1. Introduction

Near-field imaging by flat lenses has been theoretically predicted and experimentally confirmed using negative index materials, also known as double negative (DNG) materials,^1–3 photonic crystals,^4,5 paired single negative (SNG) materials,⁶ and unpaired SNG such as a silver lens.^1,7,8 The phenomenon is closely related to photon tunneling through a subwavelength layer, whose refractive index is smaller than the media that sandwich it. Frustrated total internal reflection has been known since Newton’s time and was extensively studied by Hall in 1902.⁹ While tunneling can be understood as a class of quantum phenomena, the well-established explanation of photon tunneling in classical wave optics is the coupling of two oppositely decaying evanescent waves.¹⁰ Due to the coupling, the resulting Poynting vector has a nonzero normal component, suggesting that energy transmission between the media is possible as long as the distance of the gap is smaller than the wavelength. The ray optics fails to describe such a wave because the parallel component of the wave vector for an evanescent wave is so large that no polar angle within the real space can be defined. The lateral shift of the beam accompanied with photon tunneling is commonly described as a Goos-Hänchen type of shift due to the phase differences between different incidence angles of a wave packet.^11–15 Another approach is to use numerical simulations such as the finite-difference time-domain (FDTD) method to determine the local field and Poynting vector distributions for a plane wave or Gaussian beam incidence.^6,16,17

In most of these studies, the coupling between the forward and backward waves was not completely addressed. In this study, we employ an energy streamline method based on the Poynting vector of coupled plane waves to study the lateral beam shift in layered structures when photon tunneling occurs. The direction of energy flow can be characterized by the streamline that connects the local Poynting vectors, similar to the streamline in fluid dynamics that connects the velocity vectors. Energy streamlines can be defined in media with a magnetic response as well as losses. The lateral shift could be a crucial factor in the design of devices based on nanoscale radiation heat transfer. While the energy streamline method has been used previously,^15,18,19 it has not been employed for photon tunneling in planar structures.

 

Fig. 1. Schematic of the three layer structure in this study, where media 1 and 3 are semi-infinite, and medium 2 has a thickness of d. The incidence angle θ₁ is determined by the wavevector. Each medium is linear, homogeneous, and isotropic. The complex permittivity and permeability, relative to vacuum, are given as ε’s and µ’s, whereas A’s and B’s are the coefficients of the forward and backward wave at the interface.

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2. Analysis

Consider the geometry of three layers shown in Fig. 1, where media 1 and 3 are semi-infinite. Each layer is linear, homogeneous and isotropic. The electric and magnetic responses can be characterized by the relative permittivity ε and permeability µ, which in general are complex and frequency dependent. If the incidence is a transverse magnetic (TM) wave with an angular frequency ω, the magnetic field in each region is given by Eq. (1), where the time harmonic exp(-iωt) and subscripts are omitted.

Here, A and B are the coefficients of forward and backward waves at the interface, x is relative to the origin in media 1 and 2, while in medium 3, x is relative to d, and k_x and k _y are the x (normal) and y (parallel) components of the wave vector. Note that $k_x^{\mathit{2}}$ +$k_y^{\mathit{2}}$ =εµω²/c² , where $c=1⁄\sqrt{{\mu }_0{\epsilon }_0}$ is the speed of light in vacuum, and ε ₀ and µ ₀ are the absolute permittivity and permeability of free space. Boundary conditions require that k_y be real and independent of x. The components of the time-averaged Poynting vector, $〈\mathbf{S}〉=\frac{1}{2}\mathrm{Re}\left(\mathbf{E}\times {\mathbf{H}}^{*}\right)$, can be obtained by applying Maxwell’s equations for the electric field; hence,

where η and β are the real and imaginary parts of k_x . The last terms in Eqs. (2) and (3) arise from the coupling between the forward and backward waves, and are in general nonzero. The direction of 〈S〉 of the combined wave can always be defined by a polar angle ϕ=arctan(〈S_y 〉/〈S_x 〉); in contrast, it is not always possible to define the angle of incidence or refraction, θ=arctan(k_y/k_x ) in the real space. The trajectory of 〈S〉 for given ω and k_y is a streamline, which defines the path of the net energy flow. Note that the dependence of µ is implicit in Eqs. (2) and (3), since k_x is a function of µ, and furthermore, A’s and B’s depend on k_x ’s. By setting A ₁=1 and B ₃=0, the coefficients A and B in each layer can be obtained from boundary conditions that tangential components of the magnetic and electric fields are continuous at the interface. Assuming the symmetric configuration (i.e., ε₁=ε₃), the coefficients are given by

where ξ₁=(k_1x /ε ₁+k_2x /ε₂ ), ξ₂ =(k_1x /ε₁ -k_2x /ε2), and $\zeta =\left({\xi }_1^2{e}^{-i{k}_{2x}d}-{\xi }_2^2{e}^{i{k}_{2x}d}\right)$. Note that above equations are not applicable if the incidence angle is exactly the same as the critical angle due to the mathematical singularity, and such a case will be discussed later. For transverse electric (TE) waves, the magnetic field is replaced by the electric field in Eq. (1), and ε’s are replaced by µ’s in Eqs. (2), (3), and (4). Several illustrative examples of the energy streamline method are given for TM waves in the following.

3. Results and discussion

Let us consider photon tunneling in the prism-DNG-prism configuration, as shown in Figs. 2(a) and 2(b), for different incidence angles or k_y ’s. The energy transports from the left to the right, and the trajectory of the Poynting vector in the three regions forms a zigzag path, especially when d≪λ. The x and y axes are normalized to the slab thickness d, which is λ/20 in Fig. 2(a) and λ/5 in Fig. 2(b). Here, λ is the wavelength in vacuum. All the streamlines are for positive k_y values and pass through the origin. With the dielectric prism (ε=2.25), the critical angle is θ _cr =41.81°. Causality requires that 〈S_x 〉 be positive; furthermore, when loss is neglected, 〈S_x 〉 is independent of x. Note that $〈S_y〉=\frac{k_y}{2\omega {\epsilon }_0}\mathrm{Re}\left(\frac{1}{\epsilon }\right){\mid H_z\mid }^2$ is opposite to k_y when Re(ε)<0, as in the DNG layer.

At the critical angle, the magnetic field in medium 2 propagates along the interface and does not decay in the x direction because k_x =0. In such case, tangential component of the electric field vector must be zero in medium 2 because electromagnetic wave is transverse to the wave propagation direction. It is obvious that there is no analytic solution of the electromagnetic fields that satisfy boundary conditions for both electric and magnetic fields if there is a transmitted wave in medium 3. The exact solution yields total reflection without tunneling. When the incidence angle is slightly perturbed from the critical angle, the solutions in the proximity limit approach the same result from either side as the incidence angle approaches the critical angle. Hence, the singularity at the critical angle can be removed numerically, so that transmittance is obtained as a continuous function of the incidence angle.

From Figs. 2(a) and 2(b), it is surprising to notice that at θ₁=θ _cr , when the phase refraction angle θ₂=90°, the energy refraction angle ϕ₂ is much less than 90°. Furthermore, the dash-dotted line in the slab separates the propagating-wave streamlines (inside the cone) from the evanescent-wave streamlines (outside the cone). The observation that the energy paths of propagating waves and evanescent waves are separated by a cone provides a new explanation of the photon tunneling phenomenon based on wave optics. When d=λ/5, the streamlines become wavy as seen in Fig. 2(b). The energy transmittance through the DNG slab, calculated by T=|A₃/A₂ |2, is labeled for each streamline except θ₁=θ _cr . At the critical angle, Airy’s formulae are used to obtain T by taking the limit k_x →0. The tunneling transmittance decreases rapidly as d increases.

 

Fig. 2. Energy streamlines for prism-DNG-prism and prism-SNG-prism configurations at various incidence angles: θ₁=20° (solid), 30° (dotted), 41.81° (dash-dot), and 50° (dashed). The prism has ε=2.25 and µ=₁, so θ1=41.81° corresponds to the critical angle for DNG in (a) and (b). Only evanescent waves exist in medium 2 for SNG in (c) and (d). The energy transmittance (T) from medium 1 to 3 is shown for each incidence angle. A different scale is used for the y axis in (d).

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By changing µ from -1 to 1 and plotting the energy streamline in Figs. 2(c) and 2(d), one can investigate photon tunneling through a SNG slab. Here, only evanescent waves exist in the slab because k_x is purely imaginary even at normal incidence. Energy is carried through medium 2 by coupled evanescent waves, whose path can be completely described by a streamline. The transmittance with a SNG slab is much smaller than that with a DNG slab for the same ε, and the beam shift in the y direction becomes very large in Fig. 2(d). Yet, Figs. 2(a) and 2(c) look alike. Alu and Engheta⁶ pointed out that, when d≪λ, propagating waves and evanescent waves are similar because both the sinusoidal function and the hyperbolic function are the same under the small-argument approximation. After some tedious derivations, we arrive at the following approximation for energy angles when d/λ→0:

assuming that only propagating waves exist in medium 1, and both ε₁ and ε₂ are real. Note that µ2 does not affect the TM wave results in the electrostatic limit, as noted by Pendry.¹ However, the effect of µ₂ becomes significant when d/λ>0.1, as demonstrated in Figs. 2(b) and 2(d). Here, Eq. (5) is always applicable in the proximity limit if the energy refraction angle ϕ₂ is numerically determined at the critical angle. While Poynting vector traces have been used in inhomogeneous media and around small spheres,^18,19 the application to planar layered structures here reveals some quite striking features of the energy transfer by coupled evanescent waves.

Both positive and negative phase-time shifts were noticed by Li²⁰ for an optically dense dielectric slab in air without evanescent waves. It is worthwhile to take a look at the energy streamlines for the vacuum-dielectric-vacuum configuration. For propagating waves, because the second term in Eq. (3) depends on x, the streamline (solid curve) exhibits wavelike features for d=λ, as can be seen from Fig. 3(a), where the solid curve is the streamline and the dashed lines are the traces of the wave vector. The lateral shift of the energy line is determined by point Q rather than P. When d/λ is reduced to 0.01 as shown in Fig. 3(b), the streamline is almost a straight line in each medium. However, Q becomes closer to the x axis than P, in contrary to Fig. 3(a). When d/λ≪1, Snell’s law determines θ₂ and Eq. (5) determines ϕ₂. The shift of Q with respect to P depends on the incidence angle, which can be positive or negative. Although lateral shifts for tunneling or transmission through a dielectric film have been extensively studied using beams of finite width, such as a Gaussian beam, or the derivative of the phase shift,^11–16,20 the present study uses plane waves of infinite extension to reveal a beam shift due to interference of propagating waves or coupled evanescent waves.

 

Fig. 3. The energy streamline for vacuum-dielectric-vacuum configuration at θ₁=30° when (a) d/λ=1 and (b) d/λ=0.01.

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Goos-Hänchen shift of tunneling accompanied with surface wave excitation has been studied in the attenuated total reflection configuration.^22,23 It was found that a giant lateral shift could occur due to surface wave excitation. In the following example, we apply the energy streamline method to investigate the lateral shift of beam when surface waves are excited in nanoscale radiation heat transfer. It is well known that radiation heat flux between two flat surfaces separated by a vacuum gap can be greatly enhanced due to photon tunneling when the gap thickness becomes smaller than the characteristic wavelength.²⁴ Furthermore, if the material supports surface phonon polaritons, which can be thermally excited because the resonance frequency is in infrared, the spectral heat flux distribution exhibits large peaks at certain frequencies in nearly monochromatic manner.²⁵ This is because surface polariton is a resonance phenomenon that can only be excited in a narrow spectral band for a given k_y . Alternatively, surface polariton can be excited in a narrow region of k_y for a given frequency.

Let us consider the two semi-infinite SiC slabs separated by a vacuum gap. Here, λ is set to be 10.55 µm, where the relative permittivity of SiC is ε_SiC=1.005+0.129i as calculated from the functional expression.²⁶ The spectral energy flux between two parallel surfaces via evanescent waves can be obtained using the fluctuation-dissipation theorem.^24,25 In the far field, only propagating waves with k_y <ω/c can transmit energy. In the near field, we must consider k_y from 0 to infinity because energy can be transferred by photon tunneling. An exchange function Z _evan can be used for tunneling transmittance and is given as^25,27

where $r_{23}^p$ $r_{21}^p$=(k_2x /ε₂-k_1x /ε ₁)/(k_2x /ε ₂+k_1x /ε₁) is the Fresnel reflection coefficient at the vacuum-SiC interface for the TM wave, since the contribution of the TE wave to energy transfer during tunneling is negligible. The calculated Z_evan is shown in Fig. 4(a) when the gap thickness is 1, 10, or 100 nm. At d=100 nm, for instance, Z _evan has a peak around k_y =46ω/c. As the thickness is reduced by a factor of 10, the peak position shifts to larger k_y values by approximately a factor of 10 due to similarity of Eq. (6) when k_y ≫ω/c. For the case of d=100 nm, the full-width-at-half-maximum (FWHM) of the peak is 31.4ω/c<k_y <60.8ω/c. This region is shaded on both the upper and lower panels of Fig. 4. Here, the portion of the spectral heat flux due to evanescent waves whose k_y falls in the FWHM is approximately 71%, suggesting that the enhanced energy transfer occurs with very large k_y values when surface phonon polaritons are excited. An important question is how large an area is needed for the surfaces to be considered as semi-infinite when energy is transferred from one to another medium via tunneling of surface waves. Since the energy streamline method allows the determination of the beam shift, it can be used to guide the selection of the surface area.

 

Fig. 4. Photon tunneling in the vacuum gap sandwiched by semi-infinite SiC slabs when surface waves are excited at d=1 nm, 10 nm, or 100 nm: (a) exchange function due to evanescent waves and (b) normalized lateral shift. The shaded area in (a) and (b) corresponds to 31.4ω/c<k_y <60.8ω/c.

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The lateral shift of the energy streamline (δ), normalized to gap thickness d, is plotted in Fig. 4(b) for the same structure considered in Fig. 4(a). Even for very large k_y values, the lateral shift can be obtained from the energy streamline method. Like the exchange function, the normalized lateral shifts also exhibit similarity because the energy streamline is nearly straight when en d/λ≪1 (i.e., in the proximity limit). The lateral shift, however, does not monotonically increase as k_y increases, as can be seen from Fig. 4(b). For d=100 nm, the ratio δ/d is 6.9 near the critical angle, i.e., k_y =ω/c. As k_y increases, the lateral shift goes up slightly and then decreases to the minimum of 2.4 at 35.6ω/c. When k_y further increases, the lateral shift abruptly goes to infinity. Because of the loss and surface wave in the case with SiC, the energy streamlines of coupled evanescent waves can be either inside or outside the cone formed by the streamline at the critical angle. At k_y =46ω/c where Z _evan shows a peak, the lateral shift is about 3d, and the corresponding energy refraction angle is approximately 72°. In the region covered by the FWHM of the exchange function, i.e., 31.4ω/c<k_y <60.8ω/c, the maximum lateral shift is δ=7.3d at k_y =60.8ω/c. To eliminate the edge effect, let us choose the lateral dimension of the surface to be seven times the lateral shift. This implies that the lateral dimension of the SiC slab should be approximately 50d for surface waves to tunnel through the vacuum gap with negligible edge effects. Furthermore, it should be noted that 71% of the energy is transferred at energy refraction angles between 67° (i.e., δ=2.4d) and 76° (i.e., δ=7.3d). This is also true when the vacuum gap thickness d is less than 100 nm due to similarity of the exchange function and the lateral shift.

4. Concluding remarks

For either propagating waves or evanescent waves, the Poynting vector in the near field is not parallel to the wave vector due to the coupling between forward and backward waves, and it is not a straight line even in vacuum. The energy streamline represents the direction of the net energy flow. For photon tunneling between dielectrics, propagating waves are squeezed towards the axis due to interference and form a cone around the centerline, giving an external conical space for the coupled evanescent waves to carry energy through the barrier. For photon tunneling between absorbing media, the streamlines of coupled evanescent waves can be either inside the cone or outside the cone depending on the magnitude of k_y . When d≪λ, the energy streamline is almost a straight line and the energy refraction angle ϕ₂ can be obtained from Eq. (5), allowing the determination of lateral shift of the energy path through photon tunneling. In the nanoscale radiation heat transfer between two parallel SiC slabs, the energy streamline method reveals that about 71% of energy is transferred inside a narrow φ₂ range between 67° and 76° via the tunneling of surface waves.