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Abstract
Complementary media, possessing permittivity and permeability with the exact same magnitude but of opposite sign to their counterpart media, can optically cancel out the counterpart media. This renders fascinating applications including perfect lens, illusion optics, invisible gateway, opening up virtual holes in a wall, etc. However, the realization of complementary media generally requires metallic resonating structures, leading to challenging fabrication technology and inevitable material loss. In this work, we demonstrate a type of all-dielectric unidirectional complementary media by using symmetric dielectric multilayers. We find that such symmetric dielectric multilayers can be regarded as effective media with flexible effective permittivity and permeability, which can operate as unidirectional complementary media of almost arbitrary media including metal, air, zero-index media, etc. Based on this interesting property, we show that unidirectional wave transmission through metal films, as well as extremely impedance-mismatched zero-index media, can be greatly enhanced by using the proposed all-dielectric unidirectional complementary media. Our work opens a simple but efficient approach for transmission enhancement, that may find practical applications in transparent electrodes and tunneling WiFi signals through walls of buildings.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Transparent media are the foundation of almost all optical systems. However, most natural materials are opaque because of disorder, material loss, decay of evanescent waves and large impedance mismatch with free space. For instance, metals, as the typical example of single-negative media (either permittivity or permeability is negative), strongly reflect light because light waves are evanescent and exponentially decay inside them. On the other hand, due to good electrical conductivity, transparent metals are highly desirable for many applications like displays and solar cells [1,2]. To improve light transmission through metal films, several methods have been proposed [3–20]. For example, holes are etched to gain surface-plasmon-induced extraordinary transmission [3,4]. Tamm-plasmon-induced high transmission is obtained in collaboration with dielectric photonic crystals, which however require a thickness of several wavelengths [5–8]. Based on transformation optics [21–23] and complementary media (CM) [20,24–28], in principle arbitrary virtual holes can be opened up in metal films to gain high transmission. Nevertheless, the CM of metals require a negative permeability, which is absent in natural optical materials. Usually, metallic resonating structures are needed to realize the CM, leading to challenging fabrication technology and inevitable material loss.
In this work, we demonstrate a type of all-dielectric unidirectional complementary media (UCM) of metals, so as to gain unidirectional high transmission through metal films. We find that a symmetric dielectric multilayer (SDM) can be regarded as an effective medium with flexible effective permittivity and permeability. Through engineering thickness of the components, the SDM can exhibit almost arbitrary effective parameters. We demonstrate that the SDM with negative effective permeability can operate as the UCM of metals, which can greatly enhance light transmission through metal films. On the other hand, the SDM with double-negative effective parameters can work as the UCM of air to optically cancel out a volume of air. In this way, we can enhance light transmission through metal films by using SDM stacks at a distance. Besides metal films, the light transmission through extremely impedance-mismatch zero-index media (ZIM) can also be greatly enhanced by using SDM with near-zero effective parameters.
2. SDM as effective UCM
To begin with, we consider an opaque wall that strongly reflects incident light, as illustrated in Fig. 1(a). In order to eliminate the reflection and gain perfect transmission, a SDM consisting of dielectrics A and B stacked along the z direction in a symmetric form, i.e. ABA multilayer, is placed close to the opaque wall (see Fig. 1(b)). In the following, we will show that such a ABA SDM can operate as the effective UCM of the opaque wall, so as to optically cancel out the wall and enhance light transmission. Furthermore, through stacking n number of ABA SDMs together as a SDM stack ${({\textrm{ABA}} )^n}$, we can enhance light transmission through the opaque wall with a large thickness, as illustrated in Fig. 1(c).
Fig. 1. Schematic graphs of (a) almost total reflection on an opaque wall, (b) total transmission through the wall accompanied by a ABA SDM consisting of dielectrics A and B. (c) A SDM stack ${({\textrm{ABA}} )^n}$ composed of n ABA unit cells is used to enhance light transmission through the opaque wall with a large thickness.
First, we study the effective medium model of the ABA SDM illuminated by a plane wave (electric field polarized in the x direction) under normal incidence. According to transfer matrix method [29], the total transfer matrix of the ABA SDM can be derived as,
In fact, as long as the SDM possesses mirror symmetry, it can be proved that ${M_{11}} = {M_{22}}$ and ${M_{11}}{M_{22}} - {M_{12}}{M_{21}} = 1$ based on Eq. (1). Moreover, in the absence of material loss or gain, ${M_{11}}$ and ${M_{22}}$ are real values, while ${M_{12}}$ and ${M_{21}}$ are imaginary values. These properties indicate that such a ABA SDM can be regarded as a uniform effective medium layer [30], whose transfer matrix can be expressed as,
In Eq. (3), if ${{{M_{21}}} / {{M_{12}} > 0}}$, we have ${{{\varepsilon _e}} / {{\mu _e}}} > 0$. This means that ${\varepsilon _e}$ and ${\mu _e}$ have the same sign, indicating that the SDM exhibits double-positive or double-negative effective parameters. In this case, the m can be an arbitrary integer. On the other hand, if ${{{M_{21}}} / {{M_{12}} < 0}}$, we have ${{{\varepsilon _e}} / {{\mu _e}}} < 0$. In this case, the SDM exhibits single-negative effective parameters. We note that the term $\arccos {M_{11}}$ in Eq. (3) is complex because $M_{11}^2 = 1 + {M_{12}}{M_{21}} > 1$. However, physically, the ${\mu _e}$ should be a real value in the absence of material loss or gain. Therefore, appropriate m should be chosen to satisfy the condition of $Re ({\arccos {M_{11}}} )+ 2\pi m = 0$. Thus, the ${\mu _e}$ turns to be a unique value as ${\mu _e} ={\pm} \frac{i}{{{k_0}D}}\frac{{\sqrt {{M_{12}}} }}{{\sqrt {{M_{21}}} }}{\mathop{\rm Im}\nolimits} ({\arccos {M_{11}}} )$.
Next, we analyze the condition of the effective UCM of a wall (relative permittivity ${\varepsilon _w}$, relative permeability ${\mu _w}$, thickness ${d_w}$) by using the SDM based on transformation optics [21–23]. Considering the normal incidence, we can apply the coordinate transformation ${x^{(2)}} = {x^{(1)}}$, ${y^{(2)}} = {y^{(1)}}$ and ${z^{(2)}} ={-} {z^{(1)}}{{{d_w}} / D}$ by assuming a wall in space $\{ {x^{(2)}},{y^{(2)}},{z^{(2)}}\} $ and the UCM in space $\{ {x^{(1)}},{y^{(1)}},{z^{(1)}}\} $ [26,27]. Thus, the relation between effective parameters of the SDM and parameters of the wall is obtained as,
3. Enhancement of light transmission through metal films
In the following, we take a specific optical example to show the approach to enhance light transmission through a metal film by using the SDM. We choose magnesium fluoride (MgF2, refractive index 1.38) and titanium oxide (TiO2, refractive index 2.35) as dielectrics A and B, respectively. According to Eq. (3), we calculate and plot effective parameters of the ABA SDM as functions of normalized thicknesses of A (${{{d_A}} / {{\lambda _0}}}$) and B (${{{d_B}} / {{\lambda _0}}}$) layers in Fig. 2. Here, is the wavelength in free space. It is seen from Fig. 2(a) that the ratio ${{{\varepsilon _e}} / {{\mu _e}}}$ can be both positive and negative values, and varies almost periodically as the increase of ${d_A}$ and ${d_B}$. In the case of ${{{\varepsilon _e}} / {{\mu _e}}} < 0$, the ${\mu _e}$ is unique (see Fig. 2(b)). While in the case of ${{{\varepsilon _e}} / {{\mu _e}}} > 0$, the ${\mu _e}$ is non-unique because the m in Eq. (3) can be an arbitrary integer. Figures 2(c) and 2(d) show the $- {{{\mu _e}D} / {{\lambda _0}}}$ under $m = 0$ and $m = 1$, respectively. From Figs. 2(a)–2(d), we see that ${\varepsilon _e}$ an${\lambda _0}$d ${\mu _e}$ can be both positive and negative values with a relatively large variation range, manifesting that the SDM can effectively work as the single-negative medium, double-positive medium, double-negative medium, as well as ZIM. Clearly, such a simple ABA SDM shows the flexibility and diversity of effective parameters, thus can serve as effective UCM of almost arbitrary media.
Fig. 2. Plot of effective parameters of the ABA SDM. (a) ${{{\varepsilon _e}} / {{\mu _e}}}$, (b) $- {{{\mu _e}D} / {{\lambda _0}}}$ for ${{{\varepsilon _e}} / {{\mu _e}}} < 0$, and [(c) and (d)] $- {{{\mu _e}D} / {{\lambda _0}}}$ for ${{{\varepsilon _e}} / {{\mu _e}}} > 0$ with (c) $m = 0$ and (d) $m = 1$ as functions of ${{{d_A}} / {{\lambda _0}}}$ and ${{{d_B}} / {{\lambda _0}}}$. The refractive index of the dielectric A (B) is 1.38 (2.35).
Then, we choose a Ag film with a thickness of ${d_w} = 0.024{\lambda _0}$ as the optically opaque wall. The relative permittivity of Ag is ${\varepsilon _w} ={-} 7.37 + 0.23i$ at ${\lambda _0} = 455.5\textrm{nm}$, and the relative permeability is ${\mu _w} = 1$ [36]. For simplicity, we first assume that the Ag is lossless, i.e. ${\varepsilon _w} ={-} 7.37$. From the effective parameters of the SDM in Fig. 2, we find that the SDM with ${d_A} = 0.27{\lambda _0}$ and ${d_B} = 0.13{\lambda _0}$ possesses ${\varepsilon _e} = 0.2656$ and ${\mu _e} ={-} 0.0360$, which satisfy the condition of UCM of the Ag film (i.e., Eq. (4)).
For numerical verification, in Fig. 3(a), we simulate magnetic-field distributions when light is normally incident on the Ag film (upper) and SDM-Ag multilayer (lower). The simulations are performed by using the finite-element software COMSOL Multiphysics. The magnitude of magnetic field of incident light is 1A/m. From the simulation results, we find that the transmittance through the Ag film is around 0.7 without the SDM, but is increased to near unity (>0.99) in the existence of the SDM. We notice that the phase and amplitude of the incident light on the left surface of the ABA SDM match well with those on the right surface of the Ag film. This indicates the space cancellation effect by the effective UCM.
Fig. 3. (a) Magnetic field-distributions when light is normally incident on a lossless Ag film (upper) and ABA-Ag multilayer (lower). (b) Transmittance through the Ag film (solid lines) and ABA-Ag multilayer (dashed lines) for TM (blue lines) and TE (red lines) polarizations as a function of incident angle. The upper and low figures in (b) are related to the cases without and with material loss in the Ag film, respectively. The thicknesses of A layer, B layer and Ag film are $0.27{\lambda _0}$, $0.13{\lambda _0}$ and $0.024{\lambda _0}$, respectively. The working wavelength is ${\lambda _0} = 455.5\textrm{nm}$.
We also calculate the transmittance as the function of incident angle for transverse-electric (TE, red lines) and transverse-magnetic (TM, blue lines) polarizations based on transfer matrix method [8], as shown by the upper figure in Fig. 3(b). The solid and dashed lines are related to the cases without and with the SDM, respectively. Clearly, the light transmission through the Ag film is greatly enhanced by the SDM over a wide incident angle range. Moreover, we take the material loss of Ag into account by using ${\varepsilon _w} ={-} 7.37 + 0.23i$ as its relative permittivity. In the lower figure in Fig. 3(b), the transmittance is recalculated, also showing wide-angle high transmission through the Ag film by using the SDM.
Actually, when taking the dispersion of the Ag film [36] into consideration, we find that high transmission (>0.9) can be observed in the wavelength range from 390 nm to 510 nm, showing a relatively large bandwidth of transmission enhancement. Through engineering the thickness of components of the SDM, the overall transmission can be further improved, and the central wavelength can be tuned to other desired wavelengths.
We also note that such high transmission is robust against the variation of thickness of the A and B layers. We find that when the thickness of A layer or/and B layer is off by ±5% of the desired values, high transmission (>0.9) can still be obtained under normal incidence for both lossless and lossy cases.
Now, we stack n number of ABA SDMs together as a SDM stack ${({\textrm{ABA}} )^n}$ to enhance light transmission through the Ag film with a large thickness. As illustrated in Fig. 4(a), (a) Ag film is sandwiched by SDM stacks ${({\textrm{ABA}} )^{{n_1}}}$ and ${({\textrm{ABA}} )^{{n_2}}}$. In numerical analysis, we set ${n_1} + {n_2} = 10$, thus the thickness of the Ag film is $0.24{\lambda _0} \approx 109\textrm{nm}$, much larger than its skin depth (∼10 nm). Figure 4(b) presents the incident angle-dependent transmittance through the whole structure for different ${n_1}$ and ${n_2}$ in the absence of material loss. It is seen that without SDM stacks (i.e. ${n_1} = {n_2} = 0$, cyan lines), incident light is almost totally reflected by the thick Ag film. Amazingly, when the Ag film is sandwiched by SDM stacks, near-perfect transmission is obtained under normal incidence, and obvious transmission enhancement is seen under small-angle incidence. Moreover, we notice that symmetric distribution of SDM stacks (i.e., ${n_1} = {n_2} = 5$, red lines) leads to better performance in improving light transmission. For further verification, in the upper figure in Fig. 4(c), we simulate the magnetic field-distribution through the ${({\textrm{ABA}} )^5}\textrm{ - Ag - }{({\textrm{ABA}} )^5}$ multilayer, showing near-perfect transmission. We note that the transmission is symmetric about ${n_1}$ and ${n_2}$, that is, the transmission through the multilayers ${({\textrm{ABA}} )^{{n_1}}}\textrm{ - Ag - }{({\textrm{ABA}} )^{{n_2}}}$ and ${({\textrm{ABA}} )^{{n_2}}}\textrm{ - Ag - }{({\textrm{ABA}} )^{{n_1}}}$ is the same. This is a direct result of optical transmission reciprocity [37].
Fig. 4. (a) Illustration of light transmission enhancement through a thick Ag film sandwiched by SDM stacks ${\textrm{(ABA)}^{{n_1}}}$ and ${\textrm{(ABA)}^{{n_2}}}$. (b) Plot of incidence angle-dependent transmittance through the multilayer ${({\textrm{ABA}} )^{{n_1}}}\textrm{ - Ag - }{({\textrm{ABA}} )^{{n_2}}}$ for ${n_1} = {n_2} = 5$ (red lines), ${n_1} = 6$ and ${n_2} = 4$ (green lines), ${n_1} = 7$ and ${n_2} = 3$ (blue lines), and ${n_1} = {n_2} = 0$ (cyan lines) in the absence of material loss. The relevant parameters of the SDM are the same as those in Fig. 3. The thickness of the Ag film is $0.24{\lambda _0} \approx 109\textrm{nm}$. (c) Magnetic field-distributions when light is normally incident on the multilayer ${({\textrm{ABA}} )^5}\textrm{ - Ag - }{({\textrm{ABA}} )^5}$. The upper figure is related to the lossless model, while the middle and lower figures are related to the model with material loss. In the lower figure, the refractive index of dielectric B is set as $2.35 - 0.01i$ for energy compensation. The white regions mean that the magnetic field is beyond (or below) the maximum (or minimal) value of the color bar.
Moreover, we consider the loss effect of Ag through re-simulating the magnetic field-distribution, as shown in the middle figure in Fig. 4(c). The transmittance is reduced to 0.27 due to light absorption by the Ag film. Interestingly, such absorption loss can be compensated through introducing gain to the SDM stacks. As an example, in the lower figure, the refractive index of dielectric B is set as $2.35 - 0.01i$. Consequently, the transmittance rises back to near unity (>0.97). In practice, it is possible to introduce optical gain via ion or dye doping [38–41].
4. Light transmission enhancement by SDM stacks at a distance
In the above, we have demonstrated the effective UCM of metal by using a SDM with and ${\mu _e} < 0$. In the following, we show that the SDM can also possess effective parameters satisfying ${\varepsilon _e} = {\mu _e} < 0$, which can serve as the effective UCM of air. Intriguingly, through combining the two kinds of SDMs together, we can make a metal film transparent by using SDM stacks at a distance, as illustrated in Fig. 5(a).
Fig. 5. (a) Illustration of light transmission enhancement by using SDM stacks at a distance. The SDM stack 1 and 2 are used to optically cancel out the wall and air, respectively. (b) Magnetic field-distribution when light is normally incident on the Ag film accompanied by a SDM stack of ABA-AB'A at a distance of $1.16{\lambda _0}$. The B’ layer has the same refractive index as the B layer, but the thickness is changed to $0.4606{\lambda _0}$.
From the effective parameters in Fig. 2, we find that the SDM exhibits ${\varepsilon _e} \approx {\mu _e} \approx{-} 1.16$ under m = 1 when the thickness of B layer is changed to $0.4606{\lambda _0}$, while other parameters remain unchanged. Here, we denote this SDM as the AB'A multilayer to distinguish from the ABA SDM in Figs. 3 and 4. According to Eq. (4), we know that such a AB'A SDM can cancel out phase accumulation in an air layer with a thickness of $1.16{\lambda _0}$. Thus, the AB'A-air multilayer effectively can be considered a null space to incident light. Interestingly, through combining the ABA SDM and the AB'A SDM together, we can enhance light transmission through a Ag film at a distance of $1.16{\lambda _0}$. For demonstration, in Fig. 5(b), light is normally incident on the ABA-AB'A-air-Ag multilayer, showing almost perfect transmission. In addition, we see that the exit phase and amplitude out of the Ag film match well with those at the incident end, indicating the space cancellation effect by the SDM stack.
We note that when ${\varepsilon _e}$ and ${\mu _e}$ have the same sign, their values are non-unique. Based on Eq. (3), the effective parameters are found to be ${\varepsilon _e} \approx {\mu _e} \approx{-} ({0.16 + m} )$. As a result, the air layer cancelled out by the proposed AB'A SDM can have a series of thicknesses as $({0.16 + m} ){\lambda _0}$ with $m = 0,1,2 \cdots$. This indicates that the ABA-AB'A SDM stack can make the opaque wall transparent at a large distance if $m > > 1$.
5. Enhancement of light transmission through extremely impedance-mismatched ZIM
Besides single-negative parameters and material loss, large impedance mismatch with free space is another reason for low transmission through a wall. For example, the impedance of a single-zero ZIM (either permittivity or permeability is near zero) tends to be infinitely large or zero, which is extremely mismatched with the impedance of free space. As a result, light transmission through such a single-zero ZIM is generally very low, and decreases quickly as the increase of thickness [42,43]. The traditional approach to enhance the transmission is by using dopants [44–51] or antireflection coatings [52,53], including quarter-wave and ultrathin conductive antireflection coatings. However, the doping method only works in two dimensions, and fails in three dimensions [49,54]. On the other hand, quarter-wave antireflection coatings of the ZIM would require a near-zero refractive index and a large thickness, while the ultrathin conductive antireflection coatings would lead to dramatic energy attenuation due to material loss [55,56]. Interestingly, here we find that a simple ABA SDM possessing near-zero effective permittivity or permeability can work as the effective UCM of a single-zero ZIM, so as to achieve perfect transmission of light.
As an example, we consider an effective ZIM by using a dielectric photonic crystal consisting of dielectric rods (relative permittivity 12.5) in a square lattice. The lattice constant is a, and the radius of the rods is $0.22a$. Near the edge of the second band in dispersion characteristic, the photonic crystal exhibits near-zero effective refractive index as a small negative value. At the normalized frequency ${{fa} / c} = 0.498$ ($f$ is the eigen-frequency, and c is the speed of light) near the band edge, the effective relative permittivity and permeability are found to be -0.1666 and -0.0273, respectively [43]. In Fig. 6(a), we present snapshots of electric fields when TE-polarized light (with out-of-plane electric fields) is incident on the photonic crystal slab with 10 layers of unit cells along the propagation direction (upper), and the corresponding effective medium slab (lower), showing a low transmittance as 0.56. To enhance the light transmission, in Fig. 6(b), (a) ABA SDM with ${d_A} = 0.386{\lambda _0}$ and ${d_B} = 0.104{\lambda _0}$ is placed in the left side of the photonic crystal slab (upper) and the effective medium slab (lower). As a result, the transmittance is increased to near unity (>0.99), clearly demonstrating the effectiveness of the UCM based on the simple ABA SDM.
Fig. 6. (a) Snapshots of electric fields when light is normally incident on a photonic crystal slab with 10 layers of unit cells along the propagation direction (upper), and the corresponding effective medium slab (lower). (b) Snapshots of electric fields when a ABA SDM is place in the left side of the photonic crystal slab (upper), and the effective medium slab (lower). The thicknesses of A and B layers are $0.386{\lambda _0}$ and $0.104{\lambda _0}$, respectively. The normalized working frequency is ${{fa} / c} = 0.498$.
6. Discussion and conclusion
Finally, it is worth noting that the SDMs can work as effective UCM of almost arbitrary media in principle, because of the flexibility and diversity of their effective parameters. In the above, through simply engineering thickness of components, the ABA SDM has shown great flexibility in working as UCM of metal, air and ZIM. Actually, there exist extra degrees of freedoms in tuning effective parameters in practical applications. For instance, we can change the materials, or add new materials, or rearrange the sequence of components (e.g., BAB), or add more layers in a unit cell (e.g., ABABA) as long as the mirror symmetry is unbroken.
Although only the UCM under normal incidence is demonstrated here, the design principle can actually be extended to off-normal incidence. The design principle can also be extended to low-frequency regimes, which may find applications in tunneling WiFi signals through walls of buildings, etc. Therefore, our results point to a simple but efficient design guideline of all-dielectric UCM for light and electromagnetic wave transmission enhancement.
In summary, we have proposed a simple but efficient approach to enhance light transmission through opaque walls under normal incidence by using SDMs. It is demonstrated that through simply adjusting thickness of components, the SDMs can work as effective UCM of almost arbitrary media, including metal, air and ZIM. Numerical results show that light transmission through a Ag film can be greatly enhanced by using the SDMs even at a distance. Besides metal films, we also demonstrate the enhancement of light transmission through ZIM by using the SDMs, even when the impedance of the ZIM is extremely mismatched with that of free space.
Funding
National Natural Science Foundation of China (11704271, 61802272); Natural Science Foundation of Jiangsu Province (BK20170326, BK20180834); Priority Academic Program Development of Jiangsu Higher Education Institutions.
Disclosures
The authors declare no conflicts of interest.
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