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Abstract
Perfectly matched layer (PML) is a virtual absorption boundary condition adopted in numerical simulations, capable of absorbing light from all incident angles, which however is still lacking in practice in the optical regime. In this work, by integrating dielectric photonic crystals and material loss, we demonstrate an optical PML design with near-omnidirectional impedance matching and customized bandwidth. The absorption efficiency exceeds 90% for incident angle up to 80°. Good consistence is found between our simulations and proof-of-principle microwave experiments. Our proposal paves the road to realize optical PMLs, and could find applications in future photonic chips.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Reflection, transmission and absorption of light are very common phenomena in optics. Due to inevitable impedance mismatch between natural materials and free space, unwanted reflection could cause bizarre phenomena in optical devices like ghost image [1]. To eliminate the reflection, perfectly matched layers (PMLs) [2,3], as artificial absorbing domains or boundaries usually used in finite-element method or finite-difference-time-domain simulations, were proposed to truncate electromagnetic wave propagation. A realizable device with unique PML properties capable of absorbing electromagnetic waves incident from all directions over a broad frequency band will be of great interest in terms of applications like electromagnetic shielding and energy harvest [4,5]. At microwave regimes, wedge-shaped electromagnetic interference (EMI) shielding materials like absorptive foams are commonly used as PMLs. However, the experimentally achievable PMLs in the optical regime, that is of particular interest to on-chip optical integration with which the unwanted interference of scattered light shall be eliminated, are still lacking.
In the past decades, artificial electromagnetic structures like photonic crystals (PhCs) and metamaterials have attracted enormous research attention because they have demonstrated great ability in controlling the propagation of light [6–10]. These artificial structures provide a new platform to implement the PMLs. Metamaterial-based absorbers have successfully demonstrated perfect absorption at one specific frequency or several discretized frequencies through engineering electric and magnetic resonances of meta-atoms [6,11–14]. In order to extend frequency spectrum of absorption, many methods have been proposed including stacking multiple resonant units [15–17], anomalous Brewster effect [18–20], etc. And metasurface, as the two-dimensional (2D) equivalent of metamaterial, has also been extensively studied for broadband optical absorbers [21–23]. In addition, great efforts have been devoted to widening the absorption angle range [24–26]. Currently, many works have reported wide angle and broadband metamaterial-based absorbers operating in microwave band [27,28], however, metamaterials with complicated electric and magnetic resonant meta-atoms are still challenging to be implemented in the optical regime.
Interestingly, PhCs provides a versatile platform for manipulating light-matter interactions and realizing unprecedented electromagnetic effects in the optical regime [8,29–31]. For instance, the photonic band gap effect of PhCs has triggered industrial applications such as PhC waveguides and PhC fibers [32,33]. The topological property of photonic band gap has led to topological photonics where the PhCs are exploited to realize the classical analogy to condensed matter physics [34–38]. Notably, there are also many interesting phenomena in the passband of PhCs [39–41]. By constructing different shapes of equi-frequency contours (EFCs), a variety of functionalities have been obtained. Recently, dielectric PhCs with unique EFCs centered at Brillouin zone boundary were proposed, which allow near 100% transmission of light from all incident angles with zero reflection, thus realizing the ultra-transparent media [42–44]. This unique impedance matching property inspires us to explore approaches to switch the functionality of the PhCs from perfect transparency to perfect absorption over an ultra-wide angle range, so as to realize optical PMLs.
In this work, we propose a kind of optical PML based on the integration of dielectric PhCs and material loss. By careful design of dielectric PhCs, near-omnidirectional impedance matching effect can be achieved so as to eliminate reflection over an ultra-wide angle range. Furthermore, through introducing material loss into the PhCs to gradually dissipate the transmitted light, the optical PMLs can be obtained, as illustrated in Fig. 1. We find that the absorption efficiency of the PhC-based PMLs can exceed 90% for light up to 80° with customized bandwidth. Proof-of-principle microwave experiments further demonstrate the excellent performance of the PhC-based PMLs. Our work opens an avenue for experimentally achievable optical PMLs, which are promising for applications in future photonic chips.
Fig. 1. Schematic diagram of a PhC-based optical PML exhibiting near-omnidirectional impedance matching effect and customized bandwidth, which can be exploited to absorb light from different directions.
2. Principle and methods
We begin with a 2D lossless dielectric PhC consisting of a rectangular array ($a \times b$ in dimension) of silicon cylinders (refractive index ${n_{\textrm{si}}}$= 3.95, $r = 0.375a$, painted in gold) in the background of silica (refractive index ${n_{\textrm{Si}{\textrm{O}_2}}}$=1.45, painted in pink), as illustrated by the inset in Fig. 2(a). To endow the PhC with the near-omnidirectional impedance matching effect, the lattice period b is optimized as $1.37a$ (see Supplemental Material). The band structure of the PhC for transverse magnetic polarization with electric field polarized along the cylinder axis is presented in Fig. 2(a). We use normalized frequency here, indicating based on the scaling law of Maxwell equation, the properties to be achieved can be directly scaled to other working frequency by shrinking or expanding the whole structure if material property can be fixed. The simulations throughout this work are performed using the finite-element software COMSOL Multiphysics. The horizontal dashed line denotes the central operating frequency $fa/\textrm{c} = 0.20648$, which lies in the second band. Here c is the speed of light in free space. The corresponding EFCs of the second band are further shown in Fig. 2(b), where the red and the blue dashed lines represent the EFCs of the PhC and air at $fa/\textrm{c} = 0.20648$, respectively. We find that the PhC’s EFC is nearly a part of ellipse centered at the X point, and its height (i.e., the maximum ${k_y}$) is almost the same as that of air, which are crucial for the realization of the near-omnidirectional impedance matching effect [43].
Fig. 2. (a) Band diagram for transverse magnetic polarization of a 2D dielectric PhC, whose unit cell is illustrated in the inset. (b) The EFCs of the second band. The red and the blue dashed lines denote, respectively, the EFC of the PhC and EFC of air at $fa/c = 0.20648$. (c) The impedance difference between the PhC and air, i.e., $\left|{\frac{{{Z_{\textrm{PhC}}} - {Z_{\textrm{Air}}}}}{{{Z_{\textrm{PhC}}} + {Z_{\textrm{Air}}}}}} \right|$. (d) Transmittance through the PhC slab with N ($= 3,4,10,15$) layers of unit cells along the x direction as the function of incident angle. The inset illustrates the configuration of numerical setup.
To explore the impedance characteristic of the PhC, we consider light incident onto the $yz$ surface of the PhC with a wave vector lying on the $xy$ plane, i.e., $\mathbf{k} = {k_x}\hat{x} + {k_y}\hat{y}$. The wave impedance of the PhC ${Z_{\textrm{PhC}}}$ can be evaluated by analyzing the eigen-fields of PhC, and can be expressed as ${Z_{\textrm{PhC}}} = \frac{{\langle{E_z}\rangle}}{{\langle{H_y}\rangle}}$ [43], where $\langle{E_z}\rangle$ and $\langle{H_y}\rangle$ are the averaged $z$-component of eigen-electric field and $y$-component of eigen-magnetic field on the $yz$ surface of the PhC unit cell. On the other hand, the wave impedance of air ${Z_{\textrm{Air}}}$ can be expressed as ${Z_{\textrm{Air}}} = \frac{{\omega {\mu _0}}}{{\sqrt {k_0^2 - k_y^2} }}$ [43], where $\omega $, ${\mu _0}$, ${k_0}$ are the angular frequency, the permeability of free space, the wave number in free space, respectively. For comparison, in Fig. 2(c) we plot the impedance difference between the PhC and air, i.e., $\left|{\frac{{{Z_{\textrm{PhC}}} - {Z_{\textrm{Air}}}}}{{{Z_{\textrm{PhC}}} + {Z_{\textrm{Air}}}}}} \right|$, in k-space. We observe that the impedance difference is very small within a broad frequency band for a large range of ${k_y}$, indicating broadband and wide-angle impedance matching effect. In particular, at $fa/c = 0.20648$, the impedance difference is near-zero for almost all ${k_y}$ smaller than ${k_0}$, indicating the near-omnidirectional impedance matching effect within the PhC. For further verification, we numerically calculate the transmittance through the PhC slab consisting of N ($= 3,4,10,15$) layers of unit cells assembled along the x direction, as shown in Fig. 2(d). The configuration of numerical setup is illustrated by the inset. As expected, the transmittance exceeds 0.99 over a wide range of incident angle ($0\sim 82^\circ $) at $fa/c = 0.20648$ irrespective of the layer number N due to the occurrence of the near-omnidirectional impedance matching effect.
Now, we introduce material loss into the PhC to explore the road to optical PMLs. Here, we assume a complex refractive index of silicon cylinders as $3.95 + i\kappa $. In practice, the parameter $\kappa $ can be tuned through changing the doping concentration of silicon [45]. In the following, we will investigate the overall effect of material loss to the impedance matching effect and the absorption performance through adjusting the parameter $\kappa $, and finally find out the optimal scheme of optical PMLs. Figures 3(a)–3(d) show the absorptance by the PhC slab as the function of layer number N and incident angle at $fa/c = 0.20648$ for different values of $\kappa $. We find that for relatively small $\kappa $, the impedance matching property is not disturbed, and high absorption ($A \ge 0.9$) can be obtained up to $80^\circ $ in the model with $\kappa = 0.025$ [Fig. 3(a)], which exhibits a great advantage in angular performance compared with other optical absorbing structures, e.g., optical multilayer films [46,47]. Nevertheless, the maximal incident angle for absorptance $A \ge 0.9$ would decrease with increasing $\kappa $, indicating that the impedance matching at large incident angles would be destroyed by the material loss. On the other hand, we see that the model with small $\kappa $ requires a larger thickness (i.e., larger $N$) to obtain the same amount of absorption compared with the model with large $\kappa $, which would make the PhC-based PMLs too bulky. Considering the balance of angular performance and thickness, we set $\kappa = 0.1$. We note that the wide-angle high absorption can cover a relatively broad frequency band. As we have shown in Fig. 3(e), in which the absorptance is plotted as the function of frequency and incident angle by fixing $\kappa = 0.1$ and $N = 20$, a frequency band of $0.1887 \le fa/c \le 0.2387$ (or $0.1986 \le fa/c \le 0.2266$) for absorptance $A \ge 0.9$ is obtained at normal incidence (or at the incident angle of $50^\circ $).
Fig. 3. [(a)-(d)] Absorptance by the PhC slab with respect to layer number N and incident angle for $\kappa = $0.025, 0.05, 0.1, 0.2. (e) Absorptance with respect to frequency and incident angle with fixed $\kappa = 0.1$ and $N = 20$. In (a)-(e), the dash lines denote that equi-absorptance contours of $A$=0.9. (f) Simulated electric field-distribution when a point source is placed inside an optical cavity constructed by four trapezoidal-shaped PhC-based PML walls.
Based on the above design principle, we would like to show an optical cavity constructed by four trapezoidal-shaped PhC-based PML walls, as shown in Fig. 3(f). The PhC consists of a rectangular array ($a = 200\textrm{nm}$, $b = 274\textrm{nm}$) of doped silicon cylinders ($r = 75\textrm{nm}$, ${n_{\textrm{Si}}} = 3.95 + 0.1i$) in silica background. The central operating wavelength is 969 nm. Each PhC-based PML wall is an isosceles trapezoid with a short base of $30a$ and an altitude of $12b$, thus creating a square area of free space ($30a \times 30a$ in dimension) in the center. A point source with out-of-plane electric field is placed in the central free space. The simulated electric field-distribution in Fig. 3(f) shows near-perfect cylindrical radiation pattern in free space and fast attenuation of light in the PhC region, indicating near-perfect absorption of all the emitted light by the PhC-based PML walls, with almost no reflection. These results clearly demonstrate the excellent performance of the PhC-based optical PML, which could find promising applications in photonic chips. Although the PML is only demonstrated for transverse magnetic polarization in our work, the design principle is general and can be extended to transverse electric polarization or even both polarizations.
3. Experiment and results
As a proof of principle, we experimentally fabricated a PhC-based PML at microwaves, as shown in Fig. 4(a). The PhC consists of a $33 \times 13$ rectangular array ($a = 10\textrm{mm}$, $b = 13.7\textrm{mm}$) of lossless alumina cylinders (relative permittivity 7.5, $r = 3.75\textrm{mm}$), which is designed to exhibit the near-omnidirectional impedance matching effect at 10.16 GHz (see Supplemental Material). Different from the above optical design, it is challenging to directly tune the material loss of the alumina cylinders in microwave experiments. By lasing cutting absorptive foam (relative permittivity $1.5 + 1i$) into annuli whose inner diameter equals to the diameter of alumina cylinders, alumina cylinder with loss can thus be prepared. Moreover, the material loss of PhC can be controlled by adjusting the thickness of absorptive annuli. Considering the balance of impedance matching and absorption performance, the thickness is optimized as $1\textrm{mm}$ (see Supplemental Material).
Fig. 4. (a) Photograph of a microwave PhC-based PML. The inset shows the zoom in view of the PhC unit cell. (b) Photograph of the experimental setup. [(c)-(e)] Measured ${E_z}$-distributions at 10.16 GHz for incident angles of (c) $30^\circ $, (d) $45^\circ $, and (e) $60^\circ $. The PhC region is the area marked by red lines. (f) Simulated reflectance R (blue) and absorptance A (red) of the PhC-based PML (solid lines) and the EMI material (dashed lines) at 10.16 GHz. The asterisks denote the experimentally measured absorptance A of the PhC-based PML.
The PhC-based PML is placed inside a metallic parallel-plate waveguide, and is surrounded by absorptive foams (in blue) to avoid unwanted scattering, as shown in Fig. 4(b). The parallel-plate waveguide is composed of two parallel aluminum plates separated by $10\textrm{mm}$, in which transverse magnetic polarized propagating waves with electric field in the vertical direction (i.e., parallel to the alumina cylinders) is supported. The probe antenna is fixed through a hole on the top aluminum plate and the source antenna and the sample is mounted on a computer-controlled x-y translational stage. Thus, the time harmonic electric field distributions of the whole PhC structure can be measured point by point. In our experiment, a source antenna connected to a vector network analyzer (Agilent E5071C) is placed behind an acrylic lens to generate an empirical plane-wave incidence. The incident channel has a width of $70\textrm{mm}$, and its orientation is fixed. In experiments, the orientation of the PhC is changed to obtain different incident angles. Figure 4(c)–4(e) present the measured ${E_z}$-distributions at 10.16 GHz for incident angles of $30^\circ $, $45^\circ $, and $60^\circ $, respectively. All results show barely noticeable reflection on the PhC’s surface and fast wave attenuation in the PhC region (the area marked by red lines), manifesting the good performance of the PhC-based PML.
Furthermore, in Fig. 4(f) we compare this PhC-based PML to an EMI shielding material of the same size and shape in numerical simulations. Here, absorptive foam is chosen as the EMI shielding material as an example. In Fig. 4(f), the solid and dashed lines correspond to the PhC-based PML and the EMI material, respectively. The blue and red lines denote the reflectance R and absorptance A, respectively. The asterisks denote the experimentally measured absorptance A of the PhC-based PML, showing good agreement with the simulation results. Compared with the tradition absorptive foam, we see that our PhC-based PML exhibits much better performance, i.e., lower reflection and higher absorption over a wider range of incident angles.
To further verify the good performance of the PhC-based PML, we examine the radiation pattern of a dipole source (polarized along the z direction) placed on the left side of the fabricated PhC sample. The measured ${E_z}$-distribution at 10.16 GHz is presented in Fig. 5(a), showing a cylindrical wave pattern on the source side and strong dissipation in the PhC region (the area marked by red lines). We note that the distortion of the cylindrical wave pattern on the source side is caused by the reflection on the surrounding absorptive foams [the foams in blue in Fig. 4(b)], rather than the reflection from the PhC. For verification, the measurement is performed again for free space without the PhC sample, as shown in Fig. 5(b). The field patterns on the source side are observed to be almost identical to the cases of the PhC sample and free space. This proves that the PhC has almost no reflection over a wide range of incidence angle. Moreover, Fig. 5(c) shows the simulated ${E_z}$-distribution, also showing almost no reflection on the PhC under the illumination of a nearby point source. These results clearly demonstrate the good performance of the PhC as a PML.
Fig. 5. (a) Measured ${E_z}$-distribution when a point source is placed on the left side of the PhC-based PML (the area marked by dashed lines). (b) Measured ${E_z}$-distribution in free space without the PhC-based PML. (c) Simulated ${E_z}$-distribution in the presence of the PhC-based PML. (d) Normalized ${E_z}$-distribution along the black dashed line in (a). The black, red and blue lines denote the experimental results, fitting of experimental data and the simulation results, respectively. The operating frequency is 10.16 GHz.
In addition, the attenuation coefficient of waves inside the PhC is analyzed in Fig. 5(d). The black lines denote the measured normalized ${E_z}$ along a horizontal line in the x direction [the black dashed line in Fig. 5(a)]. By fitting the experimental data, we obtain an attenuation coefficient of 16$\textrm{}{\textrm{m}^{ - 1}}$ (red lines), which coincide well with the simulation results (blue lines). This large attenuation coefficient indicates the high-efficiency absorption of our PhC-based PML.
4. Conclusion
In summary, we have proposed an approach to design experimentally achievable optical PMLs based on the integration of dielectric PhCs and material loss. Through introducing appropriate amount of material loss to PhCs with near-omnidirectional impedance matching effect, the functionality of the PhCs can be switched from perfect transparency to perfect absorption over a wide range of incident angle with customized bandwidth. Consequently, the PhCs can behave as high-efficient PMLs, which have been substantiated by full-wave simulations and proof-of-principle microwave experiments. This design principle is universal for waves, and similar approach can be applied to resolve the impedance mismatching issue for acoustic waves and elastic waves.
Funding
National Natural Science Foundation of China (Grant No. 12274315); Natural Science Foundation of Jiangsu Province (Grant No. BK20221354); Key Technologies Research and Development Program (2022YFA1404400); Priority Academic Program Development of Jiangsu Higher Education Institutions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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