1. Introduction

Resonant light scattering by nanoparticles for control and tuning of scattered light colors has lately been attracting a significant interest because of numerous potential applications and practical developments. The scattered light color manipulation can be utilized in different application domains such as color printing, displays and screens, optical filters, and optical data storage, etc. During recent years, it has been demonstrated that the surface plasmon resonance supported by metal nanoparticles can be used for generating a multitude of colors in scattered light, establishing eventually a new research field of structural (artificial) colors with various nanostructures being explored for the scattering color control [1–5].

Another approach to the scattering color control and manipulation is based on the Mie resonances in high-refractive index dielectric nanoparticles. It has been shown, for instance, that 150-200 nm nanoparticles exhibit strong light scattering with distinct colors due to excitation of the electric and magnetic dipole resonances in the visible spectral range [6–9]. Additionally, tuning the scattering color with silicon nanostructures [10] and color display effects on all-dielectric metasurfaces composed of high-refractive index building blocks have successfully been realized and demonstrated [11–13].

Recently a new approach for dynamic color control based on light scattering by high-refractive index spherical nanoparticles has been suggested [14]. Evanescent waves generated in the total internal reflection (TIR) configuration have been employed for illumination of silicon nanoparticles supporting the electric and magnetic dipole resonances. It has been shown, although only for spherical nanoparticles, that the orientation of the electric and magnetic dipoles induced in the nanoparticles can be manipulated by changing the polarization of the incident waves, resulting thereby in tuning of the scattered light colors [14].

Usage of evanescent waves for the polarization control of scattered light colors has several important advantages: (i) there is no background obscuring the effect, since, by virtue of the TIR configuration, only scattered light propagates away from the nanostructured surface (at the side opposite to the illumination), (ii) the spatial extent of evanescent waves (i.e., their overlap with scatterers) and their electric and magnetic field orientations with respect to the scatterers can be controlled by the incident angle and polarization of the incident waves, (iii) the physical system for generating evanescent waves includes a properly constructed interface that can be used and re-used with different scatterers and surface nanostructures. Note that the TIR configuration with suitable silicon nanophotonic metasurfaces can also be exploited for the light absorption control [15].

In this work, we develop the concept of color management in the resonant evanescent field scattering by high-refractive index cylindrical nanoparticles. The main idea is to advantageously exploit the well-established fact [16] that the spectral separation between electric and magnetic multipole resonances depends on the aspect ratio of cylindrical nanoparticles. For example, it was shown that, by changing the height of cylindrical silicon nanoparticles, it is possible to realize the electric and magnetic dipole resonances at the same wavelength [16], an important feature that was successfully applied for the development and experimental realization of Huygens’ metasurfaces [17,18]. Here, we also consider cylindrical silicon nanoparticles with large aspect ratio (i.e., nanodisks) by conducting numerical calculations of scattering cross sections and applying the multipole decomposition method for clarification and explanation of spectral resonances and color features. We demonstrate that the scattered light color can efficiently be controlled and tuned by the incident light polarization, with the relative contributions of nanodisk multipole moments playing a crucial role in the color formation.

The modelling approach presented in this work can be used for investigations of light scattering by single nanoparticles or small nanoparticle clusters for applications in color display and color printing technologies. In these cases, unlike the situation with infinite periodic arrays of identical nanoparticles, the far-field scattering of evanescent waves by nanoparticles will always be realized because of inhomogeneity of nanoparticle structures composed of differently sized nanoparticles and nanoparticle clusters.

 

Fig. 1 Schematic side view of the Cartesian coordinate system and physical system with TIR conditions. We consider that matter above the substrate is air.

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2. Physical system and theoretical background

2.1. Physical system

Schematic presentation of physical system considered in this paper is shown in Fig. 1. A silicon nanoparticle (nanodisk) is placed on the flat surface of a glass substrate with dielectric constant ${\epsilon }_1$. An incident light plane wave is propagated from the side of the substrate and reflected from the glass-air interface at the total internal reflection (TIR) conditions. As a result, the silicon nanoparticle interacts only with evanescent light waves located in air near the substrate surface. Incident angle is denoted by $\chi$, a scattering angle is denoted by $\theta$. TM (TE) polarization corresponds to the case when the electric field of the incident wave is parallel (perpendicular) to the incident plane. Here and in the following, we consider monochromatic fields with the time dependence defined by $\exp (-i\omega t)$, where $o\mathbf{m}ega$ is the angular frequency.

2.2. Multipole approach for light scattering near a flat substrate

Let assume that the external electric field of the incident waves induces the polarization P inside a dielectric scatterer which is considered to be a source of scattered waves [19]. If the scatterer is placed near flat substrate, the far-field scattered electric field in the above substrate region is determined by

${\mathbf{E}}_{\mathbf{r}}$ is the electric field of the scattered wave reflected from the substrate surface

Here $k_0$ is the vacuum wave number, $k_2=k_0\sqrt{{\epsilon }_2}$ (${\epsilon }_2$ is the dielectric constant of a medium above the substrate), $v_2$ is the light velocity in a medium with ${\epsilon }_2$, $\mathbf{n}=\mathbf{r}/\mathbf{r}$, $\tilde{\mathbf{n}}=\tilde{\mathbf{r}}/\mathbf{r}$, $\mathbf{r}=\left(x,y,z\right)$, $\tilde{\mathbf{r}}=\left(x,y,-z\right)$, $\mathbf{D}=p+i{k}_2\mathbf{T}/v_2$ is the total electric dipole (TED) of the scatterer [22]. The tensors $\widehat{R}\left(\mathbf{r}\right)$ and a detail consideration of transmission waves for an one-interface system can be found in Ref. [21].

The multipole moments in the above expressions are determined by the following expressions [22]:

The vector L and the value q in (2) are

The combinations $\nabla \nabla$, ${\mathbf{r}}^{″}\mathbf{P}$ and ${\mathbf{r}}^{″}{\mathbf{r}}^{\prime }\mathbf{P}$, ${\mathbf{r}}^{″}\mathbf{P}{\mathbf{r}}^{\prime }$, $\mathbf{P}{\mathbf{r}}^{\prime }{\mathbf{r}}^{″}$ represent the tensor products between corresponding vectors (reminding that ${\mathbf{r}}^{″}={\mathbf{r}}^{\prime }-{\mathbf{r}}_0$). The electric quadrupole $\widehat{Q}$, electric octupole $\widehat{O}$, and magnetic quadrupole $\widehat{M}$ tensors are used in the irreducible representations. It means that they satisfy both symmetric and traceless properties [24]. We consider multipoles located at the point ${\mathbf{r}}_0$ coinciding with the scatterer center of mass.

The scattering directivity above the substrate is determined by the differential scattering cross section ${\sigma }_d$ corresponding to the normalized power $\mathrm{d}{\mathbf{P}}_{\text{sca}}$ scattered into the solid angle $\text{dΩ}=\sin \theta \mathrm{d}\theta \mathrm{d}\phi$, where $\theta$ and $\phi$ are the polar and azimuthal angles of the spherical coordinate system, respectively [19,25]. Here, the scattered power is normalized by the incident wave intensity in free space. From this definition of ${\sigma }_d$ one obtains

2.3. Evanescent waves in TIR conditions

In the considered case the scatterer interacts with external evanescent waves generated above a glass substrate owing to the TIR effect. Polarization, phase and amplitude relations between the incident, reflection, and transmission electric field are presented elsewhere [20]. Here we use the relations from [26]. For a flat interface the transmitted electric field is

2.4. Numerical method

The distribution of polarization $\mathbf{P}\left(\mathbf{r}\right)$ inside a scatterer can be in general obtained only numerically. We apply the discrete dipole approximation (DDA) for this goal. In this approach the scattering object is presented as a cubic lattice of electric point dipoles with a polarizability ${\alpha }_p$. The corresponding dipole moment $p_j$ induced in each lattice point j (with the radius-vector ${\mathbf{r}}_j$) is found by solving coupled-dipole equations [19]. After numerical solution of the equations, the induced polarization $\mathbf{P}\left(\mathbf{r}\right)$ of the scatterer is

Inserting (20) into (1) one can calculate the electric field of the scattered waves. Moreover the discrete dipole representations of multipole moments, up to the magnetic quadrupole and electric octupole, are obtained by inserting (20) in the above integral expressions of the multipole definitions [16, 19].

Converting a spectrum to a color presentation is realized using a simple algorithm with the RGB spectral sensitivity curves: the intensity value of given spectrum at each wavelength is multiply by the sensitivity value of each RGB curve at this wavelength and then it is integrated over considered spectral range. The details of the converting procedure are the following. For a calculated spectrum $R\left(\lambda \right)$ we obtain the values $X=\int R\left(\lambda \right)x\left(\lambda \right)d\lambda$, $Y=\int R\left(\lambda \right)y\left(\lambda \right)d\lambda$, and $Z=\int R\left(\lambda \right)z\left(\lambda \right)d\lambda$, where $x\left(\lambda \right)$, $y\left(\lambda \right)$, $z\left(\lambda \right)$ are the the color-matching functions in the CIE1931 color space, an analytical approximation of which can be found in [27]. After normalization of $\left(X,Y,Z\right)$ by the maximal value from them, they are used as a color map row (with the first element specifying the intensity of red light, the secondgreen, and the third blue) in MATLAB realization.

 

Fig. 2 Spectra of the scattering cross sections into the semi-space above the glass substrate calculated for evanescent field scattering by the silicon nanodisk with diameters of 200 nm and height H. The evanescent fields are created due to the TIR configuration as shown in Fig. 1. Polarization of the incident waves and the nanodisk height are indicated in every panels. The separate (without interference) contributions of basic multipoles are also shown (TED corresponds to total electric dipole, MD - magnetic dipole, EQ - electric quadrupole, MQ - magnetic quadrupole).

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3. Results

The above theoretical approach is applied for investigations of multipole moment contributions into the resonance optical response of silicon disk (cylinder) nanoparticles located on a dielectric (glass) substrate and irradiated by evanescent waves created at TIR conditions (Fig. 1). The dielectric constant ${\epsilon }_1$ of the substrate is equal to 2.25. The dielectric constant ${\epsilon }_2$ above the substrate is equal to 1. The relative dielectric permittivity of crystal silicon is taken from [28]. In all calculations the incident angle $\chi$ is fixed and equal to ${75}^{\mathrm{o}}$ (Fig. 1). The disk geometrical parameters were chosen so as to ensure the presence of electric and magnetic dipole resonances, whose excitation depends strongly on the incident light polarization, in the visible spectral range.

Figure 2 demonstrates spectra of scattering cross sections into the semi-space above the glass substrate calculated for evanescent field scattering by the silicon nanodisk with diameters of 200 nm and height H. The separate (without interference) contributions of basic multipoles are also shown. Independently on the polarization of the incident waves and the nanodisk height, the scattering cross sections (the red curves in Fig. 2) include a broad resonance overlapping between electric (TED) and magnetic (MD) dipole contributions. With increasing of the height H the TED and MD resonances are shifted to the red side. Moreover, the magnetic quadrupole (MQ) resonant contribution increases in the scattering cross sections. In the condition of TE polarization the only in-plane (oriented along substrate surface) electric dipole moment is exited in the nanodisks, whereas magnetic dipole moment has both the in-plane and out-of-plane components. As a results, one can see two strongresonant peaks for MD contributions in Fig. 2c,e. For TM polarization MD contributions demonstrate the only one resonant peak which is shifted to the red side with increasing H (Fig. 2b,d,f). Importantly, for all cases presented in Fig. 2 (excepting Fig. 2e), the values of the scattering cross section are larger than the separate dipole and quadrupole contributions in the resonant spectral regions that corresponds to constructive interference between waves generating by the multipoles.

 

Fig. 3 The light color corresponding to the spectra presented in Fig. 2.

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Color presentations of the scattering cross section spectra of Fig. 2 are shown in Fig. 3. One can see that more color contrast due to switching between TE and TM polarizations of incident waves is realized for the nanodisk with $H=50$ nm. Such effect is explained by MD resonance behavior: for the nanodisk with small height, the in-plane MD resonance can be excited only in the short wavelength regions ($\lambda \approx 480$ nm in Fig. 2b), whereas the out-of-plane MD resonance is realized for larger wavelengths ($\lambda \approx 640$ nm in Fig. 1a), the excitation of the out-of-plane MD resonances with switching TM to TE provide considerable change of scattered light color (Fig. 2, $H=50$ nm). For nanodisks with height of 75 and 100 nm, the spectral resonant regions become broader and shift to the red side for the both polarizations (Fig. 2) and the color switching between the TE and TM cases is realized basically in yellow-orange range (Fig. 3).

 

Fig. 4 Spectra of the light scattering cross sections into the conical region with $\theta ={40}^{\mathbf{o}}$ (Fig. 1) above the glass substrate calculated for the silicon nanodisk with the diameter of 200 nm and height $H=50$ nm and for different polarization angle $\psi$ of the incident light waves.(From the right hand side) evolution of the scattered light color with changing polarization angle $\psi$ of the incident waves.

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Detail polarization dependence of the scattering cross sections into conical sector, as shown in Fig. 1, for a nanodisk with $H=50$ nm and $\mathbf{D}=200$ nm is shown in Fig. 4. One can see that with decreasing of the angle $\psi$, determining the orientation of incident electric field with respect to the incident plane, the spectral maximums change smoothly its positions from the “blue” side ($\lambda <500$ nm) to “green” side ($\lambda >500$ nm). The transformation of the scattering cross sections with the angle $\psi$ is connected with spectral behavior of TED and MD resonances (Fig. 5). At the conditions of TM polarization ($\psi ={90}^{\mathrm{o}}$) the blue color is basically determined by the in-plane MD scattering (as was discussed above). Role of out-of-plane TED moment is not so important for scattering color formation because the out-of-plane TED moment does not radiated perpendicular to the substrate surface to the far-field zone. With decreasing of $\psi$ (TM$\to$TE) the role of in-plane component of TED increases, providing appearance of broad TED resonance in the green side, the value of the in-plane MD resonance in the scattering cross section is decreased (Fig, 5b, $\lambda \approx 480$ nm), as a result the blue-green switching is realized (Fig, 4). Note that the minimum of the TED contribution at $\lambda \approx 490$ nm for $\psi =0$ in Fig. 5a corresponds to the anapole state [29] when electric-dipole scattering is suppressed due to destructive interference between waves generated by electric p and toroidal T dipole moments.

 

Fig. 5 Spectra of (a) TED and (b) MD contribution in the light scattering cross sections into the conical region with $\theta ={40}^{\mathbf{o}}$ (Fig. 1) above the glass substrate calculated for the silicon nanodisk with the diameter of 200 nm and height $H=50$ nm and for the different polarization angle $\psi$.

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4. Conclusion

Summarizing, we have applied the multipole decomposition method for studies of the resonant optical responses of cylindrical silicon nanoparticles (nanodisks) illuminated by evanescent light waves in the TIR configuration. Considering silicon nanodisks placed on a glass substrate surface in air environment, we have conducted the color analysis of scattered (into the semi-space above the substrate) light for different polarizations of incident light generating the evanescent illumination of nanodisks. It has been demonstrated that scattered light color can efficiently be controlled and tuned by changing the incident light polarization. It has also been shown that the scattered light color depends significantly not only on the incident light polarization but also on the nanodisk aspect ratio. Using the multipole decomposition method we have revealed that the width and spectral tuning of scattering resonances for different incident light polarizations are mainly determined by the contributions of the in- and out-of-plane components of the magnetic and electric dipole moments. For relatively large nanoparticles, the excitation of the corresponding quadrupole moments results in the broadening of the resonance spectra, decreasing thereby the scattered light color contrast between the cases of the TE and TM polarized illumination. We believe that the results obtained clarify the main physical mechanisms involved in the evanescent scattering color formation and suggest new avenues for and exciting prospects in the development of color printing and multicolor displays. Since the calculated scattering cross sections are determined through normalization of the scattered power by the time-averaged Poynting vector of incident radiation, they can be used to estimate the scattered power for a given incident beam power in experimental realization.

Important aspect for future investigations is related to the consideration of finite-sized nanoparticle clusters and regular arrays of nonidentical nanoparticles (metasurfaces) in the TIR configuration. Taking into account that a typical individual nanodisk considered in this work scatters about one third of the incident (directly on the disk) light power, subwavelength nanodisk structures should be very efficient in light scattering, although some modification of colors might be expected due to the near-field coupling [30]. Finally, we would like to note that the efficiency of polarization control of colors can also be influenced by the choice of the substrate material and the angle of incidence, both of which controlling the evanescent field penetration into the upper half-space, but the corresponding investigations go beyond the framework of this study.