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Abstract
We theoretically demonstrate a giant power enhancement effect for a line current source in a ε-near-zero (ENZ) two-dimensional (2D) shell with proper physical dimensions. Compared with the traditional high-ε dielectric approach, the ENZ scheme has the prominent advantage that the radiation performance is less sensitive to the outer radius of the shell, which is critically important for real applications where micro-nano fabrications are often involved. The enhancing performance is independent on the position of the source inside the ENZ shell and could be substantially strengthened by incorporating more sources, while the quasi-omnidirectional radiation pattern could be managed to have negligible variance, as evidenced by a particular example with an inner radius of the shell equal to 0.156λ0. Compared with the single source case, two identical sources with a phase difference less than 134° will raise the total radiation power more than 4 folds and the maxima will be about 30 when they are in phase. The field analysis shows that this quasi-isotropic radiation enhancement is mainly contributed by the amplification of the isotropic zeroth order mode radiation while the higher orders with anisotropic emission patterns are effectively suppressed by the specifically designed ENZ shell. In the end, a practicable device employing 4H-silicon carbide (4H-SiC) naturally available with ENZ properties in the mid-infrared regime is numerically proposed, which could provide more than 10 times of radiation enhancement through optimizing the permittivity of the inner dielectric cylinder. These results may find very important applications in the design of novel devices for mid-infrared photon sources or detectors.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
During the last decade, zero-index metamaterials (ZIMs) including epsilon-near-zero (ENZ) [1–5], mu-near-zero (MNZ) [6–8], or both ε and µ near zero [9–11] in either isotropic or anisotropic properties [12, 13] have attracted great research interests due to their abnormal electromagnetic (EM) responses that may lead to unique applications. The effective wavelength in ZIMs is much larger than that in vacuum and the EM field in both amplitude and phase will spread uniformly over the entire ZIM region with little variance. Special EM phenomena or applications enabled by ZIMs have been envisioned such as directive radiation [1, 12], light squeezing [2, 3, 7], cloaking [14], perfect absorption [15–17], etc. Jin et al. theoretically found that the radiation of a line current surrounded by a MNZ annulus can be greatly enhanced or completely suppressed dependent on the concrete geometries [8]. But the isotropic MNZ model has low practical feasibility. Utilizing a radially anisotropic MNZ (RAMNZ) shell, Cheng et al. experimentally showed a spatial power combination effect [18]. However, there is no enhancement for the radiated power carried by the zeroth order mode that is impedance matched with the free space, while the higher orders with anisotropic field patterns are trapped inside the MNZ shell when the line sources are excited far from the shell center. By introducing a dielectric rod in the RAMNZ shell, Wang et al. showed these anisotropic higher order modes could be re-scattered into isotropic zeroth order modes, thus substantially enhancing the radiation power [19]. For practice, these structures are more suitable for low frequencies such as microwaves but will be difficult for infrared or visible light. The intrinsic resonance loss of these MNZs will be another issue to inhibit their short-wavelength applications.
In this work, we propose a more practical design for enhancing the radiation power of a line current source by an ENZ shell for infrared or high-frequency regimes. The loss influence on the power enhancement is studied, which is inevitable in practice. Compared with the traditional high permittivity based scheme, our ENZ enhancement design is insensitive to the outer dimension of the radiation shell, which will greatly facilitate the implementation. For a source inside the ENZ shell deviating from the center, remarkable power enhancement with a quasi-omnidirectional radiation pattern is achieved with the inner radius of the shell equal to 0.156λ0. Introduction of a second source will enhance the total radiation energy by four times while the quasi-omnidirectional radiation pattern is negligibly disturbed if the phase difference of the two sources is less than 134°. We reveal that the ENZ shell will operate as a zeroth order mode amplifier but a suppressor for all the higher order modes, thus ensuring the quasi-isotropic radiation pattern. A feasible model using natural ENZ materials (4H-silicon carbide) that could operate at 10.3 μm is proposed. Optimization of the permittivity value of the dielectric cylinder inside the ENZ shell will give rise to a radiation enhancement factor as large as 10 compared with a single source placed in free space when the inner radius varies between 0.57λ0 and 0.67λ0.
2. Theoretical model
As shown in Fig. 1, our model depicts that a line current source is embedded in the central dielectric cylinder with permittivity ε1 surrounded by an ENZ annulus with inner radius b and outer radius a in free space. The relative permittivity of the ENZ region is denoted by ε2 and the relative permeability of all the regions are unity, i.e., nonmagnetic. A 2D EM radiation scenario is considered here with the electric field polarized along the z-axis and the magnetic components in the x-y plane. For convenience and without loss of generality, the source is supposed to be located at a polar coordinate (s1, ϕ1) with ϕ1 = 0°.
Fig. 1 Schematic of the theoretical model. Region 1 is a normal dielectric cylinder with permittivity ε1, region 2 is an ENZ ring with inner radius b and outer radius a, and region 3 is air. A line current source I (black point) with the time harmonic factor eiωt is located at (s1, ϕ1).
According to the generalized Lorenz-Mie theory, the electric field in region 2 can be expanded into the linear combination of the eigenmodes [19, 20]
where k2 = k0(με2)1/2 with k0 the wavenumber in vacuum, Jm and Hm are the m-th order Bessel functions and Hankel functions of the first kind, respectively, and the summation factor m is an integer varying from −∞ to ∞. The electric field radiated by the line source can be expressed aswhere k1 = k0(με1)1/2, ρ is the position vector and s1 denotes the position vector of the source. From the addition theorem of Hankel function, the above equation is expanded as [19, 20]with ρ = |ρ| and s1 = |s1|. Due to the reflection from the dielectric-ENZ interface, we can write the total electric field for the part s1 < ρ ≤ b of region 1 aswhere Am characterizes the reflection coefficient. In region 3, there is only the outgoing wave and the electric field is expressed as [19]where Dm describes the radiated wave coefficient.Based on the boundary conditions (i.e., the continuity for the tangential Ez and Hϕ components), we derive the partial wave expansion coefficient,
where the generalized Mie coefficients are given by [19]First, we consider a special case with the source locating at the center point (s1 = 0, ϕ1 = 0°). Due to the cylindrical symmetry, the EM fields are independent on the azimuthal angle ϕ, which vanishes all the coefficients (Am, Bm, Cm, Dm) for m ≠ 0. Hence, the radiated field is 2D isotropic and can be expressed as When the dielectric and ENZ annulus are removed, i.e., source in free space (k1 = k2 = k0, a = b), p0, p0’and D0 are equal to unit, which means E3z = H0 (k0ρ). We define the power enhancement factor P = |D0|2, which is the ratio between the radiated power when the source is placed inside the ENZ device and that in free space. The enhancement factor P is correlated with the outer radius a, the inner radius b, the permittivity ε1 of the inner dielectric and ε2 of the ENZ shell.Figure 2 presents the power enhancement factor P as a function of b when the ENZ material has different loss degrees. First we assume the inner region 1 is filled with air (ε1 = 1). For lossless case (ε2 = 0.001), when the outer radius a of the ENZ ring is 1.4λ0 (λ0 is the operation wavelength in free space), as shown in Fig. 2, there are three narrow peaks for P at the special values of b/λ0, which correspond to the first three resonance eigenmodes of the zeroth order Bessel function A0J0(k1ρ) described in Eq. (4). The power enhancement factor P has the largest peak value of 45 for the first eigenmode at a small inner radius b = 0.171λ0, where the electric field near the source position reaches maximum. When the loss of the ENZ is considered, which is inevitable in practical cases, the largest peak value decreases to 12.5 at b = 0.663λ0 (second mode) for ε2 = 0.001 + 0.01i, and decays to 2 at b = 1.201λ0 (third mode) for ε2 = 0.001 + 0.1i. The enhancement effect disappears when the loss factor increases further. The peak value is less sensitive to the loss factor for a larger inner radius, as it corresponds to a thinner ENZ shell when the outer radius is fixed.
Fig. 2 P as a function of b/λ0 for the ENZ ring with outer radius a = 1.4λ0. The inner dielectric is air and the source is at the center of the system in this case. The permittivity ε2 is 0.001, 0.001 + 0.01i and 0.001 + 0.1i for black, red and blue lines, respectively.
The parametrical dependence of P on the outer radius a and inner radius b is fully depicted in a 2D map of enhancement factor plotted in Fig. 3(a). It’s seen that the power enhancement occurs at some special combinations of b/λ0 and a/λ0, and at most 12.5 times amplification of the radiating power is obtained when the ENZ has permittivity of 0.001 + 0.01i. Traditional high-ε dielectric shell can also enhance the radiation power at the fixed source current intensity [21]. Here we compare the enhancement effect of the ENZ shell and high-ε dielectric shell (14 + 0.01i). The enhancement factor P as the function of b/λ0 and a/λ0 is presented in Fig. 3(b) for the high-ε case, where the remarkable radiation enhancements (more than 10 times) only occur at some discrete values of outer radius a. But for the ENZ structure, this enhancement for each b could be obtained with a broad distribution for the value of the outer radius a, thus offering more tolerance for the sample fabrication. The maximum power enhancement factor with different outer radius a for these two structures are compared in Fig. 3(c). When the outer radius a of the ENZ shell is larger than 0.8λ0, more than 9 times amplification can be obtained by choosing appropriate inner radius b [see Fig. 3(d)]. However, for the high-ε dielectric shell, the maximum enhancement factor shows strong oscillation as a function of outer radius a. It means that for some outer dimensions of the high-ε dielectric shell, it is impossible to obtain more than 10 times of enhancement whatever inner radius b is. The rigorous requirement of the outer radius for giant enhancement makes the traditional high-ε dielectric scheme quite difficult to be precisely implemented.
Fig. 3 A 2D map of power enhancement factor as a function of b/λ0 and a/λ0 for the (a) ENZ (0.001 + 0.01i) shell and (b) high-ε dielectric (14 + 0.01i) shell, respectively. (c) Maximum enhancement factor and (d) corresponding inner radius b of shell to get maximum the enhancement as the function of outer radius a for the ENZ (black curve) and high-ε dielectric (red curve) shells.
To get the physical insight about the roles of the ENZ and high-ε dielectric shells in the power enhancement and their differences, numerical simulations are conducted by considering three different situations: source in free space, an ENZ shell and a high-ε dielectric shell. Compared with the free space, both ENZ and high-ε dielectric shells can constructively strengthen the field (or local voltage V) inside the inner cavity and thus the radiated power (Prad = I⋅V and moreover, I and V are in phase), as shown in Figs. 4(a)-4(c). The amplitude of the electric field along the x-axis is plotted in Fig. 4(d) to give a quantitative comparison. The electric field intensity has a strong Febry-Perot oscillation in the high-ε dielectric shell, while in the ENZ shell the amplitude exhibits a monotonic change. As shown in the inset of Fig. 4(d), the radiated electric field of the source with the ENZ shell and high-ε dielectric shell is 0.463 kV/m and 0.472 kV/m at x = 2λ0 which could be regarded as a far field approximation, respectively. This means 3.38 and 3.45 times of amplification compared with the Ez for the free space case (0.137 kV/m), thus correspondingly 11.4 and 11.9 times power enhancement. The simulated power enhancement agrees well with the analytical results in Fig. 3(c). The phase of the electric field along the + x-axis is presented in Fig. 4(e), which illustrates a slow phase variation in the ENZ regime and a fast one in the high-ε dielectric shell. Both shells can arouse strong cavity resonance by their large impedance mismatch with the ambient, acting as an opaque reflective layer. For the ENZ shell, the inner resonance state is not sensitive to the thickness of the shell if b is fixed due to the zero-phase delay (equivalently the shell behaves as an ultra-thin impedance layer). While for the high-ε shell, the Febry-Perot resonance inside it will strongly modulate the transmission/reflection upon the change of outer radius a as indicated by Fig. 3(b). For practice, the ENZ approach is more favorite.
Fig. 4 Numerical simulation results of the real part of electric field Ez radiated by a current source (1 Ampere) in (a) free space, (b) an ENZ shell (a = 1.385λ0, b = 0.661λ0) and (c) a high-ε dielectric shell (a = 1.385λ0, b = 0.383λ0), respectively. (d) Amplitude and (e) phase distributions of electric fields on the + x-axis.
3. Influence of the source position and power combination of multiple sources
In the above, the source is set at the center of region 1, generating a 2D isotropic radiation pattern. When the source deviates from the central point, the radiation pattern is no longer omnidirectionally isotropic due to the excitation of other higher order wave components. Their coefficients Dm can be calculated using Eq. (6). But if the source is very close to the center (s1 << λ0), we can expect the Bessel function Jm(k1s1) will approach 1 and 0 for m = 0 and m ≠ 0, respectively, which leads to the zeroth order coefficient D0 close to p0 p0’ and the higher order coefficients Dm (m ≠ 0) close to 0, as indicated by Eq. (6). Here we choose an ENZ (ε2 = 0.001 + 0.01i) shell with inner and outer radii of 0.156λ0 and 0.948λ0, which is the minimum inner radius with more than 10 times of enhancement according to Fig. 3(a). A line current is put at the position of (0.15λ0, 0°). The coefficients of different components emitted by the source in the ENZ shell and in free space are given in Fig. 5(a). The coefficients of a source in free space are Dm = Jm(k1s1), and when the ENZ shell is added, scattering coefficients pmpm’ determined by the dimensions and permittivity of the ENZ shell are introduced to make Dm = Jm(k1s1)pmpm’. It is shown that only the zeroth order of the cylindrical EM wave is enhanced by 3.2 times, and all the higher order (m ≠ 0) modes are suppressed by the ENZ shell (see the | pm pm’| curve). The suppression results from the confinement of most of the higher order mode by the ENZ shell which is designed for the zeroth order enhancement. The giant enhancement for the zeroth order and suppression for the nonzero-order modes ensure quasi-omnidirectional radiation by the source in the ENZ shell. The power flow density along the radial direction out of the ENZ shell can be calculated as [20]
The normalized radiation patterns of a line source at (0.15λ0, 0°) with and without the ENZ shell are presented in Fig. 5(b) according to Eqs. (6) and (7). About 6.7 times of amplification of the total power and a quasi-omnidirectional radiation pattern with directional enhancement factor varying between 6.82 and 6.65 are achieved, as seen in Fig. 5(b). Although the radiation pattern is not perfectly angularly isotropic, a good omnidirectionality is still maintained. The separation s1 is no more than 0.156λ0 (inner radius of the ENZ shell), leading to |D0| > 2.55 and |Dm| < 0.06 (m ≠ 0) calculated by Eq. (6) (the isotropic zeroth order component is dominant), which ensures the quasi-omnidirectionality with slightly deviated source position.Fig. 5 (a) The coefficients of the m-th order wave components and (b) the normalized power density plotted as a function of the polar radiation angle for a source with or without the ENZ shell. The separation between the source and the shell center is 0.15λ0, and the ENZ shell has permittivity of 0.001 + 0.01i, inner and outer radius of 0.156λ0 and 0.948λ0, respectively.
Next, we will investigate the spatial power combination of multiple sources introduced in region 1. For convenience, two sources with the same current amplitude and a phase difference Δφ at positions (s1, 0°) and (s2, ϕ2) are discussed here. Based on the superposition principle, the total electric field outside the shell in Fig. 1 can be easily obtained from Eq. (5),
For two close and in-phase sources in free space, the maximum combined power will increase by four times compared with a single source. When the ENZ shell satisfies the condition |D0’| > 2, the radiated power will be greatly improved and the zeroth dominant order will maintain the quasi-omnidirectional far-field radiation pattern. Using Eq. (8) and the conditions p0 p0’ = 3.29 and s1, s2<0.156λ0, it is easy to find the phase difference of the two sources should be less than 134° in order to have a power enhance factor larger than 4 no matter where we put the sources in region 1. Figure 6(a) presents the normalized radiation patterns of the two in-phase line sources at position (0.1λ0, 0°) and (0.15λ0, 160°) with and without the ENZ shell, respectively. As indicated by the dashed line in Fig. 6(a), these two sources with the ENZ shell demonstrate nearly 30 times of power enhancement and a quasi-isotropic radiation pattern. For comparison, without the ENZ shell, the solid line indicates a directional radiation with normalized power density varying between 2 and 4. At Δφ = 130°, this directional emission in free space is more obvious, as indicated by the solid line in Fig. 6(d). When the ENZ shell is used, the omnidirectionality of radiation can be improved, as seen in Fig. 6(d), and about 5 times of power enhancement by that of a single source in free space is achieved. Numerical simulations have been carried out using the commercial software COMSOL, as shown in Figs. 6(a)-6(f), where the results agree with the calculated radiation patterns well.
Fig. 6 The polar-plot of the normalized power density by two sources (a) in phase and (b) with a 130° phase difference with and without the ENZ shell, respectively. The numerical simulation results of the real parts of electric field Ez radiated by the two sources (b, c) in phase and (e, f) with a 130° phase difference. The other parameters are (s1, ϕ1) = (0.1λ0, 0°), (s2, ϕ2) = (0.15λ0, 160°), a = 0.948λ0, b = 0.156λ0 and ε2z = 0.001 + 0.01i.
When the phase difference |Δφ| increases to 180°(antiphase), the coefficient |D0’| decreases to 0.3755, which means the power cancellation of these two sources. The coefficient |D0’| decreases to 0 when s1 = s2, independent of the azimuthal angle of each source.
It should be noted that in all our discussions, the sources are driven by currents with the same amplitudes. The giant enhancement of radiation power reflects a dramatic increase of radiation resistance in the language of antenna theory [22], or local density of states in photonics [23].
4. Radiation power enhancement by 4H-SiC with dispersion
In real implementation, loss and dispersion are two big issues of ENZ metamaterials, especially for the power enhancement application. Low-loss semi-conductive materials are promising candidates for this application in the mid- to far-infrared range. For instance, 4H-SiC naturally exhibits an ENZ point in the mid-infrared range with small Im(ϵ) attributed to the slow scattering rates of optic phonons [24]. The dielectric function of 4H-SiC can be approximated as a Lorentz oscillator [25, 26],
with a pole at ωTO, and a zero-point crossing at ωLO, which are the transverse optic (TO) and longitudinal optic (LO) phonon frequencies, respectively. The spectral dependence of the permittivity calculated for a 4H-SiC epitaxial layer reported by Tiwald et al. [25] are presented in Fig. 7(a), where ε∞ = 6.6, ωTO = 797 cm−1, ωLO = 970 cm−1, and the damping constant γ = 1.4 cm−1. The 4H-SiC exhibits the ENZ property at λ0 = 10.3 μm and the loss part can be as low as 0.03, which is key to achieve a remarkable radiation power enhancement.Fig. 7 (a) The permittivity of 4H-SiC material. The 4H-SiC dielectric function has the ENZ point at 10.3 μm with a small loss part Im(ϵ) = 0.03. (b) The analytical power enhancement factor spectrum of the 4H-SiC shell with the parameters shown in the inset. The line-current source is center-positioned.
The inner dielectric has the permittivity of 3.5 and the dimensions of the 4H-SiC shell are designed by the outer radius a = 1.21λ0 = 12.463 μm and the inner radius b = 0.614λ0 = 6.324 μm, as shown in the inset of Fig. 7(b). The maximum power enhancement (11 times) occurs at the ENZ point (10.3 μm wavelength), and more than 2 times amplification is achieved in the 10.245-10.325 μm range [see Fig. 7(b)].
Since the power enhancement effect is very sensitive to the inner radius b of the ENZ shell [see Fig. 3(a)], it is important to optimize the permittivity of the inner dielectric in order to accommodate the fabrication deviation of the inner radius b of the shell. Figure 8(a) demonstrates that when the inner radius b deviates from the expected value 0.614λ0 between 0.57λ0 and 0.67λ0, more than 10 times amplification at 10.3μm-wavelength can still be obtained by tuning the permittivity of the inner dielectric within 3 and 4. In Fig. 8(b), when the permittivity of inner dielectric increases, the power enhancement spectrum shows red shifts, which is normal as the 4H-SiC shell has fixed physical dimensions. Largest enhancement factor (> 10) occurs at 10.33 and 10.44 μm where the permittivity of 4H-SiC has near-zero and negative real parts, when the inner dielectric has permittivity of 3.6 and 3.8, respectively. Similar to the ENZ point, for permittivity ε2 with a negative real part and a small imaginary part, a large impedance mismatch to free space also exists [ z2 = (μ/ε2)1/2 with μ = 1], leading to high reflection for excited waves, which makes the formation of a resonate cavity in the inner dielectric region. The refraction index n of 4H-SiC at 10.33 or 10.44 μm has a near-zero real part and a moderate loss part [n = (με2)1/2], which results in a near-zero phase delay for propagating wave in the 4H-SiC shell. When permittivity of the inner dielectric is tuned to a value making constructive interference occur in the inner cavity at a certain wavelength, a remarkable enhancement is achieved. It is demonstrated that wavelength-tunable power enhancement can be obtained via the tunable inner dielectric coated with 4H-SiC shell, with a tuning range within 10.2-10.55 μm near the ENZ wavelength of 4H-SiC.
Fig. 8 (a) Power enhancement factor as a function of the permittivity ε1 of the inner dielectric and the inner radius b/λ0 of the ENZ shell at λ0 = 10.3 μm and a = 12.463 μm. (b) Power enhancement factor for tunable permittivity ε1 from 3 to 4, with the dimensions of the shell fixed as shown in the inset of Fig. 7(b).
5. Conclusion
In summary, we have proposed a method to realize a remarkably enhanced radiation by an ENZ shell. Based on an exact theoretical approach, the relation between the dimensions of the shell and the enhancement effects are explored. It is shown that more than 9 times amplification can be obtained by choosing appropriate inner radius when the outer radius of the ENZ shell is larger than 0.8λ0. The maximum power enhancement effect by the ENZ shell is less sensitive to the outer radius than that by the conventional high-ε approach. The ENZ shell with appropriate dimensions generates a strong field within the inner region due to constructive interferences, which plays a crucial role for the enhancement of radiation. When the source is off-center in a small region, the ENZ shell amplifies the zeroth order component and suppresses the nonzeroth orders partial waves, ensuring both giant enhancement effect and quasi-isotropy of the radiating EM wave. When two sources in phase are used, nearly 30 times of enhancement and a quasi-isotropic characteristic are realized. The combined power can be more than 4 times of single power while the phase difference of these two sources is less than 134°. Our analytical and numerical results suggest that enhancement and combination for quasi-omnidirectional radiation can be achieved with the ENZ shell despite of the positions of sources within a certain phase difference, which cannot be fulfilled in free space. Finally, we have studied a real design using the 4H-SiC ENZ materials at 10.3 μm wavelength. A tunable inner dielectric is introduced into the design, which is important as the compensation for the micromachining tolerance of the inner radius b. More than 10 times amplification can be retrieved by tuning the permittivity of the inner dielectric to a corresponding value within 3 and 4. The power enhancement and combination for quasi-omnidirectional radiation via ENZ shell proposed in this work can be feasible experimentally in infrared and optical frequencies.
Funding
Natural National Science Foundations of China (NSFC) (61701268, 61775195 and 61631012); Natural Science Foundations of Ningbo City (2016A610066) and Zhejiang Province (LR15F050001, LQ17F010003 and LZ17A040001); K. C. Wong Magna Fund in Ningbo University.
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