Bright and vivid plasmonic color filters having dual resonance modes with proper orthogonality

Hyowook Kim; Myungjoon Kim; Taeyong Chang; Arthur Baucour; Suwan Jeon; Nayoung Kim; Hak-Jong Choi; Heon Lee; Jonghwa Shin

doi:10.1364/OE.26.027403

1. Introduction

Scattering is one of the earliest topics of optics studied, with diverse scientific and practical applications. From the colors of Morpho menelaus [1] to optical coherence tomography [2], the understanding of scattering phenomena has motivated developments in many optical theories and techniques. While the scattering of continuum states by a single localized mode is a simple but important problem that occurs in many different disciplines, it has been shown that more interesting phenomena manifest if more than one resonance mode is involved in the scattering event. The Kerker forward scattering is a well-known optical example [3], which is used to obtain a peculiar spatial scattering profile at a given temporal frequency, by utilizing the precise control of two resonance modes. Also of interest, is the manipulation of the scattering spectral profile in a given spatial direction and Fano resonances are such examples [4,5]. To understand these spectral features, temporal coupled mode theory (CMT) has proven to be a valuable tool, whereby general conclusions can be drawn without restricting the problem to particular structural configurations [6,7]. Of particular interest among these works is the conclusion that the orthogonality of resonance modes may have an important consequence for the overall shapes of scattering spectra [8]. While not fully recognized previously, the potential color filter applications of the principle have far-reaching commercial impacts.

A color filter is a core part of modern color display panels and image sensors. Among the possible configurations, the red, green, and blue (RGB) transmissive color filter array is the most widely adopted design. In commercialized devices, pigment- or dye-embedded photoresists are employed for color filter arrays. However, this approach requires at least three different pigments or dyes, and three separate lithography cycles. Furthermore, the stability of the filter against heat and ultraviolet light is less than desirable because of the organic components [9]. Therefore, many researchers are exploring alternative solutions.

Structural colors are a novel way of controlling the colors as they do not solely rely on the intrinsic electronic energy levels of the materials and associated absorption resonances, and instead achieve the desired colors by introducing new artificial resonances that can be tailored by choosing structural parameters, such as the shape, size, and lattice configuration of the sub-wavelength-scale motifs. Diverse structures, such as gratings [10–18], nanohole arrays [19–27], nanoparticle or disk arrays [28–32], metal-insulator-metal (MIM) structures [33–40], and others [41–43], have been proposed. By adjusting the structural designs without changing the materials, most of these structures can possess a wide range of resonance wavelengths in the visible range, and they are often orders-of-magnitude thinner than conventional color filters. While some designs based on guided mode resonances or surface plasmon polaritons exhibit considerable dependences of the resonant wavelength on the incident angle, those based on localized surface plasmon resonances (LSPRs) can potentially retain their colors over a wide range of incidence angles [44]. However, the biggest problems for all plasmonic transmissive color filters are their low efficiency and low color saturation, both of which are due to significant ohmic loss in the metals constituting the structural color filters. Most previous transmissive filter designs, consisting of a continuous metal film with a periodic array of sub-wavelength apertures, used a resonance of the system (whether it was a guided mode resonance or an LSPR) to transmit the desired wavelengths while the unwanted wavelengths are reflected by the continuous metal film. However, if the metal is optically lossy, a simple CMT can be used to demonstrate that this approach has a fundamentally limited maximum transmission, regardless of the actual implementation [6]. In addition, the loss-induced spectral broadening limits the sharpness of the transmission wavelength band and, hence, limits the color saturation.

In this paper, we propose a set of plasmonic transmissive RGB color filters with the widest color gamuts and highest efficiencies, based on a pair of orthogonality-controlled resonance modes. First, the ideal tri-color transmission spectra with the highest possible brightness for a given color gamut are derived, which are a universal result regardless of the implementation technology, be it pigments, LSPR, or any other non-luminescent approach. Then, based on temporal CMT [7], it is shown that only two Lorentzian resonances per color filter are needed to retain a significant part of the color gamut and brightness. The key differences from many previous designs are that (1) the transmissive wavelength range is away from the resonances and that (2) both the electric and magnetic resonance modes are utilized, both of which are essential in achieving a high transmittance. Furthermore, a proper balance of the absorption and scattering coefficients results in a near-complete blockage of unwanted wavelengths, increasing the color gamut. The required dual resonances are realized by metal-insulator-metal structures for the red and blue color filters, and di-atomic structures for the green color filter. Their performances, including the transmission spectra and incident angle dependence, are numerically verified. The results show that the brightest and most saturated transmissive red, green, and blue color filters can be achieved with this approach.

2. Theoretical model

We first derive the transmission spectra of ideal color filters that can achieve the maximum brightness while satisfying a given color gamut. As an example, the DCI-P3 color gamut is chosen for the derivation in this section, and we adopt the Standard Illuminant D65, specified by the Commission Internationale de l'Eclairage (CIE). Results for other choices of color gamuts, such as sRGB and BT.2020, can be obtained in the same manner. The spectral transmittance design problem is transformed into a linear programming one, with equality (the chromaticity coordinates) and inequality (0 ≤ transmittance ≤ 1) constraints (Table 1); the solutions are plotted in Fig. 1. The results are easy to understand intuitively: the transmittance should be perfect over the desired wavelength passband, while it should be identically zero over the rejection bands.

Table 1. Conditions for the linear optimization. (x_target, y_target) are the target chromaticity coordinates of a given color gamut, S_D65 is the power spectral density of the CIE Standard Illumination D65, T(λ) is the transmittance at each wavelength, and, $\bar{x}$ , $\bar{y}$ and $\bar{z}$ are color matching functions.

View Table | View all tables in this article

Fig. 1 Ideal transmission spectra and CMT applied transmission spectra of bright and saturated color filters. Optimization results of ideal color filters spectra and CMT based spectra for the (a) red, (b) green, and (c) blue color filters. (d) Color gamut of dual resonance spectra indicated in CIE1931 color space.

Download Full Size | PDF

In reality, it would be impossible to implement such flat-top, infinite transition-slope spectral profiles. However, in this section, we show that a simple system with two Lorentzian resonances can retain a large portion of the color gamut as in the above ideal filters, while the overall brightness decreases only slightly. To derive a general conclusion, irrespective of the actual physical systems embodying such resonances, a temporal CMT is adopted for the theoretical argument [7]. The color filter is modeled as a two-port (incident and transmitted sides) system with two localized resonances. This is equivalent to an optical cavity with two modes of interest, which are coupled to two external waveguides. While the free space has infinitely many plane wave modes indexed by the wave vector and polarization, a particular incident plane wave with a known incident direction and an eigen-polarization, can only couple to a single reflected plane wave and single transmitted plane wave with the same in-plane wave vector components if the color filter does not diffract or scatter diffusively. Planar arrays of nanostructures with a period less than a half-wavelength satisfy this condition and can be considered as two-port systems. In addition, we assume that the ports are symmetric, which is the case if the substrate and superstrate of the color filter have the same refractive index, and the color filter structure itself has a mirror symmetry plane parallel to the surface. The structural symmetry requires that the radiation from the resonance mode into two waveguides, also be symmetric in magnitude and have a phase difference of 0 (even mode) or π (odd mode). The temporal CMT dictates that if two resonance modes are of the same parity (two even or two odd modes), they are necessarily non-orthogonal [8]. If the resonances are of opposite parity, they are orthogonal. With these assumptions and the theoretical background, the amplitudes of the two resonance modes under external illumination can be found in Eq. (1) and Eq. (2) for orthogonal modes, Eq. (3) and Eq. (4) for non-orthogonal modes.

a_{1} = \frac{\sqrt{γ_{1}} \exp (j θ_{11})}{j ω - j ω_{1} + γ_{1}},

a_{2} = \frac{\sqrt{γ_{2}} \exp (j θ_{12})}{j ω - j ω_{2} + γ_{2}},

a_{1} = \frac{(j ω - j ω_{2}) \sqrt{γ_{1}}}{(j ω - j ω_{1} + γ_{1}) (j ω - j ω_{2} + γ_{2})},

a_{2} = \frac{(j ω - j ω_{1}) \sqrt{γ_{2}}}{(j ω - j ω_{1} + γ_{1}) (j ω - j ω_{2} + γ_{2})},

where a_n, γ_n, and ω_n (

= 2 π c / λ_{n}

) are the complex amplitude, radiation damping rate and resonance frequency of each mode (n = 1, 2), respectively; θ₁_i is the phase of the coupling coefficient from mode 1 to each port (i = 1, 2). From these amplitudes, the transmission coefficient from the color filter can be derived as shown in Eq. (5) for the orthogonal modes and Eq. (6) for the non-orthogonal modes in non-absorptive systems [8]:

t = t_{b} - \frac{γ_{1} (r_{b} + t_{b})}{(j ω - j ω_{1} + γ_{1})} + \frac{γ_{2} (r_{b} - t_{b})}{(j ω - j ω_{2} + γ_{2})},

t = t_{b} - \frac{(r_{b} + t_{b}) [γ_{1} (j ω - j ω_{2}) + γ_{2} (j ω - j ω_{1})]}{(j ω - j ω_{1} + γ_{1}) (j ω - j ω_{2} + γ_{2}) - γ_{1} γ_{2}},

where t_b and r_b are the background transmission and reflection coefficients, respectively. CMT predicts that when the resonance frequencies of two orthogonal modes are relatively close (

| ω_{1} - ω_{2} | \leq γ_{n}

), a broad rejection band can be formed. Yet, for non-orthogonal modes, a narrow transmission peak exists between two resonance frequencies, even when they are very close. Therefore, orthogonal modes are suitable for red and blue filters because the wavelength range to be blocked is a single, broadband region, and non-orthogonal modes are appropriate for a green color filter with two rejection bands (red and blue) sandwiching a narrow passband (green). If used the other way, there is a loss of saturation or brightness: the use of non-orthogonal modes for a red or blue filter induces an unwanted additional passband that decreases the color saturation and overall color gamut, while the use of an orthogonal mode for a green filter results in the decreased transmittance of green light.

By using these expressions for the transmission coefficient, the transmittance spectra can be numerically optimized for each of the red, green, and blue color filters. Since it is a multi-objective optimization problem, in which the transmittance and color saturation are simultaneously optimized, we developed a single figure-of-merit (FOM) that can facilitate the search for balanced color filters with good performances in both aspects, and adopted a particle-swarm optimization (PSO) method (the details are provided in Section 3). Table 2 presents the optimized parameters of the non-absorbing dual-resonance systems and the resulting transmittance spectra are shown in Figs. 1(a)–1(c). The spectra suggest that, by using orthogonality-controlled Lorentzian resonances to block unwanted wavelengths, a very high transmittance can be expected at the passband. Figure 1(d) illustrates the color gamut spaces calculated from the ideal color filter spectra and CMT-optimized dual-Lorentzian spectra. The color gamut area of the optimized dual-Lorentzian spectra is 90.2% compared to that of DCI-P3 (122.3% compared to that of sRGB). These results imply that bright and vivid plasmonic color filters may be obtained with orthogonality controlled dual resonance mode systems. However, it is yet to be seen if this implication remains true even when the optical absorption is non-negligible.

Table 2. Optimized parameters of the non-absorptive dual-resonance systems

View Table | View all tables in this article

Now we propose and verify the physical structures with dual resonances, with realistic optical losses of the constituent metals considered: an MIM stacked disk array for orthogonal modes and di-atomic cross-square mixed array for non-orthogonal modes (Fig. 2). Silver is used in both designs as the metallic material. The stacked disk structure has both an electric resonance mode, in which the electric currents in the upper and lower metal disks are in the same direction, and a magnetic resonance mode, in which the currents are in opposite directions. The electric resonance mode possesses an electric dipole, while the magnetic resonance mode has conduction currents in the metal and displacement currents in the sandwiched dielectric, forming a complete current loop generating a magnetic dipole. These two modes are even and odd, respectively, and are orthogonal, making this structure a good candidate for red and blue filters. In comparison, the di-atomic structure has two electric resonance modes, each controlled by the shape and size of a different “atom” (cross or square) in a unit cell. Thus, by designing the two resonance wavelengths at red and blue, a green color filter can be obtained. We note that the principle remains the same for other nanostructure shapes, such as stacked square structures for red and blue filters and an array of small and large disks for a green filter.

Fig. 2 Schematic diagrams of (a) MIM nanodisk structures and (b) di-atomic structures.

Download Full Size | PDF

3. Numerical optimizations

The linear optimizations were performed using the dual-simplex algorithm implementation built in Matlab (Mathworks Co. Ltd). We set the transmittance for each wavelength at 1 nm spacing [T(360nm), T(361nm) … T(830nm)]. The optimization variables were defined on a closed interval [0, 1] and CIE1931 chromaticity coordinates of the DCI-P3 [(x, y) = (0.680, 0.320), (0.265, 0.690), and (0.150, 0.060), for red, green, and blue, respectively] were used as equality constraints. The optimization method, condition, and objective function are presented in Table 1. While not included, we also used the chromaticity coordinates of other common RGB color spaces, such as sRGB [(x, y) = (0.64, 0.33), (0.30, 0.60), and (0.15, 0.06)] and BT.2020 [(x, y) = (0.708, 0.292), (0.170, 0.797), and (0.131, 0.046)] and found similar results.

We used the particle-swarm optimization (PSO) algorithm as explained by J. Robinson [45] to find the optimized transmittance spectrum for concurrent high color saturation and brightness. Coupled mode theory (CMT) was used to generate the wavelength-dependent transmittance from the t_b, r_b, γ_n, and ω_n parameters. One thousand particles in each generation and a total of 1,000 generations were calculated for the optimization. The figure of merit (FOM) for the PSO was defined as follows:

(FOM) = p^{m} q^{n},

p = \frac{1}{2} (\frac{| \vec{d} | | \cos θ | \cos θ}{| \vec{d_{0}} |} + 1),

q = \frac{1}{π} arc \tan [a (T_{\max} - T_{c})] + \frac{1}{2},

where p and q are constituent FOMs related to the chromaticity and brightness, respectively; m and n are relative weights for these two FOMs;

\vec{d}

is the two-dimensional vector from the equal-energy locus to the chromaticity point of the color filter on the CIE1931 chromaticity diagram;

\vec{d_{0}}

is the vector from the equal-energy locus to the chromaticity point of the target monochromatic primary light; and θ is the angle between

\vec{d}

and

\vec{d_{0}}

. In addition, T_max is the maximum transmittance of the color filter in a narrow band around the target monochromatic primary light wavelength, T_c is the threshold transmittance, and a is a parameter that determines the slope of the FOM near T_c. For m, n, a, and T_c, we used the values 4, 1, 10, 0.7, respectively, and BT.2020 for the target monochromatic primaries.

FDTD simulations (Lumerical Co. Ltd) were performed to calculate the optical properties of actual structures. The tabulated optical properties of Ag and SiO₂ from E. D. Palik [46] and those of MgF₂ from M. J. Dodge [47] were utilized. While the Ag data reported by Johnson and Christy [48] indicate a smaller optical loss and would produce an even higher transmittance or larger color gamut, Palik's data were utilized because they could be used to better represent the realistic optical properties of silver, made by typical large-area fabrication processes that can introduce impurities and non-ideal film quality. Perfectly matched layers were utilized to terminate the z-normal boundaries of the simulation volume, while periodic boundary conditions were used for the x- and y-directions. The structure was excited by an x-polarized plane wave, propagating in the z-direction. Uniform 5 Å meshes were used for the metallic structures, and non-uniform meshes with a maximum mesh size of 10 nm were used outside the metal. For simulations illustrating the mode orthogonality, two total-field scattered-field sources which have opposite propagation directions and time-apodized monitors were used to show the difference between the non-orthogonal and orthogonal modes clearly. In the odd mode simulation, two sources had π phase difference. Broadband fixed angle source technology was used for the angled incidence simulations.

We used FDTD simulations and the aforementioned PSO algorithm with FOM to find the optimized structural parameters. For the Ag/SiO₂/Ag stacked disk structures, the period (p), diameter (d), thickness of Ag layers (t_Ag), and thickness of the SiO₂ spacer layer (t_SiO2) were set as variables. For the cross-square structures, the period (p), length of cross arms (a), width of cross arms (w), length of the side of squares (b), and thickness of the metal layer (t_Ag) were set as variables. Thirty particles in each generation with a total of 100 generations were simulated for the optimization for each color and for each of the two different illumination conditions (D65 and quantum-dot enhancement films).

4. Results and discussion

The finite-difference time-domain method is adopted to quantitatively analyze the mode properties of the actual structures; the details are given in Section 3. For the orthogonal mode system, an example of an Ag/SiO₂/Ag stacked disk array in a hexagonal lattice with p = 250 nm (period), d = 150 nm (diameter of the disk), t_Ag = 30 nm (thickness of the silver layer), and t_SiO2 = 60 nm (thickness of the SiO₂ layer) [Fig. 2(a)] is simulated to show the mode profiles. Figures 3(a) and 3(b) present the amplitude and phase profiles of x-directional electric fields when the system is excited at a 472-nm wavelength, which is the resonance wavelength of the electric resonance mode, and Figs. 3(c) and 3(d) show those at the magnetic resonance wavelength of 712 nm. The dashed box indicates the boundary of a total-field scattered-field source; therefore, the depicted fields outside the box are the scattered fields only, clearly showing the symmetry of the resonance mode. The amplitude of the scattered field is clearly symmetric above and below the color filter for both the electric and magnetic resonances. Conversely, the phase shows opposite behaviors for the two resonances, being symmetric for the electric resonance and anti-symmetric for the magnetic resonance. It is apparent that these two resonance modes have opposite symmetries and are therefore orthogonal.

Fig. 3 FDTD field profile results for the MIM structure for the excitation of orthogonal modes. (a) Amplitude and (b) phase for the $\vec{E_{x}}$ profile of the electric resonance mode (at 472 nm); (c) amplitude and (d) phase for the scattered $\vec{E_{x}}$ profile of the magnetic resonance mode (at 712 nm) near the MIM structure.

Download Full Size | PDF

For the non-orthogonal mode system, an example of a cross-square di-atomic array in a square lattice with parameters p = 250 nm, t_Ag = 30 nm, w = 50 nm, a = 200 nm, and b = 100 is chosen to clearly show the mode profiles. The simulation reveals two electric dipole modes with the electric fields of the longer-(shorter-)wavelength mode mainly localized to the cross (square) in a unit cell. As shown in Figs. 4(a)–4(d), both the amplitude and phase of the x-directional electric field are symmetric for both modes, which implies that they have the same symmetry and are thus non-orthogonal.

Fig. 4 FDTD field profile results for the di-atomic structure for the excitation of non-orthogonal modes. (a) Amplitude and (b) phase for the $\vec{E_{x}}$ profile of the electric resonance mode excited in a square-shape silver structure (at 492 nm); (c) amplitude and (d) phase for the scattered $\vec{E_{x}}$ profile of the electric resonance mode excited in a cross-shape silver structure (at 802 nm).

Download Full Size | PDF

In both types of structures, the resonant wavelength of each mode can be controlled independently by changing the structural parameters. For instance, in stacked disk structures, increasing the period (p) redshifts the electric resonance, while the magnetic resonance wavelength is almost unaffected [Fig. 5(a)]. However, increasing the thickness of the insulator (t_SiO2) blueshifts the magnetic resonance mode, while the electric resonance wavelength is less affected [Fig. 5(b)]. In cross-square mixed arrays, each mode can be easily shifted by adjusting the size of the cross or square structure [Figs. 5(c) and 5(d)].

Fig. 5 Transmittance of the MIM structure for several different (a) periods (p) and (b) thicknesses of the SiO₂ layer (t_SiO2). The transmittance of the di-atomic structures for several different sizes of the (c) square and (d) cross. The default structural parameters for (a) and (b) are p = 250 nm, d = 150 nm, t_Ag = 30 nm, and t_SiO2 = 30 nm (square lattice) and, for (c) and (d), p = 250 nm, t_Ag = 30 nm, w = 50 nm, a = 200 nm, and b = 100 nm (square lattice).

Download Full Size | PDF

By taking advantage of this tunability, a customized PSO routine was used to accumulate a database of nanostructure designs and their transmittance spectra for transmissive RGB color filters. A hexagonal array of Ag/SiO₂/Ag stacked nanodisks on a glass substrate were used for the red and blue filters while a square array of di-atomic Ag cross-squares with MgF₂ support (50 nm) on a glass substrate were used for the green filters, and their structural parameters were varied. With 6,000 records for each color, a total of 18,000 candidate designs were explored. Considering that there is a trade-off between the maximum transmittance in the pass band and color saturation, a series of optimal solutions that form a Pareto frontier with respect to the transmittance and color gamut was sought within this database. We defined the red, green, and blue passbands as centered on each of the primary color wavelengths 630, 532, and 467 nm, respectively, in ITU-R Recommendation BT.2020, with a ± 20 nm range. The T_max for each color filter is defined as the maximum transmittance within the corresponding passband.

As an example, consider a set of color filters constructed with the widest color gamut, with T_max for each color being at least 70%; the solid lines in Figs. 6(a)–6(d) shows the performances the set of color filters and the transmittance values are presented in Table 3. The resulting color gamut of the set is 90.0% of the sRGB area. If one lowers the critical value for T_max to 50%, a much wider color gamut of 117.2% of the sRGB can be obtained [dotted lines in Figs. 6(a)–6(d)]. If a narrower color gamut (45.0% of the sRGB) is tolerable, one can reach a very high T_max of 85% [dashed lines in Figs. 6(a)–6(d)]. The relevant geometric parameters for these three sets of color filters are presented in Table 4.

Fig. 6 PSO results from FDTD simulations of the (non-)orthogonality controlled structure. Transmittance of (a) red, (b) green, and (c) blue color filters with T_max higher than 50%, 70%, and 85%. (d) The white solid line presents the color gamut of the sRGB reference and black lines present the color gamuts of the simulated color filters shown in (a–c) with D65 standard illumination. In (a–d), the dashed, solid, and dotted lines represent the simulation results of color filters with T_max higher than 50%, 70%, and 85%, respectively. (e) Pareto frontier (black line) obtained by orthogonality controlled dual-resonance modes structures. Results of some of the recently proposed transmissive RGB plasmonic color filters are plotted for comparison.

Download Full Size | PDF

Table 3. Transmittance values of an optimized set of color filters with T_max≥70%. The maximum value is presented for the passband of each color filter. The minimum value is presented for the rejection bands.

View Table | View all tables in this article

Table 4. Structure parameters of the best color filters which have T_max equal to or larger than 85%, 70%, and 50%.

View Table | View all tables in this article

Figure 6(e) presents the Pareto frontier with the maximum achievable color gamut area shown for all T_max values from 5% to 95% in 5% intervals and the structural parameters of some representative points in the Pareto frontier can be found in Table 4 and Table 5. The achievable color gamut area increases as T_max decreases and saturates to slightly over 120% for T_max below 50%. The color gamut area drops below 100% of the sRGB space for T_max larger than 65%, and quickly diminishes if T_max increases further. We note that the performance of the color filters with T_max = 70% exceeds that of all previous numerical and experimental results on polarization-insensitive, transmissive color filters based on LSPRs, both in terms of the maximum transmittance and color gamut. Furthermore, other points in this Pareto frontier may serve as reference for evaluating the performances of plasmonic color filters for applications with varying emphasis on the efficiency and color purity. The color filter set can be further optimized by considering actual illumination spectra of a display and, as shown in Figs. 7(a) and 7(b), the color filters illuminated by quantum-dot based enhancement films used in the latest commercial television sets [49] show an even larger color gamut (120.8% of sRGB) with T_max = 70%. These results indicate that bright and saturated transmissive RGB color filters based on plasmonic nanostructures are indeed possible, despite the ohmic losses associated with the presence of metals.

Table 5. Structural parameters of the samples on the Pareto frontier with minimum T_max of 90%, 75%, 60%, 45%, 30%, and 15%.

View Table | View all tables in this article

Fig. 7 (a) Pareto frontier of optimized color filters with D65 and QDEF illuminations, and (b) their color gamuts in the CIE1931 chromaticity diagram for T_max = 70%.

Download Full Size | PDF

The fabrication tolerance of the proposed color filter designs can be investigated by analyzing the transmittance spectra of modified samples with fractional changes in the structural parameters from those of the optimal designs. Figure 5 already gives some information on the dependence of the transmittance on the period and thickness of the SiO₂ layer for the orthogonal-mode filters and on the dependence of the square and cross size for the non-orthogonal-mode filters. To complement this, we further conducted FDTD simulations for different disk diameters of the orthogonal-mode filters and for different metal layer thicknesses of all filters. The optimal color filters with T_max = 70% were used as the reference. As shown in Fig. 8(a), the transmittance spectra of red filter and blue filter can be affected by the diameter size considerably. In contrast, the thickness of the metal layer does not severely affect the performance of color filters [Fig. 8(b)]. Hence, it would be important to control the lateral dimensions precisely when fabricating these filters to get the intended performance.

Fig. 8 Sensitivity of the color filter spectra on the changes of the structural parameters. (a) Red and blue filters with ± 10% variation of the disk diameter. (b) All filters with ± 10% variation of the metal thickness. The color filters with T_max = 70% are used as the reference.

Download Full Size | PDF

These plasmonic color filters are based on LSPR; therefore, they are relatively insensitive to the incident angle. As an example, simulated transmittance spectra of a hexagonal array of Ag/SiO₂/Ag stacked nanodisks with p = 125 nm, d = 84 nm, t_Ag = 32 nm, and t_SiO2 = 55 nm (parameters of the optimized red filter with T_max = 70%) at several incident angles are shown in Figs. 9(a) and 9(b). As the incident angle increases from zero to sixty degrees, the transmission spectra for the s-polarization are almost unchanged for the rejection band, while the maximum transmittance of the passband (610 to 650 nm) shows some fluctuations, but stay above 70% up to 60 degrees. For the p-polarization, the passband is almost unaffected, while the transmittance of the rejection band increases slightly. Remarkably, in either case, the resonance wavelengths show negligible shifts. As a result, the chromaticity remains well controlled while the incident angle changes [Fig. 9(c)]. These results show the advantage of LSPR-based designs compared to guided mode resonance or surface plasmon polariton-based designs that are typically highly sensitive to the incident angle. While not exhaustively tested, the angular dependence of other filters in this paper with different T_max or different colors are expected not to differ significantly, except for the cases in which the unit cell period approaches the short end of the visible spectrum and diffractive effects as well as the effect of phase retardation within a unit cell start to manifest.

Fig. 9 Transmittance and color gamut results for the red filter FDTD simulations with angled incident light source. Transmittance results of the FDTD simulations with the (a) s- and (b) p-polarized angled incident light source. (c) Gradual transition of the chromaticity of the optimized color filter with the increase in incident angle occurs along the direction of the arrow.

Download Full Size | PDF

5. Conclusion

A new design principle for transmissive structural color filters is proposed. It is theoretically shown that a set of dual-resonance systems with the right mode orthogonality can achieve a very wide color gamut with only slightly decreased brightness, compared to hypothetical color filters with ideal spectral responses. Numerically optimized realizations of the concept based on actual material properties of silver and dielectrics are also presented. The appropriate choice of mode orthogonality advances the previous transmittance–color gamut Pareto frontier found in plasmonic color filters by a large amount. A particular set of new transmissive RGB filter designs exhibited simultaneous record high values among LSPR-based designs in both the brightness (above 70% maximum transmittance for all three colors) and color gamut (90.0% of the sRGB color space). Moreover, other points on the Pareto frontier with higher transmittance designs as well as larger-color gamut designs are presented. The off-resonance design, in which the passband is spectrally separate from the structural resonance wavelengths, assists in achieving the high transmittance. It is suggested, that plasmonic nanostructure-based color filters can approach the optical performance of conventional color filter materials with the advantage of reducing the thickness by orders-of-magnitude.

Funding

Ministry of Science, ICT, and Future Planning of Korea (2013M3C1A3063598, 2014M3A6B3063708, and 2017R1A2B2005702).

References

1. S. Berthier, E. Charron, and A. Da Silva, “Determination of the cuticle index of the scales of the iridescent butterfly Morpho menelaus,” Opt. Commun. 228(4-6), 349–356 (2003). [CrossRef]

2. J. J. Wild and J. M. Reid, “Application of echo-ranging techniques to the determination of structure of biological tissues,” Science 115(2983), 226–230 (1952). [CrossRef] [PubMed]

3. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic Press, 1969).

4. F. Hao, Y. Sonnefraud, P. V. Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, “Symmetry breaking in plasmonic nanocavities: subradiant LSPR sensing and a tunable Fano resonance,” Nano Lett. 8(11), 3983–3988 (2008). [CrossRef] [PubMed]

5. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010). [CrossRef] [PubMed]

6. L. Verslegers, Z. Yu, P. B. Catrysse, and S. Fan, “Temporal coupled-mode theory for resonant apertures,” J. Opt. Soc. Am. B 27(10), 1947 (2010). [CrossRef]

7. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003). [CrossRef] [PubMed]

8. W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004). [CrossRef]

9. R. W. Sabnis, “Color filter technology for liquid crystal displays,” Displays 20(3), 119–129 (1999). [CrossRef]

10. T. Xu, Y.-K. Wu, X. Luo, and L. J. Guo, “Plasmonic nanoresonators for high-resolution colour filtering and spectral imaging,” Nat. Commun. 1(5), 59 (2010). [CrossRef] [PubMed]

11. M. J. Uddin and R. Magnusson, “Efficient guided-mode-resonant tunable color filters,” IEEE Photonics Technol. Lett. 24(17), 1552–1554 (2012). [CrossRef]

12. B. Zeng, Y. Gao, F. J. Bartoli, D. J. Norris, and M. Masuda, “Ultrathin nanostructured metals for highly transmissive plasmonic subtractive color filters,” Sci. Rep. 3(1), 2840 (2013). [CrossRef] [PubMed]

13. Y.-K. R. Wu, A. E. Hollowell, C. Zhang, L. J. Guo, and R. Quidant, “Angle-insensitive structural colours based on metallic nanocavities and coloured pixels beyond the diffraction limit,” Sci. Rep. 3(1), 1194 (2013). [CrossRef] [PubMed]

14. L. Wen, Q. Chen, F. Sun, S. Song, L. Jin, and Y. Yu, “Theoretical design of multi-colored semi-transparent organic solar cells with both efficient color filtering and light harvesting,” Sci. Rep. 4(1), 7036 (2014). [CrossRef] [PubMed]

15. V. Raj Shrestha, S.-S. Lee, E.-S. Kim, D.-Y. Choi, and M. J. Bloemer, “Polarization-tuned dynamic color filters incorporating a dielectric-loaded aluminum nanowire array,” Sci. Rep. 5(1), 12450 (2015). [CrossRef] [PubMed]

16. J. K. Hyun, T. Kang, H. Baek, D. S. Kim, and G. C. Yi, “Nanoscale single-element color filters,” Nano Lett. 15(9), 5938–5943 (2015). [CrossRef] [PubMed]

17. L. Duempelmann, A. Luu-Dinh, B. Gallinet, and L. Novotny, “Four-fold color filter based on plasmonic phase retarder,” ACS Photonics 3(2), 190–196 (2016). [CrossRef]

18. A. F. Kaplan, T. Xu, and L. Jay Guo, “High efficiency resonance-based spectrum filters with tunable transmission bandwidth fabricated using nanoimprint lithography,” Appl. Phys. Lett. 99(14), 143111 (2011). [CrossRef]

19. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]

20. S. Yokogawa, S. P. Burgos, and H. A. Atwater, “Plasmonic color filters for CMOS image sensor applications,” Nano Lett. 12(8), 4349–4354 (2012). [CrossRef] [PubMed]

21. Y. J. Liu, G. Y. Si, E. S. P. Leong, N. Xiang, A. J. Danner, and J. H. Teng, “Light-driven plasmonic color filters by overlaying photoresponsive liquid crystals on gold annular aperture arrays,” Adv. Mater. 24(23), OP131–OP135 (2012). [CrossRef] [PubMed]

22. S. P. Burgos, S. Yokogawa, and H. A. Atwater, “Color imaging via nearest neighbor hole coupling in plasmonic color filters integrated onto a complementary metal-oxide semiconductor image sensor,” ACS Nano 7(11), 10038–10047 (2013). [CrossRef] [PubMed]

23. R. Rajasekharan, E. Balaur, A. Minovich, S. Collins, T. D. James, A. Djalalian-Assl, K. Ganesan, S. Tomljenovic-Hanic, S. Kandasamy, E. Skafidas, D. N. Neshev, P. Mulvaney, A. Roberts, and S. Prawer, “Filling schemes at submicron scale: Development of submicron sized plasmonic colour filters,” Sci. Rep. 4(1), 6435 (2014). [CrossRef] [PubMed]

24. L. Wen, Q. Chen, S. Song, Y. Yu, L. Jin, and X. Hu, “Photon harvesting, coloring and polarizing in photovoltaic cell integrated color filters: efficient energy routing strategies for power-saving displays,” Nanotechnology 26(26), 265203 (2015). [CrossRef] [PubMed]

25. Z. Li, A. W. Clark, and J. M. Cooper, “Dual color plasmonic pixels create a polarization controlled nano color palette,” ACS Nano 10(1), 492–498 (2016). [CrossRef] [PubMed]

26. E. Balaur, C. Sadatnajafi, S. S. Kou, J. Lin, and B. Abbey, “Continuously tunable, polarization controlled, colour palette produced from nanoscale plasmonic pixels,” Sci. Rep. 6(1), 28062 (2016). [CrossRef] [PubMed]

27. F. Gan, Y. Wang, C. Sun, G. Zhang, H. Li, J. Chen, and Q. Gong, “Widely tuning surface plasmon polaritons with laser-induced bubbles,” Adv. Opt. Mater. 5(4), 1600545 (2017). [CrossRef]

28. Y. Ohko, T. Tatsuma, T. Fujii, K. Naoi, C. Niwa, Y. Kubota, and A. Fujishima, “Multicolour photochromism of TiO2 films loaded with silver nanoparticles,” Nat. Mater. 2(1), 29–31 (2003). [CrossRef] [PubMed]

29. T. Ellenbogen, K. Seo, and K. B. Crozier, “Chromatic plasmonic polarizers for active visible color filtering and polarimetry,” Nano Lett. 12(2), 1026–1031 (2012). [CrossRef] [PubMed]

30. V. R. Shrestha, S.-S. Lee, E.-S. Kim, and D.-Y. Choi, “Aluminum plasmonics based highly transmissive polarization-independent subtractive color filters exploiting a nanopatch array,” Nano Lett. 14(11), 6672–6678 (2014). [CrossRef] [PubMed]

31. L. Wang, R. J. H. Ng, S. Safari Dinachali, M. Jalali, Y. Yu, and J. K. W. Yang, “Large area plasmonic color palettes with expanded gamut using colloidal self-assembly,” ACS Photonics 3(4), 627–633 (2016). [CrossRef]

32. R. Proietti Zaccaria, F. Bisio, G. Das, G. Maidecchi, M. Caminale, C. D. Vu, F. De Angelis, E. Di Fabrizio, A. Toma, and M. Canepa, “Plasmonic color-graded nanosystems with achromatic subwavelength architectures for light filtering and advanced SERS detection,” ACS Appl. Mater. Interfaces 8(12), 8024–8031 (2016). [CrossRef] [PubMed]

33. K. Diest, J. A. Dionne, M. Spain, and H. A. Atwater, “Tunable color filters based on metal-insulator-metal resonators,” Nano Lett. 9(7), 2579–2583 (2009). [CrossRef] [PubMed]

34. L. Frey, P. Parrein, J. Raby, C. Pellé, D. Hérault, M. Marty, and J. Michailos, “Color filters including infrared cut-off integrated on CMOS image sensor,” Opt. Express 19(14), 13073–13080 (2011). [CrossRef] [PubMed]

35. C.-J. Yu, “Transmissive color filtering using plasmonic multilayer structure,” Opt. Eng. 51(4), 44001 (2012). [CrossRef]

36. K.-T. Lee, S. Seo, J. Yong Lee, and L. Jay Guo, “Ultrathin metal-semiconductor-metal resonator for angle invariant visible band transmission filters,” Appl. Phys. Lett. 104(23), 231112 (2014). [CrossRef]

37. K.-T. Lee, S. Seo, and L. J. Guo, “High-color-purity subtractive color filters with a wide viewing angle based on plasmonic perfect absorbers,” Adv. Opt. Mater. 3(3), 347–352 (2015). [CrossRef]

38. Z. Li, S. Butun, and K. Aydin, “Large-Area, Lithography-Free Super Absorbers and Color Filters at Visible Frequencies Using Ultrathin Metallic Films,” ACS Photonics 2(2), 183–188 (2015). [CrossRef]

39. Y. J. Jung and N. Park, “Independent color filtering of differently polarized light using metal-insulator-metal type guided mode resonance structure,” J. Opt. Soc. Korea 20(1), 180–187 (2016). [CrossRef]

40. S. U. Lee, B.-K. Ju, E. S. Kim, D. Y. Choi, and H. J. Lezec, “Wide-gamut plasmonic color filters using a complementary design method,” Sci. Rep. 7(1), 40649 (2017). [CrossRef] [PubMed]

41. P. Neutens, L. Lagae, G. Borghs, and P. Van Dorpe, “Plasmon filters and resonators in metal-insulator-metal waveguides,” Opt. Express 20(4), 3408–3423 (2012). [CrossRef] [PubMed]

42. L. De Sio, G. Klein, S. Serak, N. Tabiryan, A. Cunningham, C. M. Tone, F. Ciuchi, T. Bürgi, C. Umeton, and T. Bunning, “All-optical control of localized plasmonic resonance realized by photoalignment of liquid crystals,” J. Mater. Chem. C Mater. Opt. Electron. Devices 1(45), 7483 (2013). [CrossRef]

43. K.-T. Lee, S. Y. Han, and H. J. Park, “Omnidirectional flexible transmissive structural colors with high-color-purity and high-efficiency exploiting multicavity resonances,” Adv. Opt. Mater. 5(14), 1700284 (2017). [CrossRef]

44. R. B. M. (Richard B. M.) Schasfoort and A. J. Tudos, Handbook of Surface Plasmon Resonance (Royal Society of Chemistry, 2008).

45. J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antenn. Propag. 52(2), 397–407 (2004). [CrossRef]

46. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998).

47. M. J. Dodge, “Refractive properties of magnesium fluoride,” Appl. Opt. 23(12), 1980 (1984). [CrossRef] [PubMed]

48. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

49. V. Teoh, “Samsung UE65JS9500 SUHD TV review,” (HDTV test, 2015) http://www.hdtvtest.co.uk/news/ue65js9500-201502234012.htm.

Algorithm	Dual-simplex
Variables	T(360 nm), T(361 nm), 󠆶…, and T(830 nm)
Lower boundary	0%
Upper boundary	100%
Constraint 1	$\begin{matrix} (x_{target} - 1) \sum_{λ = 360 nm}^{830nm} S_{D65} (λ) T (λ) \bar{x} (λ) \\ + x_{target} \sum_{λ = 360 nm}^{830nm} S_{D65} (λ) T (λ) \bar{y} (λ) \\ + x_{target} \sum_{λ = 360 nm}^{830nm} S_{D65} (λ) T (λ) \bar{z} (λ) = 0 \end{matrix}$
Constraint 2	$\begin{matrix} y_{target} \sum_{λ = 360 nm}^{830 nm} S_{D65} (λ) T (λ) \bar{x} (λ) \\ + (y_{target} - 1) \sum_{λ = 360 nm}^{830nm} S_{D65} (λ) T (λ) \bar{y} (λ) \\ + y_{target} \sum_{λ = 360 nm}^{830nm} S_{D65} (λ) T (λ) \bar{z} (λ) = 0 \end{matrix}$
Objective function for photometric brightness	$\sum_{λ = 360 nm}^{830nm} S_{D65} (λ) T (λ) \bar{y} (λ)$

	$\| t_{b} \|$	$∠ t_{b}$ (rad)	γ₁	γ₂	λ₁ (nm)	λ₂ (nm)
Figure 1(a)	1.00	π	0.20ω₁	0.09ω₂	428	592
Figure 1(b)	0.59	π	0.05ω₁	0.20ω₂	523	701
Figure 1(c)	1.00	−0.59π	0.07ω₁	0.20ω₂	493	647

	Red band	Green band	Blue band
Red filter	75.7%	0.6%	0.7%
Green filter	1.1%	70.2%	5.3%
Blue filter	1.1%	0.7%	70.1%

T_max	Color	p	d	w	a	b	t_Ag	t_SiO2
85%	Red	154 nm	90 nm	-	-	-	35 nm	52 nm
	Green	300 nm	-	36 nm	180 nm	76 nm	83 nm	-
	Blue	332 nm	138 nm	-	-	-	33 nm	106 nm
70%	Red	125 nm	84 nm	-	-	-	32 nm	55 nm
	Green	279 nm	-	58 nm	181 nm	82 nm	86 nm	-
	Blue	263 nm	159 nm	-	-	-	31 nm	106 nm
50%	Red	89 nm	68 nm	-	-	-	33 nm	46 nm
	Green	262 nm	-	77 nm	191 nm	89 nm	96 nm	-
	Blue	195 nm	139 nm	-	-	-	30 nm	113 nm

T_max	Color	p	d	w	a	b	t_Ag	t_SiO2
90%	Red	182 nm	93 nm	-	-	-	41 nm	53 nm
	Green	298 nm	-	31 nm	230 nm	75 nm	63 nm	-
	Blue	342 nm	136 nm	-	-	-	25 nm	107 nm
75%	Red	125 nm	84 nm	-	-	-	32 nm	55 nm
	Green	297 nm	-	56 nm	180 nm	80 nm	88 nm	-
	Blue	273 nm	143 nm	-	-	-	31 nm	108 nm
60%	Red	89 nm	68 nm	-	-	-	33 nm	46 nm
	Green	247 nm	-	58 nm	174 nm	80 nm	86 nm	-
	Blue	226 nm	148 nm	-	-	-	31 nm	109 nm
45%	Red	120 nm	88 nm	-	-	-	36 nm	50 nm
	Green	206 nm	-	53 nm	160 nm	78 nm	88 nm	-
	Blue	191 nm	161 nm	-	-	-	30 nm	107 nm
30%	Red	120 nm	88 nm	-	-	-	36 nm	50 nm
	Green	174 nm	-	53 nm	149 nm	68 nm	75 nm	-
	Blue	166 nm	138 nm	-	-	-	32 nm	118 nm
15%	Red	97 nm	77 nm	-	-	-	39 nm	55 nm
	Green	174 nm	-	53 nm	149 nm	68 nm	75 nm	-
	Blue	167 nm	140 nm	-	-	-	36 nm	123 nm

Bright and vivid plasmonic color filters having dual resonance modes with proper orthogonality

Abstract

1. Introduction

2. Theoretical model

3. Numerical optimizations

4. Results and discussion

5. Conclusion

Funding

References

Cited By

Figures (9)

Tables (5)

Equations (9)

Optics Express