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Integral equation methods in scattering theory are a set of mathematical techniques used to analyze the interaction of waves with obstacles. These methods are essential in understanding the behavior of waves in complex media and in particular, in determining the scattering properties of objects.In scattering theory, the interaction of a wave with an obstacle is typically described using integral equations. These equations express the relationship between the scattered field and the incident field, as well as the properties of the obstacle itself. The most common integral equation method in scattering theory is the Lippmann-Schwinger equation.The Lippmann-Schwinger equation is a Fredholm integral equation that relates the scattered field to the incident field and the obstacle's scattering operator. It is derived from the conservation of energy and momentum in the scattering process. The equation provides a means to calculate the scattered field efficiently, given a known incident field and obstacle's scattering operator.Another important integral equation method in scattering theory is the Born approximation. The Born approximation is a perturbative method that approximates the exact solution of the Lippmann-Schwinger equation using a series expansion. It is useful when the obstacle's scattering operator is small compared to the incident field, allowing for an analytical solution of the scattering problem.In addition to these two methods, there are other integral equation techniques that can be used in scattering theory, such as the Rayleigh-Sommerfeld diffraction formula and the Kirchhoff integral formula. These methods are derived from different physical assumptions and are suitable for different types of scattering problems.Integral equation methods in scattering theory have found applications in various fields, including acoustics, electromagnetics, and quantum mechanics. Inacoustics, for example, these methods are used to study the scattering of sound waves by obstacles such as buildings or mountains. In electromagnetics, they are used to analyze the interaction of electromagnetic waves with conducting objects or dielectrics. In quantum mechanics, integral equation methods are used to study the scattering of particles by potentials or potentials.Integral equation methods in scattering theory provide a powerful tool for understanding wave interactions with obstacles. They allow for efficient calculations of scattered fields and provide insights into the physical properties of scattering systems. As such, these methods continue to play a crucial role in various fields of applied mathematics and physics.。
法布里珀罗基模共振英文The Fabryperot ResonanceOptics, the study of light and its properties, has been a subject of fascination for scientists and researchers for centuries. One of the fundamental phenomena in optics is the Fabry-Perot resonance, named after the French physicists Charles Fabry and Alfred Perot, who first described it in the late 19th century. This resonance effect has numerous applications in various fields, ranging from telecommunications to quantum physics, and its understanding is crucial in the development of advanced optical technologies.The Fabry-Perot resonance occurs when light is reflected multiple times between two parallel, partially reflective surfaces, known as mirrors. This creates a standing wave pattern within the cavity formed by the mirrors, where the light waves interfere constructively and destructively to produce a series of sharp peaks and valleys in the transmitted and reflected light intensity. The specific wavelengths at which the constructive interference occurs are known as the resonant wavelengths of the Fabry-Perot cavity.The resonant wavelengths of a Fabry-Perot cavity are determined bythe distance between the mirrors, the refractive index of the material within the cavity, and the wavelength of the incident light. When the optical path length, which is the product of the refractive index and the physical distance between the mirrors, is an integer multiple of the wavelength of the incident light, the light waves interfere constructively, resulting in a high-intensity transmission through the cavity. Conversely, when the optical path length is not an integer multiple of the wavelength, the light waves interfere destructively, leading to a low-intensity transmission.The sharpness of the resonant peaks in a Fabry-Perot cavity is determined by the reflectivity of the mirrors. Highly reflective mirrors result in a higher finesse, which is a measure of the ratio of the spacing between the resonant peaks to their width. This high finesse allows for the creation of narrow-linewidth, high-resolution optical filters and laser cavities, which are essential components in various optical systems.One of the key applications of the Fabry-Perot resonance is in the field of optical telecommunications. Fiber-optic communication systems often utilize Fabry-Perot filters to select specific wavelength channels for data transmission, enabling the efficient use of the available bandwidth in fiber-optic networks. These filters can be tuned by adjusting the mirror separation or the refractive index of the cavity, allowing for dynamic wavelength selection andreconfiguration of the communication system.Another important application of the Fabry-Perot resonance is in the field of laser technology. Fabry-Perot cavities are commonly used as the optical resonator in various types of lasers, providing the necessary feedback to sustain the lasing process. The high finesse of the Fabry-Perot cavity allows for the generation of highly monochromatic and coherent light, which is crucial for applications such as spectroscopy, interferometry, and precision metrology.In the realm of quantum physics, the Fabry-Perot resonance plays a crucial role in the study of cavity quantum electrodynamics (cQED). In cQED, atoms or other quantum systems are placed inside a Fabry-Perot cavity, where the strong interaction between the atoms and the confined electromagnetic field can lead to the observation of fascinating quantum phenomena, such as the Purcell effect, vacuum Rabi oscillations, and the generation of nonclassical states of light.Furthermore, the Fabry-Perot resonance has found applications in the field of optical sensing, where it is used to detect small changes in physical parameters, such as displacement, pressure, or temperature. The high sensitivity and stability of Fabry-Perot interferometers make them valuable tools in various sensing and measurement applications, ranging from seismic monitoring to the detection of gravitational waves.The Fabry-Perot resonance is a fundamental concept in optics that has enabled the development of numerous advanced optical technologies. Its versatility and importance in various fields of science and engineering have made it a subject of continuous research and innovation. As the field of optics continues to advance, the Fabry-Perot resonance will undoubtedly play an increasingly crucial role in shaping the future of optical systems and applications.。
2022考研英语阅读捕获希格斯粒子Looking for the Higgs捕获希格斯粒子Enemy in sight?敌军现身?The search for the Higgs boson is closing in on its quarry希格斯玻色子的讨论接近其目标ON JULY 22nd two teams of researchers based at CERN, Europe s main particle-physicslaboratory, near Geneva, told a meeting of the European Physical Society in Grenoble thatthey had found the strongest hints yet that the Higgs boson does, in fact, exist.7月22日，驻欧洲粒子物理讨论所的两组讨论人员在格勒诺布尔欧洲物理协会的一次会议上声称，他们已经得到迄今为止最有力的线索，将力证希格斯玻色子的确真实存在。
The Higgs is thelast unobserved part of the Standard Model, a 40-year-old theory which successfullydescribes the behaviour of all the fundamental particles and forces of nature bar gravity.希格斯粒子是基础模型中最终一个尚未观测到的组件，基础模型已有40年的历史，它胜利地描述了全部基础粒子的行为及除重力以外的全部自然力。
Mathematically, the Higgs is needed to complete the modelbecause, otherwise, none of theother particles would have any mass.在数学层面上，希格斯粒子对于完成模型是必不行少的，这是由于，一旦缺少它，全部的其它粒子都将会失去质量。
a r X i v :m a t h -p h /0003006v 1 8 M a r 2000Poles and zeros of the scattering matrix associated to defectmodesD.FelbacqLASMEA UMR CNRS 6602Complexe des C´e zeaux63177Aubi`e re CedexFranceAbstract We analyze electromagnetic waves propagation in one-dimensional periodic media with single or periodic defects.The study is made both from the point of view of the modes and of the diffraction problem.We provide an explicit dispersion equation for the numerical calculation of the modes,and we establish a connection between modes and poles and zeros of the scattering matrix.Periodic media with defects have been intensively addressed in the field of Quantum Mechanics (see [1–3]and references therein)and also,since the development of photonic crystals,in Electromagnetics [4–7].On the general subject of photonic crystals,the inter-ested reader can find an impressive bibliography on the Internet [8].From the theoretical point of view,the quasi-totality of the studies are involved in the characterization of the spectrum of the infinite medium.Nevertheless,for the working physicists,the main problem is that of the finite structure.Indeed,one can only imagine experiments,such as the diffrac-tion of a plane wave,by finite devices.The main question is then to relate the modes with the behavior of the diffracted field.The simplest connection is that of the determination of the conduction bands:whenever the frequency belongs to the spectrum of the infinite structure,the finite one allows the guidance of waves while in the gap the electromagneticfield decreases exponentially.In the present paper,we study,as a model problem,the simple case of a one dimensional periodic structure with defects,which may model for instance a quantum cavity.Such a device has also been intensively addressed [9–16].Nevertheless,these studies involve the infinite structure for which a spectral analysis is given,whereas in the present communication we aim at establishing a link between the diffractive properties of a finite structure with a defect,and the known spectral properties of an infinite structure with a defect.Throughout this paper,an orthonormal triaxial cartesian coordinate system (0,x,y,z )is used.We consider a periodic structure,described by a bounded real one-periodic function ε(x ),(ε(x )=ε(x +1)),representing the relative permittivity with respect to x ,whereas the permeability is assumed to be µ0,i.e.that of vacuum.The structure is assumed to be invariant in the y and z directions and the harmonic fields (time dependence of exp (−iωt ))are invariant along z .That way,the field is described by a function u n (x )exp(iαy ),where α∈(−π,+π]is the Bloch frequency (in case of a scattering problem,the frequency αis equal to k 0sin θ,where θis the angle of incidence of an incident plane wave).When theelectricfield E is parallel to the z-axis(E||case),u N(x)exp(iαy)represents the z-component of E and when the magneticfield H is parallel to the z-axis(H||case),it represents the z-component of H.Denoting:β2(x)=k20ε(x)−α2,β20=k20−α2,and setting U=(u,q−1∂x u),the propa-gation equation takes the form:∂x U= 0q−q−1β20 U(1) with:q(x)≡1for E||polarization,q(x)≡ε(x)for H||polarization.The monodromy matrix of the equation,or Floquet operator,is the2×2matrix T k,αsuch that:U(x+ d)=T k,αU(x).When considering only n periods of the medium embedded in vacuum and illuminated by a plane wave(the device extends over[0,n]),the following boundary conditions hold:iβ0u|x=0+∂x u|x=0=2iβ0(2)iβ0u|x=n−∂x u|x=n=0from which the reflection and transmission coefficients can be derived:r n=u|x=0−1t n=u|x=nIn the infinite medium,the conduction bands are characterized by the condition|tr(T)|≤2, we thus define:G={(k,α)/|tr(T k,α)|>2}B={(k,α)/|tr(T k,α)|<2}For(k,α)∈G∪B,we denote(v,w)a basis of eigenvectors of T k,α,such that det(v,w)=11,associated to eigenvaluesµandT 0=a 0c 0b 0d 0 .Denoting:χ=(χij )= w 2−iβ0w 1w 2+iβ0w 1−v 2+iβ0v 1−v 2−iβ0v 1 ,(3)an expression of the coefficients (r n ,t n )can be easily obtained:r n (k,α)=p (µ2n )q (µ2n )(4)where:p (X )=χ21χ11a 0X 2+ χ221c 0−χ211b 0X −χ21χ11d 0q (X )=−χ21χ12a 0X 2+(−χ21χ22c 0+χ11χ12b 0)X +χ11χ22d 0.Assume now that there exists some defect mode for a couple (k 0,α0)so that |α0|<k 0,allowing to define an angle of incidence θ0by α0=k 0sin θ0.It is easily seen that the equation of Proposition 2simply writes:d 0(k 0,α0)=0,whence we get from(4):r n (k 0,θ0)=−χ211b 0+χ221c 0+χ21χ11µ2n a 0−χ21χ22c 0+χ11χ12b 0−χ21χ12µ2n a 0As n tends to infinity,we have the following limits:r n (k 0,θ0)−→χ221c 0−χ211b 0χ21χ11b 0−χ22χ21c 0r n (k,θ0)−→−χ21χ22=1,and so as a conclusion |r n (k 0,θ0)|tends pointwise towards a value that is strictly less than 1,whereas for any point different from k 0,in a small enough neighborhood of k 0,|r n (k,θ0)|tends to 1as n −→∞.This result means that the reflected energy admits a sharp minimum near k 0.It is important to note that the minimum of |r n (k,θ0)|is not a priori reached for value k 0of the wavenumber,but the sharpness is all the more important as n is important.Clearly,for a given incidence θ0,it is possible to transmit waves of wavenumber belonging to a small interval near k 0and tending to {k 0}as n tends to infinityIt is known that for a fixed incidence θ,the scattering matrix admits a meromorphic extension to the complex half-space {Im(k )<0}[17].From (4),poles and zeros of r n are solutions respectively of equations:µ4n a 0+µ2nχ21χ21b 0 =d 0µ4n χ21χ12χ11c 0−χ12χ22k −k n zχ22Im (k n z )χ21c 0−χ21χ11χ−122b 0.Remark 3:For real z ,z →−χ21z −k n p is the equation of a circle of diameterk−→ β0 −2where T= t ij .We have analyzed wave propagation in one-dimensional photonic crystals with one defect. We have shown a connection between the scattering properties and the modes:defect modes of the infinite structure give rise to a pole and a zero explaining the behavior of the refection coefficient.These results should help in the understanding of more complicated structures such as bidimensional photonic crystals,for which our present work could be used as an approximate theory[6,7].REFERENCES[1]J.Rarity,C.Weisbuch Ed.,Micro-cavities and Photonic Band Gap:Physics and Ap-plications,Kluwer Academic Publishers,1996.[2]R.Carmona,croix,Spectral Theory of Random Schr¨o dinger Operator,Birkhauser,1990.[3]A.Figotin,L.Pastur,Spectra of Random and Almost-Periodic Operators,Springer-Verlag,1992.[4]D.Joannopoulos,R.D.Meade,J.N.Winn,Photonic Crystal s,Princeton UniversityPress,1995.[5]A.Figotin,P.Kuchment,1996,Siam J.Appl.Math.,56,1561.[6]E.Centeno,D.Felbacq,1999,J.Opt.Soc.Am.A,16,2705.[7]D.Maystre,1994,Pure Appl.Opt,3,975.[8]/labs/photon/biblio2.html.[9]J.Lekner,1994,J.Opt.Soc.Am.A,11,2892.[10]D.Felbacq,B.Guizal,F.Zolla,1998,.,152,119.[11]D.Felbacq,B.Guizal,F.Zolla,1998,J.Math.Phys.,39,4604.[12]A.Figotin,V.Gorenstein,1998,Phys.Rev.B,58,180.[13]K.M.Leung,1993,J.Opt.Soc.Am.B,10,303.[14]Nian-hua Liu,1997,Phys.Rev.B,55,4097.[15]Shanhui Winn Fan,N.Joshua,J.D.Joannopoulos,1995,J.Opt.Soc.Am.B,12,1267.[16]F.G,Bass,G.Ya.Slepyan,A.V.Gurevich,1994,Phys.Rev.B,50,3631.[17]R.Reed,B.Simon,Methods of Modern Mathematical Physics,Academic Press,1979.。
