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


 To prevent the crosstalk between adjacent waveguides in photonic integrated circuits, the minimum thickness of the cladding layers is around half a wavelength, which imposes a fundamental limitation to further integration and miniaturization of photonic circuits. Here, we reveal that epsilon-near-zero claddings, either isotropic or anisotropic, can break the above bottleneck by prohibiting the crosstalk for the modes with magnetic field polarized in the z direction at a deep-subwavelength thickness (e.g., λ
 0/30, λ
 0 is the free-space wavelength), therefore bestowing ultra-compact waveguide systems. The physical origin of this remarkable effect attributes to the divergent impedance of epsilon-near-zero materials far beyond those of dielectric or epsilon-negative claddings. Through full-wave simulations and microwave experiments, we have verified the effectiveness of the ultrathin epsilon-near-zero cladding in crosstalk prohibition. Our finding reveals the significant impact of impedance difference in waveguide designs and opens a promising route toward ultra-compact photonic chips.


Dispersion relation and coupling length of two coupled slab waveguides
We consider two coupled slab waveguides with a width of  of each waveguide and an edge-to-edge separation of , as schematically shown in Figure S1(a).We assume the waveguide core (or cladding) is characterized by a relative permittivity tensor )).The field (), i.e.   (  ) for the mode with magnetic field polarized in the  direction (the mode with electric field polarized in the  direction), in different regions can be written as, where  Then, based on the   and   from Equation S3-S4 (or Equation S5-S6) for the modes with magnetic field polarized in the  direction (or the mode with electric field polarized in the  direction), the coupling length   between the two waveguides can be evaluated as (see e.g.J. Opt. Soc. Am. A 11, 963 (1994))

Suppression of crosstalk for high-order waveguide modes
In the main text, near-complete suppression of waveguide crosstalk by using ENZ claddings is demonstrated for the fundamental mode with magnetic field polarized in the  direction.Here, Figure S2 demonstrates that for high-order modes, the waveguide crosstalk can also be suppressed by isotropic ENZ or anisotropic ENZ claddings.For comparison, SiO2 and ENG claddings are compared here, showing strong waveguide crosstalk.

Suppression of crosstalk in bending waveguides
In the main text, near-zero crosstalk in straight slab waveguides with ENZ claddings is demonstrated.Here, we show that the ENZ claddings can also be exploited to significantly suppress crosstalk in bending waveguides.Figure S3(

Mu-near-zero (MNZ) and epsilon-and-mu-near-zero (EMNZ) claddings
In the main text, we find that ENZ claddings can be used to realize near-zero waveguide crosstalk for the mode with magnetic field polarized in the  direction.Here, we show that mu-near-zero (MNZ) claddings can eliminate waveguide crosstalk for the mode with electric field polarized in the  direction, and epsilon-and-mu-near-zero (EMNZ) media can eliminate waveguide crosstalk for both the two kinds of modes.First, we analyze the wave impedance of the MNZ and EMNZ media.Figure S4(a ) presents the impedance contrast between Si and MNZ medium (relative permittivity   = 1, relative permeability   = 0.01) for the polarization with electric field along the  direction (left), EMNZ medium (  =   = 0.01 ) for the polarization with electric field along the  direction (middle) and the polarization with magnetic field along the  direction (right).The solid and dashed lines denote, respectively, the real and imaginary parts of .We see that the wave impedance of the MNZ and EMNZ media for the polarization with electric field along the  direction becomes vanishingly small with increasing the   compared with that of Si, while the wave impedance of the EMNZ medium for the polarization with magnetic field along the  direction tends to be infinitely larger.Such an extreme impedance can prevent light penetrating into the MNZ and EMNZ media, thus suppressing the evanescent waves.
Due to this extraordinary property, when the MNZ (EMNZ) medium is utilized as claddings in waveguides, the waveguide crosstalk for the modes with electric field polarized in the  direction (both the modes with electric field polarized in the  direction and the mode with magnetic field polarized in the  direction) can be eliminated.
For verification, Figure S4 Here, we study the loss effect of indium tin oxide (ITO) claddings through investigating the coupling between two Si waveguides (with  = 460nm and  = 80nm).Figure S5 shows the distributions of electric-field amplitude in two coupled Si slab waveguides when the ITO cladding possesses different imaginary part of permittivity (or loss tangent).Simulation results show that the crosstalk can still be reasonably suppressed.
The similar behavior is also observed in two coupled Si strip waveguides (Figure S6).
These results manifest that the waveguide crosstalk suppression using the ITO claddings is relatively robust against the material loss.

Figure S1 .
Figure S1.(a) Illustration of two coupled slab waveguides.(b) Distributions of   for the fundamental symmetric (left) and antisymmetric (right) modes with magnetic field polarized in the  direction at  0 = 1550nm.The waveguide system consists of Si cores and SiO2 claddings with  = 460nm and  = 300nm.
,cl ), where   and   denote propagation constants of symmetric and antisymmetric modes, respectively. ( = 0,1,2, … ) refers to the order of waveguide modes.For illustration, Figure S1(b) presents the distributions of   of the fundamental symmetric (left) and antisymmetric (right) modes in two coupled waveguides composed of Si cores and SiO2 claddings with  = 460nm and  = 300nm at  0 = 1550nm.Similarly, for the mode with electric field polarized in the  direction, considering the continuity of   on the interfaces, we obtain the dispersion relations of symmetric and antisymmetric modes as, modes.Here,  = √( 0 2  ,co −  () 2 / ,co ) ,co , and  = √ ,cl (  () 2  ,cl −  0 2  ,cl ).

Figure S2 .
Figure S2.Distributions of   of different orders of the modes with magnetic field polarized in the  direction modes in a coupled-waveguide system composed of Si cores and four different types of claddings (i.e., SiO2, ENG, isotropic ENZ, anisotropic ENZ) with  = 1000nm and  = 50nm at  0 = 1550nm.
FigureS3(b).We see that transmission is low for SiO2 or ENG claddings.Interestingly, the transmission for the deep-subwavelength isotropic ENZ and anisotropic ENZ claddings is near-100%, irrespective of the bending radius .

Figure S3 .
Figure S3.(a) Distributions of   in a bending waveguide ( = 460nm,  = 50nm,  = 400nm) with isotropic ENZ claddings.(b) Transmission (port 1 to port 2) in the bending waveguide with SiO2, ENG, isotropic ENZ or anisotropic ENZ claddings.(c) Distributions of electric-field amplitude in 43.1μm-long sinusoidally curved (upper) and Z-shape (lower) waveguides with isotropic ENZ claddings at  0 = 1550nm.We set  = 460nm and  = 50nm.The light signal is sent to the left port of the upper Si waveguide.

Figure S4 .
Figure S4.(a) Wave impedance contrast between Si and MNZ medium (  = 1,   = 0.01, left) for the polarization with electric field along the  direction, Si and EMNZ medium (  =   = 0.01) for the polarization with electric field along the  direction (middle) and the polarization with magnetic field along the  direction (right).The solid and dashes lines denote, respectively, the real and imaginary parts of impedance contrast as the function of   .[(b)-(d)] Distributions of electric field for the fundamental (left,  = 460nm,  = 50nm) and 1st-order (right,  = 800nm,  = 50nm) waveguide modes in two coupled Si slab waveguides with (b) MNZ claddings for the modes with electric field polarized in the  direction, (c) EMNZ claddings for the modes with electric field polarized in the  direction, (d) EMNZ claddings for the mode with magnetic field polarized in the  direction.The light signal of  0 = 1550nm is sent to the left port of the upper Si waveguide.
Figure S5.Distributions of electric-field amplitude in two 10μm-long coupled Si slab waveguides with ITO claddings of different loss tangent.

Figure S6 .
Figure S6.Distributions of electric-field amplitude in two 10μm-long coupled Si strip waveguides with ITO claddings of different loss tangent.

Figure
Figure S7 discusses the waveguide crosstalk suppression by non-ideal ENZ claddings, whose relative permittivity   is not very close to zero.The simulation results show that excellent performance of crosstalk suppression can be obtained as long as   < 0.1.
,co ) ,co ,  cl = √ ,cl (  1 ,  2 and  3 are amplitudes of fields.The sign + ( − ) denotes the symmetric (antisymmetric) eigenmode.Here, we note that the | cl | denotes the decay rate of evanescent waves in claddings.When  ,cl ≫  ,cl , we have | cl | ≫  for non-zero .As a consequence, for anisotropic ENZ claddings with  ,cl ≫  ,cl → 0, we would have an extremely 2  ,co −  2 / ,co ) ,co ,  cl = √ ,cl ( 2  ,cl ) for the mode with electric field polarized in the  direction. 0 is the wave number in free space, and  is the propagation constant.