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Abstract
Perfect absorbers play crucial roles in optical functional devices. Among various types of absorbers, moth-eye structures are known for their excellent absorbing efficiency. In this paper, we apply an electromagnetic multipole expansion method to treat an isolated all-dielectric moth-eye structure as a large particle and calculate various electric and magnetic multipole modes within the moth-eye structure. In periodical array, the multipole modes within each particle interact with each other. These constructive or destructive interactions cause shifts in the multipole resonant peaks. The multipole modes inside the particle array introduce reflecting peaks for loss-less materials. The absorption enhancement inside moth-eye structures can be attributed to the electric field enhancement resulting from these electric and magnetic multipole modes. Based on our theoretical study, we propose a near-ideal selective absorber based on moth-eye array, which achieves near 100% absorption within wavelength range from 400 nm to 1500 nm while achieving near 0% absorption over about 1700 nm.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recent years have seen remarkable progress in the area of metamaterials. Their applications have covered substantial numbers of optical functional devices and systems such as refractive index sensors [1–3], thermal emitters [4], and light harvesting systems [5,6]. Among them, electric-magnetic wave absorbers are widely used for energy transferring, saving or collecting purposes. These absorbers work at different wavelength ranges and their absorption spectra are tailored appropriately for different functions, like thermal emitting [7–9], solar energy transforming [10], and infrared and terahertz detecting [11,12]. To achieve high absorbing performances, many absorbers are constructed with the idea of exciting plasmon resonances, like surface plasmon polaritons (SPPs) in metal structures [13,14], and phonon polariton resonances (PPRs) in photonic structures [15]. Through this way, electric fields near the structures are enhanced and localized in sub-wavelength sizes. The localized fields are then consumed and transferred into thermal heat. In literatures, the absorbing efficiencies of plasmonic absorbers can be improved by coupling various resonant modes, like localized plasmon polariton (LSP) [16,17], cavity resonance (CR) and Fabry-Perot resonance (FP) [18]. Owing to their high absorbing efficiency and diversity, a large part of present absorbers are plasmonic absorbers.
Among various kinds of absorbers, moth-eye structures are utilized for improving absorbing efficiency and realizing broadband absorption. For example, Devarapu et al. reported a SiC- micro-pyramid structure, which utilizes both phonon polariton resonances and moth-eye effect to realize 90% absorption in infrared region where the strong photon-phonon coupling yields a negative permittivity for SiC [15]. Wu et al. proposed another pyramid structure consisting of alternative Al2O3 and SiO2 layers. The structure can realize over 80% absorption/emission at the mid-infrared region and near 0% reflection at the visible and near-infrared region [19]. In these absorbers, lossy dielectric or semiconductor materials are applied as the main absorbing materials instead of metals. These materials are sculpted into pyramid- or moth-eye like structures to reduce the reflection of light [20–22], hence to improve the absorbing efficiency.
The high absorption in these moth-eye structures is caused by the moth-eye effect, which was first found from the cornea of nocturnal insects. Present explanations for the high absorbing performance of moth-eye effect are mainly based on the impendence matching theory [23–26]. According to these explanations, the moth-eye structure is equalized as a serial of thin layers with gradual increased effective refractive index. The top-down distribution of the gradual increased effective refractive indexes results in the impendence matching between air and the moth-eye structure. Once the impendence matching condition is satisfied, the reflection of a structure reaches to the minima. There are also other explanations about the absorption enhancement of moth-eye structure. For example, Cui et al. illustrated that the moth-eye structure they proposed can be akin to a waveguide, which supports slow light modes within a broadband wavelength range and leads to broadband absorption [9]. Besides, Liu et al. applied equivalent inductor-capacitor circuits to describe the perfect absorption caused by magnetic resonances in their structure [27]. These methods have been proved to be useful in their cases. However, the moth-eye structures are often designed to have large heights to support various vertical electric and magnetic modes [27]. There are little works relating to the explanation of the reflection-reduction effect [28] and the absorption-enhancement effect inside moth-eye structures in the view of the electric and magnetic multipole modes according to our best knowledge. As the electric and magnetic multipole modes, like electric dipole mode, and magnetic dipole mode, are the main electric and magnetic modes in the high-index all-dielectric particles, the study of these multipole modes can help to understand the reflection-reduction effect of moth-eye structures and pave way for the design of high-efficiency absorbers based on moth-eye structures.
In this paper, we will use electromagnetic multipole expansion (EME) method to calculate the multipole electric and magnetic multipole modes inside all-dielectric moth-eye particles. We will compare the multipole modes inside a moth-eye particle with those inside other regular shaped particles to show the radiative feature of the single moth-eye particle. Then, we will study the interaction of the multipole modes to make a connection between the multipole modes inside a single particle and the spectral reflection of its periodical array. Further, we will discuss absorbing enhancement inside the moth-eye particle in the view of the electric and magnetic multipole modes. In the end, we will propose a near-ideal selective solar absorber based on the moth-eye structure for the application of solar systems.
2. Structures and calculation
2.1 Electromagnetic multipole expansion method
The interaction between incident light with nano-particles, or called nano-antennas, can induce changes in the charges and current density distributions inside those structures. These induced charges and currents further interact with the incident field and accommodates themselves into electric and magnetic modes in those structures. These modes are called electric and magnetic multipole modes. For electric multipole modes, they result from the polarizations of spatially distributed induced charges. For magnetic multipole modes, they arise due to the existence of the induced circular currents. These multipole modes widely exist in nanoparticles. The lower-order multipole resonances, like electric dipole resonance (ED) and magnetic dipole resonance (MD), are basic multipole modes and have been already experimentally proved in the last decade [29–32]. Higher-order multipoles, which often have weaker intensities than dipoles, also need to be taken into account to evaluate the macroscopic electromagnetic characteristics of some optical devices and materials [33].
Electromagnetic multipole expansion (EME) method is an efficient solution to numerically calculate these multipole modes [34]. At normal incidence, the spatially localized induced electric charges and currents inside a particle produce induced electric and magnetic fields. The induced fields outside of the particle, which are called the scattered fields, can be represented as the superposition of a serial of fields created by a corresponding set of multipoles. The l-th ordered electric and magnetic multipole coefficients can be calculated through the EME method performed in a spherical coordinate through the following equations:
where Ψl(kr) = krjl(kr) are the Riccati-Bessel functions Ψl’(kr) and Ψl”(kr) are their first and second derivatives with respect to the argument kr. Plm are the associated Legendre polynomials. Olm, πlm, and τlm are parameters, which can be found in [34,35]. Js,j(r) describes the effective current density that creates the scattered field of the j-th particle in the self-consistent solution of Maxwell equations and can be numerically acquired through commercial software such as COMSOL, Lumerical FDTD, and CST Studio. The integrations in Eqs. (1) and (2) are over the whole space, while the integrands are equal to zero everywhere outside the particle. In our simulation process, the integrals are operated only inside the particle. The scattering cross section of the particle can be derived as the superposition of a serial of fields created by a corresponding set of electric and magnetic multipoles [34,36]:The terms of series in Eq. (3) allow one to acquire the contribution of each multipole excitation in a nanoparticle. According to Eqs. (1)-(3), the scattering performance and multipole modes are closely related with the electric and magnetic multipole coefficients aE and aM, which are physically decided by not only the shape and optical property of the particle itself, but also the outside factors like the surrounding environment and incident conditions.To start with, the scattering cross sections of a large germanic sphere with a diameter of 534nm (plotted in Fig. 1(a)) is calculated. For simulation, we use the finite element method (FEM), corresponding to the commercial software COMSOL, to calculate the scattering cross section according to Eq. (3). The background electric field is set as an x-polarized light propagating along negative z- axis. The particle is embedded in the center of a large n0 = 1 index sphere, out of which are thick spherical perfectly matched layers (PMLs) absorbing outgoing radiation. The mesh generation adopts the built-in algorithm of the software, where tetrahedral meshes are created. The calculation of the cross sections of the total scattering fields are operated at the scattered-field environment in COMSOL. The refractive indexes of the materials we used in this paper are taken from [37]. Except for the EME method, we also use Mie theory to verify the EME result. Mie theory is an analytical solution of the scattering performance of spherical particles. With the help of an open-access MATLAB code of the Mie theory [38], we calculate the total scattering cross section and plot this result in Fig. 1(b), together with the results derived from EME method. In Fig. 1(b), the total scattering cross section derived from two different methods are almost same, indicating the validity and reliability of the EME method. Mie theory is very accurate and fast, but it cannot be utilized for the calculating the scattering performance of other more complicated structures. This is the reason why we use EME method in this paper.
Fig. 1 (a) Sketch of a spherical germanic particle with a diameter of 534nm. (b) Total scattering cross section (SCS) of the spherical particle acquired through three calculating methods. For EME result, the accumulation in Eq. (3) are calculated up to l = 6.
2.2 Scattering property of a single moth-eye particle
For a normal moth-eye structure, its height should be high enough to support various vertical modes. The moth-eye particle is plotted in Fig. 2(c), and its corresponding geometric parameters are denoted in the subtitle of Fig. 2(c). We choose germanium to be the material for its high refractive index, which helps to support the stronger high order multipole modes. For the moth-eye particle in Fig. 2(c), its total scattering cross section (SCS) is plotted in Fig. 3(a) together with the SCSs of the two other particles plotted in Figs. 1(a) and 1(b). All of these particles are set to share the same volume for comparison.
Fig. 2 (a) An individual spherical particle. (b) An individual cubic particle. (c) An individual moth-eye particle (or conical particle). The size parameter of the three particles are set to have the same volume. The diameter of the sphere is 543nm. The side length of the cubic is 430nm. The diameter of the bottom of the moth-eye particle is 450nm and the height of the moth-eye particle is 1500nm.
Fig. 3 (a) Total scattering cross section (SCS) of the three particles calculated through EME method. (b) Disordered electric field distribution in the y = 0 plane inside the moth-eye particle when the incident wavelength is 850nm.
Compared with the SCS intensities of two other particles in Fig. 2(a), the SCS intensity of the moth-eye particle (red line) is relatively lower in the shorter wavelength range from 400nm to about 1200nm. The total SCS describes the spatial scattering or radiating ability of a particle, which means that the moth-eye particle is less radiative than the two other regular shaped particles in the shorter wavelength. In the particle array, the lower radiative property of each moth-eye particle leads to the lower collective radiative performance for the moth-eye array and finally cause the lower reflecting performance. This is a comprehensive understanding for the reflection-reduction performance of the moth-eye particle.
To distinguish the contribution of each multipole to the total SCS, we calculate the multipole components of the three particles in Fig. 4. The figures in the first row depict the SCS components of the electric multipole modes and the figures in the second row depict the SCS components of the magnetic multipole modes. As depicted in Fig. 4, the electric and magnetic dipole modes (ED and MD), which have the strongest SCS than other multipole resonances, are the main multipole modes existing in these particles. The amplitude of the high-order multipole modes is getting weaker with the increment of the multipole order. Compared with those in the two other particles, the high-order multipole modes in the moth-eye particle show some distinct characteristics. One is the pronounced existence of high-order multipole modes, such as EO, EH and MH, all of which are subtle in a spherical or a cubic particle. With the increment of the multipole order, this phenomenon becomes more prominent. Another characteristic is that the amount of high-order multipole resonant peaks in moth-eye particle are more than those in the cubic and spherical particle. It indicates more kinds and numbers of the high order electric and magnetic resonant modes exists in the moth-eye particle than the cubic and spherical particle.
Fig. 4 Electric and magnetic multipole components inside different shaped particles. The figures in the first row depict the electric multipole modes including electric dipole (ED, l = 1), electric quadrupole (EQ, l = 2), electric octupole (EO, l = 3), and electric hexadecupole (EH, l = 4), respectively. Those in the second row depict the magnetic multipole modes including magnetic dipole (MD, l = 1), magnetic quadrupole (MQ, l = 2), magnetic octupole (MO, l = 3), and magnetic hexadecupole (MH, l = 4), respectively.
The existence of those high order electric modes in the shorter wavelength can be attributed to the large height of the moth-eye structure. When the height of the moth-eye particle is larger than the wavelength of incident light, there will be multiple perpendicular modes locating along the z- direction as shown in the Fig. 3(b). These perpendicular modes lead to multiple spatial distributed linear polarized currents and circular currents, which results in the occurrence of the high order electric and magnetic multipole modes in the moth-eye particle. Although these electric and magnetic multipole modes are too weak to lead to strong spectral radiation in the moth-eye particle, the disordered distribution of these multipole modes make it easier to trap light inside the whole moth-eye particle.
2.3 Multipole resonances in particle array
When the equivalent particles make up a periodical array, their radiative property and resonant performance will change. Periodical arrangement can suppress the radiative loss of each single particle inside the array, called the array effect [39]. In addition, the interaction between each particle induces surface lattice resonances (SLR) in those particles [40,41]. The multipole modes inside each particle will also change due to the interaction between each other. For simplicity, we will take the spherical particle array (plotted in Fig. 5(a)) as an example to show how the multipole modes influence the spectral reflecting performance of the array. The index of the spherical particle is set as 4 in this section to eliminate the absorbing effect in the array.
Fig. 5 (a) Sketch of the periodical spherical-particle array. The periodicity of the array is 1500nm in both x- and y- direction, respectively. The index of the spherical particle is set as 4 in this section. (b) The sketch of the periodical moth-eye particle array. The periodicity of the array 500nm in both x- and y- direction. The structural parameters of the spherical and moth-eye particle stay the same as in Fig. 2.
The periodical spherical particles in Fig. 5(a) can be considered mainly as periodical electric dipoles (ED) and magnetic dipoles (MD) with electric and magnetic polarizabilities αEeff and αMeff . Under normal incidence of a TM-polarized plane wave, the ED and MD moments arising in all particles cannot interact with each other due to their orthogonality [42,43]. Therefore, the interaction modes are self-coupling modes, namely ED-ED and MD-MD coupling modes. The corresponding non-zero components of dipole moments p and m can be acquired from the coupled-dipole equations as [42]
where E0 and H0 are the external electric and magnetic field at z = 0. ks is the wave vector in the surrounding environment. G0xx and G0yy is the diagonal elements of a tensor G0, which can be found in [40]. αE and αM is the electric and magnetic polarizabilities of an isolated particle extracted from the periodical array and can be calculated through the Mie theory as [44]where a1 and b1 are the corresponding ED and MD coefficients calculated through Mie theory [38]. The reciprocal denominator of these two expressions in Eqs. (4) are often called the effective polarizabilities, namelywhich describe the effective ED and MD polarizabilities of particles in the array. The dipole-approximation method is only valid when the following self-consistent inequality is satisfied:where DL is the area of the lattice unit cell. We choose the periodicity of the spherical-particle array in Fig. 5(a) as 1500nm in both x- and y- direction to ensure the satisfaction of Eq. (7).In Figs. 6(a) and 6(b), we plot our calculating result for the ED and MD polarizabilities of the isolated spherical particle together with their effective ED and MD polarizabilities inside the array. The interactions of the EDs and MDs inside the periodical array, described respectively by the term ks2G0xx and ks2G0yy in Eq. (6), have pronounced influence on the ED and MD polarizabilities of the isolated particle. Even though the periodicity of the array is very large (1500nm), the interactions of the electric and magnetic dipoles cannot be neglected. The change in the ED and MD polarizabilities leads to the change of spectral reflecting, which we will discuss later.
Fig. 6 (a) ED polarizability of the isolated spherical particle with reflective index n = 4 and effective ED polarizability in the periodical spherical-particle array. (b) The corresponding MD polarizability and the effective MD polarizability. The periodicity of the spherical-particle array is set as Px = Py = 1500nm to fulfill the dipole-approximating condition.
For an individual particle, the ED and MD resonances occur when Re(1/αE) = 0 and Re(1/αM) = 0, respectively. In the periodical array, the ED and MD resonant conditions change to Re(1/αeffE) = 0 and Re(1/αeffM) = 0. These efficient electric and magnetic dipole resonances are also called surface lattice dipole resonances (SLDRs) [40,41]. In respect to the multipole resonances, Eqs. (6) also describe the relationship between the ED and MD resonances of the individual particle and the ED and MD resonances of the array. The interaction term ks2G0xx leads to the change of the dipole resonant wavelengths in the particle array. The ED and MD resonant conditions in the particle array are calculated in Figs. 7(a) and 7(b). Compared with the resonant wavelength of the individual particle, the ED resonant wavelength in the array redshifts due to the ED-ED interaction. However, the wavelength of MD resonance in the array has a blue shift due to MD-MD interaction. For a fixed periodicity, the interaction of the EDs and the interaction of MDs cause different resonance-shifting effects.
Fig. 7 (a) ED resonant condition for the individual particle (Re(1/αE) = 0, grey line) and the particle array (Re(1/αeffE) = 0, black line). (b) MD resonant condition for the individual particle (Re(1/αM) = 0, grey line) and the particle array (Re(1/αeffM) = 0, grey line). The operator ‘Re()’ is omitted in both figures for conciseness. The ED and MD resonant wavelengths of the particle array are marked in the figures as λED and λMD.
Indeed, this phenomenon also happens to the higher multipoles like electric quadrupole (EQ) and magnetic quadrupole (MQ).The effective polarizabilities of EQ and MQ in the particle array can be expressed as [44]
where αEQ is the EQ polarizability of the individual particle and αMQ is the MQ polarizability of the individual particle. ks2Gqjl is the quadrupole sum, describing the interaction of EQs and MQs in the particle array. The constructive and destructive interaction of quadrupoles also can lead to the spectral shift of EQ and MQ resonances.2.4 Multipole resonance and spectral reflection
Ignoring the higher order multipole modes, the reflection coefficients of a periodical array can be expressed in the term of the effective ED and MD polarizabilities as [42]
where DL is the area of the lattice unit cell. According to Eq. (8), the reflection peaks of the periodical particle array are decided by the dipole modes in terms of αeffE and αeffM. These modes, however, are caused by the multipole resonances inside an isolated particle. In other words, the reflecting peaks of a periodical array is essentially determined by the multipole resonances inside the individual particle in the particle array. To better show this phenomenon, we plot the SCS of an isolated loss-less spherical particle with the reflecting spectrum of its periodical array in Fig. 8(a). The 3-D finite difference time domain (FDTD) method is applied for the calculation of spectral reflection and absorption. As for the physical domain, the particle is embedded in the n0 = 1 environment. TM polarized plane wave is incident along the negative z- direction. Periodical boundary conditions are adopted at both the x- and y- boundary of a unit cell. Perfectly matched layers (PMLs) are imposed at both the bottom and the top of the physical domain. A minima mesh size of 5nm is adopted. The calculating result turns out to be independent with the mesh sizes.Fig. 8 (a) Normalized total SCS of an isolated spherical particle and the reflection of the spherical particle array in Fig. 5(a). The reflection spectrum of the periodical array is calculated through Lumerical FDTD. (b) High order electric and magnetic modes inside the single sphere with the refractive index n = 4.
The main reflecting peaks of the array, which we have marked on the figure, agrees well with the SCS multipole resonant peaks of an isolated particle. Besides, the ED and MD resonant wavelengths in Fig. 8(a) also agrees well with the ED and MD resonant wavelengths predicted by the dipole-coupling method plotted in Fig. 7. Compared with the ED resonance of individual particle in SCS spectrum, the reflecting peak caused by ED resonance redshifts due to the ED-ED coupling. Instead, the reflecting peak caused by MD resonance has a blue shift due to the MD-MD coupling. These results also agree well with the conclusion of Fig. 7 we mentioned before.
Except for the electric dipoles resonances, the higher order multipoles also show the same effect as we marked in Fig. 8(a). We plot the high order multipole modes of the spherical particle in Fig. 8(b) for comparison. Comparing the Figs. 8(a) and 8(b), one can find that wherever there is a multipole resonant peak in Fig. 8 (b), there will be a corresponding reflecting peak in its spectral vicinity in Fig. 8(a). This phenomenon indicates that the electric and magnetic multipole modes in an individual particle lead to spectral reflection peaks for the particle array. The interactions of these multipoles cause the shift for the reflecting peaks of the array. It indicates that the coupling theory used in this paper is not only valid for dipole resonances but also valid for quadrupole resonances and maybe even higher multipole resonances. The reflection coefficients of a periodical array should be expressed in the form:
where αeffD, αeffQ, αeffO, and αeffH is the effective dipole, quadrupole, octupole, and hexadecupole polarizabilities in the array.The multipole resonances inside a single particle will inevitably cause reflective peaks in the reflecting spectrum of its array. In loss-less moth-eye particle, there should also be various multipole resonances, which can lead to multiple reflecting peaks in the reflecting spectrum of its array. This deduction seems to be in contradiction to the moth-eye effect, which shows that moth-eye structure can have more prominent reflection-reduction performance over other structures. To figure out this problem, we calculate the reflection of the moth-eye particle array (plotted in Fig. 5(b)), which consists of loss-less material with a refractive index of 4. Its reflecting spectrum is plotted in Fig. 9(a). Instead of near-zero reflection according to the moth-eye effect, there are multiple reflective peaks extensively appearing at short wavelength range. For the high-index loss-less particle array, incident light cannot be trapped inside particles. The multipole modes inside the structure lead to inevitable reflection peaks. Broadband near-zero reflection cannot be realized in these arrays. However, these reflection peaks can be subdued by applying thick substrates beneath those arrays. The adoption of substrates changes the radiation patterns of particles [45] and affects the spectral reflection as a result.
Fig. 9 (a) Reflection spectrum of the moth-eye array that made of an imaginary loss-less material whose refractive index is nm = 4. (b) Reflection spectrum of the moth-eye array that made of an imaginary lossy material whose refractive index is nm = 4 + 0.5i.
Once we introduce loss to the material and change the extinction coefficient of the material from 0 to 0.5, the reflection and absorption of the moth-eye structure changes dramatically as we plot in Fig. 9(b). The reflection reduces to near zero while its absorption increases to near unit. Instead of reflected, most of the incident light is absorbed by the structure. The rapid spectral change indicates that the near-zero reflection in moth-eye structure is achieved at the sacrifice of large ohmic loss inside moth-eye structures. The lossy property of the structure can confine and trap the light within the moth-eye structure with the help of the multipole modes, and results in reflection-reduction effect for moth-eye particle.
2.5 Multipole resonances and spectral absorption
For an isolated particle, its absorbing ability of light can be estimated by its absorbing cross section (ACS). In our simulation, the ACS of an isolated particle is calculated by the volume integration of the ohmic loss inside the particle in COMSOL. The ACS of the individual spherical and the moth-eye particle is plotted in Fig. 10(a). The spectral absorbing spectrum of the corresponding particle array is plotted in Fig. 10(b). Shown in both of the two figures, moth-eyes array has a higher absorbing ability over the spherical array in the shorter wavelength range from 400nm to 1200nm. In return, the ACS and the spectral absorption of the spherical structures exceeds those of the moth-eye structures at around 1600nm in Fig. 10(a) and 10(b). The well coincidences indicates that the absorption performance of an individual particle plays an important role for the spectral absorption in the particle array. It can be inferred that the high absorption of the moth-eye particle array is due to the excellent absorbing performance of the individual moth-eye particle.
Fig. 10 (a) Absorbing cross sections (ACSs) of the germanic moth-eye particle and the germanic spherical particle. (b) Absorbing spectrum of the moth-eye germanic particle array and the spherical germanic particle array. The periodicity of the both arrays changes to 600nm for comparison.
The absorption enhancement of moth-eye structures can be owing to the existence of multipole mode. To make a connection between the absorption ability and the multipole modes, we use the following expression to represent the ACS of an isolated particle [46]
where ε” is the imaginary part of the dielectric constant of the material used and |Em| is the normalized electric field amplitude inside a particle. The normalized inner field intensity |Em| can be regarded as the superposition of a serial of electric fields created by a corresponding set of multipole modes inside the particle. Equation (11) indicates that the absorption inside a specific particle is directly associated with its inner electric field distribution |Em|. In other words, Eq. (11) describes the relationship between the collective intensity of multipole modes inside a particle and its absorbing ability. Considering the complexity of the electric field distribution inside a particle, we use the definition of average electric field amplitude to numerically describe the total intensity of the electric and magnetic multipole inside a particle as for absorption. This term is defined aswhere V0 is the volume of the particle. In Fig. 11, we plot the average electric field amplitudes E1 of both the spherical particle and the moth-eye particle for comparison. At short wavelength, the average electric field magnitude inside a moth-eye particle is slightly larger than that inside a spherical particle. However, it can greatly promote the absorbing ability of a particle as shown in Figs. 10(a) and 10(b). When wavelength is longer, Ge becomes loss-less and the absorption of both particle comes to zero. In Fig. 11, one can directly derive that the moth-eye particle can generate stronger electric field inside itself than spherical particle. The electric field enhancement, caused by the multipole electric and magnetic modes, is the reason why the moth-eye structure can have a better absorption and hence a lower reflection than spherical structure.2.6 Ideal absorber based on moth-eye structure
Up to now, we have explained the reflection reducing and absorption enhancing effect inside a moth-eye particle. In many functional optical systems, the absorption produces undesired heat, which can degrade the working efficiency and decrease the lifetime of optical devices. However, the absorption enhancing is required in some optical energy-transforming systems like solar steam, water desalination, thermophotovoltaic (TPV) systems, and thermal control systems [7–10]. To improve the absorption of the moth-eye array and make it applicable in real solar systems, we apply a 3-μm thick Ge- substrate to the moth-eye array as plotted in Fig. 12(a). In Fig. 12(b), we plot the absorption of the Ge- moth-eye array with and without the Ge- substrate, together with the absorption of a 3-μm Ge- substrate. For the Ge- moth-eye array with a 3-μm substrate, we use both the FEM (COMSOL) and the FDTD (Lumerical FDTD) method to ensure the accuracy of the calculating result. The adoption of Ge- substrate promotes the absorption of the light that penetrates the upper particle array. We can see that the bare Ge- substrate already has up to 60% absorption. The introduction of the Ge- substrate notably improves the absorption in wavelength range from 1000nm to 1450nm. The absorbing efficiency of the structure increases to near-unite from 400nm to 1400nm when the substrate is applied. Unlike metal substrates in which incident light penetrates only dozens of nanometers on its surface and then is mostly reflected to the top structure, semiconductor substrates enable the incident light to penetrate the substrate and to be consumed inside the substrate due to its large skin depth. We should also note that the introduction of metal substrates will cause undesired absorption over 1800nm, reducing the absorption selectivity of the structure.
Fig. 12 (a) Sketch of the germanium moth-eye array with a 3-μm Ge- substrate. The parameters of the moth-eye array upon the substrate stay unchanged. (b) Absorption spectrum of the moth-eye array with (W) or without (W/O) the substrate, and the absorption of a bare Ge- substrate. FDTD and FEM (COMSOL) method are both adopted to ensure the accuracy of the simulation result.
In solar systems, the incident angle is one of the crucial factor that needs to be considered to evaluate the absorbing performance of an absorber. For the proposed absorber, the incident angle can affect the multipole resonant modes inside a single moth-eye particle. The variation of the incident angle leads to the change of the induced currents and charges distribution. In this way, both the amplitude and the geometric center of the multipole modes change as a result. For the all-dielectric moth-eye structure, the induced currents and charges exist at any incident angle. This means that these multipole modes exist no matter how the incident angle changes. The variation of the incident angle may cause either an enhancing effect or a subduing effect for the current density at different area of the particle, which leads to the strengthen effect for some multipole modes while the weaken effect for the other. We plot the lower ordered multipole components of the isolated moth-eye particle at wavelength around 900nm varying with the incident angle in Fig. 13(a). It proves that the multipole modes exist at different incident angles. We also plot the absorbing cross section (ACS) of the particle as a function of the incident angle in Fig. 13(b). The ACSs at the other incident angles have larger amplitudes than the normal incident (θ = 0°), which is another evidence proving that the multipole modes are not suppressed when the incident angle is changed.
Fig. 13 (a) Scattering amplitude of the lower-ordered multipole modes calculated using EME method varying with the incident angle for a single moth-eye particle at wavelength 900nm. (b) Absorbing cross sections (ACSs) varying with the incident angle at wavelength 900nm.
As told before, the multipole modes can couple with each other in the infinite particle array. However, the ways in which these multipole couples are different when the incident angle changes. For example, we plot a scheme of the ED-ED coupling mode when the incident angle is 0° in Fig. 14(a). In the x-z plane, the ED moments are linearly distributed along the x- axis. The coupling mode between EDs is a kind of serial coupling mode, or called longitudinal coupling modes [47]. However, when the incident angle changes to 90°, the ED moments are along the z- axis as depicted in Fig. 14(b). The ED moments changes to parallel to each other and the coupling between EDs becomes to parallel coupling mode. The former coupling mode leads to a constructive effect for the dipole moments, while the latter one leads to a destructive effect for the dipole moments [47,48]. When the incident angle is between 0° and 90°, the two modes exist at the same time while contribute differently for the ED moment. The coupling mode of other electric multipole follow the same rule as the ED coupling mode. The incident angle variation results in either a constructive effect or a destructive one for the multipole coupling, which leads to the variation of the absorbing performance for the particle array. In Fig, 14(c), we have plotted the absorbing spectrum of the proposed absorber varying with the incident angle. The result shows that the absorption turns out to be insensitive to the change of the incident angle from 0° to 75°, where the absorptivity maintains over 80% and the spectral absorbing selectivity doesn’t degrade with the increment of the incident angle. According to Fig. 14(c), we can infer that the longitudinal coupling mode may function as the major role for the incident angle from 0° to 75°. However, when the incident angle increases to over 80°, the parallel coupling mode may take over the major role and result in a destructive effect for the multipole modes, which degrades the absorbing efficiency as a result.
Fig. 14 (a) The longitudinal coupling modes between the electric dipoles (EDs) at normal incidence. (b) Parallel coupling modes between EDs at incident angle θ = 90°. (c) Absorbing spectrum for the proposed structure varying with the incident angle.
Another crucial part to achieve the perfect absorption is the choosing of the absorbing materials. Metals are not applied in our structure due to its small skin depth, which forbids the occurrences of the high-order magnetic multipole modes. In our structure, we use Ge as the absorbing material because of its medium extinction coefficient, which enables the incident light to penetrate the structure and to be well consumed in the structure at the same time. In addition, the extinction coefficient of Ge is zero for the wavelength longer than about 1700nm, resulting a near-zero absorption over this wavelength. This property helps the structure better selectively absorb light. The absorbing selectivity, which is closely related with the solar absorption efficiency, is very important in the area of solar energy systems [17]. Indeed, we can apply other semiconductor materials to realize different selective absorption. When Si and GaSb are applied in the proposed structure, the cut-off absorbing wavelength changes around the range from 1000nm to 1700nm as plotted in Fig. 15(a). We also plot the extinction coefficients of these three semiconductor materials in Fig. 15(b) for comparison. Apparently, the cut-off absorbing wavelengths are determined by their extinction coefficients for the proposed structure. Indeed, the cut-off absorbing wavelength can be precisely decided by applying the compound of several semiconductors or using doped semiconductors according to one’s demand.
Fig. 15 (a) Absorbing spectrum of the moth-eye array made of different semiconductor materials. (b) Extinction coefficients of the three semiconductor materials.
3. Conclusion
In this paper, we use electromagnetic multipole expansion method to show the existence of multipole modes inside a moth-eye particle. Compared with other regular particles, the moth-eye particle can produce prominent higher order multipole modes, leading to the enhancement of the electric field inside the particle and the absorption enhancement for the particle. We prove that the electric and magnetic multipole resonances can introduce spectral reflection peaks. The interaction among multipoles further leads to the shift of these reflection peaks. In loss-less structures, there are multiple reflection peaks, resulting from the multipole resonances. In lossy structures, the multipole modes can well trap light and improve the absorbing ability. Based on these studies, we propose a perfect absorber based on the moth-eye structure. The absorber achieves near unit absorption from 400nm to about 1500nm while near-zero reflection over 1700nm. The proposed absorber is insensitive to the incident angle from 0°to 75° due to the occurrence of the multipole electric and magnetic modes. The excellent absorbing performance makes it promising to be used in solar systems.
Funding
National Key R&D Program of China (2016YFA0301300); National Natural Science Foundation of China (NSFC) (61671090, 61875021); Fund of State Key Laboratory of Information Photonics and Optical Communications (IPOC20172204); Guangxi Key Laboratory of Wireless Wideband Communication and Signal Processing, Natural Science Foundation of Beijing (2192036).
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