semiconductor thin films”, Phys. Rev. B 61, 10832 (2000).
[16] P. S. Hauge, “Recent developments in instrumentation in ellipsometry”, Surf. Sci. 96, 108 (1980).
[17] I. An, Y.M. Li, H.V. Nguyen, and R.W. Collins, “Spectroscopic ellipsometry on the millisecond time scale for realtime investigations of thin film and surface phenomena”, Rev. Sci. Instrum. 63, 3842 (1992).
[18] J. Lee, P.I. Rovira, I. An, and R.W. Collins, “Rotating compensator multichannel ellipsometry: applications for real time Stokes vector spectroscopy of thin film growth”, Rev. Sci. Instrum. 69, 1800 (1998).
[19] J. Lee, J. Koh, and R. W. Collins, "Dual rotatingcompensator multichannel ellipsometer: instrument development for highspeed Mueller matrix spectroscopy of surfaces and thin films", Rev. Sci. Instrum. 72, 1742 (2001).
[20] R.W. Collins, I. An, J. Li, and J.A. Zapien, “Multichannel ellipsometry”, in: H.G. Tompkins and E.A. Irene, (eds.), Handbook of Ellipsometry, (William Andrew, Norwich NY, 2005), p. 481.
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