1. Introduction

Most electromagnetic (EM) phenomena and devices result from the interactions between wave and materials, governed by Maxwell’s equations. For a material impinged by a wave, its EM properties are usually determined by two material parameters, i.e., the permittivity $\epsilon$ and the permeability $\mu$. To go beyond the limitations of natural materials and to gain the strength of manipulating EM waves or light, in the past few decades, great efforts have been devoted to metamaterials [1, 2], artificial materials composed of subwavelength engineered blocks, which can possess, in principle, arbitrary EM parameters. With them, many amazing phenomena or optical devices have been proposed and well demonstrated in experiments, such as perfect lens [3, 4] and invisibility cloaks [5–8]. At present one often utilizes a two dimensional (2D) parameter plane to summarize and classify all possible isotropic metamaterials, as shown in Fig. 1(a) [1]. This plane with two axes corresponding to the real parts of permittivity and permeability, respectively, is divided into four quadrants, with the common dielectrics (CDs) located at the first quadrant, the electric plasma (EP) materials or negative-permittivity metamaterials at the second quadrant, the negative-index metamaterials (NIMs) at the third quadrant, and the magnetic plasma (MP) materials or negative-permeability metamaterials at the fourth quadrant. The impedance-matched zero index metamaterials which have been widely studied recently [9–13], are indicated by the origin of the plane. The epsilon-near-zero (ENZ) metamaterials are denoted by the y-axis and the mu-near-zero (MNZ) metamaterials by x-axis.

 

Fig. 1 The parameter space for $\epsilon$ and $\mu$. (a) Two dimensional (2D) plane with two axes corresponding to the real parts of permittivity and permeability, respectively. (b)Three dimensional (3D) space with loss or gain as new parameters. Three axes correspond to the real parts of permittivity, permeability and the imaginary part of permittivity or permeability, respectively.

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In practice, the realizations of metamaterials, in particular NIMs, inevitably rely on lossy metals or resonant components, resulting in an intrinsic loss feature of metamaterials. As the loss may deteriorate the performance of efficient devices, it is often regarded as a drawback. To overcome such a problem, one of the strategies for loss mitigation is by introducing media with optical gain into metamaterials [14, 15], as a result the balanced gain and loss might ensure that the designed devices work well. Alternately, we are also aware that for a metamaterial or in an optical system with both loss and gain, the loss and gain together may enable some new physics phenomena. For instance, by spatially modulating loss and gain, one can design an optical parity-time (PT) symmetric system with a complex refractive index profile satisfying the following condition $n(x)=n{(-x)}^{\ast }$ [16–22]_. It has been shown that such PT systems have real eigen-value spectra at some specific situations, with numerous effects of light, such as unidirectional invisibility phenomena [16], coherent perfect absorption [17, 18], and extraordinary nonlinear effects [19, 20]. Furthermore, several groups recently have realized that the losses in metamaterials may be utilized for other applications, such as perfect absorbers [23–25] and solar (or thermal) energy harvesting [26–28]. In addition, if an EM wave passes from a medium with a high refractive index to the ENZ metamaterial, the loss will induce the EM power to enter the ENZ metamaterial at the normal direction for arbitrary incident angles [29]. A dielectric with a permittivity profile whose real and imaginary parts meeting the so-called spatial Kramers-Kronig relations, can absorb all incoming EM wave with any incidence angle [30]. Such Kramers–Kronig optical media also can trigger bidirectional invisibility [31], partly because of the loss. These significant outcomes fully indicate that in some situations, the material loss and gain are not trivial but may play an important role in controlling EM waves or light behavior. They can even be used for realizing some applications that metamaterials without any gain or loss cannot reach. Hence to some extent, the loss and gain, i.e., the imaginary parts of permittivity and permeability, may be regarded as another two parameters for manipulating EM fields with metamaterials.

With regard to the parameter space, the conventional 2D plane is obviously incomplete (see in Fig. 1(a)), which cannot embrace the loss or gain metamaterials. Based on new parameters, loss and gain, we suggest a three-dimensional (3D) space (see Fig. 1(b)) with three axes corresponding to the real parts of permittivity, permeability and the imaginary part of permittivity or permeability, respectively. Therein$\mathrm{Im}[\epsilon (\mu )]>0$ represents loss, while $\mathrm{Im}[\epsilon (\mu )]<0$ indicates gain. Compared with the 2D plane, the 3D space involves more abundant materials, most of which have not been touched in previous studies, with unknown EM properties. For instance, if $\mathrm{Im}[\epsilon ]\in (-\infty ,0)$and $\mathrm{Im}[\mu ]\in (0,\infty )$, one can find a region in the considered 3D space, where both $\epsilon$ and $\mu$ are the same in real parts, whereas are opposite in imaginary parts. In other words, the permittivity and the permeability are conjugate to each other, i.e., $\epsilon =\mu *$. This kind of metamaterials are called as conjugate metamaterials (CMs) which have not been systematically studied before [32]. It has been shown that non-attenuated propagation of EM wave is possible in these CMs in [32]. However, neither clear physics picture nor novel functionalities of these metamaterials has been proposed in literature. More importantly, although both $\epsilon$ and $\mu$ are complex, the CMs have well defined refractive index $n^2=\epsilon \mu$, which raises several significant questions. One of the interesting points is how to choose the sign in square root when we calculate the refractive index of CMs? From the formula, it seems that the index $n$ is independent of complex angle $A\text{rg}[\epsilon ]$ or $A\text{rg}[\mu ]$. Does it mean that all CMs with different complex angles have the same EM properties? If not, what's the relationship between the EM properties of CMs and complex angle? Is there any new physics expected?

In this work, we will explore in deep the EM properties of CMs. Starting from Maxwell's equations, firstly we will revisit the CMs from the view of field transformation optics (FTO). We show that the CMs can be regarded as the outcomes of FTO. Then we will investigate the properties of CMs and uncover some novel optical properties and functionalities. Amazingly we will show the CMs may serve as a subwavelength-resolution lens, with a perfect lens as a limiting case. All results of this work can fully answer the above proposed questions.

2. CMs from field transformation optics

Transformation optics (TO) is a powerful method for designing novel optical devices, such as invisibility cloaks, and is based on the invariance of Maxwell's equations under coordinate transformations (CT) [7]. As a complementary to the TO approach, alternately, a field transformation (FT) method has been proposed, providing a direct control on the impedance and polarization signature [33]. Likewise to CT method, the FT between virtual space and physics space also will induce a transformation on $\epsilon$ and $\mu$ tensors, which may be called as field transformation optics (FTO). In this section, we are going to propose a more general version of FTO and revisit the CMs using this new method.

Figure 2 shows schematically the FT operation from virtual space to physics space. Before FT, the EM space is virtual space which is supposed to be filled with a simple and isotropic medium of ${\epsilon }_V$ and ${\mu }_V$ (see Fig. 2(a)). For a given frequency $\omega$, the Maxwell's equations in such a medium can be written as:

 

Fig. 2 Phase transformation from virtual space to physics space. (a) is virtual space filled with a medium of ${\epsilon }_V$ and ${\mu }_V$. If the filled medium is located in the first quadrant in Fig. 1(a), then the electric field, magnetic field and propagating wave vector inside it form a right-handedness. For instance, the upper inset schematically shows the handedness of vacuum. (b) is physical space after the transformation of Eq. (8), and the transformed medium is $\epsilon \text{'}$ and $\mu \text{'}$ which are given by Eq. (9). In particular, when the medium in virtual space is vacuum, and when $\alpha =\pi$ and $\gamma =0$, then the electric field, magnetic field and propagating wave vector inside the transformed medium form a left-handedness.

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If the coupling between electric and magnetic fields are not considered, that is ${\kappa }_{12}\exp (i{\alpha }_{12})=0$ and ${\kappa }_{21}\exp (i{\alpha }_{21})=0$ in Eq. (1), then the FTs of Eq. (2) reduces to a simple one,

In physics, the FT in Eq. (8) changes the handedness signature of EM wave. For the ease of discussion, we assume the virtual space to be vacuum. When a plane wave propagating inside, its three vectors $\stackrel{\rightharpoonup }{E}$, $\stackrel{\rightharpoonup }{H}$and $\stackrel{\rightharpoonup }{k}$ form a right-handedness, as shown by the inset in Fig. 2(a). After FT of Eq. (8) with ${\alpha }_{22}=0$ and ${\alpha }_{11}=\pi$, then ${\stackrel{\rightharpoonup }{E}}^{\prime }=-\stackrel{\rightharpoonup }{E}$ and ${\stackrel{\rightharpoonup }{H}}^{\prime }=\stackrel{\rightharpoonup }{H}$, as a result ${\stackrel{\rightharpoonup }{E}}^{\prime }$, ${\stackrel{\rightharpoonup }{H}}^{\prime }$ and $\stackrel{\rightharpoonup }{k}$ in physical space form a left handedness (see the inset in Fig. 2(b)). As far as is known, the right-handedness usually states a material with a positive refractive index, while the left-handedness denotes that the materials are LHMs, with negative refractive indexes. Under such a FT, the transformed medium is of ${\epsilon }^{\prime }=-1$and ${\mu }^{\prime }=-1$. Therefore the handedness signatures from the FT picture and material parameters are consistent. The FT is an useful method to judge the handedness signature of a CM, i.e., by using FT, it is convenient for us to identify a CM to be a right-hand material (corresponding to a positive index) or a left-hand material (corresponding to a negative index). Apparently, such a feature is quite difficult to be obtained from the view of the refractive index as well as the impedance. On the other hand, it is well known that in the process of an EM wave propagating in a medium, the time-varying electric field and magnetic field induce each other, accompanied with the EM energy of both electric and magnetic fields exchanging. Then the physical meaning of the phase difference $\phi ={\alpha }_{11}-{\alpha }_{22}$ in CMs of Eq. (9) is exactly the delayed or advanced phase between the electric field and magnetic field at any moment. For more information about the CMs, however, the FTO picture is far in short. For instance, the scattering properties of CMs depend on their sizes, shapes, and the configurations of the CMs and other media. Therefore, it needs the help of both the refractive index and the impedance, and much effort should be devoted to further explorations.

3. The scattering properties of CMs

Next we will study the scattering properties of CMs by considering a EM wave striking into CMs from vacuum. Without loss of generality, the studied CMs are of ${\epsilon }^{\prime }=\exp (-i\alpha )$and ${\mu }^{\prime }=\exp (i\alpha )$ with $0\le \alpha \le \pi$. Here $\alpha$ is used to replace ${\alpha }_{11}$ for simplicity. It means that ${\epsilon }^{\prime }$ contains gain, while ${\mu }^{\prime }$ contains loss; both gain and loss are functions of the phase $\alpha$. The refractive index is $n_{CM}^2(\alpha )=1$ for any $\alpha$. For convenience, we label ${\epsilon }^{\prime }({\mu }^{\prime })$ as $\epsilon (\mu )$ in the following. By deducing the 3D parameter space, such CMs can be illustrated by a unit circle in a 2D complex plane, as shown by Fig. 3(a) in which $\mu$ is located at the upper part (red half circle), while $\epsilon$ is located at the lower part (blue half circle). Based on the handedness from the FT picture, the unit circle can be divided into three regions: (i) $0\le \alpha <0.5\pi$, for a plane wave in these CMs, the common thing is that its three vectors (electric field, magnetic field and wave vector) form right handedness, so that these CMs own positive refractive indexes; (ii) $0.5\pi <\alpha \le \pi$, the handedness of three vectors in this case is left, so that these CMs own negative refractive indexes; (iii) $\alpha =0.5\pi$, then $\mu =i$ and $\epsilon =-i$, which are purely imaginary CMs (PICMs). As the phase difference between the electric and magnetic field is $\phi =0.5\pi$, one field (e.g., the electric field) may be degenerated in a period, as a result only two vectors are left (see Fig. 3(b)). In this case the corresponding handedness of EM wave is ambiguous, which can be regarded as a degeneracy of the right handedness and left one. From this perspective the PICMs can support both positive and negative refractions.

 

Fig. 3 (a) The complex plane for CMs. The horizontal axis represents $\mathrm{Re}[\epsilon ,\mu ]$; for vertical axis, the upper axis indicates $\mathrm{Im}[\mu ]$, while the lower axis is $\mathrm{Im}[\epsilon ]$. The unit circle shows the studied CMs with ${\epsilon }^{\prime }=\exp (-i\alpha )$and ${\mu }^{\prime }=\exp (i\alpha )$. According to the handedness, the purple color region indicates the CMs with positive refractive index ($0\le \alpha <0.5\pi$); the blue region shows the CMs with negative refractive index ($0.5\pi <\alpha \le \pi$). (b) The degeneracy handedness of EM wave in CMs with $\alpha =0.5\pi$.

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3.1 Semi-infinite cases

Let us now confine to a 2D plane (i.e., x-y plane) with both material parameters and fields invariant in the z direction. For a linearly polarized wave, it can be decoupled into TE (transverse electric, $E_z$) and TM (transverse magnetic, $H_z$) polarizations, separately. Here we focus on TE polarization; the similar procedures can be applied to TM polarization. First we consider a semi-infinite case composed of vacuum and CMs. A TE wave is incident from air to CMs, with working frequency $\omega$ and incident angle ${\theta }_{in}$, which is expressed as ${\stackrel{\rightharpoonup }{E}}_{in}=\widehat{z}\exp (i{k}_{x}x+i\beta y)$, where $k_x=k_0\cos {\theta }_{in}$, $\beta =k_0\sin {\theta }_{in}$ and $k_0=\omega /c$ is wave vector in vacuum. At such an interface, there will be reflection and refraction occurring. After reflection by CM, the reflected wave in air is ${\stackrel{\rightharpoonup }{E}}_r=\widehat{z}r\exp (-i{k}_{x}x+i\beta y)$, where r is the reflection coefficient. For the refraction, the transmitted wave is ${\stackrel{\rightharpoonup }{E}}_t=\widehat{z}t\exp (i{k^{\prime }}_{x}x+i\beta y)$, where t is the transmission or refraction coefficient and ${k^{\prime }}_x=\pm {\left(n_{CM}^2(\alpha )k_0^2-{\beta }^2\right)}^{1/2}$ with the sign depending on $\alpha$. When $0\le \alpha <0.5\pi$, the EM wave incident from air will not change its direction but will continue to propagate in CMs to the right, which is schematically shown in Fig. 4(a), so that ${k^{\prime }}_x=k_x$ for positive refraction; when $0.5\pi <\alpha \le \pi$, the EM wave incident from air will undergo negative refraction (see Fig. 4(b)), resulting in ${k^{\prime }}_x=-k_x$. When $\alpha =0.5\pi$, due to the degeneracy handedness, $k^{\prime }=k_x$ or $-k_x$.

 

Fig. 4 The refraction and reflection of TE wave incident from vacuum to CMs. (a) For $0\le \alpha <0.5\pi$, the propagation direction of the EM wave will not change as it passes from air to CMs. For the refraction wave, the red arrow shows the direction of wave vector (k), which parallels with that of Poynting vector (S) (see the yellow arrow). (b) For $0.5\pi <\alpha \le \pi$, negative refraction happens at the interface of air and CMs. For the refraction wave, the blue arrow shows the direction of wave vector (k), which is anti-parallel with that of Poynting vector (S) (see the yellow arrow). (c) The relationship between the reflection (refraction) coefficients and $\alpha$.

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After matching the boundary conditions at the interface of $x=0$, the reflection and refraction coefficients are calculated as: for $0\le \alpha <0.5\pi$,

Based on Eqs. (10) and (12), we plot the relationship between all coefficients and the phase $\alpha$ in Fig. 4(c). It is clearly shown that for different $\alpha$, the scattering properties of CMs are different. As $\alpha$ changes from 0 to $0.5\pi$, $\left|r_{+}\right|$ or $\left|t_{+}\right|$ increases monotonously from 0 or 1; as $\alpha$ varies from $0.5\pi$ to $\pi$, $\left|r_{-}\right|$ ($\left|t_{-}\right|$) decreases monotonously from maximum values to 0 (1). Both coefficient curves are symmetric with respect to $\alpha =0.5\pi$. It means that we can obtain a CM-pair, one CM with $\alpha ={\alpha }_0\in [0,\pi /2]$ (positive refractive index) and the other with $\alpha =\pi -{\alpha }_0$ (negative refractive index), having the same scattering amplitudes. Furthermore, for transmitted wave propagating in CMs, it will neither be amplified nor absorbed, although $\epsilon$ involves gain and $\mu$ includes loss. This point can be captured by considering the Poynting's theorem: $-\nabla \cdot 〈\stackrel{\rightharpoonup }{S}〉=\Omega$, where $\stackrel{\rightharpoonup }{S}=\stackrel{\rightharpoonup }{E}\times {\stackrel{\rightharpoonup }{H}}^{*}/2$ is the complex Poynting vector and $\Omega =\mathrm{Re}[i\omega (\stackrel{\rightharpoonup }{E}\cdot {\stackrel{\rightharpoonup }{D}}^{*}-{\stackrel{\rightharpoonup }{H}}^{*}\cdot \stackrel{\rightharpoonup }{B})/2]$ is dissipative term about a medium. In general, $\Omega >0$(or $\Omega <0$) tells us that the medium has energy dissipation (or energy amplification); $\Omega =0$means the medium is purely material without any loss and gain. By applying the Poynting's theorem to studied CMs, then $\Omega =\omega [\mathrm{Im}(\epsilon ){\left|\stackrel{\rightharpoonup }{E}\right|}^2+\mathrm{Im}(\mu ){\left|\stackrel{\rightharpoonup }{H}\right|}^2]/2=0$ because of $\mathrm{Im}[\mu ]=-\mathrm{Im}[\epsilon ]$. In physics, the magnetic field energy lost in a half time period is entirely compensated by the electric field energy gained in the other half period; both loss and gain keep balance in a time period. In addition, by solving the eigen-mode problems of the air/CM interface, we know that the EM character of such a single interface is trivial. i.e., the interface doesn't amplify or absorb the incident wave. Therefore the EM energy is conserved in the process of refraction and reflection. This can be further confirmed by calculating the reflectivity and transmittivity, which are defined by $R=\left|〈S_x^r〉\right|/\left|〈S_x^{in}〉\right|$ and $T=\left|〈S_x^t〉\right|/\left|〈S_x^{in}〉\right|$, respectively, where $〈S_x〉$ is the x-component of averaged energy flow $〈\stackrel{\rightharpoonup }{S}〉=\mathrm{Re}\left[\stackrel{\rightharpoonup }{E}\times {\stackrel{\rightharpoonup }{H}}^{\ast }\right]/2$. The calculated result confirms that $R+T=1$.

To verify the above analysis, the numerical simulations are carried out by using COMSOL Multiphysics. We chose a CM-pair for illustration, one CM is $\alpha =0.25\pi$ (see Fig. 5(a)) and the other is $\alpha =0.75\pi$ (see Fig. 5(b)). The incident angle of TE polarized wave is ${\theta }_{in}={40}^{\circ }$. Both field patterns shows that the positive refraction occur in CM with$\alpha =0.25\pi$, while the negative refraction happens in CM with $\alpha =0.75\pi$. To further demonstrate the negative refraction, we also plot the distributions of power plow, as shown by the black arrows in Fig. 5(b), which indeed goes to negative direction. Comparing the power flow distributions in both the air and CM, the energy entering the considered CM is smaller than that in the air, illustrating again that during the refraction, the EM wave is not amplified, and the EM energy is conserved. From the above analysis, the phase $\alpha =0.5\pi$is a critical point where the handedness is degenerate. To examine how sensitive such a switching action is, we consider two CMs, one with $\alpha =0.49\pi$ and the other with $\alpha =0.51\pi$. Figures 5(c) and 5(d) display the corresponding simulated field patterns. It is clearly seen that the refraction in case of $\alpha =0.49\pi$is still positive; while in case of$\alpha =0.51\pi$, it is negative. The switching action around the critical point is very sensitive.

 

Fig. 5 The simulated electric field patterns for a TE plane wave obliquely incident to CMs with (a) $\alpha =0.25\pi$; (b) $\alpha =0.75\pi$; (c) $\alpha =0.49\pi$; (d) $\alpha =0.51\pi$. The incident angle is ${\theta }_{in}={40}^{\circ }$ for all cases. In simulations, a perfect matched layer is added to the right side to avoid backscattering. In (b), the black arrows indicate the averaged power flow.

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3.2 CM slab case

Further we consider a CM slab of thickness d placed in vacuum, and a TE wave ${\stackrel{\rightharpoonup }{E}}_{in}=\widehat{z}\exp (i{k}_{x}x+i\beta y)$ is obliquely incident from vacuum. After scattered, the reflected wave is ${\stackrel{\rightharpoonup }{E}}_r=\widehat{z}r\exp (-i{k}_{x}x+i\beta y)$, and the transmitted wave is ${\stackrel{\rightharpoonup }{E}}_t=\widehat{z}t\exp [i{k}_x(x-d)+i\beta y]$, where r (t) is the reflection (transmission) coefficient. Inside the CM slab, the EM wave is the superposition of the forward wave and backward wave, that is ${\stackrel{\rightharpoonup }{E}}_{CM}=\widehat{z}[a_p\exp (i{k}_{x}x+i\beta y)+a_n\exp (-i{k}_{x}x+i\beta y)]$, with $a_p$ and $a_n$ indicating the corresponding coefficients. After matching the boundary conditions at the interfaces of $x=0$ and $x=d$, we have

Here a case of thickness $d=2\lambda$ is studied to further discussion. By applying Eq. (14), both reflectance $\left|r({\theta }_{in},\alpha )\right|$ and transmittance $\left|t({\theta }_{in},\alpha )\right|$ are calculated analytically, as shown in Figs. 6(a) and 6(b), respectively. From two plots the following points can be catch. (i) Like that in semi-infinite case, both $\left|r\right|$ and $\left|t\right|$ are symmetric with respect to the case $\alpha =0.5\pi$, leading to the same scattering properties for two CMs with $\alpha ={\alpha }_0\in [0,\pi /2]$ and $\pi -{\alpha }_0$. (ii) In Fig. 6(b), the transmittance $\left|t\right|\ge 1$ for any ${\theta }_{in}$ and any $\alpha$, so that ${\left|t\right|}^2+{\left|r\right|}^2\ge 1$. In this regard the CM slabs might be used to amplify the incident EM wave or signals. (iii) Besides the Fabry–Pérot resonances (as shown by three red arrows in Fig. 6(a)), we can observe a series of new resonances in both reflectance and transmittance, which are obvious by reading Fig. 6(c) for a fixed $\alpha =0.25\pi$. It is clearly shown that in each curve (e.g., $\left|t\right|$), there are four resonance peaks, and the transmission and reflection resonances happen at the same incident angles. For instance, the first one is at ${\theta }_{in}\approx {29}^{\circ }$. More importantly, we can also see from Figs. 6(a) and 6(b) that for different $\alpha$, all these resonances still existed, with different strength but with unchanged resonance positions. When ${\theta }_{in}={29}^{\circ }$, Fig. 6(d) shows the peak values of both $\left|r\right|$ and $\left|t\right|$, as $\alpha$varies from 0 or $\pi$. As $\alpha$ increases from 0 (or $\pi$) to $0.5\pi$, the peak value$\left|r\right|$(and $\left|t\right|$) increases monotonously from 0 (and 1). In particular, when $\alpha =0.5\pi$, both $\left|r\right|$ and $\left|t\right|$ are infinite, leading to lasing phenomenon. Returning back to analyze Eq. (14), the conditions for these new resonances are derived as,

 

Fig. 6 The scattering coefficients vs ${\theta }_{in}$ and $\alpha$. (a) Reflectance $\left|r\right|$. Three arrows shows the Fabry–Pérot resonances, where $\left|r\right|=0$. (b) Transmittance $\left|t\right|$. (c) Both $\left|r\right|$ and $\left|t\right|$ vs ${\theta }_{in}$ for $\alpha =0.25\pi$. (d) Both $\left|r\right|$ and $\left|t\right|$ (in log scale) vs $\alpha$for ${\theta }_{in}={29}^{\circ }$. The values in red spots in (a) and (b) are beyond that scoped by the color bars.

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Now we examine what happens for EM wave inside CMs, by studying two coefficients $a_p$ and $a_n$, which are expressed as,

 

Fig. 7 The amplitude $A_p=\left|a_p\right|$ and $A_n=\left|a_n\right|$ vs ${\theta }_{in}$ and $\alpha$. (a) and (b) are for $A_p$ and $A_n$, respectively. (c) Both $A_p$ and $A_n$ vs ${\theta }_{in}$ for a fixed $\alpha =0.25\pi$. (d) Both $A_p$ and $A_n$ vs $\alpha$for fixed incident angles. The solid curves are for ${\theta }_{in}={29}^{\circ }$, and the dashed curves are for ${\theta }_{in}={26}^{\circ }$.

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All these analysis have been verified by simulated results. Figures 8(a)-8(c) shows the corresponding field patterns when a TE polarized wave with ${\theta }_{in}={40}^{\circ }$ strikes onto a CM slab with $\alpha =0.25\pi$, $\alpha =0.5\pi$ and $\alpha =0.75\pi$, respectively. Obviously, in case of $\alpha =0.25\pi$, the positive refraction occurs; in case of $\alpha =0.75\pi$, the negative refraction happens, with the energy flow anti-parallel with the corresponding wave vector. In case of $\alpha =0.5\pi$, the interference pattern inside the CM slab shows that the positive and negative refraction are of the same ratio. To check the lasing phenomena, we change the incident angle to ${\theta }_{in}={29}^{\circ }$, and Fig. 8(d) shows the scattering field pattern, and the white arrows show time-averaged power flow. The outgoing waves are largely amplified.

 

Fig. 8 The simulated electric field (E_z) patterns for a TE plane wave obliquely incident unto a CM slab with (a)$\alpha =0.25\pi$; (b) $\alpha =0.5\pi$; (c) $\alpha =0.75\pi$. In (a)-(c), the incident angle is ${\theta }_{in}={40}^{\circ }$. (d) The scattering electric field (ScE_z) patterns when a TE plane wave with ${\theta }_{in}={29}^{\circ }$ strikes unto the CM slab with $\alpha =0.5\pi$. In (c) and (d), the white arrows indicates the time-averaged power flow. Here $\lambda =1$ and $d=2\lambda$. The amplitude of incident wave is unity.

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3.3 The CM and perfect lens

As we have shown above, the CMs are more general than perfect lens or LHMs, to which they reduce for the special case of $\alpha =\pi$, where $\mu =\epsilon =-1$. Now we come to see whether such a CM slab can also amplify the evanescent waves and serve as a sub-wavelength lens or even a perfect lens. Following Eq. (21) in [3], the transmission coefficient of evanescent waves with $\beta \ge k_0$ becomes,

 

Fig. 9 The amplification of evanescent waves by the CMs. (a) The transmission coefficient of evanescent waves passing through a CM slab. Four curves are corresponding to four different $\alpha$. (b) The same as (a) but on a log scale as $\alpha$ is very close to $\pi$. In calculations, we set $\lambda =1$ and $d=2\lambda$.

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4. Summary

In conclusion, we have proposed a general version of field transformation optics. Based on this method, we can deduce the conjugate metamaterials (CMs) with its permittivity and permeability conjugate to each other. We systematically studied the EM properties of such CMs, and in particular found that for the transformation phases ranging from $0.5\pi$ to $\pi$, negative refractions happen at the interfaces of CMs and vacuum (or air). We also found a new kind of resonances that will support lasing phenomenon in a CM slab. Finally we found that some of CM slabs may serve as a subwavelength-resolution lens, with a perfect lens as the limiting case. Tiny deviations from $\epsilon =\mu =-1$ will produce imperfections in the imaging functionality. We note that in this work we emphatically discuss a kind of CMs with a refractive index $n_{CM}^2(\alpha )=1$. For other CMs with $n_{CM}^2(\alpha )\ne 1$, the FTO in Section 2 tells us that the virtual spaces of these CMs are not vacuum but some other media. In this way, the scattering properties would be different, thereby deserving further explorations. In addition, the CMs may extended to general cases of $\epsilon =\left|\epsilon \right|\exp (-i\alpha )$ and $\mu =\left|\mu \right|\exp (i\alpha )$ with $\left|\epsilon \right|\ne \left|\mu \right|$. Their EM properties are still unknown. Besides the isotropy, the anisotropy also can be introduced into the CMs. One interesting case is the perfectly matched layers (PMLs) absorbing all incident waves without any reflection, which is very important in computational electromagnetism electromagnetics. PMLs could also be regarded as an example of transformation optics [35], whether they can be obtained from the concept of CM is also very interesting. Therefore, we believe that CMs add to the catalog of metamaterials and the results reported here, can motivate further studies.