Crosstalk prohibition at the deep-subwavelength scale by epsilon-near-zero claddings

2.1 The critical role of impedance

The crosstalk between adjacent waveguides originates from the evanescent waves in the cladding. Traditionally, in order to guarantee negligible crosstalk, the cladding thickness is preferred to be large enough such that the amplitude of evanescent waves decays to a negligible level across the thickness. Usually, this can only be achieved by a thickness larger than λ ₀/2. Recent studies suggest that increasing the anisotropy of the claddings could further enhance the decay rate of the evanescent waves and thus efficiently suppress the crosstalk [3], [4], [5], [6], [7], [8], [9], [10], [11]. Here, we reveal that besides the decay rate, the impedance of claddings also plays an important role suppressing crosstalk. The impedance contrast between the core and cladding determines the amplitude of the evanescent waves on the interface between them. Since the ENZ cladding has a substantial impedance for the polarization with magnetic field along the z direction, the amplitude of the evanescent waves is significantly reduced on the interface between the ENZ cladding and the core. In this way, although the decay rate inside the ENZ cladding is not extremely large, the amplitude of evanescent waves can still be reduced enormously. When the relative permittivity of the ENZ cladding approaches zero, the impedance approaches infinity for all modes with magnetic field polarized in the z direction, and the mission of crosstalk prohibition can be accomplished by a near-zero thickness, which is in the deep-subwavelength scale. Our finding shows that the impedance contrast between the core and cladding has a significant impact on the required thickness of cladding for crosstalk prohibition, besides the decay rate that has been exploited extensively.

To clearly illustrate the importance of impedance in the claddings, we consider the physical model of a waveguide with a transmission core of silicon (Si) with relative permittivity ɛ _Si = 12 sandwiched by the claddings of nonmagnetic media with relative permittivity tensor ε x 0 0 ε y at the wavelength of λ ₀ = 1550 nm, as illustrated in Figure 1a. The right panel graph shows the origin of total reflection, where the evanescent waves are excited in the claddings, leading to the undesired crosstalk. Now, we consider the polarization with magnetic field along the z direction. The dispersion relation and wave impedance (i.e., the ratio of the tangential electric and magnetic fields) are expressed as k y 2 / ε x + k x 2 / ε y = k 0 2 and Z = k 0 2 − k x 2 / ε y / ( ω ε 0 ε x ) [58], respectively. Here k ₀ is the wave number in free space. k _x and k _y are the x- and y-components of wave vector, respectively. ω is the angular frequency, and ɛ ₀ is the permittivity of free space. The left panel graphs in Figure 1b–e show, respectively, the dispersion curves (top) and impedance contrast Z/Z _Si (bottom) of four types of claddings, i.e. dielectric (SiO₂, ε x = ε y = ε Si O 2 = 2.07 ), ENG media (ɛ _x = ɛ _y = ɛ _ENG = −5), isotropic ENZ media (ɛ _x = ɛ _y = 0.01) and anisotropic ENZ media (ɛ _x = 1, ɛ _y = 0.01). Here, Z _Si ( = ε Si k 0 2 − k x 2 / ( ω ε 0 ε Si ) ) is the wave impedance of Si. The solid and dashed lines denote, respectively, the real and imaginary parts of k _y (top, grey) or Z/Z _Si (bottom, red) as functions of k _x . The blue lines correspond to the dispersion curve of Si, which is a circle with a radius of ε Si k 0 .

Figure 1:

Physical properties of the ENZ claddings. (a) Schematic diagrams of a Si waveguide (left) and the mechanism of total reflection (right). The left panel graphs in (b)–(e) show the dispersion curves (top) and the impedance contrast (bottom) of four types of claddings, i.e., (b) SiO₂, (c) ENG medium, (d) isotropic ENZ medium, and (e) anisotropic ENZ medium. The solid and dashed lines denote the real and imaginary parts of k _y (top, grey) and log(Z/Z _Si) (bottom, red) as functions of k _x . The blue lines correspond to the dispersion curve of Si. The right panel graphs in (b)–(e) show the distributions of normalized magnetic fields H / | H max | inside the waveguide consisting of a 460 nm-width core of Si and four types of claddings for the fundamental mode with magnetic field polarized in the z direction. H _max is the maximal magnetic field inside the waveguide.

In the case of SiO₂ cladding (Figure 1b), the dispersion curve is a hyperbola in the regime of ε Si O 2 k 0 < | k x | ≤ ε Si k 0 , where both k _y and Z are imaginary numbers. The imaginary k _y is the signature of the total reflection as well as the formation of evanescent waves inside the cladding [1, 2], the magnitude of which determines how fast the evanescent waves decay inside the cladding. When the cladding is composed of ENG media (Figure 1c), the dispersion curve is a hyperbola of a different orientation, and the k _y is imaginary for any k _x because the ENG media cannot support any propagating wave. In the cases of the isotropic ENZ (Figure 1d) and anisotropic ENZ (Figure 1e) claddings, the dispersion curves become linear lines, which is reasonable as the parameter of ɛ = 0 or ɛ _y = 0 represents a critical transition point between the dielectric and ENG cases. It is seen that the k _y is imaginary for any k _x ≠ 0, indicating that total reflection occurs except for normal incidence. This property infers that ENZ media can serve as the claddings of waveguides with an excellent tolerance on the propagation constants of the waveguide modes, i.e., β = k _x ≠ 0. Interestingly, the imaginary k _y in Figure 1e diverges rapidly with increasing k _x . This means that the anisotropic ENZ claddings could have an extreme decay rate. This is induced by the extreme anisotropy (i.e., ɛ _x ≫ɛ _y ) and the details of derivation are presented in Supporting Information.

However, the decay rate is not the whole story of how good claddings work. Besides the decay rate, the impedance contrast between the cladding and core is another crucial factor that determines the effectiveness of claddings, but was rarely discussed in the literature. Unlike the decay rate that determines how fast the evanescent waves decay in the cladding, the impedance contrast determines the amplitude of the evanescent waves on the interface between the cladding and core. In Figure 1b–e, it is seen that the impedance contrast Z/Z _Si of both isotropic and anisotropic ENZ claddings is several orders higher than that of the SiO₂ and ENG claddings. Such a colossal impedance contrast for the polarization with magnetic field along the z direction is a unique result of near-zero ɛ or ɛ _y [59] which diverges when ɛ or ɛ _y approaches zero. Due to the large impedance contrast, despite the fact that the decay rate of isotropic ENZ cladding is comparable to that of the SiO₂ and ENG claddings, the effectiveness of isotropic ENZ cladding is much better. For anisotropic ENZ cladding, the combination of significant decay rate and large impedance contrast makes a super cladding.

The right panel graphs in Figure 1b–e show the distributions of normalized magnetic fields inside a waveguide consisting of a Si core with 460 nm width and four types of claddings (i.e., SiO₂, ENG, isotropic ENZ, and anisotropic ENZ media). The fundamental mode with magnetic field polarized in the z direction is excited. We can clearly see that the magnetic fields are stable on the Si-SiO₂ and Si-ENG interfaces, but drop abruptly on the Si-isotropic ENZ and Si-anisotropic ENZ interfaces. This is a fundamental difference between ENZ claddings and all other types of claddings. Due to this physical mechanism, the ENZ claddings in principle can eliminate the crosstalk with an ultra-thin thickness of deep-subwavelength scale, which is an impossible for the traditional dielectric/air or ENG claddings.

The functionality of the significant impedance contrast can be easily understood from the reflection coefficient (regarding to the magnetic field) at the Si-ENZ interface, which is r = (Z _Si − Z)/(Z _Si + Z). When Z ≫ Z _Si, we have r = −1. This implies that the incident and reflected waves have the same amplitude, but a π phase difference, thus leading to a near-zero total magnetic field at the interface. Consequently, the magnetic field inside the ENZ claddings must be near-zero, as implied by the boundary conditions, and therefore the evanescent waves inside the ENZ claddings almost disappear.

2.2 Analysis of waveguide coupling

We perform theoretical analysis and numerical simulations to confirm the remarkable functionality of the ENZ claddings. For simplicity, we consider two Si waveguides with the same width w and length l separated by a separation distance of g, as schematically shown in Figure 2a. The operating wavelength is chosen as λ ₀ = 1550 nm. Figure 2b shows the normalized coupling length L _c/λ ₀ of the fundamental mode with magnetic field polarized in the z direction in the case of w = 460 nm for the SiO₂, ENG, isotropic ENZ, and anisotropic ENZ claddings with different g. The coupling length L _c is defined such that the power would transfer completely from the first waveguide to the adjacent waveguide after a propagating distance of L _c. Here, L _c is calculated based on the coupled-mode theory. According to the coupled-mode theory, the mth-order mode dispersion of the coupled waveguides with a nonmagnetic cladding of ε x 0 0 ε y can be derived as (see Supporting Information),

for symmetric and antisymmetric modes, respectively. Here k = ε Si k 0 2 − β s ( a ) 2 and α = ε x β s ( a ) 2 ε y − k 0 2 , where β _s and β _a denote the propagation constants of symmetric and antisymmetric modes, respectively. m (= 0, 1, 2, …) refers to the order of waveguide modes.

Figure 2:

Prohibition of crosstalk between two Si waveguides. (a) Schematic layout of two Si waveguides with width w, separation distance g, and length l. (b) Normalized coupling length L _c/λ ₀ of the fundamental mode with magnetic field polarized in the z direction in the case of w = 460 nm for four types of claddings with different g. (c) Distributions of electric-field amplitude E for the four types of claddings. The fundamental mode of the upper Si waveguide is excited on the left port, as illustrated by the inset. Here, we set w = 460 nm, g = 50 nm (∼λ ₀/30, marked by stars in (b)) and l = 10 μm. (d) L _c/λ ₀ as a function of the relative permittivity ɛ of isotropic cladding for g = 20 nm (red), 50 nm (green), and 150 nm (blue). The inset shows the range of 10⁻⁵ ≤ ɛ ≤ 0.002.

Then, the coupling length L _c between the two waveguides can be evaluated as [60].

Based on Equations (1) and (2), we calculate and plot the L _c for different types of claddings in Figure 2b. It is clearly seen that the L _c of the ENZ claddings is several orders larger than that of the SiO₂ and ENG cladding, implying that the crosstalk between the two Si waveguides is significantly reduced by the ENZ claddings. This is because under the limit of ɛ _y → 0, we have

Equation (3) indicates that L _c → ∞, irrespective of g, which is also valid for the isotropic case of ɛ _x = ɛ _y = ɛ → 0. It is noteworthy that this conclusion is valid for any waveguide mode with magnetic field polarized in the z direction as the equality β _s = β _a is m-independent (see numerical proofs in Supporting Information).

Then, we perform full-wave finite-element simulations using the commercial software COMSOL Multiphysics to verify the prohibition of crosstalk, as shown in Figure 2c. Here, the width is w = 460 nm, the separation distance is g = 50 nm, and the length is l = 10 μm. We note that g ∼ λ ₀/30, indicating deep-subwavelength claddings. The fundamental mode with magnetic field polarized in the z direction in the upper waveguide is excited on the left input port, as illustrated by the inset. Figure 2c shows the distributions of the electric-field amplitude for the four types of claddings. Clearly, strong coupling appears between the two Si waveguides for the SiO₂ or ENG claddings. The coupling lengths are observed to be ∼ 10.7 μ m for the SiO₂ cladding and ∼ 3.2 μ m for the ENG cladding, which agree well with the theoretical prediction in Figure 2b ( ∼ 10.5 μ m and ∼ 3.3 μ m ). Remarkably, for the cases of isotropic ENZ and anisotropic ENZ claddings, the coupling almost completely disappears and there is no wave or signal in the adjacent waveguide. The theoretical coupling length is 1.8 × 10³μm for the isotropic ENZ cladding and 6.546 × 10⁴μm for the anisotropic ENZ cladding, far beyond that of the SiO₂ and ENG claddings. We should emphasize that when ɛ (isotropic case) or ɛ _y (anisotropic case) of claddings approaches zero, the coupling length goes to infinity, i.e., L _c → ∞. For verification, we plot the coupling length L _c for an isotropic cladding with different thickness g in Figure 2d, which confirms that L _c → ∞ when ɛ → 0, no matter how small g is.

2.3 Microwave experiment

In the following, we experimentally demonstrate the prohibition of crosstalk by using an isotropic ENZ cladding with deep-subwavelength thickness in the microwave regime. Here, the low-loss isotropic ENZ cladding is realized by waveguide metamaterials based on parallel-plate waveguides (PPWs) [26, 36, 44, 61], [62], [63], [64], [65]. Figure 3a shows the schematic of the experimental setup. Two parallel aluminum plates of the PPW are separated vertically by a distance of h = 10.3 mm. Now, we consider the TE₁ mode in this PPW, which is dominated by E _y , H _x , and H _z , as illustrated in the left panel of Figure 3b. On the middle plane of the PPW, there are only non-zero E _y and H _z . Under this circumstance, the TE₁ mode-dominated PPW can be viewed as a two-dimensional effective medium where the polarization of magnetic field is along the z direction (right panel of Figure 3b). In this way, we can effectively regard the experimental PPW model with TE₁ mode shown in Figure 3a as the aforementioned waveguide system where the magnetic field is polarized in the z direction. The effective relative permittivity of the waveguide metamaterial is [26, 36, 44, 61], [62], [63], [64], [65].

where f is the operating frequency, c is the speed of light in vacuum, and ɛ _r is the relative permittivity of the filling dielectric inside the PPW. From Equation (4), we see that for fixed h and f, the ɛ _eff can be adjusted to be negative, zero, and positive values through engineering the ɛ _r .

Figure 3:

Experimental verification. (a) Schematic of the experimental setup of a PPW. Inside the PPW, two alumina bars are placed along the x direction working as two dielectric waveguides. Periodic thin copper strips are vertically wrapped around the alumina bars to eliminate the fundamental TEM mode. A horizontally oriented antenna is placed nearby the left port of the upper alumina bar to excite the TE₁ mode. (b) Illustrations of TE₁ mode in a PPW (left) and an effective two-dimensional medium for the polarization with magnetic field along the z direction (right). (c) Effective relative permittivity ɛ _eff as a function of ɛ _r of the filling material inside the PPW at 9.04 GHz. (d) Photograph of the experimental sample and antenna. (e) Zoomed-in photograph of the thin copper strips. (f) Simulated distributions of |E _y | on the middle plane of the PPW for air (effectively ENG, left) and PMMA (effectively isotropic ENZ, right) claddings. (g) Experimental measured (lines) and simulated (dots) normalized |E _y | to its maximal value |E _y,max| on the exiting surface for PMMA (effectively isotropic ENZ) and air (effectively ENG) claddings with g = 5.2 mm.

To implement the waveguide system with isotropic ENZ claddings, we utilize two 280 mm-long alumina (ɛ _r = 9.2) bars along the x direction with the same width of 12 mm and a separation distance of g = 5.2 mm in the y direction. They work as two coupled dielectric waveguides. Thin copper strips of area 0.05 mm² and a pitch of 2.5 mm are placed vertically around the alumina strips to eliminate the fundamental TEM mode (with electric field along the z direction) of the PPW [36, 44, 61], [62], [63], [64], [65], as shown in Figure 3e. The working frequency is chosen to be f = 9.04 GHz. According to Equation (4), we find that the ɛ _eff of this waveguide metamaterial filled with alumina (air) are 6.6 (−1.6). This indicates that this waveguide metamaterial filled with air can effectively serve as an ENG medium, which has been proposed to realize low-loss effective surface plasmon polaritons [61, 62]. Then, we insert a PMMA bar (ɛ _r = 2.6) between the two alumina bars. The ɛ _eff of the PMMA-filled waveguide metamaterial turns out to be near-zero (Figure 3c), thus it can effectively serve as an isotropic ENZ cladding.

In the microwave experiment, we use a horizontally oriented antenna to excite the TE₁ mode of the PPW from the left port of the upper alumina bar and measure the electric field E _y on the exiting surface of area 20 mm × 80 mm, as illustrated in Figure 3a. The measurement is performed using a PNA Network Analyzer N5224B. The photos of experimental samples and antenna are presented in Figure 3d. In the experiments, in order to avoid the situation that the mode has returned back in the first channel at the end of waveguide, the length of the fabricated waveguide is chosen to be shorter than the coupling length, and at the same time, long enough to show the difference between the models with air (effectively ENG) and PMMA (effectively isotropic ENZ) claddings. Figure 3f shows the simulated distributions of |E _y | for air (left) and PMMA (right) claddings with g = 5.2 mm on the middle plane of the PPW. We can see a clear difference of field distributions at the end of the waveguides with air and PMMA claddings, which confirms the crosstalk suppression by the PMMA cladding. Figure 3g displays the experimentally measured normalized |E _y | along the exiting surface along the y direction (dots), which agrees well with the simulation results (lines), for both PMMA (effectively isotropic ENZ) and air (effectively ENG) claddings. In the case of ENG cladding, two obvious peaks appear where the two alumina waveguides are located, indicating evident crosstalk between the two waveguides. While in the case of isotropic ENZ cladding there is only one peak located in the input waveguide, indicating that the crosstalk is low. These experimental and simulation results have verified the functionality of crosstalk prohibition by the ultra-thin isotropic ENZ cladding (∼λ ₀/6).

We note that the mismatch between experimental and simulation results could be caused by two main factors. First, it is hard to excite the modes in the channel 1 only, while completely preventing the excitation of modes in the adjacent channel 2 on the waveguide port, because the two channels are very close to each other. Second, the imperfection of the alumina strips may cause scattered waves that would couple into the adjacent waveguide.

2.4 Ultra-compact waveguide arrays

We utilize the above crosstalk prohibition based on ENZ claddings to realize ultra-compact waveguide arrays. Figure 4a illustrates the schematic of an array of Si waveguides. Each waveguide has a width w, and their separation distance (edge-to-edge) is g. Based on the coupled-mode theory, the dispersion relation of the waveguide array with a pitch of a (= w + g) is expressed as [17, 66].

where β is the propagation constant for an isolated waveguide, and C = (β _s − β _a )/2 is the coupling coefficient between two waveguides. Here β _s (or β _a ) is the propagation constant of the symmetric (antisymmetric) mode in the waveguide array, which equals to k _x when k _y a = 0 (or k _y a = ±π), as marked by the arrows in Figure 4b. Equation (5) indicates that a flat dispersion curve, i.e., k _x = β, appears in the absence of coupling (i.e., C = 0), which is a criterion of zero crosstalk in a waveguide array.

Figure 4:

Ultra-compact waveguide arrays. (a) Schematic diagram of a waveguide array. (b) Dispersion relation of the waveguide array with SiO₂ (green), ENG (red), isotropic ENZ (blue) and anisotropic ENZ (black) claddings at λ ₀ = 1550 nm. The purple (brown) marks denote the symmetric (antisymmetric) mode at k _y a = 0 (k _y a = ±π). (c) Distributions of E in the waveguide arrays with the SiO₂, ENG, isotropic ENZ, and anisotropic ENZ claddings. The fundamental waveguide mode with magnetic field polarized in the z direction is excited in the central waveguide.

Figure 4b presents the exact dispersion relation of the Si waveguide array at λ ₀ = 1550 nm calculated by using the transfer matrix method [67]. Here, we set the widths of the Si waveguide and cladding as w = 460 nm and g = 50 nm, respectively. Four types of claddings, i.e., SiO₂, ENG, isotropic ENZ, and anisotropic ENZ claddings with the same parameters in Figure 1, are applied here. It is clearly seen that k _x is independent of k _y for the isotropic ENZ and anisotropic ENZ claddings, indicating that the crosstalk is nearly zero in the waveguide array with such deep-subwavelength (∼λ ₀/30) ENZ claddings. On the contrary, k _x varies with k _y for the SiO₂ or ENG claddings, indicating the existence of coupling between waveguides. For further verification, in Figure 4c, we show the simulated distributions of electric-field amplitude in a 40 μm-long waveguide array by exciting the fundamental mode with magnetic field polarized in the z direction in the central waveguide. The simulation results clearly show that the wave propagating in the central waveguide couples to the adjacent waveguides and spreads out for the SiO₂ or ENG claddings, but remains confined in the central waveguide for the isotropic ENZ and anisotropic ENZ claddings. Therefore, the crosstalk in the waveguide array is reduced to zero with the ENZ claddings.

2.5 On-chip strip waveguides

In the following, we would like to show a practical implementation of on-chip Si strip waveguides using a practical ENZ medium, i.e., indium tin oxide (ITO) claddings. A 80 nm-thick ITO film is characterized by a relative permittivity around 0.01 + 0.45i at λ ₀ = 1550 nm [68], which can approximately serve as an isotropic ENZ medium. Figure 5a illustrates the cross-section of the Si strip waveguide consisting of two Si strips (width w = 460 nm, height h) separated by a cladding layer (separation distance g = 80 nm, height h). Based on Equation (2), we calculate the normalized coupling length L _c /λ ₀ in this waveguide through evaluating β _s and β _a in simulations. In Figure 5b, we see that the coupling length L _c /λ ₀ of the fundamental TE mode in the waveguide with ITO claddings is much larger than those in the waveguides with SiO₂ or ENG claddings, and it is almost irrespective of the waveguide height h, indicating excellent performance of crosstalk suppression by the ITO claddings. Figure 5c shows the top view of simulated E _y -distribution on the xy plane for claddings of SiO₂ (upper), ENG (middle), and ITO (lower), respectively. Clearly, at the deep-subwavelength thickness of g ∼ λ ₀/20, evident crosstalk appears for SiO₂ and ENG claddings, while the crosstalk is largely suppressed by the ITO claddings.

Figure 5:

On-chip strip waveguide. (a) Schematic layout of the cross-section of two coupled on-chip Si waveguides with width w, separation distance g and height h. (b) Normalized coupling length L _c/λ ₀ as a function of h for SiO₂, ENG, and ITO (practical ENZ) claddings in the case of w = 460 nm, g = 80 nm. (c) Distributions of E _y in two coupled Si strip waveguides with SiO₂, ENG, and ITO claddings at λ ₀ = 1550 nm. The fundamental TE mode is excited in the upper Si waveguide on the left port. The relevant parameters are w = 460 nm, g = 80 nm, and h = 220 nm.