Growth rate for the expected value of a generalized random Fibonacci sequence

A random Fibonacci sequence is defined by the relation g_n = | g_{n-1} +/- g_{n-2} |, where the +/- sign is chosen by tossing a balanced coin for each n. We generalize these sequences to the case when the coin is unbalanced (denoting by p the probability of a +), and the recurrence relation is of the form g_n = |\lambda g_{n-1} +/- g_{n-2} |. When \lambda >=2 and 0 < p <= 1, we prove that the expected value of g_n grows exponentially fast. When \lambda = \lambda_k = 2 cos(\pi/k) for some fixed integer k>2, we show that the expected value of g_n grows exponentially fast for p>(2-\lambda_k)/4 and give an algebraic expression for the growth rate. The involved methods extend (and correct) those introduced in a previous paper by the second author

Pinning by a sparse potential

We consider a directed polymer interacting with a diluted pinning potential restricted to a line. We characterize explicitely the set of disorder configurations that give rise to localization of the polymer. We study both relevant cases of dimension 1+1 and 1+2. We also discuss the case of massless effective interface models in dimension 2+1.Comment: to appear in Stochastic Processes and their Application

Self-Similar Corrections to the Ergodic Theorem for the Pascal-Adic Transformation

Let T be the Pascal-adic transformation. For any measurable function g, we consider the corrections to the ergodic theorem sum_{k=0}^{j-1} g(T^k x) - j/l sum_{k=0}^{l-1} g(T^k x). When seen as graphs of functions defined on {0,...,l-1}, we show for a suitable class of functions g that these quantities, once properly renormalized, converge to (part of) the graph of a self-affine function. The latter only depends on the ergodic component of x, and is a deformation of the so-called Blancmange function. We also briefly describe the links with a series of works on Conway recursive $10,000 sequence.Comment: version to appear in Stochastics and Dynamics. We added a discussion on the links with Conway 10,000$ recursive sequenc

Tunable delay lines in silicon photonics: coupled resonators and photonic crystals, a comparison

In this paper, we report a direct comparison between coupled resonator optical waveguides (CROWs) and photonic crystal waveguides (PhCWs), which have both been exploited as tunable delay lines. The two structures were fabricated on the same silicon-on-insulator (SOI) technological platform, with the same fabrication facilities and evaluated under the same signal bit-rate conditions. We compare the frequency- and time-domain response of the two structures; the physical mechanism underlying the tuning of the delay; the main limits induced by loss, dispersion, and structural disorder; and the impact of CROW and PhCW tunable delay lines on the transmission of data stream intensity and phase modulated up to 100 Gb/s. The main result of this study is that, in the considered domain of applications, CROWs and PhCWs behave much more similarly than one would expect. At data rates around 100 Gb/s, CROWs and PhCWs can be placed in competition. Lower data rates, where longer absolute delays are required and propagation loss becomes a critical issue, are the preferred domain of CROWs fabricated with large ring resonators, while at data rates in the terabit range, PhCWs remain the leading technology

Effect of titanium dioxide (TiO₂) nanoparticle coating on the detection performance of microfiber knot resonator sensors for relative humidity measurement

In this study, the sensitivity and the linearity of the un-coated and TiO2-coated microfiber knot resonator (MKR) have been analyzed. The MKR is very sensitive to humidity changes since its refractive index is strongly humidity dependent. As a result, shifts occur in the resonance wavelength and there are also changes in output power. The un-coated MKR showed a sensitivity of 1.3 pm/%RH, in terms of the resonance wavelength, and a sensitivity of 0.0626 dB/%RH for the transmitted output power. The sensitivity increased greatly after the deposition of a porous TiO2 nanoparticle coating on the MKR. The TiO2-coated MKR showed an improved sensitivity of 2.5 pm/%RH, with respect to the resonance wavelength, and 0.0836 dB/%RH for the transmitted output power. This MKR sensor has the potential for use in a variety of humidity sensing applications

Variable Waist-Diameter Mach-Zehnder Tapered-Fiber Interferometer as Humidity and Temperature Sensor

In-line single-mode tapered-fiber Mach-Zehnder interferometer (MZI-SMTF) with average waist diameters (davg) of 4.05 and 2.89 μ m have been fabricated, and both the temperature and the humidity sensitivity of the surrounding media have been measured and compared. The humidity and the temperature were measured over the ranges from 0% to 90% and 28 °C to 40 °C, respectively. The stability of the system at 50%RH and 90%RH was investigated, while the temperature of the chamber was maintained at about 28 °C. The humidity and temperature sensitivity resolution values were 0.02 nm/%RH and 0.05 nm/0.1 °C for the MZI-SMTF-1 with an average waist diameter of 4.05 μ m , while they were 0.01 nm/%RH and 0.025 nm/0.1 °C for the MZI-SMTF-2 with an average waist diameter of 2.89 μ m

Design of a high-performance optical tweezer for nanoparticle trapping

Integrated optical nanotweezers offer a novel paradigm for optical trapping, as their ability to confine light at the nanoscale leads to extremely high gradient forces. To date, nanotweezers have been realized either as photonic crystal or as plasmonic nanocavities. Here, we propose a nanotweezer device based on a hybrid photonic/plasmonic cavity with the goal of achieving a very high quality factor-to-mode volume (Q/V) ratio. The structure includes a 1D photonic crystal dielectric cavity vertically coupled to a bowtie nanoantenna. A very high Q/V ~ 107 (λ/n)−3 with a resonance transmission T = 29 % at λR = 1381.1 nm has been calculated by 3D finite element method, affording strong light–matter interaction and making the hybrid cavity suitable for optical trapping. A maximum optical force F = −4.4 pN, high values of stability S = 30 and optical stiffness k = 90 pN/nm W have been obtained with an input power Pin = 1 mW, for a polystyrene nanoparticle with a diameter of 40 nm. This performance confirms the high efficiency of the optical nanotweezer and its potential for trapping living matter at the nanoscale, such as viruses, proteins and small bacteria