Exceptional coupling in photonic anisotropic metamaterials for extremely low waveguide crosstalk

1. INTRODUCTION

Photonic chips can confine light down to hundreds of nanometer-scale waveguide cross sections, miniaturizing bulky tabletop optical systems into a single tiny chip. A compact, chip-scale device size allows the use of photonic systems outside of the laboratory, and the integration of photonic chips with electronic circuitry leads to a broad range of applications in high-speed optical communication [1,2], biochemical sensing [3,4], high-precision spectroscopy [5,6], and light detection and ranging [7–9]. The large index contrast between silicon (Si) and silicon dioxide (${{\rm SiO}_2}$) allows the integration of highly complicated photonic integrated circuits (PICs) on a silicon-on-insulator (SOI) wafer, and its compatibility with a complementary metal-oxide-semiconductor (CMOS) manufacturing process offers an opportunity for low-cost solutions with large-scale device fabrication. Furthermore, recent advances with high-$Q$ microresonators significantly enhance the light–matter interaction and have revolutionized research in nonlinear and quantum photonics [10–13], atomic physics [14,15], and time/frequency metrology [16,17]. In many applications, realizing a high-density photonic chip integration is highly desired, as more building blocks provide more functionalities on a single chip. It also could reduce the unit cost of mass production and lower the power consumption within the chip. However, further miniaturization of photonic chips is hampered by the wave nature of light, i.e., the evanescent wave in the cladding causes the waveguide crosstalk.

The waveguide crosstalk is the power transfer of a light signal between the adjacent waveguides due to the evanescent waves. In a typical photonic chip, to avoid the crosstalk, waveguides are required to be separated large enough, and this limits the integration density of photonic chips. To overcome this limit, there have been many research efforts directed toward reducing the crosstalk [18–27]. Plasmonics have been explored with their ability to confine light in the subwavelength scale [18–21]; however, metallic losses remain to be considered. Topological approaches that use a waveguide super-lattice [22,23], inverse design [24,25], or transformations optics [26] have been proposed, yet these approaches add more complexity or phase variations in the design and often lead to higher scattering losses. Recently, an extreme skin-depth (e-skid) waveguide that utilizes all-dielectric metamaterial claddings has been proposed to reduce the crosstalk [27]. The subwavelength multilayers effectively work as an anisotropic metamaterial [27–29] and could suppress the evanescent waves in the cladding. This reduces the crosstalk, and an $\approx 30$ times longer coupling length has been demonstrated compared to typical strip waveguides. However, even with the reduced skin-depth, there still is some degree of crosstalk, and the further question remains as to if it is possible to suppress the crosstalk completely.

 

Fig. 1. On-chip coupled waveguide configurations and exceptional coupling in coupled extreme skin-depth (e-skid) waveguides. (a)–(c) Schematic cross sections, geometric parameters, and mode profiles of the coupled silicon waveguides: (a) strip, (b) practical e-skid with subwavelength multilayers, and (c) ideal e-skid with effective medium theory (EMT). (d)–(f) Numerically simulated effective indices of the symmetric ${n_{\rm s}}$ (yellow solid) and anti-symmetric ${n_{\rm a}}$ (blue dashed) modes, and (g)–(i) their corresponding normalized coupling lengths ${L_{\rm c}}/{\lambda _0} = 1/(2|{n_{\rm s}} - {n_{\rm a}}|)$ (blue dots): (d) and (g) strip, (e) and (h) e-skid with multilayers, and (f) and (i) e-skid with EMT. All the simulations are performed as a function of the core width $w$, while fixing the other parameters as $h = 220\;{\rm nm} $, $\Lambda = 100\;{\rm nm} $, $\rho = 0.5$, and $N = 5$. The free space wavelength is ${\lambda _0} = 1550\;{\rm nm} $. The inset boxes of (d)–(f) show the zoomed-in view of each mode. The red-shaded areas in e-skid couplings show the non-trivial coupling regimes, where ${n_{\rm s}} \lt {n_{\rm a}}$, which cannot be observed in a typical strip coupling. The red arrows in e-skid couplings indicate the exceptional coupling points, where ${n_{\rm s}} \approx {n_{\rm a}}$, thus causing the ${L_{\rm c}} \to \infty$.

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In this paper, we explore the limit of the waveguide crosstalk and show that there exists an exceptional coupling with anisotropic metamaterial cladding where the crosstalk can be suppressed completely. Our coupled mode analysis reveals that the anisotropic dielectric perturbation of metamaterial cladding causes a non-trivial coupled regime, where the modal index of the anti-symmetric mode is higher than that of the symmetric mode, and an exceptional coupling at the transition to the non-trivial coupling regime. We experimentally demonstrate such exceptional couplings on an SOI wafer, achieving extremely long coupling lengths. The engineering capability of the exceptional coupling is also presented both theoretically and experimentally.

2. EXCEPTIONAL COUPLING IN COUPLED EXTREME SKIN-DEPTH WAVEGUIDES

To push the limit of chip integration density, we explore the exceptional coupling phenomena in the e-skid waveguides by evaluating coupling lengths of the three different coupled waveguide schemes. Figures 1(a)–1(c) show the cross sections and geometric parameters of the three coupled waveguide configurations: (a) typical strip waveguides, (b) practical e-skid waveguides with subwavelength-scale multilayer claddings, and (c) ideal e-skid waveguides with anisotropic metamaterial claddings using the effective medium theory (EMT). All configurations are implemented on an SOI platform, i.e., Si and ${{\rm SiO}_2}$ as a core and a substrate, respectively. Throughout the paper, we explore the coupled modes between the two identical fundamental quasi-transverse-electric (quasi-${{\rm TE}_0}$) modes, and the electric field profiles ${\rm Re}({E_x})$ and ${\rm Im}({E_z})$ of the coupled symmetric (sym) and anti-symmetric (anti) modes are also plotted in Figs. 1(a)–1(c). The multilayer e-skid in Fig. 1(b) is a practical structure that can be fabricated with the current electron-beam lithography [27] and CMOS technology [30,31]. The EMT e-skid in Fig. 1(c) is an equivalent model with an anisotropic metamaterial, and its permittivities (${\varepsilon _x} = {\varepsilon _ \bot}$ and ${\varepsilon _y} = {\varepsilon _z} = {\varepsilon _\parallel}$) follow [32]

The crosstalk between the two adjacent waveguides is assessed by the coupling length ${L_{\rm c}}$, which quantifies the length that transfers the optical power completely from one waveguide to the other waveguide, i.e., the crosstalk is lower for a longer ${L_{\rm c}}$, and it is the opposite for a shorter ${L_{\rm c}}$. To compare the coupling lengths of each configuration, effective refractive indices of the coupled symmetric (${n_{\rm s}}$, yellow solid) and anti-symmetric (${n_{\rm a}}$, blue dashed) modes are simulated in Figs. 1(d)–1(f), and their corresponding normalized coupling lengths (blue dots) are plotted in Figs. 1(g)–1(i): (d,g) strip, (e,h) e-skid with multilayer, and (f,i) e-skid with EMT. Each coupling length is normalized by the free-space wavelength at ${\lambda _0} = 1550\;{\rm nm} $, and the ${L_{\rm c}}$ of the two identical waveguides can be calculated by [34,35]

3. ANISOTROPIC COUPLED MODE ANALYSIS ON THE EXCEPTIONAL COUPLING

To understand the underlying mechanism of the exceptional coupling, we analyzed each configuration using the anisotropic coupled mode analysis. In a quasi-${{\rm TE}_0}$ mode, an ${E_x}$ component is dominant, but there is an ${E_z}$ component as well. Thus, to address the coupled modes correctly, the anisotropic coupling coefficients from all the field components (i.e., ${\kappa _x},{\kappa _y}$, and ${\kappa _z}$) should be considered [34,35]:

Figures 2(a)–2(c) show the normalized coupling coefficients of each component ${\kappa _x}$ (blue dashed), ${\kappa _y}$ (orange dashed), and ${\kappa _z}$ (yellow dashed), and their corresponding overall coupling coefficient $|\kappa |$ (orange solid) and the normalized coupling length ${L_{\rm c}}/{\lambda _0}$ (blue dots) are plotted in Figs. 2(d)–2(f) and Figs. 2(g)–2(i), respectively: (a,d,g) coupled strip, and coupled e-skid with (b,e,h) the multilayer and (c,f,i) EMT. In every case, as the $w$ increases, the coupling coefficients are reduced, and the coupling lengths are increased, due to the higher confinement in the core and less overlap between the modes. In Fig. 2(a), the ${\kappa _x}$ is clearly dominant than the other components, even with a non-negligible ${\kappa _z}$. The sign of ${\kappa _z}$ is negative due to the imaginary ${E_z}$, and it counteracts with the ${\kappa _x}$ in determining the $|\kappa |$. In coupled strip waveguides, the magnitude of ${\kappa _x}$ is always greater than that of ${\kappa _z}$ (i.e., $\kappa \gt 0$) as the ${E_x}$ is dominant in the quasi-${{\rm TE}_0}$ mode. Figure 2(d) shows the overall $|\kappa |$ with the actual unit, and its corresponding normalized coupling length in Fig. 2(g) exactly matches the result from the full numerical simulation in Fig. 1(g). In case of coupled strip waveguides, the dielectric perturbation is isotropic (i.e., $\Delta {\varepsilon _x} = \Delta {\varepsilon _y} = \Delta {\varepsilon _z}$) and the magnitude of ${\kappa _z}$ would be always lower than that of ${\kappa _x}$. In cases of e-skids, as shown in the red-shaded regions of Figs. 2(b) and 2(c), there are non-trivial coupling regimes, where the magnitude of ${\kappa _z}$ is greater than that of ${\kappa _x}$ (i.e., $\kappa \lt 0$). These non-trivial coupling regimes in the coupled e-skids are due to the anisotropic dielectric perturbations $\Delta {\varepsilon _i}$ of the scheme (i.e., $\Delta {\varepsilon _x} \ne \Delta {\varepsilon _y} = \Delta {\varepsilon _z}$), allowing the ${\kappa _z}$ to compensate for the ${\kappa _x}$. The overall $|\kappa |$ approaches to zero at the transition points, resulting in infinitely long coupling lengths (i.e., ${L_{\rm c}} \to \infty$). The peaks in Figs. 2(h) and 2(i) correspond to these exceptional couplings, and these match well with the full simulation results in Figs. 1(h) and 1(i). Again, the anisotropic dielectric perturbation in the metamaterial cladding (i.e., $\Delta {\varepsilon _z} \gt \Delta {\varepsilon _x}$) is the fundamental origin of such non-trivial coupled regime ($\kappa \lt 0$) and the exceptional coupling ($|\kappa | \approx 0$), by reducing the coupling via ${E_x}$ component (${\kappa _x}$), while increasing the counteractive coupling via ${E_z}$ component (${\kappa _z}$). In the previous demonstration of e-skid in [27], this exceptional coupling was not observed due to a lower anisotropic perturbation with an upper cladding and different geometric parameters.

 

Fig. 2. Anisotropic coupled mode analysis on the exceptional coupling in coupled e-skid waveguides. (a)–(c) Normalized anisotropic coupling coefficients ${\kappa _x}$ (blue dashed), ${\kappa _y}$ (orange dashed), and ${\kappa _z}$ (yellow dashed) of the coupled (a) strip, (b) e-skid with multilayer, and (c) e-skid with EMT waveguides. Geometric parameters and the wavelength are the same as in Figs. 1. (d)–(f) Magnitude of the total coupling coefficient $|\kappa | = |{\kappa _x} + {\kappa _y} + {\kappa _z}|$ (orange solid), and (g)–(i) their corresponding normalized coupling lengths ${L_{\rm c}}/{\lambda _0} = \pi /(2|\kappa |{\lambda _0})$ (blue dots) for each configuration: (d) and (g) strip, (e) and (h) e-skid with multilayer, and (f) and (i) e-skid with EMT. The normalized coupling lengths in (g)–(i), which are obtained with anisotropic coupled mode analysis, match with those results in Figs. 1(g)–1(i) from the full numerical simulations. The red-shaded areas in e-skid couplings show the non-trivial coupling regimes where $\kappa \lt 0$, which cannot be observed in typical strip waveguide coupling. The red arrows in e-skid couplings indicate the exceptional coupling points where $|\kappa | \approx 0$, thus causing the ${L_{\rm c}} \to \infty$. As shown in (b) and (c), the anisotropic nature of e-skid waveguides can cause a larger ${\kappa _z}$, which results in the non-trivial coupling regime ($\kappa \lt 0$) and the exceptional coupling ($\kappa \approx 0$) at the transition.

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4. EXPERIMENTAL DEMONSTRATION OF THE EXCEPTIONAL COUPLING AND EXTREME SUPPRESSION OF THE CROSSTALK

To confirm our theoretical finding, we fabricated the coupled e-skid (multilayer) and strip waveguides, and the crosstalks of each configuration were measured and compared. Figures 3(a) and 3(b) show the schematic views of the coupled e-skid and strip waveguides, respectively, and Fig. 3(c) shows the scanning electron microscope (SEM) images of the fabricated devices: (top) wide-view of the scheme, and zoomed-in views of the (left) coupled e-skids and (right) adiabatic transition from strip to e-skid (see Supplement 1). The ${I_0}$ indicates the optical power at the input port, and the ${I_1}$ and ${I_2}$ are the output powers at the through and coupled ports, respectively. The crosstalk is defined as the power ratio ${I_2}/{I_1}$, and it is related to the coupling length ${L_{\rm c}}$, following [34]

 

Fig. 3. Experimental demonstration of the exceptional coupling in coupled e-skid waveguides. Schematic view of the coupled (a) e-skid (multilayer) and (b) strip waveguides. ${I_0},{I_1}$, and ${I_2}$ indicate the optical powers at input, through, and coupled ports, respectively. (c) SEM images of the fabricated devices. Zoomed-in images show (left) the coupled e-skid waveguides and (right) the adiabatic transition from strip to e-skid waveguides. (d) Experimentally measured waveguide crosstalk and (e) the corresponding normalized coupling length of the coupled e-skid (solid) and strip (dashed) waveguides: $w = 420\;{\rm nm} $ (blue), 430 nm (orange), 440 nm (yellow), and 450 nm (purple). Numerically simulated (f) crosstalk and (g) normalized coupling length that correspond to the experimental results in (d) and (e), respectively. Geometric parameters are $h = 220\;{\rm nm} $, $\rho = 0.5$, $\Lambda = 100\;{\rm nm} $, $N = 5$, and $L = 100\,\,\unicode{x00B5}$m. Map plots of the measured crosstalk as functions of $\lambda$ and $w$ for the coupled (h) e-skid and (j) strip waveguides; (i) and (k) are their corresponding simulation results, respectively. Dark regions in (h) and (i) indicate the exceptional couplings in coupled e-skid waveguides.

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The full map plots of the measured crosstalk, as functions of $\lambda$ and $w$, for the coupled e-skid and strip waveguides are plotted in Figs. 3(h) and 3(j), respectively, clearly showing much lower crosstalk with the e-skid waveguides. Figures  3(i) and 3(k) are their simulation results, respectively. The dark regions in Figs. 3(h) and 3(i) indicate the exceptional coupling, which can be observed only with the coupled e-skid waveguides. Since the exceptional coupling occurs at the point where the coupling coefficient ${\kappa _z}$ compensates the ${\kappa _x}$, the exceptional coupling can be engineered by controlling the modal overlaps between the two waveguides. For example, as shown in Figs. 3(h) and 3(i), increasing the $w$ shifts the exceptional coupling to a longer wavelength; this is because a wider $w$ increases the light confinement and reduces the modal overlap between the two coupled e-skids, while a longer wavelength works the opposite way. Similarly, changing the other geometric parameters $h,g,$ and $\rho$ shifts the exceptional coupling point, and we also observed exceptional couplings with different numbers of e-skid layers $N$ and filling fraction $\rho$, both numerically and experimentally (see Supplement 1).

5. CONCLUSION

In summary, we have presented exceptional couplings in the coupled e-skid waveguides that can achieve extremely low waveguide crosstalk. Our coupled mode analysis reveals that the unique anisotropic dielectric perturbation of the e-skid is the fundamental origin of the non-trivial coupling regime that can cause the exceptional coupling at the transition. We experimentally demonstrated the exceptional couplings on an SOI platform, which is low loss, low cost, and compatible with the CMOS foundry. At the exceptional coupling wavelength, the crosstalk suppression of $\approx 50\;{\rm dB} $ was achieved, which corresponds to $\approx 500$ times longer coupling length than the case of strip waveguides. There is a tradeoff between the bandwidth and the degree of crosstalk suppression, but within the range of practical operation. The exceptional coupling can be engineered with geometric parameters and the filling fraction of the e-skids. Our approach of using the exceptional coupling in the coupled e-skid waveguides drastically increases the photonic chip integration density and can be applied to design other ultracompact photonic devices realizing highly dense PICs.