Quintessence with Coupled Scalar Fields

费登奎斯抖动方法
费登奎斯抖动方法（Feynman-Kac path integration method）是一种数值计算方法，用于求解偏微分方程。

该方法利用费登奎斯公式（Feynman-Kac formula），通过将偏微分方程转化为积分方程的形式，从而以抽样路径（path sampling）的方式近似求解原方程。

费登奎斯公式是基于物理学家理查德·费曼（Richard Feynman）和数学家马克·卡茨（Mark Kac）的工作，将量子力学中的路径积分理论应用到概率论和偏微分方程的求解中。

费登奎斯公式给出了一个连续时间随机过程的解，它将该随机过程与一个偏微分方程联系起来。

费登奎斯抖动方法的基本思想是将偏微分方程的解表示为积分形式，其中被积函数是一个伴随方程（adjoint equation）的解和一个权重函数的乘积。

然后，通过对路径抽样并计算路径的贡献，以蒙特卡洛方法的形式逼近积分，从而求解偏微分方程。

费登奎斯抖动方法的优点包括可以处理高维问题和非线性问题，同时还可以处理带约束的问题。

然而，该方法的缺点是计算量较大和收敛速度较慢。

总而言之，费登奎斯抖动方法是一种用于数值求解偏微分方程的方法，它利用费登奎斯公式将偏微分方程转化为积分方程，并通过对路径抽样以蒙特卡洛方法逼
近积分来求解方程。

All-Optical Flip-Flop Based on Coupled Laser Diodes Martin T.Hill,Associate Editor,IEEE,H.de Waardt,G.D.Khoe,Fellow,IEEE,and H.J.S.DorrenAbstract—An all-optical set–reset flip-flop is presented that isbased on two coupled lasers with separate cavities and lasing atdifferent wavelengths.The lasers are coupled so that lasing in oneof the lasers quenches lasing in the other laser.The flip-flop stateis determined by the laser that is currently lasing.A rate-equationbased model for the flip-flop is developed and used to obtainsteady-state characteristics.Important properties of the system,such as the minimum coupling between lasers and the opticalpower required for switching,are derived from the model.Theseproperties are primarily dependent on the laser mirror reflectivity,the inter-laser coupling,and the power emitted from one of thecomponent lasers,affording the designer great control over theflip-flop properties.The flip-flop is experimentally demonstratedwith two lasers constructed from identical semiconductor opticalamplifiers(SOAs)and fiber Bragg gratings of different wave-lengths.Good agreement between the theory and experimentis obtained.Furthermore,switching over a wide range of inputwavelengths is shown;however,increased switching power isrequired for wavelengths far from the SOA gain peak.Index Terms—Flip-flop,memories,optical bistability,semicon-ductor lasers,semiconductor optical amplifiers.I.I NTRODUCTIONO PTICAL bistable and multistable devices based on laserdiodes(LDs)have been extensively studied as they havemany potential applications in optical computing and telecom-munications.The most important types of optical bistable laserdiode devices can be classified into three broad types:1)absorp-tive bistability,involving multi-section LDs,in which a sectionacts as a saturable absorber;2)two modes or polarization bista-bility,where the lasing mode suppresses the other mode via non-linear gain effects;and3)dispersive bistability,where the bista-bility is based on refractive index changes due to carrier-densitychanges in the LD.A review and explanation of these three typesof bistable LDs can be found in[1].The optical bistable system considered here is not based uponany of the above-mentioned effects.Rather,it is based on thefact that lasing at the natural lasing wavelength in a laser can bequenched when sufficient external light is injected into the lasercavity.The injected external light is not at the same wavelengthas that of the lasing sing is stopped because the gaininside the laser drops below lasing threshold due to the amplifiedexternal light.The optical flip-flop considered here consists of two coupledlasers(see Fig.1).Laser1lases at wavelengthFig.1.Master–slave arrangement of two identical lasing cavities,showing the two possible states.In state1,light from laser1suppresses lasing in laser2.In state2,light from laser2suppresses lasing in laser1.To change states,lasing in the master is stopped by injecting light not at the master laser’s lasing wavelength. structure was not suitable for integration(as it used isolators).Furthermore,the free-space optic setup did not easily allow pre-cise quantitative comparisons between theory and experiment.In[7],the rate equation model used later in this paper wastersely presented(with a slight difference in the definition of oneparameter and one rate equation).Here,we use the rate equationmodel as a base from which to derive a simplified ingthe simplified model,the flip-flop operation is explained and important properties and results about the flip-flop are derived. Furthermore,the results from the model are later quantitatively compared with the experimental results.Note that while the flip-flop does not rely on the effects of refractive index change and nonlinear gain,these effects will be present.Refractive index changes will cause changes in the wavelength of the lasers as the flip-flop switches state.Care should be taken to ensure that the wavelengths of the lasers are sufficiently different so that there is no overlap of the wave-lengths during switching of states,and also that other effects such as four-wave mixing do not become significant.Non-linear gain effects may also affect the switching thresholds calculated here.The effects of refractive index change and nonlinear gain are not considered in this paper.Furthermore,we have not considered thermal effects in the lasers due to power dissipation.However,for integrated flip-flops,this issue may need to be considered as at least two lasers will be operating on the same chip.The rest of this paper is organized as follows.In Section II, we develop a model of the flip-flop using rate equations.In Sec-tions II and III,we derive important properties of the flip-flop from the model.In Section IV,we experimentally demonstrate the system with lasers constructed from semiconductor optical amplifiers(SOAs)and fiber Bragg gratings(FBGs).Finally,the paper is concluded in Section V.II.R ATE E QUATION M ODELWe assume that each laser in the flip-flop(Fig.1)consists of a gain section surrounded by two wavelength-dependent mirrors ofreflectivity).Each set describes one of the lasers.The number ofphotons(1)(3)(4)is related to the external power injected into thelaser is the fraction of photons leaving a facet of one of the lasers that are injected into the other laser. The light power at the lasing wavelength leaving a facet of one ofHILL et al.:ALL-OPTICAL FLIP-FLOP BASED ON COUPLED LASER DIODES 407TABLE ID ESCRIPTION AND V ALUE OF S YMBOLS U SED IN EQUATIONSthelasers.4)The light injected into the laser experiences the same gain,guiding and internal loss as the light at the lasing wave-length inside the ter,we will relax the assumption on the gain.These assumptions lead to simple analytic re-sults and their accuracy is supported by experimental re-sults.A SOA model that assumes constant photon density and thus constant carrier concentration along the cavity length is givenin [8].In (1)and (2),the only effect of injected photons is toreduceandandand ,due to the factthatappear in an exponent (3).Furthermore,byincluding in the model,simplified expressions for laser couplingand switching power are obtained.The steady-state solution of the rate equations for a single laser can be found by setting the left-hand side of (1)and (2)equal to zero and solvingfor,(takingis shown in Fig.2.It can be seenfrom Fig.2that.Assuming,the slopeof.408IEEE JOURNAL OF QUANTUM ELECTRONICS,VOL.37,NO.3,MARCH2001Fig.2.Steady-state photon and carrier numbers of single laser versus injected photons.From (3),(4),(6),and(8),can be found tobe,and from (1)and (2),it is found tobeaxis of Fig.2,then the valueofaxisisexceeds,approaches zeroasis set to zero,and the only light injected isfrom the otherlaserbetween the lasers is high enoughto allow the slope of the linear regions in Fig.3to be greaterthanFig.3.Steady-state curves for the two lasers showing possible operating points.one,then the curves will intersect at three points.The require-mentondoes not satisfy (12),then the curves will only intersect at one point,leading to only one possible symmetric state.The maximum valueofHILL et al.:ALL-OPTICAL FLIP-FLOP BASED ON COUPLED LASER DIODES409 valueofindi-cates it may not be possible to use all types of semiconductorlasers.Specifically,devices withhigh.Ina small neighborhoodaround,the flip-flop,which is anonlinear system,behaves like a linear system.The linearizedsystem can be checked for stability using standard techniques[10]to determine the flip-flop stabilityfor.However,this formal approach does not yield simple analytic expressionsfor crucial flip-flop properties in terms of the flip-flop parame-ters.Nor does it offer insight into how the flip-flop operates.To obtain simple analytic expressions and more insight intothe flip-flop behavior,we employ a simplified model.In themodel,the lasers are represented by their steady-state photonnumber versus injected photon number curves(see Figs.2and3).Also in the simplified model,the light output of the laserchanges instantaneously in accordance with the input light.Atime delayof s is experienced by light travelling betweenthe lasers.In this simplified model,there are just two state vari-ables:is the gain around the loopforplane[10].Thatis,the real part of the roots are less than zero.By using a methodto obtain the roots of the characteristic equation given in[11],the flip-flop is stable,providedthat;Si s r e d u c e d a sSi s i n c r e a s e d t o w a r dS r e m a i n s a p p r o x i m a t e l yc o n s t a n t a nd c l o se t o z e r o.W h e nSs u d d e n l yi n c r e a s i n g a n dSS410IEEE JOURNAL OF QUANTUM ELECTRONICS,VOL.37,NO.3,MARCH2001Fig.5.Implementation of flip-flop.Each component laser is formed by a SOA and two FBGs.is in state 2).With these assumptions,the valueofcan be found tobe,in terms of the powerout of one of the facets of an isolatedlaser(18)To show the steady-state behavior of the flip-flop,we numer-ically found the steady-state points for the full rate equation model of the flip-flop [(1)and (2)]using a fourth-order Runge–Kutta method.The steady-state points were found for various values of external injectedlight.,the statevariables were found with the flip-flop initially in state 1and also initially in state 2.The simulation parameters are given in Table I,with the semiconductor and active region parameters coming from [12].The flip-flop action can be clearly seen when the state vari-ables,(Fig.4).If the flip-flop is initially in state 2with laser 2lasing,then it will remain in state 2untilis increasedtowardreturns tozero.If the flip-flop is initially in state 1,then it will remain in state 1for all valuesofis set to zeroand obtained fromthe simplified model can be assessed by noting when switching between states occurred in the numerical example.In Fig.4,switching occurswhenofmay require knowledge of specific laser properties suchas.Up to this point,it has been assumed that the gain experienced by light injected into a laser is the same as the gain for light at the lasing wavelength of the laser.However,in some situations,light injected into the laser may have a significantly different wavelength or polarization,resulting in a significant gain dif-ference.We assume now that the injected light has a gain per unit lengthof can be positive or negativeandis the gain difference between injected and lasing light.Our results on the properties of the flip-flop are based on the slope of the linear region in plotsoffor bistable operationandandHILL et al.:ALL-OPTICAL FLIP-FLOP BASED ON COUPLED LASER DIODES411Fig.6.Fiber-to-fiber gain characteristics of an SOA used in flip-flop.Note that the gain was measured when the SOA formed part of a laser.Hence,the carrier density and also gain were clamped at the laser threshold values.II,we did not employ models of lasers with extended cavities as are present in the experiment.However,in this paper,we only deal with the steady-state properties of the flip-flop.In theory,there is no difference between the steady-state characteristics of the lasers and flip-flop with extended cavities and those of the system we have modeled in Section II.The SOAs were supplied packaged and with fiber pigtails at-tached.The coupling efficiency between a fiber pigtail and theSOA chip is denoted herebyis not known ex-actly,but the SOA manufacturer estimates it to be 0.56(and sufficiently small so that they could be ignored.The SOAs employed a strained bulk active region and were manufactured by JDS-Uniphase.The amplification of light through one of the SOAs (from pig-tail to pigtail)while it formed part of a laser is shown in Fig.6.From Fig.6,it can be seen that there is a significant dependence of the gain on wavelength.However,from 1552to 1559nm,the gain is approximately constant.Furthermore,there is some gain dependence on the polarization of the input light.The de-pendence of gain on the polarization will affect the coupling re-quirements and switching threshold (19).This issue of polariza-tion-dependent thresholds will be important when using optical fiber inputs,as the polarization is poorly controlled.To decrease the threshold variability will require that the gain medium used in the lasers be made polarization insensitive.External light to change the state of the system was injected into the laser that was currently the master through one of the central FBGs.For the flip-flop switching measurements,the injection cur-rents for the SOAs were adjusted to give approximately equal output power for the two ser 1had an injection cur-rent of 121mA,a threshold current of 107mA,and the output power from one of the fiber pigtails of its SOA was 1.15mW.The same parameters for laser 2were 108mA,86mA,and 1.15mW,respectively.B.MeasurementsIn our experimental setup,we have access to the SOAs through their fiber pigtails,and we define parameters such as reflectivity and coupling with respect to the fiber pigtails instead of the SOA chip interface.With the new parameters,we can obtain system properties independent of the unknown cou-plingefficiency(20)(22)whereoccurs in (21)and(22).givesandand the ratio of injection current to threshold cur-rent are approximately the same in the experiment as in the sim-ulation of Section III.To verify the model of the laser with injected light given by (1)and (2),one of the lasers in the setup was set apart and in-jected with light.We measured the output power at the lasing wavelength from a fiber pigtail of the SOA for various amounts of light injected into a fiber pigtail of the SOA.The injected light came from a tunable laser source that was protected by an optical isolator.The wavelength and polarization of the injected light was set close to that of the lasing light,so that its gain through the SOA would be very close to the gain seen by the lasing light.The output power from the SOA pigtail,at the lasing wave-length,versus the injected power,is plotted in Fig.7.The shape of the curve in Fig.7can be seen to be qualitatively the same as that in Fig.2.More importantly,the slope of the initial linear region of the curve can quantitatively confirm the laser model.In Section II,the slope of the linear region was given in terms ofphoton numbers as.We measure power so we must con-vert the photon numbers to power values using (7)and (5).Fur-thermore,using (22),and noting that the power injected into the412IEEE JOURNAL OF QUANTUM ELECTRONICS,VOL.37,NO.3,MARCH2001ser output power versus injected power showing predicted and experimental curves.SOA via the pigtail has the same couplingefficiency.We measured the valueof3.56,which is within a few percent of the expected value ofand-axisisinterms of the lasing power out of a SOApigtail(24)We injected light from the isolated tunable laser source into themaster laser(laser1,in this case)through one of the centralFBGs.The wavelength of the injected light was1551.5nm andits polarization was adjusted for maximum gain through SOA1.From the experiment,the power required to change states was0.121mW.Equation(24)gives theestimate.In thesimulation(Section III),the actual power to change states wasHILL et al.:ALL-OPTICAL FLIP-FLOP BASED ON COUPLED LASER DIODES413Fig.10.Oscilloscope traces showing power at .The regular togglingbetween states can be clearly seen.also less than the estimate by a similar amount,14%.(Note thats.Each pulse was splitinto two pulses of approximately equal magnitude.One of thesepulses was injected into laser1,while the other was delayed ina fiber delay line for33。

自从上学以后管理论坛的时间少了，发言回答问题的时间也少了，主要的精力都是忙于做实验和学习新的专业理论，可能是由于转行的原因吧，要学的内容太多，不会的知识也太多了，现在转行到了土木方向，接触粘弹性问题也是越来越多了，以前本人是搞损伤断裂，对于这方面接触比较少，也不太爱学。

现在是“与时俱进”，开始新的研究方向，所以恶补了一段时间，和大家交流一下学习感受，要是有什么不对的也请大家拍砖啊！！呵呵呵首先想要澄清一下粘弹性的概念，很多人认为粘弹性就是蠕变或者松弛，这不完全对。

描述粘弹性更为准确的方式应该叫做率依赖，就是本构方程中当时刻应力不仅与当时刻的应变有关，还与当时刻应变速率有关（如果还与以往的历史相关的话，就叫做粘弹塑性了）。

而蠕变与松弛只是当应力或者应变维持在定值的时候，产生的应变增加与应力减小的现象。

分清这个概念很重要，因为在abaqus中定义这些行为的方式是截然不同的，具体来说明一下粘弹性与蠕变（松弛）吧。

1粘弹性狭义上来讲粘弹性是材料在加载过程中应力变化与应变，应变率之间关系的描述，也可以称为率依赖问题。

如果你想要实现冲击载荷作用下粘弹性材料的反应，这个问题属于率依赖问题，你可以使用两种方法定义材料的力学响应，这就是微分型与积分性本构，虽然微分型本构比较直观明了，平衡方程也好获得，但是一般常用的还是基于遗传积分的积分性本构，毕竟微分型本构在基于时间或者频率离散的有限元方法中难于准确实现。

一般的粘弹性本构模型就那几个，比如maxwell，kelvin，剩下的就是它们的串联与并联，如果你有个新模型是n 个maxwell串联的，你可以通过遗传积分公式轻易获得松弛模量与蠕变模量。

然而这里又会引出一个新的问题，学过粘弹性力学的人都知道，只要涉及到粘弹性问题势必逃不过一个数学工具——laplace变换,在这里不想多讲laplace变换的内容，大家对于这个数学工具应该都很清楚（如果是初学的话推荐两本书与粘弹性，laplace变化有关的教材，一个是周光泉的粘弹性理论，还有一本南京工学院，即东南大学出版的《积分变化这本书》），只谈谈它的物理意义吧，其实laplace变换的最核心思想在于时域与频域的转化，一个在时域内控制方程为偏微分方程的转化到频域内就是常微分方程了，对于粘弹性的松弛模量与蠕变模量也是这个道理，它存在着时域表示方法与频域表示方法。

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CONFIGURABLELFATS can operate in a stand-alone configuration or be easily integrated into the ship’s combat system.TACTICAL BISTATIC AND MULTISTATIC CAPABILITYA robust infrastructure permits interoperability with the HELRAS helicopter dipping sonar and all key sonobuoys.HIGHLY MANEUVERABLEOwn-ship noise reduction processing algorithms, coupled with compact twin line receivers, enable short-scope towing for efficient maneuvering, fast deployment and unencumbered operation in shallow water.COMPACT WINCH AND HANDLING SYSTEMAn ultrastable structure assures safe, reliable operation in heavy seas and permits manual or console-controlled deployment, retrieval and depth-keeping. FULL 360° COVERAGEA dual parallel array configuration and advanced signal processing achieve instantaneous, unambiguous left/right target discrimination.SPACE-SAVING TRANSMITTERTOW-BODY CONFIGURATIONInnovative technology achievesomnidirectional, large aperture acousticperformance in a compact, sleek tow-body assembly.REVERBERATION SUPRESSIONThe unique transmitter design enablesforward, aft, port and starboarddirectional transmission. This capabilitydiverts energy concentration away fromshorelines and landmasses, minimizingreverb and optimizing target detection.SONAR PERFORMANCE PREDICTIONA key ingredient to mission planning,LFATS computes and displays systemdetection capability based on modeled ormeasured environmental data.Key Features>Wide-area search>Target detection, localization andclassification>T racking and attack>Embedded trainingSonar Processing>Active processing: State-of-the-art signal processing offers acomprehensive range of single- andmulti-pulse, FM and CW processingfor detection and tracking. Targetdetection, localization andclassification>P assive processing: LFATS featuresfull 100-to-2,000 Hz continuouswideband coverage. Broadband,DEMON and narrowband analyzers,torpedo alert and extendedtracking functions constitute asuite of passive tools to track andanalyze targets.>Playback mode: Playback isseamlessly integrated intopassive and active operation,enabling postanalysis of pre-recorded mission data and is a keycomponent to operator training.>Built-in test: Power-up, continuousbackground and operator-initiatedtest modes combine to boostsystem availability and accelerateoperational readiness.UNIQUE EXTENSION/RETRACTIONMECHANISM TRANSFORMS COMPACTTOW-BODY CONFIGURATION TO ALARGE-APERTURE MULTIDIRECTIONALTRANSMITTERDISPLAYS AND OPERATOR INTERFACES>State-of-the-art workstation-based operator machineinterface: Trackball, point-and-click control, pull-down menu function and parameter selection allows easy access to key information. >Displays: A strategic balance of multifunction displays,built on a modern OpenGL framework, offer flexible search, classification and geographic formats. Ground-stabilized, high-resolution color monitors capture details in the real-time processed sonar data. > B uilt-in operator aids: To simplify operation, LFATS provides recommended mode/parameter settings, automated range-of-day estimation and data history recall. >COTS hardware: LFATS incorporates a modular, expandable open architecture to accommodate future technology.L3Harrissellsht_LFATS© 2022 L3Harris Technologies, Inc. | 09/2022NON-EXPORT CONTROLLED - These item(s)/data have been reviewed in accordance with the InternationalTraffic in Arms Regulations (ITAR), 22 CFR part 120.33, and the Export Administration Regulations (EAR), 15 CFR 734(3)(b)(3), and may be released without export restrictions.L3Harris Technologies is an agile global aerospace and defense technology innovator, delivering end-to-endsolutions that meet customers’ mission-critical needs. The company provides advanced defense and commercial technologies across air, land, sea, space and cyber domains.t 818 367 0111 | f 818 364 2491 *******************WINCH AND HANDLINGSYSTEMSHIP ELECTRONICSTOWED SUBSYSTEMSONAR OPERATORCONSOLETRANSMIT POWERAMPLIFIER 1025 W. 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Chapter8Atomic spectroscopyTerence H.Risby8.1INTRODUCTIONElemental analysis at the trace or ultratrace level can be performed by a number of analytical techniques and the most popular are based upon atomic spectroscopy.Atomic spectroscopy is subdivided into three fields,atomic emission spectroscopy(AES),atomic absorption spectros-copy(AAS),and atomicfluorescence spectroscopy(AFS)that differ by the mode of excitation and the method of measurement of the atom concentrations.The selection of the atomic spectroscopic tech-nique to be used for a particular application should be based on the desired result since each technique involves different measurement approaches.AES excites ground state atoms(atoms)and then quan-tifies the concentrations of excited state atoms(atoms*)by monitoring their radiative deactivation.AAS measures the concentrations of ground state atoms by quantifying the absorption of spectral radiation that corresponds to allowed transitions from the ground to excited states.AFS determines the concentrations of ground state atoms by quantifying the radiative deactivation of atoms that have been excited by the absorption of discrete spectral radiation.The following schema summarizes these three analytical methods.Basis of analytical measurementAES measures a photon emitted when an excited atom deactives to the ground state½ATOM !D EEXCITATION ½ATOMà !h mDEACTIVATION½ATOMComprehensive Analytical Chemistry47S.Ahuja and N.Jespersen(Eds)Volume47ISSN:0166-526X DOI:10.1016/S0166-526X(06)47008-2r2006Elsevier B.V.All rights reserved.227AAS measures a photon absorbed when a ground state atom is excited½ATOM !h mEXCITATION ½ATOMà !h nDEACTIVATION½ATOMAFS measures a photon emitted when an excited atom deactives to the ground state½ATOM !h nEXCITATION ½ATOMà !h mDEACTIVATION½ATOMBrief history.Analytical atomic spectroscopy has taken more than 200years to become the most widely used method for elemental ana-lysis.Thomas Melville was thefirst to describe the principles offlame AES in1752,but it took another100years before Kirchoff and Bunsen (1860)proposed the potential analytical relationship between ground or excited state atoms and the absorption or emission of discrete spectral radiation.All of these pioneering studies were performed by introducing solutions of metals by various means into alcoholflames orflames sup-ported on Bunsen burners.However,the analytical utility of these early studies were limited by reproducibility of the analytical signal and this limitation was not solved until1929when Lundegardh introduced new designs for burners,nebulizers,gas control devices,and detection systems. These instrumental advances were used in most of the earlyflame pho-tometers.Although Kirchoff and Bunsen had introduced the concept of atomic absorption in their original studies,it was not until1955that Walsh and his collaborators developed analytical AAS.This advance was due to their development of the sealed hollow cathode lamp as a spectral source of radiation that avoided the need for high-resolution monochromators to select and resolve the absorption lines.Eight years later Alkemade (1963)and Winefordner(1964)independently introduced the idea of an-alytical AFS.Finally in the1960s and1970s researchers(L’vov,Greenfield, Fassel,and West)introduced nonflame atomizers in order to mini-mize the spectral and chemical interferences that often occur inflames.8.2THEORY8.2.1Atomic emission spectroscopyAES quantifies discrete radiation that is emitted by an excited atom when it deactivates to the ground state.This energy of excitation isT.H.Risby228Atomic spectroscopyprovided by thermal,chemical,or electrical means.If the atom reser-voir is in thermodynamic equilibrium then the Boltzmann distribution law gives the concentrations of atoms in the excited and ground states:N j=N o¼ðg j=g oÞeÀE j=KTwhere N j and N o are the number densities of atoms in the excited(j th state)and ground states,g j and g o the statistical weights of these states, E j the energy difference between the j th and ground states,K the Boltzmann constant;and T the temperature(K)of the atom reservoir. The Boltzmann distribution law can only be used if excitation is pro-duced by thermal collisions;the dominant process of excitation in flames.This equation is not valid to explain excitation caused either by chemical reactions inflames or by energetic collisions with excited species(electrons,ions,metastable atoms)that occur in electrical dis-charges or plasmas.The concentration of atoms in the excited state is measured by monitoring their spectral deactivation to the ground state.The radiant power of this mechanism of deactivation is given by:P¼ðhv o=4PÞðg j=g oÞA j!o l½M eÀE j=KTwhere P is theflux of radiant energy per unit of solid angle and per unit surface area in a direction perpendicular to theflame surface.A j-o is the transition probability per unit time of the transition from the j th to the ground state,and l is the thickness of atom reservoir along the axis of observation from which the emitted photons are monitored.[M]is the concentration of metal atoms and hv o is the energy of the emitted photon.This equation demonstrates that the radiant power of the spectral deactivation is directly proportional to the concentration of the atoms and this linear relationship is followed providing that no inter-ferences such as self-absorption occur.Self-absorption is the absorption of radiation by ground state atoms and this interference increases with the concentration of atoms.Also,this equation shows that small var-iations in temperature will produce larger variations in the radiant energy.8.2.2Atomic absorption spectroscopyAAS measures the discrete radiation absorbed when ground state atoms are excited to higher energy levels by the absorption of a photon of energy.The radiant power of the absorbed radiation is related to the absorption coefficient of the ground state atoms using the229Beer Lambert equation:I ðl Þ¼I o ðl Þ10ÀK ðl Þbwhere I o (l )is the radiant power of the incident radiation of wavelength l ,I (l )the radiant power of the transmitted radiation at wavelength l ,K (l )the absorption coefficient of the ground state atom at wavelength l ,and b the path length.This equation can be expressed in terms of absorbance (A (l ))where:A ðl Þ¼log I ðl Þ=I o ðl ÞÀÁ¼K ðl ÞbThe usual method of excitation of the ground state atoms is to use an elemental spectral source (often a hollow cathode lamp)that emits the atomic spectra of the analyte element.If the width of the emission line from the spectral source is negligible compared to the absorption line of the ground state atoms,and if it is assumed that the ab-sorption profile is determined by Doppler broadening,then the ab-sorption coefficient integrated over the absorption-line profile can be approximated by the absorption coefficient at the absorption peak maximum (K max ).The relationship between K max and the number density of ground state atoms is given by the following equation:K max ¼2l 2=l D ÀÁl n 2=PÀÁ0:5P e 2=mc 2ÀÁN o f where l D is the Doppler width of the line;l the wavelength of the absorption maxima;e and m the charge and mass of an electron,res-pectively;c the velocity of light;and f the oscillator strength (average number of electrons per atom that can be excited by the incident radi-ation (l )).Therefore,the absorbance is directly proportional to the concentration of atoms,provided that the absorption profile is domi-nated by Doppler broadening.8.2.3Atomic fluorescence spectroscopyAFS quantifies the discrete radiation emitted by excited state atoms that have been excited by radiation from a spectral source.There are a number of mechanisms that are responsible for the atomic fluorescence signal:resonance fluorescence,step-wise fluorescence,direct-line fluo-rescence,and sensitized fluorescence.Generally,the lowest resonance transition (1-0)is used for AFS.If a line source is used for excitation and if the atomic vapor is dilute,then the radiant power of the atomic T.H.Risby230fluorescence signal (I f )can be related to the concentration of ground state atoms by the following equation:I f ¼e 2O f L l 2f d F I L O A N o ÀÁ6P mc 2D l D ðÞ2l n 2PÀÁwhere O f /4p and O A /4p are the solid angles of fluorescence and exci-tation that are measured by the instrument,or are incident upon the atom reservoir,respectively;L the length of the atom reservoir in the analytical direction;F the atomic fluorescence quantum efficiency;I L the integrated radiant power for the incident beam per unit area;@a correction factor that accounts for the relative line widths of the source and absorption profiles;and D l D the Doppler half width of the fluores-cence profile.On the basis of this equation it can be seen that the radiant power of atomic fluorescence signal is directly proportional to the concentration of the ground state atoms and to the radiant power of the exciting radiation.Therefore,increasing the intensity of the incident beam will improve the sensitivity of the technique.8.3INSTRUMENTATIONThe following block schemas show the essential instrumental features of the various atomic spectroscopy techniques.Clearly,there are many similarities between these techniques.The subsequent discussions will describe the instrumental components of these tech-niques.Atomic emissionspectroscopy ATOM RESERVOIRMONOCHROMATORPHOTOMULTIPLIER READ OUT SYSTEMAtomic absorptionspectroscopyATOMRESERVOIRMONOCHROMATORPHOTOMULTIPLIER READ OUT SYSTEM SPECTRAL SOURCE Atomic spectroscopy231AtomicfluorescencespectroscopyATOM RESERVOIRMONOCHROMATORPHOTOMULTIPLIER READ OUT SYSTEMSPECTRAL SOURCE8.3.1Atom reservoirsThe production of atoms is a common critical requirement of all these techniques and there are various devices that can be used to generate the atoms in analytically useful spectroscopic states.The following schema summarizes the processes that occur during atomi-zation:Atomization½SOLUTION !NEBULIZATION ½AEROSOL !DESOLVATION½SOLID !VOLATILIZATION½VAPOR !DISSOCIATION½ATOM8.3.1.1Flame atomizersFlames have been traditionally the most popular atom reservoirs for all atomic spectroscopic techniques since they provide the most convenient way to generate atoms.Typically,solutions are aspirated into the ox-idizer gas of a premixed hydrocarbonflame via a pneumatic nebulizer. Direct nebulizers pass the entire liquid aerosol that is generated into theflame,whereas indirect nebulizers pass only liquid aerosol particles of a given size and size distribution.During passage through theflame, the aerosol particles are desolvated,dissociated,and atomized.The efficiencies and reproducibilities of these processes will define the limit of detection that can be obtained and therefore considerable effort has been expended in developing efficient and quantitative nebulizers and atom reservoirs.The size and distribution of the aerosol particles will play a major role in the atomization efficiency since if the aerosol drop-lets are too large they may have insufficient residence time in theflame to be completely atomized or if the aerosol droplets are too small they can be desolvated in the nebulizer and be lost by collisions with the walls.Therefore,for a given composition offlame gases and burnerT.H.Risby232Atomic spectroscopythere will be an optimum size and size distribution for the aerosol particles.The residence time available in theflame for atomization of the sample is dependent upon theflow rates of the fuel and oxidizer, and theseflow rates are determined by the burning velocities of the particularflame gas mixture.Most atomic spectrometers employ indi-rect pneumatic nebulizers to generate a liquid aerosol of optimum size and size distribution and for these devices only about10%of the as-pirated sample reaches theflame.Theflame properties will also affect the atomization efficiency.The most popularflames used in analytical atomic spectroscopy are air–acetylene or nitrous oxide–acetylene.The former has aflame tem-perature of approximately23001C and the latter28001C.The increased atomization efficiency of the nitrous oxide–acetyleneflame is not solely the result of the increase inflame temperature since thisflame contains significant concentrations of excited cyanogen radicals(CN*)whose presence in theflame is exhibited by the emission of molecular bands in the region650nm(known as the red feather).The nitrous oxide–acetyleneflame is recommended for those elements that form stable molecular species,such as refractory oxides,which can persist through theflame without atomization(such as aluminum,barium,beryllium, calcium,scandium,silicon,tantalum,titanium,uranium,vanadium, tungsten,zirconium,the lanthanides,and the rare earths).High con-centrations of the reactive excited cyanogen radicals will reduce re-fractory oxides to atoms.The cooler air–acetyleneflame is preferred for a different group of elements that have low ionization potentials(such as lithium,sodium,potassium,rubidium,and cesium).The hotterflame could ionize these elements with the result that the atom concentra-tions of the analyte species are reduced.There are a number of dis-advantages to the use offlames as atom reservoirs and the most notable are the quantity of sample required for analysis,the brief residence time spent by the atom in the analytical zone of theflame,and the chemical environment within theflame.As a result,a number of non-flame atom reservoirs have been developed that generate atoms by electrical energy in controlled inert environments.8.3.1.2Nonflame atomizersNonflame atom reservoirs have been developed for specific atomic spectrometric techniques.Electrothermal atomizers(carbon rods,car-bon furnaces,or tantalum ribbons)have been developed for AAS or AFS since they require the generation of ground state atoms,whereas233T.H.Risbyatmospheric pressure inductively coupled argon plasmas have been used to produce excited atoms for AES.8.3.1.3Electrothermal atomizersThe use of resistively heated carbon or tantalum rods,tubes,orfila-ments to generate atoms by thermal energy has increased the sensitiv-ities of AAS and AFS by factors of40–4000.The majority of this improvement in sensitivity is the result of increased residence time that the ground state atoms spend in the analytical zone,with minor con-tributions due to the reductions in chemical or spectral interferences from theflame.Some of the materials of construction of these atomizers can also play a role in the atomization processes since for example hot carbon can increase the reduction of the analyte species.The mode of sample introduction for these atomizers is to introduce a discrete aliquot of sample into the atomizer with a microsyringe(sample sizes are ap-proximately5m l).The analyte is then dried,ashed,and atomized suc-cessively by resistive heating of the atomizer with a time-controlled ramp of low-voltage,high-current electricity.As a result of this mode of sample introduction,the analytical response is a transient pulse as op-posed to a steady-state signal that is produced when a sample is ne-bulized continuously into aflame.Theoretically,the repeatability of the response obtained by electrothermal atomization should be lower than theflame since the peak signal for the former is dependent upon the precision with which the volume of the sample can be introduced.How-ever,the electrothermal atomizers can be carefully controlled so that optimum temperatures can be obtained to dry,ash,and atomize the sample,which is not possible withflames.Also automatic syringes (usually injecting50m l)can be used to introduce the sample into the non-flame atomizer.Electrothermal atomizers can ash the sample in situ to destroy the sample matrix,which is a major advantage since flame atomizers require samples to be preashed.This advantage can also present difficulties unless precautions are taken to correct for absorption due to molecular species by background correction.The absorption profiles of molecular species are broad compared to the absorption profiles of atomic species.Currently,most instrument man-ufacturers use graphite furnace technology,which is based upon the research by L’vov from the early1970s.The major advantage of the graphite furnace is that the atom vapor is maintained in the ana-lyzer cell for significant periods of time allowing quantification to be performed.Also it is easy to generate reproducible temperature–time ramps to dry,ash,and atomize the elements of interest.234Atomic spectroscopy8.3.1.4Inductively coupled plasmasGreenfield and Fassel independently proposed the use of atmospheric pressure inductively coupled argon plasmas as atom reservoirs for atomic spectroscopy.This proposition has revolutionized thefield of AES since this atom reservoir produces large concentrations of excited atoms for most elements in the periodic table.The plasma torch is produced by inductively coupling a high level(1.5kW)of radio fre-quency energy(around30MHz)to aflowing stream of argon.This energy causes the argon to be ionized and various energetic species are produced,such as metastable argon atoms,excited argon atoms,argon ions and energetic electrons.These species will collide with the analyte to produce excited atoms and ions via various fragmentation and ex-citation mechanisms.The plasma torch has sufficient energy to excite most elements and will also populate multiple energy levels for a given element with the result that this atom reservoir is useful for multielement analysis.The population of different energy states for a given element allows more sensitive lines to be used for trace levels of analytes and less sensitive lines for higher concentrations of analyte atoms.This capability enables wide concentrations of samples to be determined without the problem of nonlinearity of response versus concentration.The analyte solutions are nebulized into the argon sup-port using an ultrasonic nebulizer(3MHz)and desolvation system.The desolvation system is necessary since this nebulizer is more efficient than pneumatic nebulizers and without predesolvation the plasma is cooled significantly.Inductively coupled plasmas have also been used successfully as the ionization source for elemental analysis by mass spectroscopy replacing arc and spark sources.8.3.2Spectral sources8.3.2.1Continuous sourcesHigh-pressure electrical discharges were thefirst sources used in atomic spectroscopy.These sources consist of a sealed tubefilled with a gas containing two electrodes.A voltage is applied between the electrodes and at a given voltage an electrical discharge is initiated.Electrons are accelerated by the potential difference between the electrodes and collide with thefiller gas to produce excited molecules,atoms,and ions.At low gas pressures,the predominant output from these lamps is atomic line spectra characteristic of thefiller gas,but as the pressure is increased the spectral output is broadened and a continuous spectra are produced. Hydrogen,deuterium,and xenon are the most widely used gases.235T.H.Risby8.3.2.2Line sourcesHollow cathode lampsThe introduction of sealed hollow cathode lamps by Walsh and his col-laborators was the single event that revolutionized thefield of atomic spectroscopy in the mid-1950s.These devices consist of a hollow cylin-drical cathode manufactured from the element of interest.This cathode and an anode are sealed inside an optically transparent envelope with a quartz front window that is inline with the cathode.The lamp isfilled with a low pressure(1–3torr)of an inert gas(usually neon or argon). The hollow cathode lamp operates by producing inert gas discharge that sputters or vaporizes the element of interest from the cathode.These atomic species are subsequently excited by collision with inert gas ions, energetic inert gas atoms,or electrons to produce excited atoms that deactivate by the emission of characteristic photons.Cathodes can even be manufactured from nonelectrically conducting materials by the judi-cious choice of alloys.Once the discharge is struck a stable glow dis-charge is produced and the hollow cathode lamps can be operated with the minimum current.This operating procedure maintains a stable discharge and ensures that the spectral line output is not broadened. Hollow cathode lamps are available for most of the elements in the periodic table and multielement sources have been made using cath-odes manufactured from mixtures of elements.Electrodeless discharge lampsIn the1960s and1970s,Rains,West,Dagnall,and Kirkbright developed electrodeless discharge tubes as intense line sources for AAS and AFS. These lamps are easy to manufacture in the laboratory and consisted of sealed quartz tubes containing the element of interest or its halide,and low pressure(1–3torr)of an inert gas usually argon.Energy in the microwave region(2.45GHz)is supplied to the tube by placing it in a resonant cavity.The discharge is initiated by supplying electrons with a Tesla coil and intense atomic spectra of the element are obtained. Typically,these lamps produce higher intensity atomic spectra than the corresponding hollow cathode lamp however often the spectral outputs of these lamps are less stable.8.3.3MonochromatorsThe requirements for wavelength dispersion are very different for AES as compared to the spectral requirements of AAS and AFS.For AES it is essential to monitor only the radiation that results from 236the desired atomic transition and high-resolution monochromators (dispersion41.6nm/mm of slit width)are required.The usual mode of operation is to monitor the intensity at a selected wavelength and then repeat the determination at a different wavelength.However,induc-tively coupled plasma-AAS with its ability to perform concurrent multi-element analysis requires a different design of monochromator.The most popular design involves monitoring thefirst-order spectrum of a concave grating spectrometer by placing suitable photosensitive devices on the Rowland circle.These photosensitive devices can be a photodiode array in which each element in the array can be monitored separately or else multiplefixed exit slits with photomultipliers placed at each slit. This latter arrangement is often called a quantometer or polychroma-tor.Obviously there are a limited number of wavelengths that can be measured with this type of spectrometer with photomultipliers al-though advances in photodetection devices are reducing this limitation. The wavelength dispersion requirements for AASs and AFS are much less demanding than AES since the hollow cathode lamp is already producing radiation that is characteristic of the element under inves-tigation.Therefore,the monochromator has only to separate the emis-sion line of interest from other nonabsorbing lines.Theoretically, interferencefilters could replace these monochromators.8.3.4Read-out systemsPhotomultipliers are generally used to convert the spectral radiation to an electrical current and often phase-sensitive lock-in amplifiers are used to amplify the resulting current.AES and AFS require similar read-out systems because both methods are measuring small signals.The diffi-culty associated with both these methods is the separation of the signal for the atomic transition of interest from the background radiation emitted by excited molecular species produced in the atom reservoir.AFS phase locks the amplifier detection circuit to the modulation frequency of the spectral source.Modulation of the source is also used in AAS.8.4GENERAL CONSIDERATIONS8.4.1Atomic emission spectroscopyAES quantifies the deactivation of excited atoms.Atom reservoirs will also produce excited molecules that could interfere with the sub-sequent analysis since emission from excited molecular species is broad237compared to the emission from excited atomic species.Multielement AES can identify and quantify concurrently all the elements contained in a sample,which is a major advantage compared to the single-element techniques,atom absorption,and AFS.Until the introduction of in-ductively coupled plasmas as atom reservoirs for AES,the number of elements that could be determined was limited by the availableflame energy.Only those elements with low excitation energies could be de-termined unless arcs or sparks atom reservoirs were used.These latter types of atom reservoirs are not suitable for trace and ultratrace ana-lysis as a result of extensive molecular spectral interferences.The cur-rent awareness of the importance of complex antagonistic and/or synergistic interactions between elements has increased the interest in rapid multielement analyses and inductively coupled plasma-AES is ideally suited for this application.Internal standards are generally added to the sample to aid identification and the selection of internal standards(such as gallium and yttrium)is based on the composition of the matrix.These elements enable chemical or spectral interferences to be subtracted.Theoretically,the inductively coupled plasma source should have sufficient energy to atomize all the elements in the sample and sample pretreatment should be minimal.8.4.2Atomic absorption spectroscopyThe instrumental requirement and cost of atomic absorption spectro-meters are considerably less than those for multiwavelength atomic emission spectrometers.AAS quantifies the concentration of the ele-ment on the basis of the absorption of radiant energy by ground state atoms and the analytical response is based on the difference between the incident radiation and the transmitted radiation,i.e.,the difference between two large signals.Therefore,it is imperative to use a spectral source with a very stable spectral output unless a double-beam spec-trometer is used.Generally,the radiation from the spectral source is electronically modulated so that it can be selectively amplified with a lock-in amplifier.This mode of detection discriminates against the continuous background radiation from other species present in the atom reservoir.The usual sources of radiation for AAS are hollow cathode lamps that are available for most elements.The major limi-tation of AAS is the need to use a different hollow cathode lamp for each element since the spectral stability of multielement hollow cathode lamps is often poorer than single-element lamps.No spectral interfer-ences from other elements are observed with AAS although spectral 238interferences can be produced by molecular species.Molecular spectral interferences can be minimized by the use of background correction with a hydrogen or deuterium continuum source or on the basis of the Zeeman effect,which occurs when energy levels are split by placing atoms in an external magneticfield.Molecular interferences are more significant with electrothermal atomizers than withflames.Since the magnitude of the absorbance is proportional to the path length,atom reservoirs have been designed so that the maximum concentration of ground state atoms are in the incident beam which is exactly opposite to the designs for atomic emission to AFS.The latter sources should be thin in the direction of measurement to avoid self-absorption.AAS can be used to quantify selectively any element that can be produced in the ground state,provided that a suitable spectral source is available. These spectral sources also allow facile optimization of the monochro-mator to the absorption maximum.The analytical response is displayed as a percent transmission or else as the logarithm of the percent trans-mission by the use of logarithmic amplifiers.8.4.3Atomicfluorescence spectroscopyThe instrumental requirements of AFS are the same as that of AAS with the exception that the incident radiation is at right angles to the analytical measurement direction.The atomicfluorescence signal is amplified with a phase-sensitive amplifier that is locked into the modu-lated incident radiation.AFS is more sensitive than AAS since the limit of detection is defined as the minimum detectable signal as opposed to minimum difference that can be measured between two large signals. Theoretically,the excitation source can be a continuum source,since only the radiation that has the energy that corresponds to the electronic transition will be absorbed and therefore the atom reser-voir is acting as a high-resolution monochromator.However,prac-tically most continuum sources do not have sufficient intensity at the wavelength of interest to produce analytically useful atomicfluo-rescence signals.AFS will quantify selectively any ground state atom that can be excited with incident radiation.The sensitivity of AFS is superior to the other atomic spectrometric techniques for a number of elements(Ag,Cu,Cd,Ni,Sb,Se,Te,Tl,and Zn).AFS is virtually free from spectral interferences although light scattering of the inci-dent radiation can occur when samples with high solid contents are analyzed.This interference only occurs if resonancefluorescence is studied.239。

第6章光源和放大器在光纤系统，光纤光源产生的光束携带的信息。

激光二极管和发光二极管是两种最常见的来源。

他们的微小尺寸与小直径的光纤兼容，其坚固的结构和低功耗要求与现代的固态电子兼容。

在以下几个GHz的工作系统，大部分（或数Gb /秒），信息贴到光束通过调节输入电流源。

外部调制（在第4、10章讨论）被认为是当这些率超标。

我们二极管LED和激光研究，包括操作方法，转移特性和调制。

我们计划以获得其他好的或理念的差异的两个来源，什么情况下调用。

当纤维损失导致信号功率低于要求的水平，光放大器都需要增强信号到有效的水平。

通过他们的使用，光纤链路可以延长。

因为光源和光放大器，如此多的共同点，他们都是在这一章处理。

1.发光二极管一个发光二极管[1，2]是一个PN结的半导体发光时正向偏置。

图6.1显示的连接器件、电路符号，能量块和二极管关联。

能带理论提供了对一个）简单的解释半导体发射器（和探测器）。

允许能带通过的是工作组，其显示的宽度能在图中，相隔一禁止区域（带隙）。

在上层能带称为导带，电子不一定要到移动单个原子都是免费的。

洞中有一个正电荷。

它们存在于原子电子的地点已经从一个中立带走，留下的电荷原子与净正。

自由电子与空穴重新结合可以，返回的中性原子状态。

能量被释放时，发生这种情况。

一个n -型半导体拥有自由电子数，如图图英寸6.1。

p型半导体有孔数自由。

当一种P型和一种N型材料费米能级（WF）的P和N的材料一致，并外加电压上作用时，产生的能垒如显示的数字所示。

重参杂材料，这种情况提供许多电子传到和过程中需要排放的孔。

在图中，电子能量增加垂直向上，能增加洞垂直向下。

因此，在N地区的自由电子没有足够的能量去穿越阻碍而移动到P区。

同样，空穴缺乏足够的能量克服障碍而移动进入n区。

当没有外加电压时，由于两种材料不同的费米能级产生的的能量阻碍，就不能自由移动。

外加电压通过升高的N端势能，降低一侧的P端势能，从而是阻碍减小。

如果供电电压（电子伏特）与能级（工作组）相同，自由电子和自由空穴就有足够的能量移动到交界区，如底部的数字显示，当一个自由电子在交界区遇到了一个空穴，电子可以下降到价带，并与空穴重组。

第41卷第2期2023年4月沈阳师范大学学报(自然科学版)J o u r n a l o f S h e n y a n g N o r m a lU n i v e r s i t y(N a t u r a l S c i e n c eE d i t i o n)V o l.41N o.2A p r.2023文章编号:16735862(2023)0216805林德勒加速度对修正史瓦西黑洞引力势及光的轨迹的影响李慧玲,黄雨萌,李瑶(沈阳师范大学物理科学与技术学院,沈阳110034)摘要:主要研究林德勒加速度对修正史瓦西黑洞周围引力势变化及光的轨迹的影响㊂引入拉格朗日方程,得到修正史瓦西黑洞周围引力势变化和光在赤道平面内的偏折规律㊂在黑洞附近,由于势垒的存在,一部分光在到达黑洞时发生反射;而光子球附近的光将围绕黑洞旋转很多次,然后逃逸到无穷远处,使得黑洞周围产生更大的亮度,这也是黑洞周围 光环 形成的原因;当碰撞参数增大时,势垒逐渐变小,直至消失,这部分光由于没有遇到势垒,进入黑洞内部,形成黑洞内部 阴影 ㊂此外,还讨论了不同林德勒加速度对修正史瓦西黑洞事件视界半径㊁光子球半径及碰撞参数的影响,林德勒加速度越大,对应的事件视界半径㊁光子球半径和碰撞参数越小㊂关键词:林德勒加速度;引力势;事件视界;碰撞参数中图分类号:140.1540文献标志码:Ad o i:10.3969/j.i s s n.16735862.2023.02.013E f f e c t o f t h eR i n d l e ra c c e l e r a t i o no nt h e g r a v i t a t i o n p o t e n t i a la n dt h et r a j e c t o r y t h el i g h ta r o u n da m o d i f i e dS c h w a r z s c h i l db l ac kh o l eL IH u i l i n g,HU A N GY u m e n g,L IY a o(C o l l e g e o f P h y s i c a l S c i e n c e a n dT e c h n o l o g y,S h e n y a n g N o r m a lU n i v e r s i t y,S h e n y a n g110034,C h i n a)A b s t r a c t:I n t h i s p a p e r,w em a i n l y s t u d y t h e i n f l u e n c e o f L i n d l e r a c c e l e r a t i o n o n t h e g r a v i t a t i o n a lp o t e n t i a l c h a n g e a n dt h e t r a j e c t o r y o f t h e l i g h t a r o u n dt h e m o d i f i e dS c h w a r z s c h i l db l a c kh o l e.B yi n t r o d u c i n g t h eL a g r a n g e e q u a t i o n,w e o b t a i n t h e g r a v i t a t i o n a l p o t e n t i a l c h a n g e a r o u n d t h em o d i f i e dS c h w a r z s c h i l db l a c kh o l e a n d t h e d e f l e c t i o n l a wo f t h e l i g h t a r o u n d t h em o d i f i e dS c h w a r z s c h i l d b l a c kh o l e i n t h e e q u a t o r i a l p l a n e.D u e t o t h e e x i s t e n c e o f t h e p o t e n t i a l b a r r i e r,p a r t o f t h e l i g h t n e a r t h eb l ac kh o l e i s r e f l e c t e dw h e n i t r e a c h e s t h eb l a c kh o l e.T h e l i g h t n e a r t h e p h o t o ns p h e r ew i l l r o t a t ea r o u n d t h eb l ac kh o l e m a n y t i m e s,a n dt h e ne s c a p et oi n f i n i t y,r e s u l t i n g i n g r e a t e rb r i g h t n e s sa r o u n d t h eb l ac kh o l e,w h i c hi sa l s ot h er e a s o nf o r t h ef o r m a t i o no f t h e p h o t o nr i n g a r o u n dt h eb l ac kh o l e.W h e n t h e c o l l i s i o n p a r a m e t e r i n c r e a s e s,t h eb a r r i e r g r ad u a l l y be c o m e ss m a l l e ru n t i l i td i s a p pe a r s.T h i s p a r t of t h e l igh t e n t e r s t h eb l a c kh o l e,b e c a u s ei t d o e sn o t e n c o u n t e r t h eb a r r i e r,f o r m i ng th e s h a d o wi n s i d et h eb l a c kh o l e.I na d d i t i o n,w ea l s od i s c u s st h e i n f l u e n c eo fd i f f e r e n tL i n d l e r a c c e l e r a t i o n s o n t h e c o r r e c t e dS c h w a r z s c h i l db l a c kh o l e e v e n t h o r i z o n r a d i u s,p h o t o n s p h e r e r a d i u s a n di m p a c t p a r a m e t e r.T h el a r g e rt h e L i n d l e r a c c e l e r a t i o n v a l u ei s,t h e s m a l l e rt h ec o r r e s p o nd i n ge v e n t h o r i z o n r a d i u s,p h o t o n s p h e r e r a d i u s a n d i m p a c t p a r a m e t e r a r e.K e y w o r d s:R i n d l e r a c c e l e r a t i o n;g r a v i t a t i o n p o t e n t i a l;e v e n t h o r i z o n;i m p a c t p a r a m e t e r收稿日期:20221028基金项目:辽宁省教育厅科学研究经费项目(L J KM20221474)㊂作者简介:李慧玲(1977 ),女,辽宁沈阳人,沈阳师范大学教授,博士㊂关于黑洞的讨论由来已久,尤其是在2019年,事件视界望远镜合作组织发布第一张黑洞图像[1],由此关于黑洞图像的相关研究引起人们广泛关注㊂G r a l l a 等[2]研究了史瓦西黑洞附近光的阴影,由于引力势的存在,使光发生偏转,导致了黑洞附近出现光环㊂在四维高斯-博纳特黑洞附近的引力势的研究中发现,不同的耦合常数也将影响引力势变化[3]㊂暗物质对黑洞附近引力势大小也有影响,由于宇宙视界的存在,不同参数会导致引力势变化,进而影响黑洞附近光的偏折[4]㊂H u 和Z h a n g [5]研究了非奇异H a y w a r d 黑洞的引力势及光子运动轨道㊂P e d r a z a 等[6]利用基塞列夫得到的解研究了被q u i n t e s s e n c e 暗能量包围的H a y w a r d 黑洞,通过分析H a y w a r d 黑洞的零测地线和引力势,得到H a y w a r d 黑洞的不同能级对应的轨道类型㊂G u e r r e r o 等[7]分析了具有薄盘吸积的b l a c kb o u n c e s 的测地线方程,并且得到了有效势变化㊂G u o 等[8]得到了K e r r 时空中引力势方程,并分析了光子运动轨迹㊂更多关于引力势的讨论在文献[911]中也有研究㊂本文考虑的林德勒修正的史瓦西黑洞[12]最初是由G r u m i l l e r 提出的,以解释恒定的径向力[13]㊂H a l i l a o y 等[14]研究了具有林德勒修正的史瓦西黑洞的测地线㊂关于林德勒加速度如何影响G r u m i l l e r 黑洞光谱学的研究在文献[15]中做了详细讨论,更多关于林德勒加速度对黑洞的影响也已经得到了深入的研究[1621]㊂在本文中,主要考虑了林德勒加速度对黑洞周围引力势及光的轨迹的影响㊂对于不同的参数,黑洞势垒发生改变,进而影响黑洞周围光的偏折轨迹,对黑洞周围光环亮度产生影响㊂1 林德勒修正的史瓦西黑洞度规和事件视界格鲁米勒构建了一个星系外中心物体引力的有效模型,称为林德勒修正的史瓦西黑洞几何㊂黑洞时空中的林德勒项导致测试粒子测地线的加速度异常㊂林德勒修正的史瓦西黑洞是由下列作用量所描述的一般有效引力理论的解[17]:S =-ʏd 2x -g [Φ2R +2(췍Φ)2+8a Φ-6ΛΦ2+2](1)这里:g 是度规张量的行列式;Φ表示标量场;Λ是宇宙常数;R 是里奇标量;a 代表林德勒加速度㊂将变分原理应用于作用量并求解相应的场方程后,可以得到以下模拟红外引力的球对称度规:d s 2=-f (r )d t 2+f (r )-1d r 2+r 2d θ2+r 2s i n 2θd φ2(2)其中f (r )=1-2M r-Λr 2+2a r (3)这里M 是黑洞质量㊂当a =Λ=0时,将回到史瓦西黑洞㊂此外,如果M =Λ=0,将变成二维林德勒度规㊂由于本文的讨论不包含Λ,故取Λ=0㊂式(3)可以改写为f (r )=1-2M r +2a r (4)解方程f (r )=0,可以得到事件视界:r +=-1+1+16a M 4a (5)2 林德勒修正的史瓦西黑洞附近的引力势及光的偏折规律首先引入拉格朗日方程:d d λ췍L 췍̇x æèçöø÷μ=췍L 췍x μ(6)这里λ是仿射参量,̇x μ是四速度,L是角动量,其形式如下:L =12g μν̇x μ̇x ν=12-f (r )̇t 2+̇r 2f (r)+r 2(̇θ2+s i n 2θ̇ψ2éëêêùûúú)(7) 一般来说,关注的是赤道平面内的光的偏折,即θ=π2和̇θ=0㊂考虑到拉格朗日方程中不包含t 和ψ,即췍L 췍t=0(8)961 第2期 李慧玲,等:林德勒加速度对修正史瓦西黑洞引力势及光的轨迹的影响췍L 췍ψ=0(9) 它们对应着守恒量E 和L ,即全部的能量和全部的角动量㊂结合式(4)㊁式(6)和式(7),可以得到时间㊁方位角和四速度下的径向分量的表达式:̇t =1b 1-2M r +2æèçöø÷a r (10)̇φ=ʃ1r 2(11)̇r 2+1r 21-2M r +2æèçöø÷a r =1b 2(12)其中方程(11)的 - 号和 + 号分别表示为方位角沿着顺时针和逆时针方向,而参数b 满足b =L E ㊂通过改写方程(12)可以得到̇r 2+V (r )=1b2(13)V ᶄ(r )=0(14)这里V ᶄ(r )表示对r 求导,基于方程(13)和(14),能得到光子球的半径r p h 和碰撞参数b ph ㊂不同的a 对应不同的r +,r p h 和b p h ,结果见表1㊂显然,r +,r p h 和b ph 都随着a 的增大而减小㊂表1 M =1时不同a 下事件视界r +,光子环半径r p h 和碰撞参数b p h T a b l e1 T h en u m e r i c a l r e s u l t s o f r +,r p h ,b ph f o r d i f f e r e n t a w i t h M =1a =0.0001a =0.001a =0.02a =0.05a =0.1a =0.4a =0.55a =0.8r +1.99921.99211.86141.70821.53111.07520.96840.8483r ph 2.99912.99112.83882.64912.41621.76041.59711.4098b p h 5.19155.14994.43873.70972.98431.56071.30201.0401(a )a =0.001(b )a =0.02图1 M =1时有效势的变化,区域1对应的是V (r )<1/b 2p h ,区域2对应的是V (r )=1/b 2p h ,而区域3对应的是V (r )>1/b 2p h F i g .1 O v e r v i e wo f t h ee f f e c t i v e p o t e n t i a l w i t h M =1.R e g i o n 1c o r r e s p o n d s t o V (r )<1/b 2p h ,r e g i o n 2c o r r e s p o n d s t o V (r )=1/b 2p h ,a n d r e g i o n 3c o r r e s p o n d s t o V (r )>1/b 2p h 图1分别表示出了a =0.001和a =0.02时的有效势㊂可以看出,在事件视界处有效势消失,而在光子球处,有效势达到最大值,随后又随着r 的增大而减小㊂当光沿着径向移动的时候,有效势将影响它的运动轨迹㊂在区域1,光线遇到势垒后向外辐射;在区域2,光渐进地接近光子球,由于角速度是非零的,光将绕着黑洞旋转很多次;在区域3,光将继续向内移动,并且没有遇见势垒,最后进入黑洞内部㊂基于运动方程,结合式(11),光的运动轨迹可以表示为d r d φ=ʃr 21b 2-1r 21-2M r +2æèçöø÷a r (15)把u =1/r 带入式(15),可以得到071沈阳师范大学学报(自然科学版) 第41卷d u d φ=1b 2-u 21-2M u +2a æèçöø÷u ʉH (u )(16)解式(16),可以得到光的轨迹,即图2㊂光从右侧进入黑洞,在区域1(b <b p ),光由于没有遇到势垒而进入到黑洞内部;在区域2(b =b p ),光围绕黑洞旋转很多次;在区域3(b >b p ),由于势垒作用,光轨迹发生偏折,不能进入黑洞㊂这与图1相对应㊂(a )a =0.001(b )a =0.02图2 设M =1时,a =0.001和a =0.02的极坐标(r ,φ)图像(所有射线的碰撞参数的间距为0.2,黑洞用黑色圆盘表示)F i g .2 T r a c k sw i t h M =1i n p o l a r c o o r d i n a t e s (r ,φ),f o r a =0.001a n d a =0.02(t h e i m p a c t p a r a m e t e r s f o r a l l r a y s h a v ea s p a c i n g o f 0.2,b l a c kh o l e s a r e r e p r e s e n t e db y b l a c kd i s k s )3 结 论本文讨论了林德勒加速度对修正史瓦西黑洞周围引力势及光的轨迹的影响㊂黑洞附近的光在引力的作用下会偏转,黑洞周围出现不同的亮度㊂由于势垒的存在,一些光被反射㊂光子球附近的光围绕黑洞旋转无数次,从而产生更大的亮度㊂当b 小于b p 时,势垒不再阻碍光的运动,光进入黑洞内部㊂通过计算,得到了不同林德勒加速度对应的事件视界半径㊁光子环半径和碰撞参数的大小㊂显然,随着林德勒加速度的增加,事件视界半径㊁光子环半径和碰撞参数都在减小,这改变了观测者与事件视界之间的距离,并影响了观测图像㊂本文只讨论了修正常数对黑洞引力势及光的轨迹的影响,有关黑洞周围阴影及光子环的其他问题,在本文并没有讨论㊂在未来的工作中,也可以进一步计算黑洞周围光子环和阴影的变化㊂参考文献:[1]A K I Y AMA K ,A L B E R D IA ,A L E F W ,e ta l .F i r s t M 87e v e n th o r i z o nt e l e s c o p er e s u l t s .Ⅳ.I m a g i n g th ec e n t r a l s u p e r m a s s i v eb l a c kh o l e [J ].A s t r o p h ys JL e t t ,2019,875(1):L 4.[2]G R A L L ASE ,HO L ZD E ,WA L D R M.B l a c kh o l es h a d o w s ,p h o t o nr i n g s ,a n d l e n s i n g r i n g s [J ].P h ysR e vD ,2019,100(2):024018.[3]Z E N G X X ,Z HA N G H Q ,Z HA N G H.S h a d o w sa n d p h o t o n s p h e r e s w i t h s p h e r i c a la c c r e t i o n si nt h ef o u r -d i me n s i o n a lG a u s s -B o n n e t b l a c kh o l e [J ].E u rP h y s JC 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