Enhancing Organic SemiconductorâSurface Plasmon Polariton

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Enhancing organic semiconductor-surface plasmon polariton coupling with molecular orientation Steven J Brown, Ryan A DeCrescent, David M Nakazono, Samuel H Willenson, Niva A. Ran, Xiaofeng Liu, Guillermo C. Bazan, Thuc-Quyen Nguyen, and Jon A Schuller Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b02767 • Publication Date (Web): 14 Sep 2017 Downloaded from http://pubs.acs.org on September 16, 2017

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Enhancing organic semiconductor-surface plasmon polariton coupling with molecular orientation Steven J. Brown✿ ,† Ryan A. DeCrescent✿ ,‡ David M. Nakazono,‡ Samuel H. Willenson,‡ Niva A. Ran,¶ Xiaofeng Liu,¶ Guillermo C. Bazan,¶ Thuc-Quyen Nguyen,¶ and Jon A. Schuller∗,§ †Department of Materials Science, University of California Santa Barbara, Santa Barbara, CA 93106, USA ‡Department of Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA ¶Center for Polymers and Organic Solids, University of California, Santa Barbara, CA 93106, USA §Department of Electrical and Computer Engineering, University of California Santa Barbara, Santa Barbara, CA 93106, USA E-mail: [email protected]

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Author Information Corresponding Author *E-mail: [email protected]

Author Contributions ✿ R.A.D. and S.J.B. contributed equally to this work.

ORCID

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Abstract Due to strong electric field enhancements, surface plasmon polaritons (SPPs) are capable of drastically increasing light-molecule coupling in organic optoelectronic devices. The electric field enhancement, however, is anisotropic, offering maximal functional benefits if molecules are oriented perpendicular to the interface. To provide a clear demonstration of this orientation dependence, we study SPP dispersion and SPP-mediated photoluminescence at a model Au/small-molecule interface where identical molecules can be deposited with two very different molecular backbone orientations depending on processing conditions. First, we demonstrate that thin films of p-SIDT(FBTTh2 )2 can be deposited with either all “in-plane” (parallel to substrate) or a 50/50 mix of in-plane/“out-of-plane” (perpendicular to substrate) optical transition dipoles by the absence or presence, respectively, of diiodooctane during spin-coating. In contrast to typical orientation control observed in organic thin films, for this particular molecule, this corresponds to films with conjugated backbones purely in-plane, or with a 50/50 mix of in-plane/out-of-plane backbones. Then, using momentum-resolved reflectometry and momentum-resolved photoluminescence, we study and quantify changes in SPP dispersion and photoluminescence intensity arising solely from changes in molecular orientation. We demonstrate increased SPP momentum and a two-fold enhancement in photoluminescence for systems with out-of-plane oriented transition dipoles. These results agree well with theory and have direct implications for the design and analysis of organic optoelectronic devices.

Keywords molecular antennae, surface plasmon polaritons, photoluminescence, reflectometry

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Main Within a simple light emitting device (LED) geometry, emitted photons are ordinarily trapped in waveguided modes. Photonic architectures — e.g., periodic microstructures or other strategic texturing — can couple these waveguided modes into desired far-field radiation. In photovoltaics (PVs), an opposite situation arises; photonic architectures can redirect normally incident light into oblique angles and waveguide modes, increasing light absorption within the active layer. As a particular example, surface plasmon polariton 1,2 (SPP) optical architectures (i.e., metal-dielectric interfaces) are now regularly employed to improve the device performance of inorganic, as well as organic, PVs 3–6 and LEDs. 7–9 The functional benefits are well described by strongly enhanced and localized electric fields associated with the SPP at the metal-dielectric interface. The electric field enhancements associated with the SPP are, however, anisotropic; the dominant electric field enhancement occurs for the component of the electric field perpendicular to the interface (e.g., Ez ). For the case of organic semiconductors, effects of such anisotropies are particularly pronounced 10 since optical transitions in organic molecules are highly anisotropic. 11,12 Still, many studies of SPP-coupled organic LEDs 9,13,14 and organic PVs 4,5,15 have inappropriately treated the organic layers without any mention of optical anisotropy. Indeed, theoretical studies have highlighted the importance of optical anisotropies at metal-dielectric interfaces, 10,16–18 but are supported by few systematic experimental studies. Moreover, a majority of these experimental studies only considered changes due to varying in-plane optical anisotropies 19–21 (i.e., optic axis in the plane of the substrate), whereas typical device morphologies have an out-of-plane (i.e., parallel to interface normal) optic axis with accompanying in-plane symmetry. 22–25 Some studies 26–30 have indeed considered such morphologies, measuring the effects of variable anisotropies using liquid crystals, or by comparing results obtained for different molecular species. However, the influence of SPPs on photoluminescence and absorption was not considered, or was done so in complex many-layered systems in a highly model-dependent way. 4


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Energy-momentum spectroscopies have emerged as a powerful approach for resolving anisotropic optical phenomena. 31–33 Here we use a combination of momentum-resolved photoluminescence (mPL) and momentum-resolved reflectometry (mR) to clearly demonstrate and quantify the effects of molecular orientation on SPP propagation and SPP-mediated photoluminescence. Distinct from previous studies, we truly isolate the effects of an “extreme” molecular reorientation using thin films of a single molecular species — p-SIDT(FBTTh2 )2 (Figure 1a) — that can adopt both in-plane and out-of-plane molecular orientations depending on processing conditions. Though the main results of this work are model-independent, we support interpretation of the results with multi-layer Fresnel models that fully incorporate the optical anisotropies appropriate for the organic thin films studied here (i.e., optically uniaxial dielectric with optic axis out-of-plane). (See Supporting Information section S1 for complete details regarding optical models and matrix methods.) Grazing incidence wide-angle X-ray scattering and transmission electron microscopy measurements 34 show that thin films of p-SIDT(FBTTh2 )2 cast directly from chlorobenzene (CB) adopt a “face-on” morphology (Figure 1b). With this morphology, all molecular faces and backbones lie parallel to the plane of the substrate; that is, all π-stacking directions are parallel to the interface normal, zˆ. When cast from CB with 0.4% by volume diiodooctane (DIO), the films instead adopt an “edge-on” morphology (Figure 1c) with all π-stacking directions perpendicular to the interface normal, zˆ. With this morphology, we expect a 50/50 mix of in-plane/out-of-plane oriented molecules. Note that, in both cases, the films are uniaxial over micrometer scales with optic axes in the zˆ-direction. Importantly, these morphologies are robust to film thicknesses down to a few tens of nanometers. (We leave complete details of chemical and structural characterization to the relevant references 34,35 and corresponding supporting information.)

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a)

Edge-on

Face-on b)

c)

z y x

π faces

π faces

Figure 1: (a) The chemical structure of p-SIDT(FBTTh2 )2 . (b-c) Molecular structures of p-SIDT(FBTTh2 )2 in films cast (b) from pure CB and (c) from CB with 0.4% diiodooctane by volume, representing “face-on” and “edge-on” morphologies, respectively. The zˆ-direction is parallel to the interface normal. Over micrometer scales, molecules exhibit no preferred in-plane orientation, and thus films are rotationally symmetric about the zˆ axis — that is, the films are uniaxial with optic axes in the zˆ-direction. Therefore, the xˆ and yˆ unit vector directions are arbitrary. The transition electric dipole moment is expected to lie parallel to the long-axis of the conjugated molecular backbone.

We begin by comparing the optical anisotropies between thin (20 nm) films of p-SIDT(FBTTh2 )2 prepared with and without DIO (i.e., edge-on and face-on films, respectively) deposited directly on quartz substrates. Specifically, the average in-plane and out-of-plane dipole moments of face-on and edge-on p-SIDT(FBTTh2 )2 films are determined here via mPL measurements. In mPL spectroscopy, 31,32,36 the momentum distribution (i.e., angular distribution) of emitted light is imaged in the back focal plane, or Fourier plane, of a microscope objective. The orientation of emitting (and absorbing) dipoles is determined by fitting analytical expressions for the radiation patterns of oriented dipoles to experimentally measured radiation patterns. (2D momentum-space radiation patterns for both face-on and edge-on films, and associated fits, are shown in Figure S2.) Results of this procedure for face-on films (Figure 2a) are consistent with optical properties typical for conjugated molecules; the absorption and emission dipoles lie along the “average” molecular backbones, 11,12 and the face-on morphology exhibits nearly pure in-plane electric dipole emission. In contrast, then, 6


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the edge-on morphology should exhibit a 50/50 mixture of in-plane/out-of-plane oriented dipoles. Specifically, in edge-on films, the in-plane oriented dipoles are distributed evenly between xˆ and yˆ directions. The density of molecules oriented along any single in-plane axis (ˆ x or yˆ) is thus half the density of out-of-plane (ˆ z ) oriented molecules, and we expect a 2:1 ratio of out-of-plane to in-plane dipole moments. Indeed, mPL measurements of edge-on films (Figure 2b) reveal a 2.08:1 ratio, confirming this simple model of the optical structurefunction relationship in p-SIDT(FBTTh2)2 thin films (see Supporting Information section S2 for complete details).

a) Rel. dipole strengths (arb. units)

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Wavelength (nm) Figure 2: Relative in-plane (blue, solid line) and out-of-plane (red, dashed line) emission dipole strengths of (a) face-on and (b) edge-on p-SIDT(FBTTh2 )2 films on quartz. Dipole strengths are normalized to peak value of unity. “In-plane” (“out-of-plane”) is perpendicular (parallel) to the interface normal. Face-on films exhibit almost pure in-plane dipole emission, while edge-on films exhibit a 2.08:1 ratio of out-of-plane to in-plane dipole strengths.

To explore the effects of molecular orientation on SPPs, we cast 20 nm thick face-on and

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edge-on films of p-SIDT(FBTTh2 )2 (prepared identically to those of the previous paragraph) on Au-coated quartz substrates. The resulting sample structure and measurement geometry are shown in Figure 3a. We map out the energy-momentum dispersion of SPPs for the two morphologies using both mPL spectroscopy and variable-wavelength momentum-resolved reflectometry (mR). 33 In mR (as in mPL), the intensity of reflected (emitted) light is measured as a function of the conserved in-plane electromagnetic wave momentum, k|| . (Since films are rotationally symmetric about zˆ, we refer to kx and ky collectively as k|| such that, in general, k|| = (kx2 + ky2 )1/2 . All data is presented in terms of the normalized in-plane momentum, k|| /k0 , where k0 = 2π/λ0 is the free-space electromagnetic wave momentum. See Methods and our previous work 32,33 for complete details on both sample preparation and experimental techniques.)

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p-pol.

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s-pol.

k||

b) 1

k|| ___ k0

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k||/k0 Figure 3: (a) Sample and measurement geometry. Thin (20 nm) organic films are cast upon a 44 nm film of Au. A 3 nm Ti layer serves as an adhesion layer between Au and quartz. The zˆ-direction is parallel to the interface normals. The system is rotationally symmetric about the zˆ axis, and thus the xˆ and yˆ unit vector directions are arbitrary. (b) mPL energymomentum spectrum of an edge-on film on Au. The emission is almost entirely mediated by coupling to SPPs, as evidenced by the sharp wavelength-dependent emission feature (maximal emission is denoted by a dashed white line). The inset shows the zoomed-out mPL spectrum, confirming negligible PL intensity outside the SPP, down to k|| = 0. (c) mR of face-on (orange up-triangles) and edge-on (green down-triangles) films on Au, as well as bare Au (blue circles), at 775 nm wavelength. The dashed lines are theory curves generated using a multi-layer uniaxial Fresnel model 31,37 with p-SIDT(FBTTh2 )2 optical constants derived from ellipsometry. Reflection minima in the region k|| /k0 > 1 are associated with coupling to an SPP resonance. 9


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A p-polarized mPL spectrum of an edge-on film on Au is shown in Figure 3b. The PL is dominated by a sharp intensity maximum beyond the critical angle (k|| /k0 = 1), with increasing in-plane momentum at smaller wavelengths. This energy-momentum emission feature is characteristic of SPP dispersion relations and shows excellent agreement with calculations of SPP dispersion using a five-layer Fresnel model (to be discussed further, below). The large emission intensity within a narrow band around the wavelength-dependent SPP momentum, and relative null elsewhere (inset of Figure 3b), demonstrates that nearly all light emission is mediated through surface plasmons. (Indeed, virtually zero emission is witnessed in the s-polarized state. See Figure S3.) Tracking the PL maxima provides a simple, high resolution approach to measuring SPP dispersion, but only within the molecular luminescence band. To extend measurements outside this band, we perform complementary p-polarized mR measurements. p-Polarized mR measurements of bare Au and organic films on Au are shown in Figure 3c, for illumination at 775 nm wavelength. Sharp reflection minima beyond the critical angle are characteristic of coupling to SPPs. Momentum-resolved reflection curves generated from an n-layer uniaxial Fresnel model 31,37 (n=4 for bare Au samples, n=5 for samples with organic films on Au) are shown as dashed lines and demonstrate excellent agreement with experimental measurements. (Though these curves are primarily intended as guides to the eye, they may also be used to verify, and aid fitting of, the material optical constants. See Methods and Supporting Information section S1.) The reflection dip appears at higher momenta for Au/organic films than for bare Au systems due to the larger refractive index of the organics as compared to air. We identify the k|| -position of the reflection minimum as the SPP (resonance) momentum. 29,38 Thus by tracking the reflectance minimum at different incident wavelengths we can trace the SPP dispersion. The SPP energy-momentum dispersion compiled from mPL (solid lines) and mR (triangles and circles) are plotted in Figure 4. Comfortably, we see nice agreement between the independent methods of mR and mPL. Furthermore, as expected, 39 edge-on films ex-

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hibit greater SPP momentum, indicating enhanced coupling of molecular excitations to the SPP in the case of edge-on films (or, equivalently, a redistribution of oscillator strength into the out-of-plane orientation 29 ). To support our interpretation of these experimental results, we also plot theoretical SPP dispersion curves (dashed lines), calculated via matrix methods, 38,40,41 for these model systems (see Methods). We note that the large vertical error bars in the mR data at small wavelengths are a consequence of large losses in the organic layer as the wavelength approaches the optical absorption resonance (centered at approximately 610 nm). This effect can also be seen in the increasing width of the mPL SPP luminescence feature on the left side of the mPL spectrum of Figure 3b.

1.10

mR Edge-on mR Face-on

1.08

k__|| k0

mR Au 720 760 Wavelength (nm)

mPL Edge-on mPL Face-on

Wavelength (nm) Figure 4: The surface plasmon polariton energy-momentum dispersion for face-on (orange) and edge-on (green) films on Au, as well as an Au reference (blue), as measured by momentum-resolved reflectometry (‘mR’, solid markers) and momentum-resolved photoluminescence (‘mPL’, solid lines). Theoretical dispersion curves generated via matrix methods 40,41 are shown as dashed lines. The large vertical errors for the mR data at small wavelengths are a consequence of large losses in the organic film as the wavelength approaches the optical absorption resonance. Edge-on films exhibit a larger SPP momentum in comparison to face-on films. 11


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Although changes in dispersion with molecular orientation are subtle here, this is largely due to the deeply subwavelength film thickness. (See Supporting Information section S3 for a detailed discussion and calculations for infinitely thick films.) Nonetheless, both mPL and mR are sensitive enough to resolve these subtle differences. The separation of the SPP resonances between face-on and edge-on films is systematic across all wavelengths; in mPL, the separation is everywhere larger than the associated uncertainties. In mR, the same holds true at most wavelengths. Further, both (independent) techniques agree well with one another and with calculated dispersions. Finally, both mPL and mR derived dispersions require no fitting; we simply identify the PL maximum (or reflection minimum) at each wavelength. The subtle changes in dispersion shown in Figure 4 are consistent with large changes in SPP-molecule coupling (e.g., Figure S4). This is particularly evident in measures of PL intensity. Figure 5 shows unpolarized, momentum-integrated PL intensity (i.e., mPL integrated over all k|| ; see Methods) for face-on (orange) and edge-on (green) films of pSIDT(FBTTh2 )2 on (a) bare quartz and (b) Au-coated quartz substrates. For organic films deposited directly on quartz (Figure 5a), both morphologies exhibit similar PL intensities. (A slight blue-shift and increased structure of the edge-on films’ PL with respect to that of face-on films is likely due to the improved crystallinity and long-range order in edge-on films. 34 ) In sharp contrast, for organic films on Au (Figure 5b), the maximum edge-on PL is a factor of two greater than the face-on PL. This is due to the emission enhancement offered by the SPP mode, as evidenced by the mPL spectra (e.g., Figure 3b). That is, strongly enhanced electric fields associated with the SPP modes leads to preferential emission from films with out-of-plane oriented dipoles.

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PL intensity (a.u.)

a)

b)

PL intensity (a.u.)

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Figure 5: Momentum-integrated unpolarized photoluminescence of p-SIDT(FBTTh2 )2 faceon (‘FO’, orange) and edge-on (‘EO’, green) films on (a) bare quartz substrates and (b) Au-coated quartz substrates. On quartz, emission intensities are nearly identical. On Au, edge-on films exhibit a substantial increase in PL intensity as compared to face-on films, due to the strong coupling to SPPs. Within each plot the PL counts are normalized to a maximal value of unity.

In conclusion, we experimentally demonstrate changes in SPP dispersion, as well as significant changes in relative PL intensity, arising solely from a redistribution of molecular backbones into the out-of-plane direction. We show, in a model-independent way, that the emission is strongly mediated by SPP modes at the Au/organic film interface. We find that

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films of edge-on p-SIDT(FBTTh2 )2 on a thin Au layer exhibit a substantial photoluminescence enhancement (a factor of two) over that of face-on films. This effect is understandable on the grounds of fundamental molecule-light interactions in the subwavelength vicinity of a metal interface. Concomitantly, we observe increased SPP momentum for films comprised of out-of-plane oriented molecules deposited on Au. These results highlight the importance of molecular orientation in the context of plasmonic device architectures and suggest that significant improvements in optoelectronic device performance may be enabled through appropriate control over thin-film morphology. Further, these results demonstrate that momentumresolved spectroscopies are particularly well-suited for studies of SPP-mediated absorption and emission phenomena in systems with anisotropic media (i.e., organic semiconductors). Additionally, with measurements of optical dipole orientation in face-on and edge-on film morphologies, we introduce p-SIDT(FBTTh2 )2 as a model system for studying orientation dependent phenomena in uniaxial media with out-of-plane optic axes. We expect future experiments to study the interplay of optical and electrical performance with molecular orientation to determine the ideal geometry for surface plasmon polariton optoelectronic devices.

Methods Sample preparation Starting with quartz coverslips, a 3 nm Ti adhesion layer, and subsequent 44 nm Au layer were deposited via thermal evaporation. On a subset of such prepared samples, we spincast both face-on and edge-on films of p-SIDT(FBTTh2 )2 . To produce face-on films, we use a solution of 10mg/mL p-SIDT(FBTTh2 )2 in chlorobenzene (CB). For edge-on films, 0.4% diiodooctane by volume was added to the CB solution. Both varieties were deposited by spinning at 3000 RPM, producing films of 20 nm thickness, as measured by atomic 14


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force microscopy in tapping mode. (Face-on films showed very smooth interfaces with RMS roughness of approximately 1nm. Edge-on films showed a higher RMS interface roughness.) Thickness was not considered as a free parameter for any of the SPP results presented here.

Calculating anisotropic dipole strengths Effective emission dipole strengths were calculated by decomposing emitted light measured via mPL into light coming from in-plane and out-of-plane dipoles using a three layer optical model that fully incorporates the optical anisotropies relevant to the case at hand (see our previous work 31,32 for complete details). To measure mPL, the sample was excited by a 425-650 nm light emitting diode (LED) (ThorLabs MCWHL5-C5) that uniformly filled momentum space. The incident light was filtered via a 633 nm shortpass filter and reflected off a 635 nm dichroic mirror to remove overlap with the emission wavelengths. The emitted PL was then filtered to remove excitation reflections by transmitting through the same dichroic as well as a 635 nm longpass filter. The PL was then passed through an analyzing polarizer such that light along the ‘y-momentum axis’ was p-polarized. The energy and y-momentum of the light was then measured by an imaging spectrometer (Princeton Instruments IsoPlane SCT320) with attached charge coupled device camera (Princeton Instruments PIXIS 1024BRX) at a back focal plane (BFP) conjugate to the objective BFP. The mPL counts were corrected for momentum-dependent apodization in post-processing. The optical constants used for this analysis are taken from ellipsometry (Supporting Information Figure S1).

Optical constants and theoretical reflection/dispersion curves Theoretical reflection and SPP dispersion curves were calculated from 2×2 transfer matrix methods using four- and five-layer Fresnel models for samples without and with an organic film, respectively. 37,38,40,41 The 2×2 matrix methods are appropriate for the anisotropies at hand, since we expect no mixing between s- and p-polarized components of the incident light. 15


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For these curves, optical constants were taken from mR or ellipsometry measurements of reference samples: Starting from literature values, 42,43 Ti and Au permittivities were further refined at each wavelength to achieve best agreement between calculated and measured mR traces of bare Au samples (i.e., samples with no organic films, e.g., Figure 3c, blue circles). The titanium thickness (3nm) and Au thickness (44nm) determined from mR measurements agree well with values derived from a crystal deposition monitor (3nm and 40nm respectively). The in-plane and out-of-plane complex permittivities of face-on and edge-on films of p-SIDT(FBTTh2 )2 were determined via ellipsometry of films deposited on Si substrates. In order to ensure material consistency, ‘control’ samples (i.e., bare metal films on quartz, and bare organic films on quartz) were produced at each stage of the sample preparation, simultaneously with the ‘test’ samples. See Supporting Information section S1 for material optical constants and complete details.

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Associated Content Supporting Information Further details regarding optical models, optical constants, and mPL spectroscopy. Expanded discussion about film thickness and calculations of SPP dispersion for infinitely thick organic films.

Acknowledgement We thank Chris Takacs and John Love for helpful discussions. This work was supported by a National Science Foundation CAREER award (DMR-1454260). N.A.R., T.-Q.N., X.F. and G.B. acknowledge the support from the Department of the Navy, Office of Naval Research (Award No. N00014-14-1-0580). AFM was 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.

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(26) Sprokel, G. J.; Santo, R.; Swalen, J. D. Molecular Crystals and Liquid Crystals 1981, 68, 29–38. (27) Welford, K. R.; Sambles, J. R. Applied Physics Letters 1987, 50, 871–873. (28) Welford, K. R.; Sambles, J. R.; Clark, M. G. Liquid Crystals 1987, 2, 91–105. (29) Wang, X.; Wang, P.; Chen, J.; Lu, Y.; Ming, H.; Zhan, Q. Applied Physics Letters 2011, 98, 021113. (30) Gruber, M.; Mayr, M.; Lampe, T.; Gallheber, B.-C.; Scholz, B. J.; Brtting, W. Applied Physics Letters 2015, 106, 083303. (31) Schuller, J. A.; Karaveli, S.; Schiros, T.; He, K.; Yang, S.; Kymissis, I.; Shan, J.; Zia, R. Nature Nanotechnology 2013, 8, 271–276. (32) Brown, S. J.; Schlitz, R. A.; Chabinyc, M. L.; Schuller, J. A. Physical Review B 2016, 94, 165105. (33) DeCrescent, R. A.; Brown, S. J.; Schlitz, R. A.; Chabinyc, M. L.; Schuller, J. A. Opt. Express 2016, 24, 28842–28857. (34) Ran, N. A.; Roland, S.; Love, J. A.; Savikhin, V.; Takacs, C. J.; Fu, Y.-T.; Li, H.; Coropceanu, V.; Liu, X.; Brdas, J.-L. et al. Nature Communications 2017, 8, 79. (35) Love, J. A.; Nagao, I.; Huang, Y.; Kuik, M.; Gupta, V.; Takacs, C. J.; Coughlin, J. E.; Qi, L.; van der Poll, T. S.; Kramer, E. J. et al. Journal of the American Chemical Society 2014, 136, 3597–3606. (36) Taminiau, T. H.; Karaveli, S.; van Hulst, N. F.; Zia, R. Nature Communications 2012, 3, 979. (37) Azzam, R.; Bashara, N. Ellipsometry and polarized light; North-Holland personal library; North-Holland Pub. Co., 1977. 20


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(38) Dawson, P.; Puygranier, B. A. F.; Goudonnet, J.-P. Physical Review B 2001, 63 . (39) Krokhin, A. A.; Neogi, A.; McNeil, D. Physical Review B 2007, 75, 235420. (40) Ward, C. A. The Journal of Chemical Physics 1975, 62, 1674. (41) Connolly, M. P.; Dawson, P. Journal of Applied Physics 1995, 78, 5522–5533. (42) Malitson, I. H. JOSA 1965, 55, 1205–1209. (43) Johnson, P. B.; Christy, R. W. Physical Review B 1972, 6, 4370–4379.

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