1. Introduction

The desire to make objects invisible has been going for hundreds of years. The transformation optics [1] and conformal mapping method [2] provide powerful approaches to design invisibility cloak [3–5]. The designed permittivity and permeability parameters of the cloak are usually anisotropic and inhomogeneous, which could be partially realized by metamaterials [6–10]. However the resonance characteristics of the metamaterials limit the bandwidth of the cloak. In order to overcome this problem, some unidirectional invisibility cloaks designed by geometry optics [11–15] and all dielectric material with graded index [16, 17] are reported. However most of them could only be used for natural incoherent light source because they did not consider the phase matching problem at the output end of the cloak. The scheme proposed in [15] is a multiband unidirectional cloak, which can be used for coherence light of multiple discrete narrow bands.

In this paper, a new method for designing optical device is derived based on the eikonal theory. Using the proposed method, we designed a scheme of broadband unidirectional cloak, which could be used for coherent wave and has continuous broadband performance.

2. Design optical device with the eikonal theory

According to the eikonal theory [18], the direction of the wave vector is determined by the eikonal gradient in the isotropic medium. Therefore carefully designing the eikonal distribution on a curved surface can control the propagation behavior of the subsequent light ray. For simplicity, we consider a two-dimension situation as shown in Fig. 1. A curved surface $S$ defined by $x=g(y)$ is located between two isotropic mediums with the refractive indexes of $n_0$ and $n_1$.

 

Fig. 1 Ray schematic of a plane wave traveling through a curve surface with extra eikonal distribution.

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Suppose a plane wave is incident upon $S$ and the angle between the propagating direction of the incident wave and the $x$ axis is ${\theta }_0$. The extra eikonal introduced by $S$ is represented by $s_e(y)$. When the plane wave passing through $S$, the complex amplitude distributed on the rear surface of $S$ can be expressed as $U\left(x,y\right)=U_0\exp \left\{i{k}_0{n}_0\left[g(y)\cos ({\theta }_0)+y\sin ({\theta }_0)\right]\right\}\exp \left[i{k}_0{s}_e(y)\right]$. Therefore, the total eikonal distribution on the rear surface of $S$ is:

3. Broadband unidirectional cloak

In order to achieve a unidirectional cloak, we consider a pair of flat surfaces $S$ and $S^{\prime }$ being vertically placed with the distance of $D$ as shown in Fig. 2(a). Same as the notations used in section 2, the refractive indexes of the mediums are noted by $n_0$ and $n_1$. A plane wave propagating along $x$ axis is normally incident upon $S$. Carefully designing the eikonal distribution $s_e(y)$ and ${s^{\prime }}_e(y)$, the light ray with desired biconvex caustic $C$ could be formed between $S$ and $S^{\prime }$. The shape of $C$ is symmetric to the line $x\text{=}D/2$. To restore the light ray to plane wave and considering the reversibility principle of optical path, the extra eikonal distributions introduced by $S$ and $S^{\prime }$ should be equal, or $s_e(y)$ = ${s^{\prime }}_e(y)$. In this case, all optical path lengths (OPLs) along various ray trajectories between $S$ and $S^{\prime }$ are equal. If the OPLs of the rays passing through $S$ and $S^{\prime }$ equal to that propagating in the outside isotropic medium (referred to as “the OPL matching condition” in the following), the objects placed in the areas $I$, $II$ or $III$ are undetectable for the outside reviewers and independent of the wavelength of the incident wave. Therefore this structure could be used as a unidirectional cloak for coherent wave and has continuous broadband performance.

 

Fig. 2 (a) The ray trajectories schematic for the unidirectional cloak. (b) One ray schematic for analyzing. The red dashed curve is the caustics $C$, the blue plates are used to introduce the extra eikonal distribution on $S$ and $S^{\prime }$.

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Accordingly, $s_e(y)$ should be derived by the shape of the caustic $C$ and the OPL matching condition. In order to produce the light ray with the desired caustic $C$, all the light rays emitting from $S$ should be tangent with the surface $C$. Suppose $C$ is defined by $y\text{=}f(x)$ and the light ray exiting from the point $(0,y_0)$ on $S$ is tangent with $C$ at point $(x_1,y_1)$ as shown in Fig. 2(b), the slope of this light ray could be expressed as:

 

Fig. 3 Computational domain and details for full-wave simulation of the unidirectional cloak.

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The thickness and the refractive index of the plate could be calculated according to the practical parameters. In the simulation, the biconvex caustic is defined by $f(x)=\pm ax(1-\frac{x}{D})$, where $0\le x\le D$ and $a$ is a coefficient. Thus the height of the transparent plate should be $h=aD$ and the extra eikonal introduced by $S$ is $s_e(y)\text{=}{n}_1\left(a\left|y\right|-\frac{4}{3}\sqrt{\frac{a}{D}}{\left|y\right|}^{3/2}\right)+c$,where the constant $c$ could be determined by Eq. (6). It can be found that the refractive index of the plate has minimum value at $y=h$. Suppose $n(h)=n_{\min }$, then we have:

As example, we set $D=60cm$ and $a\text{=}1/6$ in the simulation. The simplest case is putting the two plates in the air, or $n_0\text{=}{n}_1\text{=}1$. We set $n_{\min }\text{=}0.3$, then it can be calculated that $d=1cm$ and$n\left(y\right)=0.17\left|y\right|-0.07{\left|y\right|}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+0.85$. Assuming the transparent plates are nonmagnetic, then the permittivity and permeability are $\epsilon \left(y\right)=n^2\left(y\right)$ and $\mu \text{=}1$. Without loss of generality, a TE polarized plane wave is taken as incident wave. Numerous simulation results show that the waves match very well at the output surface of the right plate and the cloaking property is independent of the wavelength. As examples, Fig. 4(a)-(c) show the simulated electric field distributions with the incident wavelengths of ${\lambda }_0\text{=}8cm$, $12cm$ and $16cm$ respectively. The elliptical areas are objects with $\epsilon \text{=}3$, $\mu \text{=1}$ placed inside the cloaked regions. The objects placed in the cloaked regions do not influence the wave propagation property. Figure 4(d) shows the distribution of $n(y)$.

 

Fig. 4 The electric field distributions for different incident wavelengths. (a)${\lambda }_0\text{=}8cm$, (b)${\lambda }_0\text{=}12cm$, (c)${\lambda }_0\text{=}16cm$. (d) The refractive index of the transparent plates.$n_0\text{=}{n}_1\text{=}1$.

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With the present technology, realizing the graded material parameters, $n\left(y\right)$ or $\epsilon \left(y\right)$, requires artificial materials such as metamaterials or nonlinear photonic crystals [7, 19]. Thus the practical permittivity is usually frequency dependent. Our simulations show that the performance of the cloak has good stability when the permittivity of the transparent plate slightly deviates from$\epsilon \left(y\right)$. The bandwidth of the cloak should be related to the frequency-dispersive function of the material parameters. Using the Drude model [20] as example, we express the permittivity as:

When the angular frequency of the incident wave equals ${\omega }_0$, the permittivity of the transparent plates is $\epsilon \left(y\right)$. The value of ${\omega }_0$ could be selected according to the geometric size of the cloak system. For the structure used in Fig. 4, we set ${\omega }_0\text{=}9.4\times {10}^{9}rad/s$(or the frequency of $1.5GHz$). With the frequency-dependent permittivity described by Eq. (8), we simulate the spectra of the total scattering cross-width (SCW) of the unidirectional cloak. Figure 5(a) presents the simulated results for the frequency ranging from $1.5GHz$ to $10GHz$ (or the wavelength of the incident wave changing from $20cm$ to $3cm$). It can be seen that the total SCW changes slowly along with the increment of the frequency when the frequency is less than$7GHz$. The electric field distribution for the frequency of $7GHz$is shown in Fig. 5(b), which remains rather good invisibility performance. In this case, the bandwidth of the proposed unidirectional cloak could be considered at least $5.5GHz$.

 

Fig. 5 The simulated results using the frequency-dispersive permittivity with ${\epsilon }^{\prime }\left(y,\omega \right)=1-\frac{{\omega }_0^2\left[1-\epsilon \left(y\right)\right]}{{\omega }^2}$,${\omega }_0\text{=}9.4\times {10}^{9}rad/s$. (a)The spectra of the total SCW of the unidirectional cloak. (b)The electric field distribution for the frequency of $7GHz$.

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The proposed unidirectional cloak could be scaled to other frequency band to meet the practical requirement. For the incident waves near the visible band, we made simulations using the frequency dispersion described by Eq. (8) also. The simulated results show that the cloak has rather good performance for the frequencies ranging from $3.5THz$ to $14.5THz$. The bandwidth reaches about $11THz$.

4. Conclusion

A novel method for designing optical device is derived based on the eikonal theory. A continuous broadband unidirectional cloak is proposed by utilizing this method. Full-wave simulation results verify the properties of the broadband unidirectional cloak.