Optimization of sharp and viewing-angle-independent structural color

Structural coloration produces some of the most brilliant colors in nature and has many applications. However, the two competing properties of narrow bandwidth and broad viewing angle have not been achieved simultaneously in previous studies. Here, we use numerical optimization to discover geometries where a sharp 7% bandwidth in scattering is achieved, yet the peak wavelength varies less than 1%, and the peak height and peak width vary less than 6% over broad viewing angles (0--90$^\circ$) under a directional illumination. Our model system consists of dipole scatterers arranged into several rings; interference among the scattered waves is optimized to yield the wavelength-selective and angle-insensitive response. Such designs can be useful for the recently proposed transparent displays that are based on wavelength-selective scattering.

Here, however, we show that both goals can be achieved simultaneously: a narrow 7% bandwidth can be obtained for reflection at all observation angles (0-90 • ) with directional illumination of an optimized structure.Besides the bandwidth (color saturation), the reflectivity's peak wavelength (hue) and peak intensity (brightness) are also insensitive to the viewing angle.The structure consists of a collection of wavelength-scale ring scatterers amenable to fabrication by direct laser writing (multiphoton lithography) [31][32][33][34][35], and is optimized so that constructive interference occurs only in a narrow bandwidth but with a dipole-like broad-angle pattern.We perform the optimization using a simplified semi-analytical model, and validate our results via 3D boundary-element method (BEM) simulations.One immediate application of this design is for angle-independent viewing of transparent displays based on wavelength-selective light scattering [36].
Previous work on synthetic structural color has explored a wide range of designs, but none has been able to achieve narrow bandwidth (sharp color) and angular insensitivity (wide viewing angle) simultaneously.Multilayer films [11][12][13], periodically-modulated surfaces [14][15][16], and three-dimensional photonic crystals [37] reflect light only at a discrete set of angles rather than omnidirectionally.Amorphous structures scatter light to all directions [17][18][19][20][21][22][23][24][25], but their colors are not as sharp and are viewing-angle dependent under directional illumination [26,27].The dipole scattering from resonant spherical nanoparticles has a broad-angle pattern, but sharp colors cannot be achieved because dielectric particles tend to have multiple overlapping resonances [28,38], and metallic particles are limited by absorption loss which broadens the resonances [36,39,40].Topological optimization has recently been used to design structural colors [29,30] but was focused on maximizing the intensity of a prescribed color, rather than on minimizing the bandwidth or the angular dependence.From these prior studies, it appears that simple hand-designed geometries cannot achieve both narrow bandwidth and broad viewing angle.Therefore, we use numerical optimization to explore to what extent these two properties can be achieved simultaneously.We focus on the case of illumination from a directional light source, as is the case for viewing "structurally painted" objects under bright sunlight, for sensing applications, and for the display application of interest [36].
We start by considering light scattering from a collection of point scatterers where the electric dipole scattering dominates.There are a variety of ways to realize this model system; later in this Letter, we describe a simple and realistic realization using stacked dielectric rings.Here, scatterer j has electric dipole moment p j = 4π 0 α j E 0 , where α j is the electric polarizability with units of volume, 0 is the vacuum permittivity, and E 0 is the incident electric field.Neglecting multiple scattering, the differential scattering cross section from this collection is [41] where k = 2π/λ is the wavenumber, λ is the wavelength, êin and êout are the polarization vectors of the incident and the scattered light.Coherent interference among radiation from different scatterers gives rise to the "structure factor" Here, r j is the position of the j-th scatterer, and q is the momentum transfer vector with nin and nout being the propagation direction of the incident and the scattered light.Note that we include α j in the definition of S(q) to account for different types of scatterers, but for simplicity we consider frequencyindependent scalar α j (no metallic particles where α j depends strongly on frequency).The assumptions in this model-that dipole scattering dominates with a constant scalar polarizabilty and that multiple scattering is negligible-are valid for subwavelength weak scatterers and can be realized in a variety of physical systems.In this model system, the angle and wavelength dependence comes from S(q) and can be calculated efficiently for fast optimization.
The interference can give rise to coloration through a wavelength-selective S(q).However, this wavelength dependence is coupled to the incident and the viewing angles through Eq. ( 3), so structural colors are typically angle dependent.For example, periodic structures scatter strongly when q lies on the reciprocal lattice [41], and the peak wavelength depends sensitively on the angles.The S(q) of amorphous structures depends only on |q|, so the peak wavelength is a function of the angle between illumination and view [26,27]; for a fixed illumination direction, the peak wavelength varies with viewing angle, as given by Eq. ( 3).Such typical behavior of a viewing-angle-dependent structural color is schematically illustrated in Figure 1(a).
We illustrate the desired behavior in Figure 1(b), where the structural color is viewing-angle independent when illuminated from a fixed direction.This requires an S(q) that reaches its maximum at the target wavelength (corresponding to the desired color) and is low at other wavelengths, with very little variation when the viewing angle is varied.We use numerical optimization to search for such structures.We define a figure of merit (FOM) that, as we demonstrate below, captures the desired char-acteristics, where A is S(q) at the target wavelength averaged over the viewing angles, B is S(q) averaged over both the viewing angles and the wavelengths in the visible spectrum, and C is the maximum value in the visible spectrum for the standard deviation of S(q) with respect to the viewing angles.We take the illumination to be in the z direction (n in = ẑ), and the angular average and standard deviation are calculated by integrating over all solid angles nout in the backward hemisphere (i.e., all nout with nout • ẑ ≤ 0), which are the typical angles when viewing the structure's color and for applications such as displays and sensing.We search for structures in the 4N -dimensional parameter space (given by {α j , r j } N j=1 ) that maximize this FOM.
Optimization frequently led to structures where the discrete points arrange themselves into several rings aligned along the z axis to eliminate the azimuthal part of the viewing-angle dependence.With this insight, we focus our attention on structures made of rings aligned along the z axis.Instead of parameterizing coordinates of the individual point scatterers, we now directly parameterize coordinates of the rings to reduce the dimensionality of the search space and to avoid spurious local optima.The computation is further accelerated by analytically summing over point scatterers in a ring (assuming closely spaced points), as where the new summation is over the constituent rings, q z and q ρ are the z and the radial components of the q vector, and J 0 is the Bessel function.Each ring is specified by its z coordinate z j , its radius ρ j , and its weight w j (which is given by the number of point scatterers in this ring times the polarizability per scatterer).
We then optimize the ring-shaped structures.An appropriate search strategy is important; directly applying global or local optimization algorithms tends to yield very suboptimal results due to the high dimensionality of the parameter space, the unevenness of the FOM landscape, and the sensitivity on initial guess.We find that a step-wise procedure [42] is very effective for this problem.Starting from vacuum, we add rings one by one.Each time a new ring is added, a global optimization is performed on parameters {z j , ρ j , w j } of the new ring only, keeping the existing parameters fixed; then a local optimization on all of the parameters is performed.This procedure keeps the FOM high as the parameter space expands its dimensionality, and global optimization is used only for the more manageable low-dimensional searches.After some experimentation with a free optimization package [43], we chose well known local [44] and global [45] search algorithms and implemented our FOM [Eq.( 4)] to four-digit accuracy in the high-performance dynamic language Julia [46].The best structure is picked from results of several independent runs.Figure 2 shows the optimized structures and their corresponding structure factors.The wavelength window is 400-800 nm, and the target wavelength is 600 nm; structures for other target wavelengths (with corresponding shift in the window) can be obtained through scaling the structure sizes since Eq. ( 1) is scale invariant.With 10 rings (shown in the upper panel), the structure reaches FOM = 2.54; its color sharpness is characterized by A/B = 3.51, and its viewing-angle independence is characterized by A/C = 9.15.With 40 rings (shown in the lower panel), the optimization discovers a structure that reaches FOM = 5.51, with its color sharpness characterized by A/B = 7.64 and angle independence by A/C = 19.8.The full width at half maximum (FWHM) of this structure factor is only 43 nm at λ 0 = 600 nm, corresponding to a narrow 7% bandwidth.As shown in Figure 3, the hue (peak wavelength), the brightness (peak height), and the saturation (contrast between the target color and the other colors) are all nearly independent of the viewing angle.We validate this design by realistic simulations below.
Interestingly, the parameters of these optimized structures (as shown in Figure 2 and tabulated in supplementary Tables S1 and S2 [47]) do not exhibit perceivable patterns, suggesting that hand design would not be a good route for this problem.The optimization procedure discovers structures with very different parameters but comparable performance, suggesting that we can pick structures more amendable to fabrication without sacrificing much performance.
A particularly simple realization of our model system consists of dielectric rings embedded in a transparent medium, with low refractive-index contrast between the two.Such a structure may be fabricated using direct laser writing (multiphoton lithography), which has been used to fabricate waveguides [34], photonic crystals [31], and many complex high-resolution three-dimensional structures [35] with feature size as small as 40 nm [32,33].To cover a large-area surface, one may place many copies of the optimized structure at random positions on the surface to increase the overall response while avoiding inter-structure interference.To verify that all assump- tions in our model are valid in such a continuous-ring structure, we use a free-software implementation [48,49] of the boundary-element method (BEM) to solve the corresponding scattering problem in the original 3D vectorial Maxwell's equations.BEM employs no approximation aside from discretization, so it accounts for effects not considered in our model such as multiple scattering and scattering beyond the dipole approximation.Given the computation cost, we only perform the fullwave BEM calculation on the optimized 10-ring structure, at 50 wavelengths and 50 angles.The z coordinate and major radius of each ring are taken directly from the already-optimized parameters z j and ρ j .The weight w j is the polarizability per scatterer times the number of scatterers in each ring, so it is proportional to the ring volume.Therefore, we choose the thickness of each ring to have its volume proportional to the optimized w j while keeping the thinnest ring thicker than the 40-nm resolution limit of direct laser writing.We consider dielectric rings with = 1.2 in air; when the index contrast is low, a different dielectric material changes only the overall scattering strength, and a different medium only scales the overall wavelength.The BEM-calculated differential scattering cross section of this structure yields FOM = 2.51 (with A/B = 3.58 and A/C = 8.38), versus FOM = 2.54 predicted by the semi-analytical S(q).The angle-resolved scattering spectrum from BEM is shown in supplementary Figure S1 (b) [47]; it agrees very well with S(q) [shown in Figure S1 (a)], verifying that our model is appropriate for such low-index-contrast continuous-ring structures and that the optimized result has a certain degree of robustness with respect to how the model is realized.Future work could look for results even more insensitive to errors and implementation details via the "robust optimization" techniques [50][51][52][53][54][55][56][57].
The same optimization procedure can be used when a different illumination angle is considered.However, independence of both the incident and the outgoing angles is not possible; S(q) is a function of q = k(n in − nout ), so independence of both angles would require independence of the wavelength as well, meaning no color.It may be possible, however, to reduce angle dependence or enhance wavelength selectivity by intentionally going beyond dipole scattering and single scattering, or by introducing resonances in the polarizabilities or in the form factors of individual scatterers.Another interesting future direction would be to explore whether there is a fundamental lower limit on the product of the angular and frequency bandwidths per unit volume, analogous to similar limits on the delay-bandwidth product [58].
The wavelength-selective and viewing-angleindependent light scattering here can be useful for the selective-scattering transparent display recently proposed [36].In Ref. [36], the wavelength selectivity was achieved using plasmonic nanoparticles, but that approach is limited by absorption from the metal, which introduces undesirable absorption and resonance broadening; the approach here has no such limitations.Another application of this study is the creation of "structural paints" with accurately defined colors that will never fade.The sharp wavelength response can be modified by the presence of nearby molecules, so our structure may also be used for chemical and biosensing.5) in the main text] and the exact scattering response calculated using the boundary-element method (BEM) for a dielectric-ring structure.(a) Scatteringangle-resolved spectrum of the structure factor S(q) (in arbitrary units) when the optimized 10-ring structure is illuminated along the z direction.(b) Normalized differential scattering cross section, (dσ/dΩ)/k 4 (in arbitrary units), calculated using BEM.BEM employs no approximation aside from discretization, so it accounts for multiple scattering and scattering beyond dipole approximation.The incident light is E0 = E0e ikz x, and we calculate light scattered into the y-z plane using Eq.(10.93) in Ref. [41] within the BEM framework.The particular system here consists of dielectric rings ( = 1.2) in air.(c) The surface mesh we use for the BEM calculation, with a total of 3100 triangles.The thickness (minor diameter) of each ring lies in between 42 nm and 122 nm.
Supplementary Tables

FIG. 1 .
FIG. 1.(a) Schematic illustration of a typical structurallycolored object under directional white-light illumination, where its color depends strongly on the viewing angle.(b) Schematic illustration of an object with a structural color that is independent of viewing angle under directional illumination.

FIG. 2 .
FIG.2.Structures optimized for a structural color that is independent of the viewing angle.(a) Visualization of an optimized structure consisting of dipole scatterers that make up 10 rings.(b), (c) Radii and weights of the rings.(d) Calculated structure factor for light incident along the z direction (nin = ẑ) and scattered into directions in the backward hemisphere (all solid angles nout with nout • ẑ ≤ 0).The solid curve is S(q) averaged over all of these viewing angles, and the vertical bars represent the 10th and the 90th percentiles (i.e., S(q) range spanned by 80% of the viewing angles).(e)-(h) Same plots for an optimized structure with 40 rings.

FIG. 3 .
FIG. 3. Scattering-angle-resolved spectrum of the structure factor when the 40-ring structure is illuminated along the z direction.The y-axis scale accounts for the weight in the solid angle integration, | sin θdθ| = |d(cos θ)|.
FIG. S4.Comparison between the semi-analytical model using structure factor [Eqs.(1) and (5) in the main text] and the exact scattering response calculated using the boundary-element method (BEM) for a dielectric-ring structure.(a) Scatteringangle-resolved spectrum of the structure factor S(q) (in arbitrary units) when the optimized 10-ring structure is illuminated along the z direction.(b) Normalized differential scattering cross section, (dσ/dΩ)/k 4 (in arbitrary units), calculated using BEM.BEM employs no approximation aside from discretization, so it accounts for multiple scattering and scattering beyond dipole approximation.The incident light is E0 = E0e ikz x, and we calculate light scattered into the y-z plane using Eq.(10.93) in Ref.[41] within the BEM framework.The particular system here consists of dielectric rings ( = 1.2) in air.(c) The surface mesh we use for the BEM calculation, with a total of 3100 triangles.The thickness (minor diameter) of each ring lies in between 42 nm and 122 nm.

TABLE S1 .
Parameters of the optimized 10-ring structure shown in Figure2(a-c) of the main text.

TABLE S2 .
Parameters of the optimized 40-ring structure shown in Figure2(e-g) of the main text.