Relaxation of alternating iterative algorithms for the Cauchy problem associated with the modified Helmholtz equation

We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over-specified boundary, in the case of the alternating iterative algorithm of ` 12 ` 12 `$12 `&12 `#12 `^12 `_12 `%12 `~12 *Kozlov91 applied to Cauchy problems for the modified Helmholtz equation. A convergence proof of these relaxation methods is given, along with a stopping criterion. The numerical results obtained using these procedures, in conjunction with the boundary element method (BEM), show the numerical stability, convergence, consistency and computational efficiency of the proposed methods.

An alternating method for Cauchy problems for Helmholtz-type operators in non-homogeneous medium

Kozlov & Maz'ya (1989, Algebra Anal., 1, 144–170) proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems. However, in many applied problems, operators appear that do not satisfy these requirements, e.g. Helmholtz-type operators. Therefore, in this study, an alternating procedure for solving Cauchy problems for self-adjoint non-coercive elliptic operators of second order is presented. A convergence proof of this procedure is given.

Delay of light in an optical bottle resonator with nanoscale radius variation : dispersionless, broadband, and low-loss

It is shown theoretically that an optical bottle resonator with a nanoscale radius variation can perform a multinanosecond long dispersionless delay of light in a nanometer-order bandwidth with minimal losses. Experimentally, a 3 mm long resonator with a 2.8 nm deep semiparabolic radius variation is fabricated from a 19??µm radius silica fiber with a subangstrom precision. In excellent agreement with theory, the resonator exhibits the impedance-matched 2.58 ns (3 bytes) delay of 100 ps pulses with 0.44??dB/ns intrinsic loss. This is a miniature slow light delay line with the record large delay time, record small transmission loss, dispersion, and effective speed of light.

Optical microbubble resonator

We develop a method for fabricating very small silica microbubbles having a micrometer-order wall thickness and demonstrate the first optical microbubble resonator. Our method is based on blowing a microbubble using stable radiative CO2 laser heating rather than unstable convective heating in a flame or furnace. Microbubbles are created along a microcapillary and are naturally opened to the input and output microfluidic or gas channels. The demonstrated microbubble resonator has 370 µm diameter, 2 µm wall thickness, and a Q factor exceeding 10. © 2010 Optical Society of America.

Optical microbubble resonator

We develop a method for fabricating very small silica microbubbles having a micrometer-order wall thickness and demonstrate the first optical microbubble resonator. Our method is based on blowing a microbubble using stable radiative CO2 laser heating rather than unstable convective heating in a flame or furnace. Microbubbles are created along a microcapillary and are naturally opened to the input and output microfluidic or gas channels. The demonstrated microbubble resonator has 370 µm diameter, 2 µm wall thickness, and a Q factor exceeding 10. © 2010 Optical Society of America.

Delay of light in an optical bottle resonator with nanoscale radius variation:dispersionless, broadband, and low-loss

It is shown theoretically that an optical bottle resonator with a nanoscale radius variation can perform a multinanosecond long dispersionless delay of light in a nanometer-order bandwidth with minimal losses. Experimentally, a 3 mm long resonator with a 2.8 nm deep semiparabolic radius variation is fabricated from a 19??µm radius silica fiber with a subangstrom precision. In excellent agreement with theory, the resonator exhibits the impedance-matched 2.58 ns (3 bytes) delay of 100 ps pulses with 0.44??dB/ns intrinsic loss. This is a miniature slow light delay line with the record large delay time, record small transmission loss, dispersion, and effective speed of light.

Controllable group delay in a θ-shaped microfiber resonator with coupled-resonator-induced transparency

The control of Light velocity is theoretically and experimentally demonstrated in a θ-shaped microfiber resonator with coupled-resonator-induced transparency. By adjusting the structure parameters, group delays from -60ps to 200ps are achieved in the all-fiber resonator.

Transfer characteristics and propagation effects of a multi-resonance ring resonator-based optical device

We describe the linear and nonlinear optical transfer characteristics of a multi-resonance device consisting of two optical ring resonators coupled one to the other and to an optical waveguide. The propagation effects displayed by the device are compared with those of a sequence of fundamental ring resonators coupled to a waveguide.

Multifunctional, tunable and wideband all-fiber filter based on θ shaped microfiber resonator

A compact Θ shaped microfiber resonator for multifunctional, tunable and wideband filter is proposed. The filtering performance of reflection and transmission spectra depending on coupling coefficients and cavity length is theoretically investigated and experimentally demonstrated. © 2015 OSA.

Kerr nonlinear switching in a core-shell microspherical resonator fabricated from the silicon fiber platform

We investigate the Kerr nonlinearity in a core-shell microspherical resonator fabricated from a silicon fiber. By exploiting the ultrafast wavelength shifting, sub-picosecond modulation is demonstrated. © OSA 2015.

On Fractional Helmholtz Equations

MSC 2010: 26A33, 33E12, 33C60, 35R11

Kerr nonlinear switching in a hybrid silicasilicon microspherical resonator

A hybrid silicon-core, silica-clad microspherical resonator has been fabricated from the semiconductor core fiber platform. Linear and nonlinear characterization of the resonator properties have shown it to exhibit advantageous properties associated with both materials, with the low loss cladding supporting high quality (Q) factor whispering gallery modes which can be tuned through the nonlinear response of the crystalline core. By exploiting the large wavelength shift associated with the Kerr nonlinearity, we have demonstrated all-optical modulation of a weak probe on the timescale of the femtosecond pump pulse. This novel geometry offers a route to ultra-low loss, high-Q silica-based resonators with enhanced functionality.

Inverse Problems Involving Generalized Axial-Symmetric Helmholtz Equation

MSC 2010: 35J05, 33C10, 45D05

A thermally wavelength-tunable photonic switch based on silicon microring resonator

Silicon photonics is a very promising technology for future low-cost high-bandwidth optical telecommunication applications down to the chip level. This is due to the high degree of integration, high optical bandwidth and large speed coupled with the development of a wide range of integrated optical functions. Silicon-based microring resonators are a key building block that can be used to realize many optical functions such as switching, multiplexing, demultiplaxing and detection of optical wave. The ability to tune the resonances of the microring resonators is highly desirable in many of their applications. In this work, the study and application of a thermally wavelength-tunable photonic switch based on silicon microring resonator is presented. Devices with 10μm diameter were systematically studied and used in the design. Its resonance wavelength was tuned by thermally induced refractive index change using a designed local micro-heater. While thermo-optic tuning has moderate speed compared with electro-optic and all-optic tuning, with silicon’s high thermo-optic coefficient, a much wider wavelength tunable range can be realized. The device design was verified and optimized by optical and thermal simulations. The fabrication and characterization of the device was also implemented. The microring resonator has a measured FSR of ∼18 nm, FWHM in the range 0.1-0.2 nm and Q around 10,000. A wide tunable range (>6.4 nm) was achieved with the switch, which enables dense wavelength division multiplexing (DWDM) with a channel space of 0.2nm. The time response of the switch was tested on the order of 10 μs with a low power consumption of ∼11.9mW/nm. The measured results are in agreement with the simulations. Important applications using the tunable photonic switch were demonstrated in this work. 1×4 and 4×4 reconfigurable photonic switch were implemented by using multiple switches with a common bus waveguide. The results suggest the feasibility of on-chip DWDM for the development of large-scale integrated photonics. Using the tunable switch for output wavelength control, a fiber laser was demonstrated with Erbium-doped fiber amplifier as the gain media. For the first time, this approach integrated on-chip silicon photonic wavelength control.

A Thermally Wavelength-tunable Photonic Switch Based on Silicon Microring Resonator

Silicon photonics is a very promising technology for future low-cost high-bandwidth optical telecommunication applications down to the chip level. This is due to the high degree of integration, high optical bandwidth and large speed coupled with the development of a wide range of integrated optical functions. Silicon-based microring resonators are a key building block that can be used to realize many optical functions such as switching, multiplexing, demultiplaxing and detection of optical wave. The ability to tune the resonances of the microring resonators is highly desirable in many of their applications. In this work, the study and application of a thermally wavelength-tunable photonic switch based on silicon microring resonator is presented. Devices with 10µm diameter were systematically studied and used in the design. Its resonance wavelength was tuned by thermally induced refractive index change using a designed local micro-heater. While thermo-optic tuning has moderate speed compared with electro-optic and all-optic tuning, with silicon’s high thermo-optic coefficient, a much wider wavelength tunable range can be realized. The device design was verified and optimized by optical and thermal simulations. The fabrication and characterization of the device was also implemented. The microring resonator has a measured FSR of ~18 nm, FWHM in the range 0.1-0.2 nm and Q around 10,000. A wide tunable range (>6.4 nm) was achieved with the switch, which enables dense wavelength division multiplexing (DWDM) with a channel space of 0.2nm. The time response of the switch was tested on the order of 10 us with a low power consumption of ~11.9mW/nm. The measured results are in agreement with the simulations. Important applications using the tunable photonic switch were demonstrated in this work. 1×4 and 4×4 reconfigurable photonic switch were implemented by using multiple switches with a common bus waveguide. The results suggest the feasibility of on-chip DWDM for the development of large-scale integrated photonics. Using the tunable switch for output wavelength control, a fiber laser was demonstrated with Erbium-doped fiber amplifier as the gain media. For the first time, this approach integrated on-chip silicon photonic wavelength control.