1. Introduction

The electromagnetic properties of a linear isotropic homogeneous material are determined by the relative permittivity $\varepsilon $ and the relative permeability $\mu $. We shall suppress the common subscript r. These parameters of the (non-gain) medium are in general complex with a non-negative imaginary part. They depend on the angular frequency $\omega $ of the radiation under consideration. We shall assume monochromatic irradiation, so that this frequency dependence becomes irrelevant. The index of refraction n is a solution of

In the past decades, a great deal of effort has been devoted to the development of so-called double negative metamaterials. For these materials $\varepsilon $ and $\mu $ are both negative (with inevitable small positive imaginary parts), and therefore the index of refraction is approximately real and negative. These negative index of refraction materials (NIM’s) have been predicted to have peculiar properties by Vesalago [2]. A plane wave that refracts into the medium at an interface appears at the opposite side of the surface normal as compared to refraction into a regular dielectric with positive n. Later, Pendry [3] showed that incident evanescent waves could possibly be amplified by a slab of NIM, leaving a possible path to the construction of a superlens that can resolve images below the diffraction limit. Some controversies regarding the Fresnel reflection and transmission coefficients for such a layer were resolved in [4].

The Green’s function for the emission of radiation by a localized source, embedded in a medium with index of refraction n, is given by

More recently, materials with epsilon-near-zero (ENZ) have attracted a great deal of attention. We shall assume that $\mu = 1$, so that $n = \sqrt \varepsilon $ is the correct solution of Eq. (1). We then have $n \approx 0$, and we see from Eq. (2) that the Green’s function for wave propagation in an ENZ material becomes $g(r) \approx 1/r$. The time dependence for a monochromatic field is taken as $\exp ( - i\omega {\kern 1pt} t)$. The expected spherical wave becomes $\exp ( - i\omega {\kern 1pt} t)/r$. The field oscillates with angular frequency $\omega $, but the spatial oscillations have disappeared. This phenomenon is referred to as ‘static optics’. The field still oscillates with time t, but the spatial part becomes $g(r) \approx 1/r$, which is the Green’s function of electrostatics. One may then wonder whether any energy transport is possible through such a material.

ENZ materials exist in nature. Most notably, if $\omega $ of the source is close to the plasma frequency of a metal, the $\varepsilon $ of the metal is close to zero. Below the plasma frequency the metal is opaque and above the plasma frequency it is transparent. However, these plasma frequencies are in the UV (wavelength ∼130 nm for gold). Metamaterial ENZ media have been demonstrated in the microwave region [5–7], for terahertz radiation [8,9] and for the visible range of the spectrum [10,11], and have been studied theoretically [12–18]. An interesting application of ENZ media is the possibility of squeezing, funneling or tunneling electromagnetic radiation through narrow channels or bends [19–24]. Also waveform shaping and angular filtering has been proposed [25–28], and levitation of small particles near the surface of an ENZ interface has been predicted [29]. Other applications include perfect optical absorbers [30,31] and enhancement of the magneto-optical effect [32].

The electric and magnetic fields emitted by a localized source can be represented in terms of the scalar Green’s function of Eq. (2) [33,34]. When this function is represented by an angular spectrum [35], the fields become superpositions (integrals) over plane waves. These waves are either traveling or evanescent, with respect to the surface of the material, and they are either s or p polarized. The reflected and transmitted plane waves can then be constructed with the help of appropriate Fresnel coefficients, and the reflected electric and magnetic fields then follow by superposition. This approach has proven to be very successful for the study of the radiation pattern of an electric dipole near an interface [36], and for the computation of the field lines of energy flow in the near field [37,38]. In order to tackle this problem for an ENZ medium, it is imperative to study the solutions for plane waves first, and in particular include the evanescent incident plane waves.

2. Traveling and evanescent waves

We shall assume a harmonic time dependence with angular frequency $\omega $. Then the electric field is represented as

 

Fig. 1. Illustration of a travelling or an evanescent plane wave reflecting off and transmitting into a dielectric. We take the positive z direction as up.

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The wave vectors ${{\textbf k}_r}$ and ${{\textbf k}_t}$ of the reflected wave and the transmitted wave, respectively, must have the same parallel component ${{\textbf k}_{||}}$ as the wave vector ${{\textbf k}_i}$ of the incident wave. The wave vector of the incident wave is

Let us now consider what kind of wave the t wave is. Let us assume for a moment that $\varepsilon $ is positive, like in good approximation for most dielectrics. Then also n is positive. If $\alpha < n$, then ${v_2}$ is real, and the t wave is traveling. If $\alpha > n$, then ${v_2}$ is imaginary, and the t wave is evanescent. If the incident wave is traveling, then $\alpha $ is in the range $0 \le \alpha < 1$, and $\alpha = \sin {\theta _i}$ with ${\theta _i}$ the angle of incidence. If $n > 1$, then $\alpha < n$ for all $\alpha $, and thus the t wave is traveling. If $n < 1$, the t wave is traveling for $0 \le \alpha < n$ and evanescent for $n < \alpha < 1$. Borderline is $\alpha = n$, and this is $\sin {\theta _i} = n$. This is the familiar critical angle. If the incident wave is evanescent, we have $\alpha > 1$. If $n < 1$, the t wave is evanescent for all $\alpha $. If $n > 1$, the t wave is evanescent for $\alpha > n$ and traveling for $1 < \alpha < n$. In the latter case we have the unusual situation that an evanescent incident wave is converted into a traveling wave upon transmission through the interface. For a metallic medium we have $\varepsilon < 0$, and n is positive imaginary. The t wave is evanescent for all $\alpha $. When $\varepsilon $ is complex, with a positive imaginary part, then so is n. In addition, the real part of n is positive. Then the t wave is partially traveling and partially evanescent in the z direction.

3. Polarization vectors

When the incident wave is s polarized (TE), then so are the reflected and transmitted waves. The same holds for p polarization (TM). The unit vector in the ${{\textbf k}_{||}}$ direction is

4. Fields and Fresnel coefficients

The incident plane wave has wave vector ${{\textbf k}_i}$ and amplitude ${E_{\textrm{o}}}$, which may be complex. The expressions for the reflected and transmitted waves are plane waves with the corresponding wave vectors and polarization vectors. The amplitudes are ${E_{\textrm{o}}}$ times the appropriate Fresnel coefficients. We find for s polarization

5. ENZ limit of the Fresnel coefficients

For an ENZ medium we have $\varepsilon \to 0$ and $n \to 0$. This gives ${v_2} \to i\alpha $ and ${v_1}$ remains as is, Eq. (5). In Eqs. (27–29) we can immediately take this limit, with result

 

Fig. 2. The figure on the left shows the real and imaginary parts of the reflection coefficient for s waves. The dashed lines are for the small value of $\varepsilon = 0.015 \times (1 + i)$, and the solid lines are the ENZ limits, with the real part red and the imaginary part blue. The figure on the right shows the ENZ limit of the absolute value of ${R_s}$ (green curve), with the dashed line the result for the same small $\varepsilon $ as in the figure on the left.

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Fig. 3. Shown are the real and imaginary parts of ${T_s}$, and the absolute value of ${T_s}$, for small $\varepsilon $ (dashed curves). The solid curves are the ENZ limits.

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Let us now consider p waves. We see from Eq. (33) that the reflection coefficient is $- 1$ for all $\alpha $. Any p wave reflects back with only a phase shift, and this phase shift is the same for all $\alpha $. This behavior is illustrated in Fig. 4. The transmission coefficient vanishes for all $\alpha $, except $\alpha = 0$, as shown in Fig. 5. It seems that no radiation penetrates the material for $\alpha \ne 0$. This would be the same as for a perfect conductor (mirror). We shall show below that for an ENZ material this conclusion is incorrect. The dashed curves for the small value $\varepsilon = 0.015 \times (1 + i)$ in Fig. 5 are not close to the ENZ limit for traveling waves, which is zero for all $\alpha $, except $\alpha = 0$. ${T_p}{(\alpha )^{ENZ}}$ in Eq. (34) has a discontinuity at normal incidence. Such a point singularity is obviously unphysical. The interpretation of this phenomenon can be inferred from Fig. 5. For $\varepsilon \to 0$, the imaginary part vanishes, and the real part has a sharp peak near $\alpha = 0$, with a height of 2. From Eq. (9) for ${v_2}$ we see that in the ENZ limit and $\alpha \to 0$ we have ${v_2} \to 0$. However, the limits $n \to 0$ and $\alpha \to 0$ do not commute. For $n = 0$ and $\alpha \to 0$ we have ${T_p}(\alpha ) \to 0$, but for $\alpha = 0$ and $n \to 0$ we have ${T_p}(0) \to 2$. Physically, neither n nor $\alpha $ can be exactly zero. Whether ${T_p}(\alpha ) \to 0$ for $n = 0$ or ${T_p}(0) \to 2$ depends on which one is smaller: |n| or $\alpha $. As mentioned in Sec. 2, the critical angle (of incidence) is at $n = \alpha = \sin {\theta _i}$ (for n real). For any n, there will always be a small range $0 \le \alpha < \,\,|n|$ for which ${T_p}(\alpha ) \approx 2$, and the transmitted wave is (partially) traveling. Outside this range, the transmitted wave is mainly evanescent, and ${T_p}(\alpha ) \approx 0$. For $\varepsilon = 0.015 \times (1 + i)$ we have $n = 0.13 + 0.056 \times i$ and $|n|\, = 0.15$ as the upper limit on $\alpha $. This corresponds to an angle of incidence of about 8°.

 

Fig. 4. Shown are the real and imaginary parts of ${R_p}$, and the absolute value of ${R_p}$, for small $\varepsilon $ (dashed curves). The solid curves are the ENZ limits.

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Fig. 5. Shown are the real and imaginary parts of ${T_p}$, and the absolute value of ${T_p}$, for small $\varepsilon $ (dashed curves). The solid curves are the ENZ limits. For evanescent waves, the values of the real and imaginary parts of ${T_p}$ are close to their ENZ limit of zero, with the real part larger than the imaginary part. For traveling waves, the convergence to the ENZ limit is much slower. The dot represents ${T_p}{(0)^{ENZ}} = 2$. For $\alpha \ne 0$ we have ${T_p} = 0$, which is the green line in the figure on the right. In the figure on the left, the red and blue lines are on top of each other, making it look purple.

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Finally, we mention that for $\alpha \approx 0$ the distinction between s and p waves should disappear. We see from Figs. 2 and 4 that ${R_s} \approx 1$ and ${R_p} \approx - 1$. The difference in sign is due to the phase conventions for the polarization vectors. From Figs. 3 and 5 we see that ${T_s} \approx 2$ and ${T_p} \approx 2$.

6. ENZ limit of the fields

We now consider the fields from Eqs. (19–26) in the ENZ limit. Nothing simplifies for the incident and reflected waves, except that ${R_p} \to - 1$ in Eqs. (23) and (24), and ${R_s}$ in Eqs. (19) and (20) can be replaced by the right-hand side of Eq. (31). In the medium, ${v_2} \to i\alpha $, and so $\exp (i{v_2}\bar{z}) \to \exp ( - \alpha \bar{z})$. All fields in the ENZ medium are evanescent, with $\alpha = 0$ borderline.

Let us first look at the s waves. In Eq. (22) we have the product $n{{\textbf e}_{p,t}}$. From Eq. (16) we see that ${{\textbf e}_{p,t}} \propto 1/n$, so the factor n cancels. We then get $n{{\textbf e}_{p,t}} = \alpha {{\textbf e}_z} - {v_2}{{ \hat{\textbf k}}_{||}}$, and with ${v_2} \to i\alpha $ this yields $n{{\textbf e}_{p,t}} \to \alpha {\boldsymbol {\eta}}$, with

 

Fig. 6. Shown are the field lines of the magnetic field in the plane of incidence, for s polarization. Here, $2\pi $ correspond to an optical wavelength in free space. The fat line in the middle is the interface, and the ENZ medium is above it.

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For p waves we get the factor ${T_p}{{\textbf e}_{p,t}}$ in Eq. (25). From Eqs. (16) and (30) we see that again a factor of n cancels. We therefore have two cases, just as for ${T_p}$ in Eq. (34). We now find

7. Energy flow

We now return to the question whether or not energy will flow through the ENZ medium. To this end, we consider the time averaged Poynting vector

7.1 Traveling incident wave

For $0 \le \alpha < 1$ the incident wave is traveling, and parameter ${v_1}$ from Eq. (5) is real and positive. In the complex conjugate of ${\textbf B}({\textbf r})$ we can then set $v_1^\ast = {v_1}$. For the reflection and transmission coefficients in the ENZ limit we use Eqs. (37) and (38). We then obtain for s waves

7.2 Evanescent incident wave

For an evanescent incident wave, ${v_1}$ is positive imaginary, and we have $v_1^\ast = - {\kern 1pt} {v_1}$. We define

8. Normal incidence

Of practical importance is the case of normal or near normal incidence. In this case, the vector ${{\textbf k}_{||}}$ goes to zero, and ${{ \hat{\textbf k}}_{||}}$ becomes undefined. We then assume a direction for s polarization chosen, and set

With ${v_1} = 1$, ${R_s} = 1$, ${{\textbf e}_{p,i}} = - {{ \hat{\textbf k}}_{||}}$, ${{\textbf e}_{p,r}} = {{ \hat{\textbf k}}_{||}}$ and ${T_s} = 2$ we find for s polarization

For p waves near normal incidence we obtain

Finally, we consider the time averaged Poynting vector in the limit of normal incidence. By considering the complex amplitudes in this limit, we find that in vacuum the value of ${\textbf E}({\textbf r}) \times {\textbf B}({\textbf r})\ast $ is imaginary, and therefore the Poynting vector vanishes. In the ENZ medium, the magnetic field is zero. Therefore, for normal incidence we have ${\textbf S}({\textbf r}) = 0$ everywhere.

9. Conclusions

We have studied the reflection off and the transmission through an ENZ interface. The incident field is taken as a plane wave, and we consider both traveling and evanescent incoming waves. The only variable in the problem is parameter $\alpha $. For a traveling incident wave ($0 \le \alpha < 1$) this is the sine of the angle of incidence, and for an evanescent incident wave ($\alpha > 1$), $1/\alpha $ is a dimensionless measure for the penetration depth. Moreover, ${k_{\textrm{o}}}\alpha $ is the magnitude of the parallel components of all wave vectors (incident, reflected and transmitted).

First we have obtained the ENZ limit of the Fresnel reflection and transmission coefficients. We found that for p polarization of the incident wave the reflection coefficient is -1 for all $\alpha $. The transmission coefficient vanishes for all $\alpha $, except for $\alpha < < \,|n|$ with both $\alpha $ and n going to zero. Then the transmission coefficient is 2. Close to normal incidence ($\alpha = 0$) there is an abrupt but smooth transition in the transmission coefficient for p waves. There is no such behavior for s polarized incident radiation. In that case, the absolute value of the reflection coefficient is unity for traveling incident waves and for $\alpha > 1$ it drops to zero rapidly with increasing $\alpha $.

We then considered the electric and magnetic fields at both sides of the interface. At the vacuum side we have the usual interference between the incident and the reflected waves. In the ENZ medium all fields are evanescent for $\alpha \ne 0$. For an s polarized incident wave, the electric field in the ENZ medium is also s polarized. Remarkably, however, the magnetic field is exactly circularly polarized, and moves with a phase velocity of $c/\alpha $ along the surface. For p polarized incident radiation, the magnetic field in the medium vanishes identically. The electric field is linearly polarized for $0 \le \alpha < < \,|n|$ and circularly polarized for $\alpha > > \,|n|$. For $\alpha > > \,|n|$, the transmission coefficient is zero for all $\alpha $, but there is still an electric field. For normal incidence, the magnetic field in the medium is zero, both for s and p irradiation. The electric field is ‘static’, either linearly or circularly polarized. The Poynting vector in the medium for p waves is identically zero, but for s polarization there is a finite flow of energy along the surface.