On fundamental diffraction limitation of finesse of a Fabry-Perot cavity

We perform a theoretical study of finesse limitations of a Fabry-Perot (FP) cavity occurring due to finite size, asymmetry, as well as imperfections of the cavity mirrors. A method of numerical simulations of the eigenvalue problem applicable for both the fundamental and high order cavity modes is suggested. Using this technique we find spatial profile of the modes and their round-trip diffraction loss. The results of the numerical simulations and analytical calculations are nearly identical when we consider a conventional FP cavity. The proposed numerical technique has much broader applicability range and is valid for any FP cavity with arbitrary non-spherical mirrors which have cylindrical symmetry but disturbed in an asymmetric way, for example, by tilt or roughness of their mirrors.Comment: 15 pages, 10 figure

Optical Gravitational Wave Antenna with Increased Power Handling Capability

Fundamental sensitivity of an optical interferometric gravitational wave detector increases with increase of the optical power which, in turn, limited because of the opto-mechanical parametric instabilities of the interferometer. We propose to optimize geometrical shape of the mirrors of the detector to reduce the diffraction-limited finesse of unessential optical modes of the interferometer resulting in increase of the threshold of the opto-mechanical instabilities and subsequent increase of the measurement sensitivity. Utilizing parameters of the LIGO interferometer we found that the proposed technique allows constructing a Fabry-Perot interferometer with round trip diffraction loss of the fundamental mode not exceeding $5$~ppm, whereas the loss of the first dipole as well as the other high order modes exceed $1,000$~ppm and $8,000$~ppm, respectively. The optimization comes at the price of tighter tolerances on the mirror tilt stability, but does not result in a significant modification of the optical beam profile and does not require changes in the the gravity detector read-out system. The cavity with proposed mirrors is also stable with respect to the slight modification of the mirror shape.Comment: 5 pages, 4 figure

Diffraction losses of a Fabry-Perot cavity with nonidentical non-spherical mirrors

Optical cavities, with both optimized resonant conditions and high quality factors, are important metrological tools. In particular, they are used for laser gravitational wave (GW) detectors. In order to have high cavity powers in GW detectors, it is necessary to suppress the parametric instability and to reduce the loss in the arm caused by point absorbers by damping the resonant conditions of harmful higher order optical modes (HOOM). This can be achieved effectively by using non spherical mirrors in symmetric Fabry–Perot (FP) cavities by increasing roundtrip losses of HOOMs Ferdous F et al 2014 Phys. Rev. A 90 033826; Matsko A et al 2016 Phys. Rev. D 93 083010. FP cavities in most of the GW detectors have non-identical mirrors to optimize clipping losses and reduce thermal noise by reducing the beam size on one side of the cavity facing to the beam splitter and recycling cavities. We present here a general method to design non spherical non-identical mirrors in non-symmetric FP cavities to damp HOOMs. The proposed design allows us to suppress the loss of the arm power caused by point absorbers on test masses

Diffraction losses of a Fabry-Perot cavity with nonidentical non-spherical mirrors

Optical cavities with both optimized resonant conditions and high quality factors are important metrological tools. In particular, they are used for laser gravitational wave (GW) detectors. It is necessary to suppress the parametric instability by damping the resonant conditions of harmful higher order optical modes (HOOM) in order to have high cavity powers in GW detectors. This can be achieved effectively by using non spherical mirrors in symmetric Fabry-Perot (FP) cavities by increasing roundtrip losses of HOOMs. Fabry-Perot cavities in most of the GW detectors have non-identical mirrors to optimize clipping losses and reduce thermal noise by reducing the beam size on one side of the cavity facing to the beam splitter and recycling cavities. We here present a general method to design non spherical non-identical mirrors in non-symmetric FP cavities to damp HOOMs. The proposed design allows to the suppress the loss of the arm power caused by point absorbers on test masses.Comment: 12 pages, 10 figure

The noise in gravitational-wave detectors and other classical-force measurements is not influenced by test-mass quantization

It is shown that photon shot noise and radiation-pressure back-action noise are the sole forms of quantum noise in interferometric gravitational wave detectors that operate near or below the standard quantum limit, if one filters the interferometer output appropriately. No additional noise arises from the test masses' initial quantum state or from reduction of the test-mass state due to measurement of the interferometer output or from the uncertainty principle associated with the test-mass state. Two features of interferometers are central to these conclusions: (i) The interferometer output (the photon number flux N(t) entering the final photodetector) commutes with itself at different times in the Heisenberg Picture, [N(t), N(t')] = 0, and thus can be regarded as classical. (ii) This number flux is linear in the test-mass initial position and momentum operators x_o and p_o, and those operators influence the measured photon flux N(t) in manners that can easily be removed by filtering -- e.g., in most interferometers, by discarding data near the test masses' 1 Hz swinging freqency. The test-mass operators x_o and p_o contained in the unfiltered output N(t) make a nonzero contribution to the commutator [N(t), N(t')]. That contribution is cancelled by a nonzero commutation of the photon shot noise and radiation-pressure noise, which also are contained in N(t). This cancellation of commutators is responsible for the fact that it is possible to derive an interferometer's standard quantum limit from test-mass considerations, and independently from photon-noise considerations. These conclusions are true for a far wider class of measurements than just gravitational-wave interferometers. To elucidate them, this paper presents a series of idealized thought experiments that are free from the complexities of real measuring systems.Comment: Submitted to Physical Review D; Revtex, no figures, prints to 14 pages. Second Revision 1 December 2002: minor rewording for clarity, especially in Sec. II.B.3; new footnote 3 and passages before Eq. (2.35) and at end of Sec. III.B.

Polymerization of Hexene-1 and Propylene over Supported Titanium–Magnesium Catalyst: Comparative Data on the Polymerization Kinetics and Molecular Weight Characteristics of Polymers

Data are presented on the great differences of the kinetics of hexene-1 and propylene polymerization over the same supported titanium–magnesium catalyst, as well as molecular weight and molecular weight distribution of the polymers produced. It is found that the composition of cocatalysts (AlEt3 or Al(i-Bu)3 greatly affects the kinetics of hexene-1 polymerization and molecular weight distribution of polyhexene, contrary to data obtained at propylene polymerization. The presence of hydrogen at hexene-1 polymerization leads to a much higher increase of activity in comparison with propylene polymerization. Possible reasons for these differences are discussed on the basis of experimental results

Particular issues associated with underwater low density polyethylene pelletizing in tubular reactor processes

The urgency of the discussed issue is caused by the periodic underwater pelletizing unit performance degradation in the form of spontaneous increase of number of polyethylene pellets stuck together as well as practical non*controllability of this phenomenon. The main aim of the study: definition of polyethylene fractions in both high and low pressure recycling systems being able by physical and chemical properties to initiate pellet sticking process and determination of the structure of these polymers for predicting synthesis conditions for the purpose of the further minimization. The methods used in the study: IR-Fourier and NMR 13С-spectroscopy, differential scanning calorimetry, X-ray phase analysis, selective extraction, flotation density determination. The results: based on the data of physical and chemical polyethylene deposit analysis in high pressure systems the authors have made an assumption that the reason of pellet sticking is uncontrolled penetration of the polymer with such characteristics into the melt. Mixing of amorphized polymer deposit with low melting and crystallization temperatures with polyethylene of basic grade 153 was simulated; the comparison with stuck together pellets material in a contact point was made. Using the differential scanning calorimetry and X-ray phase analysis methods the material characteristic similarity was demonstrated

Ethylene Polymerization over Supported Vanadium-Magnesium Catalysts with Different Vanadium Content: The Effect of Hydrogen on Molecular Weight Characteristics of the Produced Bimodal Polyethylene

Data are presented on the activity of supported vanadium-magnesium catalysts (VMCs) with different vanadium content in ethylene polymerization and the molecular weight characteristics of the produced polyethylene. The VC1 catalyst, with a very low vanadium content (0.12 wt.%), showed a sixfold higher activity per unit weight of vanadium than the VC2 catalyst with a high-vanadium content (4.0 wt.%). Additionally, the total activity of VC2 (kg PE/g cat·h) was fivefold higher when compared to VC1. The introduction of hydrogen in polymerization leads to a considerable decrease in the activity of both catalysts. The polyethylene obtained in the presence of hydrogen over both catalysts has a broad bimodal molecular weight distribution (MWD) with a distinct shoulder in the high-molecular region (Mw ≥ 106 g/mol). Decomposition of the MWD curves of bimodal polyethylene into two fractions (high- and low-molecular fractions) made it possible to determine for the first time the ratio of the reaction rate constants of chain transfer with hydrogen (KtrH) and polymer chain propagation (Kp) for two groups of the VMC active sites producing low- and high-molecular fractions of bimodal polyethylene