The authors present a second-generation broadband 4×4 Mueller-matrix (MM) ellipsometer for ultrasensitive infrared-spectroscopic (8000800cm1) studies of complex nanometer-thin films. In a modular design, the instrument employs retractable achromatic retarders and various sets of tandem polarizers. Using high-transmittance free-standing wire-grid polarizers, the device reaches an unparalleled precision of up to 5105 in the important fingerprint region, even for block-offdiagonal MM elements. Broadband and signal-to-noise optimized access to the full 4×4 MM provides in-depth information on the sample’s polarimetric properties and opens the door for detailed explorations of depolarizing and anisotropic materials. The authors discuss examples of highly depolarizing nonuniform polyimide membranes, uniaxial-to-biaxial anisotropy changes in ultrathin polymer films, and azimuthal off-axis effects in 2D-structured silica arrays. Diverse optical modeling approaches based upon anisotropic layer stacks and rigorous coupled-wave analysis are used to quantify the optical, structural, and chemical properties of the sample.

Ellipsometry is a well-established, nondestructive, and noninvasive linear-optical spectroscopic technique for characterizing the properties of materials by means of polarized light.1–4 Ellipsometry based upon the Jones formalism addresses nondepolarizing samples. It measures ellipsometric ratios typically expressed as

ϱ=tanΨeiΔ=rp/rs,
(1)

with tanΨ and Δ being the amplitude ratio and phase difference between the complex p- and s-polarized reflection (or transmission) coefficients. Depolarization can be measured using optical retarders and is fully handled by Mueller-matrix (MM) ellipsometry. The 4×4 Mueller matrix obtained by this extended method is a general and complete description of the sample’s polarimetric properties.5 

In recent years, MM ellipsometry has gained much attention as an advanced characterization technique in science and metrology.6–16 For various reasons, usually related to insufficient signal-to-noise (S/N), the infrared (IR) range is often neglected for thin-film sample analysis. Yet, this spectral region offers detailed complementary insights into optical, physical, and chemical properties of the sample.

Investigating the IR optical response is of particular relevance for organic thin films, which exhibit characteristic vibrational bands due to material-specific absorption, rendering IR spectroscopy a powerful tool for chemical analysis. The IR response is also of interest for studying inorganic layers and structured surfaces, as it delivers information on phonon modes, free charge carriers, structure, and related properties. IR MM ellipsometry, which can measure both amplitude ratios, phase differences and depolarization, has thus become an emerging technique for probing the sample’s dielectric properties, its structure and composition, as well as numerous other characteristics, such as optical anisotropy, molecular orientation, and interactions.

IR MM measurements are challenging when aiming for thin-film sensitivity. The reasons for this are manifold. Experimental limitations arise from the low brilliance of standard IR radiation sources like globars, from the measurement technique [e.g., Fourier-transform (FT-IR)] itself, from the nonideal optical properties of polarizers and compensators, from the low detectivity of standard pyroelectric detectors, and from low optical throughput. Furthermore, the materials used for optical elements can impact sensitivity and restrict the accessible spectral range.

MM ellipsometers rely on generating and projecting a sufficient number of different polarization states in order to probe the sample’s full MM. For polarization-space sampling, many ellipsometer designs employ dual-rotating compensators or a multiple-photoelastic-modulator configuration, while polarizers are typically fixed (nonrotating) to avoid problems with source polarization and detector polarization sensitivity. In such designs, the radiation must pass through all optical elements during the whole measurement process. Hence, all 16 MM entries are affected by a nonoptimal throughput limited by the optical properties of these elements.

In a recent publication,17 we presented a novel IR MM ellipsometer that is based on a different design and measurement approach. The ellipsometer measures the upper-left 3×3 MM block via different polarizer/analyzer settings, while both retarders are retracted from the optical path. These first 9 of the 16 MM elements are, therefore, least afflicted by noise. Optional retarders are only used to measure the fourth row or column of the MM. Tandem polarizers guarantee an optimized polarization control necessary to accurately determine the block-offdiagonal MM elements. For temporal resolutions of a few minutes, the measurement scheme is centered around subsets of defined polarizer/retarder settings for generating and projecting pure polarization states, allowing one to extract quadruples of MM elements.

In this contribution, we have significantly optimized the instrument for full 4×4 Mueller-matrix measurements with increased signal-to-noise, high-throughput, and close-to-ideal optical elements. A modular polarizer design provides access to the widest spectral range of 8000800cm1 (1.2512.5μm) at sensitivities of up to 104 in the normalized MM elements. Tiniest vibrational features frequently observed in the fingerprint region below 2000cm1 can be resolved by using high-transmittance free-standing wire-grid polarizers, which provide an unprecedented sensitivity of up to 5105, even in the block-offdiagonal MM elements. Such advances in the S/N ratio enable full 4×4 MM measurements and quantitative analyses of depolarizing and/or anisotropic thin films down to the nanometer level.

For many depolarizing thin-film samples, one often encounters a 2×2 block structure of the infrared Mueller matrix that looks similar to the characteristic structure of a nondepolarizing surface. The latter can be expressed by5 

M=12(|rpp|2+|rss|2+|rps|2+|rsp|2)12(|rpp|2|rss|2+|rps|2|rsp|2)(rpprsp+rpsrss)(rpprsp+rpsrss)12(|rpp|2|rss|2|rps|2+|rsp|2)12(|rpp|2+|rss|2|rps|2|rsp|2)(rpprsprpsrss)(rpprsprpsrss)(rpprps+rsprss)(rpprpsrsprss)(rpprss+rsprps)(rpprssrsprps)(rpprps+rsprss)(rpprpsrsprss)(rpprss+rsprps)(rpprssrsprps).
(2)

Our measurement approach ensures lowest noise in the upper-left 2×2 MM block, which contains key amplitude information. The other three 2×2 MM blocks comprise phase information related to copolarization (pp, ss) and cross polarization (ps, sp). Noise is also reduced for MM elements in the third row and column. Block-offdiagonal elements usually show much weaker IR signatures. In such cases, our data-acquisition scheme enables one to specifically increase S/N for particular MM elements. A further S/N enhancement is possible when the complete 4×4 MM is measured, in which case matrix filtering methods can be applied.5 

Acquiring high-quality MM data is often merely the first step in sample characterization and quantification. The ever-growing complexity of layer stacks, structured surfaces, and depolarizing thin films also demands the advancement of theoretical models to interpret the data.

In this contribution, we apply the IR MM ellipsometer with its unique capabilities, such as the broad spectral range and unsurpassed S/N, to obtain comprehensive 4×4 MM data of three sample systems. Each system poses a different challenge regarding quantitative analysis. We describe diverse optical modeling approaches to quantify the data and to extract information concerning the optical and physical sample properties.

The first set of samples are uniaxially anisotropic and depolarizing polyimide-based microporous membrane thin films for gas separation purposes. These membranes are strongly heterogeneous, which impacts the measured polarization degree over a wide spectral range, thus requiring broadband experimental data to accurately determine their optical properties.

The second example is a 32 nm thin polyimide film (PI2611) with applications in gas sensors and electronic devices. Upon surface rubbing, the molecules inside the polymer layer can be realigned. We make use of the ellipsometer’s excellent S/N to quantify the corresponding changes in optical film properties from uniaxial to biaxial anisotropy.

Lastly, we investigate two-dimensional silica arrays suitable for use in sensor applications. These trapezoid-shaped structures have profile dimensions in the micrometer range that give rise to complicated MM spectra. Here, the wide accessible spectral range of the ellipsometer proves crucial for probing the relevant structure parameters. We employ rigorous coupled-wave analysis (RCWA) to model and quantify array profile and orientation, highlighting the possibilities of broadband IR MM ellipsometry for applications such as process control and quality management.

The infrared Mueller-matrix ellipsometer (Fig. 1) operates both in reflection, with possible incidence angles between φ0=45° and 90°, and in transmission. Naturally, highest sensitivity for characterizing nanometer-thin films is achieved in reflection. A tilt- and height-adjustable rotation stage with an autocollimation unit is used for sample alignment and azimuthal orientation. The ellipsometer’s input arm is fixed, whereas the sample stage and the analyzer arm are mounted on a motorized, independently angle-adjustable 2ϑ goniometer stage (2-Circle Goniometer 423, Huber Diffraction and Positioning Equipment, Germany). An FT-IR spectrometer (Tensor 37, Bruker, Germany) with a globar as the radiation source is coupled to the ellipsometer.

FIG. 1.

Schematic of the infrared 4×4 Mueller-matrix ellipsometer.

FIG. 1.

Schematic of the infrared 4×4 Mueller-matrix ellipsometer.

Close modal

Input and output arms of the instrument contain the optical elements of the polarization-state generator (PSG) and analyzer (PSA), respectively. PSG and PSA are modularly designed and each consist of a tandem pair of aligned polarizers as well as optional, nonrotating retarders mounted on motorized translation stages for reproducible retarder insertion into, or retraction from, the optical path.

Using tandem polarizers suppresses unwanted leaking polarization states, thereby improving the ellipsometer’s polarization control. Holographic KRS-5 wire-grid polarizers (25 mm clear aperture, 0.25μm wire spacing, Specac Ltd, England) provide access to the broadest spectral range of 8000800cm1. Free-standing Au wire-grid polarizers (P03, InfraSpecs, Germany, 5000800cm1) can be engaged when enhanced S/N is required below 2000cm1.

The retarder units consist of a rhombic KBr prism for two phase-shifting attenuated total reflections and two gold mirrors for beam repositioning.18 A phase shift between p- and s-polarization close to 90° yields an almost achromatic retarder response. The utilized polarizers and retarders enable the generation and projection of close-to-ideal purely linear and circular polarization states.

Off-axis parabolic mirrors focus the IR radiation onto the sample and detector. The latter is mounted on a translation stage for optimal focusing. A photovoltaic mercury-cadmium-telluride detector (MCT, KLD-1-J1-3/11, Kolmar Technologies, USA) is used to avoid nonlinearity issues, while providing orders of magnitude higher detectivity compared to standard pyroelectric detectors.

The whole instrument is constantly purged with dried air (RH0.1%) to ensure atmospheric stability and minimal IR absorption from CO2 and H2O vapor.

According to the MM calculus,19 the output Stokes vector Sout at the detector, which characterizes the light’s polarization state after interaction with the sample and ellipsometer optics, is derived from

Sout=PSATMPSG=DAR2M11M12M13M14M21M22M23M24M31M32M33M34M41M42M43M44R1PSin,
(3)

where M is the Mueller matrix of the sample; P and A are the polarizer and analyzer matrices; Rm is the retarder matrices; Sin=[s0,s1,s2,s3]T is the input Stokes vector in front of the first polarizer, with si/s00 implying source prepolarization; and D is the detector Mueller matrix.

In general, a minimum number of 4×4=16 sufficiently different polarization states must be generated in the PSG and projected in the PSA in order to measure a full 4×4 Mueller matrix. An optimum choice results for such combinations of (usually mixed) polarization states that inscribe regular tetrahedra on the Poincaré sphere.20 For the sake of thin-film sensitivity and noise reduction, however, we chose to work with the six pure polarization states [1,±1,0,0]T, [1,0,±1,0]T, and [1,0,0,±1]T, i.e., with octahedral sampling.21 These states can be obtained by ideal PSGs and PSAs. Four intensity measurements with such PSG/PSA vectors in Eq. (3) lead to a set of four equations solvable for a subset of four MM elements at a time.19 By using different PSG/PSA combinations, the whole 4×4 MM can be deduced.

We have adapted this approach for the presented ellipsometer and derived generalized equations that account for source polarization, polarizer diattenuation, and other nonidealities.17 

In practice, the pure polarization states are realized according to Fig. 2 by appropriate rotation of the polarizers to 0°/90° (linear polarization), to 45°/135° (linear diagonal polarization), or by combining the latter settings with a 90° phase-shifting retarder (circular polarization). In other words, for maximum optical throughput, and hence improved S/N, the upper-left 3×3 MM block is acquired without retarders in the optical train. The first three MM elements in the fourth column and row are obtained with the first or second retarder present, respectively. Only M44 requires the use of both retarders.

FIG. 2.

Quadruples of MM elements measured from different combinations of polarizer/analyzer/retarder settings. Figure from A. Furchner, C. Walder, M. Zellmeier, J. Rappich, and K. Hinrichs, Appl. Opt. 57, 7895 (2018). Copyright 2018 Optical Society of America.

FIG. 2.

Quadruples of MM elements measured from different combinations of polarizer/analyzer/retarder settings. Figure from A. Furchner, C. Walder, M. Zellmeier, J. Rappich, and K. Hinrichs, Appl. Opt. 57, 7895 (2018). Copyright 2018 Optical Society of America.

Close modal

Having the sample’s full 4×4 Mueller matrix at hand allows for the application of matrix filtering methods. Measured MMs were subjected to covariance filtering, which can further reduce polarimetric noise.5,22,23 Principal-component covariance filtering, which retains only the nondepolarizing information of the MM, was applied to MM data of the nondepolarizing, 32 nm thin PI2611 film. Routines for covariance filtering were adapted from Matlab code provided by Razvigor Ossikovski (LPICM, Ecole Polytechnique, France).

Data evaluation was performed by Levenberg–Marquardt fitting minimizing the reduced χ2, i.e., the sum of all error-weighted squared differences between measured and calculated MM elements.1 All presented MM data are normalized to their respective M11 elements.

Polymers of intrinsic microporosity24 (PIMs) are a recently discovered group of organic polymers featuring extremely rigid, contorted backbones that prevent efficient molecular packing. This packing inefficiency results in the formation of micropores in the size scale of small molecules (<2nm), rendering PIMs highly attractive for gas separation applications.

Ogieglo et al.25 used polyimide (PI) based PIMs as a precursor to fabricate gas-separating carbon molecular sieve membranes. Different membranes were created by altering the precursor film’s structural properties via pyrolysis at temperatures ranging from 500 °C to 1000°C. Polyimide pyrolysis induces chemical and structural changes leading to a unidirectional layer collapse enforced by the presence of the substrate. This collapse manifests in the film’s optical properties, namely, in marked optical anisotropy. Ellipsometry in the ultraviolet–visible–near-IR spectral range was successfully used to quantify the degree of anisotropy25 as the films progressed from being almost isotropic at low pyrolysis temperatures to being highly uniaxially anisotropic at elevated temperatures.

Here, we employ IR MM ellipsometry to follow the optical, chemical, and structural changes as evidenced by changes in film thickness and uniaxial dielectric function. We focus on two films prepared on silicon substrates: a pristine one (415 nm thick) exposed to 50°C, and a collapsed one (263 nm thick) that underwent pyrolysis at 600°C.

Figure 3 shows the PIM-PI molecular structure along with typical optical images in size relation to the measurement area of the IR spot, as well as experimental and fitted MM data of the pristine film at multiple incidence angles between φ0=45° and 80°. The challenge in extracting the films’ optical properties from the measured MM spectra is that the PIM-PI layers exhibit pronounced thickness inhomogeneities, seen already in the optical images. Thickness variations of up to ±150nm within the measured spot lead to substantial drops in the polarization degree to values as low as 11%. Standard optical models for nondepolarizing samples thus fail to work.

FIG. 3.

(a) PIM-PI molecular structure. (b) Optical images (50 and 600°C film) in relation to the spot size of the IR MM ellipsometer. (c) Experimental (turquoise) and fitted (red) MM data of the pristine film (50°C) at incidence angles between 45° and 80° in steps of 5°.

FIG. 3.

(a) PIM-PI molecular structure. (b) Optical images (50 and 600°C film) in relation to the spot size of the IR MM ellipsometer. (c) Experimental (turquoise) and fitted (red) MM data of the pristine film (50°C) at incidence angles between 45° and 80° in steps of 5°.

Close modal

In order to properly quantify the complex dielectric function, particularly the anisotropic vibrational signatures between 2000800cm1, optical modeling of the data has to account for the thickness nonuniformity. In essence, MM elements are calculated by averaging over the measured area assuming that multiple different film thicknesses contribute to the observed signals. As a consequence, the polarization degree diminishes and follows a characteristic spectral behavior depending on the incidence angle and the thickness distribution. The resulting agreement between measured and fitted MM data in Fig. 3 shows that such modeling of inhomogeneous films is in fact feasible. We now explain the modeling process in more detail.

Since the MM is block diagonal, as is expected for a uniaxially anisotropic (or isotropic) film,5,26 it is convenient for further discussions to switch to the more familiar Ψ/Δ representation of the data, given by

M=|rp|2+|rs|221cos2Ψ00cos2Ψ10000sin2ΨcosΔsin2ΨsinΔ00sin2ΨsinΔsin2ΨcosΔ.
(4)

Measured spectra were fitted with a stratified layer model (Air–PIM-PI–SiO2–Si). The anisotropic dielectric function of the uniaxial PIM-PI layer was described as a sum of Lorentzian oscillators4 associated with PIM-PI’s molecular vibrational modes. To reduce the number of fit parameters and avoid parameter correlations, oscillator frequencies and line widths for in-plane (xy) and out-of-plane (z) directions were assumed to be identical, but oscillator strengths were allowed to differ.

In a first approach, we restricted the fit’s spectral range to below 2000cm1, where depolarization is seemingly negligible (Pol.Degree>0.98). The PIM-PI layer was assumed to be uniform and fitted with a single, average thickness.

In a second, more sophisticated approach, the complete spectral range up to 8000cm1 was fitted, and the film thickness was modeled as nonuniform, following a certain thickness distribution, as proposed by Jellison et al.27 In other words, depolarization was taken into account. Besides film heterogeneity, depolarization can also arise from the ellipsometer’s instrument function, which is dominated by effects of the device’s opening angle (about ±2°). Averaged MM elements were, therefore, computed according to

M¯ij=D(d)Φ(φ0)Mijdddφ0
(5)

by integrating over both the thickness distribution D(d) and the incidence-angle distribution Φ(φ0).

The polarization degree P we define by

P=M¯122+M¯332+M¯4321
(6)

then shows a characteristic spectral dependence on thickness and angle of incidence.

In a first pass, D(d) was taken to be a linear function to approximate the observed P(φ0) dependence. Subsequently, D(d) was refined as a third-degree polynomial. Fifty thickness steps turned out to be sufficient to describe the measured data.

Measured and fitted data as well as results of thicknesses and optical constants (n and k) obtained from the two modeling approaches are contrasted in Fig. 4. With 439 nm vs 415 nm and 280 nm vs 263 nm, the uniform and nonuniform film methods yield similar average film thicknesses for the pristine and the 600°C film, respectively. Only the nonuniform film model can describe the dampening effects from the drop in polarization degree that significantly influence tanΨ and Δ above 3000cm1. Yet, for both approaches, measured and fitted data of the two PIM-PI films agree within the small spectral window between 2000800cm1. However, the corresponding optical constants differ quite substantially. Specifically for the 600°C sample, which exhibits strong out-of-plane anisotropy, n and k values are unreliable if the film heterogeneities are not accounted for.

FIG. 4.

Measured (turquoise) and fitted (black, red) PIM-PI spectra at incidence angles between 45° and 80° in steps of 5°. tanΨ and Δ are the effective ellipsometric parameters corresponding to the averaged MM elements in Eq. (5). The left two columns are data of the pristine (50°C), almost isotropic film. The right two columns are data of the pyrolyzed (600°C), highly uniaxially anisotropic film. Refractive indices and extinction coefficients were extracted from the fits assuming that the film thickness is either uniform (fit restricted to below 2000cm1) or nonuniform, with fitted thickness distributions shown below.

FIG. 4.

Measured (turquoise) and fitted (black, red) PIM-PI spectra at incidence angles between 45° and 80° in steps of 5°. tanΨ and Δ are the effective ellipsometric parameters corresponding to the averaged MM elements in Eq. (5). The left two columns are data of the pristine (50°C), almost isotropic film. The right two columns are data of the pyrolyzed (600°C), highly uniaxially anisotropic film. Refractive indices and extinction coefficients were extracted from the fits assuming that the film thickness is either uniform (fit restricted to below 2000cm1) or nonuniform, with fitted thickness distributions shown below.

Close modal

Comparing the optical constants obtained from the uniform and the nonuniform approach demonstrates that even small deviations of the polarization degree from unity can easily be mistaken for pronounced anisotropy if only a narrow spectral range is considered and depolarization is disregarded. Looking at a wider spectral range can reveal the true nature of the film’s structural properties and is often necessary for the correct determination of its optical properties. This point is best illustrated for the in-plane and out-of-plane refractive indices of the pristine (50°C) PIM-PI film. As seen in Fig. 4, the uniform model yields substantial baseline differences between nxy and nz, i.e., birefringence related to the respective high-frequency dielectric constants. These differences diminish markedly if the nonuniform model is applied. For instance, birefringence at, and above, 2250cm1 reduces by a factor of 10 from about nxynz=0.054 for the uniform to nxynz=0.006 for the nonuniform model, meaning that the contributions to the susceptibility due to higher-energetic electronic transitions are almost isotropic, in accordance with Ref. 25.

The IR optical constants reveal dramatic changes in the polymer’s vibrational fingerprint reflective of the pyrolysis-induced chemical transformations. Various chemical reactions, such as the removal of CO, CO2, H2, or N2,28 lead to a structural collapse accompanied by carbon enrichment and an increasing degree of conjugation, i.e., partial graphitization. Compared to the 50°C pristine sample, these chemical pyrolysis effects in the 600°C film significantly diminish the imide-associated carbonyl-stretching bands between 1800–1675 cm−1, as well as virtually all other bands at lower wavenumbers associated with the previously chemically intact PI molecular structure. The film’s overall dielectric function becomes rather featureless, which is typical for a turbostratic, highly conjugated carbon matrix.29 The remaining broad spectral signatures are mainly associated with vibrational modes of fused aromatic rings that occur in the microporous structure of such carbon molecular sieves after pyrolysis at 600°C.30 

We stress that the simultaneous fits on thickness distribution and dielectric function were only possible because the ellipsometer provided access to a broad spectral range. In conjunction with a sufficient number of measurements at different incidence angles, the wide spectral range allowed the accurate modeling of the polarization degree, and hence the decorrelation of film thickness distribution and anisotropic optical constants.

In this example, we investigate an ultrathin polyimide film (PI2611, DuPont) on silicon for its anisotropic Mueller-matrix vibrational fingerprint. After precursor spincoating and thermal annealing,31 PI2611 layers typically exhibit uniaxial optical anisotropy. Films of 1.81μm and 84 nm thickness have previously been thoroughly investigated using generalized IR ellipsometry,31,32 and average molecular orientations could be deduced.

Polyimide layers find wide applications in microelectromechanical systems (MEMS), sensors, and liquid-crystal devices.33–37 For the latter, correct in-plane orientation of the rodlike molecular units is often required. The alignment of PI molecules can be influenced, for instance, via defined surface rubbing.33,35,38

Here, we utilize IR MM ellipsometry to quantify such alignment effects. The aim is to test the thin-film limits of block-offdiagonal MM analysis by probing small changes from uniaxial to biaxial anisotropy in a 32 nm thin, molecularly reoriented PI layer. For this purpose, we used a polyester microfiber cloth to partially align the molecules inside the originally uniaxial polyimide film. The surface was manually rubbed several times along an axis about 5°10° off the sample’s nominal x-axis.

Figure 5 shows that rubbing has an obvious effect on the block-offdiagonal MM elements measured at different azimuthal sample orientations. For the uniaxial film before molecular realignment, the block-offdiagonal MM elements are zero, whereas they become nonzero in the aligned, biaxial case, if measured off an optical axis. This observation is in line with Eq. (2), from which one expects a block-diagonal form of the uniaxial film’s MM with zero-valued block-offdiagonals (M13=M31=M23=M32=M14=M41=M24=M42), and a fully populated form in the biaxial case.5,26

FIG. 5.

(a) Selected MM elements at φ0=45° and different azimuthal rotations of the 32 nm PI2611 film in its uniaxial state before rubbing and its biaxial state after rubbing. (b) Molecular structure of PI2611.

FIG. 5.

(a) Selected MM elements at φ0=45° and different azimuthal rotations of the 32 nm PI2611 film in its uniaxial state before rubbing and its biaxial state after rubbing. (b) Molecular structure of PI2611.

Close modal

The ellipsometer is sensitive enough to resolve even tiny vibrational signatures with MM amplitudes below 5104. Prominent vibrational features, related to the PI molecular structure given in Fig. 5, are the ν(CNC) stretching band around 1355cm1, the ν(C=C) band of the PDA (phenylenediamine) ring, and the stretching modes of the imide ring’s two C=O groups. These carbonyl vibrations are visible as in-phase and out-of-phase ν(C=O) modes at 1775cm1 and 1715cm1, respectively, with a smaller contribution at 1725cm1 that is likely to arise from packing perturbations of the imide-ring structures.39 

Similar to the microporous polymer membranes discussed before, we used an anisotropic oscillator layer model to quantify the PI film’s MM optical response. Multiple incidence and azimuth angles were fitted with a single thickness and with unconstrained oscillator parameters in x-, y-, and z-directions. The fit revealed an azimuthal orientation of 8°±2° of the film’s in-plane optical axes with respect to the ellipsometer coordinate system, which corresponds well to the nominal rubbing direction.

Exemplary experimental and fitted 4×4 MM data at four azimuths are shown in Fig. 6. The measured complex lineshapes, particularly observed in the block-offdiagonal MM elements, are reproduced by the biaxial optical model. In-plane anisotropy resulting from the rubbing procedure appears to be small, though, as the block-offdiagonal MM elements contain only relatively minute vibrational features.

FIG. 6.

Measured and fitted MM data of the rubbed, biaxially anisotropic, 32 nm thick PI2611 film at φ0=65° and four azimuthal rotations.

FIG. 6.

Measured and fitted MM data of the rubbed, biaxially anisotropic, 32 nm thick PI2611 film at φ0=65° and four azimuthal rotations.

Close modal

The change in optical anisotropy is more obvious when comparing the uniaxial and biaxial optical constants obtained from the respective fits before and after rubbing. Such a comparison is presented in Fig. 7. Indeed, the in-plane optical properties (x vs y) of the biaxial film differ only by a few percent, indicating that the polyimide molecules were only slightly realigned.

FIG. 7.

Anisotropic optical constants of the PI2611 film in its uniaxial state before and its biaxial state after molecular realignment.

FIG. 7.

Anisotropic optical constants of the PI2611 film in its uniaxial state before and its biaxial state after molecular realignment.

Close modal

A closer inspection of the optical constants reveals subtle changes in PI2611’s vibrational bands, for example, in the out-of-phase carbonyl stretching modes. Upon rubbing, a peak arises in kx and ky at 1732cm1 in the biaxial state that was only visible as a small shoulder in the uniaxial state. This band component can be assigned to imide-ring distortions or perturbations of molecular packing, i.e., steric disturbations, which also causes a shift in the ν(C=O) oscillator force constant, and hence frequency, compared to the unperturbed component at 1715cm1.39 

The observed changes in band composition suggest that rubbing not only leads to an overall small realignment of the polyimide molecules, but that it also slightly disturbs the molecular packing. Our findings are in agreement with Hietpas and Allara39 who reported similar ν(C=O) band compositions and lineshapes for aligned, stacked layers of BPDA–PDA polyimides.

More detailed investigations of rubbing effects on molecular orientations and band compositions are the objective of upcoming studies. These first quantitative results, however, already demonstrate the use of IR 4×4 MM ellipsometry for sensitively probing small changes of biaxial anisotropy in nanometer-thin organic films. Future work will aim at in-depth characterizations of anisotropic and/or depolarizing polymer and protein thin films, also in combination with liquid flow-cells for in situ IR MM studies of such sample systems.

In the third example, we use the MM ellipsometer for detailed measurements of micrometer-sized SiO2 arrays whose structure elements have a rectangular base and a trapezoidal profile. A schematic and an SEM (scanning electron microscopy) topview of such arrays are depicted in Fig. 8. The structures were prepared from an H=1000nm thick SiO2 layer on an Si wafer by photolithographic processing with subsequent wet-chemical etching.17 Their characteristic structure parameters are the trapezoid’s period lengths (Px, Py), base widths (Bx, By), and top widths (Tx,Ty).

FIG. 8.

(a) Schematic topview of the trapezoidal SiO2 arrays on Si with marked structure parameters. (b) SEM image of the investigated surface. (c) Radial plots of measured (left) and fitted (right) MM elements and polarization index at azimuths between 0° (horizontal) and 90° (vertical).

FIG. 8.

(a) Schematic topview of the trapezoidal SiO2 arrays on Si with marked structure parameters. (b) SEM image of the investigated surface. (c) Radial plots of measured (left) and fitted (right) MM elements and polarization index at azimuths between 0° (horizontal) and 90° (vertical).

Close modal

As seen in the SEM image, the structures are not perfect. They exhibit some defects, such as holes and crooked edges, at every other trapezoid. Corners and slopes are also slightly rounded. These deviations from an ideal array are expected to influence the sample’s experimental Mueller matrix. As a result of the array’s geometrical complexity, with all profile parameters being in the micrometer range, the infrared MM is correspondingly rich in spectral features.

For a quantitative evaluation, it is important that such a structured surface no longer constitutes a homogenizable medium. Therefore, we aim to understand the measured IR spectra by combining MM ellipsometry with RCWA-based optical modeling. This approach allows us to quantify the array’s profile and orientation as well as to assess its fabrication quality. The latter is related to symmetries and polarization index,40 

PI=i,jMij2/M1121/3,
(7)

of the measured Mueller matrix.

Figure 8 shows measured and fitted broadband (7000800cm1) MM data at 50° incidence angle and azimuthal sample rotations between 0° and 90°. The data are presented in polar form, with radial distance and angular position corresponding to spectral position and azimuthal sample rotation, respectively.

The MM is dominated by the vibrational signatures of the trapezoid material’s SiO2-stretching modes, ν(SiO2), around 1100cm1, as well as by structure-related broad spectral patterns that are particularly intense on the MM’s 2×2 block diagonal. These spectral features are modulated upon sample rotation by the varying space between the periodically aligned trapezoidal structure elements.

The block-diagonal MM elements (M12, M21, M22; M33, M34, M43, M44) are nonzero at all azimuths. In contrast, the block-offdiagonal MM elements vanish along symmetry axes. This pseudoisotropic case approximately occurs at the nominal azimuths of 0° and 90°, where the plane of incidence coincides with the sample’s x- and y-axis, respectively. For all other sample rotations, the MM is fully populated and shows anisotropy effects in all matrix elements, such as distinctly different, azimuth-dependent lineshapes of the ν(SiO2) band in the various MM elements. It is the nonzero block-offdiagonal MM elements that render the MM highly sensitive toward the array’s orientation and structure parameters, enabling sufficient parameter decorrelation in a fit.

Data fitting was performed in SpectraRay 3 (Sentech Instruments GmbH) based on RCWA calculations.41,42 The SiO2 trapezoid structures were sliced into three vertical sublayers to approximate their slopes. Nine diffraction orders were taken into account to accurately model the short-wavelength spectral range. A numerical aperture of NA=0.04 was simulated to reproduce the impact of the ellipsometer’s opening angle on the observed polarization index, which would otherwise be 1 for an ideal array.

Figure 8 shows that measurement and fit are in remarkable agreement. The fitted MM data follow not only the broad spectral features but also the smaller effects that arise in the block-offdiagonal elements from the array’s unequal profile parameters in the x- and y-directions. Moreover, the different ν(SiO2) lineshapes are correctly described.

The fit reveals a sample orientation of 0.5° off its nominal axis. Results for array height and structure parameters are listed in Table I and compared to nominal values obtained from SEM image analysis. Also stated are the volumes of a single trapezoid and the unit cell of the periodic structures. From these, SiO2 surface coverages of less than 10% are deduced. IR MM ellipsometry combined with RCWA can resolve profile information at such low coverage because it is the array structures themselves that generate non-negligible entries in the block-offdiagonal MM elements. Both SEM and RCWA deliver similar results for all profile parameters. This demonstrates that IR MM ellipsometry bears great potential as a nondestructive method for structure quantification, rendering the technique highly relevant for process control and in-line monitoring.

TABLE I.

Nominal and fitted array profile parameters, volumes of single trapezoid and unit cell, as well as SiO2 surface coverage.

ParameterNominal (SEM)Fitted (RCWA)
H (μm) 1.00 ± 0.05 1.00 ± 0.02 
Px (μm) 20.8 ± 0.2 20.2 ± 0.5 
Py (μm) 10.3 ± 0.3 9.9 ± 0.6 
Bx (μm) 9.8 ± 0.2 10.6 ± 0.6 
By (μm) 4.4 ± 0.5 5.3 ± 0.4 
Tx (μm) 4.7 ± 0.8 5.1 ± 0.4 
Ty (μm) 1.4 ± 0.9 2.2 ± 0.4 
Vtrap (μm322 ± 8 30 ± 5 
Vunit (μm3449 ± 52 468 ± 57 
Coverage (%) 4.9 ± 2.0 6.6 ± 1.9 
ParameterNominal (SEM)Fitted (RCWA)
H (μm) 1.00 ± 0.05 1.00 ± 0.02 
Px (μm) 20.8 ± 0.2 20.2 ± 0.5 
Py (μm) 10.3 ± 0.3 9.9 ± 0.6 
Bx (μm) 9.8 ± 0.2 10.6 ± 0.6 
By (μm) 4.4 ± 0.5 5.3 ± 0.4 
Tx (μm) 4.7 ± 0.8 5.1 ± 0.4 
Ty (μm) 1.4 ± 0.9 2.2 ± 0.4 
Vtrap (μm322 ± 8 30 ± 5 
Vunit (μm3449 ± 52 468 ± 57 
Coverage (%) 4.9 ± 2.0 6.6 ± 1.9 

Polarization index and symmetries of the Mueller matrix are important measures regarding the array’s fabrication quality. Certain symmetries are expected as a consequence of the electromagnetic reciprocity theorem for Mueller matrices of symmetric structures.40,43 In particular, M12=M21, M13=M31, M23=M32, M14=M41, M24=M42, and M34=M43. These symmetries are indeed observed in the measured Mueller matrix. However, small asymmetries are found between the lower-left and upper-right 2×2 MM blocks, especially at larger azimuths. These differences also impact the MM’s polarization index.

To a large extent, the measured deviations of the polarization index from unity are a result of the ellipsometer’s opening angle and are thus partially described by the RCWA model. Measured and fitted polarization index agree in the material-specific ν(SiO2) signatures around 1100cm1 and in the geometry-related features around 5000 cm−1, 3500 cm−1, and 2100cm1. It turns out that the array’s particular combination of profile parameters renders the 48003800cm1 range most sensitive toward structure variations. The largest deviation between the polarization indices is thus found in this region. The differences are indicative of a slight lack of structural symmetry. We, therefore, attribute them to the above-mentioned defects visible with SEM. However, the absolute differences are small, corresponding to less than 0.5% depolarization and hence suggest macroscopic homogeneity of the array structure.

Future studies will investigate in more detail the relationship between polarization index and array nonidealities. Thus far, the polarization index proves to be a valuable quantity for improving the fit and to be a potential marker for assessing the quality of the array.

Lastly, we point out that SEM image analysis covers only a limited number of structure elements, whereas IR MM ellipsometry can probe much larger footprints, up to the complete array. Regarding sensor, coating, and photonics applications, ellipsometry can thus provide more reliable information on the performance of the array as a whole. Furthermore, SEM requires destructive cutting for sideview imaging and quantification of the array’s base and top widths. In contrast, the combined approach of IR MM ellipsometry and RCWA delivers quantitative information in a nondestructive manner. Knowledge of the profile parameters is crucial for further array processing, for anisotropy analysis of the used etching process, and for judging the optical performance of the 2D structure. Broadband infrared MM ellipsometry, therefore, proves promising as a versatile, nondestructive characterization tool.

IR Mueller-matrix ellipsometry provides detailed information on chemical composition, molecular interactions, and structural properties of organic thin-film and patterned surfaces by probing the material-specific vibrational bands and structure-related baselines found in the infrared. In this article, we have demonstrated in-depth sample analyses by combining IR MM ellipsometry with advanced optical simulations. For this purpose, we have developed a high-throughput-optimized broadband ellipsometer that achieves an unprecedented MM sensitivity of up to 5105. The instrument enables a tailored approach for optimizing measurement time and signal-to-noise by selectively focusing on quadruples of MM elements, thus resolving even weak IR signatures often observed in block-offdiagonal elements. We have presented different optical modeling approaches for quantitative MM data evaluation based upon anisotropic layer stacks and rigorous coupled-wave analysis.

We used the MM ellipsometer to investigate profile and orientation of 2D arrays of trapezoidal silica microstructures. This example illustrates the usefulness of IR MM ellipsometry for process control, optimization, and quality monitoring. RCWA-based theoretical calculations enabled us to verify the SiO2 chemical MM signatures and to quantify key structural parameters like profile height, periods, and widths. Analysis of the polarization index obtained from the MM data suggested macroscopic homogeneity of the structure. For future applications, the use of low-energy IR radiation combined with IR fingerprint analysis could allow nondestructive investigations and quality control of photosensitive polymer masks prior to further processing. IR MM ellipsometry thus holds great potential for identifying both chemical and structural problems like distortion effects during manufacturing and processing.

Various complex polymer thin films were investigated. Polyimide membranes used for gas separation exhibited substantial thickness heterogeneity and pronounced depolarization. The broad accessible spectral range of 8000800cm1 was fundamental to decorrelating and quantifying film thickness profile and uniaxial optical constants in a depolarization optical model. Furthermore, the ellipsometer’s high S/N enabled a sensitive block-offdiagonal MM analysis of a 32 nm thin polymer film with biaxial anisotropy, demonstrating thin-film sensitivity even for complex anisotropic samples. These examples show the potential of IR MM ellipsometry for investigations of polymer, protein, and related thin films. Future projects aim at in situ IR MM ellipsometry studies of solid–liquid interfaces.

Overall, IR MM ellipsometry provides new analysis possibilities in the fields of polymer, catalytic, and biomedical materials, as well as optoelectronic and sensing devices. Comprehensive characterization of thin films and complex surface structures are becoming feasible by tapping the vast, but so far almost unused, potential of this technique.

The authors are indebted to Matthias Zellmeier (HZB Berlin) for the preparation of the trapezoidal SiO2 arrays, to Timur Shaykhutdinov for expertise in RCWA simulations, to Uwe Richter (Sentech Instruments GmbH), as well as Ilona Engler, Cordula Walder, and Özgür Savaş for technical assistance, to Margarita Günther (TU Dresden) and Karin Sahre and Klaus-Jochen Eichhorn (IPF Dresden) for providing the PI2611 samples, and to Sentech Instruments GmbH for construction of the retarder units. Financial support through European Union grants (EFRE 1.8/13, ProFIT Nos. 10160255, 10160265, and 10160256) as well as by the Ministerium für Innovation, Wissenschaft und Forschung des Landes Nordrhein-Westfalen, the Regierende Bürgermeister von Berlin—Senatskanzlei Wissenschaft und Forschung, and the Bundesministerium für Bildung und Forschung is gratefully acknowledged. I.P. acknowledges support from KAUST (baseline funding No. BAS/1/1323-01-01).

1.
H. G.
Tompkins
and
E. A.
Irene
,
Handbook of Ellipsometry
(
William Andrew
,
Norwich
,
2005
).
2.
H.
Fujiwara
,
Spectroscopic Ellipsometry: Principles and Applications
(
Wiley
,
Chichester
,
2007
).
3.
M.
Losurdo
and
K.
Hingerl
,
Ellipsometry at the Nanoscale
(
Springer
,
Berlin
,
2013
).
4.
K.
Hinrichs
and
K.-J.
Eichhorn
,
Ellipsometry of Functional Organic Surfaces and Films
(
Springer
,
Berlin
,
2018
).
5.
J. J.
Gil Pérez
and
R.
Ossikovski
,
Polarized Light and the Mueller Matrix Approach
(
CRC
,
2016
).
6.
T. W. H.
Oates
,
T.
Shaykhutdinov
,
T.
Wagner
,
A.
Furchner
, and
K.
Hinrichs
,
Opt. Mater. Express
4
,
2646
(
2014
).
7.
F.
Carmagnola
,
J. M.
Sanz
, and
J. M.
Saiz
,
J. Quant. Spectrosc. Radiat. Transfer
146
,
199
(
2014
).
8.
A.
Arwin
,
A.
Mendoza-Galván
,
R.
Magnusson
,
A.
Andersson
,
J.
Landin
,
K.
Järrendahl
,
E.
Garcia-Caurel
, and
R.
Ossikovski
,
Opt. Lett.
41
,
3293
(
2016
).
9.
W.
Li
,
H.
Jiang
,
C.
Zhang
,
X.
Chen
,
H.
Gu
, and
S.
Liu
,
J. Vac. Sci. Technol. B
34
,
020602
(
2016
).
10.
A.
Peinado
,
M.
Kildemo
,
L. M. S.
Aas
,
C.
Martella
,
M. C.
Giordano
,
D.
Chiappe
,
F.
Buatier de Mongeot
,
F.
Borondics
, and
E.
Garcia-Caurel
,
Appl. Surf. Sci.
421
,
728
(
2017
).
11.
12.
M.
Wang
,
A.
Löhle
,
B.
Gompf
,
M.
Dressel
, and
A.
Berrier
,
Opt. Express
25
,
6983
(
2017
).
13.
A. T.
Martin
,
S. M.
Nichols
,
M.
Tan
, and
B.
Kahr
,
Appl. Surf. Sci.
421
,
578
(
2017
).
14.
B.
Song
,
H.
Gu
,
S.
Zhu
,
H.
Jiang
,
X.
Chen
,
C.
Zhang
, and
S.
Liu
,
Appl. Surf. Sci.
439
,
1079
(
2018
).
15.
A.
Mendoza-Galván
,
L. F.
del Río
,
K.
Järrendahl
, and
H.
Arwin
,
Sci. Rep.
8
,
6456
(
2018
).
16.
B.
Gompf
,
M.
Gill
,
M.
Dressel
, and
A.
Berrier
,
J. Opt. Soc. Am. A
35
,
301
(
2018
).
17.
A.
Furchner
,
C.
Walder
,
M.
Zellmeier
,
J.
Rappich
, and
K.
Hinrichs
,
Appl. Opt.
57
,
7895
(
2018
).
18.
J.-C.
Cigal
, “
A novel spectroscopic ellipsometer in the infrared
,” Ph.D. thesis (
Technische Universiteit Eindhoven
,
Eindhoven
,
2002
).
19.
R. A.
Chipman
, “Polarimetry,” in Handbook of Optics, edited by M. Bass (McGraw Hill, New York, 1995).
20.
R. M. A.
Azzam
,
I. M.
Elminyawi
, and
A. M.
El-Saba
,
J. Opt. Soc. Am. A
5
,
681
(
1988
).
21.
D.
Layden
,
M. F. G.
Wood
, and
I. A.
Vitkin
,
Opt. Express
20
,
20466
(
2012
).
22.
S. R.
Cloude
,
Proc. SPIE
1166, 177 (1990).
23.
R.
Ossikovski
,
M.
Anastasiadou
,
S.
Ben Hatit
,
E.
Garcia-Caurel
, and
A.
De Martino
,
Phys. Status Solidi (a)
205
,
7020
(
2008
).
24.
N. B.
McKeown
and
P. M.
Budd
,
Chem. Soc. Rev.
35
,
675
(
2006
).
25.
W.
Ogieglo
,
A.
Furchner
,
X.
Ma
,
K.
Hazazi
,
A. T.
Alhazmi
, and
I.
Pinnau
,
ACS Appl. Mater. Interfaces
11
,
18770
(
2019
).
26.
O.
Arteaga
,
Thin Solid Films
571
,
584
(
2014
).
27.
G. E.
Jellison
and
J. W.
McCamy
,
Appl. Phys. Lett.
61
,
512
(
1992
).
28.
H.
Hatori
,
Y.
Yamada
,
M.
Shiraishi
,
M.
Yoshihara
, and
T.
Kimura
,
Carbon
34
,
201
(
1996
).
29.
J. N.
Barsema
,
S. D.
Klijnstra
,
J. H.
Balster
,
N. F. A.
van der Vegt
,
G. H.
Koops
, and
M.
Wessling
,
J. Membr. Sci.
238
,
93
(
2004
).
30.
K.
Hazazi
,
X.
Ma
,
Y.
Wang
,
W.
Ogieglo
,
A.
Alhazmi
,
Y.
Han
, and
I.
Pinnau
,
J. Membr. Sci.
585
,
1
(
2019
).
31.
K.
Hinrichs
,
D.
Tsankov
,
E.-H.
Korte
,
A.
Röseler
,
K.
Sahre
, and
K.-J.
Eichhorn
,
Appl. Spectrosc.
56
,
737
(
2002
).
32.
K.
Hinrichs
,
J.
Rappich
, and
T.
Shaykhutdinov
, “Field manipulation of infrared absorption properties in thin films,”
Phys. Status Solidi B
(published online, 2019).
33.
N. A. J. M.
van Aerle
and
A. J. W.
Tol
,
Macromolecules
27
,
6520
(
1994
).
34.
R.
Buchhold
,
A.
Nakladal
,
U.
Büttner
, and
G.
Gerlach
,
Meas. Sci. Technol.
9
,
354
(
1998
).
35.
J.
Stöhr
and
M. G.
Samant
,
J. Electron Spectros. Relat. Phenomena
98–99
,
189
(
1999
).
38.
M.
Yamahara
,
M.
Nakamura
,
N.
Koide
, and
T.
Sasaki
,
Liq. Cryst.
34
,
381
(
2007
).
39.
40.
E.
Garcia-Caurel
,
R.
Ossikovski
,
M.
Foldyna
,
A.
Pierangelo
,
B.
Drévillon
, and
A.
De Martino
, “Advanced Mueller ellipsometry instrumentation and data analysis,” in Ellipsometry at the Nanoscale, edited by M. Losurdo and K. Hingerl (Springer, Berlin, 2013).
41.
M. G.
Moharam
and
T. K.
Gaylord
,
J. Opt. Soc. Am.
71
,
811
(
1981
).
42.
M. G.
Moharam
and
T. K.
Gaylord
,
J. Opt. Soc. Am.
72
,
1385
(
1982
).
43.
L.
Li
,
J. Opt. Soc. Am. A
17
,
881
(
2000
).