Tackling the inverse problem in ellipsometry: analytic expressions for supported coatings with nonuniform refractive index profiles in the thin film and weak contrast limits

Ellipsometry is a powerful tool for the evaluation of the refractive index profile of a film or coating supported on a solid substrate. A well-acknowledged problem, however, is the inverse problem: Given a set of data, under what conditions can a refractive index profile be determined unambiguously? To this end, a series expansion of the Abeles matrix method has been applied to an arbitrary refractive index profile to determine analytic expressions of the ellipsometric ratio ρ. Two types of expressions are found: The thin film limit in which the film thickness L is much less than the wavelength of the incident light beam ( L≪λ ) and the weak contrast limit in which the refractive index of the coating is very near the refractive index of the supporting substrate. In the thin film limit, the first two terms in the series expansion are relatively straight-forward, and they depend on two types of integrals involving the difference between the dielectric profile of the coating and the dielectric constant of the substrate. While higher order terms are possible, they are quite convoluted and do not assist in the inverse problem. In the weak contrast limit, however, the series expansion of ρ depends on the moments of the difference between the dielectric profile of the coating and the dielectric constant of the substrate, allowing an analytic expression that applies to coatings that are even much larger than the incident light beam. The expressions associated with both limits are verified through comparison to the numerical evaluation of ρ with the Abeles matrix method. The results demonstrate that through judicious selection of the substrate refractive index and incident wavelength, conditions can be created that permit critical insights into the inverse problem for either thin coatings or for coatings that are very near the refractive index of the substrate.


Introduction
The key use of ellipsometry is to evaluate the refractive index and thickness of a supported coating or film [1][2][3].Due to historic reasons, the change in the polarization ellipse r r where r p ( ) and r s ( ) are the p and s reflection amplitudes, which can be directly calculated through numerical solution of the Fresnel and Maxwell's equations [4,5].The angles Y and D represent the measurement values provided by the ellipsometer, which can be taken over multiple angles of incidence and/or multple wavelengths.Normally, to fit Y and , D a specific model for the refractive index profile is assumed and confirmed through minimization of the difference between the model (as evaluated through the Abeles matrix formulation) and the experimental data [6,7].Despite ellipsometry being a mature characterization technique, confidence in fitting remains challenging.This is especially crucial in the general areas absorbed macromolecules [8][9][10], polymer brushes [11][12][13], swelling of polymer films [14], and the evaluation of glass transition and thermal expansion in thin polymer films [15].In the latter case, significant deviations in refractive index have been reported between bulk and sub 100 nm thick films [16][17][18][19].It is commonly understood that while the forward numerical solution of the Abeles matrix formulation is exact, a significant amount of insight is also lost.Specifically, does agreement between experimental data and a model prove that the model is correct especially with measurement precision taken into consideration [20,21]?Due to this challenge, most studies assume a uniform refractive index profile whether warranted or not [14,16].
In the past few years, there has been renewed interest in addressing the so-called inverse problem through the development of advanced optimization methods [20,[22][23][24][25][26][27][28], machine learning [29,30], and sophisticated ellipsometry methods that characterize the full Mueller matrix [31,32].In this paper, we are most concerned with the ability to unambiguously determine the sensitivity of r to a variation in refractive index across the thickness of a transparent coating using multiple-angle-of-incidence ellipsometry [33].
For roughly nanometer thick coatings, Drude established that the refractive index and thickness are correlated, and the two cannot be independently determined [4].In this limit, all coatings can be described with a constant or single refractive index.For slightly thicker coatings, Lekner expanded on these results using perturbation theory and showed that r could be expressed in terms of integral invariants up to second order in the interface thickness [34][35][36][37].Integral invariants are combinations of the integrals of the difference between a step profile between the incident and transmitted mediums and the actual dielectric profile.Significantly, Lekner's perturbation analysis established a closed form relationship that permits estimation of the confidence intervals in generating a refractive index profile from ellipsometry data in the thin film limit.Recently, Helm et al performed a Taylor series expansion of Drude's reflection coefficients to second order in interfacial thickness assuming a constant refractive index [25].While a completely different approach than Lekner, both results are equivalent given an isotropic and homogenous thin film.Helm et al reported that for an Si-SiO 2 -air interface, the second order analysis is valid up to approximately 30 nm in SiO 2 thickness.
Several other efforts have also focused on analytic approximations to r under a variety of conditions [23,  38-44].Most notably, Charmet and de Gennes [45], investigated model independent fitting of ellipsometry data and showed that in the limit where the refractive index is close to the incident medium (such as an adsorbed polymer or polymer brush) there are two limiting cases: 1. Thin layers in which the ellipsometry data can be inverted to determine the first and second moments of the refractive index file; and 2. Thick layers in which the Fourier transform of the refractive index profile can be reconstructed from ellipsometry data.The latter was used by Biesalski et al to evaluate segment density profiles in polyelectrolyte brushes [46].
Herein, and for the first time, a Taylor series expansion is applied to the Abeles matrix formalism for an arbitrary refractive index profile to evaluate both the p and s reflection amplitudes in terms of interfacial thickness.While the Abeles matrix formalism is a strictly discrete method, (i.e. the interfacial region is divided into n slabs of constant refractive index and thickness z D ), in the limit that the number of slabs goes to infinity (n  ¥ and z 0 D  ), the series expansion reduces to the Riemann sums of the squares of the refractive index profile, which can be substituted with integrals of the dielectric profile.The significance of this approach is that it allows fundamental insights into the inverse problem.In the thin film limit, the reflection ratio r is fully described by two types of integrals of the interfacial dielectric profile to second order in coating thickness.Through judicious selection of the substrate and the incident wavelength, experimental conditions can be established to help discriminate between coatings with a uniform versus a nonuniform refractive index profile.Here, the uniformity of the refractive index profile refers to the distribution of the refractive index across the thickness of the coating.
In the weak contrast limit, where the refractive index profile of the coating approaches the refractive index of the substrate, r is fully specified by the moments of the difference between the dielectric profile of the coating and the dielectric constant of the substrate, providing the opportunity to reconstruct the dielectric profile of the coating limited only by the number of moments that can be resolved.For coating thicknesses less than 100 nm, only the first few moments can be reliably determined depending on the precision in the measurement; however, through taking measurements on multiple types of substrates (of close but slightly different refractive index), the moments associated with each substrate can be compared to assess the validity of an assumed refractive index profile.The applicability of both these limits are validated through numerical evaluation of r as functions of the angle of incidence, the substrate refractive index, and the incident wavelength, providing a strategy not only for interpreting the numerical solution, but also for providing constraints on candidate refractive index profiles to be used in the numerical solution.
This paper is organized as follows.In section 2, the basic mathematical formulation of the Abeles matrix method is outlined along with how the series expansion of r is performed.In section 3, the analytic thin film approximation to second order in coating thickness is given for r along with an investigation of its limits of applicability.The sensitivity to small variations in the refractive index of the coating is also considered.In section 4, the results are given for the complex reflection ratio r in the weak contrast limit as well as strategies for accepting or rejecting candidate refractive index profiles.Sections 2 and 3 rely on the explicit series expansion of both the s and p reflection amplitudes, which are provided in Appendices A and B, respectively.In particular, the first two orders of this expansion are given for a nonuniform refractive index profile, and the first three orders are given for a uniform refractive index profile.

Mathematical formulation
The typical experimental situation is a polarized light beam traveling from an incident medium of uniform dielectric constant , i e through an interfacial region of a varying dielectric constant z , ( ) e and finally a transmitted medium of uniform dielectric constant , To begin the analysis, the interfacial region is divided into n slabs of uniform dielectric constant and thickness z.
D The incident wavevector can be decomposed into a parallel component, k, and a perpendicular component, q, with respect to the interface.If j represents the jth slab from the incident medium, the perpendicular and parallel components of the wavevector in each slab are Where l is the wavelength and j q is the angle between the propagating light beam and the unit normal.Because of Snell's Law, the parallel component of the wavevector is the same for each slab and is equal to its value in the incident medium.Hence, it can be represented with the constant K.
The perpendicular value of the wavevector, on the other hand, depends on the dielectric constant of the slab and can be related to the value of the incident medium via For the incident and transmitted media, the values of the perpendicular wavevector are q 2 cos incident 6 To determine the overall reflection amplitudes of the s and p polarizations, or r s ( ) and r , p ( ) the Abeles transfer matrix is Where the s and p reflection amplitudes at the j j , 1 + interface are defined as r q q q q 9 j j s j j j j = - e e e e = -+ = - It should be noted that the interface between slab n and n 1 + is the interface between slab n and the transmitted medium with a dielectric constant of .t e From the Abeles transfer matrix, both r s ( ) and r p ( ) are calculated as

=
A Taylor Series expansion can be performed with respect to z D and both the s and p reflection amplitudes can be expressed in orders of z The series expansion of both r s ( ) and r p ( ) are shown in appendices A and B, respectively.The ellipsometry ratio r can also be expressed in terms of the series expansion of r s p Or, in general terms, the kth order term in r for k 1 The zeroth order term in r is the ratio of the p and s reflection amplitudes for reflection between the incident and transmitted media (also known as the Fresnel reflection coefficients) in the absence of a coating.
Results for the complex reflection ratio r in the thin film limit and sensitivity to small variations in refractive index throughout the coating From equation ( 14) the first order term in the expansion of r is r r r 1 16 Where the reflection amplitudes r s 1 ( ) and r p 1 ( ) are found in appendix A, equation (A.4), and appendix B, equation (B.4), respectively.The integrals are written so that z 0 = is the outermost position of the coating and z L = is the position at the interface between the coating and the transmitted medium (see figure 1).The functions z ( ) d and z ( ) g represent 'excess' functions that depend on the dielectric constant of the coating, the incident medium, and the transmitted medium: the first order correction to r ultimately reduces to Which is also the same result as provided by Lekner, who showed that the integral is independent of the particular location of z 0 = [34].If the integral over the coating thickness is expressed as Then the first order term reduces to r e e = -+ D The second order term in the expansion of r from equation ( 14) is r r r r 1 24 After substitution of second order reflection amplitudes from appendices A and B, 2 r ultimately reduces to It should be noted that the integrals in the above equation must be evaluated by the specific placement of z 0 = at the interface between the incident medium and the coating and z L = at the interface between the coating and the transmitted medium.These integrals represent a special case of the invariant formulations as reported by Lekner [34].
Hence, to second order in z, Higher orders of r(i.e. 3 r and higher) are largely inaccessible, however, the analytical expressions describing these orders have a common form: They are polynomials of integrals containing z ( ) d where the degree of the polynomial is the same as the order of .
r Only the leading term in each of these expressions has a straightforward analytical expression: ) with two distinct refractive index profiles each with the same average refractive index of n 1.5 avg = . In figure 2(a), 'Profile 1' represents a bilayer of two equal thickness films with a refractive index of 1.4 and 1.6, respectively, and 'Profile 2' represents a single layer film with a constant refractive index of 1.5.Figures 2(b) and (c) show both the real and imaginary parts of , 0 r r respectively, where solid lines refer to the Abeles numerical evaluation and the dashed lines refer to the analytic thin film approximation to second order in thickness.The incident wavelength is 632.8 nm.
Figure 3 shows the same comparison except for a 20 nm thick coating supported on silicon.Here the analytic approximation does not hold as well, especially at angles of incidence between 60°and 80°.To investigate the upper limit of applicability of both the LaSFN9 and silicon substrates, figure 4 shows 0 r r -at two angles of incidence (55°and 75°) for a coating with a constant refractive index of n 1.5 f = , wherein the thickness of the coating is varied from 0 nm to 50 nm.Although the magnitude of 0 r r -is significantly larger for silicon than for LaSFN9 (a consequence of the greater difference between the dielectric constant of the coating and the dielectric constant of the substrate), the range of applicability is appreciably greater on the LaSFN9 substrate.For example, at a coating thickness of 50 nm on silicon the error between the analytic approximation and the numerical solution is 20% and 27% (representing the real and imaginary components of , 0 r r respectively) at an angle of incidence of 75°.In comparison, at the same angle of incidence, the error between the analytic approximation and the numerical solution for LaSFN9 is 9% and −1% (representing the real and imaginary components of , 0 r r respectively).A similar result is also obtained at an angle of incidence of 55°, where the error between analytic approximation and the numerical solution is systematically larger for silicon substrate than the LaSFN9 substrate.
While the above simulations were run at a wavelength of 632.8 nm (the wavelength of a standard He-Ne laser), a key question is the advantage, if any, of spectroscopic ellipsometry [15,26,28,47], in the thin film limit.
To develop an idea of how 1 0 r r depends on the wavelength, we consider the response of 1 r only at the Brewster angle , B q which corresponds approximately to the angle of incidence in which 1 0 r r is maximized in the thin film limit.At the Brewster angle, the following relationships are true:  Which provides a direct relationship between the wavelength, the substrate dielectric constant, and the response in .
1 r Interestingly, this expression also suggests that 1 r will diverge as i e approaches .t e This is clearly non- physical and is a consequence of the series expansion in r given by equation (13).When i e approaches , t e the  supported on an LaSFN9 substrate compared to the Abeles numerical evaluation at three wavelengths: 488 nm, 1000 nm, and 1650 nm.The Brewster angle of LaSFN9 is roughly 61°( the refractive index of LaSFN9 varies between 1.87 and 1.81 over this range of wavelengths).As suggested by equation (30), the magnitude of Im 1 0 l At a wavelength of 488 nm, however, a clear deviation between the analytical and numerical solutions is observed in both Im 1 0 whereas at 1000 and 1650 nm, the magnitude of Im 0 ( ) r r increases as well.That said, above an approximate refractive index value of 2.0, the agreement between the analytic thin film expression and the numerical solution breaks down, which is consistent with the previous finding that the upper coating thickness of the approximation depends on the refractive index of the substrate.Here, it is found that LaSFN9 is a more natural choice of substrate over silicon for application of the analytic thin film expression.
A question that obviously arises, nevertheless, is the sensitivity of r to a small gradient in refractive index over the thickness of the coating.Here, it is assumed that the refractive index profile is linear, or Where n f is the average refractive index of the coating and n D is the change in the refractive index across the thickness of the coating.In the case of a perfectly isotropic coating of constant dielectric constant n f f 2 e = and thickness L, the integrals involving z ( ) d in equation (27) become   Hence, to first order in n, D the difference between the real and imaginary parts of r with respect to an equivalent film of constant refractive index (specified with the subscript f ) is r r e e e e e e e e -» +

+ -D
To validate this finding, figure 6 shows three refractive index profiles corresponding to the linear refractive index profile of equation (31), where n 1.5 f = and the layer thickness L is 40 nm.The substrate is LaSFN9 and the incident medium is air.Figures 7(b   (dashed lines).Therefore, it can be seen that in the thin film limit, the imaginary part of r is insensitive to a small gradient in the refractive index profile; however, the real part of r could potentially validate whether a coating is has a uniform refractive index profile or not.While equation ( 37) is model specific, the result is generic in nature.Using the known average bulk value of the refractive index will unambiguously give the coating thickness in the fitting of Im . 0 can be used to assess the validity and confidence limit of fitting with a nonhomogeneous refractive index profile.
4. Results for the complex reflection ratio r in the weak contrast limit and validation of the analysis The integrals associated with the analytic thin film approximation are a bit abstract, however, the expression can be significantly simplified if the refractive index of the coating is very near the refractive index of the transmitted medium, or substrate.Taking the series expansion of z ( ) And retaining only the linear term, a straight-forward expression is found in the expansion of r for all orders of k: In order in to validate the weak contrast limit as 0, j t e e - refractive index profiles of the following form are used.
Where s is a measure of the interface width.Taking the substrate to be glass (n ), the incident medium to be air (n ), and the layer thickness L to be 800 nm, figure 7 shows both the numerical evaluation of Im 0 ( ) r r and Re 0 ( ) r r using the Abeles matrix formulism (solid lines) and the analytic weak contrast approximation (dashed lines) for n 1.5, f = 200 s = nm, and n 0.02.

D = 
The almost mirror images of the r profiles can be understood with regards to the moments of the difference between the dielectric constant of the coating and the substrate, which have opposite signs with respect to one another as the refractive index profile is inverted (i.e.n 0.02 D = versus n 0.02 D = - ). Figure 8 further illustrates the applicability of the weak contrast approximation.Figure 8 for an 800 nm thick coating where s is varied from 0 to 800 nm.Here the angle of incidence is also 55°a nd n 0.02.

D = -
The ability to determine the moments will come down to the standard error [48][49][50] in both Y and D as well as the roughness of the substrate [51].That said, strategies can also be conceived to aid in the acceptance or rejection of candidate refractive index profiles.For instance, If the refractive index of the coating is perfectly matched to the substrate it would be expected that 0. 0 r r -= Hence, if such a substrate is used and associated with each refractive index profile and the three substrates along with the analytic weak contrast approximation.Clearly, if there is the existence of an inner layer, all three measurements will be distinct from the situation in which the refractive index has a uniform value of 1.5.
In terms of an optimal wavelength, unlike in the thin film approximation, there is no lower bound on the wavelength as long as both the coating and substrate are transparent to the incident radiation.The magnitude of 0 r r -scales as various powers of 1/l (depending on the number of moments needed) and therefore smaller ), the angle of incidence is 55°, and the incident wavelength is 632.8 nm.wavelengths will give a better signal-to-noise ratio.For example, if the measurement uncertainty in both Y and D is +/− 0.01°, the corresponding confidence intervals [52], in Im substrate for two wavelengths: 300 nm and 632.8 nm.It can be clearly observed that the 300 nm wavelength gives a larger response than at 632.8 nm.Again, multiple wavelengths provide more certainty in the fit with the caveat that wavelengths that are much longer than the extent of the coating will be insensitive to the presence of the coating.For instance, in figure 10(b), within the estimated uncertainty in the measurement, the response in Re 0 ( ) r r at 632.8 nm may not provide a reliable determination of the moments associated with the fit.Hence, care must be exercised with regards to measurement precision in the weak contrast limit.

Conclusions
While ellipsometry is a mature field, confidence in data reduction remains challenging.This paper sets out to establish straight-forward analytic relationships that can be used to construct refractive index profiles from experimental measurements.Herein, we express an approximation of the Abeles matrix formalism in terms of a Taylor series expansion of the s and p reflection amplitudes in the limit where z 0, D  which in principle can be expanded to any number of orders in z.
D The ellipsometry ratio r r p s ( ) ( ) / r = therefore can also be expanded about the limit where z 0, D  or ..., 0 1 2 r r r r -= + + where 0 r r -represents the difference between a coating supported on a substrate and the interface without the coating.In this analysis, two types of analytic approximations were identified: The thin film limit and the weak contrast limit.In the thin film limit, 0 r r can be easily expressed with the first two orders in the expansion.While higher order terms are possible, they are quite convoluted and do not aid in understanding of the inverse problem.In the weak contrast limit, where the refractive index profile of the coating approaches the refractive index of the substrate, r can be readily expanded to any order term.The applicability of both these limits were validated through the numerical evaluation of r as functions of the angle of incidence, the substrate refractive index, and the incident wavelength, providing a strategy not only for interpreting the numerical solution in these limits, but also for providing constraints on candidate refractive index profiles to be used in the numerical solution.
can be related to the difference in the dielectric constants between slab j and the transmitted medium: Moreover, in the limit that n ,  ¥ the summation can be replaced with the integral Therefore, the first order term can be expressed as The second and third order terms in z D for the s reflection amplitude are r q q q z dz q q q q j z 2 2 2 2 2 1 A.5 The missing term in r s 3 ( ) is rather lengthy in form.For a single layer of constant dielectric constant, , f e and thickness, L, the third order term becomes r i q q q L i q q q q q L i q q q q L 2 2 2 3 g Both the leading (or highest) order and lowest order terms of this polynomial have simple analytical representations.The highest order term is And the lowest order term is Where the summation in the above equation can also be replaced with the integral equivalent r i k q q q q z z dz

-
The first order term in z D is ( ) e can be split into the following two contributions: ( ) e as , j j e x the first three terms in the Taylor Series expansion of z D are ...  x Only the leading (or highest) order and lowest order terms of this polynomial have simple analytical representations.The highest order term is ò e x = -+ + And the lowest order term is

t e as shown in figure 1 . 2 e
The dielectric constant of a material and its refractive index are related via n .=

Figure 1 .
Figure 1.Reflection of polarized light from a stratified interface.An arbitrary layer of varying dielectric of height L is divided into n slabs of unform thickness.The component of the polarization parallel to the plane of incidence is the p polarization and the component of the polarization perpendicular to the plane of incidence is the s polarization.After reflection, the polarizations are represented as p¢ and s , ¢ respectively.

Figure 2
Figure2shows the comparison between the Abeles numerical evaluation of Hence, for a coating of constant thickness L and dielectric constant , f e the first order term becomes i L 30

Figure 2 .
Figure 2. Given the refractive index profile in (a), figures (b) and (c) represent the numerical evaluation of Im

Figure 3 .
Figure 3.Given the refractive index profiles in (a), figures (b) and (c) represent the numerical evaluation of Im

Figure 4 .
Figure 4. Numerical evaluation of first order variation of the integrals with respect to n D is ) and (c) show both the numerical evaluation of Im 0 matrix formulism (solid lines) and the analytic thin film approximation

Figure 6 .
Figure 6.Given the index profiles in (a), figures (b) and (c) represent the numerical evaluation of Im (a) shows 0 r r -over a range of thicknesses from 0 to 800 nm at an angle of incidence 55°, where 40 s = nm and n 0.02.D = -Figure 8(b) shows 0 r r -over a range of interface widths

Figure 7 .
Figure 7.Given the refractive index profiles in (a), figures (b) and (c) show the numerical evaluation of Im I m 0 ( ) ( ) r r and Re Re , 0 ( ) ( ) r r respectively, versus the analytic weak contrast approximation.The substrate is glass (n 1.5 t =), and the wavelength of the incident light beam is 632.8 nm.

Figure 8 .
Figure 8. Numerical of Im I m 0 ( ) ( ) r r and Re Re 0 ( ) ( ) r r versus the versus the analytic weak contrast approximation.Figure (a) shows the variation of r with regards to coating thickness given n 1.5, f = n 0.02 D = -40 s = nm.Figure (b) shows the variation of r with regards to the interface width s given n 1.5, f = n 0.02, D =and L 800 = nm.The substrate is glass (n 1.5 t =), the angle of incidence is 55°, and the incident wavelength is 632.8 nm.

Figure 9 .
Figure 9.Given the refractive index profiles in (a) labeled 'Profile 1' and 'Profile 2', figure (b) shows the numerical evaluation of Im 0 ( ) r r -(solid lines) versus the weak contrast limit (dashed lines) for three different substrates n 1.46, 1.50, 1.54 .t ( ) = The wavelength of the incident light is 632.8 nm.
figure 10 for 'Profile 1' on the n 1.50 t =

Figure 10 .
Figure 10.Figures (a) and (b) show the numerical evaluation of Im

)
It should be noted that r k s ( ) always produces a polynomial of degree k in .

- 3 (
Similar to the case of the s reflection amplitude, the missing term in r p ) is second order in .xFor a layer of constant dielectric constant f e and thickness L, r p 3 ( ) becomes produces a polynomial of degree k in .
https:/ /orcid.org/0000-0003-4401-5688 (16)tituting equations(17)and (18) into equation(16), and noting from equations (5) and (10 .1.Series expansion of the p reflection amplitude The analysis for the p reflection amplitudes is analogous to the s reflection amplitudes, except that the p reflection amplitudes are expressed in integer powers of Q Q . B . 1 2