The synthesis of (Co and Ni) ferrite nanoparticles followed the methods reported in the literature, 1, 2 with the following modifications. Regarding the materials, Benzyl ether (99%), 1,2hexadecanediol (90%), oleic acid (OA) (90%), oleylamine (OAm) (70%), cobalt (II) acetylacetonate, nickel (II) acetylacetonate, iron (III) acetylacetonate were purchased from Sigma-Aldrich Inc. Absolute ethanol and hexane (95%) were purchased from Fisher Chemical.
We introduce a glide symmetric optical waveguide exhibiting a stationary inflection point (SIP) in the Bloch wavenumber dispersion relation. An SIP is a third-order exceptional point of degeneracy (EPD) where three Bloch eigenmodes coalesce to form a so-called frozen mode with vanishing group velocity and diverging amplitude. We show that the incorporation of chirped distributed Bragg reflectors and distributed coupling between waveguides in the periodic structure facilitates the SIP formation and greatly enhances the characteristics of the frozen mode regime. We confirm the existence of an SIP in two ways: by observing the flatness of the dispersion diagram and also by using a coalescence parameter describing the separation of the three eigenvectors collapsing on each other. We find that, in the absence of losses, both the quality factor and the group delay at the SIP grow with the cubic power of the cavity length. The frozen mode regime can be very attractive for light amplification and lasing in optical delay lines, sensors, and modulators.
The concept of lasers based on the frozen mode regime in active periodic optical waveguides with a 3rd-order exceptional point of degeneracy (EPD) is advanced. The frozen mode regime in a lossless and gainless waveguide is associated with a stationary inflection point (SIP) in the Bloch dispersion relation, where three Bloch eigenmodes coalesce forming the frozen mode. As a practical example, we consider an asymmetric serpentine optical waveguide (ASOW). An ASOW operating near the SIP frequency displays a large group delay of a non-resonant nature that scales as the cube of the waveguide length, leading to a strong gain enhancement when active material is included. Therefore, a laser operating in the close vicinity of an SIP has a gain threshold that scales as a negative cube of the waveguide length. We determine that this scaling law is maintained in the presence of small distributed losses, such as radiation associated with waveguide bends and roughness. In addition, we show that although gain causes a distortion in the modes coalescing at the SIP, the properties of the frozen mode are relatively resistant to such small perturbations and we still observe a large degree of exceptional degeneracy for gain values that bring the system above threshold. Finally, our study also reveals that lasing near an SIP is favored over lasing near a photonic band edge located in close proximity to the SIP. In particular, we observe that an SIP-induced lasing in an ASOW displays lower gain threshold compared to lasing near the photonic regular band edge (RBE), even though the SIP resonance has a lower quality factor than the RBE resonance.
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