On the correlation of absorption cross-section with plasmonic color generation

Soroosh Daqiqeh Rezaei; Jinfa Ho; Ray Jia Hong Ng; Seeram Ramakrishna; Joel K. W. Yang

doi:10.1364/OE.25.027652

1. Introduction

Nanoparticles can possess absorption cross-sections that are several times larger than their geometrical cross-sections. For example, a 13 nm diameter spherical aluminum particle can have an absorption cross-section as large as eighteen times its physical size [1]. This phenomenon is observed in numerical simulations as the appearance of “funneling” field lines of incident light into the particle within a large area beyond the physical boundaries of the particle. The significance of this phenomenon is in the possibility of absorbing all incident light within a target wavelength range with sparsely distributed particles, and has relevance in photovoltaic devices [2], glazing materials [3,4], and plasmonic color printing [5].

Plasmonic colors possess attractive attributes such as fade-resistant colors and printing at resolutions beyond the light diffraction limit, which are enabled by light manipulation at sub-wavelength dimensions [6–8]. Various structures are used in plasmonic color generation such as nanoantennas above a back-reflector [8], nanoantennas on transparent substrates [9,10], all-metallic nanostructures [9,11,12], hole arrays [13–16] and plasmonic diffraction gratings [17]. Here, we investigate a gap-plasmon structure [18,19] which has an enormous absorption cross-section up to twelve times the geometrical cross-sectional area. We investigate the relationship between the absorption cross-section and the optimum pitch size to obtain saturated colors. In addition, such large absorption cross-sections enable color mixing at the individual nanostructure level. We use aluminum (Al) due to its low cost and high stability relative to silver (Ag) and gold (Au) [20–22], though the approach is applicable to other materials.

Unlike dye-based printing, where a broad range of colors can be achieved by half-toning and overlaying techniques, plasmonic colors requires that the constituent nanostructured pixels be printed side-by-side. Attempts at stacking plasmonic colors [23] show that plasmon coupling between the individual layers produce colors that are not simply subtractive in nature. For instance, while overlaying cyan, magenta, and yellow inks on a printed substrate is one technique for producing black in desktop printers, one finds that sequentially stacking layers of metallic nanostructures cycles through red, blue, and green colors [23]. Hence, variation of nanostructure geometry and combination of nanostructures within a given pixel area [8,24] are approaches to produce a broad color range. Though it has been shown [24] that diluting the area fraction of color vs black and white sub-pixels of 10 micrometer dimensions can produce variation in hue, saturation and brightness, it is less well understood how diluting the area fraction of individual nanostructures within a dense array will affect plasmonic colors. It is this question that we aim to address in this paper.

The reflectance and transmittance spectra, and colors of most of the structures used in color printing rely heavily on structural geometry and in some cases structural pitch [13–16]. Consequently, sub-10-nm variations in structure size would result in observable color change. Although this sensitive size dependence enables a continuously size-tunable color production, in practice, one would prefer to use disks varying in diameter with at least ~10-nm steps. In particular, if large area fabrication methods are to be employed, the requirements on critical-dimension control would be significant. On the other hand, if we are to tune the pitch to generate colors, spatial resolution uniformity is affected. For instance, in a printed image that employs pitch variations to tailor colors, the resolution varies spatially throughout the printed image. We use a fixed pitch and employ only eight fixed disk diameters to realize 260 new colors. Instead of continuously varying disk diameters in sub-10-nm steps, we utilized mixing strategies to create these new colors. In addition, colors generated with our structure show slight changes with pitch size variations in a wide range. We also establish a relationship between absorption cross-section and optimum pitch size to obtain saturated colors. Using the obtained optimum pitch, mixing strategies, and enabled by large absorption cross-section, we predict that a wide color gamut including red, green, blue, and dark pixels that exceed the standard RGB (sRGB) gamut (standard color space used in displays and printers) for some colors, is achievable with a fixed pitch only with eight different nanodisk diameters.

2. Absorption cross-section and optimum pitch in periodic arrays

When a beam of light interacts with matter, energy is removed from the beam path by absorption and scattering. The absorption cross-section of a particle is the effective area through which a given plane wave would transmit the same power that is absorbed by the particle [25]. Some nano-sized particles can possess very large absorption cross-sections relative to their physical cross-sectional area [26]. For instance, a 13 nm diameter spherical aluminum particle can have an absorption cross-section as large as 18 times of its geometric cross-sectional area at the wavelength of 140 nm [1]. This phenomenon could allow sparse arrangement of particles yet exhibit perfect absorption of incident light [2]. One can determine the absorption cross-section as follows [1]:

σ_{a b s} = \frac{P_{a b s}}{I}

where

σ_{a b s}

,

P_{a b s}

,

I

and are the absorption cross section, power absorbed by particle, and incident irradiance respectively.

Another useful metric is the absorption efficiency which is the ratio of the absorption cross-section to geometric cross-section. For a cylindrical particle illuminated along its axis, the absorption efficiency is given by [1]:

Q_{a b s} = \frac{4 σ_{a b s}}{π d^{2}}

where

d

is the diameter of the cylinder. We calculate the absorption cross-section and the absorption efficiency using FDTD simulations (LUMERICAL software package [27]). In the following, we establish a relationship between the absorption cross-section of an individual nanoantenna and pitch of the corresponding periodic array to achieve saturated colors.

Figure 1 shows a schematic of different packing densities of a nanoantenna with a given absorption cross-section indicated as a purple disk. For simplicity, we consider only a square array and assume a circular absorption cross-section profile with a diameter given by: (3)In order to obtain saturated colors in subtractive structures, we conjecture that the absorption cross-section should have a large overlap with the unit cell area. Three unit cell lattice constants or pitch are considered: We could choose the pitch to be equal to the diameter of the absorption circle, according to Fig. 1(a), i.e. $P_{u} = d_{a b s}$ , hence:

P_{u} = 2 \sqrt{\frac{σ_{a b s}}{π}}

where

P_{u}

is the upper bound for optimum pitch. This approach oversimplifies the problem but still provides a useful estimate of the optimum pitch. In this case, the ratio of the absorption cross section to the unit size area, i.e. unit cell coverage, is ~78% (high absorption), and consequently the nanoantenna can still absorb a significant amount of the incoming light. This pitch assignment would be an indicator for the upper bound for the optimum pitch. If the pitch is chosen such that absorption area is equal to the unit cell area (green square), from Fig. 1(b) the pitch is given as

P_{1} = \frac{\sqrt{π}}{2} d_{a b s}

and

P_{1} = \sqrt{σ_{a b s}}

where

P_{1}

is the lower bound for the optimum pitch, as the absorption will be even higher than the previous case.

P_{u}

and

P_{1}

are, respectively, the upper and lower bounds to the optimum pitch region to produce colors with unchanging saturation. In the last scenario shown in Fig. 1(c), the pitch is the average of the absorption circle diameter and nanoantenna diameter

P_{m} = \frac{d + d_{a b s}}{2}

, therefore

P_{m} = \sqrt{\frac{σ_{a b s}}{π}} + \frac{d}{2}

In the above equation

P_{m}

is the minimum pitch. In this case, absorption circle reaches the adjacent nanoantennas; hence it can be an indicator of the onset of coupling between nanoantennas. Pitch sizes smaller than

P_{m}

should be avoided to minimize inter-structure interactions, i.e. plasmon hybridization [25], which result in unwanted change in hue and saturation. The peak in the absorption cross-section plot corresponds to the plasmonic resonance wavelength of the nanostructure. The absolute value of the absorption cross-section is taken literally to correspond to the physical area onto which incident photons at the resonant wavelength will be absorbed. Hence, the coverage of this physical area relative to the unit cell area will determine the color saturation. For instance, an absorption cross-section that occupies the entire unit cell of the disk will correspond to near-perfect absorption, a large dip in the reflection spectrum and high color saturation. Conversely, a small absorption cross-section relative to the unit cell area results in low saturation or “diluted” colors. In the following, we will calculate the absorption cross-section for a gap plasmon structure numerically using finite-difference time-domain (FDTD).

Fig. 1 Periodic arrays with different absorption cross-sections and pitches. d_abs is the absorption cross-section diameter assuming a circular profile. (a) Pitch (P_u) is equal to the absorption cross-section diameter (b) Pitch (P_l) is smaller than the absorption cross-section diameter (c) Pitch (P_m) of the array is the average diameter of absorption cross-section plus nanostructure diameter (d).

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Throughout this paper, the results for the absorption cross-sections, reflection spectra, Poynting vector and electric field distributions were calculated using the LUMERICAL FDTD software package [27]. To calculate the absorption cross-section, efficiency, electric field distribution, and Poynting vector field of a single nanoparticle, the total field scattered field (TFSF) analysis was used. For periodic array simulations, a standard plane wave was employed with periodic boundary conditions along the x- and y-axes. Due to structural symmetry, symmetry conditions were used to reduce the computational time. Perfectly matched layers (PML) were used along the propagation direction at the top and bottom boundaries. The source spans the wavelength range of 350 nm to 800 nm. A mesh override region was added to aluminum and aluminum oxide (Al₂O₃) disks. The mesh size was set to 2 nm. The simulation time was set to 1000 fs in the time domain solver to give the optical pulse enough time to pass through the simulation area and get absorbed by PML layers. Simulation was automatically stopped when the remaining energy in the simulation setup reduced to 1e-5 of the initial value. The permittivity data of aluminum and aluminum oxide (Al₂O₃) were obtained from Palik [28]. It should be highlighted that Al permittivity data were extracted from polycrystalline films which have closer properties to evaporated Al in experiments.

Figure 2(a) shows the schematic of the individual gap-plasmon structure investigated in this study, consisting of three functional layers. A 25-nm thick Al₂O₃ disk is sandwiched between a 40-nm thick Al nanodisk and an Al backreflector on a silicon substrate. Light is incident along the z-axis upon the structure and polarized along the x-axis. The use of this structure that supports gap-plasmons is multifold: It is fairly straightforward to fabricate (usually only a single lithographic step) [18,29], its resonances are narrower compared to standalone aluminum structures, and most importantly it has a large absorption cross-section which enables us to investigate the correlation of pitch with absorption cross-section. In addition, we will be able to employ color mixing strategies which will be shown in the next section.

Fig. 2 (a) Schematic of a unit cell for the nanostructure under study, the normal incident light is polarized along the x-axis. (b) Absorption cross-section map for individual nanodisks of various diameters. (c) Absorption efficiency map for different diameters. (d) The electric field distributions and Poynting-vector field lines at wavelength of 600 nm in x-z plane and (e) in y-z plane.

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The absorption cross-section and efficiency maps calculated from FDTD method are presented in Figs. 2(b) and 2(c). The absorption cross-section map shows that absorption cross-section increases with increasing nanodisk diameters. Conversely, the absorption efficiency decreases as the diameter is increased. These trends are consistent with the behavior of silver, gold, and aluminum nanodisks where absorption is the dominant mechanism for localized surface plasmon (LSP) decay process [30,31]. Although the absolute absorption cross-section increases with diameter, its rate of increase is smaller than the rate of increase of the geometrical area of the particle. By changing the diameter, the absorption peak can be tuned over the whole visible spectrum. A second peak (at shorter wavelengths) can be observed in Figs. 2(b) and 2(c), which is due to higher order resonance modes.

In order to understand the mechanism for absorption and investigate the interaction of the structure and its large absorption cross-section with the incident light, we plot the electric field distribution and Poynting vector field lines in Figs. 2(d) and 2(e) for d = 130 nm at resonance points A and B. As observed in both plots, the electric field is tightly confined in the dielectric layer which results in high absorption at the particular wavelength. In this metal-insulator-metal structure, the resonance is dominated by the gap plasmon mode [32]. As can be seen from the field lines in this figure, the gap-plasmon structure is absorbing light from a projected distance of ~3x its diameter, i.e. much beyond its physical boundaries. The structure therefore appears to the incoming photons as an absorber with an area several times larger than its geometrical cross-section. In Fig. 2(d) the Poynting vector field lines are directed towards the dielectric gap, however in Fig. 2(e) field lines end on the aluminum disk. The structure funnels light far beyond its physical boundaries and confines it in the gap. We will show that high confinement and light capturing will be useful mainly because the confinement helps to minimize the coupling between adjacent structures in periodic arrays while light capturing enhances color mixing.

Figure. 3(c) shows that by changing the diameter of the disk and keeping the pitch at 340 nm, the near-unity absorption, i.e. near-zero reflectance dip, can span the entire visible spectrum. To investigate realizable colors, we studied structures with nanodisk diameters ranging from (80 – 250 nm) and pitches ranging from (200 – 400 nm) systematically. The minimum gap between disks should be larger than 50 nm to minimize significant changes in hue due to coupling. Doing so enables color saturation to be investigated for a given hue. The color palette in Fig. 3(d) shows the calculated colors in reflection mode. The simulated reflectance spectra was converted to these colors using the color-matching functions of the human eye defined by the International Commission on Illumination CIE 1931 standard [33]. Most of the colors in the range of P = 280 to 360 nm possess high saturation and brightness. Although the resonance wavelengths span the entire visible spectrum, saturated green and red are not present in the color palette due to the subtractive nature of generated colors. Instead, magenta, cyan, and yellow are realizable, enabling the generation of all other colors including red and green by color mixing. It is worth mentioning that we expect a self-limiting native oxide layer to form around Al disk, with a thickness of ~3 nm [31]. Negligible color change was observed in the simulated color palette for oxide shells with thicknesses up to 3 nm, due to the balance between blueshifting reduction of the actual Al disk diameter, and the redshifting increase in index of the nano-environment.

Fig. 3 (a) HSB color coordinate. HSB refers to the color hue (H), saturation (S) and brightness (B). (b) Top view of the periodic nanostructure. (c) Calculated reflectance spectra for various disk diameters at P = 340 nm. (d) Simulated color palette in reflection mode. Minimum pitch, lower bound and upper bound pitch are calculated and highlighted on the color palette for d = 90, 110, 130 nm. (e) HSB plot vs. pitch with calculated P_m, P_l and P_m for d = 90 nm. (f) d = 110 nm. (g) d = 130 nm. (h-j) Calculated reflectance for d = 90, 110, 130 nm and various periods.

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Intuitively, one expects colors to become less saturated with minimal change in hue as the disks are spaced further apart. However, as the absorption cross section is larger than the physical dimension of the disks, it is likely that a range of periodicities would exist where the saturation remains constant despite increased spacing. To investigate the effect of pitch on the color saturation and validate our optimum pitch model, it would be more convenient to map the spectra to the cylindrical-coordinate HSB color space (also known as HSV). The HSB color model is a cylindrical-coordinate representation of RGB color model and defines the color space in terms of the hue (H), saturation (S), and brightness (B) [34] as shown in Fig. 3(a). The color points are specified by azimuthal coordinate (H), radial coordinate (S), and height (B) axis. The angle H determines the color, while color saturation and brightness are determined by S, and B respectively. By moving radially away from the axis along S, the color saturation is increased while by moving up along B increases the color brightness.

In Figs. 3(e)–3(g) simulated colors are converted to the HSB parameters for three different diameters with various periodicities. $P_{u}$ , $P_{1}$ , and $P_{m}$ are calculated based on Eq. (4) to Eq. (6) and indicated in the plots. In Figs. 3(e) and 3(f), $P_{u}$ _{and $P_{1}$ are accurate upper and lower bounds for the highly-saturated region while in Fig. 3(g), they overestimate the optimum pitch region. By investigating Figs. 3(e)–3(g) we notice that at the minimum pitch point the saturation is rapidly changing due to coupling interactions with neighboring structures, which is correctly estimated by Eq. (6). The simulated results show good quantitative agreement with Eq. (4) to Eq. (6).}

The results shown in Figs. 3(h)–3(k) demonstrate the insensitivity to pitch, unlike hole array structures that are highly dependent on pitch [13–16]. This insensitivity to pitch, is due to the tight field localization at the disks as shown in Fig. 2, also pointed out in [35], and the minimal interaction between adjacent nanodisks. This feature enables the employment of a single pitch which results in constant density of structures across the entire printing area. The sharp dip in calculated reflection spectra in Fig. 3(j) at ~400 nm is due to diffractive effects, as the pitch of the structure matches the dip position.

Next, by averaging the upper bound of the optimum pitch sizes for d = 90, 110, 130 nm which results in the closest pitch size P = 340 nm in our color palette, we will carry out color mixing with a P = 340 nm unit cell. The advantage of choosing the upper bound to determine our unit cell size is having more room for mixing; consequently, we can avoid inter-structure interactions even though we use a wide range of disk sizes.

3. Color mixing

Here we investigate color mixing strategies due to the overlapping absorption cross sections of different color disks. We expect that combining structures that have absorption cross sections (at their respective resonances) that encompass the total area occupied by all structures would allow one to mix colors without reduction in saturation, and generate colors that are more challenging to realize such as black. Black has been previously achieved via different strategies, such as using lossy metals [36] to trap light and damp its energy, and by varying structure geometries [18,19,24,37].

In traditional subtractive colors, the mixture of cyan, magenta, and yellow colors should generate black when overlapped in space. However, if placed side-by-side, the produced color is gray at best, as light incident on the any of the sub-pixels at off-resonance conditions would be reflected. Structures with large absorption cross sections could produce black if the total area of the composite pixel is comparable to the absorption cross section of any one of the constituent structures. Figure 4(a) illustrates the pixel layout for color mixing and a schematic of light “funneling” to realize black. Each color is separately “funneled” into its corresponding nanodisk and absorbed. As we have used four disks in this design, total number of combinations (without taking into account the symmetry of the design with respect to the polarization direction) would be 5632 states. To reduce the number of possible combinations for simulation, we have fixed two disk diameters as d₁. We used the same approach as the one used to simulate the individual nanodisks in a periodic array, except that the no-symmetry conditions were used due to lack of structural symmetry about the x- and y-axes. The E-field is polarized along x-axis. In Fig. 4(b), the second black color represented by solid line is the result of cyan (d₁ = 140 nm), magenta (d₂ = 110 nm), and yellow (d₃ = 90 nm) which produces black, as expected based on color theory. This result shows that the respective resonant wavelength components of incident light are indeed separately funneled to the corresponding nanodisk and absorbed.

Fig. 4 (a) Schematic of sub-pixel layout for color mixing. Each color is separately funneled into its corresponding nanodisk and absorbed. The incident light is polarized along x-axis. (b) Calculated reflectance spectra for a near-black pixel. (c) Calculated reflectance for red and green colors. (d) Simulated Color palettes for d₁ = 0 (e - l) d₁ = 80 - 150 nm (m-n) Chromaticity coordinates corresponding to basic (individual periodic disks) and extended color gamut on the CIE 1931 chromaticity diagram.

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Due to the optical losses in Al, its resonances are broad [30,31], and thus it is challenging to realize high-purity and saturated colors. Specifically, green and red [12,18,38–41] are challenging to produce even with previous color mixing strategies [8,24] and hole-arrays [14,42]. Here, by mixing yellow and cyan a saturated green color is generated as shown in Fig. 4(c) which has a high purity and is not achieved in previous works. The generated red is close to the sRGB red vertex, which was not realized before. Next, we use our CMY (cyan, magenta, yellow) color basis set and other colors to realize a wider range of colors. Simulated color palettes for d₁ = 0 nm and d₁ = 80 – 150 nm are depicted in Figs. 4(d)–4(l). First, we have used only two disks to perform color mixing. Chromaticity coordinates corresponding to basic color gamut (individual periodic disks) and color mixing by two disks are plotted on CIE 1931 chromaticity diagram shown in Fig. 4(m). As observed in this figure, colors overlap the sRGB region. To populate the diagram with more points, we perform color mixing using three disks. Accumulated results for two and three disks color mixing is shown in Fig. 4(n). It can be seen that considerably many new color points are added to the diagram using only eight disk sizes. Although we have used only three degrees of freedom of the system, one can further carry on by employing full four degrees of freedom (i.e. four disks), to create new points on the CIE chromaticity diagram.

Finally, we use our color gamut to reproduce a painting and demonstrate the diversity of the palette. We use a digital copy of Vincent van Gogh’s Café Terrace at Night painting, Fig. 5(a), obtained from a public domain resource (https://en.wikipedia.org/wiki/Café_Terrace_at_Night). The digital copy was input into MATLAB and the color of each pixel was matched to the simulated color. Reproduced image is shown in Fig. 5(b). As can be seen, reproduced images show high color consistency which highlights the effectiveness of our color mixing. Moreover, dark and bright colors are closely matched.

Fig. 5 (a) Digital copy of Vincent van Gogh’s Café Terrace at Night painting obtained from a public domain resource. (b) Simulated painting by extended color gamut.

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4. Conclusions and future work

In this work, aluminum was used in a structure that supports gap plasmon resonances with large absorption cross-sections to realize a wide color gamut. A relationship was established to determine the optimum pitch to maximize the saturation of colors based on the absorption cross-section of an individual unit cell. The electric field distribution of the structure was considered at resonance, and near-unity absorption was achieved. By investigating the Poynting vector field, significant manipulation of light was demonstrated far beyond the physical boundaries of the structure. It was shown that resonance dips could cover the whole visible spectrum by varying the disk diameter while keeping the pitch constant for normal incidence and viewing angles. The generated colors were insensitive to pitch variation in a considerably wide range which is the result of the structure’s large absorption cross-section. In addition, diffractive effects were observed for the large pitch structure. The insensitivity of the colors to nanostructure separation could minimize the effects that stem from fabrication errors and ease the adoption of large area fabrication methods with lower precision which can be of great practical utility.

By color mixing, an extended color gamut was realized that covered most of the sRGB color gamut and even exceeded it for some colors. The generated colors included a dark color that was similar to black along with saturated red and green. Color mixing was carried out using only eight different disk diameters (d = 80, 90, 100, 110, 120, 130, 140, 150 nm) to realize 260 new colors which demonstrates the practicality of mixing strategies instead of small size variations of the structure which are mainly limited by the resolution constraints of the fabrication methods. Finally, a painting was reproduced with high color consistency which further demonstrated the visual performance of the color mixing method. Results showed that fundamentally, plasmonic colors can achieve high color saturation with good overlap with sRGB. We should point out that experimental results might not work as well due to imperfections in fabrication, surface roughness, and larger damping from surface scattering in nanostructures. We believe that the optimum pitch equation can be employed as a simple tool to design structures with high color saturation. Utilization of large absorption cross-section structures is an alternative approach toward wide color gamut realization instead of halftoning and overlaying. In addition, our easy to fabricate and cost efficient platform will be an excellent candidate to realize a wide color gamut with subwavelength spatial resolution.

Funding

S.D.R acknowledges financial support provided by Agency for Science, Technology and Research (A*STAR) through Singapore International Graduate Award (SINGA) program.

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On the correlation of absorption cross-section with plasmonic color generation

Abstract

1. Introduction

2. Absorption cross-section and optimum pitch in periodic arrays

3. Color mixing

4. Conclusions and future work

Funding

References and links

Cited By

Figures (5)

Equations (5)

Optics Express