Pure circular dichroism by curved rows of plasmonic nanoparticles

Meng Wang; Bruno Gompf; Martin Dressel; Nathalie Destouches; Audrey Berrier

doi:10.1364/OME.8.001515

Introduction

The search for chiroptical effects in plasmonic nanostructures is a hot topic in nanooptics [1–6]. Unfortunately, the physical origin of the observed optical response is often difficult to identify. This lack of understanding is hidden in terms like chirality, circular dichroism or optical activity, implying that these terms were interchangeable. In reality they are connected to totally different physical properties. Whereas chirality is a geometric property, which strictly speaking does not exist in two-dimensions, circular dichroism is an optical property which initially is not related to any physical property of the sample, and optical activity is a property of molecules exhibiting no inversion center. The optical response of a sample can always be decomposed into its seven basic optical properties, namely linear dichroism (LD), linear birefringence (LB) at (0°,90°), linear dichroism (LD’), linear birefringence (LB’) at ( ± 45°), circular dichroism (CD) and circular birefringence (CB) and the overall absorption [7]. In complex nanostructures these properties are often interweaved and to disentangle them is not straightforward. This can simply be illustrated by the fact that optical active molecules and the combination of two non-orthogonally oriented uniaxial optical elements, say, a polarizer and a quarter waveplate, both exhibit circular birefringence, but only the optical active molecules are subjected to the Cotton-effect. Like LD and LB, CD and CB are connected to each other by Kramers-Kronig relations. The Cotton effect is therefore a direct consequence of causality and locality, the two explicit conditions for Kramers-Kronig relations. This is, what we want to show with the example of combined polarizer with quarter waveplate compared to optical active molecules. Chiral biomolecules exhibit a local optical response, which can be directly calculated by their electric and magnetic dipole transitions. Whether chiral nanostructures exhibit pure circular dichroism or whether their optical response is a (non-local) combination of linear effects, has to be analyzed carefully. To avoid misinterpretations of the used basic physical terms, we want to lead the attention of the reader to two older references, the seminal paper of Schellman [8] on this topic and the nice collection of papers on optical activity by Lakhtakia [9].

It is clear that absorption is an optical quantity per unit length that is distributed along the beam path and therefore Lambert-Beer’s law applies. This approach is less common for the other basic optical properties, although a generalized framework of Lambert-Beer’s law for the entire optical response does exist: measuring all 16 elements with Mueller-matrix (MM) spectroscopic ellipsometry [10] followed by analysis with the differential matrix decomposition [11–13]. However, with a few exceptions^{[4], this method was hitherto not really applied to the complex optical response of plasmonic chiroptical structures leading to the above mentioned confusions. In other words, it is crucial to distinguish between optical active materials with pure circular optical effects (i.e. all the decomposed MM elements representing linear effects vanish), from a combination of polarizer and quarter waveplate based on pure linear effects [14,15] (i.e. all decomposed elements representing circular optical effects vanish).}

In the case of plasmonic designs, many concepts discussing the presence of CD consider structures where two (or more) main optical axes can be defined. Examples of such designs include the V-shaped particles, the superposition of rods, crosses, gammadia, etc. Some reports even underline the tunability of the CD signal as a function of the respective angle formed by the axes of resonance. Although the definition of CD is simple, its origin is not straightforward to determine experimentally. Without investigating each reported case individually, here we want to call for attention when interpreting measured CD signals. First of all, one trivial way to generate CD, is to break the orthogonality of the optical axes in the sample. Let us take the simple example of two plasmonic rods forming an angle ϕ = 90°. We can define two orthogonal optical axes following the orientation of the rods, determining the value of the Euler angle ϕ (Fig. 1).

Fig. 1 Calculated circular dichroism (CD) and circular birefringence (CB) as a function of the angle ϕ between two plasmonic nanorods simulated each by a Lorentz oscillator along the optical axes defined by the orientation of the rods.

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The modelling of this situation with the ellipsometry software from J.A. Woollam Co. yields a CD signal equal to 0. However when ϕ≠90°, the simulations show the presence of an isotropic CD signal, which can be tuned by the value of ϕ. The CD signal increases in the region of the plasmonic resonance when the angle between the optical axes is away from 90° until reaching a maximum at ϕ = 45°. Then it decreases further until ϕ = 0°. The circular birefringence (CB) presents a similar variation with its maximum slightly shifted in angle ϕ. This example illustrates the difficulty in determining the real origin of a measured CD signal. It is easy to mistake pure CD signal with signals of trivial origin. Therefore a robust method for the identification of CD signal, and especially its disentanglement, is necessary.

The problem is that most characterizations measure only the differences in intensity, and are not able to unambiguously determine the origin of the measured signal. A simple intensity measurement does not allow the separation between contributions from the linear effects and intrinsic CD [4]. Additionally we note that effects appearing only at non-zero angles of incidence are due to the mixing of linear optical effects and are not included within real CD. Furthermore, plasmonic designs are often complex as they can support various physical phenomena and interactions. Plasmonic resonances [16–18], inter-particle coupling effects [19–22] coupling to photonic modes, periodicity and anisotropy are commonly encountered. All depend on the wavelength, the light polarization, the azimuthal orientation of the sample in case of an anisotropic arrangement and on the angle of incidence, when dispersion coming from periodicities or coupling to the neighbors is important. The interplay of these different contributions (anisotropy, dispersion, periodicity, diffraction, interferences with photonics modes, etc.) often leads to a complex optical response, where the influence of the different ingredients cannot be separated easily. On the other hand, these effects have to be perfectly understood, in order to tailor the optical response for the desired application. A clear procedure is still lacking that allows an easy separation of the effects.

Contrary to pure intensity based measurement techniques, MM spectroscopic ellipsometry enables us to measure both intensity and phase responses as a function of the input polarization [10], as a function of the parameters wavelength λ, azimuthal orientation α (sample rotation) and angle of incidence θ in order to take dispersion effects into account. Although already used in the case of complex samples [23–25], the interpretation of the information hidden in experimentally obtained MM is still challenging.

In this work we demonstrate the necessity of the comprehensive approach as well as its potential to identify the nature of the measured CD signals. We present a versatile method systematically disentangling the intricate optical response of an elaborate nanoparticle array by spectroscopic Mueller matrix ellipsometry [25]. The investigated nanoparticle array is produced by a self-organized growth process of Ag nanoparticles in a TiO₂ thin film loaded with metallic precursors by continuous laser light excitation (Fig. 2). This type of sample is known to produce bright and robust colors [26–29]. The optical characterization starts with the angle of incidence dependent transmission measurements revealing the dispersion coming from waveguide excitations due to the periodic arrangement of the Ag nanoparticles embedded into the thin TiO₂-film as well as the azimuthal dependent transmission showing the anisotropy of the sample. From these intensity measurements, a first simple effective medium model is developed. Comparing the full MM measurements with the simulated MMs calculated from this simple model and subsequently decomposing the obtained MMs allows us to decouple the different contributions to the optical response and learn about the special properties of this sample design.

Fig. 2 a) Schematic of the fabricated sample; (b) SEM image of the Ag nanoparticle region; (c) Definition of azimuthal angle α as well as orientations x and y with respect to the nanoparticle lines; (d) schematic drawing illustrating the different physical phenomena at play in the investigated sample; (e) and (f) represent schematic drawings of the cross-section of the sample illustrating the modes supported by the structure and their interactions.

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1. Results and discussion

1.1 Sample description and transmittance

We consider a curved grating, which lines are a succession of silver nanoparticles formed by a self-assembly process (Figs. 2(a)-2(c)); for details about the fabrication procedure see Appendix 1). The nanoparticle array is located at the interface between the glass substrate and the titania layer, which acts as a photonic waveguide. Figures 2(d)-2(f) illustrate the physical interactions within the structure: when light impinges on the sample, part of the light is transmitted through the layer stack. This light is absorbed by the presence of the Ag nanoparticles, and hence it is spectrally shaped by the plasmonic resonance which decreases the transmittance accordingly. Another part of the incident light beam is diffracted by the presence of the particle grating. This diffracted light is able to excite the waveguide mode, which is a leaky mode. The leaky waveguide mode interferes destructively with the transmitted light, leading to an effective absorbance of the transmitted light. However, in the spectral region affected by the plasmonic absorbance, the superposition of both effective absorbance signals leads to a transparency window within the plasmonic resonance. This is what we call the destructive interference modes (DIM ±1) in the Appendix 2 at the end of the paper, where ± 1 refers to the respective diffraction orders.

The identification of these modes can be confirmed only by analyzing the specific dispersion behavior of the sample in transmittance, over the whole measured wavelength (λ) range, as a function of the angle of incidence θ as well as a function of the sample rotation, via the azimuthal angle α. The measured intensity variations for both polarization states, p-polarization and s-polarization, are well reproduced over the whole accessible space (λ,θ,α) by the analytical expressions for the DIM.

Finally, the individual plasmonic nanoparticles forming the curved grating rows can couple within the same row or across the rows. This inter-particle coupling is intrinsically polarization dependent, as illustrated in the inset of Fig. 2(d). Figure 3 displays the transmittance spectra (Figs. 3(a) and 3(b)) and transmittance contour plots as a function of the incidence angle (Figs. 3(c)-3(g)) and of the azimuthal angle (Figs. 3(i)-3(l)). It is interesting to note that the splitting of the plasmonic resonance by the DIM(-1) and DIM(+1) are both polarization and sample orientation dependent. This originates from the anisotropy introduced by the one-dimensional grating. In particular, away from normal incidence, the degeneracy of the DIM is lifted and the splitting pattern becomes more complex. The azimuthal dependent transmittance spectra measured at normal incidence and at θ = 10° are shown at the bottom of Fig. 3, as well as the comparison with the simulated spectra (Figs. 3(j) and 3(l)). Indeed, from the measured intensity spectra, it is possible to create an ellipsometry model based on a biaxial layer that is able to reproduce the transmittance for both polarizations (Fig. 3(j)). Figures. 3(e) and 3(h) illustrate the oscillators used for the ellipsometry models. More details on the model are given in the next section and in Appendix 3. It is important to note that the model reproduces the transmittance over the whole azimuthal range. The comparison of the linecuts of the transmittance plots are shown in Fig. 3(l).

Fig. 3 Measured transmittance T_ss and T_pp for (a) α = 90° θ = 0° and (b) α = 0° θ = 0°; (c) dispersion plots of the measured transmittance between θ = 0° and 20° in steps of 2° in the spectral range between 370 and 700 nm with s-polarized light at α = 90° and (d) at α = 0°; (e) The Lorentz oscillators used in general oscillator model for fitting the transmittance with p-polarized light at α = 90°. All contour plots are shown together with dashed lines indicating DIM(±1); (h) The Lorentz oscillators used in general oscillator model for fitting the transmittance at α = 0°; (i-k) Transmittance contour plots as a function of the wavelength and the azimuthal angle (i) at normal incidence for s-polarized light and p-polarized light; (j) simulated transmittance at normal incidence for both polarizations; (k) contour plot at angle of incidence 10° for s-polarized light and p-polarized light, and (l) comparison of linecuts at azimuthal angle 0°. The calculated DIM lines are shown only on half-space to preserve the visibility of the raw measurement on the other half.

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1.2 Mueller matrix characterization

We apply now Mueller matrix spectroscopy in order to get more information than already gathered by intensity measurements only. The 4x4 Mueller matrix was measured at normal incidence and at higher angles of incidence. For the sake of readability, only selected Mueller Matrix Elements (MMEs) will be considered here.

The first step in the interpretation of Mueller matrix patterns is to create an ellipsometry model able to reproduce both the intensity data in the whole measured space (λ, θ, α). Our model, detailed in Appendix 3, is formed by a biaxial layer where the axis normal to the sample interface is represented by a Cauchy function, and the in-plane axes X and Y are respectively obtained by fitting the intensity spectra along to the corresponding direction. This model reproduces the intensity plots very well (Fig. 3(j)). We use the same model to generate all the MMEs.

In order to extract the pure values of the optical properties, linear dichroism LD/LD’ along X and Y respectively, linear birefringence LB/LB’, circular dichroism CD and circular birefringence CB, we use the differential decomposition method [12,29,30]. We apply the differential decomposition to both the measured MM as well as the model generated MM and obtain the matrix Lm.

Let concentrate our attention to selected MMEs: M12 (linked to LD), M34 (linked to LB), M14 and M41 (linked to CD) and M23 (linked to CB). Figure 4 presents a comparison of the measured and simulated, before and after differential decomposition, over four quadrants for each of the five selected elements. This representation allows us to directly compare, from quadrant to quadrant, the quality of the simulated data, as well as the difference brought by the decomposition step. We note here that it is enough to show only one quadrant owing to the inversion symmetry of the MM elements. The first observation is that the linear properties LD (M12) and LB (M34) are perfectly reproduced by our model. As expected the decomposition does not introduce any change in these patterns, since the corresponding MME represents the pure optical property without superposition. The three other elements however, represented in the bottom row of Fig. 4, are very different in all of the four quadrants. If we compare the first quadrant for M14 and M41, we can see that their data differ greatly. This is due to the fact that they are superposing the true CD signal and a combination of linear signals: $M 14 = {CD}_{i n t} + \frac{1}{2} ({LB LD}^{'} - {LB}^{'} LD)$ and $M 41 = {CD}_{i n t} - \frac{1}{2} ({LB LD}^{'} - {LB}^{'} LD)$ only the differential decomposition allows extracting the true CD signal [12]. Effectively, the bottom left quadrant is identical for M14 and M41. Both right side quadrants, representing the simulated elements, in M14 and M41 are equal to 0. This is expected since in the simulations ${CD}_{i n t}$ = 0 and the couples $LB / {LB}^{'}$ and $LD / {LD}^{'}$ are perfectly symmetric and shifted respectively by 45°. The optical axes are perfectly perpendicular. In the case of non-perpendicular axes, presented in the introduction, it is found that both M14 and decomposed Lm14 are azimuthal independent and form an isotropic circle with the spectral dependence of Fig. 1 (not shown). Hence the azimuthal dependent four-fold symmetry obtained experimentally does not correspond to skew optical axes. Instead we find that the origin of the non-vanishing linear CD and CB signal originates from spectral, angular and strength differences between the measured LB and LB’. M23 does not present much difference between measured and simulated element (only the intensity differs while the symmetry is conserved), while simulated Lm23 vanishes contrary to measured Lm23. This is due to the fact that measured M23 is similarly to M14 superposing the pure optical element CB with the linear combination of the optical properties, and moreover the simulation does not include CB, as CB and CD are intrinsically linked. That is to say, the ellipsometry model, although very powerful to represent the linear elements, is unable to reproduce the circular elements. Let us discuss the experimental circular elements in details. Figure 5(a) shows a linecut through the elements M14 and M23, as well as their decomposed version Lm14 and Lm23, at azimuthal angle 0°. The comparison between M14 and Lm14 indicates that the true CD, the intrinsic part, amounts to the half of the apparent, non-decomposed signal. This part is significant and it is confirmed by the corresponding CB signal (Lm23). Figure 5(b) brings a comparison of the extracted intrinsic CD and CB obtained from decomposition, and the part that can be calculated as the linear superposition of the linear dichroism and linear birefringence.

Fig. 4 Comparison of measured and simulated selected Mueller Matrix Elements M as well as the corresponding decomposed matrix elements Lm (a) labels of the four quadrants for the representations of (b) M12, (c) M34, (d) M41, (e) M14 and (f) M23. The azimuthal dependence of the elements shown in this figure are made from the concatenation of four quadrants taken from the full elements while respecting their original quadrant place.

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Fig. 5 Comparison of linecuts at azimuthal angle 0° through matrix elements related to the circular dichroism or circular birefringence. a) Linecut along α = 0° for M14 (and its decomposed version Lm14) related to CD compared to the linecut of M23 (and its decomposed version Lm23) related to CB, the inset represents the color-coded contour plots of the respective elements plotted over 1/3 of the azimuthal range for sake of comparison; b) Spectra of CDlin, CBlin, CBint and CDint obtained from the decomposition of the experimental Mueller matrix.

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3. Origin of intrinsic CD

The origin of the intrinsic CD signal is to be found in an element of symmetry breaking in the sample design: here it is the curvature of the grating lines that follow the shape of the beam waist of the laser during exposure (see Appendix 3). In the investigated sample, there are no optically active materials, i.e. there is no circular dichroism at the molecular level; instead the observed effect is created at the mesoscopic level, arising from the geometrical arrangement of the dipoles. We note here that this happens at normal incidence and it is different from anisotropy effects or other effects due to non-orthogonality of the optical axes discussed in the introduction of this article. The interactions of the nanoparticles in the nearest-neighbor range can be tailored by the precise arrangement of the metallic nanoparticles. Hence, the arc-shaped nanoparticle chains can, not only break the symmetry of left and right circular polarized light, but also generate an azimuth dependent optical response due to the varying environment that neighboring particles see as a function of the rotation of the electric field along the arc. In other words, one part of the true (intrinsic) CD originates from a broken symmetry on a mesoscopic level due to the curvature of the lines. We note also that the linear CD originates from a non-symmetric and spectral varying couple LB/LB’, which is not reproduced by the simulations.

An additional source of CD originates from the interaction with the DIM. In order to explicit this dependence, we plot the position of the DIM lines on top of the two-dimensional spectra versus azimuthal angle representing the transmittance, for p- and s-polarizations (Figs. 6(a), 6(b) and 6(d), 6(e)), as well as that of the three CD signals, the measured apparent, the extracted intrinsic and finally the calculated linear parts at normal incidence and away from normal at θ = 20° (Figs. 6(g)-6(l)). The comparison of the linecuts of the transmittance plots with the corresponding linecut of the apparent CD, the linear CD and the intrinsic CD at relevant wavelengths (λ = 492nm, DIM wavelength, at normal incidence and λ = 575nm at θ = 20°) are shown in Figs. 6(c) and 6(f). The transmittance plots illustrate the effect of the DIMs on the intensity: the DIM lines generate an azimuthal dependent splitting of the transmittance. At normal incidence, the splitting lines for p- and s-polarization are azimuthally shifted by 90°. The main difference of the DIM with Rayleigh-Wood anomalies (RWAs) is apparent now: RWAs introduce a splitting in the plasmonic resonances when the polarization of light is perpendicular to the grating lines. In the case of the DIMs, the intermediate coupling to the waveguide mode introduces a 90° azimuthal shift, leading to a splitting when the light impinging on the sample is parallel to the grating. When the incident angle increases, the lift of degeneracy reshapes the transmittance dip in a complex fashion and „pushes“ it to both lower and higher wavelengths. In particular, the azimuthal oscillations around 600 nm and around 400 nm are to be explained by the effect of the DIM. The transmittance plots being so dramatically affected, let us turn our attention to the CD plots to identify the role of the DIM there. Figures. 6(g)-6(i) present the apparent, intrinsic and linear parts of the CD signal at normal incidence. The CD signal oscillates as a function of the azimuthal angle and the wavelength. The positive lobes coincide with dips of T_ss. The (λ, α) contour plots of CD_app, CD_lin and CD_int are similar at first sight, as far as pattern and sign are concerned. Strength-wise, the intrinsic CD signal is almost as intense as the linear one. This can be quantitatively seen in the linecut plot of Fig. 6(c). A more careful observation of the contour plot of CD_int (Fig. 6(h)) reveals that the positive lobes (yellow areas) are connected with each other forming a zigzag pattern. This is due to the different physical origin of the CD_int and the CD_lin signal.

Fig. 6 Comparison of the (λ,α) contour plots representing the transmittance plots, and the CD signal distinguished in apparent CD, intrinsic CD and linear CD at normal incidence and at incident angle 20°. The comparison of selected linecuts is also shown in (c) and (f).

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The situation at non-zero incident angle is interesting but more complex. The degeneracy of the DIM lines is lifted and dispersion is introduced. The transmittance plots are shaped by the DIM lines. In particular an extra DIM-induced splitting is visible in T_ss, where the transmittance dips are pushed towards higher wavelengths in a pattern following the shape of the DIM(+1) line. The (λ, α) contour plots of CD are also instructive. In this case the CD_int signal gain in strength, becoming more intense than the linear part (visible in the linecuts of Fig. 6(f)). Moreover we can see that the lobes of CD_int split, following both DIM(+1) towards higher wavelengths and DIM(−1) towards lower wavelengths. This is a strong hint for the fact that part of the CD_int originates from the presence of the DIM modes.

4. Summary

To conclude, a comprehensive approach combining intensity measurements, Mueller matrix spectroscopy and appropriate decomposition techniques was applied to a sample combining plasmonic resonances, photonic waveguiding and diffraction effects. It is found that the sample not only generates linear-based circular dichroism but it produces also intrinsic circular dichroism. The origin of the intrinsic CD is found to be related to the mesoscopic arrangement of the interacting plasmonic nanoparticles and to the influence of the interference with the waveguide mode. In particular the arc-shape of the self-organized nanoparticle lines is identified to be the main reason for the generation of intrinsic CD. This opens the possibility for a new, self-generated way to create CD, which is expected to be tunable by the curvature of the arc. The demonstration of this tunability will be the topic of future work.

Appendix 1 Experimental

A mesoporous film of amorphous titania loaded with silver salt and small silver nanoparticles was deposited on a glass substrate. The laser beam was slightly focused on the sample by a 10x microscope objective (Olympus MPlan N, N.A. 0.25) under normal incidence, the diameter of the circular spot in the focal plane can be varied by changing the focus. The incident power on the sample is 300 mW for a laser wavelength of 488 nm. During exposure, the samples are translated at a constant speed of 0.6 mm/s to draw 4 mm long lines with the laser beam. Under such illumination conditions, silver nanoparticles spontaneously grow along periodically spaced chains parallel to the laser polarization that form periodic Ag nanoparticle ensemble with periodicity 270 nm measured from diffraction measurements in a back-scattering configuration. Here the laser polarization is set perpendicularly to the laser path. The nanoparticles are located at the TiO₂/glass interface. The size of silver nanoparticles ranges from 20 to 100 nm and the interparticle distance along a chain varies. The sample topography was measured with profilometer (Dektak). It is found that the thickness of the laser exposed area is 85 ± 35 nm thinner (the surface roughness induces the error bar) than the non-illuminated TiO₂ area, which is expected due to the collapse of the mesoporosity of the initial TiO₂ layer during the laser-induced heating. The thickness of the non-illuminated TiO₂ area is 160 nm as determined by ellipsometry measurement. The Ag nanoparticle chains are not straight but arc-shaped lines that mainly follow the shape of the laser spot in the front edge of the translating elliptical beam. The radius of curvature of the nanoparticle lines is around 15 µm by analyzing the arc in SEM images. This value is confirmed by the optical microscopy pictures that shows the starting area of the laser beam exposure. The angle α=0° is defined when the plane of incidence is perpendicular to the particle lines.

Two different sets of measurements were performed on the sample: (1) angular dependent transmittance measurements with p-polarized and s-polarized light varying both the incident and azimuthal angles, and (2) spectroscopic Mueller matrix ellipsometry measurements in transmittance in the spectral range between 210 nm and 1690 nm at angle of incidence θ varying between 0° and 20° in steps of 2° with full azimuthal rotation α from 0° to 360° in steps of 5° in order to fully map the optical response of the sample. Both measurements were performed with a variable angle dual rotating compensator spectroscopic Mueller matrix ellipsometer (RC2) from J. A. Woollam Company.

Appendix 2 Analytical expressions for the DIM

The phase difference between transmitted light and leaky waveguide mode can be expressed from the ray optics approximation as $2 k_{T i O_{2}} h - (2 φ_{1} + 2 φ_{2})$ , where $2 k_{{TiO}_{2}} h = 2 n_{{TiO}_{2}} \frac{2 π}{λ} h \cos θ_{d}$ approximates the phase of the waveguide mode, and $2 φ_{1} s$ and $2 φ_{2}$ are the two polarization-dependent phase shifts introduced at the two interfaces air/TiO₂ and TiO₂/glass. $2 φ_{1} s$ and $2 φ_{2}$ for TE polarization are expressed as:

2 φ_{1} = 2 \tan^{- 1} {\frac{{(n_{T i O_{2}}^{2} \sin^{2} θ_{d} - n_{a i r}^{2})}^{\frac{1}{2}}}{n_{T i O_{2}} \cos θ_{d}}}, 2 φ_{2} = 2 \tan^{- 1} {\frac{{(n_{T i O_{2}}^{2} \sin^{2} θ_{d} - n_{g l a s s}^{2})}^{\frac{1}{2}}}{n_{T i O_{2}} \cos θ_{d}}} f o r T E p o l a r i z a t i o n

2 φ_{1} = 2 \tan^{- 1} {\frac{n_{T i O_{2}}^{2}}{n_{a i r}^{2}} \frac{{(n_{T i O_{2}}^{2} \sin^{2} θ_{d} - n_{a i r}^{2})}^{\frac{1}{2}}}{n_{T i O_{2}} \cos θ_{d}}}, 2 φ_{2} = 2 \tan^{- 1} {\frac{n_{T i O_{2}}^{2}}{n_{g l a s s}^{2}} \frac{{(n_{T i O_{2}}^{2} \sin^{2} θ_{d} - n_{g l a s s}^{2})}^{\frac{1}{2}}}{n_{T i O_{2}} \cos θ_{d}}} f o r T M p o l a r i z a t i o n

where $n_{T i O_{2}}$ (to be determined), $n_{g l a s s} = 1.52$ and $n_{a i r} = 1$ are refractive indexes of the TiO₂ matrix, glass and air, $θ_{d}$ is the diffracted angle.

In case of destructive interference, the phase difference $Δ φ$ between the leaky waveguide mode and the transmitted incident light beam, which crossed both the air/TiO₂ and the TiO₂/glass interfaces, should be equal to a multiple of π, which is expressed by the following equation:

Δ φ = 2 k_{T i O_{2}} h - (2 φ_{1} + 2 φ_{2}) = (2 f + 1) π, f = 0, \pm 1, \pm 2 \dots

where f indicates the order of the DIM.

The diffracted angle $θ_{d}$ in Eq. (3) can be further calculated from the grating equation in both x and y directions as:

X axis: $\sin θ_{inc} \cos α \pm n_{{TiO}_{2}} \sin θ_{d} \cos α_{d} = m \frac{λ_{0}}{P}, m = \pm 1, \pm 2 \dots$ ; Y axis:

n_{{TiO}_{2}} \sin θ_{d} \sin α_{d} = \sin θ_{inc} \sin α

where $α$ is the azimuthal angle, $θ_{i n c}$ is the angle of incidence, P is the grating periodicity. Overall, grating Eqs. (4) and Eq. (3) should be combined to calculate the DIM. In this work, only $Δ φ = π$ (f=0) with $m = \pm 1$ order of grating is calculated, corresponding to the DIM( $\pm$ 1) discussed all along the paper..

Refractive index $n_{T i O_{2}}$ (1.9) and thickness h (160nm) of the TiO₂ waveguide before illumination are obtained by spectroscopic ellipsometry. Then we estimate the range of $n_{T i O_{2}}$ and h due to the collapse of the mesoporous TiO₂ after laser exposure: $n_{T i O_{2}}$ is comprised in the range [1.9; 2.33], where 2.33 is the lower range for the refractive index of bulk TiO₂ obtained from tabulated values, while h is estimated in the range [50; 120 nm] according to profilometer measurements on the sample. In the next step, the uncertainty on the values $n_{T i O_{2}}$ and h are reduced. Experimentally the position of the DIM at θ=0°, α=0° is located at λ=492 nm. This DIM position can be calculated by the Eqs. (3) and (4) with various pairs ( $n_{T i O_{2}}$ , h) within the above mentioned restricted ranges, which lead to a variation in the slopes of the calculated dispersion lines. The slope of the dispersion line for the pair (h=100 nm, $n_{T i O_{2}}$ =2.21) fits the measured intensity modulation.

At normal incidence, the forward- and backward- propagating leaky modes have the same propagation constant β. At oblique incidence, these two modes cannot be excited with the same wavelength. The condition for waveguide eigenmode can be expressed as Eq. (5):

Δ φ = 2 k_{T i O_{2}} h - 2 φ_{1} - 2 φ_{2} = 2 f π, f = 0, \pm 1, \pm 2 \dots

where f indicates the number of the modes which can propagate in the waveguide layer. The waveguide eigenmode would induce the absorption of light and would be measured in transmittance. However, the waveguide eigenmode is outside the measured wavelength range, and it is not discussed here.

Appendix 3 Ellipsometry model

The ellipsometry model is a simple anisotropic layer model based on Fresnel’s equations. The fitting procedure uses the mean square error (MSE) to quantify the difference between experimental and calculated model data. A lower MSE implies a better fit. The Levenberg-Marquardt algorithm is used to quickly determine the minimum MSE, which indicates the best model fit to the data [31,10]. This model allows us to understand the anisotropy and other underlying physical features of the observed optical behavior at normal incidence. A 100 nm thick biaxial layer, representing effectively the active TiO₂ layer with the embedded Ag NPs, is modelled on top of a glass substrate. The biaxial layer is defined by the oscillators along the three directions X, Y and Z. In the Z direction, which is normal to the layer interface, a Cauchy oscillator representing the TiO₂ matrix is used. The parameters of the oscillators along the two main optical axes in the X-direction (perpendicular to the grating lines) and Y-direction (parallel to the grating lines) are obtained by fitting the experimental transmittance measured at normal incidence along X and Y using two different general oscillator (Genosc) layer models. Four Lorentz oscillators are used in both X and Y directions in order to obtain a perfect fit of the transmittance spectra. The three Lorentz oscillators, L1 to L3, fully located inside the measured range 400-700 nm correspond to plasmonic excitations within the sample.

We note here that the created model is a purely biaxial model only valid at normal incidence since it is not able to reproduce dispersion.

The nanoparticles are very close to spheres and the small deviation due to the arrangement of the NP into lines cannot explain the split of the resonances. Due to the size variation of the NP, the resulting plasmonic resonance is inhomogeneously broadened. When the incident polarization is along X the particles do not couple strongly in the near-field. However, then the incident polarization is exciting the resonances along the grating lines, the near-field coupling between particles along the line is stronger and broadens the overall plasmonic resonance of the particle ensemble. The presence of the DIM mode induces a split into the plasmonic resonance by inducing an extra absorption feature inside the plasmonic absorbance. Therefore this results in an enhanced transmittance at the position of the DIM mode. Since our model does not take the destructive interference mode into account, the coupling strength between the DIM mode and the plasmonic absorbance results in the two modelled resonances L1 and L2, which do not represent independent resonances within the sample.

Funding

Chinese Scholarship Council (CSC) (201307040036).

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Pure circular dichroism by curved rows of plasmonic nanoparticles

Abstract

Introduction

1. Results and discussion

1.1 Sample description and transmittance

1.2 Mueller matrix characterization

3. Origin of intrinsic CD

4. Summary

Appendix 1 Experimental

Appendix 2 Analytical expressions for the DIM

Appendix 3 Ellipsometry model

Funding

References and links

Cited By

Figures (6)

Equations (5)

Optical Materials Express