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Abstract
Metatronics, as a feasible paradigm of nanocircuits using effective electronic elements (e.g., nanocapacitors, nanoinductors, and nanoresistors), provides the possibility for light manipulation in subwavelength scales assisted by the circuit-related technologies in electronics. As a representative technique in electronics, Smith Chart provides a fast, less-computation and graphical approach to solve the problems related to impedance matching. Here, we transplant the Smith Chart into the paradigm of optical metatronics to develop an analytical approach for impedance matching for light propagation and coined the name of graphical metatronics. In this approach, the impedance characteristics of four basic types of ultrathin metatronic layers are creatively mapped into each rotation trace on the complex Γ mathematical plane (Γ means the reflection coefficient). The impedance matching problems can be graphically solved by searching for feasible rotation traces on the Γ plane without full-wave simulations. Based on this approach, various applications related to impedance matching (e.g., antireflection coating, perfect transmission, absorber, etc.) are developed analytically and validated by numerical results. The proposed approach constructs the bridge among Smith Chart, plasmonics and photonics, providing a fast, visualized and less-computation route and guideline to develop various nanophotonic structures and devices for impedance-matching applications.
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1. Introduction
Electronics has developed rapidly over several decades and has exerted a far-reaching influence on modern science and technology. The wide applicable range of electronics is not only caused by its flexibly controllable ability on the motion of electrons but also benefits from subwavelength components (i.e., inductors, capacitors, and resistors). Conventional subwavelength components are utilized in electronics with unique merits such as modularization and simplification and are challenging to apply to optical regions [1–4]. Recently, several researches have successfully applied the circuit theorem to guide the design of the optical nanoantennas and composite nanostructures [5–8], significant promoting the development of nanocircuit theory. Nonlinear materials [5,6], plasmonic nanoparticles [7], and hyperbolic metamaterial [8] are utilized in these works to improve the performance of various optical devices. Interestingly, inspired by the concepts of metamaterials and plasmonics, metatronics (i.e., metamaterial-inspired nanoelectronics) has been devised creatively at optical frequencies, providing a feasible circuit paradigm for light manipulation, as an extension of electronic circuits. Subwavelength and uniform nanolayers whose material permittivities have a negative real part, positive real part, or nonzero imaginary part can act as effective nanoinductors, nanocapacitors or nanoresistors, respectively [9–12]. Combining the concept of displacement currents and nanoelements, circuit theory in electronics can be transplanted to the optical metatronics. The notion of metatronics provides methodologies to develop various optical applications, including microcavity lasers [13], nanofilters [14], computational devices [15], and impedance matching layers [16–18]. Experiments have been successfully carried out to validate the realizability of metatronics [19–23]. The merits of metatronic elements are listed as follows: broad operating ranges from microwave to optical domains; not only applicable for normal incident wave but also applicable for oblique incident wave; easily integrated with nanocircuits. To sum up, the paradigm of metatronics builds the bridge between electronics and optics and thus, offers an avenue for light manipulation in subwavelength scales assisted by circuit-related techniques in electronics.
Impedance matching is an important subject aiming to reduce the power reflection and enhance the transmission. Smith Chart, as an essential and well-known graphical technology in electronics, provides a fast, less-computation and visualized way to solve various problems related to impedance matching [24]. Impedance characteristics of inductors, capacitors and transmission lines are mapped to the chart and are investigated without full-wave simulations [25–28]. Smith Chart can be utilized to describe not only the impedance trends at a certain frequency with the length variation of transmission lines, but also the impedance trends of a certain component with the variation of operating frequencies. In this paper, we mainly focus on the latter. On this basis, various interesting studies are presented to validate the practicability of this graphical technique. Smith Chart is utilized to design switchable impedance matching networks comprised of serval components, realizing high-performance antennas for broadcasting applications and mobile terminals [29,30]. It is also used to improve the surface impedance and reflection coefficient of a tunable active impedance metasurface for the manipulation of electromagnetic waves [31,32]. Performances of metamaterial-based absorbers are optimized according to the Smith Chart [33,34]. Extensions to the conventional Smith Chart (such as generalized ZY Smith Chart [35], spherical Smith Chart [36]) have been proposed in the past decades. Besides, Smith Chart is an integral part of much of the current computer-aided design (CAD) software and modern measurement devices like vector network analyzers. To summarize, Smith Chart plays an irreplaceable role in electronics and makes impedance-matching design accessible to all levels of electronic engineers. However, due to the lack of optical lumped components, the design philosophy of low-frequency lumped circuits cannot directly be transplanted to optical regimes.
In this work, we transplant the notion of the Smith Chart into the paradigm of optical metatronics, providing an analytical way to solve impedance matching problems via ultrathin metatronic layers. Smith Chart is employed as the fast design tool for optical metatronics, exhibiting advantages of visualization and substantially reducing computation complexity. Here, this approach is coined the name of graphical metatronics. Four basic types of metatronic layers are analyzed, and their impedance properties are creatively mapped into four rotation traces on the complex reflection (Γ) plane. The angle and direction of the rotation trace are determined by the material characteristics (i.e., permittivity and thickness) of the metatronic layer. On this basis, impedance-matching problems can be figured out by finding feasible transformation traces on the Smith Chart without full-wave simulations. Therefore, various optical requirements related to impedance matching can be analytically and graphically designed via ultrathin metatronic layers assisted by Smith Chart, e.g., electromagnetic transparency, invisible cloaking, anti-reflection coating, absorber, etc. Compared with previous optical matching structures based on metamaterials or metasurfaces that are composed of periodical or non-uniform unit cells [37–44], the metatronic-based matching structures show potential merits as follows: they are composed of multiple uniform layers and are easily compatible with existing optical fabrication processes [19,45–47]; metatronic layer are with deep subwavelength size and are suitable for ultrathin optical applications. Moreover, previous metatronic structures [13–23] are usually optimized through numerical solutions. Our work provides an analytical solution assisted by Smith Chart, showing the advantages of fast, visualized and less-computation. Besides, this work presents a general impedance matching approach that can be utilized to address multiple optical problems. The notion of graphical metatronics applies Smith Chart into plasmonic and nanophotonics regions, guiding the design of optical devices related to impedance matching. It exhibits a broad range of potential applications in various research fields, such as optical, nanophotonics, electronics, and material science.
2. Results and discussions
2.1 Theory of graphical metatronics
As shown in Fig. 1(a), to investigate the characteristics of metatronic elements, they are positioned between Medium 1 (M1, with intrinsic impedance of Z1 = R1 + jX1) and Medium 2 (M2, with intrinsic impedance of Z2 = R2 + jX2). There are four basic types of metatronic elements in the paradigm of graphical metatronics. Each type is a uniform and ultrathin layer (blue, brown, green, and pink) with relative permittivity ɛi and width along the propagation direction of wi (i = 1, 2, 3, or 4). The value of wi is less than λ0/10 with negative value of ɛi in Type 1 and Type 3 and positive value of ɛi in Type 2 and Type 4. Relative permittivities of M1 and M2 are ɛM1 and ɛM2.
Fig. 1. Theoretical analysis of graphical metatronics. (a) Four basic types of metatronic layers, indicating four rotation traces on the Γ-plane in (b). The material parameters of permittivities and width of metatronic elements determine the rotation direction and angle.
As demonstrated in previous investigations [16,21], when the incident plane wave illuminates the metatronic layer with normal incidence, the individual metatronic element behaves as an effective shunt-circuit component, with the effective impedance ZP = -j/(2πfɛ0ɛiwi). Its impedance is effective capacitive when the layer has a positive permittivity (e.g., Si or SiO2). On the other hand, effective inductive impedance is obtained when the layer has a negative permittivity (e.g., plasmonic materials, such as Ag or Au). Thus, the metatronic layers of Type 1 and Type 2 behave as an effective shunt component. For Type 3 and Type 4, the sandwich structure comprising a dielectric layer inside the host medium (gray parts; relative permittivity is ɛh and the length along propagation direction is λh/2) is constructed as the effective series component [23]. Therefore, the effective impedances ZS of Type 3 and Type 4 are expressed as ZS = ZH2/ZP, where ZH is the intrinsic impedance of the host medium.
Then, in Fig. 1(b), according to the reflection coefficient ΓH between one of the two Media (M1 or M2) and the host medium, we build a mathematical graph, i.e., ΓH-plane, to emulate the Smith Chart in electronics. The ΓH-plane is designed based on the intrinsic impedance ZH of the host medium. We assume that the intrinsic impedance of an arbitrary medium is ZM = RM+jXM. Thus, the reflection coefficient ΓH is expressed as a complex value, i.e., ΓH = (ZM-ZHost)/(ZM+ZHost) = ΓReal+jΓImag. We utilize the real part ΓReal and the imaginary part ΓImag of ΓH to represent the abscissa axis and the vertical axis. Then, the required ΓH-plane is constructed. For any passive system, ΓH satisfies 0≤|ΓH |≤1. When ZM = ZHost, i.e., |ΓH|=0, the impedance ZM of this medium is located at the origin point of the ΓH -plane. When |ZM| is infinity, i.e., |ΓH| = 1, the impedance of this medium is located on the outermost circle (black circle in Fig. 1(b)). Therefore, the effective region of the ΓH-plane is from the origin point to the circle of |ΓH| = 1, and an arbitrary media impedance ZM can be mapped to one point within this region. More detailed descriptions of ΓH-plane are depicted in Supplemental Section 1 of Supplement 1.
Next, the proposed graphical metatronics constructs the mapping relation between four rotation traces on the ΓH-plane and four basic metatronic types. As shown in Fig. 1, the incident wave illuminates with normal incidence, and we suppose M1, M2, and the host medium are all air and have impedances of Z1 = Z2 = ZHost = 377 Ω. Z2 is located at the origin point in the ΓH-plane as the starting point, i.e., the black point. There are two significant circles (i.e., Circle-G2 and Circle-R2) going through point Z2. Circle G2 is the constant G circle, the impedance points on which have different susceptances (B) but identical conductances (G). Circle R2 is the constant R circle, the impedance points on which have different reactances (X) but identical resistances (R). Detailed descriptions of these two circles are provided in Supplemental Section 1 of Supplement 1. When metatronic layer Type 1 or 2 (as the effective shunt susceptance) is positioned on the interface, it only affects the susceptance (B) and does not affect the conductance (G). Thus, when changing ɛi or wi, the impedance Z2 always rotates on Circle-G2. For Type 1 with negative ɛ1, the rotation is in a counterclockwise direction with an increment of |ɛ1| or |w1| (i.e., the blue trace in Fig. 1(b)). For Type 2 with positive ɛ2, the rotation is in a clockwise direction with an increment of |ɛ2| or |w2| (i.e., the brown trace). When metatronic element Type 3 or 4 (as the effective series reactance) is positioned on the interface, it only affects the reactance (X) and does not affect the resistance (R). Thus, when changing ɛi or wi, the impedance Z2 rotates on Circle-R2. For Type 3 with negative ɛ3, the rotation is in a counterclockwise direction with an increment of |ɛ3| or |w3| (i.e., the green trace). For Type 4 with positive ɛ4, the rotation is in a clockwise direction with an increment of |ɛ4| or |w4| (i.e., the pink trace). The effect of material characteristics on the impedance Z2 is observed through the rotation traces in the ΓH-plane. In Supplemental Section 2 of Supplement 1, theoretical derivations and numerical validations are carried out to validate this rule with detailed results. Therefore, the investigations on metatronic layers are simplified to the combinations of different rotation traces in the ΓH-plane. Parameters of each metatronic layer can be determined through this analytical and graphical approach without full-wave simulations. It should be noted that the homogeneous metatronic layer with negative permittivity is made of plasmonic materials, which may be limited from the natural material selection. As an implementation of metatronic layer, the mixed medium theory [48] can be adopted to synthesize the desired permittivity by using natural materials. Detailed descriptions and one example are presented in Supplement Section 3, Fig. S3 and Fig. S4 of Supplement 1.
2.2 Impedance-transformation regions of graphical metatronics
As shown in Fig. 1(a), the wave impinges on the interface between M1 and M2 at normal incidence. We suppose that the intrinsic impedances of M1 and M2 are complex numbers of Z1 = R1 + jX1 and Z2 = R2 + jX2. The variation regions of impedance Z2 of Medium 2 in the ΓH-plane are studied when metatronic structures containing different numbers of metatronic layers are positioned respectively. Without loss of generality, the initial Z2 is set as Z2 = ZH = 377 Ω, i.e., at the origin point of the ΓH-plane. As shown in Fig. 2(a), for the single-element metastructure, Z2 can only rotate on Circle-G2 (Type 1 or Type 2) or Circle-R2 (Type 3 or Type 4). For the double-element structure containing an effective shunt component and an effective series component, there are two feasible routes. For route-1 shown in Fig. 2(b), the effective series component (L1) is placed at the left of the effective shunt component (L2). Except for all points located inside Circle-R2, Z2 can be moved to any position inside the ΓH-plane (i.e., the green region). On the other hand, for route-2 in Fig. 2(c), the effective series component (L1) is placed at the right of the effective shunt component (L2). Except for all points located inside Circle-G2, Z2 can be moved to any position inside the ΓH-plane (the purple region). It is observed that by adequately selecting the above-mentioned single-element or double-element metatronic routes, impedance-transformation areas can cover all impedance points in the ΓH-plane. In summary, arbitrary impedance matching can be realized based on the notion of graphical metatronics. The material characteristics of metatronic layers can be determined by the graphic traces in the ΓH-plane. Theoretical proofs of these conclusions are presented in Supplemental Section 4 of Supplement 1. We assume metatronic layers L1, and L2 are with the material characteristics of (ɛ1, w1) and (ɛ2, w2), respectively. After positioning these metatronic layers, the impedance Z2 is transformed to Z2’, and Γ12 is transformed to Γ12’ = |(Z2'-Z1)/(Z2'+Z1)|. According to the rotation traces in the ΓH-plane from Z2 to Z2’, the values of (ɛ1, w1) and (ɛ2, w2) can be determined to complete impedance matching (i.e., Z2’ = Z1 and Γ12’ = 0).
Fig. 2. Impedance-transformation regions of metatronic structures with different number of metatronic layers. a)Single-element metatronic structure, b) double-element metatronic structure with route-1, and c) double-element metatronic structure with route-2.
2.3 Three representative cases based on graphical metatronics
Here, three representative cases related to impedance matching are designed to validate the theory of graphical metatronics. Figure 3(a) and 3(b) present sketches of the stratified media without or with the double-element metatronic structure. We take the transverse electric (TE) wave with normal incidence as an example in these investigations. The design frequency is f0 = 200 THz, and the free space wavelength is λ0 = 1500 nm. The intrinsic impedance, permeability, and relative permittivity of M1 are Z1, μ0 and ɛr1, and those of M2 are Z2, μ0 and ɛr2. The host medium is with a relative permittivity of 1. As shown in Fig. 3(c-e), Case 1 matches the real impedance (M1) and real impedance (M2), where ɛr1 = 2.25, ɛr2 = 49, Z1 = 251.3 Ω and Z2 = 53.9 Ω. Z1 and Z2 are located on the real axis (green line) in the ΓH-plane of Fig. 3(e). In Fig. 3(c), a strong standing wave appears within M1 when the system is without the metastructure, indicating that strong reflection exists on the interface. Then, according to the feasible rotation trace from Z2 to Z1 in Fig. 3(e), we design a metastructure whose metatronic layers L1 and L2 have permittivities of 6.5 and 69, respectively. The uniform electrical field distribution of the two media in Fig. 3(d) demonstrates that the reflected wave in M1 is removed and that energy is totally transmitted to M2. The different magnitudes of the electrical field in M1 and M2 are due to their different permittivities, and this phenomenon does not conflict with the law of conservation of energy. Then, Case 2 in Fig. 3(f-h) matches the real impedance (M1) and complex impedance (M2), where ɛr1 = 2.25, ɛr2 = 49-j98, Z1 = 251.3 Ω and Z2 = (30.6 + j18.9) Ω. In the ΓH-plane of Fig. 3(h), Z1 is located on the real axis, and Z2 deviates from the real axis. In Fig. 3(f), there also appears to be a strong reflected wave in M1 when the system is without the metastructure. Then, according to the feasible rotation trace from Z2 to Z1 in Fig. 3(h), we design a metastructure whose metatronic layers L1 and L2 have permittivities of 4 and 96, respectively. Figure 3(g) demonstrates that the reflected wave in M1 is removed and that energy is totally consumed in M2. Finally, Case 3 in Fig. 3(i-k) discusses the matching between complex impedance (M1) and complex impedance (M2), where ɛr1 = 2.25-j0.675, ɛr2 = 49-j14.7, Z1 = (243.4 + j35.7) Ω and Z2 = (52.2 + j7.7) Ω. In the ΓH-plane of Fig. 3(k), Z1 and Z2 both deviate from the real axis. In Fig. 3(i), there also appears to be a strong reflected wave in M1 when the system is without the metastructure. The standing wave here is slightly smaller than those in Fig. 3(c) and 3(f). It is because M1 in Case 3 is a lossy medium and partial energy is consumed. Then, according to the feasible rotation trace from Z2 to Z1 in Fig. 3(k), we design a metastructure whose metatronic layers L1 and L2 have permittivities of 6 and 64.5, respectively. Figure 3(j) demonstrates that the reflected wave in M1 is removed and that the transmitted wave is totally consumed in M2. The widths of the metatronic elements in these three cases are all 10 nm (i.e., 1/150 λ0), and detailed structures are presented in Fig. S6 of Supplement 1. To conclude, regardless of the media with real or complex impedances, we can always use the concept of graphical metatronics to realize the required impedance transformation, showing the applicable potential for the design of applications related to impedance matching, e.g., electromagnetic transparency, cloaking, anti-reflection coating, absorber, etc. Here, we would like to emphasize that the generality of the proposed approach is validated in Section 2.2 by using Smith Chart. Three proof-of-concept examples presented in Section 2.3 are without loss of generality, and they represent three general situations for optical impedance matching. Permittivities of materials M1, M2, L1 and L2 selected in these cases are just one example set of values. The proposed approach also works when these materials are with other suitable values of permittivity, including real materials. Besides, in these cases, ‘plane wave in open space’ is chosen as a general situation to demonstrate and validate the concept of graphical metatronics. This proposed concept is also effective for other applicable situations, for instance, guide waves in a metallic waveguide. Based on the waveguide effective medium in [21,49], we verify the feasibility of the graphical metatronics in a metallic waveguide operating at guide-wave mode. Detailed descriptions and results are presented in Supplement Section 5, Fig. S7 and Fig. S8 of Supplement 1.
Fig. 3. Three Cases related to impedance matching, by using a double-element metatronic structure. Sketches of the stratified media (a) without or (b) with the metastructure. (c-d) Numerical results when M1 and M2 are both with the real impedances: the average magnitude distributions of electrical field (c) before and (d) after inserting the metastructure, and (e) the transformation traces from Z2 to Z1 on the mathematical Γ-plane. (f-h) Numerical results when M1 is with the real impedance and M2 is with the complex impedance. (i-k) Numerical results when M1 and M2 are both with the complex impedances.
2.4 Loss analysis on graphical metatronics
For the sake of simplicity, the previous analysis and numerical validations of the graphical metatronics are based on lossless metatronic layers. For metatronic structures in actual optical applications, the loss of the metatronic layer, especially plasmonic materials, cannot be negligible. Here, systematic investigations on the loss effect of the graphical metatronics are carried out. An optical metatronic structure constructed with realistic materials is also presented to validate the proposed concept. The incident electromagnetic wave is the TE mode with electrical fields along the y-axis and propagation direction along the z-axis. The designed frequency is f0 = 200 THz. Without loss of generality, dual-element structure with two metatronic layers (L1, L2) is utilized, as shown in Fig. 4(a). Two kinds of metatronic layers can be utilized: one kind is the normal layer with a positive real part of the relative permittivity. Its relative permittivity is expressed as ɛ = ɛ'-jɛ'' and dielectric loss depends on loss tangent tanδ = ɛ''/ɛ'. The other kind is the plasmonic layer with a negative real part of the relative permittivity. Its relative permittivity is expressed using a lossy Drude model ɛ(f)=ɛinf -fp2/(f 2-jf γ) (where the epsilon infinity ɛinf and plasmonic frequency fp are constant and γ represents the collision frequency) and its dielectric loss depends on γ. First, the effect of the normal layer’s loss is studied, and the relative permittivity of M1, M2, L1, and L2 is set as 1, 9, 11.75-j(11.75tanδ), and 35.1-j(35.1tanδ), respectively. When the loss tangent tanδ of normal metatronic elements has different values of 0, 0.01, and 0.1, the reflection coefficients are shown in Fig. 4(b). With increasing tanδ, the reflections at f0 are −56 dB, −37 dB and −20 dB. Then, the effect of plasmonic layer loss is studied, and the relative permittivities of M1, M2, L1, and L2 are set as 1, −2-j(0.25), 2-fp2/(f 2-jf γ), and 112.8, respectively. Here, fp is 737.7 THz. When γ of the plasmonic layer L1 has different values of 0, fp/200, and fp/20, the reflection coefficients are shown in Fig. 4(c). With increasing γ, the reflections at f0 are −64 dB, −35 dB and −16 dB. In summary, compared with the reflection when the system is without the dual-element metastructure (i.e., −4.4 dB in Fig. 4(b) and −0.5 dB in Fig. 4(c) at f0), the reflection can be significantly improved after positioning the metastructure even though material loss is considered. The effect of loss on the rotation traces in the mathematical Γ-plane is presented in Fig. 4(d-i), and a slight discrepancy is observed with the increment of loss. Then, as shown in Fig. 4(j), we provide an example based on realistic materials. The materials for M1, M2, L1 and L2 are air, silicon dioxide, silver, and silver, respectively. From the results in Fig. 4(k), the reflection is reduced from −13.5 dB to −41.4 dB at f0 = 200 THz through the two-element structure, illustrating that the approach has the potential to be utilized in actual optical metatronics. Dimensions of Fig. 4(j) and another example using silver as M2 are given in Fig. S9-S10 of Supplement 1. Besides, the effects of a double-layer metatronic and a triple-layer metastructures on the operating bandwidth are investigated and compared in Supplement Section 6 of the Supplement 1. Results indicate that the impedance bandwidth of the proposed metastructure can be enhanced by changing the number of metatronic layers and substantially finding low-Q (quality factor) traces on the Smith Chart.
Fig. 4. Loss analysis and an example for optical metatronics. Sketches of the stratified media (a) Structure to investigate the influence of material losses of metatronic layers. (b) When the value of ɛ''/ɛ’ varies, the reflection coefficients are shown. (c) When the value of γ varies, the reflection coefficients are shown. (d-f) At f0 = 200 THz, rotation traces on the Γ-plane from Z2 to Z1 when the value of ɛ''/ɛ’ is 0, 0.01 and 0.1. (g-i) At f0 = 200 THz, rotation traces on the Γ-plane from Z2 to Z1 when the value of γ is 0, fp/200 and fp/20. (j-k) Simulation structure and reflection coefficient when realistic materials are utilized in the dual-stratified media.
2.5 Investigation on the oblique incidence
Here, investigations on oblique incidence are carried out to broaden the applicable region of graphical metatronics. As shown in Fig. 5, the light propagates from M1 to M2, and the permittivities of M1 and M2 are ɛ1ɛ0 and ɛ2ɛ0, respectively. The incident angle is θi (i.e., the angle between the propagation ki and the normal direction z of the interface), and the transmitted angle is θt. The relationship between θi and θt is sinθi/sinθt = sqrt(ɛ2/ɛ1). Triple-element metatronic structures are utilized to realize the impedance matching. When the incident wave is TE polarized, the individual metatronic layer with relative permittivity of ɛi and width of wi has an effective impedance ZP expressed as ZP = -j/(2πfɛ0ɛiwi(1-(sinθi)2ɛ1/ɛi)). If θi = 0°, the effective impedance is simplified as ZP = -j/(2πfɛ0ɛiwi). When incident wave is TM polarized, the effective impedance is expressed as ZP = -j/(2πfɛ0ɛiwi) whatever the incident angle is. Without loss of generality, two examples are studied and the results are shown in Fig. 5, including (I) TE wave with incident angle of θi = 70° in Fig. 5(a) and (II) TM wave with incident angle of θi = 30° in Fig. 5(f). We select ɛ1 = 1 and ɛ2 = 2.25 for both two examples. For example (I) of TE wave, the intrinsic impedance of tangential electrical fields in M1 and M2 are Z1 = η1/cosθi = sqrt(μ0/(ɛ1ɛ0))/cosθi and Z2 = η2/cosθt = sqrt(μ0/(ɛ2ɛ0))/cosθt. Z1 and Z2 are marked as the black circle point and blue triangular point in the ΓH-plane, as shown in Fig. 5(b). A feasible rotation trace (Z2→L1→Z2'→L2→Z2''→L3→Z1) is found. By utilizing the approach of graphical metatronics, the relative permittivities of metatronic layers L1, L2 and L3 in Fig. 5(a) are designed to be −14, −0.94, and −9.6, and their widths are all λ0/75. As shown in Fig. 5(c), after positioning the triple-element structure, the reflection at f0 is reduced from −5.2 dB to −52 dB, and the −10-dB bandwidth is from 0.98 f0 to 1.02 f0. In Fig. 5(d), an obvious standing wave is observed in M1 without the metastructure. In Fig. 5(e), perfect suppression of the reflected wave is observed with the metastructure and all the energy is transmitted from M1 to M2. For example (II) of TM wave, the intrinsic impedance of tangential fields in M1 and M2 are Z1 = η1cosθi = sqrt(μ0/(ɛ1ɛ0))/cosθi and Z2 = η2cosθt = sqrt(μ0/(ɛ2ɛ0))/cosθt. According to the feasible rotation trace, i.e., Z2→L1→Z2'→L2→Z2''→L3→Z1 in Fig. 5(g), the relative permittivities of metatronic layers L1, L2 and L3 in Fig. 5(f) are designed to be −23, −10, and −21.5, and their widths are all λ0/75. As shown in Fig. 5(h), after positioning the metastructure, the reflection at f0 is reduced from −16 dB to −36 dB, and the −10-dB bandwidth is from 0.965 f0 to 1.038 f0. The field distributions without and with metastructure are compared in Fig. 5(i) and 5(j), indicating the reflected wave in M1 is also successfully eliminated. Detailed dimensions of Fig. 5(a) and 5(f) are presented in Fig. S12 and S13 of Supplement 1. These two examples indicate that the concept of graphical metatronics has the potential to tackle variously applicable scenarios. It should be noted that, although metatronic-based matching structures may have the limitation on relative bandwidth due to their ultrathin sizes, they can still be utilized in many application scenarios, e.g., angle measurement, biosensor and optical filter [50–52].
Fig. 5. Studies on the oblique-incidence situation. (a) TE wave with θi = 70°. (b) Rotation traces on the Γ-plane from Z2 to Z1; (c) Reflection coefficients. (d-e) Numerical snapshots on the distribution of y-component of electrical fields at f0 (d) without or (e) with the metastructure. (f) TM wave with θi = 30°. (g) Rotation traces on the Γ-plane from Z2 to Z1. (h) Reflection coefficients. (i-j) Numerical snapshots on the distribution of y-component of magnetic fields at f0 (i) without or (j) with the metastructure.
3. Methods
Numerical Validation. The numerical results in this paper are obtained using the commercial software CST STUDIO SUITE 2016. Specifically, the frequency-domain solver is utilized with tetrahedral meshing. The maximum cell size is calculated by dividing the largest edge of the model bounding box by 90. Along the x-direction and y-direction, unit cell boundary conditions are applied to construct infinite lengths. In these structures, Floquet ports are used to excite the desired modes. For the investigation on plasmonic material, the Drude model is utilized in the setting of material parameters.
4. Conclusion
In this work, we propose to design the light-manipulation structure via metatronic nanocircuits with lumped and ultrathin properties, and the Smith Chart is an assistant tool to simplify and guide the design procedure. Representative cases, loss analysis and oblique incidence, are investigated to show the broadly applicable scenarios of the graphical metatronics. By far, reported works based on optical metatronic are usually designed and optimized via computer and simulation software. Different from them, our work provides an intuitional physical picture for the designers. The structure and parameters of the optical metatronic devices can be directly determined under the guidance of the Smith Chart, exhibiting advantages of visualization and substantially reducing computation complexity. The proposed approach is applicable for a variety of subwavelength nanophotonic structures, exhibiting potential values in optics, electronics, and nanophotonics.
Funding
National Natural Science Foundation of China (62022045); Beijing Nova Program of Science and Technology (Z191100001119082); Beijing National Research Center For Information Science And Technology.
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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