I. Introduction

Structured materials have become ubiquitous in wave physics due to their ability to manipulate electromagnetic waves. Periodic dielectric structures known as photonic crystals (PhCs) are usually designed to stop the propagation of electromagnetic waves in a narrow band of frequencies [1–3]. However, limited symmetry imposed by crystallographic restriction theorem and thus reciprocal space designs of PhCs impede their ability to be utilized in several other potential applications [4]. Due to their richer Fourier space symmetries and patterns, many aperiodic and disordered photonic structures have been designed for such applications as photovoltaics [5–7], optical lenses [8–10], holograms [11–13], broadband absorption in solar cells [14,15], and light extraction from light emitting diodes [16,17]. These structures often utilize tailored reciprocal space such as hyperuniform disordered structures [18–20], photonic glasses [21,22], quasicrystals [23–25] and Moiré lattices [26].

In a weak scattering regime, PhCs show scattering in discrete directions defined by the Bragg condition, while random and aperiodic structures have broadband and wide-angle light control [27–30]. Particularly the correlated disorder leads to distinct features in reciprocal space [31–33]. This can be obtained by short range order [34], as for example happens in particles agglomerates [35–37]. Some sharp changes in reciprocal space can also be obtained by suppression of certain long distance density fluctuations, such as in hyperuniform structures [38,39]. To obtain structures with tailored distribution of scattering directions is challenging. Many inverse design methods have been reported in literature which employs optimization techniques [40,41]. These methods converge to complex distributions of material that give the desired functionality. Many techniques are dependent on manipulation of the real space structure by introducing disorder while aiming to change the reciprocal space towards desired properties [42–44]. An alternative to real space design is to work directly in reciprocal space and later obtain the final structure in the real space [45–48]. The direct manipulation of reciprocal space provides the desirable optical response of the structure.

Recently, we have proposed quasiperiodic structures based on manipulation of reciprocal space to open a complete 2D-photonic bandgap and suppress emission in 3D-media at arbitrarily small refractive index contrast [49,50]. Also, the effect of total internal reflection of light in structures with isotropic selective scattering was investigated [48]. These designs are based on engineered distribution in reciprocal space, which is Fourier transformed into real space and binarized to obtain structures consisting of two distinct materials. Optical filtering plays a pivotal role in controlling the propagation and manipulation of light in various photonic devices. There are numerous applications, ranging from spectroscopy and imaging to telecommunications and sensing, of such filter by selectively transmitting or absorbing light at specific wavelengths or angles [51–53].

In this paper, we propose a low refractive index 3D quasiperiodic structure based on reciprocal space design to obtain direction and wavelength selective trapping. Recently, the group at ITMO university published an independent study of 2D structures with the same functionality [54]. Here we go significantly beyond this work by considering 3D structures and developing an analytical theory. The engineered reciprocal space contains spherical belts offset from the center of reciprocal space, to achieve scattering and trapping only for designed incident light. This is different to usual correlated disorder systems, where disorder correlation function in reciprocal space is spherically symmetric around the center of reciprocal space [31,32] and is similar to anisotropic stealthy hyperuniform structures with non-circular symmetric disorder correlation function reported recently [33]. In contrast to resonant structures the trapping disappears for other wavelengths and does not reappear for a different incident direction. Incident light is designed to be scattered into directions trapped inside the slab of structured material by total internal reflection. On the other hand, the trapped light is only weakly scattered and is attenuated by weak absorption in the slab. We derived an analytical model to estimate the scattering mean free paths for incident and trapped light, based on first order approximation, and to find the absorption in the structure before the light escapes the slab. The structures are optimized to provide strong absorption of the trapped light with only weak absorption of light of different wavelength or direction. It is an intricate optimization problem where absorption length should be greater than the scattering mean free path for trapped light but much smaller than the slab thickness. The prediction of the model is confirmed by full wave numerical simulations of a slab with a 3D quasiperiodic structure. Since the trapped light is absorbed, we practically propose a transparent structure that is able to take out light of a specific wavelength and coming from a specific direction, while light of other wavelengths coming from that specific direction is transmitted and light from any wavelength coming from others than the specific direction is also transmitted. Such structures with dual functionality of direction and wavelength selectivity can find their potential applications in head-up and other transparent displays, for selective color filtering or as wavelength and direction selective optical sensors and detectors (see Fig. S(6) in Supplement 1 for schematic illustration of two examples: selective detectors and transparent displays).

II. Proposed light trapping mechanism

We design here our structures for normal incidence of light, but the same approach can be applied for any other direction. When light of a specific wavelength is incident normally upon a structured dielectric slab of thickness L scattering should occur in the directions illustrated in Fig. 1(a), which are trapped in the slab by total internal reflection. The scattering should be provided by fulfilling Bragg conditions in the disordered structure. The required structure in reciprocal space can be identified by the construction of the Ewald sphere [48]. Light is scattered only in the directions that satisfy the total internal reflection condition at the slab interface (see Fig. 1(a)). The tailored reciprocal space representation of the permittivity of our structure will resemble a spherical belt offset from the center. The light will be then scattered into directions defined by the intersection of the Ewald sphere and the spherical belt (see Fig. 3(a)). The height of the belt is defined by the angle of total reflection.

 

Fig. 1. (a) Schematic representation of the scattering in a slab of thickness L when wave vector ${\vec{k}_{in}}$ is incident in the normal direction. The incident beam enters through the top surface and undergoes scattering within the volume. The scattered light experiences total internal reflection as it interacts with upper and lower surfaces of the slab. Only the first scattering event is shown here, as light would traverse longer propagation distance within the slab for a second scattering event to occur. (b) The same wavelength shown with a red arrow does not scatter and is fully transmitted (c-d) The different wavelength, shown with a green arrow, incident normally or obliquely is transmitted through slab without any scattering.

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For these disordered structures, exhibiting small permittivity perturbation $\mathrm{\Delta }\varepsilon ({\vec{r}} )$, the scattering can be quantitatively estimated by integrating the square of the absolute value of the Fourier transform (FT) of the permittivity distribution across the surface of the Ewald sphere, utilizing the first-order Born approximation [48]. This is equivalent to the consideration utilizing the disorder correlation function [31] if the Fourier transform is evaluated on a large representative volume V. The scattering mean free path can then be calculated from this integral as [31]:

Once the Bragg condition has been satisfied and light is scattered into designed directions, the scattered light will eventually encounter the media-air interface, where it is reflected back and confined within the slab due to total internal reflection. A small imaginary component of permittivity is introduced which does not substantially absorb the transmitted light, which spends only a short time or path length in the slab, and significantly absorbs trapped light that stays in the slab for longer time and path length. The light that is trapped may scatter again and exit the slab; however, the probability of such secondary scattering is significantly reduced for light traveling in the horizontal direction due to diminished overlap of Bragg peaks with Ewald sphere as seen in Fig. 3(b).

The absorption of trapped light can be achieved through a delicate balance among the absorption length ${l_a}$ attributed to the minor imaginary part of effective permittivity and scattering mean free paths covered by both normally incident light ${l_ \bot }$ and trapped light ${l_\parallel }$. Similarly, when light of a comparable wavelength is incident at oblique angles, as depicted in Fig. 1(b), or even when light of a different wavelength is incident either normally or obliquely upon the structured material Fig. 1(c-d), light is transmitted through the structure with minimal or no scattering due to non-compliance with Bragg conditions and only weak absorption.

III. Structure generation from reciprocal space

Quasiperiodic 3D structures were numerically synthesized in reciprocal space on a 3D grid. The design within this reciprocal space features two spherical belts characterized by central radius of ${k_0}$ and radial thickness of $2\Delta k$. This implies that the inner radius of the belt is ${k_0} - \Delta k$ while the outer radius is ${k_0} + \Delta k$ encompassing the entire thickness of the belt, as illustrated in Fig. 3. The belts in reciprocal space are filled with constant amplitude and random phase on the grid $dk = 2\pi /L$, where L is the edge length of cubic structure in real-space. The belt thickness defines the spectral selectivity of the scattering. But it should not be much smaller than the resolution $dk$ in reciprocal space, as in this case homogeneous filling of the belts is problematic.

Spherical belt sections originate from the sphere overlapping with the Ewald sphere for the incident light, which is subsequently modified through the removal of its spherical caps as shown in Fig. 2(a). Removal of spherical caps is dictated by a condition of critical angle ensuring total internal reflection of scattered light. We have two spherical belts on opposite sides of the origin O of reciprocal space to maintain central symmetry, thus, to get a real-valued refractive index distribution in real-space. We already know that two diametrically opposite Bragg peaks with same distance from center O of reciprocal space constitute a single grating in reciprocal space. Therefore, real space consists of superposition of a large number of gratings depending on the number of Bragg peaks in the belt.

 

Fig. 2. (a) A schematic of the spherical belts with central radius ${k_0}$ shown with purple color in reciprocal space. (b) Binarized quasiperiodic structure with single cuboidal unit cell with edge length of $L$. The structure in the cube is periodic in all directions.

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Once the reciprocal space design has been generated, we take the inverse Fourier transform of the belt function to get the quasiperiodic structure in real space with continuous distribution of refractive indices. The structure in real space is a result of superposition of many 1D sinusoidal gratings with different grating periods because the distance from center of reciprocal space to any of Bragg peak in spherical belt is different. Eventually, continuous refractive index distribution is mathematically binarized assigning two materials to indices above and below average level [49,50]. This way we obtain a mixed structure out of two different materials with refractive indices ${n_1}$ and ${n_2}$ possessing equal volume fractions as shown in Fig. 2(b).

We can mathematically express the binarized refractive index distribution as follows:

In above equation, $\Delta n$ is the refractive index perturbation from the mean value $\bar{n}$ such that ${n_1} = \bar{n} + \Delta n$ and ${n_2} = \bar{n} - \Delta n$, N is number of sinusoidal gratings used to generate the quasiperiodic structure, ${\vec{g}_i}$ are the wave vectors defining the grating periods which, except the mirror symmetry, are different for each grating in this case and ${\phi _i}$ are the corresponding random phases assigned to each grating. The phase randomization allows avoiding singularity in the center of the simulation volume and helps to analytically estimate the strength of the original gratings in the binarized structure. The same as in the Ref. [49] after the inverse Fourier transform from the reciprocal space to real space, we obtained a Gaussian random field with Gaussian distribution of the amplitude. The binarization of Gaussian random field reduces the strength of the original gratings to $2/\pi $ as shown in SI of Ref. [49]. We confirm that by applying Fourier transform on binarized structures and comparing it to the original structure in reciprocal space. The binarization introduces additional homogenous noise around the belts, without creation of any secondary belts (see Figure S2 in Supplement 1).

IV. Theory

When light of a specific wavelength is incident on the quasiperiodic structure such that the Ewald sphere completely overlaps with the spherical belt in $k$-space, then the length of incident wave vector ${k_{in}} = {k_0}$. We can then express the equation for trapped wavelength ${\lambda _t}$:

We take here the mean refractive index as a simple arithmetic mean because the two materials have equal volume fractions, and the refractive index contrast is small. The spherical belt section is deliberately shifted by ${k_0}$ from the center of $k$-space such that the Ewald sphere with the radius equal to the length of the incident wave vector ${\vec{k}_{in}}$ and pointing towards the origin O completely overlaps with it to maximally scatter and trap the incident light as shown in Fig. 3(a). To design trapping for obliquely incident light, the belts would have to be laterally shifted in reciprocal space. Ideally, thickness of the spherical belt should be infinitely thin to trap a single wavelength but numerical generation of such structures would require infinite volumes. Thus, we define the belts of finite thickness.

 

Fig. 3. (a) Ewald sphere, shown by the black circle, completely overlaps with the belt of thickness $2\Delta k$ for the normally incident light. Exemplary scattering directions are shown by dashed arrows. The plane ${k_x} = 0$ is shown. (b) Ewald sphere for the trapped light propagating along ${k_z}$ is drawn which has only a small overlap with the belts. Exemplary scattering directions are shown by dashed arrows.

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The scattering mean free path for normally incident light ${{l}_ \bot }$ can be estimated from the first order approximation. We can calculate the average intensity of the normalized square FT in the belts b from the real space refractive index distribution employing the Parseval theorem (see section S1 in Supplement 1). Integration on the surface of the Ewald sphere will be simplified then to the area of the spherical belt ${A_b}$ multiplied with intensity b. Once b and ${A_b}$ are known, we can estimate the integral as:

We find that the predominant scattering angle ${\theta _s}$ is 90°.

Then the scattering mean free path can be calculated as:

In similar manner, the scattering mean free path for the trapped light ${l_\parallel }$ inside the medium is also calculated (see section S2 in Supplement 1). We can draw a cross section of the Ewald sphere for trapped light inside medium which cuts through the spherical belt section as shown in Fig. 3(b). ${l_\parallel }$ is the mean distance traveled by photon inside quasiperiodic structure in the direction perpendicular to slab normal before the first scattering happens. Light which is trapped would cover a larger mean free path due to the smaller intersection area between the Ewald sphere and the spherical belt as opposed to the complete overlap in case of the normally incident light (see Fig. 3(a)). After simplification, we end up with the relation where ${l_\parallel }$ can be calculated if ${l_ \bot }$ is known (see section S2 in Supplement 1), where ${K_{arc}}$ is the length of the arc that Ewald sphere cuts through the belt.

It can be shown that ${l_\parallel }$ is always independent of belt thickness. ${l_ \bot }$ is proportional to the belt thickness and the average intensity b is inverse proportional to belt thickness (see S1 in Supplement 1). In Eq. (6), the belt thickness $2\mathrm{\Delta }k$ is in the denominator and ${l_ \bot }$ in the numerator so the effect of belt thickness for ${l_\parallel }$ would be canceled out. Though ${l_\parallel }\; $ is larger than ${l_ \bot }$, the light which is trapped can still scatter once again and exit through the slab interfaces. But even after a possible second scattering, part of trapped light will still stay inside the slab. Thus, effectively the light would travel even further before exiting the structure. The intersection of Ewald sphere with the spherical belt results in two arcs on top and bottom of Ewald sphere surfaces. Upon scattering, the probability of escape is ${P_{esc}} = $ $K$/${K_{arc}}$ which is the ratio of the length of the arc in the exit cone K to the length of the arc ${K_{arc}}$. Therefore, light before escaping will cover larger path length than ${l_\parallel }$: ${l_{esc}} = {l_\parallel }/{P_{esc}}$ (see section S2 for detailed derivation in Supplement 1).

Absorption length ${l_a}$ is another important parameter which should be optimized such that the trapped light is significantly absorbed but transmitted light is only marginally absorbed. Absorption length ${l_a}$ can be calculated from following expression (see section S3 in Supplement 1),

Here, $\mathrm{{\cal K}}$ is the coefficient which is related to the mean refractive index and loss tangent of the materials given in supplementary equation S(14). We can now find out the maximum absorption ${A_{max}}$ due to trapping in quasiperiodic structures (see section S3 in Supplement 1 for derivation):

In above equation, the term in the first bracket corresponds to the incident light that is scattered or absorbed and does not reach the bottom of the slab. The second term describes which part of this missing light is absorbed. The first term in the second bracket corresponds to absorption of the incident light on its initial propagation direction. The second term in the second bracket corresponds to the absorption of the scattered light. The scattered light is trapped by total internal reflection and has probability of absorption equal to ${l_{esc}}/({{l_{esc}}\textrm{\; } + {l_a}\textrm{}} )$. But, we are also interested to minimize the absorption for light of different wavelength or different angle of incidence which we call ${A_{min}}$:

It also results from Eq. (8) by assumption of infinite ${{l}_ \bot }$.

To find out the absorption due to trapping only, we define the absorption contrast $\Delta \textrm{A}$ which is given by:

For the application, the proposed structured films should maximize absorption at trapped wavelength and minimize it elsewhere. Thus, we need to maximize the absorption contrast.

With that we can estimate the maximal absorption contrast for varying structural parameters (Fig. 4). As expected, we find maxima of $\Delta A$ for optimized absorption length and fixed other parameters. Too much absorption strongly increases ${A_{min}}$ and too weak absorption reduces ${A_{max}}$. The absorption contrast approaches 0.7 for optimal slab thickness of 4 µm and spherical belt with radial thickness of 2% of belt radius. We define here refractive index contrast as difference of refractive index of two media ${n_2} - {n_1} = 2\mathrm{\Delta }n$, where ${n_2}$ is always greater than ${n_1}$. $\Delta A$ of nearly 0.7 can be obtained for any given refractive index contrast with fixed belt thickness of 2% of belt radius but large structures are then required for smaller index contrasts to reach same amount of scattering. We can also theoretically maximize $\Delta A$ to approach 1 for any given refractive index contrast with extremely thin spherical belts (see Fig. S5 in Supplement 1). However, it is difficult to get smaller belt thicknesses as resolution of $k$-space is dependent on the thickness of quasiperiodic structure given by $dk = 2\pi /L$. To conclude, a delicate balance between ${l_ \bot }$, $\textrm{}{l_{esc}}$ and ${l_a}$ is required such that ${l_a}$ is $\Delta \textrm{A}$ much larger than ${l_ \bot }$ to not disturb normally incident light and is smaller than ${l_{esc}}$ to absorb the trapped light before it leaves the slab. At the same time the incident light of designed wavelength should be trapped within the slab thickness. Only then we can maximize absorption contrast in Eq. (10).

 

Fig. 4. Absorption contrast $\Delta \textrm{A}$ plotted with respect to the absorption length ${l_a}$ for the different slab thicknesses L to find the maxima of $\Delta \textrm{A}$. The belt thickness is fixed to 2% of the belt radius and refractive index contrast of $2\mathrm{\Delta }n = 0.1$.

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V. Numerical simulations and results

To confirm the predicted absorption contrast $\Delta \textrm{A}$ in real quasiperiodic structures, we conducted simulations in time domain solver of CST Studio Suite 2023 [55]. An antireflection coating with thickness defined by $t = {\lambda _t}/4\sqrt {\bar{n}} $ was applied to the top and bottom of the quasiperiodic structure to minimize Fresnel reflections and Fabry Perot oscillations. Air volume was added at the top and bottom of the structure with open boundary conditions. The structure was excited by a plane wave with wavelength incident from air normally in the $y$-direction. The lateral boundaries were treated as perfect electric and magnetic mirrors to mimic infinite structure in these directions. Power loss density monitors were utilized at various frequencies to obtain the absorption spectra. The purpose of utilizing power loss monitors is to calculate the total loss within 3D structure across various frequencies.

A single cubic unit cell possessing a volume of 6 × 6 × 6 µm³ was selected with lateral dimensions x and z assumed to extend infinitely due to mirror boundary conditions. For a slab thickness L of 6 µm and trapped wavelength of 600 nm we chose a belt thickness constituting 2% of the belt radius, ${k_0}.$ Though this thickness is smaller than the resolution in the reciprocal space, we still have sufficient number of Bragg peaks in the belts to assume a homogeneous distribution. A relatively small loss tangent $\mathrm{tan\delta }$ of 2.5 × 10⁻³ was employed corresponding to absorption length of 25 µm which is adequate to absorb the trapped light and can be simulated easily. Using a loss tangent smaller than 2.5 × 10⁻³ would be computationally challenging as simulation would need a significantly extended simulation time to absorb the trapped light. Refractive indices of 1.48 and 1.58 were chosen, resulting in a small index contrast of 0.1, which also fulfills the prerequisite of first order approximation. We have deliberately chosen the refractive index contrast of 0.1 to provide enough scattering within the 6 µm slab thickness and with the belt thickness of 0.02 ${k_0}$. Further reduction in the index contrast would necessitate an extremely thin belt to achieve comparable scattering which poses a challenge for numerical generation and simulation of such structures. Our numerical simulations with aforementioned refractive indices and belt thickness demonstrated an absorption difference of 0.63 where ${A_{max}}$ is close to 0.85 and ${A_{min}}$ is 0.22 which is also theoretically expected absorption at wavelengths other than the trapped one (see Fig. 5(b)). We can clearly see the absorption peak at 611 nm which is also close to the designed wavelength. Absorption difference demonstrated in simulation is comparable to analytical model predictions which is 0.66 and indicated with red circle in Fig. 5(a). Similarly, simulation results for increased belt thickness of $0.2{k_0}$ are also in good agreement with our analytical model (see Fig. S4 in Supplement 1). The width of the absorption peak is approximately 40 nm and can be further reduced by decreasing the belt thickness. We can also verify the directional scattering of incident light in numerical simulation by implementing the far-field monitors at the desired wavelength (see Figure S1 in Supplement 1).

 

Fig. 5. (a) Absorption contrast $\Delta \textrm{A}$ plotted with respect to absorption length ${l_a}$ for a fixed slab thickness L of 6 µm and 0.02${k_0}$ belt thickness, red circle shows the case for which quasiperiodic structure has been simulated (b) Absorption inside the 3D quasiperiodic structure using several power loss monitors for ${n_1} = 1.58$ and ${n_2} = 1.48$.

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To examine the angular selectivity of the quasiperiodic structure, we conducted additional simulations by rotating the spherical belts anticlockwise by 5° and 10° along ${k_y}$ axis around the origin of the reciprocal space (see Fig. S3 in Supplement 1). We chose the same belt thickness of 0.02${k_0}$ and simulation volume of 6 × 6 × 6 µm³ for rotated spherical belt with refractive indices set at 1.48 and 1.58. Mirror boundary conditions in transient solver confine the planewave to be incident in normal direction only, prompting the rotation of structure in $k $-space for our analysis. We expect the results obtained from this approach would not diverge much from the case when the incident wave is rotated.

Even a small rotation of 5° results in disappearance of absorption peak at the designed wavelength because the Ewald sphere would only slice through the spherical belt creating little overlap, thus very weak scattering of incident wave. We see weak broadband increase of the absorption ranging from 580 nm to 700 nm due to weak scattering and trapping at these wavelengths (see Fig. 6(a)). Rotation of the spherical belt by 10° leads to the split of the absorption into two maxima (see Fig. 6(b)). This can be explained from the intersection of the Ewald sphere with the spherical belt. For small rotation angles, the Ewald sphere would touch the belt at slightly larger and smaller wavelengths exhibiting a partial overlap. As the rotation angle is increased, partial overlap between the Ewald sphere and the spherical belt would occur only for smaller wavelengths. It can be seen already at 10° that the short wavelength peak is smaller than the long wavelength peak.

 

Fig. 6. Absorption spectra for the quasiperiodic structure rotated in reciprocal space by 5° (a) and 10° (b)

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VI. Conclusion

In summary, we presented here a 3D quasiperiodic structure whose reciprocal space is engineered to efficiently scatter the incident light with designed wavelength and into the directions which are trapped by total internal reflection. Reciprocal space of such structure contains two spherical belts of well-defined thickness to efficiently scatter in all allowed directions. The spectral width of the absorption peak is dependent upon the belt thickness. Any spectral and angular selectivity can be achieved by reducing the belt thickness. Such structures might even achieve selectivity of volume Bragg gratings [56], but would still trap scattered light instead of reflecting it. The proposed structures can be 3D printed for microwave or optical applications. In this case the structural integrity should be evaluated and if necessarily additional connections should be added as mechanical supports. Such random connections would not significantly change the scattering response of the structure.

An analytical estimation containing scattering mean free paths and absorption length based on first order approximation are derived here which predicts the maximal absorption contrast for a given belt and slab thicknesses at a given small RI contrast. We show theoretically that optimized structures are possible with absorption contrast approaching one. Thus, transparent structures can be envisaged which absorb light only of specific direction and wavelength. We have conducted simulations for 6 µm films in the visible wavelength range with a belt thickness limited by the computational constraints and demonstrated absorption contrast of 0.63, in close agreement with the theoretical prediction. We envisage a higher absorption difference in our structure with thinner belt thicknesses.

The presented effect allows trapping and absorption of light coming from specific direction with specific wavelength. This effect might be useful when transparent structures are required with a filter option. Traditional multilayer, holographic or metasurface filters would show also response at other wavelengths and directions, thus being not fully transparent. If absorption is absent in the structure, the proposed concept can be used also as direction and wavelength selective diffuser. Only the light coming from the designed direction with the designed wavelength will be scattered into the film and then diffusely scatter from the film. This way transparent display can be envisaged (see Fig. S6 for illustration).