Abstract

Determining optical constants of thin material films is important for characterizing their electronic excitations and for the design of optoelectronic devices. Spectroscopic ellipsometry techniques have emerged as the predominant approach for measuring thin-film optical constants. However, ellipsometry methods suffer from complications associated with highly model-dependent, multi-parameter spectral fitting procedures. Here, we present a model-blind, momentum-resolved reflectometry technique that yields accurate and precise optical constants, with quantifiable error estimates, even for film thicknesses less than 50 nm. These capabilities are demonstrated by interrogating an optical absorption resonance in films of the polymer P(NDI2OD-T2). We show that this approach produces exceptional agreement with UV-Vis-NIR absorption measurements, while simultaneously avoiding the need to construct complicated multi-oscillator spectral models. Finally, we use this procedure to resolve subtle differences in the out-of-plane optical properties of different film morphologies that were previously obscured in ellipsometry measurements.

© 2016 Optical Society of America

1. Introduction

Thin films are a fundamental component of modern optoelectronic devices, and have found prolific applications in industrial, military, R&D, civil, and consumer settings [1–5]. Knowing their optical constants is important for performance simulation and design [6–8]. To date, variable angle-of-incidence spectroscopic ellipsometry (VASE), is the predominant method for determining the optical constants of thin films [9–13]. (For a broad overview of the topic, see [14]). However, determination of thin film optical constants with VASE can be a complicated and difficult task for numerous reasons: (i) The analysis procedure requires the construction of complicated spectral models that may involve dozens, or even hundreds, of model parameters [15,16]; (ii) Basic VASE analyses present high correlation of fit parameters, and thus poor or unknown reliability of fit parameters and possibly unphysical results [15–19]; (iii) In the thin-film limit, analyses suffer from intrinsic theoretical and experimental sensitivity limitations [15,20]; (iv) Incidence from air produces predominantly in-plane (IP) fields (parallel to the film interface) in the bulk of the film. This provides weak coupling to out-of-plane (OP) excitations and extraction of OP parameters suffers [16].

Although variations of the basic VASE technique (e.g., interference-enhanced, transmission-mode, total-internal- reflection, multiple-sample, etc...) can help alleviate correlation concerns and limitations associated with thin films [15,20–22], all spectroscopic ellipsometry approaches suffer from complications associated with multi-parameter fits based on spectral models that are not known a priori. The measurements are analyzed against an unknown number of electronic models that can take a number of distinct forms (e.g. Lorentz and Gaussian oscillators, Drude, Cauchy, etc...), each of which is characterized by several free parameters. Least-squares minimization is used not only to determine the free parameters, but also to refine the model itself. As a result, determining optical constants with spectroscopic ellipsometry is time consuming, requires finesse, and produces results with unknown errors and uncertainties.

An alternative to spectral fitting is the measurement and fitting of optical properties (e.g. transmittance, reflectance) as a function of angle, rather than wavelength. In contrast to ellipsometry, a universal “Fresnel model” is employed and the system consists of a minimal number of free parameters. For instance, an isotropic film is characterized at each wavelength by just three unknowns — the film thickness, refractive index, and absorption coefficient. Such approaches may be considered “model-blind”, or “deterministic”, since the data is fit to a priori Fresnel reflection equations. However, the analyses used in previous angle-resolved techniques are limited to films with thicknesses on the order of a wavelength or larger; that is, researchers have inferred complex refractive indices by analyzing successive Fabry-Perot interference fringes [23–26] or prism-coupled waveguide modes [27–29]. Here, we determine the complex refractive index of polymer thin films, of known thickness (∼50 nm), with a high degree of certainty using model-blind analysis of angle-resolved reflection measurements extending beyond the critical angle of total internal reflection (TIR).

Using Fourier-imaging techniques, we measure the wavelength-dependent angle-resolved reflectance from thin polymer films deposited on quartz substrates. The film thickness is determined in advance from atomic force microscopy measurements of samples co-deposited under identical processing conditions. Experiments are performed in a high-numerical-aperture (NA=1.3) imaging system, enabling measurements well beyond the critical angle, i.e. the onset of TIR. Fits of s-polarized reflectance measurements converge to a unique solution of the complex ordinary refractive index (no, E-field in the plane of the film) with quantified small uncertainty estimates. By comparing results to independent UV-Vis-NIR absorption measurements, we show that these results are substantially more accurate than optical constants determined via ellipsometry; that is, with our technique, determination of the ordinary complex refractive index is not complicated by unique out-of-plane properties. We subsequently determine the complex extraordinary refractive index (ne, E-field perpendicular to the substrate) by performing the same procedure for p-polarized reflectance. These results reveal morphology-dependent optical anisotropies that are obscured in ellipsometry measurements. Analysis of parameter sensitivity and correlation (see Appendix C for further discussion) demonstrate the robustness of this technique and the importance of measuring reflectance beyond the critical angle. These results demonstrate a “turn-key” approach for measuring optical constants that obviates the uncertainties and modeling challenges inherent to ellipsometry.

Although the techniques described here are materials agnostic, we demonstrate our technique on films of P(NDI2OD-T2) [30–33]. This material provides a good test case due to its interesting processing-dependent structural and optical properties [34]. When spin-coated from solution, the polymer molecules self-assemble into highly-ordered morphologies with molecular back-bones generally aligned parallel to the substrate surface. The molecular orientation is further influenced through thermal annealing; annealing at 150°C yields a ‘face-on’ morphology with the π-stacking direction perpendicular to the substrate, while annealing at 305°C yields an ‘edge-on’ morphology, as in Fig. 1(a). In both cases the optical transition dipole of this polymer is oriented mostly along the molecular back-bone, within the plane of the film (see Figs. 1b,c). Both morphologies thus possess uniaxial optical properties (extraordinary axis perpendicular to the substrate) with stronger polarizability and absorption along the in-plane ordinary axes. Such optical anisotropies are common to many small molecule and polymer thin films [15,16,35–39] and can significantly impact device performance and design [40–43]. The ability to change molecular orientation with annealing provides a means to subtly vary optical properties of thin films without changing the molecular constituents or other system parameters. As we will demonstrate, the measurements and analysis described here can resolve these subtle differences that were not witnessed in ellipsometric studies.

 figure: Fig. 1

Fig. 1 (a) ‘Face-on’ and ‘edge-on’ morphologies assumed by P(NDI2OD-T2) for low-and high-temperature thermal annealing, respectively; (b) P(NDI2OD-T2) molecule; (c) polymer segment with associated transition dipole moment; (d) the macroscopic EM system representing the sample on substrate.

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2. Measurement overview and geometry

The film is interrogated via angle-resolved reflection measurements using a Nikon Eclipse Ti-U inverted microscope with a 100×/1.3 NA oil-immersion objective. Using back focal plane (BFP) imaging (‘Fourier imaging’) techniques [34, 44–49], we control the incidence angle of our illumination source. Specifically, the single-mode optical fiber output of a wavelength tunable laser is placed in the BFP where it acts as a point source in Fourier space (i.e., momentum space). By translating the fiber within the BFP we control the incident momentum vector, k⃗, while a linear polarizer allows control of incident polarization. A schematic of the basic experimental geometry is shown in Fig. 1(d). The light is incident from the substrate and reflected back through the substrate and microscope objective. The reflected intensity is measured as a function of in-plane momentum, k, which relates to the incident angle, θ, within the quartz substrate according to k=nQk0sin θ, where nQ is the refractive index of quartz (1.4553 at 700 nm) and k0 is the free-space wave momentum. The reflection profiles are then normalized to those of a quartz-air interface at the same incident intensity in order to correct for angle- and polarization-dependent collection efficiencies in the microscope, yielding the s- and p-polarized reflectances, Rs and Rp, respectively (Fig. 7).

3. Results and discussion

3.1. Determining in-plane optical constants

s-Polarized reflection measurements (λ=700 nm) of bare quartz substrates (blue triangles) and substrates coated with 46nm thick PNDI2OD-T2 films (red circles) are shown in Fig. 2. Figure 2(a) corresponds to face-on (150° anneal) films and Fig. 2(b) to edge-on (305° anneal) films. Theoretical curves (solid lines) for the single quartz-air interface (blue) and the two-interface quartz-film-air system with the best-fit film index (red) are superimposed on the data. For small momentum values (near normal incidence), the reflection coefficients with and without films exhibit small but measurable differences (inset in Figs. 2(a),2(b)). These differences increase substantially as the in-plane momentum approaches and then surpasses the critical angle. The reflection intensity at the critical angle (k=k0) is nearly half as large for face-on films (∼20%) in comparison to edge-on films (∼40%), suggesting an approximately 50% larger value of the imaginary part of the refractive index. As we will discuss later, measuring and fitting the reflection intensity over this large range of momenta is critical for generating accurate, high-confidence deductions of the optical constants.

 figure: Fig. 2

Fig. 2 s-Polarized reflection measurements for bare quartz substrates (blue triangles) and film-on-substrate (red circles) at 700 nm for 46 nm (a) face-on and (b) edge-on films. Theoretical curves (solid lines) for single quartz-air interface (blue) and the two-interface quartz-film-air system with best-fit film index (red) are superimposed on the data. The edge-on film has a smaller absorption coefficient (Im[no]), manifested most evidently as higher reflectance at large momentum values (kk0). Best fit values for the real and imaginary parts of the ordinary index and associated 99% confidence values are displayed in the upper left hand corner of each plot.

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Following the procedures described in Sect. 2, we perform the same analysis at seven evenly spaced wavelength intervals between 500–900 nm. The wavelength range is chosen to span the low-energy absorption peak of P(NDI2OD-T2). Resulting values of the real (red, left axis) and imaginary (blue, right axis) components of the refractive index for face-on and edge-on films are shown in Fig. 3(a) and 3b, respectively. Both films display a strong absorption band peaking near 700nm, with a corresponding “derivative” line-shape of the real part of the refractive index, as expected for a Lorentz oscillator. The face-on film exhibits a larger absorption peak and greater variation in the real part of the index as compared to edge-on films. It is important to note that fits at each wave- length are independent of each other; we make no assumptions about the nature of the electromagnetic oscillators in the material system, and the optical parameters are determined without fitting the data to a forward simulation. The experiment/fitting procedure is, in this sense, “model-blind” and produces optical constants without the use of spectral models.

 figure: Fig. 3

Fig. 3 [(a) and (b)] Variation of the real (red, left axis) and imaginary (blue, right axis) parts of the ordinary index for (a) face-on and (b) edge-on films. The curves in each plot show the expected lineshapes associated with an optical absorption resonance around λ=700 nm. 99% confidence intervals are included, but those for the imaginary parts lie within the data markers and are not visible. [(c) and (d)] Normal-incidence transmission spectra through the 46 nm film on fused-silica substrate for (c) face-on and (d) edge-on films. Red circles show the prediction using values determined from Rs measurements; Green triangles show the prediction using values determined from ellipsometry; Blue dashed lines show UV-Vis-NIR transmission measurement. In agreement with the molecular model, smaller IP absorption is visible in the edge-on film, attributed to the higher OP dipole strength. This subtlety is well-resolved by our reflectometry procedure, whereas ellipsometry incorrectly suggests a larger IP absorption for edge-on films.

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Included in the plots (Figs. 3a,b) are 99% confidence intervals derived via the “bootstrap” method [50], assuming reasonable variations in the measured data (see Appendix D). Those for the imaginary component lie within the scale of the data markers and thus are not visible. Confidence intervals for the imaginary part of the refractive index are generally 0.1–1% the height of the data markers; the optical constants derived here are clearly precise, but an independent measure is needed to demonstrate their accuracy. For a given set of optical constants, at each wavelength, we can produce an expected transmission spectrum based on a five-layer (air-oil-quartz-film-air) Fresnel model. These predicted transmission spectra are shown in Fig. 3 (red circles) for face-on (c) and edge-on (d) films. UV-Vis-NIR transmission measurements performed upon the same sample set are also displayed in Figs. 3(c),3(d) (blue dashed lines). To compare techniques, a unique set of optical constants for identically-prepared films (on oxidized Si substrates) was determined independently via VASE (Appendix B) using a spectroscopic model consisting of ten Gaussian oscillators (and 35 total fit parameters). The expected transmission from ellipsometry-derived optical constants is also shown in Figs. 3(c),3(d) (green triangles).

Transmission values derived from our model-blind procedure show excellent agreement with the measured UV-Vis-NIR data for both samples across the entire wavelength band; the derived optical constants are both precise and accurate. In contrast, our best ellipsometry results are in poor agreement with UV-Vis-NIR transmission, especially for edge-on films. Even the qualitative differences in absorptance between the face-on and edge-on films are obscured, demonstrating the complications introduced by optical anisotropies in VASE. Interestingly, the fits of ellipsometry parameters psi and delta appear quite good (Appendix B), and reasons for the inaccuracy of the derived results are not obvious due to the complexities of the constructed model.

3.2. Demonstrating uniqueness and sensitivity to large momenta

The robustness of these fit results can be better appreciated by a quantitative examination of the fitting procedure. The fitting algorithm (the “trust-region reflective”, TRR, algorithm [51,52]) minimizes, within specified bounds, the sum of squared residuals:

S(en,𝕀mn)=Σi[yiR(k,i;en,𝕀mn)]2
where yi is the measured reflectance at in-plane momentum k‖,i, and R(k‖,i; ℝe n, 𝕀m n) is the predicted reflectance at k‖,i with index values ℝe(n) and 𝕀m(n). The bounds chosen here [1≤ℝe(no)≤2 and 0≤𝕀m(no)≤1] include reasonable values for virtually all organic films. A 2D color map of the error function for face-on films at 700 nm is shown in Fig. 4(a) (edge-on films show similar behavior and an error map is presented in Appendix C). The maps exhibit a clearly defined global minimum, and the ordinary index is consequently determined with high confidence.

 figure: Fig. 4

Fig. 4 (a) 2D map of the error FOM, S (Eqn. (1)), for real and imaginary components of the ordinary index of the face-on film as fit against Rs data at 700 nm. In order to reveal details in the neighborhood of the minimum, the vertical range is limited to 0.1≤𝕀m(no)≤0.8 in this visualization. The specified fit bounds [1≤ℝe(no)≤2 and 0≤𝕀m(no)≤1] cover index values for nearly all organic films. (b) Sensitivity of the reflection function, Rs, with respect to real (blue) and imaginary (red) components of the ordinary index, as a function of in-plane momentum, k. The curves are calculated at typical index values: ℝe(no)=1.5; 𝕀m(no)=0.2. Rs shows significantly increased sensitivity to the fit parameters for in-plane momenta beyond the critical angle of TIR. This is especially so for 𝕀m(no), and thus accurate estimates of film absorption benefit significantly from measurements in the region of TIR.

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Local minima in the error function are prevalent in ellipsometry, and their absence here highlights another attractive feature of this model-blind approach. The precision demonstrated here results in large part from the access to large momentum values afforded by the high NA imaging system. This statement is supported by numerical calculations of the partial derivatives ∂Rs/ℝe(no) (blue) and ∂Rs/𝕀m(no) (red), shown in Fig. 4(b). The reflectance exhibits significantly increased sensitivity to both the real and imaginary parts of the refractive index for in-plane momenta near and beyond the critical angle of TIR (k/k0>1.0). This suggests that the reliability of the angle-resolved procedure demonstrated here is greatly enhanced by accessing the region of TIR.

3.3. Determining out-of-plane optical constants

Measuring reflectance beyond the critical angle provides an additional advantage of great importance for analyzing organic thin-films: enhanced sensitivity to out-of plane optical properties. Resolving out-of-plane optical constants is a notorious challenge for ellipsometry [15,16,36,38,53]. In the case of P(NDI2OD-T2), momentum-resolved photoluminescence excitation measurements demonstrate a clear difference in the optical anisotropies for face-on and edge-on films. Edge-on films exhibit a substantial increase in light emission and absorption from out-of-plane oriented dipoles [34]. One would expect an associated difference in 𝕀m(ne), but no such behavior is evident in our ellipsometry results (Appendix B). To investigate the extraordinary index, we measured and fit p-polarized reflection profiles (Fig. 5) for (a) face-on and (b) edge-on films. The pseudo-Brewster’s angle and subsequent onset of TIR are highlighted in the figure insets. The most striking difference between the two films occurs right at the critical angle, where the E-field within the thin-film is purely oriented in the out-of-plane direction [48]. Here, the face-on film exhibits nearly unity reflectance whereas the edge-on reflectance is only ∼84%. The associated fits reveal a significantly larger value of 𝕀m(ne) for edge-on films (0.09) as compared to face-on films (0.01), as expected from complementary photoluminescence studies [34]. Wavelength-dependent values of 𝕀m(ne) and ℝe(ne) are presented in Figs. 6(a) and 6(b), respectively. Though the determined out-of-plane indices exhibit unexpected fluctuations and greater uncertainty, the edge-on film (blue) displays a clear increase in absorption coefficient. Indeed, the reflectance is again highly sensitive to the imaginary part the extraordinary index for in-plane momenta beyond the critical angle (see Appendix C) and, consequently, differences in out-of-plane absorption between face-on and edge-on films are well resolved.

 figure: Fig. 5

Fig. 5 p-Polarized reflection measurements for bare quartz substrates (blue triangles) and film-on-substrate (red circles) at 700nm for 46 nm (a) face-on and (b) edge-on films. Theoretical curves (solid lines) for single quartz-air interface (blue) and the two-interface quartz-film-air system with best-fit film index (red) are superposed on the data. The enhanced OP absorption is apparent in the edge-on film, manifest as substantially lower reflectance at TIR. The inset highlights both data around the pseudo-Brewster angle. Best fit values and 99% confidence values are printed in each plot.

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 figure: Fig. 6

Fig. 6 (a) The imaginary component of the extraordinary refractive index for face-on (red circles) and edge-on (blue triangles) films, determined via p-polarized reflection measurements. Though uncertainties are relatively large, the edge-on film exhibits a clear increase in OP absorption, relative to the face-on film. The wavelengths between face-on and edge-on films have been slightly offset to avoid overlapping error bars. (b) The real component of the extraordinary refractive index for face-on (red circles) and edge-on (blue triangles) films. 99% confidence bars are included in each plot.

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The determinations for ne demonstrate some of the challenges associated with measuring out-of-plane optical constants. In particular, the real and imaginary parts of the index exhibit a correlated impact on the reflection coefficient. This results in large error estimates in ℝe(ne), and complicates interpretation of the confidence intervals for (ne) (see Appendix C for further discussion). Regardless, our model-blind approach resolves differences in ne that were previously obscured in ellipsometry analysis, and these preliminary results suggest approaches for future refinements and improvements of the technique.

4. Summary and outlook

We have demonstrated a “turn-key” approach for the determination of the in-plane refractive index and absorption coefficient of organic thin films through model-blind fitting of momentum-resolved reflectance. Comparison with independent UV-Vis-NIR absorption experiments demonstrate high accuracy and precision, far surpassing ellipsometry results. Analysis of error maps and parameter sensitivity curves validate the fitting procedure and demonstrate the importance of measuring reflectivity beyond the critical angle. We further extend the technique and analysis to characterize the out-of-plane optical constants by analyzing p-polarized reflectance curves. We resolve subtle differences in the imaginary part of ne for two films comprising the same molecular constituents but different morphologies, in agreement with angle-resolved photoluminescence excitation measurements. In contrast to conventional analyses, our turn-key technique and analysis does not require constructing or refining complex physical models, and provides immediate model-blind results with quantifiable error estimates.

Appendix A: Determining reflectance of films

In this section, we describe and illustrate the procedure for acquiring the film reflectance from the raw data. We measure the reflection intensity of film samples according to the procedure described in the main manuscript. Due to polarization- and angle-dependent transmission efficiencies through the optical components of the system, and since these measurements are in units of “counts”, the measured reflection intensities (Fig. 7, red) for (a) s- and (b) p-polarization are not immediately meaningful. To determine the film reflectance, R, the film reflection intensities are compared to those of a bare quartz substrate (Fig. 7a,b, blue). The quartz data allows immediate determination of the critical angle of total internal reflection, k/k0=1. Further, the bare quartz substrate acts as a single-interface, non-absorbing reference system of known reflectance, R(k), with R(kk0)=1.

The angle-dependent film reflection intensities are then vertically scaled, according to the reflection intensity of the quartz at kk0, defining unity reflectance. That is, we define an overall vertical scale factor, vs,p, defined by the maximum intensity of the quartz data for s- and p-polarized data. An angle-dependent correction factor, cs,p(k), is then determined by directly comparing the vertically-scaled quartz profiles to the single-interface Fresnel’s equations for each polarization, i.e.,

cs,p(k)=Rs,p(k)yQuartz(k)/vs,p.
where yQuartz(k) is the measured intensity of the bare quartz substrate at position k. The profiles are then subjected to this correction factor, yielding the “reflectance”. The final results are presented in Fig. 7 (identical to those of Figs. 2b and 5b of main manuscript) for bare quartz substrates (blue) and substrates with edge-on films (red) for s- (c) and p-polarized (d) incidence. This is the procedure we, in the main manuscript, refer to as “normalization”.

Appendix B: Comparison with ellipsometry

Figure 8 shows the measured ellipsometric quantities psi (blue, left axis) and delta (red, right axis) for (a) face-on and (b) edge-on films of P(NDI2OD-T2) at three unique angles of incidence. Superimposed on this data are the best-fit curves (dashed black line) resulting from the spectroscopic model. The model consists of ten in-plane Gaussian oscillators, each of which introduces three free parameters: central energy, width, and amplitude. The out-of-plane oscillators are restricted to the same energies, widths, and relative amplitudes as the in-plane oscillators, and thus introduce a single amplitude ratio that was allowed to vary. A “UV pole” introduces two free parameters to get the real part of the refractive index in the right place. Additionally, the thickness of the film (P(NDI2OD-T2)) and substrate (SiO2) are allowed to vary slightly around 46 nm and 200 nm, respectively. This gives a grand total of (3×10)+1+2+2 = 35 total fit parameters.

 figure: Fig. 7

Fig. 7 [(a) and (b)] Reflected intensity for (a) p-polarized and (b) s-polarized incidence at 700 nm. [(c) and (d)] Efficiency-corrected and normalized data (i.e., reflectance, R) for (c) p-polarized and (d) s-polarized incidence at 700 nm. Blue triangles: bare quartz substrates. Red circles: quartz substrates with edge-on films.

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 figure: Fig. 8

Fig. 8 Variable-angle spectroscopic ellipsometry measurements of Psi (blue, left axis) and Delta (red, right axis) for (a) face-on and (b) edge-on films of P(NDI2OD-T2). The constructed model necessary for a quality fit introduces 35 free parameters. Though the fits of Delta and Psi appear to be very good, the quality of the model remains unknown, and we find the resulting refractive index estimations to be flawed.

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The resulting real (red, left axis) and imaginary (blue, right axis) parts of the ordinary (a,b) and extraordinary (c,d) refractive index are shown in Fig. 9 for (a,c) face-on and (b,d) edge-on films. Though the fits of delta and psi appear to be of high quality, the resulting index suggest variations between different film morphologies that are qualitatively and quantitively contrary to UV-Vis absorption, angle-resolved photoluminescence excitation, and model-blind reflectometry measurements. Specifically ellipsometry measurements suggest a smaller in-plane absorptance (Im[no]) for face-on films in comparison to edge-on films. In actuality UV-Vis-NIR transmission and model-blind reflectometry display larger absorptances for face-on films. Similarly, ellipsometry measurements also suggest larger out-of-plane absorptance (Im[ne]) for face-on films in comparison, whereas model-blind reflectometry and momentum-resolved photoluminescence excitation [34] measurements both show the opposite trend.

 figure: Fig. 9

Fig. 9 Spectroscopic ellipsometry results for the real (red, left axis) and imaginary (blue, right axis) parts of the (a,b) ordinary and (c,d) extraordinary refractive index for (a,c) face-on and (b,d) edge-on films. Contrary to UV-Vis transmission, angle-resolved photoluminescence, and our reflectometry measurements, the edge-on film shows higher IP absorption.

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Appendix C: Error maps, fit sensitivities, and parameter correlation

In the main manuscript, we show a 2D map of the error figure-of-merit for s-polarized reflection at 700 nm for face-on films. The map shows a deep, unique minimum of the error within the presented bounds, justifying the use of the simple gradient-descent least-squares minimization procedure. Figure 10 shows an analogous map for edge-on films. The curvature has lessened in the ℝe-direction relative to the face-on film, but a unique minimum clearly remains. It is a general trend that higher absorption leads to stronger minima, and thus ℝe(no) is determined with higher confidence where 𝕀m(no) is greater. (This effect is clearly visible in Appendix D Figs. 13). This turns out to be true also for the extraordinary index determinations (see Figs. 11 and 14).

 figure: Fig. 10

Fig. 10 2D map of the error FOM, S (Eqn. (1) of main manuscript), for real and imaginary components of the ordinary index of edge-on films as fit against Rs data at 700 nm. In order to reveal details in the neighborhood of the minimum, the vertical range is limited to 0.1≤𝕀m(no)≤0.8 in this visualization. A clear unique minimum exists, validating the use of the gradient-descent method.

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 figure: Fig. 11

Fig. 11 [(a) and (b)] 2D maps of the error FOM, S (Eqn. (1) of main manuscript), for real and imaginary components of the extraordinary index of (a) face-on and (b) edge-on films as fit against Rp data at 700 nm. [(c) and (d)] The same maps as in (a) and (b), in which values greater than a maximum value (specified by color bars) have been oversaturated (gray). This serves to show fine details in the neighborhood of the minima.

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The determined real and imaginary components of the extraordinary axis exhibit relatively high uncertainty and obvious fluctuations, compared with those of the ordinary index. We gain insight into the problem by looking at the 2D error maps for real and imaginary components of the extraordinary index as fit against Rp data. These maps are presented in Fig. 11 for (a,c) face-on and (b,d) edge-on films at 700 nm. The maps are presented at (a,b) ‘low’ and (c,d) ‘high’ contrasts in order to show details in the neighborhood of the minima.

A unique minimum exists, but it exhibits very small curvature in the ℝe(ne)-dimension. In contrast, the maps exhibit relatively high curvature in the 𝕀m(ne)-dimension in the neighborhood of the minimum. Consequently, differences in out-of-plane absorption are well-resolved between face-on and edge-on films. However, the fits now witness a broad “valley”, over which the error is minimal. This is particularly evident for face-on films (Figs. 11a,c) and, in general, when the out-of-plane absorption is low. Consequently, for any given estimate of 𝕀m(ne), the error is quite insensitive to changes in ℝe(ne) and the final determinations become extremely sensitive to measurement and calibration errors. This valley minimum is, in general, sloped in the ℝe-𝕀m plane, and it is in this sense we say the extraordinary parameters exhibit a correlated effect upon the fit.

These statements are supported by numerical calculations of the partial derivatives ∂Rp/ℝe(ne) (blue) and ∂Rp/𝕀m(ne)(red), as presented in Fig. 12(a). As out-of-plane absorption increases the reflection function Rp grows more sensitive to changes in ℝe(ne). This effect is demonstrated in the inset of Fig. 12(a), and is also apparent upon comparison of Figs. 11a and 11b. Certainly, this is why the dispersion behavior of ℝe(ne) is much more clearly resolved in the edge-on film, and the confidence intervals in ℝe(ne) are generally smaller at the absorption peak. The sensitivity curves further suggest that determination of out-of-plane optical parameters again benefits substantially by accessing the region of total internal reflection (TIR), kk0. In fact, whereas Rs shows significant sensitivity to the in-plane optical parameters for all k/k0 (see Fig. 4(b) of main manuscript), Rp shows negligible sensitivity for k/k0≲0.98 and sharply increased sensitivity beyond this point.

 figure: Fig. 12

Fig. 12 (a) Sensitivity of the reflection function, Rp, with respect to real (blue) and imaginary (red) components of the extraordinary index, as a function of in-plane momentum. The curves are calculated at typical index values: ℝe(no)=1.5; 𝕀m(no)=0.2; ℝe(ne)=1.5; 𝕀m(ne)=0.05. The reflectance shows drastically increased sensitivity to the fit parameters for in-plane momenta beyond the critical angle of TIR. The inset demonstrates increased sensitivity of Rp with respect to the real part of the extraordinary index in films with greater out-of-plane absorption. (b) The correlation function, C (Eqn. 3) as a function of in-plane momentum, again assuming typical index values specified for (a). The behavior for k/k0>1 suggests substantial correlation between the real and imaginary components of the extraordinary index in the region of TIR.

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 figure: Fig. 13

Fig. 13 Real (red, left axis) and imaginary (blue, right axis) components of the ordinary index for (a) face-on and (b) edge-on films. Values are determined from a single fit using the trust-region reflective algorithm, and 99% confidence intervals are determined here from the covariance matrix.

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Lastly, we look at a criterion for parameter correlation between parameters ℝe(ne) and 𝕀m(ne) [17]:

C=|Rpe(ne)Rp𝕀m(ne)|.
Parameters ℝe(ne) and 𝕀m(ne) are said to be correlated if C is constant over a given range of k; that is, changes in Rp due to variation in one parameter can be compensated by a corresponding variation in the other. Of course, C varies with k and one must consider C over the entire range of measurements. This ‘correlation function’, C is plotted in Fig. 12(b) over the angular range 0<k/k0<1.2 and suggests that ℝe(ne) and 𝕀m(ne) are substantially correlated in the TIR region. Unfortunately, this region is where Rp is most sensitive to both OP parameters, as seen in Fig. 12(a). We believe this to be the reason for the “valley” in the error maps.

Appendix D: Confidence Intervals

These features complicate the interpretation of the confidence intervals deduced via the covariance matrix. This motivates us to consider an additional approach for the determination of the index values and errors − the “bootstrap” method [50]. A series of 10,000 fits of resampled data is performed assuming normally distributed variations in both k and measured intensity. Errors in measured intensity exhibit a negligible effect, and the distribution of fit results is dominated by variations in the measured value of k, and thus also the horizontal scale factor determining the critical angle, k=k0. The final values and 99% confidence intervals are determined from the mean and variance of the fit distributions, respectively. The results presented in the main manuscript are those determined from the bootstrap method.

For completeness, we present results determined from single-fits using the trust-region reflective algorithm [52]. In this case, the confidence intervals are determined from the covariance matrix. Fig. 13 shows the real (red) and imaginary (blue) components of the ordinary index for (a) face-on and (b) edge-on films. Fig. 14 shows the (a) imaginary and (b) real components of the extraordinary index for (red) face-on and (blue) edge-on films. The results of both procedures are, as expected, comparable in scale, with the bootstrap method typically yielding smaller estimates of uncertainty, especially for the imaginary components of the index.

 figure: Fig. 14

Fig. 14 (a) Imaginary and (b) real components of the extraordinary index for (red) face-on and (blue) edge-on films. Values are determined from a single fit using the trust-region reflective algorithm, and 99% confidence intervals are determined here from the covariance matrix.

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Funding

This work was supported by a National Science Foundation CAREER award (DMR-1454260).

Acknowledgments

AFM and UV-Vis spectroscopy were performed in the MRL Shared Experimental Facilities which are supported by the MRSEC Program of the NSF under Award No. DMR 1121053; a member of the NSF-funded Materials Research Facilities Network (www.mrfn.org). A portion of this work was performed in the UCSB Nanofabrication Facility.

References and links

1. G. Hass, “preparation, properties and optical applications of thin films of titanium dioxide,” Vacuum , 2(4), 331–345 (1952). [CrossRef]  

2. B. G. Lewis and D. C. Paine, “Applications and Processing of Transparent Conducting Oxides,” MRS Bulletin 25(08), 22–27 (2000). [CrossRef]  

3. P. Siciliano, “Preparation, characterisation and applications of thin films for gas sensors prepared by cheap chemical method,” Sensors and Actuators B: Chemical 70(1–3), 153–164 (2000). [CrossRef]  

4. C. R. Kagan and P. Andry, Thin-Film Transistors (CRC Press, 2003). [CrossRef]  

5. K. N. Chopra and A. K. Maini, “Thin Films and Their Applications in Military and Civil Sectors,” Defence Research and Development Organization (2010).

6. P. Peumans, A. Yakimov, and S. R. Forrest, “Small molecular weight organic thin-film photodetectors and solar cells,” J. Appl. Phys. 93(7), 3693 (2003). [CrossRef]  

7. C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, Light-Emitting Diodes: Research, Manufacturing, and Applications XII, 691006 (2008) [CrossRef]  

8. C. W. Chen, S. Y. Hsiao, C. Y. Chen, H. W. Kang, Z. Y. Huang, and H. W. Lin, “Optical properties of organometal halide perovskite thin films and general device structure design rules for perovskite single and tandem solar cells,” J. Mater. Chem. A 3(17), 9152–9159 (2015). [CrossRef]  

9. F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Nat. Bur. Sec. A 67, 363 (1963). [CrossRef]  

10. R. A. Synowicki, “Spectroscopic ellipsometry characterization of indium tin oxide film microstructure and optical constants,” Thin Solid Films 313–314, 394–397 (1998). [CrossRef]  

11. G. E. Jellison Jr., V. I. Merkulov, A. A. Puretzky, D. B. Geohegan, G. Eres, D. H. Lowndes, and J. B. Caughman, “Characterization of thin-film amorphous semiconductors using spectroscopic ellipsometry,” Thin Solid Films 377–378, 68–73 (2000). [CrossRef]  

12. G. E. Jellison Jr., “Generalized ellipsometry for materials characterization,” Thin Solid Films 450(1), 42–50 (2004). [CrossRef]  

13. H. G. Tompkins, A User’s Guide to Ellipsometry (Dover Publications, 2006.)

14. T. E. Jenkins, “Multiple-angle-of-incidence ellipsometry,” Journal of Physics D: Applied Physics 32(9), R45 (1999). [CrossRef]  

15. M. Campoy-Quiles, P. G. Etchegoin, and D. D. C. Bradley, “On the optical anisotropy of conjugated polymer thin films,” Phys. Rev. B 72, 045209 (2005). [CrossRef]  

16. M. Campoy-Quiles, M. I. Alonso, D. D. C. Bradley, and L. J. Richter, “Advanced Ellipsometric Characterization of Conjugated Polymer Films,” Adv. Funct. Mater. 24(15), 2116–2134 (2014). [CrossRef]  

17. G. H. Bu-Abbud, “Parameter correlation and precision in multiple-angle ellipsometry,” Appl. Opt. , 20(17), 3020–3026 (1981). [CrossRef]   [PubMed]  

18. G. H. Bu-Abbud, “Variable Wavelength, Variable Angle Ellipsometry Including a Sensitivities Correlation Test,” Thin Solid Films 138(1), 27–41 (1986). [CrossRef]  

19. G. E. Jellison, “The calculation of thin film parameters from spectroscopic ellipsometry data,” Thin Solid Films 290–291, 40–45 (1996). [CrossRef]  

20. W. A. McGahan, B. Johs, and J. A. Woollam, “Techniques for ellipsometric measurement of the thickness and optical constants of thin absorbing films,” Thin Solid Films 234(1), 443–446 (1993). [CrossRef]  

21. C. M. Ramsdale and N. C. Greenham, “Ellipsometric Determination of Anisotropic Optical Constants in Electroluminescent Conjugated Polymers,” Adv. Mater. 14(3), 212–215 (2002). [CrossRef]  

22. H. Arwin, M. Poksinski, and K. Johansen, “Total internal reflection ellipsometry: principles and applications,” Appl. Opt. 43(15), 3028–3036 (2004). [CrossRef]   [PubMed]  

23. J. I. Cisneros, “Optical characterization of dielectric and semiconductor thin films by use of transmission data,” Appl. Opt. 37(22), 5262–5270 (1998). [CrossRef]  

24. Y. Laaziz, A. Bennouna, N. Chahboun, A. Outzourhit, and E. L. Ameziane, “Optical characterization of low optical thickness thin films from transmittance and back reflectance measurements,” Thin Solid Films 372(1–2), 149–155 (2000). [CrossRef]  

25. D. Poelman and P. F. Smet, “Methods for the determination of the optical constants of thin films from single transmission measurements: a critical review,” Journal of Physics D: Applied Physics 36(15), 1850 (2003). [CrossRef]  

26. M. Flämmich, N. Danz, D. Michaelis, A. Bräuer, M. C. Gather, H.-W. K. Jonas, and K. Meerholz, “Dispersion-model-free determination of optical constants: application to materials for organic thin film devices,” Appl. Opt. 48(8), 1507–1513 (2009). [CrossRef]   [PubMed]  

27. Metricon Corporation, “Model 2010/M Overview,” http://www.metricon.com

28. W. Knoll, “Optical Characterization of Organic Thin Films and Interfaces with Evanescent Waves,” MRS Bulletin 16(07), 29–39 (1991). [CrossRef]  

29. S. J. Bai, R. J. Spry, D. E. Zelmon, U. Ramabadran, and J. Jackson, “Optical anisotropy of polymeric films measured by waveguide propagation mode determination,” J. Polym. Sci. B Polym. Phys. 30(13), 1507–1514 (1992). [CrossRef]  

30. C. J. Takacs, N. D. Treat, S. Krämer, Z. Chen, A. Facchetti, M. L. Chabinyc, and A. J. Heeger, “Remarkable Order of a High-Performance Polymer,” Nano Letters 13(6), 2522–2527 (2013). [CrossRef]   [PubMed]  

31. E. Giussani, D. Fazzi, L. Brambilla, M. Caironi, and C. Castiglioni, “Molecular Level Investigation of the Film Structure of a High Electron Mobility Copolymer via Vibrational Spectroscopy,” Macromolecules , 46(7), 2658–2670 (2013). [CrossRef]  

32. T. Schuettfort, L. Thomsen, and C. R. McNeill, “Observation of a Distinct Surface Molecular Orientation in Films of a High Mobility Conjugated Polymer,” J. Am. Chem. Soc. 135(3), 1092–1101 (2013). [CrossRef]   [PubMed]  

33. M. Brinkmann, E. Gonthier, S. Bogen, K. Tremel, S. Ludwigs, M. Hufnagel, and M. Sommer, “Segregated versus Mixed Interchain Stacking in Highly Oriented Films of Naphthalene Diimide Bithiophene Copolymers,” ACS Nano 6(11), 10319–10326 (2012). [CrossRef]   [PubMed]  

34. S. J. Brown, R. A. Schlitz, M. L. Chabinyc, and J. A. Schuller, “Morphology dependent optical anisotropies in the n-type polymer P(NDI2OD-T2),” Phys. Rev. B 94, 165105 (2016). [CrossRef]  

35. W. M. Prest Jr and D. J. Luca, “The origin of the optical anisotropy of solvent cast polymeric films,” J. Appl. Phys. 50(10), 6067–6071 (1979). [CrossRef]  

36. M. K. Debe, “Variable angle spectroscopic ellipsometry studies of oriented phthalocyanine films. II. Copper phthalocyanine,” Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films 10, 2816 (1992). [CrossRef]  

37. L. A. A. Pettersson, S. Ghosh, and O. Inganäs, “Optical anisotropy in thin films of poly(3,4-ethylenedioxythiophene)—poly(4-styrenesulfonate),” Org. Electron. 3(3–4), 143–148 (2002). [CrossRef]  

38. O. D. Gordan, M. Friedrich, and D. R. T. Zahn, “The anisotropic dielectric function for copper phthalocyanine thin films,” Org. Electron. 5(6), 291–297 (2004). [CrossRef]  

39. C. J. Takacs, S. D. Collins, J. A. Love, A. A. Mikhailovsky, D. Wynands, G. C. Bazan, T.-Q. Nguyen, and A. J. Heeger, “Mapping Orientational Order in a Bulk Heterojunction Solar Cell with Polarization-Dependent Photoconductive Atomic Force Microscopy,” ACS Nano 8(8), 8141–8151 (2014). [CrossRef]   [PubMed]  

40. R. R. Grote, S. J. Brown, J. B. Driscoll, R. M. Osgood, and J. A. Schuller, “Morphology-dependent light trapping in thin-film organic solar cells,” Opt. Express 21(S5), A847–A863 (2013). [CrossRef]   [PubMed]  

41. M. E. Sykes, A. Barito, J. A. Amonoo, P. F. Green, and M. Shtein, “Broadband Plasmonic Photocurrent Enhancement in Planar Organic Photovoltaics Embedded in a Metallic Nanocavity,” Adv. Energy Mater. 4, 1301937 (2014). [CrossRef]  

42. F. Wang, K. Hashimoto, and K. Tajima, “Optical Anisotropy and Strong H-Aggregation of Poly(3-Alkylthiophene) in a Surface Monolayer,” Adv. Mater. 27(39), 6014–6020 (2015). [CrossRef]   [PubMed]  

43. C. E. Petoukhoff, Z. Shen, M. Jain, A. Chang, and D. M. O’Carroll, “Plasmonic electrodes for bulk-heterojunction organic photovoltaics: a review,” J. Photon. Energy 5(1), 057002 (2015). [CrossRef]  

44. M. A. Lieb, J. M. Zavislan, and L. Novotny, “Single-molecule orientations determined by direct emission pattern imaging,” J. Opt. Soc. Am. B 21(6), 1210–1215 (2004). [CrossRef]  

45. A. L. Mattheyses and D. Axelrod, “Fluorescence emission patterns near glass and metal-coated surfaces investigated with back focal plane imaging,” J. Biomed. Opt 10(5), 054007 (2005). [CrossRef]   [PubMed]  

46. T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, “Optical antennas direct single-molecule emission,” Nat Photon , 2(4), 234–237 (2008). [CrossRef]  

47. K. Hassan, A. Bouhelier, T. Bernardin, G. Colas-des-Francs, J.-C. Weeber, A. Dereux, and R. Espiau de Lamaestre, “Momentum-space spectroscopy for advanced analysis of dielectric-loaded surface plasmon polariton coupled and bent waveguides,” Phys. Rev. B , 87(19), 195428 (2013). [CrossRef]  

48. J. A. Schuller, S. Karaveli, T. Schiros, K. He, S. Yang, I. Kymissis, J. Shan, and R. Zia, “Orientation of luminescent excitons in layered nanomaterials,” Nature Nanotechnology , 8(4), 271–276 (2013). [CrossRef]   [PubMed]  

49. J. A. Kurvits, M. Jiang, and R. Zia, “Comparative analysis of imaging configurations and objectives for Fourier microscopy,” Journal of the Optical Society of America A 32(11), 2082 (2015). [CrossRef]  

50. B. Efron and R. Tibshirani, “Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy,” Statistical Science 1(1), 54–75 (1986). [CrossRef]  

51. SciPy.org, “SciPy v0.17.1 Reference Guide: scipy.optimize.least_squares,” http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least_squares.html

52. M. Branch, T. Coleman, and Y. Li, “A Subspace, Interior, and Conjugate Gradient Method for Large-Scale Bound-Constrained Minimization Problems,” SIAM J. Sci. Comput. 21(1), 1–23 (1999). [CrossRef]  

53. M. C. Gurau, D. M. Delongchamp, B. M. Vogel, E. K. Lin, D. A. Fischer, S. Sambasivan, and L. J. Richter, “Measuring Molecular Order in Poly(3-alkylthiophene) Thin Films with Polarizing Spectroscopies,” Langmuir 23(2), 834–842 (2007). [CrossRef]   [PubMed]  

References

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  • |
  • |

  1. G. Hass, “preparation, properties and optical applications of thin films of titanium dioxide,” Vacuum,  2(4), 331–345 (1952).
    [Crossref]
  2. B. G. Lewis and D. C. Paine, “Applications and Processing of Transparent Conducting Oxides,” MRS Bulletin 25(08), 22–27 (2000).
    [Crossref]
  3. P. Siciliano, “Preparation, characterisation and applications of thin films for gas sensors prepared by cheap chemical method,” Sensors and Actuators B: Chemical 70(1–3), 153–164 (2000).
    [Crossref]
  4. C. R. Kagan and P. Andry, Thin-Film Transistors (CRC Press, 2003).
    [Crossref]
  5. K. N. Chopra and A. K. Maini, “Thin Films and Their Applications in Military and Civil Sectors,” Defence Research and Development Organization (2010).
  6. P. Peumans, A. Yakimov, and S. R. Forrest, “Small molecular weight organic thin-film photodetectors and solar cells,” J. Appl. Phys. 93(7), 3693 (2003).
    [Crossref]
  7. C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, Light-Emitting Diodes: Research, Manufacturing, and Applications XII, 691006 (2008)
    [Crossref]
  8. C. W. Chen, S. Y. Hsiao, C. Y. Chen, H. W. Kang, Z. Y. Huang, and H. W. Lin, “Optical properties of organometal halide perovskite thin films and general device structure design rules for perovskite single and tandem solar cells,” J. Mater. Chem. A 3(17), 9152–9159 (2015).
    [Crossref]
  9. F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Nat. Bur. Sec. A 67, 363 (1963).
    [Crossref]
  10. R. A. Synowicki, “Spectroscopic ellipsometry characterization of indium tin oxide film microstructure and optical constants,” Thin Solid Films 313–314, 394–397 (1998).
    [Crossref]
  11. G. E. Jellison, V. I. Merkulov, A. A. Puretzky, D. B. Geohegan, G. Eres, D. H. Lowndes, and J. B. Caughman, “Characterization of thin-film amorphous semiconductors using spectroscopic ellipsometry,” Thin Solid Films 377–378, 68–73 (2000).
    [Crossref]
  12. G. E. Jellison, “Generalized ellipsometry for materials characterization,” Thin Solid Films 450(1), 42–50 (2004).
    [Crossref]
  13. H. G. Tompkins, A User’s Guide to Ellipsometry (Dover Publications, 2006.)
  14. T. E. Jenkins, “Multiple-angle-of-incidence ellipsometry,” Journal of Physics D: Applied Physics 32(9), R45 (1999).
    [Crossref]
  15. M. Campoy-Quiles, P. G. Etchegoin, and D. D. C. Bradley, “On the optical anisotropy of conjugated polymer thin films,” Phys. Rev. B 72, 045209 (2005).
    [Crossref]
  16. M. Campoy-Quiles, M. I. Alonso, D. D. C. Bradley, and L. J. Richter, “Advanced Ellipsometric Characterization of Conjugated Polymer Films,” Adv. Funct. Mater. 24(15), 2116–2134 (2014).
    [Crossref]
  17. G. H. Bu-Abbud, “Parameter correlation and precision in multiple-angle ellipsometry,” Appl. Opt.,  20(17), 3020–3026 (1981).
    [Crossref] [PubMed]
  18. G. H. Bu-Abbud, “Variable Wavelength, Variable Angle Ellipsometry Including a Sensitivities Correlation Test,” Thin Solid Films 138(1), 27–41 (1986).
    [Crossref]
  19. G. E. Jellison, “The calculation of thin film parameters from spectroscopic ellipsometry data,” Thin Solid Films 290–291, 40–45 (1996).
    [Crossref]
  20. W. A. McGahan, B. Johs, and J. A. Woollam, “Techniques for ellipsometric measurement of the thickness and optical constants of thin absorbing films,” Thin Solid Films 234(1), 443–446 (1993).
    [Crossref]
  21. C. M. Ramsdale and N. C. Greenham, “Ellipsometric Determination of Anisotropic Optical Constants in Electroluminescent Conjugated Polymers,” Adv. Mater. 14(3), 212–215 (2002).
    [Crossref]
  22. H. Arwin, M. Poksinski, and K. Johansen, “Total internal reflection ellipsometry: principles and applications,” Appl. Opt. 43(15), 3028–3036 (2004).
    [Crossref] [PubMed]
  23. J. I. Cisneros, “Optical characterization of dielectric and semiconductor thin films by use of transmission data,” Appl. Opt. 37(22), 5262–5270 (1998).
    [Crossref]
  24. Y. Laaziz, A. Bennouna, N. Chahboun, A. Outzourhit, and E. L. Ameziane, “Optical characterization of low optical thickness thin films from transmittance and back reflectance measurements,” Thin Solid Films 372(1–2), 149–155 (2000).
    [Crossref]
  25. D. Poelman and P. F. Smet, “Methods for the determination of the optical constants of thin films from single transmission measurements: a critical review,” Journal of Physics D: Applied Physics 36(15), 1850 (2003).
    [Crossref]
  26. M. Flämmich, N. Danz, D. Michaelis, A. Bräuer, M. C. Gather, H.-W. K. Jonas, and K. Meerholz, “Dispersion-model-free determination of optical constants: application to materials for organic thin film devices,” Appl. Opt. 48(8), 1507–1513 (2009).
    [Crossref] [PubMed]
  27. Metricon Corporation, “Model 2010/M Overview,” http://www.metricon.com
  28. W. Knoll, “Optical Characterization of Organic Thin Films and Interfaces with Evanescent Waves,” MRS Bulletin 16(07), 29–39 (1991).
    [Crossref]
  29. S. J. Bai, R. J. Spry, D. E. Zelmon, U. Ramabadran, and J. Jackson, “Optical anisotropy of polymeric films measured by waveguide propagation mode determination,” J. Polym. Sci. B Polym. Phys. 30(13), 1507–1514 (1992).
    [Crossref]
  30. C. J. Takacs, N. D. Treat, S. Krämer, Z. Chen, A. Facchetti, M. L. Chabinyc, and A. J. Heeger, “Remarkable Order of a High-Performance Polymer,” Nano Letters 13(6), 2522–2527 (2013).
    [Crossref] [PubMed]
  31. E. Giussani, D. Fazzi, L. Brambilla, M. Caironi, and C. Castiglioni, “Molecular Level Investigation of the Film Structure of a High Electron Mobility Copolymer via Vibrational Spectroscopy,” Macromolecules,  46(7), 2658–2670 (2013).
    [Crossref]
  32. T. Schuettfort, L. Thomsen, and C. R. McNeill, “Observation of a Distinct Surface Molecular Orientation in Films of a High Mobility Conjugated Polymer,” J. Am. Chem. Soc. 135(3), 1092–1101 (2013).
    [Crossref] [PubMed]
  33. M. Brinkmann, E. Gonthier, S. Bogen, K. Tremel, S. Ludwigs, M. Hufnagel, and M. Sommer, “Segregated versus Mixed Interchain Stacking in Highly Oriented Films of Naphthalene Diimide Bithiophene Copolymers,” ACS Nano 6(11), 10319–10326 (2012).
    [Crossref] [PubMed]
  34. S. J. Brown, R. A. Schlitz, M. L. Chabinyc, and J. A. Schuller, “Morphology dependent optical anisotropies in the n-type polymer P(NDI2OD-T2),” Phys. Rev. B 94, 165105 (2016).
    [Crossref]
  35. W. M. Prest and D. J. Luca, “The origin of the optical anisotropy of solvent cast polymeric films,” J. Appl. Phys. 50(10), 6067–6071 (1979).
    [Crossref]
  36. M. K. Debe, “Variable angle spectroscopic ellipsometry studies of oriented phthalocyanine films. II. Copper phthalocyanine,” Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films 10, 2816 (1992).
    [Crossref]
  37. L. A. A. Pettersson, S. Ghosh, and O. Inganäs, “Optical anisotropy in thin films of poly(3,4-ethylenedioxythiophene)—poly(4-styrenesulfonate),” Org. Electron. 3(3–4), 143–148 (2002).
    [Crossref]
  38. O. D. Gordan, M. Friedrich, and D. R. T. Zahn, “The anisotropic dielectric function for copper phthalocyanine thin films,” Org. Electron. 5(6), 291–297 (2004).
    [Crossref]
  39. C. J. Takacs, S. D. Collins, J. A. Love, A. A. Mikhailovsky, D. Wynands, G. C. Bazan, T.-Q. Nguyen, and A. J. Heeger, “Mapping Orientational Order in a Bulk Heterojunction Solar Cell with Polarization-Dependent Photoconductive Atomic Force Microscopy,” ACS Nano 8(8), 8141–8151 (2014).
    [Crossref] [PubMed]
  40. R. R. Grote, S. J. Brown, J. B. Driscoll, R. M. Osgood, and J. A. Schuller, “Morphology-dependent light trapping in thin-film organic solar cells,” Opt. Express 21(S5), A847–A863 (2013).
    [Crossref] [PubMed]
  41. M. E. Sykes, A. Barito, J. A. Amonoo, P. F. Green, and M. Shtein, “Broadband Plasmonic Photocurrent Enhancement in Planar Organic Photovoltaics Embedded in a Metallic Nanocavity,” Adv. Energy Mater. 4, 1301937 (2014).
    [Crossref]
  42. F. Wang, K. Hashimoto, and K. Tajima, “Optical Anisotropy and Strong H-Aggregation of Poly(3-Alkylthiophene) in a Surface Monolayer,” Adv. Mater. 27(39), 6014–6020 (2015).
    [Crossref] [PubMed]
  43. C. E. Petoukhoff, Z. Shen, M. Jain, A. Chang, and D. M. O’Carroll, “Plasmonic electrodes for bulk-heterojunction organic photovoltaics: a review,” J. Photon. Energy 5(1), 057002 (2015).
    [Crossref]
  44. M. A. Lieb, J. M. Zavislan, and L. Novotny, “Single-molecule orientations determined by direct emission pattern imaging,” J. Opt. Soc. Am. B 21(6), 1210–1215 (2004).
    [Crossref]
  45. A. L. Mattheyses and D. Axelrod, “Fluorescence emission patterns near glass and metal-coated surfaces investigated with back focal plane imaging,” J. Biomed. Opt 10(5), 054007 (2005).
    [Crossref] [PubMed]
  46. T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, “Optical antennas direct single-molecule emission,” Nat Photon,  2(4), 234–237 (2008).
    [Crossref]
  47. K. Hassan, A. Bouhelier, T. Bernardin, G. Colas-des-Francs, J.-C. Weeber, A. Dereux, and R. Espiau de Lamaestre, “Momentum-space spectroscopy for advanced analysis of dielectric-loaded surface plasmon polariton coupled and bent waveguides,” Phys. Rev. B,  87(19), 195428 (2013).
    [Crossref]
  48. J. A. Schuller, S. Karaveli, T. Schiros, K. He, S. Yang, I. Kymissis, J. Shan, and R. Zia, “Orientation of luminescent excitons in layered nanomaterials,” Nature Nanotechnology,  8(4), 271–276 (2013).
    [Crossref] [PubMed]
  49. J. A. Kurvits, M. Jiang, and R. Zia, “Comparative analysis of imaging configurations and objectives for Fourier microscopy,” Journal of the Optical Society of America A 32(11), 2082 (2015).
    [Crossref]
  50. B. Efron and R. Tibshirani, “Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy,” Statistical Science 1(1), 54–75 (1986).
    [Crossref]
  51. SciPy.org, “SciPy v0.17.1 Reference Guide: scipy.optimize.least_squares,” http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least_squares.html
  52. M. Branch, T. Coleman, and Y. Li, “A Subspace, Interior, and Conjugate Gradient Method for Large-Scale Bound-Constrained Minimization Problems,” SIAM J. Sci. Comput. 21(1), 1–23 (1999).
    [Crossref]
  53. M. C. Gurau, D. M. Delongchamp, B. M. Vogel, E. K. Lin, D. A. Fischer, S. Sambasivan, and L. J. Richter, “Measuring Molecular Order in Poly(3-alkylthiophene) Thin Films with Polarizing Spectroscopies,” Langmuir 23(2), 834–842 (2007).
    [Crossref] [PubMed]

2016 (1)

S. J. Brown, R. A. Schlitz, M. L. Chabinyc, and J. A. Schuller, “Morphology dependent optical anisotropies in the n-type polymer P(NDI2OD-T2),” Phys. Rev. B 94, 165105 (2016).
[Crossref]

2015 (4)

C. W. Chen, S. Y. Hsiao, C. Y. Chen, H. W. Kang, Z. Y. Huang, and H. W. Lin, “Optical properties of organometal halide perovskite thin films and general device structure design rules for perovskite single and tandem solar cells,” J. Mater. Chem. A 3(17), 9152–9159 (2015).
[Crossref]

F. Wang, K. Hashimoto, and K. Tajima, “Optical Anisotropy and Strong H-Aggregation of Poly(3-Alkylthiophene) in a Surface Monolayer,” Adv. Mater. 27(39), 6014–6020 (2015).
[Crossref] [PubMed]

C. E. Petoukhoff, Z. Shen, M. Jain, A. Chang, and D. M. O’Carroll, “Plasmonic electrodes for bulk-heterojunction organic photovoltaics: a review,” J. Photon. Energy 5(1), 057002 (2015).
[Crossref]

J. A. Kurvits, M. Jiang, and R. Zia, “Comparative analysis of imaging configurations and objectives for Fourier microscopy,” Journal of the Optical Society of America A 32(11), 2082 (2015).
[Crossref]

2014 (3)

M. E. Sykes, A. Barito, J. A. Amonoo, P. F. Green, and M. Shtein, “Broadband Plasmonic Photocurrent Enhancement in Planar Organic Photovoltaics Embedded in a Metallic Nanocavity,” Adv. Energy Mater. 4, 1301937 (2014).
[Crossref]

C. J. Takacs, S. D. Collins, J. A. Love, A. A. Mikhailovsky, D. Wynands, G. C. Bazan, T.-Q. Nguyen, and A. J. Heeger, “Mapping Orientational Order in a Bulk Heterojunction Solar Cell with Polarization-Dependent Photoconductive Atomic Force Microscopy,” ACS Nano 8(8), 8141–8151 (2014).
[Crossref] [PubMed]

M. Campoy-Quiles, M. I. Alonso, D. D. C. Bradley, and L. J. Richter, “Advanced Ellipsometric Characterization of Conjugated Polymer Films,” Adv. Funct. Mater. 24(15), 2116–2134 (2014).
[Crossref]

2013 (6)

C. J. Takacs, N. D. Treat, S. Krämer, Z. Chen, A. Facchetti, M. L. Chabinyc, and A. J. Heeger, “Remarkable Order of a High-Performance Polymer,” Nano Letters 13(6), 2522–2527 (2013).
[Crossref] [PubMed]

E. Giussani, D. Fazzi, L. Brambilla, M. Caironi, and C. Castiglioni, “Molecular Level Investigation of the Film Structure of a High Electron Mobility Copolymer via Vibrational Spectroscopy,” Macromolecules,  46(7), 2658–2670 (2013).
[Crossref]

T. Schuettfort, L. Thomsen, and C. R. McNeill, “Observation of a Distinct Surface Molecular Orientation in Films of a High Mobility Conjugated Polymer,” J. Am. Chem. Soc. 135(3), 1092–1101 (2013).
[Crossref] [PubMed]

R. R. Grote, S. J. Brown, J. B. Driscoll, R. M. Osgood, and J. A. Schuller, “Morphology-dependent light trapping in thin-film organic solar cells,” Opt. Express 21(S5), A847–A863 (2013).
[Crossref] [PubMed]

K. Hassan, A. Bouhelier, T. Bernardin, G. Colas-des-Francs, J.-C. Weeber, A. Dereux, and R. Espiau de Lamaestre, “Momentum-space spectroscopy for advanced analysis of dielectric-loaded surface plasmon polariton coupled and bent waveguides,” Phys. Rev. B,  87(19), 195428 (2013).
[Crossref]

J. A. Schuller, S. Karaveli, T. Schiros, K. He, S. Yang, I. Kymissis, J. Shan, and R. Zia, “Orientation of luminescent excitons in layered nanomaterials,” Nature Nanotechnology,  8(4), 271–276 (2013).
[Crossref] [PubMed]

2012 (1)

M. Brinkmann, E. Gonthier, S. Bogen, K. Tremel, S. Ludwigs, M. Hufnagel, and M. Sommer, “Segregated versus Mixed Interchain Stacking in Highly Oriented Films of Naphthalene Diimide Bithiophene Copolymers,” ACS Nano 6(11), 10319–10326 (2012).
[Crossref] [PubMed]

2009 (1)

2008 (2)

C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, Light-Emitting Diodes: Research, Manufacturing, and Applications XII, 691006 (2008)
[Crossref]

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, “Optical antennas direct single-molecule emission,” Nat Photon,  2(4), 234–237 (2008).
[Crossref]

2007 (1)

M. C. Gurau, D. M. Delongchamp, B. M. Vogel, E. K. Lin, D. A. Fischer, S. Sambasivan, and L. J. Richter, “Measuring Molecular Order in Poly(3-alkylthiophene) Thin Films with Polarizing Spectroscopies,” Langmuir 23(2), 834–842 (2007).
[Crossref] [PubMed]

2005 (2)

A. L. Mattheyses and D. Axelrod, “Fluorescence emission patterns near glass and metal-coated surfaces investigated with back focal plane imaging,” J. Biomed. Opt 10(5), 054007 (2005).
[Crossref] [PubMed]

M. Campoy-Quiles, P. G. Etchegoin, and D. D. C. Bradley, “On the optical anisotropy of conjugated polymer thin films,” Phys. Rev. B 72, 045209 (2005).
[Crossref]

2004 (4)

G. E. Jellison, “Generalized ellipsometry for materials characterization,” Thin Solid Films 450(1), 42–50 (2004).
[Crossref]

H. Arwin, M. Poksinski, and K. Johansen, “Total internal reflection ellipsometry: principles and applications,” Appl. Opt. 43(15), 3028–3036 (2004).
[Crossref] [PubMed]

M. A. Lieb, J. M. Zavislan, and L. Novotny, “Single-molecule orientations determined by direct emission pattern imaging,” J. Opt. Soc. Am. B 21(6), 1210–1215 (2004).
[Crossref]

O. D. Gordan, M. Friedrich, and D. R. T. Zahn, “The anisotropic dielectric function for copper phthalocyanine thin films,” Org. Electron. 5(6), 291–297 (2004).
[Crossref]

2003 (2)

D. Poelman and P. F. Smet, “Methods for the determination of the optical constants of thin films from single transmission measurements: a critical review,” Journal of Physics D: Applied Physics 36(15), 1850 (2003).
[Crossref]

P. Peumans, A. Yakimov, and S. R. Forrest, “Small molecular weight organic thin-film photodetectors and solar cells,” J. Appl. Phys. 93(7), 3693 (2003).
[Crossref]

2002 (2)

C. M. Ramsdale and N. C. Greenham, “Ellipsometric Determination of Anisotropic Optical Constants in Electroluminescent Conjugated Polymers,” Adv. Mater. 14(3), 212–215 (2002).
[Crossref]

L. A. A. Pettersson, S. Ghosh, and O. Inganäs, “Optical anisotropy in thin films of poly(3,4-ethylenedioxythiophene)—poly(4-styrenesulfonate),” Org. Electron. 3(3–4), 143–148 (2002).
[Crossref]

2000 (4)

Y. Laaziz, A. Bennouna, N. Chahboun, A. Outzourhit, and E. L. Ameziane, “Optical characterization of low optical thickness thin films from transmittance and back reflectance measurements,” Thin Solid Films 372(1–2), 149–155 (2000).
[Crossref]

B. G. Lewis and D. C. Paine, “Applications and Processing of Transparent Conducting Oxides,” MRS Bulletin 25(08), 22–27 (2000).
[Crossref]

P. Siciliano, “Preparation, characterisation and applications of thin films for gas sensors prepared by cheap chemical method,” Sensors and Actuators B: Chemical 70(1–3), 153–164 (2000).
[Crossref]

G. E. Jellison, V. I. Merkulov, A. A. Puretzky, D. B. Geohegan, G. Eres, D. H. Lowndes, and J. B. Caughman, “Characterization of thin-film amorphous semiconductors using spectroscopic ellipsometry,” Thin Solid Films 377–378, 68–73 (2000).
[Crossref]

1999 (2)

T. E. Jenkins, “Multiple-angle-of-incidence ellipsometry,” Journal of Physics D: Applied Physics 32(9), R45 (1999).
[Crossref]

M. Branch, T. Coleman, and Y. Li, “A Subspace, Interior, and Conjugate Gradient Method for Large-Scale Bound-Constrained Minimization Problems,” SIAM J. Sci. Comput. 21(1), 1–23 (1999).
[Crossref]

1998 (2)

R. A. Synowicki, “Spectroscopic ellipsometry characterization of indium tin oxide film microstructure and optical constants,” Thin Solid Films 313–314, 394–397 (1998).
[Crossref]

J. I. Cisneros, “Optical characterization of dielectric and semiconductor thin films by use of transmission data,” Appl. Opt. 37(22), 5262–5270 (1998).
[Crossref]

1996 (1)

G. E. Jellison, “The calculation of thin film parameters from spectroscopic ellipsometry data,” Thin Solid Films 290–291, 40–45 (1996).
[Crossref]

1993 (1)

W. A. McGahan, B. Johs, and J. A. Woollam, “Techniques for ellipsometric measurement of the thickness and optical constants of thin absorbing films,” Thin Solid Films 234(1), 443–446 (1993).
[Crossref]

1992 (2)

S. J. Bai, R. J. Spry, D. E. Zelmon, U. Ramabadran, and J. Jackson, “Optical anisotropy of polymeric films measured by waveguide propagation mode determination,” J. Polym. Sci. B Polym. Phys. 30(13), 1507–1514 (1992).
[Crossref]

M. K. Debe, “Variable angle spectroscopic ellipsometry studies of oriented phthalocyanine films. II. Copper phthalocyanine,” Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films 10, 2816 (1992).
[Crossref]

1991 (1)

W. Knoll, “Optical Characterization of Organic Thin Films and Interfaces with Evanescent Waves,” MRS Bulletin 16(07), 29–39 (1991).
[Crossref]

1986 (2)

G. H. Bu-Abbud, “Variable Wavelength, Variable Angle Ellipsometry Including a Sensitivities Correlation Test,” Thin Solid Films 138(1), 27–41 (1986).
[Crossref]

B. Efron and R. Tibshirani, “Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy,” Statistical Science 1(1), 54–75 (1986).
[Crossref]

1981 (1)

1979 (1)

W. M. Prest and D. J. Luca, “The origin of the optical anisotropy of solvent cast polymeric films,” J. Appl. Phys. 50(10), 6067–6071 (1979).
[Crossref]

1963 (1)

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Nat. Bur. Sec. A 67, 363 (1963).
[Crossref]

1952 (1)

G. Hass, “preparation, properties and optical applications of thin films of titanium dioxide,” Vacuum,  2(4), 331–345 (1952).
[Crossref]

Alonso, M. I.

M. Campoy-Quiles, M. I. Alonso, D. D. C. Bradley, and L. J. Richter, “Advanced Ellipsometric Characterization of Conjugated Polymer Films,” Adv. Funct. Mater. 24(15), 2116–2134 (2014).
[Crossref]

Ameziane, E. L.

Y. Laaziz, A. Bennouna, N. Chahboun, A. Outzourhit, and E. L. Ameziane, “Optical characterization of low optical thickness thin films from transmittance and back reflectance measurements,” Thin Solid Films 372(1–2), 149–155 (2000).
[Crossref]

Amonoo, J. A.

M. E. Sykes, A. Barito, J. A. Amonoo, P. F. Green, and M. Shtein, “Broadband Plasmonic Photocurrent Enhancement in Planar Organic Photovoltaics Embedded in a Metallic Nanocavity,” Adv. Energy Mater. 4, 1301937 (2014).
[Crossref]

Andry, P.

C. R. Kagan and P. Andry, Thin-Film Transistors (CRC Press, 2003).
[Crossref]

Arwin, H.

Axelrod, D.

A. L. Mattheyses and D. Axelrod, “Fluorescence emission patterns near glass and metal-coated surfaces investigated with back focal plane imaging,” J. Biomed. Opt 10(5), 054007 (2005).
[Crossref] [PubMed]

Bai, S. J.

S. J. Bai, R. J. Spry, D. E. Zelmon, U. Ramabadran, and J. Jackson, “Optical anisotropy of polymeric films measured by waveguide propagation mode determination,” J. Polym. Sci. B Polym. Phys. 30(13), 1507–1514 (1992).
[Crossref]

Barito, A.

M. E. Sykes, A. Barito, J. A. Amonoo, P. F. Green, and M. Shtein, “Broadband Plasmonic Photocurrent Enhancement in Planar Organic Photovoltaics Embedded in a Metallic Nanocavity,” Adv. Energy Mater. 4, 1301937 (2014).
[Crossref]

Bazan, G. C.

C. J. Takacs, S. D. Collins, J. A. Love, A. A. Mikhailovsky, D. Wynands, G. C. Bazan, T.-Q. Nguyen, and A. J. Heeger, “Mapping Orientational Order in a Bulk Heterojunction Solar Cell with Polarization-Dependent Photoconductive Atomic Force Microscopy,” ACS Nano 8(8), 8141–8151 (2014).
[Crossref] [PubMed]

Bennouna, A.

Y. Laaziz, A. Bennouna, N. Chahboun, A. Outzourhit, and E. L. Ameziane, “Optical characterization of low optical thickness thin films from transmittance and back reflectance measurements,” Thin Solid Films 372(1–2), 149–155 (2000).
[Crossref]

Bernardin, T.

K. Hassan, A. Bouhelier, T. Bernardin, G. Colas-des-Francs, J.-C. Weeber, A. Dereux, and R. Espiau de Lamaestre, “Momentum-space spectroscopy for advanced analysis of dielectric-loaded surface plasmon polariton coupled and bent waveguides,” Phys. Rev. B,  87(19), 195428 (2013).
[Crossref]

Bogen, S.

M. Brinkmann, E. Gonthier, S. Bogen, K. Tremel, S. Ludwigs, M. Hufnagel, and M. Sommer, “Segregated versus Mixed Interchain Stacking in Highly Oriented Films of Naphthalene Diimide Bithiophene Copolymers,” ACS Nano 6(11), 10319–10326 (2012).
[Crossref] [PubMed]

Bouhelier, A.

K. Hassan, A. Bouhelier, T. Bernardin, G. Colas-des-Francs, J.-C. Weeber, A. Dereux, and R. Espiau de Lamaestre, “Momentum-space spectroscopy for advanced analysis of dielectric-loaded surface plasmon polariton coupled and bent waveguides,” Phys. Rev. B,  87(19), 195428 (2013).
[Crossref]

Bradley, D. D. C.

M. Campoy-Quiles, M. I. Alonso, D. D. C. Bradley, and L. J. Richter, “Advanced Ellipsometric Characterization of Conjugated Polymer Films,” Adv. Funct. Mater. 24(15), 2116–2134 (2014).
[Crossref]

M. Campoy-Quiles, P. G. Etchegoin, and D. D. C. Bradley, “On the optical anisotropy of conjugated polymer thin films,” Phys. Rev. B 72, 045209 (2005).
[Crossref]

Brambilla, L.

E. Giussani, D. Fazzi, L. Brambilla, M. Caironi, and C. Castiglioni, “Molecular Level Investigation of the Film Structure of a High Electron Mobility Copolymer via Vibrational Spectroscopy,” Macromolecules,  46(7), 2658–2670 (2013).
[Crossref]

Branch, M.

M. Branch, T. Coleman, and Y. Li, “A Subspace, Interior, and Conjugate Gradient Method for Large-Scale Bound-Constrained Minimization Problems,” SIAM J. Sci. Comput. 21(1), 1–23 (1999).
[Crossref]

Bräuer, A.

Bräuer, A. H.

C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, Light-Emitting Diodes: Research, Manufacturing, and Applications XII, 691006 (2008)
[Crossref]

Brinkmann, M.

M. Brinkmann, E. Gonthier, S. Bogen, K. Tremel, S. Ludwigs, M. Hufnagel, and M. Sommer, “Segregated versus Mixed Interchain Stacking in Highly Oriented Films of Naphthalene Diimide Bithiophene Copolymers,” ACS Nano 6(11), 10319–10326 (2012).
[Crossref] [PubMed]

Brown, S. J.

S. J. Brown, R. A. Schlitz, M. L. Chabinyc, and J. A. Schuller, “Morphology dependent optical anisotropies in the n-type polymer P(NDI2OD-T2),” Phys. Rev. B 94, 165105 (2016).
[Crossref]

R. R. Grote, S. J. Brown, J. B. Driscoll, R. M. Osgood, and J. A. Schuller, “Morphology-dependent light trapping in thin-film organic solar cells,” Opt. Express 21(S5), A847–A863 (2013).
[Crossref] [PubMed]

Bu-Abbud, G. H.

G. H. Bu-Abbud, “Variable Wavelength, Variable Angle Ellipsometry Including a Sensitivities Correlation Test,” Thin Solid Films 138(1), 27–41 (1986).
[Crossref]

G. H. Bu-Abbud, “Parameter correlation and precision in multiple-angle ellipsometry,” Appl. Opt.,  20(17), 3020–3026 (1981).
[Crossref] [PubMed]

Caironi, M.

E. Giussani, D. Fazzi, L. Brambilla, M. Caironi, and C. Castiglioni, “Molecular Level Investigation of the Film Structure of a High Electron Mobility Copolymer via Vibrational Spectroscopy,” Macromolecules,  46(7), 2658–2670 (2013).
[Crossref]

Campoy-Quiles, M.

M. Campoy-Quiles, M. I. Alonso, D. D. C. Bradley, and L. J. Richter, “Advanced Ellipsometric Characterization of Conjugated Polymer Films,” Adv. Funct. Mater. 24(15), 2116–2134 (2014).
[Crossref]

M. Campoy-Quiles, P. G. Etchegoin, and D. D. C. Bradley, “On the optical anisotropy of conjugated polymer thin films,” Phys. Rev. B 72, 045209 (2005).
[Crossref]

Castiglioni, C.

E. Giussani, D. Fazzi, L. Brambilla, M. Caironi, and C. Castiglioni, “Molecular Level Investigation of the Film Structure of a High Electron Mobility Copolymer via Vibrational Spectroscopy,” Macromolecules,  46(7), 2658–2670 (2013).
[Crossref]

Caughman, J. B.

G. E. Jellison, V. I. Merkulov, A. A. Puretzky, D. B. Geohegan, G. Eres, D. H. Lowndes, and J. B. Caughman, “Characterization of thin-film amorphous semiconductors using spectroscopic ellipsometry,” Thin Solid Films 377–378, 68–73 (2000).
[Crossref]

Chabinyc, M. L.

S. J. Brown, R. A. Schlitz, M. L. Chabinyc, and J. A. Schuller, “Morphology dependent optical anisotropies in the n-type polymer P(NDI2OD-T2),” Phys. Rev. B 94, 165105 (2016).
[Crossref]

C. J. Takacs, N. D. Treat, S. Krämer, Z. Chen, A. Facchetti, M. L. Chabinyc, and A. J. Heeger, “Remarkable Order of a High-Performance Polymer,” Nano Letters 13(6), 2522–2527 (2013).
[Crossref] [PubMed]

Chahboun, N.

Y. Laaziz, A. Bennouna, N. Chahboun, A. Outzourhit, and E. L. Ameziane, “Optical characterization of low optical thickness thin films from transmittance and back reflectance measurements,” Thin Solid Films 372(1–2), 149–155 (2000).
[Crossref]

Chang, A.

C. E. Petoukhoff, Z. Shen, M. Jain, A. Chang, and D. M. O’Carroll, “Plasmonic electrodes for bulk-heterojunction organic photovoltaics: a review,” J. Photon. Energy 5(1), 057002 (2015).
[Crossref]

Chen, C. W.

C. W. Chen, S. Y. Hsiao, C. Y. Chen, H. W. Kang, Z. Y. Huang, and H. W. Lin, “Optical properties of organometal halide perovskite thin films and general device structure design rules for perovskite single and tandem solar cells,” J. Mater. Chem. A 3(17), 9152–9159 (2015).
[Crossref]

Chen, C. Y.

C. W. Chen, S. Y. Hsiao, C. Y. Chen, H. W. Kang, Z. Y. Huang, and H. W. Lin, “Optical properties of organometal halide perovskite thin films and general device structure design rules for perovskite single and tandem solar cells,” J. Mater. Chem. A 3(17), 9152–9159 (2015).
[Crossref]

Chen, Z.

C. J. Takacs, N. D. Treat, S. Krämer, Z. Chen, A. Facchetti, M. L. Chabinyc, and A. J. Heeger, “Remarkable Order of a High-Performance Polymer,” Nano Letters 13(6), 2522–2527 (2013).
[Crossref] [PubMed]

Chopra, K. N.

K. N. Chopra and A. K. Maini, “Thin Films and Their Applications in Military and Civil Sectors,” Defence Research and Development Organization (2010).

Cisneros, J. I.

Colas-des-Francs, G.

K. Hassan, A. Bouhelier, T. Bernardin, G. Colas-des-Francs, J.-C. Weeber, A. Dereux, and R. Espiau de Lamaestre, “Momentum-space spectroscopy for advanced analysis of dielectric-loaded surface plasmon polariton coupled and bent waveguides,” Phys. Rev. B,  87(19), 195428 (2013).
[Crossref]

Coleman, T.

M. Branch, T. Coleman, and Y. Li, “A Subspace, Interior, and Conjugate Gradient Method for Large-Scale Bound-Constrained Minimization Problems,” SIAM J. Sci. Comput. 21(1), 1–23 (1999).
[Crossref]

Collins, S. D.

C. J. Takacs, S. D. Collins, J. A. Love, A. A. Mikhailovsky, D. Wynands, G. C. Bazan, T.-Q. Nguyen, and A. J. Heeger, “Mapping Orientational Order in a Bulk Heterojunction Solar Cell with Polarization-Dependent Photoconductive Atomic Force Microscopy,” ACS Nano 8(8), 8141–8151 (2014).
[Crossref] [PubMed]

Danz, N.

M. Flämmich, N. Danz, D. Michaelis, A. Bräuer, M. C. Gather, H.-W. K. Jonas, and K. Meerholz, “Dispersion-model-free determination of optical constants: application to materials for organic thin film devices,” Appl. Opt. 48(8), 1507–1513 (2009).
[Crossref] [PubMed]

C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, Light-Emitting Diodes: Research, Manufacturing, and Applications XII, 691006 (2008)
[Crossref]

Debe, M. K.

M. K. Debe, “Variable angle spectroscopic ellipsometry studies of oriented phthalocyanine films. II. Copper phthalocyanine,” Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films 10, 2816 (1992).
[Crossref]

Delongchamp, D. M.

M. C. Gurau, D. M. Delongchamp, B. M. Vogel, E. K. Lin, D. A. Fischer, S. Sambasivan, and L. J. Richter, “Measuring Molecular Order in Poly(3-alkylthiophene) Thin Films with Polarizing Spectroscopies,” Langmuir 23(2), 834–842 (2007).
[Crossref] [PubMed]

Dereux, A.

K. Hassan, A. Bouhelier, T. Bernardin, G. Colas-des-Francs, J.-C. Weeber, A. Dereux, and R. Espiau de Lamaestre, “Momentum-space spectroscopy for advanced analysis of dielectric-loaded surface plasmon polariton coupled and bent waveguides,” Phys. Rev. B,  87(19), 195428 (2013).
[Crossref]

Driscoll, J. B.

Efron, B.

B. Efron and R. Tibshirani, “Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy,” Statistical Science 1(1), 54–75 (1986).
[Crossref]

Eres, G.

G. E. Jellison, V. I. Merkulov, A. A. Puretzky, D. B. Geohegan, G. Eres, D. H. Lowndes, and J. B. Caughman, “Characterization of thin-film amorphous semiconductors using spectroscopic ellipsometry,” Thin Solid Films 377–378, 68–73 (2000).
[Crossref]

Espiau de Lamaestre, R.

K. Hassan, A. Bouhelier, T. Bernardin, G. Colas-des-Francs, J.-C. Weeber, A. Dereux, and R. Espiau de Lamaestre, “Momentum-space spectroscopy for advanced analysis of dielectric-loaded surface plasmon polariton coupled and bent waveguides,” Phys. Rev. B,  87(19), 195428 (2013).
[Crossref]

Etchegoin, P. G.

M. Campoy-Quiles, P. G. Etchegoin, and D. D. C. Bradley, “On the optical anisotropy of conjugated polymer thin films,” Phys. Rev. B 72, 045209 (2005).
[Crossref]

Facchetti, A.

C. J. Takacs, N. D. Treat, S. Krämer, Z. Chen, A. Facchetti, M. L. Chabinyc, and A. J. Heeger, “Remarkable Order of a High-Performance Polymer,” Nano Letters 13(6), 2522–2527 (2013).
[Crossref] [PubMed]

Fazzi, D.

E. Giussani, D. Fazzi, L. Brambilla, M. Caironi, and C. Castiglioni, “Molecular Level Investigation of the Film Structure of a High Electron Mobility Copolymer via Vibrational Spectroscopy,” Macromolecules,  46(7), 2658–2670 (2013).
[Crossref]

Fischer, D. A.

M. C. Gurau, D. M. Delongchamp, B. M. Vogel, E. K. Lin, D. A. Fischer, S. Sambasivan, and L. J. Richter, “Measuring Molecular Order in Poly(3-alkylthiophene) Thin Films with Polarizing Spectroscopies,” Langmuir 23(2), 834–842 (2007).
[Crossref] [PubMed]

Flämmich, M.

M. Flämmich, N. Danz, D. Michaelis, A. Bräuer, M. C. Gather, H.-W. K. Jonas, and K. Meerholz, “Dispersion-model-free determination of optical constants: application to materials for organic thin film devices,” Appl. Opt. 48(8), 1507–1513 (2009).
[Crossref] [PubMed]

C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, Light-Emitting Diodes: Research, Manufacturing, and Applications XII, 691006 (2008)
[Crossref]

Forrest, S. R.

P. Peumans, A. Yakimov, and S. R. Forrest, “Small molecular weight organic thin-film photodetectors and solar cells,” J. Appl. Phys. 93(7), 3693 (2003).
[Crossref]

Friedrich, M.

O. D. Gordan, M. Friedrich, and D. R. T. Zahn, “The anisotropic dielectric function for copper phthalocyanine thin films,” Org. Electron. 5(6), 291–297 (2004).
[Crossref]

Gather, M. C.

M. Flämmich, N. Danz, D. Michaelis, A. Bräuer, M. C. Gather, H.-W. K. Jonas, and K. Meerholz, “Dispersion-model-free determination of optical constants: application to materials for organic thin film devices,” Appl. Opt. 48(8), 1507–1513 (2009).
[Crossref] [PubMed]

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M. Brinkmann, E. Gonthier, S. Bogen, K. Tremel, S. Ludwigs, M. Hufnagel, and M. Sommer, “Segregated versus Mixed Interchain Stacking in Highly Oriented Films of Naphthalene Diimide Bithiophene Copolymers,” ACS Nano 6(11), 10319–10326 (2012).
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C. J. Takacs, S. D. Collins, J. A. Love, A. A. Mikhailovsky, D. Wynands, G. C. Bazan, T.-Q. Nguyen, and A. J. Heeger, “Mapping Orientational Order in a Bulk Heterojunction Solar Cell with Polarization-Dependent Photoconductive Atomic Force Microscopy,” ACS Nano 8(8), 8141–8151 (2014).
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T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, “Optical antennas direct single-molecule emission,” Nat Photon,  2(4), 234–237 (2008).
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Thomsen, L.

T. Schuettfort, L. Thomsen, and C. R. McNeill, “Observation of a Distinct Surface Molecular Orientation in Films of a High Mobility Conjugated Polymer,” J. Am. Chem. Soc. 135(3), 1092–1101 (2013).
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H. G. Tompkins, A User’s Guide to Ellipsometry (Dover Publications, 2006.)

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M. Brinkmann, E. Gonthier, S. Bogen, K. Tremel, S. Ludwigs, M. Hufnagel, and M. Sommer, “Segregated versus Mixed Interchain Stacking in Highly Oriented Films of Naphthalene Diimide Bithiophene Copolymers,” ACS Nano 6(11), 10319–10326 (2012).
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M. C. Gurau, D. M. Delongchamp, B. M. Vogel, E. K. Lin, D. A. Fischer, S. Sambasivan, and L. J. Richter, “Measuring Molecular Order in Poly(3-alkylthiophene) Thin Films with Polarizing Spectroscopies,” Langmuir 23(2), 834–842 (2007).
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C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, Light-Emitting Diodes: Research, Manufacturing, and Applications XII, 691006 (2008)
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F. Wang, K. Hashimoto, and K. Tajima, “Optical Anisotropy and Strong H-Aggregation of Poly(3-Alkylthiophene) in a Surface Monolayer,” Adv. Mater. 27(39), 6014–6020 (2015).
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K. Hassan, A. Bouhelier, T. Bernardin, G. Colas-des-Francs, J.-C. Weeber, A. Dereux, and R. Espiau de Lamaestre, “Momentum-space spectroscopy for advanced analysis of dielectric-loaded surface plasmon polariton coupled and bent waveguides,” Phys. Rev. B,  87(19), 195428 (2013).
[Crossref]

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C. J. Takacs, S. D. Collins, J. A. Love, A. A. Mikhailovsky, D. Wynands, G. C. Bazan, T.-Q. Nguyen, and A. J. Heeger, “Mapping Orientational Order in a Bulk Heterojunction Solar Cell with Polarization-Dependent Photoconductive Atomic Force Microscopy,” ACS Nano 8(8), 8141–8151 (2014).
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ACS Nano (2)

M. Brinkmann, E. Gonthier, S. Bogen, K. Tremel, S. Ludwigs, M. Hufnagel, and M. Sommer, “Segregated versus Mixed Interchain Stacking in Highly Oriented Films of Naphthalene Diimide Bithiophene Copolymers,” ACS Nano 6(11), 10319–10326 (2012).
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Appl. Opt. (4)

J. Am. Chem. Soc. (1)

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C. E. Petoukhoff, Z. Shen, M. Jain, A. Chang, and D. M. O’Carroll, “Plasmonic electrodes for bulk-heterojunction organic photovoltaics: a review,” J. Photon. Energy 5(1), 057002 (2015).
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Journal of the Optical Society of America A (1)

J. A. Kurvits, M. Jiang, and R. Zia, “Comparative analysis of imaging configurations and objectives for Fourier microscopy,” Journal of the Optical Society of America A 32(11), 2082 (2015).
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Langmuir (1)

M. C. Gurau, D. M. Delongchamp, B. M. Vogel, E. K. Lin, D. A. Fischer, S. Sambasivan, and L. J. Richter, “Measuring Molecular Order in Poly(3-alkylthiophene) Thin Films with Polarizing Spectroscopies,” Langmuir 23(2), 834–842 (2007).
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Macromolecules (1)

E. Giussani, D. Fazzi, L. Brambilla, M. Caironi, and C. Castiglioni, “Molecular Level Investigation of the Film Structure of a High Electron Mobility Copolymer via Vibrational Spectroscopy,” Macromolecules,  46(7), 2658–2670 (2013).
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Nano Letters (1)

C. J. Takacs, N. D. Treat, S. Krämer, Z. Chen, A. Facchetti, M. L. Chabinyc, and A. J. Heeger, “Remarkable Order of a High-Performance Polymer,” Nano Letters 13(6), 2522–2527 (2013).
[Crossref] [PubMed]

Nat Photon (1)

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, “Optical antennas direct single-molecule emission,” Nat Photon,  2(4), 234–237 (2008).
[Crossref]

Nature Nanotechnology (1)

J. A. Schuller, S. Karaveli, T. Schiros, K. He, S. Yang, I. Kymissis, J. Shan, and R. Zia, “Orientation of luminescent excitons in layered nanomaterials,” Nature Nanotechnology,  8(4), 271–276 (2013).
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Opt. Express (1)

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M. Campoy-Quiles, P. G. Etchegoin, and D. D. C. Bradley, “On the optical anisotropy of conjugated polymer thin films,” Phys. Rev. B 72, 045209 (2005).
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S. J. Brown, R. A. Schlitz, M. L. Chabinyc, and J. A. Schuller, “Morphology dependent optical anisotropies in the n-type polymer P(NDI2OD-T2),” Phys. Rev. B 94, 165105 (2016).
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K. Hassan, A. Bouhelier, T. Bernardin, G. Colas-des-Francs, J.-C. Weeber, A. Dereux, and R. Espiau de Lamaestre, “Momentum-space spectroscopy for advanced analysis of dielectric-loaded surface plasmon polariton coupled and bent waveguides,” Phys. Rev. B,  87(19), 195428 (2013).
[Crossref]

Proc. SPIE (1)

C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, Light-Emitting Diodes: Research, Manufacturing, and Applications XII, 691006 (2008)
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G. E. Jellison, “Generalized ellipsometry for materials characterization,” Thin Solid Films 450(1), 42–50 (2004).
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H. G. Tompkins, A User’s Guide to Ellipsometry (Dover Publications, 2006.)

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Figures (14)

Fig. 1
Fig. 1 (a) ‘Face-on’ and ‘edge-on’ morphologies assumed by P(NDI2OD-T2) for low-and high-temperature thermal annealing, respectively; (b) P(NDI2OD-T2) molecule; (c) polymer segment with associated transition dipole moment; (d) the macroscopic EM system representing the sample on substrate.
Fig. 2
Fig. 2 s-Polarized reflection measurements for bare quartz substrates (blue triangles) and film-on-substrate (red circles) at 700 nm for 46 nm (a) face-on and (b) edge-on films. Theoretical curves (solid lines) for single quartz-air interface (blue) and the two-interface quartz-film-air system with best-fit film index (red) are superimposed on the data. The edge-on film has a smaller absorption coefficient (Im[no]), manifested most evidently as higher reflectance at large momentum values (kk0). Best fit values for the real and imaginary parts of the ordinary index and associated 99% confidence values are displayed in the upper left hand corner of each plot.
Fig. 3
Fig. 3 [(a) and (b)] Variation of the real (red, left axis) and imaginary (blue, right axis) parts of the ordinary index for (a) face-on and (b) edge-on films. The curves in each plot show the expected lineshapes associated with an optical absorption resonance around λ=700 nm. 99% confidence intervals are included, but those for the imaginary parts lie within the data markers and are not visible. [(c) and (d)] Normal-incidence transmission spectra through the 46 nm film on fused-silica substrate for (c) face-on and (d) edge-on films. Red circles show the prediction using values determined from Rs measurements; Green triangles show the prediction using values determined from ellipsometry; Blue dashed lines show UV-Vis-NIR transmission measurement. In agreement with the molecular model, smaller IP absorption is visible in the edge-on film, attributed to the higher OP dipole strength. This subtlety is well-resolved by our reflectometry procedure, whereas ellipsometry incorrectly suggests a larger IP absorption for edge-on films.
Fig. 4
Fig. 4 (a) 2D map of the error FOM, S (Eqn. (1)), for real and imaginary components of the ordinary index of the face-on film as fit against Rs data at 700 nm. In order to reveal details in the neighborhood of the minimum, the vertical range is limited to 0.1≤𝕀m(no)≤0.8 in this visualization. The specified fit bounds [1≤ℝe(no)≤2 and 0≤𝕀m(no)≤1] cover index values for nearly all organic films. (b) Sensitivity of the reflection function, Rs, with respect to real (blue) and imaginary (red) components of the ordinary index, as a function of in-plane momentum, k. The curves are calculated at typical index values: ℝe(no)=1.5; 𝕀m(no)=0.2. Rs shows significantly increased sensitivity to the fit parameters for in-plane momenta beyond the critical angle of TIR. This is especially so for 𝕀m(no), and thus accurate estimates of film absorption benefit significantly from measurements in the region of TIR.
Fig. 5
Fig. 5 p-Polarized reflection measurements for bare quartz substrates (blue triangles) and film-on-substrate (red circles) at 700nm for 46 nm (a) face-on and (b) edge-on films. Theoretical curves (solid lines) for single quartz-air interface (blue) and the two-interface quartz-film-air system with best-fit film index (red) are superposed on the data. The enhanced OP absorption is apparent in the edge-on film, manifest as substantially lower reflectance at TIR. The inset highlights both data around the pseudo-Brewster angle. Best fit values and 99% confidence values are printed in each plot.
Fig. 6
Fig. 6 (a) The imaginary component of the extraordinary refractive index for face-on (red circles) and edge-on (blue triangles) films, determined via p-polarized reflection measurements. Though uncertainties are relatively large, the edge-on film exhibits a clear increase in OP absorption, relative to the face-on film. The wavelengths between face-on and edge-on films have been slightly offset to avoid overlapping error bars. (b) The real component of the extraordinary refractive index for face-on (red circles) and edge-on (blue triangles) films. 99% confidence bars are included in each plot.
Fig. 7
Fig. 7 [(a) and (b)] Reflected intensity for (a) p-polarized and (b) s-polarized incidence at 700 nm. [(c) and (d)] Efficiency-corrected and normalized data (i.e., reflectance, R) for (c) p-polarized and (d) s-polarized incidence at 700 nm. Blue triangles: bare quartz substrates. Red circles: quartz substrates with edge-on films.
Fig. 8
Fig. 8 Variable-angle spectroscopic ellipsometry measurements of Psi (blue, left axis) and Delta (red, right axis) for (a) face-on and (b) edge-on films of P(NDI2OD-T2). The constructed model necessary for a quality fit introduces 35 free parameters. Though the fits of Delta and Psi appear to be very good, the quality of the model remains unknown, and we find the resulting refractive index estimations to be flawed.
Fig. 9
Fig. 9 Spectroscopic ellipsometry results for the real (red, left axis) and imaginary (blue, right axis) parts of the (a,b) ordinary and (c,d) extraordinary refractive index for (a,c) face-on and (b,d) edge-on films. Contrary to UV-Vis transmission, angle-resolved photoluminescence, and our reflectometry measurements, the edge-on film shows higher IP absorption.
Fig. 10
Fig. 10 2D map of the error FOM, S (Eqn. (1) of main manuscript), for real and imaginary components of the ordinary index of edge-on films as fit against Rs data at 700 nm. In order to reveal details in the neighborhood of the minimum, the vertical range is limited to 0.1≤𝕀m(no)≤0.8 in this visualization. A clear unique minimum exists, validating the use of the gradient-descent method.
Fig. 11
Fig. 11 [(a) and (b)] 2D maps of the error FOM, S (Eqn. (1) of main manuscript), for real and imaginary components of the extraordinary index of (a) face-on and (b) edge-on films as fit against Rp data at 700 nm. [(c) and (d)] The same maps as in (a) and (b), in which values greater than a maximum value (specified by color bars) have been oversaturated (gray). This serves to show fine details in the neighborhood of the minima.
Fig. 12
Fig. 12 (a) Sensitivity of the reflection function, Rp, with respect to real (blue) and imaginary (red) components of the extraordinary index, as a function of in-plane momentum. The curves are calculated at typical index values: ℝe(no)=1.5; 𝕀m(no)=0.2; ℝe(ne)=1.5; 𝕀m(ne)=0.05. The reflectance shows drastically increased sensitivity to the fit parameters for in-plane momenta beyond the critical angle of TIR. The inset demonstrates increased sensitivity of Rp with respect to the real part of the extraordinary index in films with greater out-of-plane absorption. (b) The correlation function, C (Eqn. 3) as a function of in-plane momentum, again assuming typical index values specified for (a). The behavior for k/k0>1 suggests substantial correlation between the real and imaginary components of the extraordinary index in the region of TIR.
Fig. 13
Fig. 13 Real (red, left axis) and imaginary (blue, right axis) components of the ordinary index for (a) face-on and (b) edge-on films. Values are determined from a single fit using the trust-region reflective algorithm, and 99% confidence intervals are determined here from the covariance matrix.
Fig. 14
Fig. 14 (a) Imaginary and (b) real components of the extraordinary index for (red) face-on and (blue) edge-on films. Values are determined from a single fit using the trust-region reflective algorithm, and 99% confidence intervals are determined here from the covariance matrix.

Equations (3)

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S ( e n , 𝕀 m n ) = Σ i [ y i R ( k , i ; e n , 𝕀 m n ) ] 2
c s , p ( k ) = R s , p ( k ) y Quartz ( k ) / v s , p .
C = | R p e ( n e ) R p 𝕀 m ( n e ) | .

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