1. Introduction

The color of metal surfaces is determined by their absorption and reflection properties. For example, uniform gold surfaces reflect in the yellow part of the visible spectrum more strongly than in the blue and green ones due to the onset of significant electronic absorption. As a result, color modifications in metals are difficult to achieve and have been primarily obtained by modifying the metal absorption properties using alloys or by coating them with thin layers of paints or dyes [1]. On the other hand, the strong enhancement of scattered light intensity associated to the resonant excitation of localized surface plasmons (i.e., LSPs) in nanostructured metal particles offers novel opportunity to engineer coloration effects in noble-metal films without the need of incorporating any extrinsic pigments or non-metallic materials. Recently, bright coloration of aluminum metal films has been demonstrated by nano and micro-patterning induced by femtosecond laser surface structuring [2, 3]. However, most structural coloration processes are limited to the fabrication of periodic or quasi-periodic nanostructures, which produce angularly dependent structural colors (i.e., iridescence) because of coherent Bragg scattering. This limitation, which prevents the applicability of metal nanostructures to display, optical identification and tagging technologies, can be overcome by engineering aperiodic nanostructured metal surfaces that simultaneously produce angle-insensitive (i.e., isotropic) and spatially uniform coloration in metal films. Recently, angle-independent structural colors have been observed in amorphous colloidal arrays [4, 5]. Several studies, inspired by the beautiful structural colors often encountered in Nature, have shown that amorphous nanostructures featuring short-range order in circularly symmetric Fourier space can result in vivid and angularly-insensitive chromatic responses such as the ones observed in the coloration of butterflies [6–10], beetles [11–13], birds [14], and seashells [15]. Two dimensional densely-packed ligament fibers in seashells [15] and random-close packing nanoparticles embedded in the scale structures of beetles [11] have been found to produce direction-independent structural colors due to the existence of isotropic photonic pseudo-gaps. Recently, Noh et. al. [14] discovered isotropic structural colors induced by multiple scattering in isotropic nanostructures with short-range order in the feather barb of the Coracias benghalensis. Moreover, detailed studies performed by Kinoshita et. al [7] and Yoshioka et al. [8] showed that the strong blue coloration of the Morpho butterflies’ wings (Morpho blue) originates from the interplay between coherent Bragg scattering in regularly arranged multilayers of “shelf” microstructures and incoherent scattering caused by the random height distribution of neighboring ridges. Artificial Morpho blue structures have been reproduced experimentally using three dimensional structures made of regularly arranged dielectric pillars with random height distribution [16, 17], and mimicking the scales of butterflies wings [18].

In this paper, by combining plasmonic resonances of metallic nanoparticles and incoherent light scattering from aperiodic arrays with isotropic and diffuse Fourier space, we demonstrate angularly insensitive structural coloration of metal films in deterministic (i.e., reproducible) metallic nanostructures. Recently, investigations of hyperuniform point patterns [19] and plasmonic aperiodic spirals [20] have provided approaches for the optimization and the generation of “amorphous-like” nanostructures featuring full rotational invariance in their Fourier space. The distinctive optical scattering properties of deterministic photonic-plasmonic structures with isotropic Fourier space are still largely unexplored, and provide unique opportunities for the engineering of structural coloration of metal-dielectric materials. In this paper, we will explore for the first time the angular light scattering properties of isotropic nanoparticle arrays based on a deterministic Pinwheel tiling and its more uniform counterpart (Delaunay-triangulated Pinwheel centroid, DTPC), which are homogeneous and statistically isotropic patterns [21]. Numerical design of angular light scattering from these novel plasmonic structures will be performed by a recently developed coupled-dipole method [22], which is suitable to model large multi-particles systems, in partnership with three dimensional (3D) Finite Difference Time Domain (FDTD) simulations.

The experimental demonstration of isotropic and homogeneous structural coloration in gold films with Pinwheel nanostructures is provided by angle-resolved reflection spectroscopy and dark-field scattering image analysis.

2. Coherent and incoherent scattering mechanisms

In order to understand how to engineer the angular scattering properties of complex arrays of nanoparticles, we will first introduce the important concepts of coherent and incoherent light scattering within the simplified kinematic theory of arbitrary spatial point patterns. This qualitative approach, which we will later substantiate by rigorous electrodynamical calculations in section 3, provides a physically transparent connection between the geometrical and the angular scattering properties of complex nanoparticle arrays.

When light waves interact with a nanostructured metal surface, they can be directly absorbed, reflected, or scattered in all directions. The scattered wave components, which are responsible for the coloration of most biological structures, can interact coherently or incoherently depending on their phase relationships, which depend on the spatial arrangement of the scatterers. In fact, if we consider for simplicity elastic wave scattering from an array of point particles within the single scattering approximation, we can immediately appreciate that the scattered intensity along a given direction is determined by the local spatial correlations in the positions of the particles, as described by the static structure factor:

3. Electromagnetic design of isotropic plasmon arrays

In this section, we will present the electromagnetic design of isotropic angular scattering from arrays of Au nanoparticles. Our simulation strategy consists of two steps. First, we will discuss angular scattering from arrays of Au spherical nanoparticles in the absence of a metallic substrate. This first step, which is computationally very efficient, will provide physical insights into the angular behavior of the electromagnetic scattering from arrays of different geometries. In a second step, we will consider the additional effects of non-spherical particle shape and of the presence of a metallic substrate by performing 3D-FDTD analysis of individual Au nanoparticles on a substrate, since isotropic scattering occurs in the incoherent scattering regime (i.e., dominated by the contribution of single particles). Clearly, this last step will determine the final structural color of the designed isotropic metal surface.

We will now discuss the out-of-plane angular distribution of the scattered radiation from arrays of Au nanoparticles by calculating their differential scattering cross section maps and angular radiation diagrams. Our computational study will be based on the Coupled Dipole Approximation (CDA), a numerical method that models each spherical scatterer of the array by a single electric dipole. The CDA is particularly suited to efficiently treat large-scale plasmonic systems made of small and well separated nanoparticles, and it has been previously validated against semi-analytical methods [22] used to describe complex nanoparticle arrangements [25]. In this paper, all the calculations are performed on Au nanoparticles with metallic dispersion modeled according to Johnson-Cristy data [26]. All the nanoparticles diameters are 100nm, and the minimum center-center inter-particle separation is kept fixed to 800nm. Moreover, all the scattering calculations presented in the paper, are performed, unless otherwise stated, using a linearly polarized plane wave normally incident on the array at the frequency of the maximum scattering efficiency.

The principal parameter of interest for the numerical analysis of angular scattering in complex plasmonic arrays is the full differential scattering cross section $\frac{d{C}_{sca}}{d\Omega }\left(\vartheta ,\phi ,\lambda \right)$, which describes the angular distribution of the electromagnetic power scattered at a given wavelength into a unit solid angle around the angular direction $\left(\vartheta ,\phi \right)$, per unit intensity.

In order to introduce the relevant concepts, we will first review the angular scattering behavior of a periodic square array and of a random array of Au nanoparticles. Specifically, we will calculate the scattering maps of the arrays, which are derived from the full differential scattering cross sections and represent the magnitude of the scattered fields on the far-field hemisphere projected into a scattering plane parallel to the array plane. In Fig. 1(a) and Fig. 1(b) we show the periodic array under investigation and its calculated scattering map [25], respectively. This map contains all the angular information in the backscattering of the array, as displayed in Fig. 1(b). The strong coherent Bragg scattering of the periodic array is manifested by the presence of sharp peaks in the scattering map, corresponding to the angular directions of constructive interference (i.e., grating orders). It is also evident from Fig. 1(b) that the scattering map closely resembles the lattice Fourier transforms of the array, even for array consisting of metal nanoparticles, as long as they can be considered within the “photonic regime” [25], where the particle separation is large enough so to prevent near-field interparticle coupling.

 

Fig. 1 Numerically investigated periodic (a) and random (d) arrays made of N = 3600 and N = 3500 spherical nanoparticles with 100nm diameter and 800nm center-center interparticle separation (b) Scattering maps (logarithmic scale) of the periodic (b) and random (c) arrays when they are excited by a linearly polarized plane wave at normal incidence.. The scattering angles can be calculated from the horizontal and the vertical axes values as cos²θ = max [0,1-x²-y²]; φ = tan⁻¹(y/x).

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In contrast to the case of periodic arrays, isotropic and incoherent light scattering is best exemplified by considering the array of randomly positioned nanoparticles shown in Fig. 1(d). This array is constructed by randomly placing 3500 particles into a square box of edge 10μm, under the constraint that the minimum distance between two particles is 800nm. The corresponding scattering map is shown in Fig. 1(c), and clearly features a high density of scattering directions with almost equal scattering intensity, consistently with the expected isotropic behavior of uncorrelated random systems.

We will now discuss the effects of metallic dispersion in the differential scattering from Au arrays. In the case of arrays composed of dispersive metal nanoparticles, the direction of light scattered from a particular structure is in general a function of the geometrical parameters of the array and of the wavelength of the incident light. This information is captured by calculating the averaged differential scattering cross section maps, ${〈\frac{d{C}_{sca}}{d\Omega }〉}_{\phi \in \left[0,2\pi \right]}\left(\vartheta ,\lambda \right)$, where the average is taken along the azimuthal angle φ and the intensity is normalized to the maximum value. These maps display very clearly the frequency dependent differential cross sections as a function of the observation angle (zenith) $\vartheta \in \left[0^{\circ },{90}^{\circ }\right]$, which varies between 0° (backscattering direction) and 90° (grazing scattering direction). The calculated scattering cross section maps for the periodic and the random array of Au nanoparticles are displayed in Figs. 2(a) and 2(d), in a wavelength range between 300nm and 900nm. The differential scattering maps vividly demonstrate for the random array the effect of plasmonic enhanced angular scattering, and show a much broader (i.e. angularly) scattering behavior in the spectral region around the plasmonic resonance of an isolated Au nanoparticle (i.e. between 500nm and 600nm). From the same analysis, it is also possible to extract the single wavelength angular scattering profiles of the arrays, or their radiation diagrams, which are shown in Figs. 2(b) and 2(c) at three different wavelengths λ_B = 475nm,, λ_G = 550nm and λ_R = 610nm (i.e corresponding to the blue, green, and red colors) for the periodic and the random structures, respectively. These radiation diagrams, plotted in dB scale, provide useful insights into the angular scattering resonant behavior of plasmonic arrays.

 

Fig. 2 Averaged differential scattering cross section${〈\frac{d{C}_{sca}}{d\Omega }〉}_{\phi \in \left[0,2\pi \right]}\left(\vartheta ,\lambda \right)$(dB) as a function of the observation angle (zenith) $\vartheta$ and of the incident wavelength λ for (a) Periodic and (d) Random structures, normalized to its maximum, when the arrays are excited by a linearly polarized plane wave at normal incidence. ${〈\frac{d{C}_{sca}}{d\Omega }〉}_{\phi \in \left[0,2\pi \right]}\left(\vartheta ,\lambda \right)$ (dB) as function of $\vartheta$ for three fixed values of$\lambda$: λ_B = 475nm,, λ_G = 550nm and λ_R = 610nm for (b) Periodic and (c) Random arrays, respectively.

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It is clear from this analysis that periodic plasmonic arrays give rise to colorimetric responses in the far-field are dominated by interference (Bragg-scattering) effects along certain directions. As a result, their structural color is always iridescent and angularly sensitive [27]. On the other hand, the lack of long-range order and particle correlations in random plasmonic arrays results in more isotropic colorimetric responses. However, the major drawback of random systems is their fundamental lack of reproducibility, which limits engineering applications. Moreover, the frequent occurrence of clusters of neighboring particles in a random system, which can be described by the methods of spatial statistics [28], reduces the spatial uniformity of the arrays, as can be observed in Fig. 1(d). These limitations can be overcome by developing isotropic plasmon scattering arrays based on deterministic aperiodic geometries, which offer the unique opportunities to tune both the structural correlations and to reduce particle clustering by well-defined homogenization procedures [28].

Deterministic aperiodic arrays are designed by mathematical rules, which interpolate in a tunable fashion between periodicity and randomness [29–31]. In particular, their reciprocal Fourier space (i.e., Fraunhofer diffraction pattern) ranges from a discrete set of δ-like Bragg peaks (i.e., pure-point spectrum), such as for periodic and quasiperiodic crystals, to a continuous spectrum (i.e., absence of Bragg peaks), as for random media. Moreover, they can encode rotational symmetries in either discrete or continuous Fourier spectra, as encountered in amorphous systems featuring isotropic scattering responses [20]. In this paper, we will focus on the angular scattering properties of two plasmonic arrays with Pinwheel geometry, obtained by a simple procedure that iteratively decomposes a triangle into five congruent copies. The regular Pinwheel arrays are generated by performing decompositions and inflation operations on a prototile, which is a right angle triangle with sides of length 1,2, and $\sqrt{5}$. In the first step, the prototile is divided into five copies and then these new triangles are expanded to the size of the original triangle. These decomposition and inflation operations are repeated ad-infinitum until the triangles completely cover the plane [21]. The resulting tiling, called Pinwheel tiling, has triangular elements (i. e., tiles) which appear in infinitely many orientations, and in the infinite-size limit, its diffraction pattern displays continuous (“infinity-fold”) rotational symmetry. Radin has shown that there is no discrete component in the regular Pinwheel diffraction spectrum [21].

The second isotropic array, which is called Delaunay triangulated Pinwheel centroid (DTPC), is obtained from a regular Pinwheel lattice by a homogenization procedure that performs a Delaunay triangulation [32] of the array and positions additional nanoparticles in the center of mass (i.e., baricenter) of the triangular elements. The DTPC, which is an example of a deterministic isotropic and homogeneous particle array, shares the same rotational symmetry of the regular Pinwheel array but features a more uniform spatial distribution of nanoparticles with strongly reduced clustering, as evident from Fig. 3 . In Figs. 3(a) and 3(d) we show the regular Pinwheel and the DTPC arrays, respectively. The corresponding calculated scattering maps are shown in Figs. 3(b) and 3(c). We notice the higher degree of spatial uniformity of the DTPC when compared to both the random (Fig. 1(d)) and the regular Pinwheel (Fig. 3(a)) arrays. Moreover, local structural correlations, which are due to the finite size of the regular Pinwheel array, results in well-defined scattering peaks in its scattering maps shown in Fig. 3(b). These correlations are absent in the scattering map of the DTPC array, shown in Fig. 3(c), due to its higher degree of spatial uniformity.

 

Fig. 3 Numerically investigated Pinwheel (a) and DTPC (d) arrays made of N = 3467 and N = 3604 spherical nanoparticles with 100nm diameter and 800nm center-center interparticle separation (b) Scattering map of the Pinwheel (b) and DTPC (logarithmic scale), when the arrays are excited by a linearly polarized plane wave at normal incidence. The scattering angles can be calculated from the horizontal and the vertical axes values as cos²θ = max [0,1-x²-y²]; φ = tan⁻¹(y/x).

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The wavelength dependent azimuthally-averaged differential scattering cross section maps and the corresponding single-wavelength radiation diagrams for the regular Pinwheel and the DTPC arrays are shown in Figs. 4(a) and 4(b), and 4(c) and 4(d), respectively.

 

Fig. 4 Averaged differential scattering cross section${〈\frac{d{C}_{sca}}{d\Omega }〉}_{\phi \in \left[0,2\pi \right]}\left(\vartheta ,\lambda \right)$(dB) as a function of the observation angle (zenith) $\vartheta$ and of the incident wavelength λ for Pinwheel (a) and DTPC (d) structures, normalized to its maximum, when the arrays are excited by a linearly polarized plane wave at normal incidence. ${〈\frac{d{C}_{sca}}{d\Omega }〉}_{\phi \in \left[0,2\pi \right]}\left(\vartheta ,\lambda \right)$ (dB) as function of $\vartheta$ for three fixed values of$\lambda$: λ_B = 475nm,, λ_G = 550nm and λ_R = 610nm for Pinwheel (b) and DTPC (c) arrays, respectively.

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The arrays simulated in Figs. 3 and 4 are made of 3467 and 3604 gold nanoparticles with 100nm diameter and 800nm minimum center-center inter-particle separation. The numerical results shown in Fig. 4(d) show that, with the exception of the backscattering direction, the ${〈\frac{d{C}_{sca}}{d\Omega }〉}_{\phi \in \left[0,2\pi \right]}$ of the DTPC array is angularly insensitive over a broad frequency range determined by the plasmonic response of the isolated Au particle, similarly to the behavior of the random array. On the contrast, the angular response of the regular Pinwheel array, shown in Fig. 4(a), features several well-defined angular scattering directions associated to the short-range structural correlations discussed above. The scattering radiation diagrams shown in Figs. 4(b) and 4(c) at the three different wavelengths of λ_B = 475nm,, λ_G = 550nm and λ_R = 610nm demonstrate that DTPC-type plasmonic arrays give rise to almost isotropic scattering, similarly to a random medium, but are deterministic and spatially more uniform.

Finally, having identified the array geometries that result in incoherent scattering from Pinwheel isotropic arrays, we can conveniently optimize their colorimetric responses by performing 3D FDTD simulations on a single Au cylindrical nanoparticle deposited on a gold film. FDTD simulations of the scattering efficiency of a 80nm-thick Au nano-disk with varying diameter in air and on a gold substrate are shown in Fig. 5(a) and Fig. 5(b), respectively. In all the presented simulations, a plane wave excitation was used at normal incidence to the substrate. Perfectly matched layers (PML) boundary conditions (absorption coefficient of 0.25) were imposed to ensure absorption of the electromagnetic radiation at the simulation boundaries. A grid spacing of 2.5 nm in x and y (in-plane) and 2 nm in z (out-of plane) has been used. As show in Fig. 5(b), the scattering peak shifts into the green spectral region for cylindrical nanoparticles deposited on a gold substrate, with a weak dependence on the particle diameters. Figure 5(c) demonstrates experimentally structural coloration in the green region by a dark-field scattering image obtained on nanofabricated Au nanoparticles with 300nm diameter atop a 100nm-thick gold film. These nanoparticles have been arranged in a periodic square array with 2.5μm lattice constant in order to prevent near-field interparticle coupling, and therefore avoid a shift in the single particle plasmonic peak. In the next section, we will discuss our experimental results on the isotropic green structural color of plasmonic arrays of Au nanoparticles with FDTD-optimized parameters for green scattering.

 

Fig. 5 (a) FDTD calculated scattering efficiency spectra of 80nm-thick Au nano-disks with varying diameter in air (a) and on a gold substrate (b). Measured dark-field scattering image (c) obtained on nanofabricated Au nanoparticles with 300nm diameter atop a 100nm-thick gold film.

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4. Experimental results

In order to experimentally demonstrate the predicted angularly insensitive coloration of Pinwheel plasmonic arrays on Au metal films, a numbers of devices were fabricated and studied by variable angle reflectance. The devices were fabricated on silicon substrates uniformly coated with 100nm of gold film deposited by electron-beam (ebeam) evaporation. 580nm of PMMA 950 (Poly Methyl Meth Acrylate) ebeam resist was spun on the gold-coated substrate and two-dimensional aperiodic arrays along with a periodic and a random reference arrays were written on the PMMA resist using a Zeiss SUPRA 40VP SEM equipped with Raith beam blanker and NPGS for nanopatterning. The total size of all the fabricated arrays is about 1x1mm² stitched by 16 (4x4) pixels of 250x250μm². The devices developed in 1:3 MIBK:IPA (Methyl isobutyl ketone: Isopropanol) solution for 70sec were then subjected to a second ebeam evaporation of 80nm of gold deposition. After lift-off in acetone solution, nanostructures consisting of 80nm-thick gold nanoparticles were obtained, as shown in Fig. 6 . All the nanoparticles in the arrays have 300nm diameter. We summarize in Table 1 the main structural parameters of the fabricated structures, namely their particle number, array area and density, minimum (d_min) and average (<d>) interparticle separations.The angular scattering properties of the fabricated nanostructures on metal surfaces were experimentally characterized by angle-resolved scattering spectra under normal incident excitation. The collection angle was varied within a range spanning from 20° to 80°, with a resolution of 5°. The samples were mounted in the center of a goniometer stage, normally illuminated by white light, and the scattered radiation was collected in the far-field zone using a long working distance lens (focal length = 30cm, acceptance angle of the collecting lens ± 2°), as illustrated by the schematics of Fig. 7(a) , using a spectrometer (Ocean Optics Jaz Module) coupled to an optical fiber. The fiber was aligned to the arrays using a low power HeNe laser prior to each angular measurement.

 

Fig. 6 Scanning electron microscope images of nanofabricated (a) periodic; (b) random; (d) Pinwheel; and (c) DTPC arrays of 300nm diameter and 80nm high Au nanoparticles with (a) 840nm interparticle separation; (b) 870nm; (d) 884nm; and (c) 800nm average interparticle separations on 100nm thick gold film.

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Table 1. Geometric Parameters Describing the Main Characteristics of Investigated Arrays

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Fig. 7 (a) Schematic of the experimental setup used for angle-resolved reflection. Microscope images of (b) periodic, (c) random, (e) Pinwheel, and (d) DTPC arrays collected in dark-field configuration under white light illumination.

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In the case of the periodic array, only half of the array (about 0.5x1mm²) was illuminated in order to keep approximately constant the total number of excited nanoparticles for all the investigated structures. Spatial filtering in the image plane was utilized to reject scattered light contributions from the surrounding unpatterned areas of the metal films. In Figs. 7(b)–7(d) we show representative images of the fabricated arrays collected under white light illumination using a dark-field microscope (custom made, 20 × objective) and a CCD digital camera (Media Cybernetics Evolution VF). In the case of the periodic grating shown in Fig. 7(b), the green coloration results from the collection of the first grating order has been optimized for maximum scattering efficiency in the 550nm spectral region. The dark-field scattering images collected for the aperiodic arrays in Figs. 7(b)–7(d) also demonstrate predominant green coloration for all types of arrays, irrespective of their geometries, proving the incoherent character of the scattering process.

In Fig. 8 we show the measured frequency spectra of the far-field reflected radiation from all the arrays, as well as their radiation diagrams. We notice that for the periodic array shown in Fig. 8(a) the peak wavelengths of the reflection spectra shift considerably to longer wavelengths by increasing the detection angle, as predicted for Bragg scattering by scalar diffraction theory. In contrast, the reflection spectra measured for the three aperiodic arrays, which are shown in Figs. 8(b)–8(d), feature a much broader (i.e. in frequency) response at all the detection angles, which are approximately centered in the 550nm-650nm spectral region. Moreover, unlike the periodic array, this broadband frequency response of the aperiodic arrays is almost insensitive to the detection angle, proving the incoherent nature of the plasmon-enhanced metal coloration mechanism discussed in section 3. This effect can be more immediately displayed by the measured radiation diagrams of the arrays, which are plotted as inserts in Fig. 8, at the three representative wavelengths of 475nm, 550nm, and 610nm. The incoherent nature of the plasmon-enhanced green coloration of Pinwheel arrays is further evidenced by their lower reflection intensity compared to periodic arrays. The maximum reflected intensity of the periodic array is approximately 10 times higher than the light intensity scattered by the aperiodic structures, which lack long-range spatial correlations. However, the measured radiation diagrams of the engineered Pinwheel arrays, displayed in Figs. 8(c) and 8(d), demonstrate an almost ideal isotropic scattering behavior originating from incoherent scattering in deterministic plasmonic nanostructures with high spatial uniformity and continuous Fourier spectra.

 

Fig. 8 Angle-resolved reflection spectra of the (a) periodic; (b) random; (d) Pinwheel; and (c) DTPC arrays of gold nanoparticles and their corresponding radiation diagrams in response to different detection angles (inert). All the gold nanoparticles are 300nm diameter and 80nm high fabricated on a 100nm gold film.

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5. Conclusions

Our results demonstrate angularly insensitive structural color from engineered aperiodic nanoparticle arrays with isotropic Fourier space on metal surfaces. In sharp contrast to the directional scattering response of periodic patterns, Pinwheel and homogenized DTPC gold nanostructures deposited on top of a uniform gold thin film display angularly insensitive green coloration based on plasmon-enhanced incoherent light scattering in deterministic systems. Our experimental results, obtained by dark-field scattering and angle-resolved reflection spectroscopy under white light illumination, are consistent with numerical simulations using the couple-dipole method. The engineered coloration mechanism of Au thin films presented in this paper promises to advance plasmonic applications to display, tagging and colorimetric sensing technologies.