Monochromatic Components Research Articles

THz Pulse Holography

The main idea of the suggested THz holographic system is to measure amplitude-phase diffraction patterns of an object at the spectrum range of 0.5..4 THz (8 octaves) and further digital processing of obtained data. The ultrabroadband THz electric field of diffraction patterns of an object at the distance can be recorded by electro-optic sampling. Using optical delay line we change THz and fs pulses superposition in electro-optic crystal and thus we can obtain THz field time dependence at each image point E (x, y, t). Using Fourier transform of E (x, y, t) we can obtain spectrum at every image point. Due to reversibility of Maxwell equations we can consider wavefront conjugation in time by changing monochromatic components into their complex conjugates. Thus, we can obtain original electric field in object plane for each wavelength component. As we can obtain complete (holos) information of the object by this approach, it can be named a THz pulse holography. To check this method verifying numerical calculations have been carried out and the holograms of amplitude and phase arbitrary shaped transparencies were obtained and reconstructed numerically.

Deciding Relaxed Two-Colourability: A Hardness Jump

We study relaxations of proper two-colourings, such that the order of the induced monochromatic components in one (or both) of the colour classes is bounded by a constant. A colouring of a graph G is called (C1, C2)-relaxed if every monochromatic component induced by vertices of the first (second) colour is of order at most C1 (C2, resp.). We prove that the decision problem ‘Is there a (1, C)-relaxed colouring of a given graph G of maximum degree 3?’ exhibits a hardness jump in the component order C. In other words, there exists an integer f(3) such that the decision problem is NP-hard for every 2 ≤ C < f(3), while every graph of maximum degree 3 is (1, f(3))-relaxed colourable. We also show f(3) ≤ 22 by way of a quasilinear time algorithm, which finds a (1, 22)-relaxed colouring of any graph of maximum degree 3. Both the bound on f(3) and the running time greatly improve earlier results. We also study the symmetric version, that is, when C1 = C2, of the relaxed colouring problem and make the first steps towards establishing a similar hardness jump.

Effect of laser beam filamentation on second harmonic spectrum in laser plasma interaction

AbstractThis paper presents the laser beam filamentation at ultra relativistic laser powers, when the paraxial restriction on the beam is relaxed during the filamentation process. On account of laser beam intensity gradient and background density gradients in filamentary regions, the electron plasma wave (EPW) at pump wave frequency is generated. This EPW is found to be highly localized because of the laser beam filaments. The interaction of the incident laser beam with the EPW leads to the second harmonic generation. The second harmonic spectrum has also been studied in detail, and its correlation with the filamentation of the laser beam has been established. Starting almost with a monochromatic component of laser beam propagation, the second harmonic spectrum becomes more complicated, and broadened when the laser beam propagates further, and filamentation takes place. For the typical laser beam and plasma parameters: laser beam with wave length of 1064 nm, power flux of 1018 W/cm2, and plasma with temperature 1 KeV, we found that the conversion efficiency equals about (E2/E0) = 8 × 10−3, and the spectrum is quite broad, which depends upon the laser beam propagation distance. The results (specifically, the second harmonic spectral feature) presented here may be used for the diagnostics of laser produced plasmas.

Effect of locking of photon echo signals in multichannel data recording

The effect of locking of a long-lived photon echo is investigated in the case where nonresonant laser radiation (a standing or traveling wave with an artificially created spatial inhomogeneity) serves as an inhomogeneous external perturbation, which results in random shifts or splitting of the initial monochromatic components of the inhomogeneously broadened line. The use of the mutual spatial orientation of the gradients of electric fields as an associative key of access to recorded information allows one to produce a large number of independent channels of data recording. It is demonstrated that the proposed scheme for recording and associative sampling of information has the advantage that, in each channel, information is recorded simultaneously at all optical centers (the whole inhomogeneously broadened resonance line), which does not decrease the intensity of the response from each channel. Moreover, the echo holographic information coded in the wavefronts of the exciting pulses can also be recorded in each channel, which does not affect the independence of individual channels of data recording.

Space-time coherence of polychromatic propagation-invariant fields

Polychromatic stationary propagation-invariant fields with complete spatial coherence in the space-frequency domain are considered. In general, the field is shown to be spatially partially coherent in the space-time domain, apart from transversely achromatic fields with complete transverse coherence. Particular attention is paid to fields that possess the same cone angle at each frequency; these are stationary counterparts of pulsed conical fields known as X waves. It is shown that, for such fields, the radius of the space-time-domain transverse coherence area depends critically on the bandwidth of the power spectrum and can be comparable to the central-lobe radius of the monochromatic field component at the peak frequency of the spectrum.

Large monochromatic components in colorings of complete 3-uniform hypergraphs

Let f ( n , r ) be the largest integer m with the following property: if the edges of the complete 3-uniform hypergraph K n 3 are colored with r colors then there is a monochromatic component with at least m vertices. Here we show that f ( n , 5 ) ≥ 5 n 7 and f ( n , 6 ) ≥ 2 n 3 . Both results are sharp under suitable divisibility conditions (namely if n is divisible by 7, or by 6 respectively).

Graph Colouring with No Large Monochromatic Components

For a graphGand an integertwe letmcct(G) be the smallestmsuch that there exists a colouring of the vertices ofGbytcolours with no monochromatic connected subgraph having more thanmvertices. Letbe any non-trivial minor-closed family of graphs. We show thatmcc2(G) =O(n2/3) for anyn-vertex graphG∈. This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such, and every fixedtwe show thatmcct(G)=O(n2/(t+1)). On the other hand, we have examples of graphsGwith noKt+3minor and withmcct(G)=Ω(n2/(2t−1)).It is also interesting to consider graphs of bounded degrees. Haxell, Szabó and Tardos provedmcc2(G) ≤ 20000 for every graphGof maximum degree 5. We show that there aren-vertex 7-regular graphsGwithmcc2(G)=Ω(n), and more sharply, for every ϵ > 0 there existscϵ> 0 andn-vertex graphs of maximum degree 7, average degree at most 6 + ϵ for all subgraphs, and withmcc2(G) ≥cϵn. For 6-regular graphs it is known only that the maximum order of magnitude ofmcc2is between$\sqrt n$andn.We also offer a Ramsey-theoretic perspective of the quantitymcct(G).

Quantum-classical correspondence in the phase control of multiphoton dissociation by two-color laser pulses

The quantum and the classical multiphoton dissociation dynamics of diatomic Morse molecules driven by intense and ultrashort two-color laser pulses are investigated and compared. Special attention is given to the role of the relative phase of the monochromatic components of the pulses. We study the excitation of the system from the vibrational ground state and from excited states for several values of amplitude, frequency, and duration of the external pulses. Similar overall sensitivity of the dissociation threshold on the phase is observed in both quantum and classical approaches, provided the excitation frequency is sufficiently far from quantum resonances. In addition, we analyze the correspondence between the effects of the relative phase on the classical and quantum dynamics.

Stopping light in metamaterials: the trapped rainbow

A global web of optical fibers forms the backbone of the Internet. This network relies on routers that switch the transmitted pulses from one fiber to another, so that information ends up at the desired destination. Currently, data traffic at major interconnections requires routers to first convert the travelling optical pulse into an electrical one, perform a switching operation, and then convert the electrical pulse back to its original form. This process can slow-down systems by a factor of 1000. Until now, the prime difficulty in designing an all-optical network router was finding a means to temporarily store or buffer the packets of information. Researchers have recently proposed a variety of methods for completely stopping light. Unfortunately, inherent limitations prevent their deployment. For instance, previous approaches employed ultra cold or hot gases1 and did not use solid-state materials. Other methods stored the data pulses as acoustical disturbances, which still travelled slowly down the fiber2. We have proposed3 a light-guiding structure that does not require cryogenic temperatures and can efficiently bring an optical pulse to a complete halt. In this approach, light travels inside an engineered solid-state meta-material (MM), that has a negative refractive index. This material was developed4 in 2001 and the scheme is conceptually simple. The device we envisioned is a three-layer waveguiding heterostructure, in which the middle MM layer is surrounded by two regular dielectrics. The center portion tapers from wide to narrow. Because the power-flow direction inside theMM layer is opposite to the one in the dielectric regions, the guided electromagnetic waves are markedly slowed. Figure 1 shows the propagation of a monochromatic ( f = 1THz) p-polarized magnetic-field component. The light enters the device from the wide end and stops at the prearranged critical thickness. Because of theMM’s anomalous frequency dispersion, the longest (red) wavelength components stop at the Figure 1. A white light beam hits the tapered negative refractive index layer (shown in white), which is surrounded by two regular dielectric layers (shown in blue and green). Upon entering the structure, each frequency component, or color, of the beam stops at different thicknesses, forming a trapped rainbow.

Large Monochromatic Components in Two-Colored Grids

Let $D^d_n$ denote the d-dimensional grid with diagonals, that is, the graph with vertex set $\{1,2,\ldots,n\}^d$ and with edges connecting every two vertices that differ by at most 1 in every coordinate. We prove that for an arbitrary coloring of the vertices of $D^d_n$ by two colors there exists a monochromatic connected subgraph with at least $n^{d-1}-d^2n^{d-2}$ vertices; and thus the “horizontal layer” coloring (by the parity of the first coordinate) is almost optimal. We also consider a d-dimensional triangulated grid; this is the graph (1-skeleton) of a triangulation of the solid cube $[1,n]^d$ that refines the subdivision of $[1,n]^d$ into the grid of unit cubes. Here every two-coloring has a monochromatic connected subgraph with $\Omega(n^{d-1}/\sqrt d)$ vertices. These results are proved by combining combinatorial and topological arguments with suitable isoperimetric inequalities, and they can be viewed as d-dimensional generalizations of the planar HEX lemma.

Search for a monochromatic component of solar axions using Fe–57

I have searched for a monochromatic component of solar axions by using resonant absorptions of 57 Fe nuclei. No evidence related with the solar axions is observed, and the upper limit on axion mass of 216 eV is obtained at 95% confidence level.

Graph coloring with no large monochromatic components

Abstract For a graph G and an integer t we let m c c t ( G ) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed family of graphs. We show that mcc 2 ( G ) = O ( n 2 / 3 ) for any n-vertex graph G ∈ F . This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such F , and every fixed t we show that mcc t ( G ) = O ( n 2 / ( t + 1 ) ) . On the other hand, we have examples of graphs G with no K t + 3 minor and with mcc t ( G ) = Ω ( n ( 2 / 2 t − 1 ) ) . It is also interesting to consider graphs of bounded degrees. Haxell, Szabo, and Tardos proved mcc 2 ( G ) ⩽ 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with mcc 2 ( G ) = Ω ( n ) , and more sharply, for every e > 0 there exists c e > 0 and n-vertex graphs of maximum degree 7, average degree at most 6 + e for all subgraphs, and with mcc 2 ( G ) ⩾ c e n . For 6-regular graphs it is known only that the maximum order of magnitude of mcc 2 is between n and n. We also offer a Ramsey-theoretic perspective of the quantity m c c t ( G ) .

Large Monochromatic Components in Two-colored Grids

Abstract Let D n d denote the d-dimensional grid with diagonals, that is, the graph with vertex set { 1 , 2 , … , n } d and with edges connecting every two vertices that differ by at most 1 in every coordinate. We prove that for an arbitrary 2-coloring of the vertices of D n d there exists a monochromatic connected subgraph with at least n d − 1 − d 2 n d − 2 vertices. We also consider a d-dimensional triangulated grid; this is the graph of a triangulation of the solid cube [ 1 , n ] d that refines the subdivision of [ 1 , n ] d into the grid of unit cubes. Here every 2-coloring has a monochromatic connected subgraph with Ω ( n d − 1 / d ) vertices.

Size of monochromatic components in local edge colorings

An edge coloring of a graph is a local r coloring if the edges incident to any vertex are colored with at most r distinct colors. We determine the size of the largest monochromatic component that must occur in any local r coloring of a complete graph or a complete bipartite graph.

Zero‐resistance states induced by bichromatic microwaves

AbstractWe have studied the bichromatic photoresistance states of a two dimensional electron gas in the regime of microwave induced resistance oscillations. Zudov and coworkers found clear experimental evidence of zero‐resistance states by measuring the bichromatic resistance in a bidimensional gas of electrons. They found that the bichromatic resistance closely replicates the superposition of the two monochromatic components provided that both contributions are positive. However, the superposition principle is no longer valid if one of the two contributions give rise to a zero‐resistance state. The experiments by Zudov and coworkers confirm that negative resistance states are rapidly driven into zero‐resistance states by an instability. In this work we present a model for the bichromatic‐photoconductivity of a two dimensional electron system subjected to a uniform magnetic field. Our model includes both components of the microwave radiation, a uniform magnetic field and impurity scattering effects. The conductivity is calculated from a Kubo‐like formula. Our calculations reproduce the main features of Zudov's experimental results. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A relaxed Hadwiger's Conjecture for list colorings

Hadwiger's Conjecture claims that any graph without K k as a minor is ( k − 1 ) -colorable. It has been proved for k ⩽ 6 , and is still open for every k ⩾ 7 . It is not even known if there exists an absolute constant c such that any ck-chromatic graph has K k as a minor. Motivated by this problem, we show that there exists a computable constant f ( k ) such that any graph G without K k as a minor admits a vertex partition V 1 , … , V ⌈ 15.5 k ⌉ such that each component in the subgraph induced on V i ( i ⩾ 1 ) has at most f ( k ) vertices. This result is also extended to list colorings for which we allow monochromatic components of order at most f ( k ) . When f ( k ) = 1 , this is a coloring of G. Hence this is a relaxation of coloring and this is the first result in this direction.

Relaxed two-coloring of cubic graphs

We show that any graph of maximum degree at most 3 has a two-coloring such that one color-class is an independent set, while the other color-class induces monochromatic components of order at most 750. On the other hand, for any constant C, we exhibit a 4-regular graph such that the deletion of any independent set leaves at least one component of order greater than C. Similar results are obtained for coloring graphs of given maximum degree with k + ℓ colors such that k parts form an independent set and ℓ parts span components of order bounded by a constant. A lot of interesting questions remain open.

Excitation of a hydrogen atom by bichromatic radiation with the frequency ratio 1:2 in the presence of electric field

In the framework of perturbation theory, a probability of excitation of a hydrogen atom from the ground to the first excited state under bichromatic laser radiation with the frequency ratio 1:2 in the presence of electric field is calculated, taking into account quantum interference of two process channels. In the case under study, a coherent superposition of the 2s-and 2p-states, necessary for quantum interference to occur, is formed under the action of the external electric field. The excitation probability is analyzed depending on the relative phase of monochromatic radiation components (the coherent phase control) for two extreme cases: for a strong electric field, where the Stark splitting of an upper level essentially exceeds the spin-orbit one, and for the contrary limit of a weak field.

Understanding the concept of resolving power in the Fabry–Perot interferometer using a digital simulation

The resolution concept in connection with the Fabry–Perot interferometer is difficult to understand for undergraduate students enrolled in physical optics courses. The resolution criterion proposed in textbooks for distinguishing equal intensity maxima and the deduction of the resolving power equation is formal and non-intuitive. In this paper, we study the practical meaning of the resolution criterion and resolution power using a computer simulation of a Fabry–Perot interferometer. The light source in the program has two monochromatic components, the wavelength difference being tunable by the user. The student can also adjust other physical parameters so as to obtain different simulation results. By analysing the images and graphics of the simulation, the resolving power concept becomes intuitive and understandable.

Transmission of Evanescent Wave Modes Through a Slab of DNG Material II

We study the transmission of evanescent plane-wave components through a slab of double-negative (DNG) material (i.e., a medium with relative electromagnetic material parameters /spl epsiv/<0,/spl mu/<0). As is well-known by now, the monochromatic wave components that emerge on the far side have regained the energy apparently attenuated in the DNG slab (which would have been lost in "tunneling" of an evanescent wave component through an ordinary medium). However, there is a minimal slab width d of any DNG medium beyond which the field amplitude of any signal with nonzero bandwidth will decay with increasing d. The decay appears to be unavoidable in far-field imaging through a slab of the previously studied medium. This study extends the validity of this effect to an arbitrary DNG medium.