V-Groove Channel Waveguides and Mach–Zehnder Interferometer in Hyperbolic van der Waals MoOCl₂

1. Introduction

One of the main challenges in modern optical engineering is the spatial scaling of photonic integrated circuit elements to sizes comparable to the characteristic base sizes of nanoelectronic components. A fundamental obstacle to the ultra-dense integration of all-optical devices is the diffraction limit, which sets the minimum size of the electromagnetic field localization region in traditional dielectric guide structures. The classical approach to overcoming the diffraction limit in visible and near-infrared ranges is based on the use of surface plasmon polaritons [1,2,3,4,5] excited at the interface between a dielectric and a noble metal. However, the use of isotropic plasmonic platforms still does not provide the required dimensions for an integrating circuit.

Highly anisotropic layered van der Waals crystals [6,7], which provide compatibility with planar technology, are currently being intensively studied as an alternative material base. The presence of hyperbolic in-plane dispersion of permittivity in such media allows the existence of electromagnetic eigenmodes [8] with large wave vectors—hyperbolic polaritons, characterized by deep subwavelength spatial confinement of the field. However, the practical application of most of the studied hyperbolic media, such as hBN [9] or α-MoO₃ [10,11], is spectrally limited to the mid-infrared range due to the phonon nature of their resonances. In this regard, MoOCl₂ [12,13,14,15,16,17,18,19,20,21,22], which is capable of supporting hyperbolic plasmonic polaritons in the technologically in-demand visible and near-infrared parts of the spectrum, has attracted particular attention.

Despite the intense study of the fundamental electrodynamic properties of MoOCl₂, the problem of efficient signal propagation in waveguiding structures [23] based on this material requires separate consideration. The influence of the transverse geometry of the waveguide on the dispersion and transport characteristics of guided modes in hyperbolic media remains insufficiently studied. The formation of traditional strip waveguides from layered crystals using nanolithography and reactive ion etching is accompanied by material degradation at the structure boundaries, which can lead to signal scattering.

In this context, the use of channel-type guiding structures—waveguides with a v-shaped cross-sectional profile—seems to be an electrodynamically and technologically viable solution. As in classical studies [24,25] examining channel plasmon modes in metals, the v-shaped geometry facilitates localization of the electromagnetic field within the channel, minimizing the influence of edge surface defects on the attenuation of plasmon-polaritons in MoOCl₂.

In this paper, we numerically study the guiding properties of v-shaped optical subwavelength channel waveguides based on the hyperbolic material MoOCl₂ and propose a computational model for a subwavelength Mach-Zehnder interferometer based on coupled channel waveguides.

2. Materials and Methods

In this work, we start with simulation of a v-shaped straight groove of width w and depth t, formed in a MoOCl₂ layer with a thickness of d = 50 nm and oriented along the [100] crystallographic axis. We consider the wavelength range of 360–1700 nm, where the dielectric permittivity tensor has a negative component along the [100] axis and positive components along the [010] and [001] axes, giving rise to a hyperbolic in-plane dispersion law.

Numerical calculations were prepared in COMSOL Multiphysics. MoOCl₂ was modeled as a biaxial dispersive medium with a diagonal dielectric tensor [15] in the crystallographic basis [100], [010], and [001]. The tensor orientation was changed throughout the simulations by in-plane rotation matrixes, while the propagation direction of the v-groove channel was kept fixed in order to find its eigenmodes.

3. Results

The groove plays a dual role: it reduces the characteristic transverse size of the mode and defines its direction of propagation within the plane of the structure. The field distribution in the cross-section (Figure 1a) demonstrates significant localization of the mode near the groove.

on the wavelength $\lambda$ for a layer of thickness $d$ = 50 nm (black solid line for full-wave simulations, blue dashed line for analytical calculations) and for channel mode (green stars, waveguide dimensions are d = 50 nm, w = 30 nm and t = 30 nm). Right-hand scale: dependency of the decay length $L_{dec}$ in a layer of thickness $d$ = 50 nm on the wavelength of incident light $\lambda$ (solid red line).

The real part of the normalized wave number $q$ and the decay length $L_{dec}$ were considered as the main characteristics of the mode, parameters are given by:

$$q={\displaystyle \frac{k}{k_0}}, L_{dec}={\displaystyle \frac{\lambda }{2\pi Im\left(q\right)}}$$

(1)

From the dependency of an effective mode index $q$ on wavelength $\lambda$ (Figure 1b), we see that this value is approximately identical for the fundamental mode of the layer [8] and for the channel mode within it, yielding values ​​slightly greater than 1, indicating the subwavelength nature of the channel mode. Furthermore, calculation of the channel mode decay length for the telecommunication range of 1260–1625 nm shows that these waves propagate over distances of 8 – 15.5 μm, which are sufficient for engineering applications.

However, to develop functional elements of optical circuits, such as splitters, elliptical resonators and interference elements, it is necessary to form waveguides at arbitrary in-plane angles to the optical axes of the crystal. By the wavelength $\lambda$ = 1550 nm, we studied the dependence of the channel mode effective index $q$ on the angle $\alpha$between the [100] axis of the crystal and the waveguide axis (Figure 2a). Having plotted an isofrequency curve for the same wavelength $\lambda$ = 1550 nm (Figure 2b), we found that the maximum channel mode propagation angle $\alpha$ is 10°. It should be noted that channel modes also exhibit stronger light localization with increasing of angle α than the fundamental mode in the layer.

on the angle α between [100] axis of MoOCl₂ and channel axis. (b) Isofrequency curve for the channel mode (red stars and interpolated dash line) and the fundamental slab mode (full-wave calculations, black points and interpolated dash line). Schematic image for channel in-plane direction (grey dashed line) defined by α in the inset. Wavelength λ = 1550 nm, parameters of the structure d = 50 nm, w = 30 nm and t = 30 nm for both (a) and (b).

To demonstrate its practical applicability, we simulated a model of a waveguide and a Mach-Zehnder interferometer based on it, performing a full-wave calculation at a wavelength of $\lambda$ = 1550 nm. It was performed via numerical simulations to find the eigenmodes of the waveguide and calculate the normal component of the electric field Re($E_z$) (Figure 3a, c) of waves propagating in the interferometer to visualize the interference pattern.

Efficient signal propogation along the interferometer requires adiabatic changes in the curvature of its arms, for which they are given an S-shape, defined by the function:

$$y(x)={\displaystyle \frac{L_{gap}}{2}}(1-\cos {\displaystyle \frac{\pi x}{L_s}}),$$

(2)

where $x$,$y$ are the coordinates along the [100] and [010] axes. In addition, due to in-plane hyperbolicity, it is necessary that the maximum angle formed by the tangent to the function $y\left(x\right)$ and the $x$-axis does not exceed 10°. At this angle, the channel mode remains guided in the waveguide at a wavelength of $\lambda$ = 1550 nm. Finding the derivative of this function and its maximum value, we see that the selected parameters (L_in = 500 nm, L_gap = 200 nm, L_arm = 2 μm, L_s = 2.5 μm, L_out = 500 nm) (Figure 3b) give the permissible value of the angle:

$$tg\alpha ={\displaystyle \frac{dy}{dx}}={\displaystyle \frac{{\pi L}_{gap}}{2L_s}}\sin {\displaystyle \frac{\pi x}{L_s}},{\alpha }_{max}=arctg\left({\displaystyle \frac{{\pi L}_{gap}}{2L_s}}\right)\approx 7°$$

(3)

The waveguide parameters w = 30 nm and t = 30 nm were chosen by varying them (Table 1A in the Appendix A section) in order to minimize the channel width and maximize the effective mode index q and its path length L_dec. To achieve the interference effect, L_dec must be guaranteed to exceed the interferometer arm length L_total = L_in + L_s + L_arm + L_s + L_out. In addition, the condition for the transfer coefficient ${\displaystyle \frac{\Delta q}{q}}$ << 1 is satisfied everywhere on the bends.

4. Discussion

The comparative analysis (Table 1) demonstrates that the proposed MoOCl₂ platform instead of gold one enables to reduce (10-fold in in-plane dimensions, 25-fold in out-of-plane dimensions) the linear dimensions of the base elements while maintaining the efficiency of subwavelength signal propagation in the telecommunications range. To demonstrate the functional potential of the developed architecture, a computational model of a Mach-Zehnder interferometer based on coupled channel waveguides is proposed.

5. Conclusions

In conclusion, the obtained results demonstrate that the combination of the optical anisotropy of MoOCl₂ and the v-shaped geometry leads to significant approach of the photonic devices sizes to the standards of modern semiconductor electronics. More broadly, given recent advances in the fabrication [26,27,28] and nanostructure processing [29,30,31] of van der Waals materials, we anticipate that our findings will be used in next-generation integrated circuits, where van der Waals materials will complement traditional material platforms such as silicon [32,33,34,35].

Table 1. Comparison of the liner geometrical parameters and propagation wavelength with other works (n = 1.02, λ = 1550 nm, v-shaped waveguide).

Criterion	Bozhevolnyi, S. I. et al. [24] Isotropic Model (Au)	This Work Anisotropic Model (MoOCl2)
w and t, nm Ldec, μm	277/1250 30	30/50 13.7

Table 1. Comparison of the liner geometrical parameters and propagation wavelength with other works (n = 1.02, λ = 1550 nm, v-shaped waveguide).

Criterion	Bozhevolnyi, S. I. et al. [24] Isotropic Model (Au)	This Work Anisotropic Model (MoOCl2)
w and t, nm Ldec, μm	277/1250 30	30/50 13.7