Matrix approach to the Shapley value and dual similar associated consistency

Replacing associated consistency in Hamiache's axiom system by dual similar associated consistency, we axiomatize the Shapley value as the unique value verifying the inessential game property, continuity and dual similar associated consistency. Continuing the matrix analysis for Hamiache's axiomatization of the Shapley value, we construct the dual similar associated game and introduce the dual similar associated transformation matrix $M_\lambda^{DSh}$ as well. In the game theoretic framework we show that the dual game of the dual similar associated game is Hamiache's associated game of the dual game. For the purpose of matrix analysis, we derive the similarity relationship $M_\lambda^{DSh}=QM_\lambda Q^{-1}$ between the dual similar associated transformation matrix $M_\lambda^{DSh}$ and associated transformation matrix $M_\lambda$ for Hamiache's associated game, where the transformation matrix $Q$ represents the duality operator on games. This similarity of matrices transfers associated consistency into dual similar associated consistency, and also implies the inessential property for the limit game of the convergent sequence of repeated dual similar associated games. We conclude this paper with three tables summarizing all matrix results

Matrix approach to consistency of the additive efficient normalization of semivalues

In fact the Shapley value is the unique efficient semivalue. This motivated Ruiz et al. to do additive efficient normalization for semivalues. In this paper, by matrix approach we derive the relationship between the additive efficient normalization of semivalues and the Shapley value. Based on the relationship, we axiomatize the additive efficient normalization of semivalues as the unique solution verifying covariance, symmetry, and reduced game property with respect to the $p-$reduced game

Reduction of hydrogen-induced optical losses of plasma-enhanced chemical vapor deposition silicon oxynitride by phosphorus doping and heat treatment

Plasma enhanced chemical vapor deposition phosphoros-doped silicon oxynitride (SiON) layers with a refractive index of 1.505 were deposited from $N_{2}O$, 2% $SiH_{4}/N_{2}$, and 5% $PH_{3}/Ar$ gaseous mixtures. The $PH_{3}/Ar$ flow rate was varied to investigate the effect of the dopant to the layer properties. We studied the compositions and the chemical environment of phosphorus, silicon, oxygen, nitrogen and hydrogen in these layers by using x-ray photoelectron spectroscopy and Fourier transform infrared spectroscopy. The number of N-H and O-H bonds, which are responsible for optical losses around 1.55 and 1.3 μm, decreases in the as-deposited layers with increasing phosphorus concentration. Furthermore, the bonded hyrogen in all P-doped layers has been eliminated after annealing at a temperature significantly lower than required for undoped silicon oxynitride layers, that is so to say 1000°C instead of 1150°C. The resulting optical loss in the entire third telecommunication window was well below 0.2dB/cm, making P-doped SiON an attractive material for demanding integrated optics applications

Deposition and characterization of PECVD phosphorus doped silicon oxynitride layers for integrated optics applications

Phosphorus-doped silicon oxynitride layers have been deposited by a Plasma Enhanced Chemical Vapor Deposition process from $N_20$, 2% $SiH_4/N_2$ and 5% $PH_3/Ar$ gaseous mixtures. The $PH_3/Ar$ flow rate was varied to investigate the effect of the dopant to the layer properties. As deposited and annealed (600, 800, 900 and 1000 °C) layers were characterized by Fourier transform infrared spectroscopy, Rutherford backscattering spectroscopy and spectroscopic ellipsometry. In this way the refractive index could be determined as well as the amount of hydrogen that is responsible for enhanced absorption in the 3rd telecommunication window around 1550 nm. The N-H bonds concentration was found to decrease with the phosphorus concentration. Furthermore the bonded hydrogen in the entire P-doped layers have been eliminated after annealing at 1000 °C, while undoped SiON layers require annealing at 1150 °C

Metal mask free dry-etching process for integrated optical devices applying highly photostabilized resist.

Photostabilization is a widely used post lithographic resist treatment process, which allows to harden the resist profile in order to maintain critical dimensions and to increase selectivity in subsequent process steps such as reactive ion etching. In this paper we present the optimization of deep UV-curing of 0,3-3.3 μm thick positive resist profiles followed by heat treatment up to 280 °C. The effectiveness of this resist treatment allows for metal mask free reactive ion etching with selectivity up to 6 for silicon structures, thermal silicon oxide and silicon oxynitride. This procedure is demonstrated by the results obtained in etching of various integrated optical structures

Influence of phosphorus doping on hydrogen content and optical losses in PECVD silicon oxynitride

PECVD Phosphorus-doped silicon oxynitride layers (n=1.5) were deposited from N2O, 2%SiH4/N2, NH3 and 5%PH3/Ar gaseous mixtures. Chemical bonds were determined by Fourier transform infrared spectroscopy. N–H bond concentration of the layers decreased from 3.29×10-21 to 0.45×10-21 cm−3, as the 5%PH3/Ar flow rate increased from 0 to 60 sccm. A simultaneous decrease of O–H related bonds was also observed within the same phosphine flow range. The optical loss of slab-type waveguides at λ=1505 nm was found to decrease from 14.1 to 6.2 dB/cm as the 5%PH3/Ar flow rate increased from 0 to 30 sccm, respectively. Moreover, the optical loss values around λ=1400 and 1550 nm were found to decrease from 4.7 to below 0.2 dB/cm and from 1.8 to 1.0 dB/cm respectively. These preliminary results are very promising for applications in low-loss integrated optical devices

Matrix analysis for associated consistency in cooperative game theory

Hamiache's recent axiomatization of the well-known Shapley value for TU games states that the Shapley value is the unique solution verifying the following three axioms: the inessential game property, continuity and associated consistency. Driessen extended Hamiache's axiomatization to the enlarged class of efficient, symmetric, and linear values, of which the Shapley value is the most important representative. In this paper, we introduce the notion of row (resp. column)-coalitional matrix in the framework of cooperative game theory. Particularly, both the Shapley value and the associated game are represented algebraically by their coalitional matrices called the Shapley standard matrix $M^{Sh}$ and the associated transformation matrix $M_\lambda,$ respectively. We develop a matrix approach for Hamiache's axiomatization of the Shapley value. The associated consistency for the Shapley value is formulated as the matrix equality $M^{Sh}=M^{Sh}·M_\lambda.$ The diagonalization procedure of $M_\lambda$ and the inessential property for coalitional matrices are fundamental tools to prove the convergence of the sequence of repeated associated games as well as its limit game to be inessential. In addition, a similar matrix approach is applicable to study Driessen's axiomatization of a certain class of linear values. Matrix analysis is adopted throughout both the mathematical developments and the proofs. In summary, it is illustrated that matrix analysis is a new and powerful technique for research in the field of cooperative game theory

Light Turning Mirrors in SiON Optical Waveguides for Hybrid Integration with CMOS Photo-detectors

A new method is proposed for hybrid integration of SiON optical waveguides and standard CMOS photo-detectors based on anisotropic etching of 45° facets in a Si substrate. After removal of anisotropically etched Si structures in cladding SiO2, the fabricated total-internal-reflection mirrors can direct the output of the waveguides to photo-detectors placed on top of the chip. The metal-free fabrication process, designed to create these mirrors, is convenient for batch production. Fourier optics based simulations predict that the reflection efficiency of the mirrors is 68.5 %. The far field pattern obtained from the fabricated device is similar to the simulated one

Waveguides, bends and Y-junctions with improved transmission and bandwidth in hexagon-type SOI photonic crystal slabs

This paper presents novel ways of implementing waveguide components in photonic crystal slabs based on silicon-on-insulator (SOI). The integration platform we consider consists of hexa¬gonal holes arranged in a triangular lattice (‘hexagon-type’ photonic crystal). The waveguides are made of one missing row of holes (W1) with triangular air inclusions symmetrically added on each side of the waveguide. \ud Size and position of these inclusions are tuning parameters for the band diagram and can be used for minimizing the distributed Bragg reflection (DBR) effect. The waveguides show single-mode behavior with reasonably high group velocity and large transmission window, inside the gap between H-like modes**. These waveguides, closely resembling conventional ridge waveguides, can be combined to form efficient bends and Y-junctions. The bends and Y-junctions include intermediate short waveguide sections at half the bend angle playing the role of corner ‘mirrors’. Qualitative design rules were obtained from 2D calculations based on effective index approximation.\u

Direct measurement of the on-chip insertion loss of high finesse microring resonators in Si3N4-SiO2 technology.

Microring resonators show the possibility for designing Very Large Scale Integrated (VLSI) photonic circuits by cascading them. In order to realize the devices, the on-chip insertion loss becomes an important parameter. The direct measurement of the on-chip insertion loss of a high finesse microring resonator will be presented. Its value (0.1 ± 0.1) dB is low, in agreement with calculations