## Introduction

One of the long-standing goals of nanophotonics is the integration of electronic and photonic circuitry on a single CMOS chip for applications ranging from information processing and data centers to massively parallel sensing1,2,3,4,5,6,7,8,9,10,11. This necessarily requires miniaturization with low power consumption in optical interconnects, active as well as passive photonic devices. There are two major figures of merit in designing photonic devices for a densely integrated circuit. One is the cross-talk, which occurs due to the field overlap of two adjacent photonic waveguides, and the second is the radiation loss at sharp bends which limits the integration density6.

Plasmonic waveguides can strongly reduce cross-talk and bending loss owing to the sub-diffraction nature of light coupling to the free electrons of metals12,13,14,15,16,17,18,19. However, the large ohmic loss of metals restricts the application of plasmonic structures for photonic integration20,21. Over the last decade, many efforts have been made to miniaturize photonic components using all-dielectric structures22,23,24,25,26,27,28,29,30,31,32,33,34,35,36. Figure 1 illustrates a few classes of dielectric waveguides for light confinement in photonic chips. Strip waveguides, the most common type of waveguides for routing light in a silicon chip, are composed of a silicon channel surrounded by silicon oxide6 (Fig. 1a). Owing to the high contrast between the refractive index of the core and the cladding, light is confined inside the core as a result of total internal reflection (TIR). However, the mode size is seen to increase as we reduce the core size which hampers the use of strip waveguides to further miniaturize photonic circuits37. Photonic crystal waveguides can confine light inside a line defect due to Bragg reflection40 (Fig. 1b). These waveguides perform efficiently at very sharp bends41, however, the integration density is limited as the periodicity of Bragg reflectors is on the order of the wavelength and it cannot be perturbed by another waveguide nearby42. Additionally, slot waveguides have been proposed to confine light inside a sub-wavelength low-index gap surrounded by high-index dielectric rods43 (Fig. 1c). To satisfy the continuity of the normal component of the displacement current at the high-contrast interface, the electric field peaks inside the gap, leading to light confinement but at the cost of skin-depth expansion in the cladding. This causes cross-talk between adjacent waveguides and radiation loss at sharp bends in dense photonic-integrated circuits.

Several alternative approaches inspired from atomic physics44,45, transformation optics46, and inverse design algorithms47,48 have also been proposed to minimize the cross-talk and the bending loss. However, these techniques add design complexity and often require modification of the core as well as cladding46. They are not implementable on a large-scale and cause propagation loss fundamentally limiting device performance. Thus, a new low-loss and scalable platform is needed for CMOS-foundry compatible dense photonic integration with low cross-talk and reduced bending loss.

In this paper, we demonstrate a platform that is fundamentally different from the existing approaches for designing CMOS compatible, ultra-compact, and low-loss waveguides using all-dielectric anisotropic metamaterials. Our approach works based on photonic skin-depth engineering of evanescent waves in the cladding using a recently proposed degree of freedom in TIR37,49. To describe the light confinement mechanism in our waveguides, first, we experimentally demonstrate the phenomenon of relaxed TIR in anisotropic metamaterials. These relaxed conditions allow the control of evanescent wave decay, which is the fundamental origin of cross-talk and bending loss in silicon photonic devices. As illustrated in Fig. 1d, we use these anisotropic metamaterials as a cladding for on-chip dielectric waveguides fabricated on a monolithic silicon-on-insulator (SOI) platform. As a result, cross-talk is reduced down to −30 dB in the photonic circuit. Furthermore, we use a transformation optics approach to show that the radiation loss at sharp bends is strongly influenced by the skin-depth in the cladding. We experimentally show that the anisotropic metamaterial cladding can simultaneously reduce the bending loss at sharp bends up to three times compared with conventional silicon strip waveguides. We clarify the counterintuitive nature of light confinement in our approach compared with existing photonic crystal, slot waveguide, and graded index waveguide methods38,43,50,51,52,53. Our work shows that all-dielectric anisotropy on-chip presents a scalable route to simultaneously improve cross-talk and bending loss with propagation loss as low as ≈3.67 dB/cm. For completeness, we show the improvements in figures of merit of our achieved platform with recent state-of-the-art photonic designs (see Table 1).

## Results

### Relaxed TIR

We demonstrate that only a single component of the dielectric tensor governs TIR in anisotropic media opening a new degree of freedom to control evanescent waves. In conventional TIR, if n1>n2 (n1 and n2 are the refractive index of the medium 1 and medium 2, respectively) and the incident angle is greater than the critical angle ($$\theta_{\rm{c}} = \mathop {{\mathrm {sin}}}\nolimits^{ - 1} \left( {{{n}}_2/{{n}}_1} \right)$$), light is reflected back to the first medium and decays evanescently in the second medium. However, we have recently found that the TIR conditions at the interface of an isotropic and an anisotropic dielectric are relaxed to37,54:

$$n_1 > \sqrt {\varepsilon _{2x}} ,\;p\;{\mathrm{polarization}}$$
$$n_1 > \sqrt {\varepsilon _{2y}} ,\;s\;{\mathrm{polarization}}$$
(1)

where $$\left[ {\varepsilon _{2x}\varepsilon _{2y}\varepsilon _{2z}} \right]$$ is the permittivity tensor of the second medium, and the interface between the two media lies on the yz plane. As a result, the critical angle for s and p polarizations differ:

$$\theta_{\rm{c}} = \mathop{{\mathrm {sin}}}\nolimits^{-1} \left({\sqrt {\varepsilon_{2x}}/{n}_1} \right),\;p\;{\mathrm{polarization}}$$
$$\theta_{\rm{c}} = \mathop{{\mathrm {sin}}}\nolimits^{-1} \left({\sqrt {\varepsilon _{2y}} /n_1} \right),\;s\;{\mathrm{polarization}}$$
(2)

We emphasize that these relaxed conditions for p polarized incident light allow us to arbitrarily increase or decrease the permittivity in the z direction, whereas still preserving TIR. This hitherto un-utilized degree of freedom can thus be used to control the skin-depth of evanescent waves. If $$\varepsilon _{2z} \gg 1$$, evanescent waves decay faster than in vacuum allowing for strong light confinement inside dielectric waveguides37,49. In this limit, note that the averaged index in anisotropic medium 2 can be larger than the refractive index in isotropic medium 1 yet the light will be totally reflected above the critical angle.

We implement this anisotropy using Si/SiO2 multilayers with subwavelength thicknesses (Fig. 2a) at the telecommunication wavelength (λ = 1550 nm). Figure 2 highlights the contrast between relaxed TIR and conventional TIR, showing the measured light reflection at the interface of a hemi-cylindrical Si prism and the anisotropic multilayer metamaterial. The periodicity of the multilayer is Λ = 100 nm and five periods have been deposited on the prism. The Maxwell–Garnett effective medium theory (EMT)55 predicts that the multilayer with subwavelength periodicity demonstrates strong anisotropy37,56,57,58,59 at the operating wavelength ($$\varepsilon _{2y} = \varepsilon _{2z} = \varepsilon _{{\mathrm{Si}}}\rho + \varepsilon _{{\mathrm{SiO}}_2}(1 - \rho )$$ and $$1/\varepsilon _{2x} = \rho /\varepsilon _{{\mathrm{Si}}} + (1 - \rho )/\varepsilon _{{\mathrm{SiO}}_2}$$ where ρ is the fill fraction of silicon). Note we are in the effective medium metamaterial limit $$\left( {\Lambda \ll \lambda } \right)$$ away from the photonic crystal regime $$\left( {\Lambda \sim \lambda } \right)$$. Reflection occurs on two interfaces, the primary silicon-metamaterial interface and the secondary metamaterial-air interface. We have deposited a thin tungsten layer on top of the multilayer metamaterial to attenuate the unwanted reflection from the secondary metamaterial/air interface. Figure 2b displays different measured critical angles for s and p polarizations. Various samples with different fill fractions have been fabricated. As ρ deviates from 0 and 1, the multilayer displays effective anisotropy and the critical angle for s and p polarized incidences are separated in agreement with relaxed TIR theory in Eq. (2). The effective permittivity of the multilayer structure (extracted from the critical angle) shows strong anisotropy and is in good agreement with EMT (Fig. 2c). It is clearly seen that the critical angle depends on only one component of the permittivity tensor for each polarization irrespective of the fill fraction. With the extra degree of freedom afforded by the other components of the dielectric tensor, this anisotropic structure can be used as a cladding for conventional dielectric waveguides to control the evanescent waves and skin-depth in the cladding37,60. Figure 2d displays the effect of the multilayer anisotropic cladding on reducing of the evanescent wave skin-depth as compared to a conventional slab waveguide. Note that this evanescent wave engineering approach in extreme skin-depth (e-skid) waveguides is owing to the fast variation of the evanescent fields in a subwavelength high-index-contrast superlattice (Fig. 2d). This is fundamentally different from other light confinement strategies which function on interference of propagating waves (photonic crystals) or field enhancement in a slot38,43,51,52,53.

### On-chip e-skid waveguides

To impact the field of silicon photonics, the one-dimensional e-skid waveguide (Fig. 2d) has to be implemented on an SOI platform. This places stringent restrictions on monolithic fabrication, minimum feature sizes, and deviations in field profiles due to quasi-2D behavior. Our design to adapt relaxed TIR and skin-depth engineering $$\left( {\delta \ll \lambda } \right)$$ on-chip is shown in Fig. 3. Here the conventional quasi-2D strip waveguide is surrounded by a cladding made of alternating vacuum-silicon sub-wavelength ridges which achieves strong anisotropy ($$\varepsilon _{2x} < \varepsilon _{{\mathrm{core}}}$$ and $$\varepsilon _{2z} \gg 1$$). Our goal is to confine the TE-like mode (its dominant electric field component is E x ) used in conventional silicon strip waveguides. Since the TM-like mode is polarized in the y direction, the electric field does not probe/feel the anisotropy of the cladding. Hence, the metamaterial cladding does not play any role in TM mode confinement. For more details about the polarization effect, see Supplementary Figures 6 and 7.

Figure 3a emphasizes the counterintuitive nature of the confinement caused by anisotropic media which beats the confinement of conventional high index-contrast interface consisting of silicon surrounded by vacuum (blue curve in Fig. 3a). As it is seen in Fig. 3a, if we increase the index of the cladding only in the z direction, whereas the index in the other directions are fixed, the evanescent wave decays faster in the cladding and the skin-depth reduces drastically. Note, the average cladding index increases but the mode decay length decreases. This behavior is in fundamental contrast to confinement in conventional strip waveguides with isotropic claddings or graded index claddings53 where decreasing the contrast between the core index and cladding index results in slow decay of evanescent waves, not faster decay49.

Figure 3b shows the field profile of a practical e-skid waveguide with the multilayer metamaterial cladding $$\left( {\mathit{\Lambda} \ll \mathit{\lambda} } \right)$$. We can see the practical multilayer cladding shows strong effective anisotropy to ensure the fast decay of the evanescent wave in the cladding. The 2D field profiles in a plane perpendicular to the propagation direction and comparison of the practical e-skid waveguide with a homogeneous anisotropic cladding are shown in Figs. 3c and 3d. A good agreement with the effective medium approximation is seen. Figures 3e-h illustrate the effective modal index and the decay constant in the cladding for a wide range of core sizes and the fill fractions of silicon in the multilayer cladding. First, a strong agreement is seen between EMT and practical multilayer structures, indicating that the cladding indeed achieves effective all-dielectric anisotropy. Second, the effective modal index is below the core index indicative of relaxed TIR as the mechanism of confinement. As expected, simulations prove that the primary role of the on-chip multilayer cladding is to increase the decay constant of evanescent waves $$\left( {k_x \hskip -3.5pt {\prime \prime }/k_0} \right)$$, as shown in Figs. 3f, h, and it does not significantly change the effective modal index of propagating waves $$\left( {k_z \hskip -3.5pt \prime /k_0} \right)$$, as shown in Figs. 3e, g. It is seen that the largest decay constant is achieved at $$\rho \approx 0.5$$ where the strongest anisotropy is attained. Note that ρ = 0 corresponds to the case of no cladding. We also emphasize that the power in the core is not compromised by the presence of the skin-depth engineering cladding and increased anisotropy causes an enhancement in power confinement.

This light confinement strategy in e-skid waveguides is fundamentally different from that in photonic crystal waveguides. Our multilayer structures function in the deep subwavelength limit $$\left( {\Lambda \ll \lambda } \right)$$, not the photonic crystal limit $$\left( {\Lambda \sim \lambda } \right)$$ and the performance of the e-skid waveguide is independent of the multilayer periodicity or disorder (see Supplementary note 2). Furthermore, there are no Dyakonov wave solutions in e-skid waveguides as the optical axis of the anisotropic cladding is perpendicular to the interface and the direction of propagation22,61,62,63,64,65. It is also important to note that the main goal of the anisotropic multilayer cladding is to control the decay constant of evanescent waves on-chip, not to achieve birefringence in the effective modal index of propagating waves as achieved previously in sub-wavelength grating structures26,50,51,66, multi-slotted waveguides52, or nonlinear phase matching applications67.

### Cross-talk in e-skid waveguides

The skin-depth of evanescent waves is the fundamental origin of cross-talk between waveguides that hinders dense photonic integration. Cross-talk, or power coupling between photonic devices, arises owing to the perturbation of the optical mode when the evanescent tail of one waveguide overlaps with a nearby waveguide. However, if we control the skin-depth in the cladding, we can reduce the perturbation due to the adjacent waveguide, subsequently reducing the cross-talk to surpass the integration limit in current silicon photonics.

We now demonstrate this drastic reduction in the cross-talk made possible with the anisotropy in the cladding (Fig. 4a). The top view image of coupled e-skid waveguides fabricated on an SOI chip is shown in Fig. 4b. The power exchanged between two identical lossless coupled waveguides is:

$$I_1 = I_0\mathop {{\mathrm {cos}}}\nolimits^2 \left( {\frac{\pi }{{2L_{\mathrm{c}}}}L} \right),$$
$$I_2 = I_0\mathop{{\mathrm {sin}}}\nolimits^2 \left( {\frac{\pi }{{2L_{\mathrm{c}}}}L} \right),$$
(3)

where I0 is the input power, L is the distance for which the two waveguides are coupled, and Lc is the coupling length. Note Lc is defined such that if the waveguides are coupled for a distance L=Lc, complete power transfer occurs from the first waveguide to the adjacent waveguide. Figure 4c illustrates the experimental setup to measure the coupling length. Light is in-coupled to the first waveguide through the grating coupler68 and the second waveguide is coupled to the first waveguide over a distance of L. The center-to-center separation between the two waveguides is only 1000 nm. The output power of the two waveguides is out-coupled by the two grating couplers at the ends of each waveguide. Note that we can ignore the propagation loss in this experiment as the total length does not exceed a few hundred microns, whereas the propagation loss is 3.67 dB/cm at the operating wavelength λ=1550 nm (see Supplementary Note 6 for details about the characterization of the propagation loss). The ratio between the measured powers for strip waveguides and e-skid waveguides are shown in Fig. 4d. This highlights that the analytical expression $$\left({I_2/I_1 = \mathop{{\mathrm {tan}}}\nolimits^2 \left( {\pi L/2L_{\mathrm{c}}} \right)} \right)$$ matches the experimental data. It is clearly seen that the coupled power to the second waveguide is almost two orders of magnitude lower for the e-skid waveguide. The measured and simulated69 coupling length normalized to the wavelength for different core sizes is plotted in Fig. 4e for waveguides with the same separation distance of 1000 nm. It is seen that the coupling length for the TE-like mode of e-skid waveguides is an order of magnitude higher than that for strip waveguides allowing miniaturization of photonic-integrated circuits without considerable cross-talk between the waveguides. Note that increasing the core size in conventional strip waveguides, which corresponds to higher power confinement inside the core, does not guarantee the reduction in cross-talk (unshaded region Fig. 4e) as the evanescent tails are not controlled. These drastically increased Lc are due to the reduced skin-depth with a higher anisotropy, reducing the waveguide cross-talk significantly in comparison with other dielectric structures, which have been proposed to reduce the cross-talk (Table 1). Note that as the TM-like mode has a negligible electric field component in the x direction, it does not probe/feel the dielectric anisotropy of the cladding. As expected, this results in higher cross-talk for TM-modes compared with strip waveguides (see Supplementary Figures 6 and 7 for more details about the polarization effect and the proposed design to control the skin-depth of the TM-like mode).

### Bending loss in e-skid waveguides

The critical phenomenon of bending loss70,71,72 can also be reduced by adding the anisotropic metamaterial cladding. The counterintuitive connection of bending loss and skin-depth is revealed by an approach adapted from transformation optics70,73. Here, we consider a curved waveguide in the xz plane with a 90° bend and a bending radius of R $$\left( {R \gg \delta } \right)$$. The center of the curved waveguide is at the origin (Figs. 5b, d (inset)). If we apply the transformation: $$u + iw = R{\mathrm l}{\mathrm n}\frac{{x + iz}}{R},\;v = y$$70,73, the curved waveguide is mapped to a straight waveguide in the uw plane (Figs. 5b, d). This causes the refractive index of the transformed waveguide to become inhomogeneous in the new coordinate system as70,73: $$n{\prime}\left( u \right) = n\left( {x(u)} \right){\mathrm e}^{u/R}$$, where $$n\left( {x\left( u \right)} \right)$$ is the refractive index of the straight waveguide as shown in Figs. 5a, b. Bending loss or radiative power leakage from the core occurs in this straight inhomogeneous index waveguide when the local index of the cladding exceeds the effective modal index (neff). Hence, if we suppress the field near and beyond this radiation condition point (Figs. 5b, d), we can reduce the bending loss. Note that due to the spatial transformation of coordinates, the electromagnetic fields are also transformed causing the expansion of the skin-depth (δ) on the right-hand side of the waveguide and shrinkage of it on the left-hand side (Figs. 5b, d):

$$\delta _{{\mathrm{right}}} \approx \delta + 2\delta ^2/R,$$
$$\delta _{{\mathrm{left}}} \approx \delta - 2\delta ^2/R.$$
(4)

However, if we add the anisotropic metamaterial cladding, we can reduce the skin-depth. As a result, less power will be radiated at the radiation condition point, leading to reduced bending loss.

To confirm the effect of the anisotropic metamaterial claddings on the bending loss, we have investigated the bending losses of the e-skid and strip waveguides, both experimentally and numerically74. To characterize the bending losses, we have cascaded 90° bent waveguides sequentially with different numbers of turns (Fig. 6a and Supplementary note 4), then characterized the bending loss per turn by comparing the transmissions. Figures 6b, c shows the measured bending losses vs. the core size and the bending radius. In all cases, the bending losses with the e-skid waveguide is lower than that with the strip waveguide, which is due to the reduced skin-depth with the anisotropic metamaterial claddings. Note that if the fabrication advancements on the CMOS platform allowed us to reduce the feature size and approach an ideal anisotropic cladding $$\left( {\Lambda \to 0} \right)$$, we would achieve further reduction of the bending loss for TE-like modes in e-skid waveguides (See Supplementary note 4 for more details as well as the bending loss calculation for TM-like modes). Our simulations account for all sources of bending loss including radiation and mode-mismatch.

## Discussion

In summary, we have introduced a photonic platform that can add the critical but overlooked functionality of controlling evanescent waves to the CMOS foundry. We have shown that high-index contrast grating structures in the deep sub-wavelength limit can act as an all-dielectric metamaterial cladding for simultaneously achieving TIR and controlling the skin-depth of evanescent waves. The coupling length is improved more than an order of magnitude and the bending loss is improved three times compared with conventional on-chip waveguides with an average propagation loss of 3.67 dB/cm at telecommunication wavelengths. The decreased photonic skin-depth regime has not been realized till date but has been attempted in initial studies75. This is because an anisotropic all-dielectric metamaterial response requires more than one period of the unit cell to control evanescent wave decay37,49,54. Although we use electron-beam lithography (EBL) as a convenient prototyping technique for the sub-wavelength structures, these devices can in principle be fabricated using deep ultraviolet CMOS foundries; specifically, advanced 193 nm immersion lithography technology has been used to fabricate silicon photonic devices with feature sizes down to 50 nm76. Our work paves the way for all-dielectric metamaterials to enter the practical realm of CMOS-foundry photonics to achieve improved photonic-integrated circuits.

## Methods

### Relaxed total internal reflection

Si/SiO2 multilayers were fabricated using magnetron sputtering and magnetron reactive sputtering, respectively. Silicon and silica were both deposited at a power of 150 W using a pulsed power supply at a frequency of 150 kHz and off time of 0.5 μs for silicon and 0.8 μs for silica. A special substrate holder was built to hold the prisms during deposition. With the new substrate holder and the large thickness of the prism, the film properties would change as the substrate was much closer to the target, producing films with higher loss. Reducing the deposition power produced lower loss films. A 200 nm thick layer of tungsten was deposited at 300 W, 150 kHz, and 0.5 μs on top of the multilayer structure at each fill fraction. The prism was illuminated with a 1530 nm narrow line width laser. A broadband linear polarizer placed in the beam path created s and p polarized light. The incident angle was increased in increments of 2° from 10° to 80°. A Newport Optics Optical Power Meter calibrated to a wavelength of 1530 nm was then used to measure the reflected power. The two reflections at the prism/air interface in the optical path were accounted for when comparing experiment with EMT simulations.

### On-chip platform 1

The on-chip devices for measuring cross-talk in the main text and bending loss in Supplementary Figure 9 were fabricated using a JEOL JBX-6300FS EBL system77 operated at 100 keV energy, 8 nA beam current, and 500 μm exposure field size. A silicon-on-insulator wafer (220 nm thick silicon on 3 μm thick silicon dioxide) has been used. A solvent rinse and hot-plate dehydration baked. Then, hydrogen silsesquioxane resist (HSQ, Dow-Corning XP-1541-006) was spin-coated at 4000 rpm, and then hot-plate baked at 80 °C for 4 min Shape placement by the machine grid, the beam stepping grid, and the spacing between dwell points during the shape writing, were 1 nm, 6 nm, and 6 nm, respectively. An exposure dose of 2800 μC/cm2 was used. The resist was developed by immersion in 25% tetramethylammonium hydroxide for 4 min., followed by a flowing deionized water rinse for 60 s, an isopropanol rinse for 10 s. Then blown dry with nitrogen. The inductively coupled plasma etching in an Oxford Plasmalab System 100 was used to remove silicon from unexposed areas, with a chlorine gas flow of 20 sccm, ICP power of 800 W, pressure of 12 mT, bias power of 40 W, and a platen temperature of 20 °C, resulting in a bias voltage of 185 V. During etching, perfluoropolyether vacuum oil was used to mount chips on a 100 mm silicon carrier wafer. Cladding oxide was deposited using plasma enhanced chemical vapor deposition in an Oxford Plasmalab System 100 with nitrous oxide flow of 1000.0 sccm, a silane flow of 13.0 sccm, high-purity nitrogen flow of 500.0 sccm, high-frequency RF power of 120 W, pressure at 1400 mT, and a platen temperature of 350 °C. Chips rest directly on a silicon carrier wafer during deposition and are buffered by silicon pieces on all sides to aid uniformity.

To characterize the on-chip devices, a custom-built automated test setup was used6. An Agilent 81635A optical power sensor was used as the output detector and Agilent 81600B tunable laser as the input source. The wavelength was swept in 10 pm steps from 1500 to 1600 nm. To maintain the polarization state of the light, a polarization maintaining fiber was used for coupling the TE polarization into the grating couplers68. A polarization maintaining fiber array was used to couple light in/out of the chip.

### On-chip platform 2

For the bending loss experiment in the main text, the cross-talk experiment in Supplementary Figure 8, the insertion loss experiments in Supplementary Figure 13, and the propagation loss experiment in Supplementary Figure 14, a standard SOI wafer is used as a substrate with 2 μm buried oxide and 220 nm top silicon layer. Diluted HSQ with methyl isobutyl ketone was spun on the substrate as a negative-tone electron-beam resist layer. The resist layer was exposed by a 100 kV EBL system, VB6-UHR (Raith), which is capable of 2 nm beam step resolution. After the development of the resist, the top silicon layer was etched by Cl2/O2 based reactive-ion plasma etching tool (Panasonic P610) to transfer the waveguide pattern from the resist to the silicon layer.

### Data availability

The data that support the findings of this study are available from the corresponding authors upon request. Some examples are available at https://github.com/lukasc-ubc/SiEPIC_EBeam_PDK/tree/master/Examples/eskid.