Problems on product of formations

 In formation theory, one of the interesting problems is the existence of a saturated formation which is a product of non-$p$-saturated formations. In this paper, we shall give an interesting example of saturated formation ${\cal F}$ which is expressible by ${\cal F}={\cal M}{\cal H},$ where ${\cal M}$ and ${\cal H}$ are both non-$p$-saturated formations for all $p\in \pi ({\cal F}).$ We then prove that if the product formation ${\cal F}={\cal M}{\cal H}$ of two formations ${\cal M}$ and ${\cal H}$ is a one-generated $w$-saturated formation with ${\cal F}\not ={\cal H}$, then ${\cal M}$ is also a $w$-saturated formation. By using this result, we shall answer two problems proposed by Skiba and Shemetkov. Received: 12 October 2001 / Revised version: 11 February 2002     

Improved air-silica photonic crystal with a triangular airhole arrangement for hollow-core photonic bandgap fiber design

We propose an improved photonic crystal (PC) cladding design for existing air-guiding photonic bandgap (PBG) fibers whose cladding airholes are arranged in a triangular lattice pattern. By increasing the sizes of concentrated silica regions in the cladding PC, we can have a larger degree of freedom in controlling the cladding bandgap regions. We predict that a fiber made from this type of cladding would perform better in terms of the PBG-guiding wavelength range, radiation loss owing to finite cladding size, and the ability to avoid surface mode problems.     

Thermally switchable and discretely tunable comb filter with a linearly chirped fiber Bragg grating

We propose and experimentally demonstrate a novel switchable and discretely tunable comb filter based on a thermally induced linearly chirped fiber Bragg grating. Experimentally we achieved a thermally induced optical bandpass filter that has eight switchable passband peaks with peak separations of 1.6 and 3.2 nm, a very narrow bandwidth (as small as 10 pm) of each peak, a tunable range of 16.5 nm, and a greater than 25 dB rejection ratio. Two spectral peaks separated by only 0.8 nm were also obtained with this comb filter. This filter provides the unique advantages of an all-fiber structure, switchable spectral peaks, independent tuning of the center wavelength and wavelength spacing of the spectral peaks, low polarization sensitivity, simple structure, ease of fabrication, and low cost.     

Deep-notch, ultracompact long-period grating in a large-mode-area photonic crystal fiber

An extremely short long-period grating (LPG) with strong resonance has been developed in a large-mode-area photonic crystal fiber (PCF) by use of the heat source of a CO2 laser. We believe that such a LPG in pure silica PCF is the first example to be obtained with the point-by-point technique. The fabrication method is simple and repeatable. The resulting LPG has 8 periods, written by a CO2 laser, within a 2.8-mm length of fiber, which yields a deep notch of core-cladding mode coupling of -31.5 dB at the telecommunication wavelength of 1529.2 nm, with a FWHM of approximately 0.7 nm. The principal advantages of this LPG are that it is practical, cost effective, and compact.     

Permutability of minimal subgroups and<Emphasis Type="Italic">p</Emphasis>-nilpotency of finite groups

In this paper it is proved that ifp is a prime dividing the order of a groupG with (|G|,p − 1) = 1 andP a Sylowp-subgroup ofG, thenG isp-nilpotent if every subgroup ofP ∩G ^N of orderp is permutable inN _G (P) and whenp = 2 either every cyclic subgroup ofP ∩G ^N of order 4 is permutable inN _G (P) orP is quaternion-free. Some applications of this result are given. The research of the first author is supported by a grant of Shanxi University and a research grant of Shanxi Province, PR China. The research of the second author is partially supported by a UGC(HK) grant #2160126 (1999/2000).     

Investigation of all-solid photonic bandgap fiber with low losses in low-order bandgaps

We report synoptically an investigation of design, fabrication and characterization of a new all-solid photonic bandgap fiber. By introducing an index depressed layer around a high index core in every unit cell of photonic crystal cladding, a novel all-solid bandgap fiber is predicted to obtain low confinement and bend losses within low-order bandgaps. After optimizing the structure parameters, we fabricate a batch of rods used for cladding cells, select a pure-silica rod for core cell and an inner-hexagonal jacket tube. We demonstrate an all-solid bandgap fiber with the transmission loss as low as 2 dB/km at 1,310 nm and a bandwidth of over 700 nm within the first bandgap. The guiding properties are also measured, respectively, such as transmission spectrum, attenuation spectrum, bend loss, mode field intensity profile, and chromatic dispersion.     

A generalized 2D FDTD model for photonic crystal fibers with frequency dependent media

A generalized model based on the two-dimensional finite-difference time-domain (2D FDTD) method for photonic crystal fibers (PCF) with frequency dependent media is presented. The Maxwell’s curl equations are formulated using flux density and the magnetic field. Auxiliary differential equations are used with complex-conjugate pole-residue pairs which incorporate the material dispersion of the dispersive media. The model is demonstrated to be a unified approach for arbitrary dispersive materials; therefore, it definitely reduces implementation cost when dealing with different frequency-dependent materials.     

Controlled buckling and crumpling of nanoparticle-coated droplets

We introduce a new experimental approach to study the structural transitions of large numbers of nanoparticle-coated droplets as their volume is reduced. We use an emulsion system where the dispersed phase is slightly soluble in the continuous phase. By adding a fixed amount of unsaturated continuous phase, the volume of the droplets can be controllably reduced, causing them to buckle or crumple, thereby becoming nonspherical. The resultant morphologies depend both on the extent of volume reduction and the average droplet size. The buckling and crumpling behavior implies that the droplet surfaces are solid.     

Properties of a vector soliton laser passively mode-locked by a fiber-based semiconductor saturable absorber operating in transmission

We experimentally demonstrate a passively mode-locked fiber laser employing a fiber-based semiconductor saturable absorber (SSA) operating in transmission. Polarization rotation locked vector solitons are observed in the laser. Due to the intrinsic dynamic feature of the laser, period-doubling of these vector solitons has also been observed. Furthermore, extra spectral sidebands are formed on the optical spectrum, caused by the energy exchange between the two orthogonal polarization components of the vector solitons. By careful reduction of the pump power together with fine adjustment to the cavity birefringence, period-one state can further be obtained. Additionally, the phase noise properties of the vector soliton fiber laser have also been characterized experimentally and analytically.