Corrigendum: Inhomogeneous field in cavities of zero index metamaterials

In common media, electromagnetic wave always possesses a fluctuant field variation, analogous to an undulant surface of sea. While electromagnetic wave in the media with zero index metamaterials (ZIMs), whose refractive indices are near zero, homogeneous or constant field distribution will emerge, resembling a tranquil surface of lake. Such impression almost could be found in all previous literatures related to ZIMs. However, in this letter, we theoretically and numerically find that, in a cavity structure with ZIMs, when higher order modes (e.g., dipole modes) are excited inside cavity, inhomogeneous field could take place in ZIMs. Such a finding challenges the common perception in ZIMs: It is generally considered that homogeneous or constant field is generated in ZIMs. In addition, the proposed cavity structure herein could be used to manipulate radiation of light, such as enhancing or suppressing radiation, controlling radiation pattern and achieving isotropic or directive radiation, thereby potential applications are expected. These effects are well confirmed by numerical simulations.


Supplementary Figures
Supplementary Figure S1. A schematic plot of 2D cylindrical cavity structure in free space.
Region 2 represents the shell of MNZ metamaterials, with its inner and outer radii R 1 and R 2 respectively. Inside the shell, it is the filled medium. An electric line source is positioned at a coordinate of (d, 0), as shown by the yellow point.

Supplementary Notes
Supplementary Note 1

Theoretical analysis
The two-dimensional (2D) cylindrical cavity structure is placed in free space, as shown schematically in Supplementary Fig.S1. The shell with MNZ metamaterials is uniform in the z direction, with its inner and outer radii R 1 and R 2 respectively. Suppose that the effective permittivity and permeability of MNZ are 2  and 2  respectively. For the core region, the parameters of filled medium are denoted as 3  and 3  . An infinite electric line source is located at a position of (d, 0), as indicated by the yellow point in Supplementary Fig.S1. Under 2D cylindrical coordinate system denoted by r ,  and z, only three components z E , r H and H  are involved, and the EM wave in each region is governed by Helmholtz equation: where  is the angular frequency, and c is velocity of light in vacuum, and N = 1, 2 and 3, labeling three different regions. By analyzing equation (1), we can obtain the general solution z E in each region as follows: in region 1, and in region 2, For region 3, the EM wave should be the superposition of all the cavity modes and the source radiation.
Generally, the field distribution produced by an infinite line source is an angular-independent uniform cylindrical wave which is described by the zero-th order Hankel function of first kind, kk   is the wave vector in region 3 and  is the radial coordinate under another cylindrical coordinate whose origin is centered at the source. In order to analyze conveniently, it is necessary to expand such afield distribution under the coordinate with its origin at the center of region 3. By applying the translational addition theorem, we have,    (see Supplementary   Fig.S2). Hence, the electric field in region 2 could be written as, When one of the cavity modes is at resonance, such mode is the most dominative one. Then Eq.(S7) could be approximately expressed as, where 0 m  .

Supplementary Note 3
Radiation from the cavity In fact, the two coefficients m a and m b , related to the excited cavity modes inside region 3 and the radiated EM wave in free space respectively, are enough for us to fully uncover the radiation issue.
Based on the above analytical formulas, Supplementary Fig.S3(a) Supplementary Fig.S3(a) and Fig.S3(b) respectively, we can find the amplitudes of m b are less than these of m a ,which is caused by impendence mismatching between the MNZ shell and air.

Enhancing or suppressing radiation
In order to exhibit intuitively the total EM energy radiating from the cavity structure, we need to compute analytically the summation of the power flow of all modes from m   to  , by using  is impedance of air. However, in a selected frequency range (e.g., 5 GHz to 30 GHz), the number of possible resonant cavity modes are finite.
Therefore, we choose the required order from m=-5 to 5 to calculate the power flow in such a frequency range, which is also enough to make the result convergent to an exact value. Supplementary By observing the results of Supplementary Fig.S3(c) and (d), we can find that when the resonance of cavity modes are obtained, i.e., at the resonant peaks, the enhancing EM radiation could be achieved, otherwise the EM radiation will be suppressed.

Supplementary Note 4 Cavity modes in a narrow band of MNZ
In the main text, we assume a dispersionless MNZ metamaterials in order to conveniently and clearly explain the potential physics of inhomogeneous field in the cavity structure. However, by changing the filled media in region 3, similar results could be obtained. For example, we assume that the working frequency of MNZ is at 10 GHz, and by changing filled media in the core region, we can still observe the corresponding resonances of cavity modes, as shown in Supplementary Fig.S4. Supplementary   Fig.S4(a) shows the analytical results of power flow radiating from the cavity and the enhancing EM energy could be obtained when they are at resonance of cavity modes, which are marked by the order m one by one in plot. The corresponding numerical result is displayed in Supplementary Fig.S4(b), which matches well with the analytical one. In addition, when the required media are filled in core region 3, the resonance of cavity modes will occur. Especially, when the higher order modes are at resonance, the inhomogeneous field will happen in ZIMs. Supplementary Fig.S4(c) shows the real part of the electric field for the case of dipole mode resonance, and we can read the information of inhomogeneous field in ZIMs. Supplementary Fig.S4(d) shows the real part of the electric field , where the quadrupole mode is at resonance. The inhomogeneous field in ZIMs could still be observed, and the radiation pattern with numbers of outgoing directions determined by the angular momentum m=2. Therefore, although MNZ metamaterials usually work for a narrow band of frequencies, if the required media are filled in the core region, all similar results could be achieved.