This paper presents an analytical study of bound states in the continuum (BICs) in a photonic step-ladder waveguide structure. We demonstrate that BICs can arise in a cavity formed by one vertical and two horizontal waveguides inserted between two semi-infinite leads. Using the Green’s function method, we derive exact analytical expressions for the system’s eigenmodes, transmission, reflection, and the conditions required for BIC formation under both Neumann and Dirichlet boundary conditions. We show that when the horizontal waveguide lengths are commensurate, BICs are generated independently of the vertical guide length, allowing resonance control through geometric design. Depending on the vertical guide length, the system operates in either weak or strong coupling regimes, producing Friedrich-Wintgen BICs (FW-BICs). Breaking the symmetry of the structure leads to electromagnetically induced transparency (EIT) or reflection (EIR) resonances with sharp transmission peaks and high quality factors, making the proposed design promising for highly sensitive photonic sensing applications.

Bound states in the continuum (BICs) has emerged as a significant research focus in electronics due to its exceptionally high quality factor (Q-factor). BICs (known also as trapped modes) are not observable from the spectrum due to their non-radiative property. However, they can exist only under a specific choice of the materials or geometrical parameters of the structure. In this paper a BIC eigenfunction is defined to be strictly localized within a subspace of the cavity structure under study and has no leakage behaviour. Its eigen wavelength can be within state continua. BICs and long-lived resonances (LLR) have become a unique way to produce the extreme confinement of electronic waves. We present a theoretical and numerical demonstration of semi-infinite bound states in the continuum (SIBICs) and LLR in a two ring-like electronic micro-cavity coupled to two electronic rib/ridge wave-guides, together with their existence conditions. This structure is composed of two tangent closed loops of lengths $$L_1$$ and $$L_2$$, and two semi-infinite leads. SIBICs are localized in a semi-infinite subspace domain induced transmission zeros. Other induce transmission ones in the middle of long-lived resonances. The BICs correspond to localized resonances of infinite lifetime inside the cavity, without any leakage into the surrounding leads. When BICs exist within state continua, they induce Fano resonances exhibiting sharp peaks in the transmittance spectra and in the variation of the density of states (VADOS) for specific values of the geometrical parameters $$L_1$$ and $$L_2$$. We demonstrate that the condition for the existence of the BICs is to make the lengths $$L_1$$ and $$L_2$$ commensurate with each other. This enables to control the resonances by engineering these lengths. Finally, such a two-tangent loops cavity can be designed to realize near-perfect absorption for some frequencies. The results obtained take due account of the state number conservation between the final system and the reference one. This conservation rule enables to find all the states of the final system and among them the BIC ones. The analytical results are obtained by means of the Green’s function technique. The cavity structure and the LLR presented in this work may have potential applications due to their high sensitivities to weak perturbations, in particular in sensing and wave filtering.

A. bound state eigenfunction is defined here to be strictly localized within a subspace of the structure under study and has no decreasing behavior. Its eigenwavelength can be within state continua. Bound states in the continuum (BICs) and long-lived resonances have become a unique way to produce the extreme localization of electronic waves. We present a theoretical and numerical demonstration of semi-infinite bound states in the continuum (SIBICs) and long-lived resonances in a ringlike electronic microresonator coupled to a finite stub and to two electronic rib/ridge / ridge waveguides, together with their existence conditions. This structure is composed of a closed loop of length L , a finite stub of length L 1 and two semi-infinite leads. SIBICs localized in a semi- infinite subspace domain induce transmission zeros. Others induce transmission ones in the middle of longlived resonances. The BICs correspond to localized resonances of infinite lifetime inside the structure, without any leakage into the surrounding leads. When BICs exist within state continua, they induce Fano resonances exhibiting sharp peaks in the transmission spectra and in the variation of the density of states for specific values of the stub length L 1 . This enables one to regulate these resonances by means of this length. The obtained results take due account of the state number conservation between the final system and the reference one. This conservation rule enables one to find all the states of the final system and among them the bound in the continuum ones. The analytical results are obtained by means of the Green's function technique. The structures and the long-lived resonances presented in this paper may have potential applications due to their high sensitivities to weak perturbations, in particular in sensing, wave filtering, and microelectronic devices.

We present new hybrid long-lived resonances in one photonic closed loop with two semi-infinite leads. The loop is composed of two connected wires (of lengths L1 and L2). This symmetry break cannot disentangle the robust closed-loop twin states. However, stubs of length L/4, where L=L1+L2 is the closed-loop length, are able to do this and induce two new long-lived resonances. Two different excitations injected through the two ports can cross within this system with little interference. The results obtained in this work take into account the state number conservation thanks to the state phase between the final system and reference one.

We present long-lived resonances in photonic spheres with leads. The spheres are assumed to have a photo-conducting surface. Such a surface is modeled by 2N, where N is a positive integer, optical fibers of length L/2, interacting together through two interface points. 2N−1 new bound in continuum (BIC) states exist in this model system once a semi-infinite lead is connected to each of the two interface points. These BIC states induce long-lived transmission resonances for 2L/λ=2n, where n is a positive integer, and λ is the wavelength. These bound states induce shifted new long-lived resonances when perturbing stubs lift their degeneracy with the active states.

Using the reference elements recalled in Chapter 2, we present many new bound in continuum (BIC) states and long-lived resonances for two tangent closed loops, first, with one interface point and then two leads attached to this point. When the two loops are of different lengths, new long-lived resonances appear.

A bound state eigenfunction is defined here to be strictly localized within a subspace of a system and has no decreasing behavior. Its eigenvalue can be within state continua. Bound states in the continuum (BICs) and long-lived resonances have become a unique way to produce the extreme localization of light waves. In this communication, we present a theoretical and numerical demonstration of infinite bound states in the continuum (IBIC) and long-lived resonances in photonic ladder-like structures with four semi-infinite leads, together with their existence conditions. IBIC states are localized in an infinite subspace domain. Some of these states are inactive in the scattering and induce output zeros. Other are scattering active and induce output ones in the middle of long-lived resonances. Several continua of IBIC states remain hidden. They are bound in one of two infinite guides and some of them also in the closed loop. These discoveries are reported here for the first time to our knowledge. The ladder reference structure is composed of connected open loops of lengths L1 and L2 and four semi-infinite leads. The obtained results take due account of the state number conservation between the final system and the reference one. This conservation rule enables to find all the states of the final system and among them the bound in the continuum ones. In addition, we show the existence of long-lived resonances in each of two parallel output lines and long-lived anti-resonances in each of the two other output lines, when one uses two identical parallel simultaneous inputs. The analytical results are obtained by means of the Green’s function technique. The structures and the long-lived resonances presented in this work may have potential applications, in particular in sensing, filtering and communications.

Bound states in the continuum (BICs) and long-lived resonances have become a unique way to produce the extreme localization of light waves. In this paper, we present a theoretical demonstration of BICs and long-lived resonances in photonic comblike structures with two semi-infinite leads, together with their existence conditions. The comb structure is composed of connected guides of length L. The BICs correspond to localized resonances of infinite lifetime inside the comb, without any leakage into the surrounding leads. When BICs exist within state continua, they induce long-lived resonances for specific values of some modified lengths of the guides constitut-ing the comb structure. This enables to regulate these resonances by means of these lengths. The obtained results take due account of the state number conservation between the final system and the reference one constituted by the independent comb and semi-infinite leads. This conservation rule enables to find all the states of the final system and among them the bound in continuum ones. In addition, we present a comb structure with highly directional outputs through two different lines. In each output line two different long-lived resonances enable to transmit two different signals. This system enables to demultiplex two different signals through each of two output lines. The analytical results are obtained by means of the Green's function technique. The structures and the long-lived resonances presented in this paper may have potential applications due to their high sensitivities to weak perturbations, in particular in filtering, sensing, and communication technology improvements.

We present new bound in continuum states and long-lived resonances in one photonic triangular pyramid with two semi-infinite leads. The pyramid is composed of connected open loops of length L. When bound in continuum states exist within state continua, they induce long-lived resonances for specific values of some modified lengths of the six open loops constituting the pyramid and the addition of L/4 stubs.