In this work, a silicon nitride-based slotted ring resonator refractive index (RI) sensor is modeled using a coupled-mode theory approach. An analytical model for the bent slot waveguide is derived by solving fields with Hankel and Bessel functions. The continuity of ${H_y}$ across interfaces yields a system of eight equations forming an $8 times 8$ matrix (${Z_1}$). Solving $det ({Z_1}) = 0$ for a given frequency ($omega$) allows us to determine the bent slot propagation constant ($gamma$) for the transverse magnetic mode. The scattering matrix is constructed to accurately model the 2D slotted ring resonator and is validated using both 2D and 3D finite-difference time-domain methods, demonstrating superior computational efficiency in terms of simulation time. The device exhibits high sensitivity of 750 nm/RIU and a maximum quality factor of 7536. The impact of key parameters, including bent waveguide width, slot width, bent radius, and slot position, on sensing performance is analyzed to optimize the device dimensions. A comparative analysis with coupled-mode-theory-based disk and ring resonator RI sensors, along with other similar devices, underscores the efficacy and advancement of the proposed sensor.

In this work, we investigate the turning points where the model's behavior diverges significantly from other points within the domain. These points play a crucial role in shaping the overall dynamics and characteristics of the system. We provide a comprehensive study of turning points in optical waveguides with three-layer constant step-index profiles, addressing both their mathematical and physical aspects. A general mathematical formula is introduced to determine the exact locations of all possible turning points for different TE and TM modes in one-dimensional straight and bent waveguides. The number and positions of these turning points depend on parameters such as the bent radius, refractive index profiles, and mode properties, which are analyzed in detail.

This work investigates a mathematical model of the propagation of lightwaves in bent optical waveguides. This modeling leads to a non-self-adjoint eigenvalue problem for differential operator defined on the unbounded domain. Through detailed analysis, a relationship between the real and imaginary parts of the complex-valued propagation constants was constructed. Using this relation, it is found that the real and imaginary parts of the propagation constants are bounded, meaning they are limited within certain region in the complex plane. The orthogonality of these bent modes is also proved. By the asymptotic analysis of these modes, it is proved that as $r tends to infinity$ the behavior of the eigenfunctions is proportional to $frac{1}{r}.$

This paper presents an analytical model of a silicon nitride-based 2D ring resonator refractive index (RI) sensor using coupled mode theory (CMT). The proposed model decomposes the ring resonator into two coupling regions and employs coupled-mode equations to describe input and output amplitudes via scattering matrix analysis. The proposed sensor, operating with varying refractive indices in the background cladding, demonstrates a sensitivity of 218 nm/RIU and a total quality factor of 1198. A comprehensive analysis of the bending loss in the proposed sensor is conducted, elucidating its impact on sensitivity, coupling quality factor, and intrinsic quality factor. This analysis aids in the selection of optimal ring resonator parameters, including radius, width, and gap, to achieve superior sensing performance. Furthermore, the paper examines the effect of dispersion on sensitivity and quality and compares the results with those obtained from CMT-based silicon core ring resonator and disk resonator RI sensors. This study provides valuable insights for the design and optimization of high-performance silicon nitride-based RI sensors for various applications.

Non-self adjoint operators describe problems in science and engineering that lack symmetry and unitarity. They have applications in convection–diffusion processes, quantum mechanics, fluid mechanics, optics, wave-guide theory, and other fields of physics. This paper reviews some important aspects of the eigenvalue problems of non-self-adjoint differential operators and discusses the spectral properties of various non-self-adjoint differential operators. Their eigenvalues can be computed for ground and perturbed states by their spectra and pseudospectra. This work also discusses the contemporary results on the finite number of eigenvalues of non-self-adjoint operators and the implications it brings in modeling physical problems.

A silicon microring resonator-based refractive index sensor is proposed using the coupled mode theory (CMT). The ring resonator is decomposed into two bent-straight waveguide coupling regions to obtain a mathematical model. To derive coupled mode equations for the interaction between bent and straight waveguides, the bent mode fields are converted from cylindrical coordinate systems to cartesian coordinates and are solved by using numerical integration. Coupled mode equations between bent and straight waveguides are derived, describing the input and output amplitude related by the scattering matrix. For the fixed dimensions and parameters, the resonant wavelength of the silicon micro-ring resonator structure is computed. The proposed CMT-based ring resonator results are validated with high accuracy with simulation results of FDTD and 2D FEM methods. Compared to FDTD and 2D FEM methods, the CMT-based ring resonator shows a significant reduction in computer resource requirements (time, speed, and memory). The ring resonator-based refractive index sensor for cancer detection applications is proposed with a high sensitivity of 146 nm/Refractive index unit and a Q factor of 3459. Finally, various parameters of the ring resonator are varied to improve sensitivity and Q factor.

For detecting skin cancer, a silicon nitride based ring resonator refractive index sensor is modelled using coupled mode theory. This device has a high sensitivity of 249nm/RIU and Q factor of 1310.

In the literature, the mathematical problem of optical wave propagation in dielectric straight waveguides has been systematically studied as a self-adjoint eigenvalue problem with real eigenvalues. In terms of the underlying physics, such real eigenvalues meant no losses during the wave propagation. However, when the waveguides were bent, experiments showed that the wave propagation became lossy. In this paper, optical wave propagation in dielectric bent waveguides is mathematically analyzed. It is shown that the corresponding eigenvalue problem is a non-self-adjoint eigenvalue problem and has complex-valued eigenvalues. The imaginary part of the eigenvalues is a measure of loss. For large-bend radii, the eigenvalue problem for bent waveguides behaves as an eigenvalue problem for straight waveguides, and the complex-valued eigenvalues approach the real-valued eigenvalues of the straight waveguide problem. By expressing the bent waveguide eigenvalue operator as a sum of the self-adjoint operator and the non-self-adjoint operator, asymptotic behaviour of guided modes and their lossy nature are investigated.