1. Introduction

In order to reduce the inherent reflection occurring on the surface of absorbers, impedance matching methods such as antireflection coatings [1] and graded index structures [2] have been developed and widely applied in many important applications such as solar cells and microwave anechoic chambers. However, these methods rely on the surface decoration of specific microstructures or materials on the surface of absorbers, which could be, sometimes, fragile and inconvenient in practice.

Recently, artificial electromagnetic structures such as photonic crystals (PhCs) [3,4], metamaterials [5–7] and plasmonic structures have afforded unprecedented ways to achieve near-perfect absorption [8–49]. For example, it has been shown that surface decoration by PhCs can enormously increase the efficiency of solar cells [44–46]. Besides surface decoration, various micro-structured materials have been designed to surpass the limits of natural absorptive materials [42,43]. However, in most cases either the bandwidth or the range of incident angles for near-perfect absorption are severely limited [9,13,32]. The pursue of perfectly matched layer absorbers in computational electromagnetics has inspired a variety of methods to broadband and omnidirectional perfect absorbers [50–53], which, however, usually require frequency-dependent, anisotropic and magnetic materials. This reduces the feasibility in practical applications. Very recently, novel methods such as transformation optics technique [54–56] and spatial Kramers–Kronig profiles [57–59] have been proposed to realize omnidirectional absorption, but these approaches require complex inhomogeneity design and even material gain [60], which hinders practical implementation.

In this work, we propose to realize near-perfect absorption by using lossy dielectric PhCs with a broadband and omnidirectional impedance-matching property. Recently, a type of PhCs has been designed and experimentally verified to be ultratransparent, i.e. allowing nearly 100% transmission of light for all incident angles [61–64]. The principle behind this PhC lies in the realization of omnidirectional impedance matching with free space, which is crucial for the design of omnidirectional near-perfect absorbers. However, loss effects and absorption performances have not been considered in the previous works [61–64], which are, however, inevitable in real structures. Therefore, investigation of the loss effects in such PhCs is crucial. Here, we find that when only a small amount of loss is introduced in such media, the omnidirectional impedance matching effect can still be well maintained, leading to non-reflection for a wide range of incident angles. On the other hand, the light energy will be gradually absorbed by the loss in the PhC. Therefore, omnidirectional near-perfect absorption can be realized. Theoretical analysis shows that the absorption efficiency can approach 100% efficiency for all incident angles within ± 80°. Moreover, such near-perfect absorption can be achieved in a broad frequency spectrum. For example, we have designed an optical absorber consisting of alternatively stacked silicon (Si) and titanium dioxide (TiO₂) films, which exhibits wide-angle significant absorption efficiency for wavelengths between 440nm and 640nm. By tuning the microstructure and unit scale of the PhCs, the frequency spectrum and range of angles for near-perfect absorption can be conveniently controlled. Our work reveals a simple and efficient method for realizing broadband and omnidirectional absorption of waves, which can be applied to PhCs composed of a large variety of materials and a broad spectrum from microwaves to optics.

2. Design principles

The concept of the proposed optical absorber based on omnidirectional impedance-matched media is illustrated by Fig. 1, in which near-perfect absorption can be realized for all optical beams emitted from a point source, indicating an omnidirectional behavior. Since the medium is impedance matched with free space for all directions, there would be no reflection for all incident waves. When there is loss in the impedance-matched media, light energy will be gradually absorbed. Therefore, this concept provide a straightforward way to realize omnidirectional near-perfect absorption by bulk materials without surface decoration. The key lies in the design of PhCs serving as optical impedance-matched media. On the other hand, an appropriate amount of loss is also important, as we shall demonstrate in the following.

 

Fig. 1 Illustration of an omnidirectional near-perfect absorber consisting of omnidirectional impedance-matched media with a small amount of material loss.

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Firstly, we demonstrate an optical impedance-matched medium realized by a multi-layer PhC composed of two different dielectric materials A and B, which are lossless and nonmagnetic. The unit cell is constructed in a symmetric way, i.e., ABA structure, as illustrated by the inset in Fig. 2(a). The unit cell is of lattice constant $a$. Each layer of material A (with relative permittivity ${\epsilon }_A=2$) has a thickness of $d_A=0.327a$, and the layer of material B (with relative permittivity ${\epsilon }_B=5$) has a thickness of $d_B=0.346a$.

 

Fig. 2 (a) Band structure of a dielectric multilayer. The green dashed, red solid and blue dotted lines denote the EFCs at the frequencies $fa/c=0.44$, $fa/c=0.451$ and $fa/c=0.46$, respectively. (b) The impedance difference between the multilayer and free space. [(c), (d)] Incident angle-dependent transmittance through the multilayer with 10 unit cells under the incidence of (c) TE- and (d) TM-polarized waves for the frequencies $fa/c=0.44$ (green dashed lines), $fa/c=0.451$ (red solid lines) and $fa/c=0.46$ (blue dotted lines). (e) Effective relative permittivity ${\epsilon }_{eff}$ (red solid lines) and permeability ${\mu }_{eff}$ (green dashed lines) of the multilayer structure under normal incidence, showing that ${\epsilon }_{eff}\approx {\mu }_{eff}$ in a broad frequency band.

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Figure 2(a) shows the equal frequency contours (EFCs) of the second band of the periodic multilayer for transverse electric (TE) polarization (with out-of-plane electric fields). The calculation is performed by using software COMSOL Multiphysics. In Fig. 2(a), the red solid line denotes the EFC of the selected central working frequency $fa/c=0.451$, where $f$ and $c$ are the eigen-frequency and light velocity in free space, respectively. It is seen that the EFC of the central working frequency is almost an ellipse centered at the $X$ point. It has been pointed out in [61–64] that despite the shift of EFC in the $k_x$ direction, the eigenstates in the EFC can be perfectly matched to those in free space, enabling the omnidirectional impedance matching property.

In addition, the impedance difference $\left|\left(Z_{eff}-Z_0\right)/\left(Z_{eff}+Z_0\right)\right|$ between the periodic multilayer and free space is presented in Fig. 2(b). Here, $Z_0$, as the ratio of the electric and magnetic fields, is the wave impedance of free space for TE-polarized lights incident onto the $yz$ plane, which is expressed as $Z_0=\omega {\mu }_0/\sqrt{k_0^2-k_y^2}$. Here, $\omega$, ${\mu }_0$, $k_0$ and $k_y$ are the angular frequency, the permeability and wave number of free space, and the $y$-component of wave vector, respectively. On the other hand, the wave impedance of the multilayer can be effectively calculated by $Z_{eff}=〈E_z〉/〈H_y〉$ [61–63,65], where $〈E_z〉$ and $〈H_y〉$ denote the average of electric fields and $y$-component of magnetic fields of the eigen-field along the boundary of the multilayer unit cell on the incident side, respectively.

From Fig. 2(b), we find that the impedance difference between the designed multilayer structure and free space is very small within a broadband frequency band for a large range of $k_y$, indicating broadband and wide-angle impedance matching effect for the TE polarization. In particular, at the frequency $fa/c=0.451$, the impedance difference is almost zero for all $k_y$ smaller than $k_0$, demonstrating an extremely omnidirectional impedance matching effect, which is also applied in the ultratransparency effect [61].

For verifications, the transmittance through such a periodic multilayer consisting of 10 unit cells is plotted in Fig. 2(c) as a function of the incident angle for the TE polarization. We note here that loss has not been introduced yet. It is seen that almost perfect transmission is achieved at the frequency $fa/c=0.451$ (red solid lines) for all incident angles $\theta \le {83}^{\circ }$. The angular performance is much improved compared with the previous works [61,62]. Moreover, for the frequencies $fa/c=0.44$ (green dashed lines) and $fa/c=0.46$ (blue dotted lines), near-unity transmission can be seen for all incident angles $\theta \le {60}^{\circ }$. Such high transmission is the result of impedance matching effect, which is irrespective of the number of the unit cells. Interestingly, although the multilayer is optimized for the TE polarization, nearly perfect transmission is also seen for the transverse magnetic (TM) polarization (with out-of-plane magnetic fields) within the frequency range $0.44\le fa/c\le 0.46$ for all incident angles $\theta \le {40}^{\circ }$, as shown in Fig. 2(d).

To further demonstrate the broadband impedance matching effect, we have also calculated the effective parameters of the multilayer structure, as shown in Fig. 2(e). The effective relative permittivity and permeability are calculated as ${\epsilon }_{eff}=-k_x/{\epsilon }_0\omega Z_{eff}$ and ${\mu }_{eff}=-k_x{Z}_{eff}/{\mu }_0\omega$ under normal incidence, i.e. $k_y=0$ [61]. Here, $k_x$ is the $x$-component of Bloch wave vector. In Fig. 2(e), the condition ${\epsilon }_{eff}\approx {\mu }_{eff}$ is satisfied in a relatively broad frequency band, manifesting the broadband impedance matching effect.

It is worth noting that the omnidirectional impedance matching effect can also be comprehended from the perspective of effective medium model. It is found in [61–63] that such a multilayer structure can regarded as an effectively nonlocal (i.e. wave-vector-dependent) and anisotropic medium for a certain frequency. It is known that non-reflection only occurs at a particular incident angle for a specific polarization due to impedance matching, which is known as the Brewster angle effect. Here, by adjusting the thickness of each component in the multilayer structure, appropriate nonlocality can be obtained to extend the Brewster angle effect from a particular angle to almost all angles, so that there is no reflection for almost any incident angle.

Now, with the designed impedance-matched multilayer structure, the second step is to realize the near-perfect absorber by introducing material loss into the system. In the following, we will add different amount of losses in different components of the impedance-matched multilayer structure, so as to investigate the overall effect of loss to the impedance matching effect as well as the absorption behavior of the whole system.

We first assume that only the material A is absorptive, whose relative permittivity is rewritten as ${\epsilon }_A={\epsilon }_A^{\text{'}}+i{\epsilon }_A^{\text{'}\text{'}}$. The real part ${\epsilon }_A^{\text{'}}$ is 2, and the imaginary part ${\epsilon }_A^{\text{'}\text{'}}$ corresponds to the material loss. In Figs. 3(a) and 3(b), we plot the absorptance of the multilayer structure consisting of $N$ unit cells at the frequency $fa/c=0.451$ for the cases of loss tangent ${\epsilon }_A^{\text{'}\text{'}}/{\epsilon }_A^{\text{'}}=0.02$ and ${\epsilon }_A^{\text{'}\text{'}}/{\epsilon }_A^{\text{'}}=0.1$, respectively. It is seen that when $N$ is small, the absorption is small. When $N$ increases, both cases can realize almost perfect absorption (absorptance $A\ge 0.99$). However, as the increase of the loss in material A, the maximal incident angle for absorptance $A\ge 0.9$ ($A\ge 0.99$) decreases from 80° to 70° (70° to 50°), indicating the omnidirectional impedance matching is weakened. On the other hand, when the material loss is large, much less unit cells are required. Actually, to obtain the same amount of absorption, the required number of unit cells $N$ is inversely proportional to the loss tangent ${\epsilon }_A^{\text{'}\text{'}}/{\epsilon }_A^{\text{'}}$ under normal incidence, as shown by the linear relationship between $\log \left({\epsilon }_A^{\text{'}\text{'}}/{\epsilon }_A^{\text{'}}\right)$ and $\log \left(N\right)$ in Fig. 3(e). When $N$ is large enough, there exists an upper bound for the loss tangent. For instance, to obtain $A\ge 0.99$ for all incidence angles $\le$70°, the loss tangent of material A should be less than 0.025.

 

Fig. 3 [(a)-(d)] Absorptance as functions of the incident angle and unit cell number $N$ when there exists loss in [(a), (b)] material A with (a) ${\epsilon }_A^{\text{'}\text{'}}/{\epsilon }_A^{\text{'}}=0.02$ and (b) ${\epsilon }_A^{\text{'}\text{'}}/{\epsilon }_A^{\text{'}}=0.1$, or [(c), (d)] material B with (c) ${\epsilon }_B^{\text{'}\text{'}}/{\epsilon }_B^{\text{'}}=0.02$ and (d) ${\epsilon }_B^{\text{'}\text{'}}/{\epsilon }_B^{\text{'}}=0.1$. [(e)-(f)] Absorptance with respect to (e) $\log \left({\epsilon }_A^{\text{'}\text{'}}/{\epsilon }_A^{\text{'}}\right)$ and $\log \left(N\right)$, and (f) $\log \left({\epsilon }_B^{\text{'}\text{'}}/{\epsilon }_B^{\text{'}}\right)$ and $\log \left(N\right)$ under normal incidence, showing a linear relationship for the same amount of absorption. The solid lines and dashed lines in (a)-(f) show the equal-absorptance contours at $A=0.99$ and $A=0.9$, respectively. The incident waves are of $fa/c=0.451$ with TE polarization.

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Next, we assume that only the material B is absorptive, whose relative permittivity is rewritten as ${\epsilon }_B={\epsilon }_B^{\text{'}}+i{\epsilon }_B^{\text{'}\text{'}}$ with ${\epsilon }_B^{\text{'}}=5$. The absorptance of the multilayer structure as functions of the incident angle and the unit cell number $N$ at the frequency $fa/c=0.451$ is plotted. Figures 3(c) and 3(d) show the cases of loss tangent ${\epsilon }_B^{\text{'}\text{'}}/{\epsilon }_B^{\text{'}}=0.02$ and ${\epsilon }_B^{\text{'}\text{'}}/{\epsilon }_B^{\text{'}}=0.1$, respectively. Similarly, we can clearly see a tradeoff situation. When the material loss increases, the maximal incident angle for near-perfect absorption is decreased, but less unit cells can obtain the maximal absorption. And the required number of unit cells is inversely proportional to the loss tangent to obtain the same amount of absorption under normal incidence, as shown by the linear relationship between $\log \left({\epsilon }_B^{\text{'}\text{'}}/{\epsilon }_B^{\text{'}}\right)$ and $\log \left(N\right)$ in Fig. 3(f). The upper bound also exists for the loss tangent of material B when $N$ is large enough. To obtain $A\ge 0.99$ for all incidence angles $\le$70°, the loss tangent of material B should be less than 0.13.

One may notice that the case in Fig. 3(c) shows an amazing absorptance of 99% for all angles within 80°. By comparing the cases of material loss in A and B, we find that the material loss in B has a weaker influence on the omnidirectional impedance matching effect, while the material loss in A can be relatively more efficient in the energy absorption. These analyses provide the design guidance for different purposes. For instance, if we want a thin optical absorber, a large loss should be added into the material A. On the other hand, if we want to design an omnidirectional near-perfect absorber, a small amount of loss should be introduced into the material B.

Considering the balance of the angular performances and the thickness of the absorber, the relative permittivity of material B is set to be ${\epsilon }_B=5+0.26i$, and the number of unit cells is $N=50$. Figure 4(a) presents the absorptance of the multilayer absorber as the function of the incident angle for both TE (red solid lines) and TM (green dashed lines) polarizations at the frequency $fa/c=0.451$. Clearly, the absorptance is almost unity within the incident angle range of $0\le \theta \le {80}^{\circ }$ ($0\le \theta \le {50}^{\circ }$) for the TE (TM) polarization. For comparison, such high absorption is absent for a single slab of the material B with the same thickness (blue dotted lines in Fig. 4(a)). Furthermore, Fig. 4(b) shows the simulated distribution of electric fields when a TE-polarized point source is placed in front of the multilayer absorber. It is seen that the radiated wave in the left side (air) is a well-defined cylindrical wave with no interference pattern induced by any reflection. This again confirms the omnidirectional characteristic of the mutilayer absober.

 

Fig. 4 (a) Incident-angle-dependent absorptance of the multilayer absorber (Red solid lines for the TE polarization, and green dashed lines for the TM polarization) and the lossy material B alone having the same thickness (blue dotted lines for the TE polarization). (b) The simulated distribution of electric fields when a TE-polarized point source is placed in front of the multilayer absorber. [(c), (d)] Incident-angle-dependent absorptance of the multilayer absorber when the thickness of (c) material A, (d) material B is varied for the TE polarization. The black solid, dashed, dotted and dash-dotted lines correspond to the cases with thickness decreased by 10%, decreased by 5%, increased by 5% and increased by 10%, respectively. The relevant parameters are $fa/c=0.451$, ${\epsilon }_A=2$, ${\epsilon }_B=5+0.26i$ and $N=50$.

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It is worth noting that such wide-angle near-perfect absorption is robust against small variation of the thicknesses of the components. In Fig. 4(c), under TE polarization, we change the thickness of materials A to $0.9d_A$ (black solid lines), $0.95d_A$ (black dashed lines), $1.05d_A$ (black dotted lines) and $1.1d_A$ (black dash-dotted lines), where $d_A$ is the original thickness, i.e. $d_A=0.327a$. It is seen that the maximal angle of near-perfect absorption can still reach 60°~70° when there is a relatively large variation of the thickness. Similarly, from Fig. 4(d), we can see that the near-perfect absorption is also robust against the varition of the thickness of material B.

Moreover, we also note that the component materials of the near-perfect absorber designed in the above are not particularly chosen. Actually, almost arbitrary dielectric materials with a relatively large refractive index contrast can be utilized to achieve the broadband and omnidirectional absorption in free space. As an example, in Fig. 5, we have redesigned the omnidirectional near-perfect absorber in Fig. 4 by choosing different material B with ${\epsilon }_B={\epsilon }_B^{\text{'}}+0.26i$, where ${\epsilon }_B^{\text{'}}$ as the real part of ${\epsilon }_B$ is a variable. Here, the material A with ${\epsilon }_A=2$ is unchanged. We find that when the ${\epsilon }_B^{\text{'}}$ is varied in the range of 4 to 6, the omnidirectional near-perfect absorption can still be obtained for certain working frequencies by simply adjusting the thickness of the component materials A and B, as shown in Fig. 5(a). For demonstration, in Fig. 5(b), we have calculated the absorptance for two cases, i.e., ${\epsilon }_B=4.5+0.26i$ (red solid lines) and ${\epsilon }_B=6+0.26i$ (green dashed lines) for the TE polarization. Clearly, nearly omnidirectional perfect absorption is still observed when the thickness of the component materials is appropriately adjusted.

 

Fig. 5 (a) The required thickness of the material B (solid lines with symbols) and the corresponding working frequency (dashed lines with symbols) to achieve the omnidirectional near-perfect absorption as the variation of the real part of ${\epsilon }_B$. The relative permittivity of material A is fixed as ${\epsilon }_A=2$. (b) Incident-angle-dependent absorptance of the mutilayer absorber with 50 unit cells for the TE polarization. The red solid lines and green dashed lines correspond to the cases with ${\epsilon }_B=4.5+0.26i$, $d_B=0.32a$, $fa/c=0.466$ and ${\epsilon }_B=6+0.26i$, $d_B=0.375a$, $fa/c=0.423$, respectively.

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Here we select the thickness of each component and working frequency based on the following procedures. First, we focus on the second frequency band of the multilayer structure, which exhibits an elliptical EFC centered at the $X$ point, as shown in Fig. 2(a). Because it is pointed out in [61–63] that the eigenstates in such an EFC are capable of realizing omnidirectional impedance matching property. Then, we engineer the thickness of each component of the multilayer structure based on a trial-and-error iterative optimal algorithm, so as to ensure impedance matching of the eigenstates with propagating waves in free space, in an angle range as large as possible. During the optimization, we mainly focus on the minimization of the sum of reflection over all incident angles.

3. Si-based broadband and omnidirectional optical absorbers

The above results reveal a general and effective way to realize the broadband and omnidirectional optical absorbers. Based on the design principle, we will show an example of optical absorber by using Si and TiO₂ films as follows.

In Fig. 6(a), we show the design of a TiO₂-Si multilayer. The unit cell of the mutilayer is TiO₂-Si-TiO₂ with the thickness of the Si film (each TiO₂ film) being 33.6nm (50nm). The refractive index of TiO₂ and Si at the wavelength of 600nm is set to be $n_{Ti{O}_2}=2.4$ and $n_{Si}=3.95+0.026i$, respectively. In Fig. 6(a), the absorptance of such an optical absober ($N=500$) is calculated as the function of the incident angle for both TE (red solid lines) and TM (green dashed lines) polarizations, showing that the maximal angle of almost perfect absoption can reach 82° (45°) for the TE (TM) polarization. For comparison, the absorptance of a Si slab having the same thickness alone is also calculated for the TE polarization (blue dotted lines in Fig. 6(a)), showing a low absorption smaller than 0.7 due to the strong impedance mismatch between the Si and free space.

 

Fig. 6 (a) Incident-angle-dependent absorptance of the multilayer absorber consisting of TiO₂ and Si films (red solid lines for the TE polarization, and green dashed lines for the TM polarization), and the Si slab having the same thickness alone (blue dotted lines for the TE polarization). (b) The real part (solid lines) and imaginary part (dashed lines) of the Si. [(c), (d)] Absorptance of the (c) Si slab alone and (d) TiO₂-Si-TiO₂ multilayer absorber as functions of incident angle and working wavelength for the TE polarization. The black solid lines in (d) show the equal-absorptance contour at $A=0.9$.

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In order to investigate the frequency-dependent absorptance behavior, the dispersion of Si is considered (Fig. 6(b)) [66,67]. We note that the dispersion of TiO₂ is not strong within the visible frequency [68,69]. In Fig. 6(c), we plot the absorptance of the Si slab as functions of the incident angle and working wavelength for the TE polarization, showing a maximal absorptance <0.7. Amazingly, the designed multilayer absorber exbihits almost perfect absorption in a wide incident angle range for TE-polarized lights with wavelength from 440nm to 640nm, as shown in Fig. 6(d). Therefore, such a simple TiO₂-Si multilayer structure can function as a broadband and omnidirectional optical near-perfect absorber. Such a Si-based absorber may find applications in many fields like solar cells and photodetectors.

4. Summary

In conclusion, we have demonstrated a general approach to realize broadband and omnidirectional optical near-perfect absorbers composed of bulk dielectric PhCs. The PhCs are designed to possess an omnidirectional and broadband impedance matching property such that reflection can be eliminated without utilizing surface decorations. With a suitable amount of material loss, light energy can be gradually absorbed in the PhC. Such optical broadband and omnidirectional near-perfect absorbers can be realized with almost any two dielectric materials with relative large contrast and small loss. An TiO₂-Si multilayer structure, as a practical example, is designed to demonstrate the omnidirectional near-perfect absorption behavior in the spectrum of wavelength between 440nm and 640nm. The principle of realizing broadband and omnidirectional near-perfect absorption by designed PhCs with the impedance matching property is general and feasible in a wide frequency spectrum from microwaves, THz waves, to optics. The potential applications include micro-bolometers, solar cells, selective thermal emitters and structure color.