Abstract

We present a high-optical-throughput infrared Mueller-matrix (MM) ellipsometer for the characterization of structured surfaces and ultrathin films. Its unprecedented sensitivity of about 104 in the normalized MM elements enables studies of the complex vibrational fingerprint of thin organic films under different ambient conditions. The ellipsometer acquires quadruples of MM elements within a few 10 s to min, rendering it interesting for process and in-line monitoring. It uses retractable achromatic retarders for increased signal to noise, and tandem wire-grid polarizers for improved polarization control. We demonstrate several scientific and industry-related applications. First, we determine the 3D profile of μm-sized trapezoidal SiO2 gratings on Si from azimuth-dependent MM measurements. Data modeling based on rigorous coupled-wave analysis is employed to quantify grating structure and orientation. We then monitor polymer relaxation processes with a time resolution of 47 s. Measurements of polymer films as thin as 7.7 nm illustrate the sensitivity of the device. We finally couple a liquid flow cell to the ellipsometer, highlighting the prospects for in situ infrared MM studies of thin films at solid–liquid interfaces.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Ellipsometry is a well-established, nondestructive and noninvasive linear-optical technique for characterizing the properties of materials by means of polarized light [14]. The method has been extended over the years to cover the infrared (IR) spectral range, and is nowadays successfully used to measure a plethora of ambient- and stimuli-dependent physical and chemical sample properties. These include film thicknesses, refractive and absorption indices, free-carrier concentration, mobility and effective masses, chemical composition, interdiffusion and cross-linking, as well as molecular adsorption, orientation, and molecular interactions [514].

Due to the ever-growing demand in the scientific community and from industry to investigate samples of increasing complexity, such as structured, biomimetic, and thin-film-covered surfaces, there is an increased need for optical methods such as ellipsometry to provide access to the sample’s dielectric, structural, anisotropy, and depolarizing properties. Depolarizing samples in particular cannot be addressed by classical ellipsometry in the Jones formalism. Within the Stokes formalism, though, all of the aforementioned properties are accessible via the sample’s 4×4 Mueller matrix (MM)—a general description of the sample’s polarimetric properties that fully characterizes its polarization response [15]. MM ellipsometry has therefore gained recent attention as an advanced characterization technique in science and metrology [1624].

MM ellipsometry is especially interesting in the IR where excitations of vibrational modes can be studied. IR MM ellipsometry thus opens up new analytical possibilities for investigations of thin films and structured surfaces, in particular with regard to molecular orientation, binding, and interactions at interfaces. However, laboratory IR ellipsometry is intrinsically challenging due to the low optical throughput, the nonideal properties of optical components such as wire-grid polarizers, the low brilliance of radiation sources such as globars, and the low specific detectivity of frequently used room-temperature-operated broadband IR detectors.

Our premise for this paper was to design and build a laboratory IR MM ellipsometer with as high an optical throughput as possible, capable of performing highly sensitive MM measurements of nanometer-thin films and surfaces with μm-sized structures. Commercially available IR MM ellipsometers mostly use pyroelectric detectors, and are not optimized for high optical throughput. Furthermore, common MM ellipsometers operate with fixed polarizers and rotating retarders in order to sufficiently sample the polarization space with mixed polarization states. Conveniently, these ellipsometers can measure the complete 4×4 Mueller matrix, or the partial MM if only one retarder is used. However, typical IR retarders based on total internal reflections exhibit maximum transmittances of merely 50% to 70%. In a dual-compensator design, all measurements are made in the presence of the two retarders, and hence all MM elements are affected by low throughput. While such a design can be advantageous for spreading experimental noise across the various MM elements [25,26], it comes at the expense of sensistivity. This is especially relevant for MM elements in the upper-left 2×2 block, which, for non-depolarizing samples, are associated with the amplitudes of the anisotropic complex reflection coefficients [27]. These MM elements usually contain key information in the IR for many samples, and can be measured most sensitively without retarders.

For these reasons, we opted for a different design approach, namely, to work with a liquid-nitrogen-cooled photovoltaic detector for increased detectivity, with non-continuously rotating tandem polarizers for optimized polarization control, and with optional and retractable retarders for enhanced throughput. Furthermore, all optical elements were chosen or designed large enough not to act as beam-restricting apertures. This novel IR ellipsometer generates and projects almost pure polarization states. It acquires subsets of four MM elements at a time. In detail, the upper-left 3×3 block of the MM is obtained without retarders for maximum signal-to-noise ratio, whereas the fourth row and/or column is measured with retarder(s) inserted into the optical path. Besides reduced spectral noise, this approach offers increased temporal resolution (<1  min) if only parts of the Mueller matrix are of interest. It also allows one to selectively spend less or more measurement time on specific MM elements, which is very useful when certain elements show stronger or weaker IR signatures, respectively.

The ellipsometer covers the spectral range of 7000800  cm1, and can reach root mean square (RMS) noise levels below 104 in the normalized MM elements. IR MM ellipsometry with such sensitivity is of high interest for research of thin-film materials with complex anisotropy and/or depolarizing properties. With sub-minute time resolutions for partial MM measurements, the ellipsometer bears great potential for industrial applications such as in-line monitoring or process and quality control. It is also highly relevant for numerous scientific investigations, for instance, of adsorption or growth processes at solid–liquid interfaces, which demand both thin-film sensitivity and sufficiently short measurement times.

This paper is outlined as follows: in the first part, we give a brief overview of the optical set-up, introduce the measurement principle and necessary equations for obtaining the sample’s MM, and then address issues of instrument alignment, calibration, and accuracy. In the second part, we demonstrate the capabilities of the new MM ellipsometer and provide several examples of measurements and modeling of complex samples. These include structured gratings on silicon wafers and thin polymer films under various ambient conditions.

2. EXPERIMENTAL SET-UP AND CHARACTERIZATION

A. Optical Configuration

The IR MM ellipsometer is schematically depicted in Fig. 1. It operates both in reflection, with possible incidence angles between ϕ0=45° and 90°, and in transmission with arbitrary ϕ0. Transmission mode is useful for measuring transparent bulk and thick-film samples, but also for instrument calibration, as will later be explained in detail. Reflection mode provides the highest sensitivity for characterizing nanometer-thin films.

 

Fig. 1. Schematic of the IR Mueller-matrix ellipsometer operating in reflection mode (top) with autocollimator (AC) and adjustable sample stage, or in transmission mode (bottom) with rotatable sample mount.

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The input arm of the ellipsometer is fixed, whereas the sample stage and analyzer arm are mounted on a motorized 2ϑ goniometer stage independently angle-adjustable within the plane of incidence (2-Circle Goniometer 423, Huber Diffraction and Positioning Equipment, Germany). A standard Fourier-transform IR (FT-IR) spectrometer (Tensor 37, Bruker, Germany) with a globar as radiation source is coupled to the ellipsometer.

The ellipsometer itself consists of two sets of rotating tandem polarizers (polarizer and analyzer), a tilt- and depth-adjustable sample rotation stage with autocollimation unit, optional and retractable non-rotating retarders, and a highly linear photovoltaic mercury-cadmium-telluride (MCT) detector (KLD-1-J1-3/11, Kolmar Technologies, USA).

The two tandem polarizers are each comprised of two properly aligned KRS-5 wire-grid polarizers (25 mm clear aperture, 0.25 μm wire spacing, Specac Ltd, England) in order to suppress unwanted leaking polarization states, thereby improving the polarization control of the ellipsometer.

Our retarder design consists of a KBr prism for two phase-shifting attenuated total reflections, and two gold mirrors to bring the transmitted light back onto the axis of the ellipsometer arm [28]. The prism’s facet angles (51.5°) and the incidence angles on the gold mirrors (25°) were chosen to achieve an almost achromatic response very close to a 90° phase shift between p- and s-polarization over the entire spectral range. The retarder unit is mounted on a motorized translation stage that reproducibly moves the retarder into, or out of, the optical path.

An off-axis parabolic mirror focuses the IR globar radiation exiting the spectrometer onto the sample, resulting in a range of incidence angles and a small spot size dependent on the FT-IR spectrometer’s Jacquinot aperture. For a maximum aperture setting of 1.5 mm, which still guarantees a linear detector response, the spot size perpendicular to the incidence plane is about 2 mm. Incidence angles range between ϕ0±2.4°, as is typical for IR ellipsometers with non-brilliant radiation sources. Optical modeling of experimental data can, if necessary, easily take into account, such as angle distributions.

The outgoing light passing through the analyzer is focused onto the detector’s 1  mm2 large detection element. Importantly, the detector is also mounted on a translation stage to ensure optimal focusing with maximum photon collection.

The whole instrument, including the FT-IR spectrometer, is constantly purged with dried air (r.H.0.1%) to provide atmospheric stability with minimal IR absorption due to CO2 and H2O vapor.

B. Measurement Principle

The ellipsometer works with a polarization-state generator (PSG) in the input arm and a polarization-state analyzer (PSA) in the output arm that sample the polarization space with a sufficient number (four) of polarization states in order to calculate quadruples of MM elements. PSG and PSA comprise a polarizer and a retractable retarder. Data collection is based on a rotating-polarizer/rotating-analyzer configuration. For maximum optical throughput we opted to acquire the upper-left 3×3 MM block without retarder, whereas the fourth row and/or column is obtained with retarder in the optical train.

Using the MM calculus [27], the output Stokes vector Sout at the detector, characterizing the light’s polarization state after interaction with sample and ellipsometer optics, is derived from

Sout=PSAT·M·PSG=D·A·R2·[M11M12M13M14M21M22M23M24M31M32M33M34M41M42M43M44]·R1·P·Sin,
where M is the Mueller matrix of the sample; P and A are the polarizer and analyzer matrices; Rm are the retarder matrices, ideally those of a linear achromatic retarder [15]; 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 MM. Ideally, D is polarization insensitive with its first row being D1j=[1,0,0,0].

Ideal PSGs and PSAs can generate and analyze the pure orthogonal polarization states [1,±1,0,0]T, [1,0,±1,0]T, and [1,0,0,±1]T. As illustrated in Fig. 2, corresponding intensity measurements are always selective towards four MM elements. Hence, four intensity measurements with two of these ± vectors in the PSG and PSA lead to a set of four equations that are solvable for a subset of four MM elements [27].

 

Fig. 2. Quadruples of MM elements obtaintable via different combinations of polarizer/analyzer/retarder settings.

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These equations have to be generalized when source prepolarization, polarizer diattenuation, and other nonidealities come into play. For this, we parameterized the tandem polarizers as linear diattenuators [27] with diattenuation D=1c1+c, where c is the inverse extinction ratio (c=r/q in Ref. [27]).

We first evaluate Eq. (1) without retarders (R1=R2 = identity matrix), for negligible circular source prepolarization (s3=0), keeping only terms with expected contributions >0.0001. Quadrupels of MM elements are then acquired via

M11=[X1/(1+c)2si/s0X2/(1c2)]/[1si2/s02]/X˜,M1b=[X2/(1c2)si/s0X1/(1+c)2]/[1si2/s02]/X˜,Ma1=[X3/(1c2)si/s0X4/(1c)2]/[1si2/s02]/X˜,Mab=[X4/(1c)2si/s0X3/(1c2)]/[1si2/s02]/X˜,
with X˜=X˜1/(1+c)2, with the intensity sums
X1(P,A)=IP,A+IP,A+90°+IP+90°,A+IP+90°,A+90°,X2(P,A)=IP,A+IP,A+90°IP+90°,AIP+90°,A+90°,X3(P,A)=IP,AIP,A+90°+IP+90°,AIP+90°,A+90°,X4(P,A)=IP,AIP,A+90°IP+90°,A+IP+90°,A+90°,
and corresponding sums X˜n=Xn(Air) determined for air in transmission without sample, as well as with
i=1forP=0°,i=2for  P=45°.
These sums are obtained from intensity measurements Iα1,α2 at four different polarizer/analyzer settings α1/α2. This way, the following combinations [M11, M1b, Ma1, Mab] can be measured:
a=2&b=2for  P=0°,A=0°,a=3&b=3for  P=45°,A=45°,a=2&b=3for  P=45°,A=0°,a=3&b=2for  P=0°,A=45°.
The remaining elements in the fourth column/row of the MM are accessible by inserting the first and/or second retarder into the light path. Measurements are performed in analogy to Eq. (2). Instead of the MM elements Mij of the sample, this yields the MM combinations [M·R1]i3, [R2·M]3j, and [R2·M·R1]33 of sample with retarder(s). For well-aligned, close-to-ideal achromatic retarders, one then receives the sample MM elements from
Mi4=[M·R1]i3/R1,43,i=1,2,3,
M4j=[R2·M]3j/R2,34,j=1,2,3,
M44=[R2·M·R1]33/(R1,43R2,34),
where Rm,34=Rm,43=(Rm,112Rm,122Rm,332) is obtained from calibration measurements of the retarders acting as a sample. The case of nonideal retarders is discussed in Appendix A.

The above formulas require knowledge of polarizer diattenuation and source prepolarization, both of which can be determined from transmission measurements X˜n of air:

D=1c1+c=X˜4(0°,0°)X˜1(0°,0°)=X˜4(45°,45°)X˜1(45°,45°),s1s0=1D·X˜2(0°,0°)X˜1(0°,0°),s2s0=1D·X˜2(45°,45°)X˜1(45°,45°).
These pre-characterization measurements should be performed regularly because the globar radiation, and hence the interferometer illumination, may vary over time. The left panel of Fig. 3 shows that the light exiting the FT-IR spectrometer exhibits significant linear prepolarization (s1/s0, s2/s0) and negligible circular polarization (s3/s0). Obviously, such amounts of prepolarization have to be accounted for in the MM calculation, since the polarizer is not stationary. Simplifications for fully polarized light sources are addressed in Appendix B. The right panel of Fig. 3 displays the diattenuation of the tandem polarizers, which is greatly improved compared to that of standard single wire-grids, enabling accurate measurements above 3500  cm1.

 

Fig. 3. Left: intensity-normalized Stokes-vector components of the FT-IR source, as measured in front of the first polarizer. Right: diattenuation of single and tandem wire-grid polarizers.

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A crucial ellipsometer component is the detector. Its response must be highly linear because of the extreme dynamic range of the detected signals. Otherwise, the MM will suffer from unrecoverable errors. Nonlinearity manifests itself as a nonzero baseline in Iα1,α2 below the detector cutoff (here at 800  cm1) and, in extreme cases, as obvious modulation artifacts at the high-wavenumber end of the Iα1,α2 spectrum [29]. Since the ellipsometer operates with a rotating analyzer, the detector should also be polarization insensitive, obeying D1j=[1,0,0,0].

Linearity of the detector was tested via transmission measurements of air by restricting the IR beam waist using different iris diaphragms. The detector was found to be linear when operated below 90% of its maximum dynamic range. Measuring the detector’s polarization sensitivity via D1j=[1,D12,D13,0], with

D12=s1/s0+(1+c)2·X˜3(0°,0°)X˜1(0°,0°),
D13=s2/s0+(1+c)2·X˜3(45°,45°)X˜1(45°,45°),
we found that |D12| and |D13| were smaller than 2·106 over the entire spectral range, which is well below the noise level of measured thin-film MM spectra.

C. Alignment and Calibration of Optical Components

A visible (VIS) laser inserted in the optical path was used to determine the ellipsometer’s central axis, the plane of incidence perpendicular to that axis, and the alignment of all optical elements with respect to that plane [30]. A major cause of systematic errors related to misalignments is beam wander. It occurs when the light’s path through the set-up is disturbed upon interaction with the optical parts, or because of environmental changes such as temperature drift. Beam wander can result in differently illuminated areas on the detector element, leading to baseline errors in the calculated MM. We used the VIS laser to meticulously adjust polarizer and retarder alignment for minimal beam wander and maximum transmission of IR radiation.

Deviations in sample tilt from the plane of incidence also induce beam-wander-like effects on the detector. The same holds true if the sample plane does not cross the central ellipsometer axis. While Ψ/Δ measurements are pretty robust against those misalignments, we found that proper sample alignment is mandatory for reliably measuring the block-offdiagonal MM elements. The correct sample tilt (to within 1 arcmin) is set utilizing an autocollimation stage with VIS light under normal incidence. Sample depth is then adjusted maximizing the photon count at the detector. The additional VIS laser can be exploited to ensure that light reflected off the sample travels along the same path as during instrument calibration in transmission.

With all optical elements properly aligned, the polarizer offsets with respect to the plane of incidence could be determined. This was achieved to ±0.05° accuracy via calibration and sample measurements [Eqs. (9) and (2)] of isotropic thin polymer films. In a minimization procedure, polarizer and analyzer offset were varied until the block-offdiagonal MM elements showed baselines of zero with vanishing polymer vibrational bands. Retarder alignment was performed in a similar fashion orienting the fast axes to 0° for vanishing block-offdiagonals. Experimentally, the phase shifts were within sinΔRet=0.997 and 0.999.

D. Instrument Operation and Characterization

One advantage of using tandem polarizers is working with much better defined, almost pure polarization states. A drawback is that fast intensity measurements with crossed polarizers can become difficult because the centerburst of the interferogram is buried within the noise, which leads to baseline errors at the high- and low-wavenumber end of the Fourier-transformed spectrum. We therefore advise to measure cross polarization by accumulating a larger number N of interferogram scans.

In the following, we consider M11-normalized MM elements. In order to assess accuracy, noise propagation, and measurement time, we performed MM measurements of air in dependence of N. Noise is minimized because the ellipsometer works with almost pure polarization states inscribing an octahedron in the Poincaré sphere [25]. The ellipsometer achieves remarkable and unprecedented signal-to-noise levels in the mid-IR range between 4000  cm1 and 800  cm1, as shown in Fig. 4. The noise follows a 1/N behavior, as is expected for additive Gaussian noise. For air, as well as for samples with high reflectivities such as thin films on metals, an RMS noise of about 0.0001 (0.00005) is reached for N=32 (128) at 4  cm1 spectral resolution and 40 kHz scanner velocity. Compared to MM elements in the fourth row and column, elements in the upper-left 3×3 block exhibit 1.5–1.8 times less noise, as they are measured without retarders.

 

Fig. 4. RMS noise (top) between 4000  cm1 and 1000  cm1 of selected MM elements (bottom) of dry air at two different spectral resolutions. The insets show examplary intensity spectra measured in an unpurged (humid) and purged (dry) ellipsometer chamber.

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With the above settings, acquisition of a MM takes about 13 min (32 min). For N64, measurement times are dominated by the time it takes to position polarizers and retarders. Using faster polarizer rotation and retarder positioning velocities, temporal resolutions of 2 min seem feasible for a complete MM acquisition with acceptable RMS noise; 15 s should be possible for a partial measurement of four MM elements.

3. RESULTS AND DISCUSSION

A. Structured Trapezoidal SiO2 Gratings

A first application of the new MM ellipsometer is the profile determination of μm-sized trapezoidal SiO2 gratings on silicon substrates, schematically depicted in Fig. 5. The gratings were prepared by chemical vapor deposition of a 1000 nm thick SiO2 layer on a Si wafer, followed by photolithography processing (transfer of the mask’s shadow cast, 7.5 s illumination with UV light, development of photo resist, and buffered oxide etch with HFNH4F=12.5%87.5% to remove exposed oxide).

 

Fig. 5. Structure parameters of trapezoidal SiO2 gratings on Si.

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Structure parameters of the gratings are their height (H), the trapezoid lengths at the base (Bx, By) and top (Tx, Ty), and their period lengths (Px, Py). All parameters are on the order of μm and therefore give rise to complex MM spectra in the IR spectral region, especially if the gratings have unequal structure sizes in x and y directions.

Figure 6 shows measured and fitted MM data of a SiO2 grating in dependence of azimuthal sample rotation. If the plane of incidence is parallel or perpendicular to a symmetry axis (e.g., at 0° azimuth), the MM looks pseudo-isotropic, i.e., all block-offdiagonal elements are zero. Upon azimuthal rotation, anisotropy effects are rotated through the various matrix elements rendering the block-offdiagonals highly sensitive towards the grating’s profile parameters and orientation.

 

Fig. 6. Measured azimuth-dependent MM data (ϕ0=50°) of a trapezoidal SiO2 grating on Si compared to fitted data according to the optical model in Fig. 5. The fourth column shows several symmetries of the Mueller matrix. Measured spectra were smoothed at the high-wavenumber end because of noise due to the grating’s low reflectivity (<0.2). Gray and colored lines are raw and smoothed data, respectively. The legend lists nominal azimuths (0.5° offset).

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The MM of symmetric structures is expected to exhibit various symmetries [15,31] as a consequence of the electromagnetic reciprocity theorem, e.g., M12M21=0, M13+M31=0, and M23+M32=0. Broken symmetries in measured MM data can be regarded as markers for deviations from an ideal grating. As shown in the fourth column in Fig. 6, the symmetries are observed up to differences from zero of about 0.3% at larger azimuths. We attribute these deviations to defects also seen with scanning electron microscopy (SEM).

In order to quantify grating profile and orientation, we performed rigorous coupled-wave analysis (RCWA) simulations [32,33] (SpectraRay software, SENTECH Instruments GmbH) to fit the measured MM data. As seen in Fig. 6, the experimental MMs are well described by the fitted ones. An overall offset in sample azimuth of 0.5° from the nominal orientation was determined from the fit. Profile parameters as a result of RWCA are listed in Table 1 and compared to nominal values obtained from SEM measurements. The two methods agree remarkably well, yielding very similar results. Ellipsometry, however, is clearly advantageous, as it is nondestructive, much faster than SEM, and theoretically deployable for process control and in-line monitoring in various environmental conditions.

Tables Icon

Table 1. Nominal and Fitted Profile Parameters of the Measured SiO2 Grating: Height (H), Periods (Px,y), Base, and Top Lengths (Bx,y, Tx,y)a

B. Time-Resolved Monitoring of Polymer Relaxation

To demonstrate the achievable time resolution of the ellipsometer, we monitored changes in the transmission MM of a multidirectionally stretched, 11 μm thick low-density polyethylene foil (Papstar GmbH, Germany). The measurement geometry is depicted in Fig. 7 at the bottom-right. The data in the figure show measured MM differences between the foil’s non-equilibrium states under tension and the equilibrium state after 1585 s of relaxation. We chose to acquire selected representative MM elements, in this case [M11, M12, M31, M32], resulting in a temporal resolution of 47 s.

 

Fig. 7. Time-resolved (47 s) transmission MM difference spectra measured between tensioned and relaxed states of a stretched low-density polyethylene (LDPE) foil. The schematic (bottom-right) shows the measurement geometry of the foil stretched across the sample stage.

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In a simplistic view, the MM elements M12 and M31 can be associated with xy and ±45° linear dichroism, respectively, while M32 can be related to circular birefringence [27]. All three elements are nonzero, which is expected because the polymer foil was stretched in multiple directions within the sample plane. The foil’s effective optical axes are thus oriented in some oblique angles with respect to the ellipsometer coordinate system.

The vibrational signatures around 2900  cm1 and 1460  cm1 correlating with the foil’s vibrational fingerprint have distinct shapes in the various entries of the MM, and decrease in amplitude as the polymer foil relaxes and becomes less anisotropic. Upon relaxation, the foil thickness increases, causing shifting and diminishing interference oscillations in δM12 and δM31. The observed nonzero baselines of the different MM elements in the non-equilibrium states could be a result of stretching-induced directional differences in polymer density leading to anisotropy of the complex refractive index. Optical modeling can, in principle, quantify such differences but is beyond the scope of this paper. Similarly, optical modeling could reveal in detail which of the above-discussed MM elements are affected by rotation, retardation, and diattenuation, as well as by depolarization.

Sub-minute time resolution and access to a sample’s anisotropy and depolarization properties render IR MM ellipsometry a promising technique for rheology, relaxation, and related studies, again with applications in process and quality control.

C. Ex Situ Studies of Thin and Ultrathin Polymer Films

We now utilize the ellipsometer to measure, and consequently characterize, vibrational MM spectra of nanometer-thin polymer films. As model surfaces, we investigate silicon substrates covered with thin films of spincoated and annealed poly(glycidylmethacrylate) [PGMA] [13,34], a widespread polymer finding applications in surface functionalizations and as a molecular linker. We prepared three PGMA films of 118.0 nm, 68.0 nm, and 7.7 nm thicknesses, as schematically depicted in Fig. 8, and measured them in reflection at ϕ0=65° incidence angle. Figure 9 shows experimental and fitted MMs. All block-offdiagonal MM elements are zero, indicating that the films are isotropic. The remaining nonzero elements are connected to the classical amplitude ratio tanΨ and phase difference Δ via M12=M21=cos2Ψ, M33=+sin2ΨcosΔ, and M43=sin2ΨsinΔ.

 

Fig. 8. Thin PGMA films (118 nm, 68 nm, and 7.7 nm) on Si substrates studied in reflection at the solid–air interface.

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Fig. 9. Measured (colored) and fitted (black) MM spectra of thin, isotropic PGMA films of various thicknesses.

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The PMGA spectra are dominated by the carbonyl-stretching vibrational band [ν(CO)] around 1735  cm1 as well as several bands between 1300  cm1 and 1100  cm1 associated with vibrations of the polymer’s ester and epoxy groups. Other, less intense, bands between 1500  cm1 and 1300  cm1 and below 1000  cm1 are related to alkyl bending and epoxy ring-deformation modes, respectively. All PGMA bands are resolved in the measured data of the two thicker films. For the ultrathin film, the carbonyl and ester bands are well above the limit of detection (3σ level), and so are the vibrational features around 1200  cm1 from the 1.2 nm thin native SiO2 layer. The weaker PGMA fingerprint bands are slightly above the noise level, which is about η(Mij)=±0.00015, η(Ψ)=±0.005°, and η(Δ)=±0.02° between 2000  cm1 and 1000  cm1. Those weak bands could, nonetheless, be resolved by accumulating more interferogram scans.

For all three films, we used the MM data to calculate polarization degree [35] and phase polarization degree [36]:

P=i,jMij2M1123M112,Pph=M332+M4321M122/M222.
While the polarization degree is very close to 1 in all cases, the phase polarization degree equals 1 only for the 7.7 nm ultrathin film but slightly drops below unity with increasing film thickness. This observation correlates well with atomic force microscopy (AFM) measurements, which show thickness variations of about ±15  nm for the thickest film, and a homogeneously smooth surface for the ultrathin one.

The measured data were fitted with an isotropic layer model describing PGMA’s dielectric function by a sum of vibrational oscillators associated with the various molecular vibrations [37]. Film thickness as well as oscillator shapes and amplitudes were fitted simultaneously for the thickest film. The obtained dielectric function was then used to determine the thicknesses of the two thinner films. One might expect that the annealing process during film preparation induces certain ordering or reorientation effects of the polymer segments. Some uniaxial anisotropy might therefore be present, especially for the thinnest film, in which anchoring to the substrate leads to the formation of tail–train–loop conformations [34]. However, employing a uniaxial model did not improve the fit results, indicating that the films are indeed isotropic.

The fact that the vibrational spectra of the 7.7 nm thin film can be modeled with the isotropic dielectric function of the thicker film implies that both films are structurally and chemically identical. Otherwise, effects such as band shifts or changes in band composition would be observed. Position and shape of the ν(CO) band, for instance, are prominent markers for molecular interactions and chemical environment [14], which seem to be very similar for the three films.

These first IR MM studies of thin films show that the technique has great potential for thin-film analysis, particularly with respect to film homogeneity via the depolarization information contained in the MM, and to film anisotropy and molecular interactions correlated with the vibrational fingerprint.

D. In Situ Studies of Polymer–H2O Interfaces

The previously discussed 68.0 nm thick PGMA film now serves as a test surface for the first in situ MM study of a thin organic film at the solid–liquid interface to water. The film was prepared on a wedge-shaped (1.5°) silicon substrate and is hence suitable for usage in our in situ flow cells [38]. With the cell tilted within the plane of incidence to select the inner reflex (Fig. 10), the wedge’s backside was illuminated under ϕ0=57.3° resulting in a non-attenuated total reflection (nATR) incidence angle of 12.6° at the Si–PGMA interface.

 

Fig. 10. Left: liquid flow cell with thin polymer film on Si wedge measured in backside reflection. Right: measured and fitted in situ ellipsometric data of a 68 nm thick, isotropic PGMA film at the polymer–air and polymer–water interface.

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Measured and fitted spectra of the film, first in contact with air and then with water at room temperature, are plotted in Fig. 10. Being an isotropic sample, only the block-diagonal MM elements are nonzero, and it is thus equivalent to calculate and analyze conventional Ψ and Δ spectra. In absolute ellipsometric spectra, the strong vibrational bands of bulk water domininate and mask the polymer bands [37]. The figure therefore shows referenced spectra with respect to a clean silicon-wedge surface. As a result, upward-pointing PGMA bands and downward-pointing residual water bands are observed. The bending mode δ(H2O) at 1650  cm1 and the libration modes νL(H2O) below 1000  cm1 [39] are a consequence of the difference in optical contrast at the substrate–film–ambient and the substrate–ambient interfaces. Amplitudes and shapes of the water bands are related to PGMA film thickness and hydration [14].

Optical modeling based on an effective-medium approach [14] confirms the hydrophobic nature of the film. There is no substantial amount of swelling (<1  nm thickness change, <5% water content) and no significant redshift of the carbonyl stretching mode, ν(CO). The latter would point towards the presence of hydrogen-bond interactions between the polymer’s ester group and surrounding water molecules.

These first in situ measurements of isotropic films with the new MM ellipsometer compare well with data obtained with established in situ IR ellipsometry [37]. The instrument thus proves to be promising for studies of more complex samples with thin-film anisotropy and/or structure, which is the aim of ongoing and future projects.

4. CONCLUSIONS AND OUTLOOK

We developed a broadband IR MM ellipsometer with high optical throughput for sensitive MM measurements of ultrathin films and structured surfaces. The ellipsometer was successfully used to obtain high-quality MM data of complex samples, such as trapezoidal oxide gratings, anisotropic polymer foils, and ultrathin polymer films. In conjunction with optical calculations based on RCWA and different layer models, we demonstrated the ellipsometer’s applicability for various research questions from the thin-film community but also for industry-related problems such as quality and process control.

The device achieves sensitivities of less than 104 in the normalized MM elements, which opens up new possibilities for polarimetric measurements. It is expected that this unrivaled sensitivity can be even further increased by at least 30% by using BaF2 wire-grid polarizers, which have better transmissivity and diattenuation properties than KRS-5 polarizers. Furthermore, a tunable IR laser could be coupled to the instrument [40] providing additional noise reduction with simultaneously increased temporal resolution.

We anticipate that the ellipsometer will help to satisfy the growing demand from science and industry for a sensitive and accurate IR MM ellipsometer to characterize complex samples.

APPENDIX A: FORMULAS FOR NONIDEAL RETARDERS

If the phase shift of the retarders is not exactly ΔRet=±90°, their MM element R33=sin2ΨRetcosΔRet will deviate from zero. In this case, it is no longer possible to acquire a fourth-row or fourth-column sample MM element from only four intensity measurements. Equations (6)–(8) have to be modified accordingly, now requiring additional prior measurements of the MM elements Mi3 and M3j:

Mi4=[M·R1]i3Mi3R1,33R1,43,i=1,2,3,
M4j=[R2·M]3jM3jR2,33R2,34,j=1,2,3,
M44=[R2·M·R1]33M33R1,33R2,33R1,43R2,34M43R1,33R1,43M34R2,33R2,34.
If the retarders exhibit substantial linear dichroism (R12=R21=cos2ΨRet0), the elements M12, M21, and M22 have to be measured first. It is also advised to acquire normalized MM elements from M11 measurements at the respective retarder settings. Corresponding equations are as follows.

Fourth-column elements:

Mi4M11=[M·R1]i3[M·R1]11·(1+M12M11R1,21R1,11)Mi3M11R1,33R1,11R1,43R1,11,i=1,2,3.
Fourth-row elements:
M4jM11=[R2·M]3j[R2·M]11·(1+M21M11R2,12R2,11)M3jM11R2,33R2,11R2,34R2,11,j=1,2,3.
44-element:
M44M11=[R2·M·R1]33[R2·M·R1]11·ZM33M11R1,33R1,11R2,33R2,11R1,43R1,11R2,34R2,11M43M11R1,33R1,43M34M11R2,33R2,34
with
Z=1+M12M11R1,21R1,11+M21M11R2,12R2,11+M22M11R1,21R1,11R2,12R2,11.

APPENDIX B: LINEARLY POLARIZED LIGHT SOURCES

The MM calculation after Eq. (2) can be simplified when fully linearly polarized light sources such as IR lasers are coupled to the ellipsometer. For this, the input linear polarization must be oriented at ±45° with respect to the first polarizer’s eigenpolarization directions. This means that the laser has to be rotated to 45° for P=0°, and to 0° for P=45°, depending on which two rows of the MM are to be projected out by the PSG after Eq. (4). Alternatively, another tandem polarizer oriented at 45° and 0°, respectively, could be used between the laser and first polarizer. The correct orientation can be tested with calibration measurements via Eq. (9) showing zero prepolarization either in linear (s1/s0) or in diagonal direction (s2/s0), respectively.

With the above measurement scheme, the prepolarization sensitivity in Eq. (2) vanishes, leaving only correction factors arising from the polarizers’ diattenuation:

M11=X1/X˜1,M1b=X2/X˜1/D,Ma1=X3/X˜1/D,Mab=X4/X˜1/D2.

Funding

European Union EFRE program (10153595); EFRE ProFIT grant (10160255, 10160265).

Acknowledgment

The authors are indebted to Christoph Kratz for fruitful discussions and preparation of a gold-coated glass calibration sample, to Sebastian Rauch (IPF Dresden) for preparation of PGMA films, to Timur Shaykhutdinov for expertise in RCWA simulations, to Ilona Engler and the ISAS workshop team for technical support, and to SENTECH Instruments GmbH for construction of the retarder unit. Financial support 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.

REFERENCES

1. H. G. Tompkins and E. A. Irene, Handbook of Ellipsometry (William Andrew, 2005).

2. H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007).

3. M. Losurdo and K. Hingerl, Ellipsometry at the Nanoscale (Springer Berlin Heidelberg, 2013).

4. K. Hinrichs and K.-J. Eichhorn, Ellipsometry of Functional Organic Surfaces and Films (Springer-Verlag Berlin Heidelberg, 2018).

5. A. Furchner and D. Aulich, “Common polymers and proteins,” in Ellipsometry of Functional Organic Surfaces and Films, K. Hinrichs and K.-J. Eichhorn, eds. (Springer International Publishing, 2018), pp. 521–527.

6. A. Furchner and D. Aulich, “Organic materials for optoelectronic applications,” in Ellipsometry of Functional Organic Surfaces and Films, K. Hinrichs and K.-J. Eichhorn, eds. (Springer International Publishing, 2018), pp. 529–538.

7. M. Schubert, H. Tino, and C. M. Herzinger, “Generalized far-infrared magneto-optic ellipsometry for semiconductor layer structures: determination of free-carrier effective-mass, mobility, and concentration parameters in n-type GaAs,” J. Opt. Soc. Am. A 20, 347–356 (2003). [CrossRef]  

8. S. Schöche, T. Hofmann, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, P. Kühne, K. Lorenz, M. Schubert, and V. Darakchieva, “Infrared dielectric functions, phonon modes, and free-charge carrier properties of high-Al-content AlxGa1−xN alloys determined by mid infrared spectroscopic ellipsometry and optical Hall effect,” J. Appl. Phys. 121, 205701 (2017). [CrossRef]  

9. K. Hinrichs, M. Gensch, N. Nikonenko, J. Pionteck, and K.-J. Eichhorn, “Spectroscopic ellipsometry for characterization of thin films of polymer blends,” Macromol. Symp. 230, 26–32 (2005). [CrossRef]  

10. L. Ionov, A. Sidorenko, K.-J. Eichhorn, M. Stamm, S. Minko, and H. Karsten, “Stimuli-responsive mixed grafted polymer films with gradually changing properties,” Langmuir 21, 8711–8716 (2005). [CrossRef]  

11. K. Lin, Y. Gu, H. Zhang, Z. Qiang, B. D. Vogt, and N. S. Zacharia, “Accelerated amidization of branched poly(ethylenimine)/poly(acrylic acid) multilayer films by microwave heating,” Langmuir 32, 9118–9125 (2016). [CrossRef]  

12. O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010). [CrossRef]  

13. A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015). [CrossRef]  

14. A. Furchner, A. Kroning, S. Rauch, P. Uhlmann, K.-J. Eichhorn, and K. Hinrichs, “Molecular interactions and hydration states of ultrathin functional films at the solid-liquid interface,” Anal. Chem. 89, 3240–3244 (2017). [CrossRef]  

15. 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, M. Losurdo and K. Hingerl, eds. (Springer Berlin Heidelberg, 2013).

16. T. W. H. Oates, T. Shaykhutdinov, T. Wagner, A. Furchner, and K. Hinrichs, “Mid-infrared gyrotropy in split-ring resonators measured by Mueller matrix ellipsometry,” Opt. Mater. Express 4, 2646–2655 (2014). [CrossRef]  

17. F. Carmagnola, J. M. Sanz, and J. M. Saiz, “Development of a Mueller matrix imaging system for detecting objects embedded in turbid media,” J. Quant. Spectrosc. Radiat. Transfer 146, 199–206 (2014). [CrossRef]  

18. A. Arwin, A. Mendoza-Galván, R. Magnusson, A. Andersson, J. Landin, K. Järrendahl, E. Garcia-Caurel, and R. Ossikovski, “Structural circular birefringence and dichroism quantified by differential decomposition of spectroscopic transmission Mueller matrices from Cetonia aurata,” Opt. Lett. 41, 3293–3296 (2016). [CrossRef]  

19. W. Li, H. Jiang, C. Zhang, X. Chen, H. Gu, and S. Liu, “Characterization of curved surface layer by Mueller matrix ellipsometry,” J. Vac. Sci. Technol. B 34, 020602 (2016). [CrossRef]  

20. 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, “IR-Mueller matrix ellipsometry of self-assembled nanopatterned gold grid polarizer,” Appl. Surf. Sci. 421, 728–737 (2017). [CrossRef]  

21. T. A. Germer, “Full four-dimensional and reciprocal Mueller matrix bidirectional reflectance distribution function of sintered polytetrafluoroethylene,” Appl. Opt. 56, 9333–9340 (2017). [CrossRef]  

22. B. Song, H. Gu, S. Zhu, H. Jiang, X. Chen, C. Zhang, and S. Liu, “Broadband optical properties of graphene and HOPG investigated by spectroscopic Mueller matrix ellipsometry,” Appl. Surf. Sci. 439, 1079–1087 (2018). [CrossRef]  

23. A. Mendoza-Galván, L. F. del Río, K. Järrendahl, and H. Arwin, “Graded pitch profile for the helicoidal broadband reflector and left-handed circularly polarizing cuticle of the scarab beetle Chrysina chrysargyrea,” Sci. Rep. 8, 6456 (2018). [CrossRef]  

24. P. Kühne, N. Armakavicius, V. Stanishev, C. M. Herzinger, M. Schubert, and V. Darakchieva, “Advanced terahertz frequency-domain ellipsometry instrumentation for in situ and ex situ applications,” IEEE Trans. Terahertz Sci. Technol. 8, 257–270 (2018). [CrossRef]  

25. D. Layden, M. F. G. Wood, and I. A. Vitkin, “Optimum selection of input polarization states in determining the sample Mueller matrix: a dual photoelastic polarimeter approach,” Opt. Express 20, 20466–20481 (2012). [CrossRef]  

26. T. Mu, Z. Chen, C. Zhang, and R. Liang, “Optimal configurations of full-Stokes polarimeter with immunity to both Poisson and Gaussian noise,” J. Opt. 18, 055702 (2016). [CrossRef]  

27. R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, ed. (McGraw Hill, 1995).

28. J.-C. Cigal, “A novel spectroscopic ellipsometer in the infrared,” Ph.D. thesis (Technische Universiteit Eindhoven, 2002).

29. J. M. Chalmers and P. R. Griffiths, eds., Handbook of Vibrational Spectroscopy (Wiley, 2006).

30. R. W. Collins, I. An, and C. Chen, “Rotating polarizer and analyzer ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds. (William Andrew, 2005).

31. L. Li, “Symmetries of cross-polarization diffraction coefficients of gratings,” J. Opt. Soc. Am. A 17, 881–887 (2000). [CrossRef]  

32. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981). [CrossRef]  

33. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982). [CrossRef]  

34. K. S. Iyer and I. Luzinov, “Effect of macromolecular anchoring layer thickness and molecular weight on polymer grafting,” Macromolecules 37, 9538–9545 (2004). [CrossRef]  

35. J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986). [CrossRef]  

36. A. Röseler, “Spectroscopic infrared ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds. (William Andrew, 2005).

37. A. Furchner, “Structure and interactions of polymer thin films from infrared ellipsometry,” in Ellipsometry of Functional Organic Surfaces and Films, K. Hinrichs and K.-J. Eichhorn, eds. (Springer International Publishing, 2018), pp. 145–171.

38. Y. Mikhaylova, L. Ionov, J. Rappich, M. Gensch, N. Esser, S. Minko, K.-J. Eichhorn, M. Stamm, and K. Hinrichs, “In situ infrared ellipsometric study of stimuli-responsive mixed polyelectrolyte brushes,” Anal. Chem. 79, 7676–7682 (2007). [CrossRef]  

39. J.-J. Max and C. Chapados, “Isotope effects in liquid water by infrared spectroscopy,” J. Chem. Phys. 116, 4626–4642 (2002). [CrossRef]  

40. A. Furchner, C. Kratz, D. Gkogkou, H. Ketelsen, and K. Hinrichs, “Infrared-spectroscopic single-shot laser mapping ellipsometry: proof of concept for fast investigations of structured surfaces and interactions in organic thin films,” Appl. Surf. Sci. 421, 440–445 (2017). [CrossRef]  

References

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  1. H. G. Tompkins and E. A. Irene, Handbook of Ellipsometry (William Andrew, 2005).
  2. H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007).
  3. M. Losurdo and K. Hingerl, Ellipsometry at the Nanoscale (Springer Berlin Heidelberg, 2013).
  4. K. Hinrichs and K.-J. Eichhorn, Ellipsometry of Functional Organic Surfaces and Films (Springer-Verlag Berlin Heidelberg, 2018).
  5. A. Furchner and D. Aulich, “Common polymers and proteins,” in Ellipsometry of Functional Organic Surfaces and Films, K. Hinrichs and K.-J. Eichhorn, eds. (Springer International Publishing, 2018), pp. 521–527.
  6. A. Furchner and D. Aulich, “Organic materials for optoelectronic applications,” in Ellipsometry of Functional Organic Surfaces and Films, K. Hinrichs and K.-J. Eichhorn, eds. (Springer International Publishing, 2018), pp. 529–538.
  7. M. Schubert, H. Tino, and C. M. Herzinger, “Generalized far-infrared magneto-optic ellipsometry for semiconductor layer structures: determination of free-carrier effective-mass, mobility, and concentration parameters in n-type GaAs,” J. Opt. Soc. Am. A 20, 347–356 (2003).
    [Crossref]
  8. S. Schöche, T. Hofmann, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, P. Kühne, K. Lorenz, M. Schubert, and V. Darakchieva, “Infrared dielectric functions, phonon modes, and free-charge carrier properties of high-Al-content AlxGa1−xN alloys determined by mid infrared spectroscopic ellipsometry and optical Hall effect,” J. Appl. Phys. 121, 205701 (2017).
    [Crossref]
  9. K. Hinrichs, M. Gensch, N. Nikonenko, J. Pionteck, and K.-J. Eichhorn, “Spectroscopic ellipsometry for characterization of thin films of polymer blends,” Macromol. Symp. 230, 26–32 (2005).
    [Crossref]
  10. L. Ionov, A. Sidorenko, K.-J. Eichhorn, M. Stamm, S. Minko, and H. Karsten, “Stimuli-responsive mixed grafted polymer films with gradually changing properties,” Langmuir 21, 8711–8716 (2005).
    [Crossref]
  11. K. Lin, Y. Gu, H. Zhang, Z. Qiang, B. D. Vogt, and N. S. Zacharia, “Accelerated amidization of branched poly(ethylenimine)/poly(acrylic acid) multilayer films by microwave heating,” Langmuir 32, 9118–9125 (2016).
    [Crossref]
  12. O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
    [Crossref]
  13. A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015).
    [Crossref]
  14. A. Furchner, A. Kroning, S. Rauch, P. Uhlmann, K.-J. Eichhorn, and K. Hinrichs, “Molecular interactions and hydration states of ultrathin functional films at the solid-liquid interface,” Anal. Chem. 89, 3240–3244 (2017).
    [Crossref]
  15. 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, M. Losurdo and K. Hingerl, eds. (Springer Berlin Heidelberg, 2013).
  16. T. W. H. Oates, T. Shaykhutdinov, T. Wagner, A. Furchner, and K. Hinrichs, “Mid-infrared gyrotropy in split-ring resonators measured by Mueller matrix ellipsometry,” Opt. Mater. Express 4, 2646–2655 (2014).
    [Crossref]
  17. F. Carmagnola, J. M. Sanz, and J. M. Saiz, “Development of a Mueller matrix imaging system for detecting objects embedded in turbid media,” J. Quant. Spectrosc. Radiat. Transfer 146, 199–206 (2014).
    [Crossref]
  18. A. Arwin, A. Mendoza-Galván, R. Magnusson, A. Andersson, J. Landin, K. Järrendahl, E. Garcia-Caurel, and R. Ossikovski, “Structural circular birefringence and dichroism quantified by differential decomposition of spectroscopic transmission Mueller matrices from Cetonia aurata,” Opt. Lett. 41, 3293–3296 (2016).
    [Crossref]
  19. W. Li, H. Jiang, C. Zhang, X. Chen, H. Gu, and S. Liu, “Characterization of curved surface layer by Mueller matrix ellipsometry,” J. Vac. Sci. Technol. B 34, 020602 (2016).
    [Crossref]
  20. 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, “IR-Mueller matrix ellipsometry of self-assembled nanopatterned gold grid polarizer,” Appl. Surf. Sci. 421, 728–737 (2017).
    [Crossref]
  21. T. A. Germer, “Full four-dimensional and reciprocal Mueller matrix bidirectional reflectance distribution function of sintered polytetrafluoroethylene,” Appl. Opt. 56, 9333–9340 (2017).
    [Crossref]
  22. B. Song, H. Gu, S. Zhu, H. Jiang, X. Chen, C. Zhang, and S. Liu, “Broadband optical properties of graphene and HOPG investigated by spectroscopic Mueller matrix ellipsometry,” Appl. Surf. Sci. 439, 1079–1087 (2018).
    [Crossref]
  23. A. Mendoza-Galván, L. F. del Río, K. Järrendahl, and H. Arwin, “Graded pitch profile for the helicoidal broadband reflector and left-handed circularly polarizing cuticle of the scarab beetle Chrysina chrysargyrea,” Sci. Rep. 8, 6456 (2018).
    [Crossref]
  24. P. Kühne, N. Armakavicius, V. Stanishev, C. M. Herzinger, M. Schubert, and V. Darakchieva, “Advanced terahertz frequency-domain ellipsometry instrumentation for in situ and ex situ applications,” IEEE Trans. Terahertz Sci. Technol. 8, 257–270 (2018).
    [Crossref]
  25. D. Layden, M. F. G. Wood, and I. A. Vitkin, “Optimum selection of input polarization states in determining the sample Mueller matrix: a dual photoelastic polarimeter approach,” Opt. Express 20, 20466–20481 (2012).
    [Crossref]
  26. T. Mu, Z. Chen, C. Zhang, and R. Liang, “Optimal configurations of full-Stokes polarimeter with immunity to both Poisson and Gaussian noise,” J. Opt. 18, 055702 (2016).
    [Crossref]
  27. R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, ed. (McGraw Hill, 1995).
  28. J.-C. Cigal, “A novel spectroscopic ellipsometer in the infrared,” Ph.D. thesis (Technische Universiteit Eindhoven, 2002).
  29. J. M. Chalmers and P. R. Griffiths, eds., Handbook of Vibrational Spectroscopy (Wiley, 2006).
  30. R. W. Collins, I. An, and C. Chen, “Rotating polarizer and analyzer ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds. (William Andrew, 2005).
  31. L. Li, “Symmetries of cross-polarization diffraction coefficients of gratings,” J. Opt. Soc. Am. A 17, 881–887 (2000).
    [Crossref]
  32. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [Crossref]
  33. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [Crossref]
  34. K. S. Iyer and I. Luzinov, “Effect of macromolecular anchoring layer thickness and molecular weight on polymer grafting,” Macromolecules 37, 9538–9545 (2004).
    [Crossref]
  35. J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
    [Crossref]
  36. A. Röseler, “Spectroscopic infrared ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds. (William Andrew, 2005).
  37. A. Furchner, “Structure and interactions of polymer thin films from infrared ellipsometry,” in Ellipsometry of Functional Organic Surfaces and Films, K. Hinrichs and K.-J. Eichhorn, eds. (Springer International Publishing, 2018), pp. 145–171.
  38. Y. Mikhaylova, L. Ionov, J. Rappich, M. Gensch, N. Esser, S. Minko, K.-J. Eichhorn, M. Stamm, and K. Hinrichs, “In situ infrared ellipsometric study of stimuli-responsive mixed polyelectrolyte brushes,” Anal. Chem. 79, 7676–7682 (2007).
    [Crossref]
  39. J.-J. Max and C. Chapados, “Isotope effects in liquid water by infrared spectroscopy,” J. Chem. Phys. 116, 4626–4642 (2002).
    [Crossref]
  40. A. Furchner, C. Kratz, D. Gkogkou, H. Ketelsen, and K. Hinrichs, “Infrared-spectroscopic single-shot laser mapping ellipsometry: proof of concept for fast investigations of structured surfaces and interactions in organic thin films,” Appl. Surf. Sci. 421, 440–445 (2017).
    [Crossref]

2018 (3)

B. Song, H. Gu, S. Zhu, H. Jiang, X. Chen, C. Zhang, and S. Liu, “Broadband optical properties of graphene and HOPG investigated by spectroscopic Mueller matrix ellipsometry,” Appl. Surf. Sci. 439, 1079–1087 (2018).
[Crossref]

A. Mendoza-Galván, L. F. del Río, K. Järrendahl, and H. Arwin, “Graded pitch profile for the helicoidal broadband reflector and left-handed circularly polarizing cuticle of the scarab beetle Chrysina chrysargyrea,” Sci. Rep. 8, 6456 (2018).
[Crossref]

P. Kühne, N. Armakavicius, V. Stanishev, C. M. Herzinger, M. Schubert, and V. Darakchieva, “Advanced terahertz frequency-domain ellipsometry instrumentation for in situ and ex situ applications,” IEEE Trans. Terahertz Sci. Technol. 8, 257–270 (2018).
[Crossref]

2017 (5)

T. A. Germer, “Full four-dimensional and reciprocal Mueller matrix bidirectional reflectance distribution function of sintered polytetrafluoroethylene,” Appl. Opt. 56, 9333–9340 (2017).
[Crossref]

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, “IR-Mueller matrix ellipsometry of self-assembled nanopatterned gold grid polarizer,” Appl. Surf. Sci. 421, 728–737 (2017).
[Crossref]

A. Furchner, A. Kroning, S. Rauch, P. Uhlmann, K.-J. Eichhorn, and K. Hinrichs, “Molecular interactions and hydration states of ultrathin functional films at the solid-liquid interface,” Anal. Chem. 89, 3240–3244 (2017).
[Crossref]

S. Schöche, T. Hofmann, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, P. Kühne, K. Lorenz, M. Schubert, and V. Darakchieva, “Infrared dielectric functions, phonon modes, and free-charge carrier properties of high-Al-content AlxGa1−xN alloys determined by mid infrared spectroscopic ellipsometry and optical Hall effect,” J. Appl. Phys. 121, 205701 (2017).
[Crossref]

A. Furchner, C. Kratz, D. Gkogkou, H. Ketelsen, and K. Hinrichs, “Infrared-spectroscopic single-shot laser mapping ellipsometry: proof of concept for fast investigations of structured surfaces and interactions in organic thin films,” Appl. Surf. Sci. 421, 440–445 (2017).
[Crossref]

2016 (4)

T. Mu, Z. Chen, C. Zhang, and R. Liang, “Optimal configurations of full-Stokes polarimeter with immunity to both Poisson and Gaussian noise,” J. Opt. 18, 055702 (2016).
[Crossref]

W. Li, H. Jiang, C. Zhang, X. Chen, H. Gu, and S. Liu, “Characterization of curved surface layer by Mueller matrix ellipsometry,” J. Vac. Sci. Technol. B 34, 020602 (2016).
[Crossref]

K. Lin, Y. Gu, H. Zhang, Z. Qiang, B. D. Vogt, and N. S. Zacharia, “Accelerated amidization of branched poly(ethylenimine)/poly(acrylic acid) multilayer films by microwave heating,” Langmuir 32, 9118–9125 (2016).
[Crossref]

A. Arwin, A. Mendoza-Galván, R. Magnusson, A. Andersson, J. Landin, K. Järrendahl, E. Garcia-Caurel, and R. Ossikovski, “Structural circular birefringence and dichroism quantified by differential decomposition of spectroscopic transmission Mueller matrices from Cetonia aurata,” Opt. Lett. 41, 3293–3296 (2016).
[Crossref]

2015 (1)

A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015).
[Crossref]

2014 (2)

F. Carmagnola, J. M. Sanz, and J. M. Saiz, “Development of a Mueller matrix imaging system for detecting objects embedded in turbid media,” J. Quant. Spectrosc. Radiat. Transfer 146, 199–206 (2014).
[Crossref]

T. W. H. Oates, T. Shaykhutdinov, T. Wagner, A. Furchner, and K. Hinrichs, “Mid-infrared gyrotropy in split-ring resonators measured by Mueller matrix ellipsometry,” Opt. Mater. Express 4, 2646–2655 (2014).
[Crossref]

2012 (1)

2010 (1)

O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
[Crossref]

2007 (1)

Y. Mikhaylova, L. Ionov, J. Rappich, M. Gensch, N. Esser, S. Minko, K.-J. Eichhorn, M. Stamm, and K. Hinrichs, “In situ infrared ellipsometric study of stimuli-responsive mixed polyelectrolyte brushes,” Anal. Chem. 79, 7676–7682 (2007).
[Crossref]

2005 (2)

K. Hinrichs, M. Gensch, N. Nikonenko, J. Pionteck, and K.-J. Eichhorn, “Spectroscopic ellipsometry for characterization of thin films of polymer blends,” Macromol. Symp. 230, 26–32 (2005).
[Crossref]

L. Ionov, A. Sidorenko, K.-J. Eichhorn, M. Stamm, S. Minko, and H. Karsten, “Stimuli-responsive mixed grafted polymer films with gradually changing properties,” Langmuir 21, 8711–8716 (2005).
[Crossref]

2004 (1)

K. S. Iyer and I. Luzinov, “Effect of macromolecular anchoring layer thickness and molecular weight on polymer grafting,” Macromolecules 37, 9538–9545 (2004).
[Crossref]

2003 (1)

2002 (1)

J.-J. Max and C. Chapados, “Isotope effects in liquid water by infrared spectroscopy,” J. Chem. Phys. 116, 4626–4642 (2002).
[Crossref]

2000 (1)

1986 (1)

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
[Crossref]

1982 (1)

1981 (1)

Aas, L. M. S.

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, “IR-Mueller matrix ellipsometry of self-assembled nanopatterned gold grid polarizer,” Appl. Surf. Sci. 421, 728–737 (2017).
[Crossref]

An, I.

R. W. Collins, I. An, and C. Chen, “Rotating polarizer and analyzer ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds. (William Andrew, 2005).

Andersson, A.

Armakavicius, N.

P. Kühne, N. Armakavicius, V. Stanishev, C. M. Herzinger, M. Schubert, and V. Darakchieva, “Advanced terahertz frequency-domain ellipsometry instrumentation for in situ and ex situ applications,” IEEE Trans. Terahertz Sci. Technol. 8, 257–270 (2018).
[Crossref]

Arwin, A.

Arwin, H.

A. Mendoza-Galván, L. F. del Río, K. Järrendahl, and H. Arwin, “Graded pitch profile for the helicoidal broadband reflector and left-handed circularly polarizing cuticle of the scarab beetle Chrysina chrysargyrea,” Sci. Rep. 8, 6456 (2018).
[Crossref]

Aulich, D.

A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015).
[Crossref]

O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
[Crossref]

A. Furchner and D. Aulich, “Common polymers and proteins,” in Ellipsometry of Functional Organic Surfaces and Films, K. Hinrichs and K.-J. Eichhorn, eds. (Springer International Publishing, 2018), pp. 521–527.

A. Furchner and D. Aulich, “Organic materials for optoelectronic applications,” in Ellipsometry of Functional Organic Surfaces and Films, K. Hinrichs and K.-J. Eichhorn, eds. (Springer International Publishing, 2018), pp. 529–538.

Bernabeu, E.

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
[Crossref]

Bittrich, E.

A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015).
[Crossref]

O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
[Crossref]

Borondics, F.

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, “IR-Mueller matrix ellipsometry of self-assembled nanopatterned gold grid polarizer,” Appl. Surf. Sci. 421, 728–737 (2017).
[Crossref]

Buatier de Mongeot, F.

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, “IR-Mueller matrix ellipsometry of self-assembled nanopatterned gold grid polarizer,” Appl. Surf. Sci. 421, 728–737 (2017).
[Crossref]

Carmagnola, F.

F. Carmagnola, J. M. Sanz, and J. M. Saiz, “Development of a Mueller matrix imaging system for detecting objects embedded in turbid media,” J. Quant. Spectrosc. Radiat. Transfer 146, 199–206 (2014).
[Crossref]

Chapados, C.

J.-J. Max and C. Chapados, “Isotope effects in liquid water by infrared spectroscopy,” J. Chem. Phys. 116, 4626–4642 (2002).
[Crossref]

Chen, C.

R. W. Collins, I. An, and C. Chen, “Rotating polarizer and analyzer ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds. (William Andrew, 2005).

Chen, X.

B. Song, H. Gu, S. Zhu, H. Jiang, X. Chen, C. Zhang, and S. Liu, “Broadband optical properties of graphene and HOPG investigated by spectroscopic Mueller matrix ellipsometry,” Appl. Surf. Sci. 439, 1079–1087 (2018).
[Crossref]

W. Li, H. Jiang, C. Zhang, X. Chen, H. Gu, and S. Liu, “Characterization of curved surface layer by Mueller matrix ellipsometry,” J. Vac. Sci. Technol. B 34, 020602 (2016).
[Crossref]

Chen, Z.

T. Mu, Z. Chen, C. Zhang, and R. Liang, “Optimal configurations of full-Stokes polarimeter with immunity to both Poisson and Gaussian noise,” J. Opt. 18, 055702 (2016).
[Crossref]

Chiappe, D.

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, “IR-Mueller matrix ellipsometry of self-assembled nanopatterned gold grid polarizer,” Appl. Surf. Sci. 421, 728–737 (2017).
[Crossref]

Chipman, R. A.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, ed. (McGraw Hill, 1995).

Cigal, J.-C.

J.-C. Cigal, “A novel spectroscopic ellipsometer in the infrared,” Ph.D. thesis (Technische Universiteit Eindhoven, 2002).

Collins, R. W.

R. W. Collins, I. An, and C. Chen, “Rotating polarizer and analyzer ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds. (William Andrew, 2005).

Darakchieva, V.

P. Kühne, N. Armakavicius, V. Stanishev, C. M. Herzinger, M. Schubert, and V. Darakchieva, “Advanced terahertz frequency-domain ellipsometry instrumentation for in situ and ex situ applications,” IEEE Trans. Terahertz Sci. Technol. 8, 257–270 (2018).
[Crossref]

S. Schöche, T. Hofmann, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, P. Kühne, K. Lorenz, M. Schubert, and V. Darakchieva, “Infrared dielectric functions, phonon modes, and free-charge carrier properties of high-Al-content AlxGa1−xN alloys determined by mid infrared spectroscopic ellipsometry and optical Hall effect,” J. Appl. Phys. 121, 205701 (2017).
[Crossref]

De Martino, A.

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, M. Losurdo and K. Hingerl, eds. (Springer Berlin Heidelberg, 2013).

del Río, L. F.

A. Mendoza-Galván, L. F. del Río, K. Järrendahl, and H. Arwin, “Graded pitch profile for the helicoidal broadband reflector and left-handed circularly polarizing cuticle of the scarab beetle Chrysina chrysargyrea,” Sci. Rep. 8, 6456 (2018).
[Crossref]

Drévillon, B.

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, M. Losurdo and K. Hingerl, eds. (Springer Berlin Heidelberg, 2013).

Eichhorn, K.-J.

A. Furchner, A. Kroning, S. Rauch, P. Uhlmann, K.-J. Eichhorn, and K. Hinrichs, “Molecular interactions and hydration states of ultrathin functional films at the solid-liquid interface,” Anal. Chem. 89, 3240–3244 (2017).
[Crossref]

A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015).
[Crossref]

O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
[Crossref]

Y. Mikhaylova, L. Ionov, J. Rappich, M. Gensch, N. Esser, S. Minko, K.-J. Eichhorn, M. Stamm, and K. Hinrichs, “In situ infrared ellipsometric study of stimuli-responsive mixed polyelectrolyte brushes,” Anal. Chem. 79, 7676–7682 (2007).
[Crossref]

K. Hinrichs, M. Gensch, N. Nikonenko, J. Pionteck, and K.-J. Eichhorn, “Spectroscopic ellipsometry for characterization of thin films of polymer blends,” Macromol. Symp. 230, 26–32 (2005).
[Crossref]

L. Ionov, A. Sidorenko, K.-J. Eichhorn, M. Stamm, S. Minko, and H. Karsten, “Stimuli-responsive mixed grafted polymer films with gradually changing properties,” Langmuir 21, 8711–8716 (2005).
[Crossref]

K. Hinrichs and K.-J. Eichhorn, Ellipsometry of Functional Organic Surfaces and Films (Springer-Verlag Berlin Heidelberg, 2018).

Esser, N.

Y. Mikhaylova, L. Ionov, J. Rappich, M. Gensch, N. Esser, S. Minko, K.-J. Eichhorn, M. Stamm, and K. Hinrichs, “In situ infrared ellipsometric study of stimuli-responsive mixed polyelectrolyte brushes,” Anal. Chem. 79, 7676–7682 (2007).
[Crossref]

Foldyna, M.

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, M. Losurdo and K. Hingerl, eds. (Springer Berlin Heidelberg, 2013).

Fujiwara, H.

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007).

Furchner, A.

A. Furchner, A. Kroning, S. Rauch, P. Uhlmann, K.-J. Eichhorn, and K. Hinrichs, “Molecular interactions and hydration states of ultrathin functional films at the solid-liquid interface,” Anal. Chem. 89, 3240–3244 (2017).
[Crossref]

A. Furchner, C. Kratz, D. Gkogkou, H. Ketelsen, and K. Hinrichs, “Infrared-spectroscopic single-shot laser mapping ellipsometry: proof of concept for fast investigations of structured surfaces and interactions in organic thin films,” Appl. Surf. Sci. 421, 440–445 (2017).
[Crossref]

A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015).
[Crossref]

T. W. H. Oates, T. Shaykhutdinov, T. Wagner, A. Furchner, and K. Hinrichs, “Mid-infrared gyrotropy in split-ring resonators measured by Mueller matrix ellipsometry,” Opt. Mater. Express 4, 2646–2655 (2014).
[Crossref]

A. Furchner and D. Aulich, “Common polymers and proteins,” in Ellipsometry of Functional Organic Surfaces and Films, K. Hinrichs and K.-J. Eichhorn, eds. (Springer International Publishing, 2018), pp. 521–527.

A. Furchner and D. Aulich, “Organic materials for optoelectronic applications,” in Ellipsometry of Functional Organic Surfaces and Films, K. Hinrichs and K.-J. Eichhorn, eds. (Springer International Publishing, 2018), pp. 529–538.

A. Furchner, “Structure and interactions of polymer thin films from infrared ellipsometry,” in Ellipsometry of Functional Organic Surfaces and Films, K. Hinrichs and K.-J. Eichhorn, eds. (Springer International Publishing, 2018), pp. 145–171.

Garcia-Caurel, E.

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, “IR-Mueller matrix ellipsometry of self-assembled nanopatterned gold grid polarizer,” Appl. Surf. Sci. 421, 728–737 (2017).
[Crossref]

A. Arwin, A. Mendoza-Galván, R. Magnusson, A. Andersson, J. Landin, K. Järrendahl, E. Garcia-Caurel, and R. Ossikovski, “Structural circular birefringence and dichroism quantified by differential decomposition of spectroscopic transmission Mueller matrices from Cetonia aurata,” Opt. Lett. 41, 3293–3296 (2016).
[Crossref]

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, M. Losurdo and K. Hingerl, eds. (Springer Berlin Heidelberg, 2013).

Gaylord, T. K.

Gensch, M.

Y. Mikhaylova, L. Ionov, J. Rappich, M. Gensch, N. Esser, S. Minko, K.-J. Eichhorn, M. Stamm, and K. Hinrichs, “In situ infrared ellipsometric study of stimuli-responsive mixed polyelectrolyte brushes,” Anal. Chem. 79, 7676–7682 (2007).
[Crossref]

K. Hinrichs, M. Gensch, N. Nikonenko, J. Pionteck, and K.-J. Eichhorn, “Spectroscopic ellipsometry for characterization of thin films of polymer blends,” Macromol. Symp. 230, 26–32 (2005).
[Crossref]

Germer, T. A.

Gil, J. J.

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
[Crossref]

Giordano, M. C.

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, “IR-Mueller matrix ellipsometry of self-assembled nanopatterned gold grid polarizer,” Appl. Surf. Sci. 421, 728–737 (2017).
[Crossref]

Gkogkou, D.

A. Furchner, C. Kratz, D. Gkogkou, H. Ketelsen, and K. Hinrichs, “Infrared-spectroscopic single-shot laser mapping ellipsometry: proof of concept for fast investigations of structured surfaces and interactions in organic thin films,” Appl. Surf. Sci. 421, 440–445 (2017).
[Crossref]

Gu, H.

B. Song, H. Gu, S. Zhu, H. Jiang, X. Chen, C. Zhang, and S. Liu, “Broadband optical properties of graphene and HOPG investigated by spectroscopic Mueller matrix ellipsometry,” Appl. Surf. Sci. 439, 1079–1087 (2018).
[Crossref]

W. Li, H. Jiang, C. Zhang, X. Chen, H. Gu, and S. Liu, “Characterization of curved surface layer by Mueller matrix ellipsometry,” J. Vac. Sci. Technol. B 34, 020602 (2016).
[Crossref]

Gu, Y.

K. Lin, Y. Gu, H. Zhang, Z. Qiang, B. D. Vogt, and N. S. Zacharia, “Accelerated amidization of branched poly(ethylenimine)/poly(acrylic acid) multilayer films by microwave heating,” Langmuir 32, 9118–9125 (2016).
[Crossref]

Herzinger, C. M.

P. Kühne, N. Armakavicius, V. Stanishev, C. M. Herzinger, M. Schubert, and V. Darakchieva, “Advanced terahertz frequency-domain ellipsometry instrumentation for in situ and ex situ applications,” IEEE Trans. Terahertz Sci. Technol. 8, 257–270 (2018).
[Crossref]

M. Schubert, H. Tino, and C. M. Herzinger, “Generalized far-infrared magneto-optic ellipsometry for semiconductor layer structures: determination of free-carrier effective-mass, mobility, and concentration parameters in n-type GaAs,” J. Opt. Soc. Am. A 20, 347–356 (2003).
[Crossref]

Hingerl, K.

M. Losurdo and K. Hingerl, Ellipsometry at the Nanoscale (Springer Berlin Heidelberg, 2013).

Hinrichs, K.

A. Furchner, A. Kroning, S. Rauch, P. Uhlmann, K.-J. Eichhorn, and K. Hinrichs, “Molecular interactions and hydration states of ultrathin functional films at the solid-liquid interface,” Anal. Chem. 89, 3240–3244 (2017).
[Crossref]

A. Furchner, C. Kratz, D. Gkogkou, H. Ketelsen, and K. Hinrichs, “Infrared-spectroscopic single-shot laser mapping ellipsometry: proof of concept for fast investigations of structured surfaces and interactions in organic thin films,” Appl. Surf. Sci. 421, 440–445 (2017).
[Crossref]

A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015).
[Crossref]

T. W. H. Oates, T. Shaykhutdinov, T. Wagner, A. Furchner, and K. Hinrichs, “Mid-infrared gyrotropy in split-ring resonators measured by Mueller matrix ellipsometry,” Opt. Mater. Express 4, 2646–2655 (2014).
[Crossref]

O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
[Crossref]

Y. Mikhaylova, L. Ionov, J. Rappich, M. Gensch, N. Esser, S. Minko, K.-J. Eichhorn, M. Stamm, and K. Hinrichs, “In situ infrared ellipsometric study of stimuli-responsive mixed polyelectrolyte brushes,” Anal. Chem. 79, 7676–7682 (2007).
[Crossref]

K. Hinrichs, M. Gensch, N. Nikonenko, J. Pionteck, and K.-J. Eichhorn, “Spectroscopic ellipsometry for characterization of thin films of polymer blends,” Macromol. Symp. 230, 26–32 (2005).
[Crossref]

K. Hinrichs and K.-J. Eichhorn, Ellipsometry of Functional Organic Surfaces and Films (Springer-Verlag Berlin Heidelberg, 2018).

Hofmann, T.

S. Schöche, T. Hofmann, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, P. Kühne, K. Lorenz, M. Schubert, and V. Darakchieva, “Infrared dielectric functions, phonon modes, and free-charge carrier properties of high-Al-content AlxGa1−xN alloys determined by mid infrared spectroscopic ellipsometry and optical Hall effect,” J. Appl. Phys. 121, 205701 (2017).
[Crossref]

Hoy, O.

O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
[Crossref]

Ionov, L.

Y. Mikhaylova, L. Ionov, J. Rappich, M. Gensch, N. Esser, S. Minko, K.-J. Eichhorn, M. Stamm, and K. Hinrichs, “In situ infrared ellipsometric study of stimuli-responsive mixed polyelectrolyte brushes,” Anal. Chem. 79, 7676–7682 (2007).
[Crossref]

L. Ionov, A. Sidorenko, K.-J. Eichhorn, M. Stamm, S. Minko, and H. Karsten, “Stimuli-responsive mixed grafted polymer films with gradually changing properties,” Langmuir 21, 8711–8716 (2005).
[Crossref]

Irene, E. A.

H. G. Tompkins and E. A. Irene, Handbook of Ellipsometry (William Andrew, 2005).

Iyer, K. S.

K. S. Iyer and I. Luzinov, “Effect of macromolecular anchoring layer thickness and molecular weight on polymer grafting,” Macromolecules 37, 9538–9545 (2004).
[Crossref]

Janzén, E.

S. Schöche, T. Hofmann, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, P. Kühne, K. Lorenz, M. Schubert, and V. Darakchieva, “Infrared dielectric functions, phonon modes, and free-charge carrier properties of high-Al-content AlxGa1−xN alloys determined by mid infrared spectroscopic ellipsometry and optical Hall effect,” J. Appl. Phys. 121, 205701 (2017).
[Crossref]

Järrendahl, K.

A. Mendoza-Galván, L. F. del Río, K. Järrendahl, and H. Arwin, “Graded pitch profile for the helicoidal broadband reflector and left-handed circularly polarizing cuticle of the scarab beetle Chrysina chrysargyrea,” Sci. Rep. 8, 6456 (2018).
[Crossref]

A. Arwin, A. Mendoza-Galván, R. Magnusson, A. Andersson, J. Landin, K. Järrendahl, E. Garcia-Caurel, and R. Ossikovski, “Structural circular birefringence and dichroism quantified by differential decomposition of spectroscopic transmission Mueller matrices from Cetonia aurata,” Opt. Lett. 41, 3293–3296 (2016).
[Crossref]

Jiang, H.

B. Song, H. Gu, S. Zhu, H. Jiang, X. Chen, C. Zhang, and S. Liu, “Broadband optical properties of graphene and HOPG investigated by spectroscopic Mueller matrix ellipsometry,” Appl. Surf. Sci. 439, 1079–1087 (2018).
[Crossref]

W. Li, H. Jiang, C. Zhang, X. Chen, H. Gu, and S. Liu, “Characterization of curved surface layer by Mueller matrix ellipsometry,” J. Vac. Sci. Technol. B 34, 020602 (2016).
[Crossref]

Kakanakova-Georgieva, A.

S. Schöche, T. Hofmann, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, P. Kühne, K. Lorenz, M. Schubert, and V. Darakchieva, “Infrared dielectric functions, phonon modes, and free-charge carrier properties of high-Al-content AlxGa1−xN alloys determined by mid infrared spectroscopic ellipsometry and optical Hall effect,” J. Appl. Phys. 121, 205701 (2017).
[Crossref]

Karsten, H.

L. Ionov, A. Sidorenko, K.-J. Eichhorn, M. Stamm, S. Minko, and H. Karsten, “Stimuli-responsive mixed grafted polymer films with gradually changing properties,” Langmuir 21, 8711–8716 (2005).
[Crossref]

Ketelsen, H.

A. Furchner, C. Kratz, D. Gkogkou, H. Ketelsen, and K. Hinrichs, “Infrared-spectroscopic single-shot laser mapping ellipsometry: proof of concept for fast investigations of structured surfaces and interactions in organic thin films,” Appl. Surf. Sci. 421, 440–445 (2017).
[Crossref]

Kilbey, S. M. I.

A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015).
[Crossref]

Kildemo, M.

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, “IR-Mueller matrix ellipsometry of self-assembled nanopatterned gold grid polarizer,” Appl. Surf. Sci. 421, 728–737 (2017).
[Crossref]

Kratz, C.

A. Furchner, C. Kratz, D. Gkogkou, H. Ketelsen, and K. Hinrichs, “Infrared-spectroscopic single-shot laser mapping ellipsometry: proof of concept for fast investigations of structured surfaces and interactions in organic thin films,” Appl. Surf. Sci. 421, 440–445 (2017).
[Crossref]

Kroning, A.

A. Furchner, A. Kroning, S. Rauch, P. Uhlmann, K.-J. Eichhorn, and K. Hinrichs, “Molecular interactions and hydration states of ultrathin functional films at the solid-liquid interface,” Anal. Chem. 89, 3240–3244 (2017).
[Crossref]

A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015).
[Crossref]

Kühne, P.

P. Kühne, N. Armakavicius, V. Stanishev, C. M. Herzinger, M. Schubert, and V. Darakchieva, “Advanced terahertz frequency-domain ellipsometry instrumentation for in situ and ex situ applications,” IEEE Trans. Terahertz Sci. Technol. 8, 257–270 (2018).
[Crossref]

S. Schöche, T. Hofmann, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, P. Kühne, K. Lorenz, M. Schubert, and V. Darakchieva, “Infrared dielectric functions, phonon modes, and free-charge carrier properties of high-Al-content AlxGa1−xN alloys determined by mid infrared spectroscopic ellipsometry and optical Hall effect,” J. Appl. Phys. 121, 205701 (2017).
[Crossref]

Landin, J.

Layden, D.

Li, L.

Li, W.

W. Li, H. Jiang, C. Zhang, X. Chen, H. Gu, and S. Liu, “Characterization of curved surface layer by Mueller matrix ellipsometry,” J. Vac. Sci. Technol. B 34, 020602 (2016).
[Crossref]

Liang, R.

T. Mu, Z. Chen, C. Zhang, and R. Liang, “Optimal configurations of full-Stokes polarimeter with immunity to both Poisson and Gaussian noise,” J. Opt. 18, 055702 (2016).
[Crossref]

Lin, K.

K. Lin, Y. Gu, H. Zhang, Z. Qiang, B. D. Vogt, and N. S. Zacharia, “Accelerated amidization of branched poly(ethylenimine)/poly(acrylic acid) multilayer films by microwave heating,” Langmuir 32, 9118–9125 (2016).
[Crossref]

Liu, S.

B. Song, H. Gu, S. Zhu, H. Jiang, X. Chen, C. Zhang, and S. Liu, “Broadband optical properties of graphene and HOPG investigated by spectroscopic Mueller matrix ellipsometry,” Appl. Surf. Sci. 439, 1079–1087 (2018).
[Crossref]

W. Li, H. Jiang, C. Zhang, X. Chen, H. Gu, and S. Liu, “Characterization of curved surface layer by Mueller matrix ellipsometry,” J. Vac. Sci. Technol. B 34, 020602 (2016).
[Crossref]

Lokitz, B. S.

A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015).
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Lorenz, K.

S. Schöche, T. Hofmann, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, P. Kühne, K. Lorenz, M. Schubert, and V. Darakchieva, “Infrared dielectric functions, phonon modes, and free-charge carrier properties of high-Al-content AlxGa1−xN alloys determined by mid infrared spectroscopic ellipsometry and optical Hall effect,” J. Appl. Phys. 121, 205701 (2017).
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M. Losurdo and K. Hingerl, Ellipsometry at the Nanoscale (Springer Berlin Heidelberg, 2013).

Lupitskyy, R.

O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
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Luzinov, I.

A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015).
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O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
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K. S. Iyer and I. Luzinov, “Effect of macromolecular anchoring layer thickness and molecular weight on polymer grafting,” Macromolecules 37, 9538–9545 (2004).
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Martella, C.

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, “IR-Mueller matrix ellipsometry of self-assembled nanopatterned gold grid polarizer,” Appl. Surf. Sci. 421, 728–737 (2017).
[Crossref]

Max, J.-J.

J.-J. Max and C. Chapados, “Isotope effects in liquid water by infrared spectroscopy,” J. Chem. Phys. 116, 4626–4642 (2002).
[Crossref]

Mendoza-Galván, A.

A. Mendoza-Galván, L. F. del Río, K. Järrendahl, and H. Arwin, “Graded pitch profile for the helicoidal broadband reflector and left-handed circularly polarizing cuticle of the scarab beetle Chrysina chrysargyrea,” Sci. Rep. 8, 6456 (2018).
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A. Arwin, A. Mendoza-Galván, R. Magnusson, A. Andersson, J. Landin, K. Järrendahl, E. Garcia-Caurel, and R. Ossikovski, “Structural circular birefringence and dichroism quantified by differential decomposition of spectroscopic transmission Mueller matrices from Cetonia aurata,” Opt. Lett. 41, 3293–3296 (2016).
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Mikhaylova, Y.

Y. Mikhaylova, L. Ionov, J. Rappich, M. Gensch, N. Esser, S. Minko, K.-J. Eichhorn, M. Stamm, and K. Hinrichs, “In situ infrared ellipsometric study of stimuli-responsive mixed polyelectrolyte brushes,” Anal. Chem. 79, 7676–7682 (2007).
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Minko, S.

A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015).
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O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
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Y. Mikhaylova, L. Ionov, J. Rappich, M. Gensch, N. Esser, S. Minko, K.-J. Eichhorn, M. Stamm, and K. Hinrichs, “In situ infrared ellipsometric study of stimuli-responsive mixed polyelectrolyte brushes,” Anal. Chem. 79, 7676–7682 (2007).
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L. Ionov, A. Sidorenko, K.-J. Eichhorn, M. Stamm, S. Minko, and H. Karsten, “Stimuli-responsive mixed grafted polymer films with gradually changing properties,” Langmuir 21, 8711–8716 (2005).
[Crossref]

Moharam, M. G.

Mu, T.

T. Mu, Z. Chen, C. Zhang, and R. Liang, “Optimal configurations of full-Stokes polarimeter with immunity to both Poisson and Gaussian noise,” J. Opt. 18, 055702 (2016).
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Müller, M.

O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
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Nikonenko, N.

K. Hinrichs, M. Gensch, N. Nikonenko, J. Pionteck, and K.-J. Eichhorn, “Spectroscopic ellipsometry for characterization of thin films of polymer blends,” Macromol. Symp. 230, 26–32 (2005).
[Crossref]

Nilsson, D.

S. Schöche, T. Hofmann, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, P. Kühne, K. Lorenz, M. Schubert, and V. Darakchieva, “Infrared dielectric functions, phonon modes, and free-charge carrier properties of high-Al-content AlxGa1−xN alloys determined by mid infrared spectroscopic ellipsometry and optical Hall effect,” J. Appl. Phys. 121, 205701 (2017).
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Oates, T. W. H.

Ossikovski, R.

A. Arwin, A. Mendoza-Galván, R. Magnusson, A. Andersson, J. Landin, K. Järrendahl, E. Garcia-Caurel, and R. Ossikovski, “Structural circular birefringence and dichroism quantified by differential decomposition of spectroscopic transmission Mueller matrices from Cetonia aurata,” Opt. Lett. 41, 3293–3296 (2016).
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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, M. Losurdo and K. Hingerl, eds. (Springer Berlin Heidelberg, 2013).

Peinado, A.

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, “IR-Mueller matrix ellipsometry of self-assembled nanopatterned gold grid polarizer,” Appl. Surf. Sci. 421, 728–737 (2017).
[Crossref]

Pierangelo, A.

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, M. Losurdo and K. Hingerl, eds. (Springer Berlin Heidelberg, 2013).

Pionteck, J.

K. Hinrichs, M. Gensch, N. Nikonenko, J. Pionteck, and K.-J. Eichhorn, “Spectroscopic ellipsometry for characterization of thin films of polymer blends,” Macromol. Symp. 230, 26–32 (2005).
[Crossref]

Qiang, Z.

K. Lin, Y. Gu, H. Zhang, Z. Qiang, B. D. Vogt, and N. S. Zacharia, “Accelerated amidization of branched poly(ethylenimine)/poly(acrylic acid) multilayer films by microwave heating,” Langmuir 32, 9118–9125 (2016).
[Crossref]

Rappich, J.

Y. Mikhaylova, L. Ionov, J. Rappich, M. Gensch, N. Esser, S. Minko, K.-J. Eichhorn, M. Stamm, and K. Hinrichs, “In situ infrared ellipsometric study of stimuli-responsive mixed polyelectrolyte brushes,” Anal. Chem. 79, 7676–7682 (2007).
[Crossref]

Rauch, S.

A. Furchner, A. Kroning, S. Rauch, P. Uhlmann, K.-J. Eichhorn, and K. Hinrichs, “Molecular interactions and hydration states of ultrathin functional films at the solid-liquid interface,” Anal. Chem. 89, 3240–3244 (2017).
[Crossref]

A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015).
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Röseler, A.

A. Röseler, “Spectroscopic infrared ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds. (William Andrew, 2005).

Saiz, J. M.

F. Carmagnola, J. M. Sanz, and J. M. Saiz, “Development of a Mueller matrix imaging system for detecting objects embedded in turbid media,” J. Quant. Spectrosc. Radiat. Transfer 146, 199–206 (2014).
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Sanz, J. M.

F. Carmagnola, J. M. Sanz, and J. M. Saiz, “Development of a Mueller matrix imaging system for detecting objects embedded in turbid media,” J. Quant. Spectrosc. Radiat. Transfer 146, 199–206 (2014).
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Schöche, S.

S. Schöche, T. Hofmann, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, P. Kühne, K. Lorenz, M. Schubert, and V. Darakchieva, “Infrared dielectric functions, phonon modes, and free-charge carrier properties of high-Al-content AlxGa1−xN alloys determined by mid infrared spectroscopic ellipsometry and optical Hall effect,” J. Appl. Phys. 121, 205701 (2017).
[Crossref]

Schubert, M.

P. Kühne, N. Armakavicius, V. Stanishev, C. M. Herzinger, M. Schubert, and V. Darakchieva, “Advanced terahertz frequency-domain ellipsometry instrumentation for in situ and ex situ applications,” IEEE Trans. Terahertz Sci. Technol. 8, 257–270 (2018).
[Crossref]

S. Schöche, T. Hofmann, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, P. Kühne, K. Lorenz, M. Schubert, and V. Darakchieva, “Infrared dielectric functions, phonon modes, and free-charge carrier properties of high-Al-content AlxGa1−xN alloys determined by mid infrared spectroscopic ellipsometry and optical Hall effect,” J. Appl. Phys. 121, 205701 (2017).
[Crossref]

M. Schubert, H. Tino, and C. M. Herzinger, “Generalized far-infrared magneto-optic ellipsometry for semiconductor layer structures: determination of free-carrier effective-mass, mobility, and concentration parameters in n-type GaAs,” J. Opt. Soc. Am. A 20, 347–356 (2003).
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Seeber, M.

A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015).
[Crossref]

Shaykhutdinov, T.

Sheparovych, R.

O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
[Crossref]

Sidorenko, A.

L. Ionov, A. Sidorenko, K.-J. Eichhorn, M. Stamm, S. Minko, and H. Karsten, “Stimuli-responsive mixed grafted polymer films with gradually changing properties,” Langmuir 21, 8711–8716 (2005).
[Crossref]

Song, B.

B. Song, H. Gu, S. Zhu, H. Jiang, X. Chen, C. Zhang, and S. Liu, “Broadband optical properties of graphene and HOPG investigated by spectroscopic Mueller matrix ellipsometry,” Appl. Surf. Sci. 439, 1079–1087 (2018).
[Crossref]

Stamm, M.

O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
[Crossref]

Y. Mikhaylova, L. Ionov, J. Rappich, M. Gensch, N. Esser, S. Minko, K.-J. Eichhorn, M. Stamm, and K. Hinrichs, “In situ infrared ellipsometric study of stimuli-responsive mixed polyelectrolyte brushes,” Anal. Chem. 79, 7676–7682 (2007).
[Crossref]

L. Ionov, A. Sidorenko, K.-J. Eichhorn, M. Stamm, S. Minko, and H. Karsten, “Stimuli-responsive mixed grafted polymer films with gradually changing properties,” Langmuir 21, 8711–8716 (2005).
[Crossref]

Stanishev, V.

P. Kühne, N. Armakavicius, V. Stanishev, C. M. Herzinger, M. Schubert, and V. Darakchieva, “Advanced terahertz frequency-domain ellipsometry instrumentation for in situ and ex situ applications,” IEEE Trans. Terahertz Sci. Technol. 8, 257–270 (2018).
[Crossref]

Tino, H.

Tompkins, H. G.

H. G. Tompkins and E. A. Irene, Handbook of Ellipsometry (William Andrew, 2005).

Uhlmann, P.

A. Furchner, A. Kroning, S. Rauch, P. Uhlmann, K.-J. Eichhorn, and K. Hinrichs, “Molecular interactions and hydration states of ultrathin functional films at the solid-liquid interface,” Anal. Chem. 89, 3240–3244 (2017).
[Crossref]

A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015).
[Crossref]

O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
[Crossref]

Vitkin, I. A.

Vogt, B. D.

K. Lin, Y. Gu, H. Zhang, Z. Qiang, B. D. Vogt, and N. S. Zacharia, “Accelerated amidization of branched poly(ethylenimine)/poly(acrylic acid) multilayer films by microwave heating,” Langmuir 32, 9118–9125 (2016).
[Crossref]

Wagner, T.

Wang, J.

O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
[Crossref]

Wood, M. F. G.

Zacharia, N. S.

K. Lin, Y. Gu, H. Zhang, Z. Qiang, B. D. Vogt, and N. S. Zacharia, “Accelerated amidization of branched poly(ethylenimine)/poly(acrylic acid) multilayer films by microwave heating,” Langmuir 32, 9118–9125 (2016).
[Crossref]

Zdyrko, B.

O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
[Crossref]

Zhang, C.

B. Song, H. Gu, S. Zhu, H. Jiang, X. Chen, C. Zhang, and S. Liu, “Broadband optical properties of graphene and HOPG investigated by spectroscopic Mueller matrix ellipsometry,” Appl. Surf. Sci. 439, 1079–1087 (2018).
[Crossref]

T. Mu, Z. Chen, C. Zhang, and R. Liang, “Optimal configurations of full-Stokes polarimeter with immunity to both Poisson and Gaussian noise,” J. Opt. 18, 055702 (2016).
[Crossref]

W. Li, H. Jiang, C. Zhang, X. Chen, H. Gu, and S. Liu, “Characterization of curved surface layer by Mueller matrix ellipsometry,” J. Vac. Sci. Technol. B 34, 020602 (2016).
[Crossref]

Zhang, H.

K. Lin, Y. Gu, H. Zhang, Z. Qiang, B. D. Vogt, and N. S. Zacharia, “Accelerated amidization of branched poly(ethylenimine)/poly(acrylic acid) multilayer films by microwave heating,” Langmuir 32, 9118–9125 (2016).
[Crossref]

Zhu, S.

B. Song, H. Gu, S. Zhu, H. Jiang, X. Chen, C. Zhang, and S. Liu, “Broadband optical properties of graphene and HOPG investigated by spectroscopic Mueller matrix ellipsometry,” Appl. Surf. Sci. 439, 1079–1087 (2018).
[Crossref]

ACS Appl. Mater. Interfaces (1)

A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K.-J. Eichhorn, M. Seeber, I. Luzinov, S. M. I. Kilbey, B. S. Lokitz, S. Minko, and K. Hinrichs, “In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment,” ACS Appl. Mater. Interfaces 7, 12430–12439 (2015).
[Crossref]

Adv. Funct. Mater. (1)

O. Hoy, B. Zdyrko, R. Lupitskyy, R. Sheparovych, D. Aulich, J. Wang, E. Bittrich, K.-J. Eichhorn, P. Uhlmann, K. Hinrichs, M. Müller, M. Stamm, S. Minko, and I. Luzinov, “Synthetic hydrophilic materials with tunable strength and a range of hydrophobic interactions,” Adv. Funct. Mater. 20, 2240–2247 (2010).
[Crossref]

Anal. Chem. (2)

A. Furchner, A. Kroning, S. Rauch, P. Uhlmann, K.-J. Eichhorn, and K. Hinrichs, “Molecular interactions and hydration states of ultrathin functional films at the solid-liquid interface,” Anal. Chem. 89, 3240–3244 (2017).
[Crossref]

Y. Mikhaylova, L. Ionov, J. Rappich, M. Gensch, N. Esser, S. Minko, K.-J. Eichhorn, M. Stamm, and K. Hinrichs, “In situ infrared ellipsometric study of stimuli-responsive mixed polyelectrolyte brushes,” Anal. Chem. 79, 7676–7682 (2007).
[Crossref]

Appl. Opt. (1)

Appl. Surf. Sci. (3)

B. Song, H. Gu, S. Zhu, H. Jiang, X. Chen, C. Zhang, and S. Liu, “Broadband optical properties of graphene and HOPG investigated by spectroscopic Mueller matrix ellipsometry,” Appl. Surf. Sci. 439, 1079–1087 (2018).
[Crossref]

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, “IR-Mueller matrix ellipsometry of self-assembled nanopatterned gold grid polarizer,” Appl. Surf. Sci. 421, 728–737 (2017).
[Crossref]

A. Furchner, C. Kratz, D. Gkogkou, H. Ketelsen, and K. Hinrichs, “Infrared-spectroscopic single-shot laser mapping ellipsometry: proof of concept for fast investigations of structured surfaces and interactions in organic thin films,” Appl. Surf. Sci. 421, 440–445 (2017).
[Crossref]

IEEE Trans. Terahertz Sci. Technol. (1)

P. Kühne, N. Armakavicius, V. Stanishev, C. M. Herzinger, M. Schubert, and V. Darakchieva, “Advanced terahertz frequency-domain ellipsometry instrumentation for in situ and ex situ applications,” IEEE Trans. Terahertz Sci. Technol. 8, 257–270 (2018).
[Crossref]

J. Appl. Phys. (1)

S. Schöche, T. Hofmann, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, P. Kühne, K. Lorenz, M. Schubert, and V. Darakchieva, “Infrared dielectric functions, phonon modes, and free-charge carrier properties of high-Al-content AlxGa1−xN alloys determined by mid infrared spectroscopic ellipsometry and optical Hall effect,” J. Appl. Phys. 121, 205701 (2017).
[Crossref]

J. Chem. Phys. (1)

J.-J. Max and C. Chapados, “Isotope effects in liquid water by infrared spectroscopy,” J. Chem. Phys. 116, 4626–4642 (2002).
[Crossref]

J. Opt. (1)

T. Mu, Z. Chen, C. Zhang, and R. Liang, “Optimal configurations of full-Stokes polarimeter with immunity to both Poisson and Gaussian noise,” J. Opt. 18, 055702 (2016).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

J. Quant. Spectrosc. Radiat. Transfer (1)

F. Carmagnola, J. M. Sanz, and J. M. Saiz, “Development of a Mueller matrix imaging system for detecting objects embedded in turbid media,” J. Quant. Spectrosc. Radiat. Transfer 146, 199–206 (2014).
[Crossref]

J. Vac. Sci. Technol. B (1)

W. Li, H. Jiang, C. Zhang, X. Chen, H. Gu, and S. Liu, “Characterization of curved surface layer by Mueller matrix ellipsometry,” J. Vac. Sci. Technol. B 34, 020602 (2016).
[Crossref]

Langmuir (2)

L. Ionov, A. Sidorenko, K.-J. Eichhorn, M. Stamm, S. Minko, and H. Karsten, “Stimuli-responsive mixed grafted polymer films with gradually changing properties,” Langmuir 21, 8711–8716 (2005).
[Crossref]

K. Lin, Y. Gu, H. Zhang, Z. Qiang, B. D. Vogt, and N. S. Zacharia, “Accelerated amidization of branched poly(ethylenimine)/poly(acrylic acid) multilayer films by microwave heating,” Langmuir 32, 9118–9125 (2016).
[Crossref]

Macromol. Symp. (1)

K. Hinrichs, M. Gensch, N. Nikonenko, J. Pionteck, and K.-J. Eichhorn, “Spectroscopic ellipsometry for characterization of thin films of polymer blends,” Macromol. Symp. 230, 26–32 (2005).
[Crossref]

Macromolecules (1)

K. S. Iyer and I. Luzinov, “Effect of macromolecular anchoring layer thickness and molecular weight on polymer grafting,” Macromolecules 37, 9538–9545 (2004).
[Crossref]

Opt. Acta (1)

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Opt. Mater. Express (1)

Sci. Rep. (1)

A. Mendoza-Galván, L. F. del Río, K. Järrendahl, and H. Arwin, “Graded pitch profile for the helicoidal broadband reflector and left-handed circularly polarizing cuticle of the scarab beetle Chrysina chrysargyrea,” Sci. Rep. 8, 6456 (2018).
[Crossref]

Other (13)

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, ed. (McGraw Hill, 1995).

J.-C. Cigal, “A novel spectroscopic ellipsometer in the infrared,” Ph.D. thesis (Technische Universiteit Eindhoven, 2002).

J. M. Chalmers and P. R. Griffiths, eds., Handbook of Vibrational Spectroscopy (Wiley, 2006).

R. W. Collins, I. An, and C. Chen, “Rotating polarizer and analyzer ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds. (William Andrew, 2005).

A. Röseler, “Spectroscopic infrared ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds. (William Andrew, 2005).

A. Furchner, “Structure and interactions of polymer thin films from infrared ellipsometry,” in Ellipsometry of Functional Organic Surfaces and Films, K. Hinrichs and K.-J. Eichhorn, eds. (Springer International Publishing, 2018), pp. 145–171.

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, M. Losurdo and K. Hingerl, eds. (Springer Berlin Heidelberg, 2013).

H. G. Tompkins and E. A. Irene, Handbook of Ellipsometry (William Andrew, 2005).

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007).

M. Losurdo and K. Hingerl, Ellipsometry at the Nanoscale (Springer Berlin Heidelberg, 2013).

K. Hinrichs and K.-J. Eichhorn, Ellipsometry of Functional Organic Surfaces and Films (Springer-Verlag Berlin Heidelberg, 2018).

A. Furchner and D. Aulich, “Common polymers and proteins,” in Ellipsometry of Functional Organic Surfaces and Films, K. Hinrichs and K.-J. Eichhorn, eds. (Springer International Publishing, 2018), pp. 521–527.

A. Furchner and D. Aulich, “Organic materials for optoelectronic applications,” in Ellipsometry of Functional Organic Surfaces and Films, K. Hinrichs and K.-J. Eichhorn, eds. (Springer International Publishing, 2018), pp. 529–538.

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Figures (10)

Fig. 1.
Fig. 1. Schematic of the IR Mueller-matrix ellipsometer operating in reflection mode (top) with autocollimator (AC) and adjustable sample stage, or in transmission mode (bottom) with rotatable sample mount.
Fig. 2.
Fig. 2. Quadruples of MM elements obtaintable via different combinations of polarizer/analyzer/retarder settings.
Fig. 3.
Fig. 3. Left: intensity-normalized Stokes-vector components of the FT-IR source, as measured in front of the first polarizer. Right: diattenuation of single and tandem wire-grid polarizers.
Fig. 4.
Fig. 4. RMS noise (top) between 4000    cm 1 and 1000    cm 1 of selected MM elements (bottom) of dry air at two different spectral resolutions. The insets show examplary intensity spectra measured in an unpurged (humid) and purged (dry) ellipsometer chamber.
Fig. 5.
Fig. 5. Structure parameters of trapezoidal SiO 2 gratings on Si.
Fig. 6.
Fig. 6. Measured azimuth-dependent MM data ( ϕ 0 = 50 ° ) of a trapezoidal SiO 2 grating on Si compared to fitted data according to the optical model in Fig. 5. The fourth column shows several symmetries of the Mueller matrix. Measured spectra were smoothed at the high-wavenumber end because of noise due to the grating’s low reflectivity ( < 0.2 ). Gray and colored lines are raw and smoothed data, respectively. The legend lists nominal azimuths (0.5° offset).
Fig. 7.
Fig. 7. Time-resolved (47 s) transmission MM difference spectra measured between tensioned and relaxed states of a stretched low-density polyethylene (LDPE) foil. The schematic (bottom-right) shows the measurement geometry of the foil stretched across the sample stage.
Fig. 8.
Fig. 8. Thin PGMA films (118 nm, 68 nm, and 7.7 nm) on Si substrates studied in reflection at the solid–air interface.
Fig. 9.
Fig. 9. Measured (colored) and fitted (black) MM spectra of thin, isotropic PGMA films of various thicknesses.
Fig. 10.
Fig. 10. Left: liquid flow cell with thin polymer film on Si wedge measured in backside reflection. Right: measured and fitted in situ ellipsometric data of a 68 nm thick, isotropic PGMA film at the polymer–air and polymer–water interface.

Tables (1)

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Table 1. Nominal and Fitted Profile Parameters of the Measured SiO 2 Grating: Height (H), Periods ( P x , y ), Base, and Top Lengths ( B x , y , T x , y ) a

Equations (20)

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S out = PSA T · M · PSG = D · A · R 2 · [ M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 ] · R 1 · P · S in ,
M 11 = [ X 1 / ( 1 + c ) 2 s i / s 0 X 2 / ( 1 c 2 ) ] / [ 1 s i 2 / s 0 2 ] / X ˜ , M 1 b = [ X 2 / ( 1 c 2 ) s i / s 0 X 1 / ( 1 + c ) 2 ] / [ 1 s i 2 / s 0 2 ] / X ˜ , M a 1 = [ X 3 / ( 1 c 2 ) s i / s 0 X 4 / ( 1 c ) 2 ] / [ 1 s i 2 / s 0 2 ] / X ˜ , M a b = [ X 4 / ( 1 c ) 2 s i / s 0 X 3 / ( 1 c 2 ) ] / [ 1 s i 2 / s 0 2 ] / X ˜ ,
X 1 ( P , A ) = I P , A + I P , A + 90 ° + I P + 90 ° , A + I P + 90 ° , A + 90 ° , X 2 ( P , A ) = I P , A + I P , A + 90 ° I P + 90 ° , A I P + 90 ° , A + 90 ° , X 3 ( P , A ) = I P , A I P , A + 90 ° + I P + 90 ° , A I P + 90 ° , A + 90 ° , X 4 ( P , A ) = I P , A I P , A + 90 ° I P + 90 ° , A + I P + 90 ° , A + 90 ° ,
i = 1 for P = 0 ° , i = 2 for    P = 45 ° .
a = 2 & b = 2 for    P = 0 ° , A = 0 ° , a = 3 & b = 3 for    P = 45 ° , A = 45 ° , a = 2 & b = 3 for    P = 45 ° , A = 0 ° , a = 3 & b = 2 for    P = 0 ° , A = 45 ° .
M i 4 = [ M · R 1 ] i 3 / R 1 , 43 , i = 1 , 2 , 3 ,
M 4 j = [ R 2 · M ] 3 j / R 2 , 34 , j = 1 , 2 , 3 ,
M 44 = [ R 2 · M · R 1 ] 33 / ( R 1 , 43 R 2 , 34 ) ,
D = 1 c 1 + c = X ˜ 4 ( 0 ° , 0 ° ) X ˜ 1 ( 0 ° , 0 ° ) = X ˜ 4 ( 45 ° , 45 ° ) X ˜ 1 ( 45 ° , 45 ° ) , s 1 s 0 = 1 D · X ˜ 2 ( 0 ° , 0 ° ) X ˜ 1 ( 0 ° , 0 ° ) , s 2 s 0 = 1 D · X ˜ 2 ( 45 ° , 45 ° ) X ˜ 1 ( 45 ° , 45 ° ) .
D 12 = s 1 / s 0 + ( 1 + c ) 2 · X ˜ 3 ( 0 ° , 0 ° ) X ˜ 1 ( 0 ° , 0 ° ) ,
D 13 = s 2 / s 0 + ( 1 + c ) 2 · X ˜ 3 ( 45 ° , 45 ° ) X ˜ 1 ( 45 ° , 45 ° ) ,
P = i , j M i j 2 M 11 2 3 M 11 2 , P ph = M 33 2 + M 43 2 1 M 12 2 / M 22 2 .
M i 4 = [ M · R 1 ] i 3 M i 3 R 1 , 33 R 1 , 43 , i = 1 , 2 , 3 ,
M 4 j = [ R 2 · M ] 3 j M 3 j R 2 , 33 R 2 , 34 , j = 1 , 2 , 3 ,
M 44 = [ R 2 · M · R 1 ] 33 M 33 R 1 , 33 R 2 , 33 R 1 , 43 R 2 , 34 M 43 R 1 , 33 R 1 , 43 M 34 R 2 , 33 R 2 , 34 .
M i 4 M 11 = [ M · R 1 ] i 3 [ M · R 1 ] 11 · ( 1 + M 12 M 11 R 1 , 21 R 1 , 11 ) M i 3 M 11 R 1 , 33 R 1 , 11 R 1 , 43 R 1 , 11 , i = 1 , 2 , 3 .
M 4 j M 11 = [ R 2 · M ] 3 j [ R 2 · M ] 11 · ( 1 + M 21 M 11 R 2 , 12 R 2 , 11 ) M 3 j M 11 R 2 , 33 R 2 , 11 R 2 , 34 R 2 , 11 , j = 1 , 2 , 3 .
M 44 M 11 = [ R 2 · M · R 1 ] 33 [ R 2 · M · R 1 ] 11 · Z M 33 M 11 R 1 , 33 R 1 , 11 R 2 , 33 R 2 , 11 R 1 , 43 R 1 , 11 R 2 , 34 R 2 , 11 M 43 M 11 R 1 , 33 R 1 , 43 M 34 M 11 R 2 , 33 R 2 , 34
Z = 1 + M 12 M 11 R 1 , 21 R 1 , 11 + M 21 M 11 R 2 , 12 R 2 , 11 + M 22 M 11 R 1 , 21 R 1 , 11 R 2 , 12 R 2 , 11 .
M 11 = X 1 / X ˜ 1 , M 1 b = X 2 / X ˜ 1 / D , M a 1 = X 3 / X ˜ 1 / D , M a b = X 4 / X ˜ 1 / D 2 .

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