Localized surface plasmons (LSPs) of metallic nanostructure arrays have been recognized as an optical tool in the design of color filters by improving color gamut, saturation, and mechanical stability. In the reflectance spectra of these types of arrays, which define color, LSP and bulk contributions co-exist, and even though there have been numerous reports in this field, the chromatic limits of both contributions have not been reported. In this study, we investigate the relative contributions of LSP and bulk to the color of arrays. Using numerical simulations, the reflectance spectra of hexagonal arrays of gold nanodisks are investigated in terms of the dimensionality of the array. With a phenomenological approach based on the fitting of reflectance spectra with Gaussian and baseline functions, LSP and bulk contributions to color are separated and quantified. The results unveil the crossover between the yellowish color of gold, the signature of bulk concentration, and the reddish color of nanostructures.

Gold and noble metals are materials that exhibit a strong-light absorbance in the UV and part of the visible region of the spectra (from 380 to 460 nm, which include the violet and blue colors) due to strong inter-band transitions.1 Therefore, gold reflects a yellowish color under white light illumination. Recently, extending the color gamut of this metal toward shorter wavelengths of the visible band has attracted significant research interest2 as it sets the scene for the synthesis of color filters with improved color saturation, enhanced brightness, and mechanical stability.3,4

Noble metal nanostructures, such as nanodisk or nanosphere arrays, are used to extend the color of metals. Metallic nanostructures are resonant systems that couple light through the oscillations of surface electrons called localized surface plasmons (LSPs). They introduce resonant reflectivity modes different from those of the bulk,5 which lead to new ways of manipulating the optical properties. Previous studies using nanostructures have been reported; for instance, Zhao et al. proposed a transmissive filter with enhanced brightness properties by coupling the resonance of silver nanodisks to the extraordinary transmissive resonances of a nano-holed silver film.6 Wang et al. synthetized reflective color filters that return high saturated colors based on two arrays of silver nanodisks coupled through a tandem structure.7 James et al. fabricated large areas of reflective filters that produce a wide color gamut using aluminum nanorods coupled to their Babinet counterpart screen.8 

Despite numerous reports of reflective color filters based on plasmonic metallic nanostructure arrays, there has not been reported yet a technical contribution in which the relative contribution of LSP and bulk to the reflective color of single-metal nanostructures is identified and quantified (although their localized resonances are well understood).5,9 Understanding how much the relative contribution of both these mechanisms changes the color of samples is a key issue that deserves to be specifically addressed, particularly those of the gold nanostructures that exhibit high quality LSP resonances.

In this article, the authors investigate the color gamut, which displays a hexagonal array of gold nanodisks, and analyze the reasons behind their response. To achieve this goal, we first investigated the LSP contribution by obtaining the reflectance spectra of hexagonal arrays. The LSP contribution is manifested through the reflective modes associated with LSPs, which together with the bulk contribution defines the color of the arrays. The optical properties of the arrays, reflectance, and absorbance are obtained with numerical simulations based on the finite-element method. The reflectance spectra were analyzed and fitted to identify and separate the surface plasmon and bulk contributions. The spectral position of plasmons can be tuned by modifying the height, diameter, and pitch of disks; thus, the effect of these parameters on reflective colors can be determined. In particular, the natural yellowish color of gold, the signature of bulk contribution, and the reddish color related to gold nanostructures present clear trade-offs, which establish the chromaticity color limits of the arrays.

The array of gold nanodisks proposed in this study is shown in Fig. 1. It consists of a hexagonal array on the top of a silicon dioxide substrate. This type of geometry is addressed due to its high coordination number, which plays a role in the tuning of LSPs. The substrate is considered as a half-space silicon dioxide region. The arrangement is specified by the height t, the diameter d, and the center-to-center distance a of disks. We have chosen this dielectric as a substrate due to its transparency in the visible band10 to allow for the detection of LSP resonances. It is worth noting that the air–substrate interface contributes to the total reflectance and the silicon dioxide shifts the resonance wavelength of disks toward longer wavelengths.11 

FIG. 1.

(a) Schematic representation of the periodic lattice hexagonal array of gold nanodisks and (b) schema of the unit cell used to build the numerical model. For this study, the parameters t and a were kept fixed at 15 and 200 nm, respectively, while the d parameter was varied in the range from 10 to 180 nm (steps of 10 nm). For those array dimensions, the LSPs appear in the visible range.

FIG. 1.

(a) Schematic representation of the periodic lattice hexagonal array of gold nanodisks and (b) schema of the unit cell used to build the numerical model. For this study, the parameters t and a were kept fixed at 15 and 200 nm, respectively, while the d parameter was varied in the range from 10 to 180 nm (steps of 10 nm). For those array dimensions, the LSPs appear in the visible range.

Close modal

To determine the wavelengths reflected and absorbed by the arrays, numerical simulations were run using the COMSOL Multiphysics® software (ver. 5a, COMSOL, Inc., MA, USA) based on the finite-element method12 and a numerical model was built to enable the simulation of the large number of nanostructures. This was achieved using a single hexagonal unit cell, in which symmetric periodic Floquet boundary conditions (PBCs) were defined at the lateral surfaces.13 The optical properties of the silicon dioxide and gold defined in the platform consist of complex refractive indices of materials reported in the literature within the 200–1000 nm range.10,14 We have defined an emitting periodic port (with an arbitrary power of 1 [W]) at the unit cell top surface to illuminate the nanostructures with monochromatic plane waves. In addition, a receiving port was defined at the unit cell bottom surface to sense the arriving light, 100 nm below the metal–substrate interface. It should be noted that the position of this port does not affect the measurement of transmitted light since the substrate behaves as a non-absorbing material.

The reflectance of the systems can thus be obtained by performing a two-port analysis.15 From this analysis, the platform generates scattering parameters or S-parameters,12 which are directly related to reflectance and transmittance, as follows:
R=S112,
(1)
T=S212.
(2)

The color that samples of nanostructures exhibit can be accurately determined from their reflectance spectra with a chromatic analysis. This analysis consists of calculating the coordinates of chromaticity x and y related to the two-dimensional CIE color space, a conceptual frame developed by the International Commission on Illumination (CIE).16 With these coordinates, it is possible to describe all colors related to the perception of the human eye.

To obtain the chromaticity coordinates of an array, we first evaluate the tri-stimulus numbers X, Y, and Z.17 These numbers are the relative contribution of light to the three spectral functions to which an observer is sensitive, called matching functions x(λ), y(λ), and z(λ). The matching functions represent the amount of light from the red, green, and blue primaries. The matching functions used to conduct the chromaticity analysis were taken from the values reported in the literature.17 Any reflectance spectra can be decomposed into tri-stimulus numbers by integrating the matching functions as follows:
X=380780R(λ)P0x(λ)dλ,
(3)
Y=380780R(λ)P0yλdλ,
(4)
Z=380780R(λ)P0zλdλ,
(5)
where the parameter P0 denotes the power of the source of illumination (in this article, P0 corresponds to the power of the emitting port). Once the tri-stimulus numbers are obtained, the chromaticity coordinates are directly evaluated with the following relationships:
x=XX+Y+Z,
(6)
y=YX+Y+Z.
(7)

To begin, we present the change of color exhibited by gold films on SiO2 substrates as their thickness t is decreased along the nanometer scale (in the range from 5 to 200 nm) beyond the skin depth, when considered as thin films (e.g., δAu ∼ 100 nm at λ = 550 nm). In this case, the reflectance of gold is dominated by the volume contribution, allowing us to show how much its color can change just by this article. The aim is to provide elements to identify the volume contribution when we address the case of gold nanostructure arrays (in Secs. III B–III D).

Gold films thicker than 200 nm are considered as a bulk material as they exhibit the same optical properties as bulk, including the yellowish color. The numerical reflectance of these films is shown in Fig. 2(a), in a black solid line. For wavelengths longer than 550 nm, the numerical reflectance is higher than 80%. This is because the real part n of the complex refractive index decreases (n* ≈ 0.4 + 2.4j, at 550 nm), while the imaginary part k increases, introducing a strong refractive index mismatch between the air and the metal. In this regime, the layer absorbance is less than 10% [see the blue dashed line in Fig. 2(a)]. For wavelengths shorter than 550 nm, the film reflectance is maintained below 40%, and the light passing through the air–gold interface is absorbed by the film (A > 40%). Numerical values are accurately described by the Fresnel equations, extended for the case of three media and two interfaces.18 

FIG. 2.

Optical properties of gold thin films on the top of SiO2 substrates. (a) Reflectance of a 200 nm thick gold layer considered optically opaque. Complete characterization of the (b) reflectance and (c) absorbance of thin gold layers, as a function of thickness and light wavelength. The gray and black lines on the surface color maps correspond to the films with thickness t = 10 nm and t = 100 nm, respectively.

FIG. 2.

Optical properties of gold thin films on the top of SiO2 substrates. (a) Reflectance of a 200 nm thick gold layer considered optically opaque. Complete characterization of the (b) reflectance and (c) absorbance of thin gold layers, as a function of thickness and light wavelength. The gray and black lines on the surface color maps correspond to the films with thickness t = 10 nm and t = 100 nm, respectively.

Close modal

The color of these thick layers is described with the chromaticity coordinates x = 0.403 and y = 0.388, related to the CIE space. These coordinates were obtained by using the reflectance spectra of the 200 nm thick film presented in Fig. 2(a) and through Eqs. (3)(7). From a phenomenological point of view, these numbers have their origin in the step-function shape of the reflectance curve. For this shape of reflectance, the weight of the matching function x(λ) · R(λ), representing the red primary, dominates over the green and blue contributions y(λ) · R(λ) and z(λ) · R(λ), respectively; the matching function peaks appear at ∼600, 542, and 446 nm, respectively.17 

The color and chromaticity coordinates of a gold layer can be changed by reducing the thickness beyond the skin depth of gold δAu. The reflectance of films with a thickness below 200 nm is shown in Fig. 2(b). A surface color map is used to show the dependence of reflectance on both parameters: the thickness of the film and the wavelength of the incident light. The results show that layers thicker than 100 nm keep their bulk properties. However, for thinner layers, the step-like function change in amplitude and the related red-to-violet asymmetry decrease. The amplitude of the thinnest layers changes since light transmission through the gold–substrate interface is enhanced by evanescence waves, before being fully absorbed by films; it should be noted that films are much thinner than the metal skin depth. These changes are noticeable for red wavelengths since the absorption coefficient is lower in such a wavelength regime. The absorbance spectra of layers shown in Fig. 2(c) confirm that a small amount of light is trapped by the thinnest films.

The changes in the chromaticity coordinates with the film thickness t are shown in Fig. 3(a). The results show that films thicker than 100 nm display the bulk coordinate numbers. However, the coordinates decrease as the thickness t decreases, reaching values around x = 0.34 and y = 0.31, for the 5 nm thick film. The chromaticity changes can be described in terms of the step-like reflectance curve. At those dimensions, the red-to-violet asymmetry of the reflectance spectra decreases, increasing the relative weight of the green y(λ) · R(λ) and blue z(λ) · R(λ) primaries’ matching functions to the color of samples.

FIG. 3.

(a) Chromaticity coordinates and red-to-violet asymmetry of gold layers with different thicknesses and (b) color evolution of gold layers on the CIE space.

FIG. 3.

(a) Chromaticity coordinates and red-to-violet asymmetry of gold layers with different thicknesses and (b) color evolution of gold layers on the CIE space.

Close modal

To quantify the red-to-violet spectral asymmetry, we evaluate the changes in reflectance between the 780 nm red and 380 nm violet wavelengths as follows: δR/R0(%) = (R780 nmR380 nm)/R780 nm × 100. The dependence of asymmetry on the film thickness shown in Fig. 3(a) matches those of the chromaticity coordinates. Finally, the CIE color space shows how layers shift their yellowish color toward a natural reddish chromatic limit, as shown in Fig. 3(b).

When gold films are nanostructured into arrays of nanodisks, additional mechanisms of light scattering and absorption are introduced by LSPs. These mechanisms co-exist with the volume contribution. In this section, we identify and separate the impact that these mechanisms have on the color of gold nanostructures.

To do so, it is convenient to first analyze the case of dilute systems of nanodisks since the optical properties of plasmons can be clearly identified. To meet the dilute conditions, we addressed the case of disks with a diameter d and a height t of 80 and 15 nm, respectively, integrated into a hexagonal array with an inter-disk distance a of 200 mm. A schema of the array is shown in Fig. 4(a). The absorbance and reflectance of this system are shown in Fig. 4(a) in red and black solid lines. For comparison purposes, the properties of a 15 nm thick film are also shown in dashed lines.

FIG. 4.

Optical properties of dilute arrays of dilute gold nanodisks. (a) Reflectance (black line) and absorbance (red line) of an array with parameters d = 80 nm, t = 15 nm, and a = 200 nm; the dashed lines indicate the optical properties of a continuous 15 nm thick film. Phenomenological fitting of the (b) absorbance and (c) reflectance of a dilute array, separating the contribution of the bulk (baseline) and surface plasmon resonant modes (Gaussian curve).

FIG. 4.

Optical properties of dilute arrays of dilute gold nanodisks. (a) Reflectance (black line) and absorbance (red line) of an array with parameters d = 80 nm, t = 15 nm, and a = 200 nm; the dashed lines indicate the optical properties of a continuous 15 nm thick film. Phenomenological fitting of the (b) absorbance and (c) reflectance of a dilute array, separating the contribution of the bulk (baseline) and surface plasmon resonant modes (Gaussian curve).

Close modal

In the dilute regime, the disks behave as a system of point dipoles with low bulk absorption, promoting the transmission toward the substrate (the transmittance is close to that of an air–SiO2 interface) for most wavelengths. However, due to the localized plasmons, light is absorbed by the effects of surface charge oscillations at the resonance wavelength. The resonance wavelength is thus indirectly determined from the peak of the absorbance in Fig. 3(a) as λLSP ∼ 635 nm. This value agrees with the results published in previous reports.19 The reflectance exhibits an amplitude peak known as resonant reflectance, which is defined as a resonant mode of light reflection that is assisted by the localized plasmons. The reflectivity mode appears at ∼640 nm. This interaction is reported for a variety of systems, such as metallic nanospheres or nanowires.1 Due to the resonant reflectivity modes, the dilute array exhibits a reddish-like color. Their chromaticity coordinates in the CIE spaces are determined by the values x = 0.48 and y = 0.34.

It should be noted that the volume contributions coexist with that of plasmons, despite their low amplitude. To separate both contributions, we consider a simplified phenomenological point of view based on the fitting of the absorbance and reflectance spectra using trend functions. For instance, the absorbance fitting was performed as follows: (i) the optical power coupled to the volume of disks was related to a baseline fitting, while (ii) the optical power coupled to the plasmonic resonance was related to a Gaussian fitting. The fitting in Fig. 4(b) shows that the baseline exhibits the same step-like absorbance trends that of a 15 nm film, which is strong for wavelengths <600 nm (due to the intra-band transitions of electrons), whereas the Gaussian function matches the damped harmonic oscillation of the electrons.

The impact of the LSPs on the color of samples can be understood by fitting the reflectance in a similar way. A step-like baseline is used to fit the bulk reflections, whereas a Gaussian function is related to the resonant reflectivity modes. The baseline by itself weakly reflects the yellow and red bands, whereas the resonant reflectivity mode strongly reflects the orange and red bands. The relative contribution of mechanisms can be weighted by adjusting the array geometrical parameters (t, d, and a) as will be shown below, given rise to the chromaticity limits of a gold array of nanostructures.

As mentioned above, the impact of bulk and LSPs on the reflected color can be adjusted by tuning the diameter of disks. There is a narrow band of diameter sizes in which the plasmons determine the color of the sample. This occurs only for dense arrays. To demonstrate that, the absorbance and reflectance of arrays with different diameters tuned in the 15–200 nm range are shown in Fig. 5 (t and a are fixed at 15 and 200 nm, respectively).

FIG. 5.

Optical properties of dense arrays of gold nanodisks: (a) absorbance and (b) reflectance as a function of the diameter of disks and the wavelength of the incident light. (c) Chromaticity coordinates of a set of arrays with different diameters of disks, and (d) the color evolution of the arrays on the CIE space.

FIG. 5.

Optical properties of dense arrays of gold nanodisks: (a) absorbance and (b) reflectance as a function of the diameter of disks and the wavelength of the incident light. (c) Chromaticity coordinates of a set of arrays with different diameters of disks, and (d) the color evolution of the arrays on the CIE space.

Close modal

The absorbance spectra unveil how LSPs and volume couple light as the disk’s diameter is increased. From the spectra shown in Fig. 5(a), the light with wavelengths shorter than <520 nm is absorbed by the disks’ volume, regardless of the diameter. LSPs appear at wavelengths longer than 520 nm; thus, the resonance of the smaller disks approaching the bulk plasma frequency is λ(ωP) ∼ 520 nm. The LSPs shift toward longer wavelengths as the diameter increases; for chosen d sizes, they shift along the visible band, from ∼540 to 720 nm. The width of the plasmon spectra becomes wider as the diameter increases as well. The amplitude of LSPs increases with the diameter; however, this occurs up to d = 80 nm. For wider diameters, the LSPs’ amplitude decreases being overridden by the volume absorption, meaning that the reflectance recovers the response of a 15 nm thick gold film. Then, it should be noted that the localized plasmons reside in the 520–720 nm wavelength band.

The related reflectance spectra of the arrays are shown in Fig. 5(b) to identify the resonant reflectivity modes and their impact on color. First, the spectra show a clear peak related to the localized plasmons that shifts toward red wavelengths as the diameter is increased. This peak shifts along the visible band, from 520 to 720 nm, matching the absorbance peaks. Second, the amplitude of the reflectivity modes always increases with the diameter, reaching values around 80% for the wider disks. However, as the diameter increases, the reflectance of volume also increases. Since the bulk/volume reflectance is maximum for wavelengths >520 nm, its contribution will override that of plasmons for the wider disks, as shown in Fig. 5(b). Finally, there is no efficient reflective mechanism in the 200–520 nm wavelength range as light is absorbed inside the volume and the reflective modes by plasmons appear for wavelengths >520 nm.

To quantify the chromaticity limits of the arrays, the coordinates related to their reflectance spectra are shown in Fig. 5(c). The modulation of the diameter of the arrays has a notorious impact on the chromaticity coordinate x, which differs from that of a continuous 15 nm thick film (∼0.39). The x coordinate of the arrays covers the range of values from 0.33 to 0.48, when the diameter is increased from 10 to 90 nm. For larger diameters, the x coordinate decreases toward a continuous film. In contrast, the y coordinate shows small variations in the range of values from 0.33 to 0.36, remaining close to the y value of a 15 nm thick film (∼0.34). The CIE color scale in Fig. 5(d) shows the color that the arrays can exhibit under the illumination of white sources, changing from red to orange and, then, from orange to yellow.

Finally, the optical properties of the array of nanodisks change with the thickness of the disks t. In particular, the localized plasmons occur at shorter/longer wavelengths as the height of the disks increases/decreases since the dipolar interaction between their faces gets weaker/stronger. This fact is, indeed, a useful strategy for tuning the chromaticity limits of the arrays previously discussed. The reflectance spectra, the chromaticity coordinates, and the CIE color scale of nanodisk arrays with parameters t = 30 nm and a = 200 nm are shown as a function of the diameter of disks in Figs. 6(a)6(c), respectively.

FIG. 6.

Optical properties of thicker arrays of gold nanodisks: (a) reflectance spectra as a function of the diameter of disks and the wavelength of the incident light. (b) Chromaticity coordinates of a set of arrays with different diameters of disks and (c) the color evolution of the arrays on the CIE space.

FIG. 6.

Optical properties of thicker arrays of gold nanodisks: (a) reflectance spectra as a function of the diameter of disks and the wavelength of the incident light. (b) Chromaticity coordinates of a set of arrays with different diameters of disks and (c) the color evolution of the arrays on the CIE space.

Close modal

The reflectance spectra of the thicker arrays exhibit reflective modes that shift toward shorter wavelengths. However, they occur in the wavelength range of 540–595 nm, covering a shorter band of the visible spectra. The chromaticity coordinates related to the reflectance spectra show that the y coordinate can be modulated in a wider range of values than those of thicker arrays. This fact results in an extended color gamut as shown in the CIE space in Fig. 6(c).

To conclude, we present in Table I a summary of the main results obtained in this article. There, the changes in the chromaticity coordinates with the geometrical parameters are presented for the case of gold thin films and arrays of nanostructures. From summary, it can be highlighted that the coordinates of the films undergo slight changes with thickness. In contrast, when nanostructure arrays are used, LSPs introduce a peak in the reflectance spectra of samples, which has more noticeable effects than thin films on the change in the x coordinate. It can also be identified the dimensions of the arrays for which the LSP contribution becomes relevant.

TABLE I.

Change in chromaticity coordinates with the geometry dimension of gold films and arrays.

SystemParameterRangex rangey range
Thin films Thickness t From 5 to 100 nm From 0.3457 to 0.403 From 0.348 to 0.388 
From 100 to 180 nm 0.403 0.388 
Array of nanodisks Diameter d (a = 200 nm, t = 15 nm) From 10 to 50 nm From 0.331 to 0.368 (LSP appears) From 0.331 to 0.362 (LSP appears) 
From 50 to 80 nm From 0.368 to 0.48 From 0.362 to 0.341 
 (LSP dominates coordinate) (LSP dominates coordinate) 
From 80 to 120 nm From 0.48 to 0.451 From 0.341 to 0.338 
 (LSP dominates coordinate) (LSP appears) 
From 120 to 180 nm From 0.451 to 0.427 0.34 (volume dominates coordinate) 
 (volume dominates coordinate)  
SystemParameterRangex rangey range
Thin films Thickness t From 5 to 100 nm From 0.3457 to 0.403 From 0.348 to 0.388 
From 100 to 180 nm 0.403 0.388 
Array of nanodisks Diameter d (a = 200 nm, t = 15 nm) From 10 to 50 nm From 0.331 to 0.368 (LSP appears) From 0.331 to 0.362 (LSP appears) 
From 50 to 80 nm From 0.368 to 0.48 From 0.362 to 0.341 
 (LSP dominates coordinate) (LSP dominates coordinate) 
From 80 to 120 nm From 0.48 to 0.451 From 0.341 to 0.338 
 (LSP dominates coordinate) (LSP appears) 
From 120 to 180 nm From 0.451 to 0.427 0.34 (volume dominates coordinate) 
 (volume dominates coordinate)  

In summary, the optical properties of an array of gold nanodisks were investigated by simulations to quantify their chromaticity limits and understand the reasons behind them. From the reflectance spectra of the arrays, the contributions of the localized surface plasmons and the volume of disks to the color of the systems were identified and separated by using a simple Gaussian and baseline phenomenological fitting. The results unveil that the reflectivity modes assisted by the LSPs occur at wavelengths longer than 540 nm (above the plasma frequency of gold), and they define the color of the arrays when their diameter is about 90 nm. In contrast, the color of arrays with different diameters is dominated by the volume/bulk contributions, overriding that of the LSPs. In addition, the chromaticity analysis of the reflectance spectra shows that the modulation of the diameter has a notorious impact on the chromaticity coordinate x, while the y coordinate remains close to the bulk value.

The authors acknowledge the support provided by the Western Institute of Technology and Higher Education, ITESO AC.

The authors have no conflicts to disclose.

Anahí Gutiérrez: Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal). Mayra Tapia-Contreras: Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal). Edgar Briones: Conceptualization (lead); Investigation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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