Poles and zeros of the scattering matrix associated to defect modes

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。

SPARSE BAYESIAN SAR IMAGING OF MOVING TARGETVIA THE EXCOV METHODWuge Su, Hongqiang Wang, Bin Deng and Ruijun WangCollege of Electronic Science and Engineering, National University of Defense TechnologyChangsha, ChinaABSTRACTThis paper presents a method for imaging of moving targets via the compress sensing by treating the imaging as a problem of signal representation in an over-complete dictionary. The essential idea behind sparse signal representation models comes from the fact that SAR ground moving targets are sparsely distributed in the observation scene and the received SAR echo is decomposed into the sum of basis sub-signals, which are generated by discretizing the target spatial domain and velocity domain.A sparse Bayesian recovering method named the expansion-compression variance-component based method (ExCoV) is used for image reconstruction since it is automatic and demands no prior knowledge about signal-sparsity or measure-noise levels, which is significantly faster than sparse Bayesian learning, particularly in large-scale problems. The numerical experiments using ExCoV method have estimated moving-targets at different velocities in the case of low SNR, and the target image has higher resolution and lower side-lobe as the number of measurements is small compared with traditional algorithms.Index Terms— Synthetic aperture radar, moving target imaging, sparse Bayesian recovery, expansion-compression variance-component1. INTRODUCTIONGround moving target imaging is attracting a great interest from the synthetic aperture radar (SAR) community since they can provide additional information for understanding SAR images [1]. However, in the azimuth direction, the image of moving targets in the observed scene may be displaced or blurred because of the motion of targets. To deal with this problem, most of the existed algorithms manipulate a single range cell and take target azimuthal echoes as linear or sinusoidal frequency modulated (SFM) signals [2, 3]. All of these methods based on the FFT method, such as time-frequency analysis, Keystone transform, Fractional Fourier transform and so on. There are variants improvement to these basic algorithm via pre and post processes[4], the attractiveness of these algorithms is that they are simple to implement and computationalefficiently but they suffer from high sidelobe levels and thus yield low resolutions.Recently, there has been a growing interest in compressed sensing (CS), which can be used to recover sparse signals [5]. Sparse signals are just the sparsest solution by solving an optimization equation. In this paper, we will focus on sparse recover algorithm which can promote sparsity for SAR ground moving target imaging. The conventional sparse recover algorithms, such as compressive sampling matching pursuit (CoSaMP) [6], can be used to recover sparse radar images with high resolution. Nevertheless, their performance is sensitive to user parameters which are typically functions of the noise or signal sparsity levels and require tuning. Setting the user parameters is not trivial and the reconstruction performance depends crucially on their choices. On the other hand, the sparse recovery schemes can be interpreted using the Bayesian philosophy, such as sparse Bayesian learning (SBL) [7] and Bayesian compressive sensing (BCS) [8]. These sparse Bayesian recover algorithm employs a sparse prior on the signal [9,10], and estimates the parameter based on the Bayesian inference. Moreover, the sparse Bayesian recover algorithm is automatic and does not require tuning or knowledge of signal sparsity or noise levels [11]. In this paper, we use a sparse Bayesian recovering algorithm named ExCoV [12] to SAR imaging of moving target, which generalizes the SBL model and has a much smaller number of parameters than SBL [13]. Unlike SBL, which assign a distinct variance component on each signal element, the ExCoV partitions the signal into the significant and the complimentary insignificant signal elements, and only assign distinct variance components to the candidates for significant signal elements and a single variance component to the rest of the signal. Because of its parsimony of the probabilistic model, the ExCoV is typically significantly faster than SBL, particularly in large-scale problems [12,13].Based on the scattering center model, we recast the SAR moving target imaging problem as a problem of signal representation in an over-complete dictionary [14, 15] andthen exploit ExCoV methods to reconstruct the SAR images and motion parameters. By using the concepts provided by CS theory, we propose a reliable imaging method that is highly desirable. Our approach is better than the traditional DFT-based algorithm and other sparse reconstruct algorithm. Also, our method can estimate the location of multiple targets with different speeds and obtain higher resolution and lower side-lobe SAR image with fewer measurements.2. WAVENUMBER-DOMAIN SIGNAL MODEL OFSAR MOVING TARGETSThe typical side-looking SAR geometry is illustrated in Fig.1, suppose the radar works in the spotlight mode and the radar moves at velocity of a V . The X-axis and Y-axis represent the range and azimuth direction respectively. Then, with the small observation angle θ and slow-time η, the radar moves to tan a c c y V t R R θθ′==≈. The θ has the similar meaning as slow-time η. We consider an arbitrarymoving target (,)P x y in the SAR imaging field and let vector ϑ represent the invariant target motion parameter, such as the initial position (),x y , velocity V etc. As shown in Fig.1, the radar is located at a y V η= and suppose thetarget moves to (,)x y . ()σϑ is the scattering coefficient of a target with respect to motion parameter ϑ. Thus thedistance model of the target can be represented by,,() cos sin R x y ϑϑθϑθθθθ=≈+ (1)The spotlight SAR imaging could be represented in the wavenumber domain as shown in Fig.2. Let 24/R K k f c π==, 4/c c K πλ=, the echo of the targetcan be obtained by,,(,;)()exp exp (cos sin )s K P K jK x y ϑθϑθθϑθθª=−¬ªº−+¬¼< (2) where K is the two way wavenumber, and ()P K is the Fourier transform of the transmitted signal. The first two terms of (,;)s K θϑ in (2) are disappeared through rangecompression and motion compensation, and the total target signal model becomes,,(,)()exp (cos sin )G K jK x y d ϑθϑθθσϑθθϑªº=−+¬¼³< (3)where ()σϑ denotes the reflectivity of the target. Suppose the point target (,)P x y moves with constant range speed xv and azimuth speedyv , andthen ,x x x v ϑθη=+, ,y y y v ϑθη=+. The (,)x y constitutesthe target space T D . Consider the extended target space ,T V D which lies in the product space [,][,]i f i f x x y y × ,,,,[,][,]x i x f y i y f v v v v ××. Here (,)i f x x and (,)i f y y denotes the initial and final positions of the target space of interest along each axis respectively, ,,(,)x i y i v v and ,,(,)x f y f v v denotes the minimum and maximum velocity candidates of the extended target space along range and azimuth velocity axis respectively. Considering the errors ε (modeling and measurement), the extended target space ,T V D is discretized as a four dimensional space of size x y N N P Q ×××. x N ,y N , P and Q are the numbers of discretized values of x ,y ,x v and y v . Note K and θcan also be discretized into M and N values respectively, then the SAR receivedecho can be expressed as the sum of all sub-signals(,,,)(,,,)0000(,)(,)yx x y x y x y N N QP n n p q n n p q m n n n p q G m n ahK θε=====+¦¦¦¦ (4)(,,,)(,)exp[()cos ()sin ]x y x p y q n n p q m n m n x n n m n y n n h K jK x v jK y v θηθηθ−+−+ (5)and (,,,)((),(),(),())x y n n p q x y x y a x n y n v p v q σ=. Vectorize all(,,,)(,)x y n n p q m n K θh for 0,,1x x n N =−", 0,,1y y n N =−",0,,1p P =−", 0,,1q Q =−", 0,,1m M =−", 0,,1n N =−", and arrange all of them as a matrix(0,0,0,0)(,,,)(1,1,1,1)12[,,,]n np q N N P Q x yvec vecvec −−−− ""h h h Φ, where (,,,)n np q xy vech is avector of length MN , then the size of Φ isx y MN N N PQ ×.Suppose that (,,,)x y n n p q a is the (,,,) th x y n n p q element of a and a is a x y N N P Q ××× reflectivity profile matrix.Let b is the vectorization of a , thus the equation (4) can be expressed in a matrix form as=+g b Φε (6)here,111212[(,),,(,),(,),,(,),N N G K G K G K G K θθθθ=""g (,)]T M N G K θ".y tFig.1. The Spotlight SAR moving target geometryFig.2. The wave number domain3. EXCOV FOR SAR MOVING TARGET IMAGING According to the CS theory, if b is a L-sparse vector, [log()]x y L O L N N PQ = measurements are enough to reconstruct b . Consider L g as randomly selected Lelements of g and LΦ means the corresponding L rowsfrom Φ, then the equation (6) can be solved by using the ExCoV method.We model the pdf of SAR moving target echo vectorL g given b and 2σ using the standard additive Gaussiannoise model 22(|,)(;,)L L L p N σσ=g b g b C Φ. Here, C is a known positive definite symmetric matrix of size MN MN ×, 2σ is an unknown noise-variance parameter and 2σC is the noise covariance matrix. The SBL employs a Gaussian prior on the signal with a distinct variance component on each signal element (;)(0,)i i i p b N γγ=, for 1,,x y i N N PQ =". This prior didn’t capture the signal keyfeature: sparsity. Therefore, the ExCoV method assigns distinct variance components to the candidates for significant signal elements and use only one common variance-component parameter to account for the variabilityof the rest of signal coefficients. Set Λ is the full index setin the signal and the size is m Λ, here x y m N N PQ Λ=. Denote A is the set of indices of the signal elements with distinct variance components and the size is A m , define the complementary index set =\B A Λ with cardinality B A m m m Λ=−. Accordingly, we partition L Φ and b intosubmatrices L A Φ and LB Φ, and subvectors A b and B b , then the ExCoV method adopt the following prior model for the signal coefficients22211(|,)(|)(|)(;0,())(;0,())q q p p P N D N D γγγ××=⋅=⋅A B A A A B A A A B B b b b b b δδδ(7) here 222,1,2,(){,,,}q D diag δδδ="A A A A A A δ and 22()q D γγ=B B Iare the signal covariance matrices, the variance components 222,1,2,,,,q δδδ"A A A A for A bare distinct and the common variance 2γ accounts for the variability of B b . Assume that the signal variance components A δ and 2γ are unknown and define the set of all unknown 22(,,,)A γσ=A θδ, thenthe marginal pdf of the observations L g is (|)L p =g θ2211(|,)(|,)(;0,())LL N p p d N P σγ−×⋅=³A gb b b g δθwhere()P θ is the inverse covariance matrix of L g given θ. Under this probabilistic model, we use the generalized maximum-likelihood (GML) rule to select the most efficientparameter assignment [12,13]. Finally, ˆθ will be approximated via applying the ExCoV algorithm tomaximize the GML objective function. 4. NUMERICAL EXAMPLES We present the experiment results using simulated data toverify the feasibility and effectiveness of the proposedmethod. As shown in Fig.1, suppose the range c R from the original to the center of the target is 10 km and the velocityof the platform aV is 200 m/s. We also assume the centralfrequency is 10 GHz with bandwidth 400 MHz, and theangular extent of azimuth is 8D . The size of the observed scene is 10m 10m ×, and the velocity in each direction islimited in (10m/s,10m/s)−. The bins of the dimensions of the extended target space are 0.5m, 0.5m, 1m/s, 1m/s, respectively. There are ten targets with the reflectivity of 1in the scene. Their movements have different types: the first is static; The second is moving along the range direction with 2m/s speed; The third one is moving with range andazimuth speed 4m/s; The fourth one is moving with 2m/range speed and -2m/s azimuth speed; For the rest of thetargets, their range speed and azimuth speed are both 2m/s. These targets are chosen randomly in experiment. When 0η=, the scatters are located in the observed scene as illustrated in Fig.3(a). Add the Gaussian noise SNR=10dBto the radar echo, then the imaging result of the Range-Doppler algorithm is illustrated in Fig. 3(b), it can be seen that the image is blurred due to the poor resolution and the high side-lobe. Based on the compressive sensing principle, Fig.3 (c) and (d) show the reconstruction results of theCoSaMP and the ExCoV method. Note that the targetnumber should be known for the CoSaMP method but notfor ExCoV. The target images are well focused and exhibit the advantages in high resolution, reduced speckle andsidelobes. To analyze the effect of versus (additive) noise level, Fig.4 depicts the estimation root mean square (RMS) error varies with SNR relation to the target motion parameters,such as the location of the target (,)x y and the target velocity (,)x y v v . As shown in the Fig.4 (a-d), compared tothe CoSaMP, it can be observed that both methods arrive at high precision estimations after 10 dB, and it alsodemonstrates the ExCoV method can recover the targets signature parameters accurately when in the low SNR, which indicate the robustness of the ExCoV method to loss and noise of measurement.5A z i m u t h (m )Range (m)102030405060Imaging by FFTA z i m u t h (m )Range (m)-505-5050.10.20.30.4(a) (b)A z i m u t h (m )Range (m)Imaging by CoSaMP-505-505050100150200250A z i m u t h (m )Range (m)Imaging by ExCoV-505-505050100150200250(c) (d) Fig.3. (a)The observed scene. (b)The RD algorithm. (c)TheCoSaMP algorithm. (d)The ExCoV algorithm.-1001000.20.40.60.811.21.41.61.82SNR M S ExCoSaMP ExCoV -1001000.20.40.60.811.21.41.61.82SNR M S EyCoSaMPExCoV(a) (b) SNR M S EVxSNR M S EVy(c) (d)Fig.4. (a)The MSE vusus SNR for the target range location. (b) The MSE vusus SNR for the target azimuth location. (c) The MSE vusus SNR for the target range velocity. (d) TheMSE vusus SNR for the target azimuth.5. CONCLUSIONSIn this paper, based on the CS theory, we establish the sparse SAR moving target imaging model and utilize a sparse Bayesian reconstruction method named ExCoV to reconstruct the SAR moving targets. Compared totraditional algorithms (CBP and CoSaMP), the proposed method can exactly recover the reflectivity distribution andmotion parameters as demonstrated by the simulation results, which achieving good performance.ACKNOWLEDGMENTWe acknowledge the support from the National Natural Science Foundation for Young Scientists of China (No. 61302148 No.61101182), National Science Fund (No. 61171133) and Natural Science Fund for Distinguished Young Scholars of Hunan Province (No. 11JJ1010).REFERENCES[1] R. P. Perry, R. C. Dipietro, and R.L. Fante, “SAR imaging of moving targets,” IEEE Trans. on Aerosp. Electron. Syst.,vol.35, no.1, pp.188-200, Jan.1999. [2] T Sparr, “Moving target motion estimation and focusing inSAR images”, Proc. IEEE RADAR, pp. 290–294, 2005 [3] G. Li, X. G. Xia, and Y. N.Peng, “Doppler KeystoneTransform for SAR imaging of moving Targets,” IEEE Geosci. Remote Sens. Lett., vol. 5, no. 4, pp. 573-577, 2008. [4] X. Li, B. Deng, Y. L. Qin, Y. P. Li, “The influence of target micromotion on SAR and GMTI”, IEEE Trans. Geosci.Remote Sens., vol. 49, no.7, pp.2738-2751, 2011. [5] E. J. Candes, J. Romberg, and T. Tao, “Robust uncertanity principles: exact signal reconstruction from highly incompletefrequency information,” IEEE Trans. Inf. Theory, vol.52, no.2, pp. 489-509, 2006. [6] D. Needell and J. A Tropp, “CoSaMP: Iterative signal recovery from incomplete and inaccurate samples,” Appl Comput Harmonic Anal., vol.26, no.3, pp.301-321, 2009. [7] D. P. Wipf and B. D. Rao, “Sparse Bayesian learning for basis selection”, IEEE Trans. Signal Process., vol.52, no.8, pp.2153-2164, 2004. [8] S. H. J, Y. Xue, and L. Carin, “Bayesian compressivesensing,” IEEE Trans. Signal process., vol. 56, no. 6, pp.2346–2356, 2008. [9] A. M. Djafari, “Bayesian approach with prior models whichenforce sparsity in signal and image processing”, EURASIP J. Adv. Signal Process., vol.52, pp.1-19, 2012. [10] S. Zhu, A. D., H. Q. Wang, B. Deng, X. Li and J. J. Mao,“Parameter estimation for SAR micromotion target based on sparse signal representation”, EURASIP J. Adv. SignalProcess., vol.13, pp.1-19, 2012. [11] H. C. Liu, B. Jiu, H. W. Liu, Z. Bao, “Superresolution ISAR imaging based on sparse Bayesian learning”, IEEE Trans Geosci Remote Sens., in press.[12] A. Dogandzic and K. Qiu, “ExCoV: Expansion-compressionvariance-component based sparse-signal reconstruction from noisy measurements”, Proc. 43rd Annu. Conf. Inform. Sci.Syst., pp.186–191, 2009. [13] K. Qiu, A. Dogandzic, “Variance-component based sparse signal reconstruction and model Selection”, IEEE Trans. Signal Process., vol.58, no.6, pp.2935-2952, 2010. [14] Q. S. Wu, M. D. Xing, C. W. Qiu, B. C. Liu, Z. Bao, and T. S. Yeo, Motion parameter estimation in the SAR system with low PRF sampling, IEEE Geosci. Remote Sens. Lett., vol. 7,no. 3, pp.450-454, 2010.[15] I. Stojanovic, W. C. Karl. Imaging of moving targets with multi-static SAR using an overcomplete dictionary, IEEE J.Sel. Topics Signal Process., vol. 4, no. 1, pp.164-176, 2010.。

TPO1-L2 确定石头年代的方法TPO4 L3 ＊＊沙漠类生词预览1.Death Valley死谷：加利福尼亚州东部和内华达州西部的干旱盆地。

西半球低于海平面86米(282英尺）的最低点位于该谷2.bulldozer ['buldәuzә] n. 推土机3.slippery ['slipәri] a. 滑的, 光滑的, 靠不住的, 圆滑的, 不稳固的4.vibrate ['vaibreit] vi. 振动, 颤动, 激动, 摇摆, 踌躇5.tilt [tilt] n. 倾斜, 倾向, 船篷, 车篷vt. 使倾斜6.magnetic [mæg'netik] a. 有磁性的, 有吸引力的, 催眠术的7.eliminate [i'limineit] vt. 除去, 排除, 剔除, 消除8.meteorology [,mi:tiә'rɒlәdʒi] n. 气象学, 气象状态9.prospective [prәs'pektiv] a. 预期的, 将来的10.physics ['fiziks] n. 物理学, 物理过程, 物理现象6. What does the professor mainly discuss?A. His plans for research involving moving rocks.B. A difference between two geological forces that cause rocks to move.C. Theories about why desert rocks move.D. Reasons why geologists should study moving rocks.7. According to the professor, what have the researchers agreed on?A. The rocks cannot move after ice storms.B. The rocks do not move at night.C. The rocks never move in circles.D. The rocks are not moved by people.8. The professor mentions experiments on the wind speed necessary to move rocks. What is the professor’s attitude toward theexperiments?A. Their results were decisive.B. They were not carried out carefully.C. They were not continued long enough to reach a conclusion.D. The government should not have allowed the experiments.9. What important point does the professor make about the area where the rocks are found?A. It has been the site of Earth’s highest wind speeds.B. It is subject to laws that restrict experimentation.C. It is accessible to heavy machinery.D. It is not subject to significant changes in temperature.10. What is the professor’s purpose in telling the students about moving rocks?A. To teach a lesson about the structure of solid matter.B. To share a recent advance in geology.C. To give an example of how ice can move rocks.D. To show how geologists need to combine information from several fields.11. Replay: What does the professor imply when he says this:A. The movement pattern of the rocks was misreported by researchers.B. The rocks are probably being moved by people.C. The movement pattern of the rocks does not support the wind theory.D. There must be differences in the rocks’ composition.TPO6 L4 ＊＊＊1.Sahara Desert 撒哈拉沙漠：北非的一个大沙漠，东至大西洋沿岸，西至尼罗河河谷，南至亚特拉斯山脉，北至苏丹境内。

Visibility depth improvement in active polarization imaging in scattering mediaJohn G.Walker,Peter C.Y.Chang,and Keith I.HopcraftA simple image-subtraction technique for further enhancement of the visibility depth in polarizedimaging of surfaces immersed in scattering media is proposed and assessed.The technique is based onactive illumination with circular or linear polarization states and image detection in the original and theopposite,or orthogonal,states.Contrast enhancement is achieved by subtraction of a fraction of theimage recorded in the original state from that recorded in the opposite state.Results demonstrating theeffectiveness of this method,obtained with Monte Carlo techniques,show that the visibility depth can beincreased by as much as a mean free path.The results obtained are compared with those obtained byuse of two alternative methods.©2000Optical Society of AmericaOCIS codes:100.2980,110.7050,120.2130.1.IntroductionThe ability to image object surfaces immersed in scat-tering media is of benefit in a number of applications including imaging through the atmosphere and opti-cal and near-infrared tissue imaging.1–7The applica-tion of interest here is underwater imaging,8–13in which active illumination is usually required and quite modest increases in visibility depth can have significant advantages.In most underwater imag-ing systems the object is illuminated with a light source in proximity to the imaging optics.An im-portant cause of loss in visibility is a lack of contrast that is due to glare from light scattered back into the image-forming optics by the intervening medium. One approach is to employ short-pulsed illumina-tion14and a time-gated detection system,with the time gate set at the return propagation time from the transmit–receive optics to the object of interest. This approach discriminates effectively against light backscattered by the medium,which has mostly trav-eled shorter distances but has the disadvantages of complexity and expense.Experimental results8,7and Monte Carlo simula-tions15have indicated the effectiveness of employing simple polarization-discrimination techniques that could be applied with low cost by modest modification of conventional active-illumination,direct-view,or detector-based imaging systems.The Monte Carlo results showed that a significant increase in image contrast could be obtained for an object that could be modeled as a surface that randomizes the polariza-tion state of reflected light.This was achieved by illumination with either linearly or circularly polar-ized light and detection,or viewing,of the light re-ceived in the orthogonal,or the opposite,polarization state.The effectiveness was shown to depend only weakly on particle shape and on the form of the illu-mination geometry.15The technique proved to be most effective when the intervening medium com-prised small particles;the effectiveness is reduced with increasing particle size and shows little benefit for particles with diameters larger than the wave-length of the light.16We can explain this behavior by noting that,for active illumination,most of the light that degrades the image quality is scattered directly back into the imaging optics by the medium.For small particles a significant fraction of this is caused by single scatter-ing and hence is in a polarization state similar to that of the illumination.By contrast,the light that has reached the object has had its polarization state ran-domized and so contains an equal component in the orthogonal state.Hence viewing in this orthogonal state provides an effective means of discriminating in favor of the light that has reached the object.How-The authors are with the University of Nottingham,UniversityPark,Nottingham NG72RD,UK.J.G.Walker and P.C.Y.Chang͑peter.chang@͒are with the School ofElectrical and Electronic Engineering K.I.Hopcraft is with theDivision of Theoretical Mechanics,School of Mathematical Sci-ences.Received18November1999;revised manuscript received13June2000.0003-6935͞00͞04933-09$15.00͞0©2000Optical Society of America20September2000͞Vol.39,No.27͞APPLIED OPTICS4933ever,as the particle size increases,the light entering the imaging optics contains an increasing fraction of light scattered by both the object and the medium that has a more randomized polarization state and is not so effectively discriminated against.The simple polarization-discrimination method,al-though effective,provides only a limited increase in visibility depth,typically an increase of a factor of less than2compared with conventional viewing. Our purpose in this paper is to assess the effective-ness of a simple image-subtraction technique to ex-tend the visibility depth further than is possible with simple polarization discrimination.2.Image-Subtraction MethodThe basis of the image-subtraction approach is founded on the following observations.In the sim-ple polarization method the light viewed is that re-turned in the opposite state,because it contains a larger proportion of light that has interacted with the object than does the light in the copolarized state. However,even for the opposite-polarized state the principal mechanism for loss of contrast is still the light backscattered directly by the medium.Fur-ther enhancement should be possible,given a means of estimating and then partially removing this con-tribution.Such a means is provided when the image formed in the same polarization state is taken as the illumination.This copolarized image is biased in favor of the light that has not reached the object.As such,it possesses poor image contrast,but it does provide a means of estimating the spatial form of the light backscattered by the medium.The proposed method is to record in addition a copolarized image and to subtract a suitable fraction of this from the opposite-polarized image.This is predicated on the basis that the opposite-polarized image is degraded mainly by additive backscattered light and that the copolarized image provides an es-timate of the form of this addition.This method differs from that proposed and used by Jacques et al.7In their papers they look at the so-called polarization image,which is defined as the difference between the copolarized and the opposite-polarized images divided by the sum of these images. Their method examines light scattering from the su-perficial skin layers and is designed to discriminate against light scattered from deeper tissue layers, whereas the method discussed in this paper attempts to discriminate against scattering from material be-tween the object and the viewer.3.Monte Carlo Computer SimulationMonte Carlo simulations have been used for many ͑more than30͒years to study light scattering—especially through atmospheres,rain or dust clouds, and fogs.17–28Fully polarized implementations are less widespread than scalar Monte Carlo simula-tions,but details of the methodology can be found in Refs.21–28.In the Monte Carlo simulations used to assess the effectiveness of the image-subtraction method the scattering medium is modeled as a slab infinite in thex–y plane containing randomly positioned,nonab-sorbing spherical Mie scatterers,with a relative re-fractive index of 1.2and specified by the sizeparameter ka,where k is the wave number of thelight and a is the particle radius.The Stokes vector ͑I,Q,U,V͒29is used to carry the full polarization in-formation of the rays under simulation.The Muel-ler matrix29–31is used to transform the Stokes vectoron each scattering event.Further details on the im-plementation may be found in Refs.15,16,32,and33.As with most Monte-Carlo-based simulations,coherence effects and particle interactions are ne-glected.By way of validation of the simulation code somecomparisons will be shown between results obtainedfrom the code and results determined by variousother means of solving the radiative transfer equa-tion.31For a plane-parallel slab offinite optical thick-ness Chandrasekhar31derived an exact solution forRayleigh scatterers,and this was tabulated in Ref.34Other numerical schemes to solve the radiativetransfer equation for more general scatterers includethe adding–doubling͑AD͒method35and the fast in-variant embedding method.36Reviews and compari-sons between the various numerical solutions of theradiative transfer problem can be found in Refs.37parisons have been made between thesimulation used in this paper,published data,and apublicly available FORTRAN code.35Excellent agree-ment has been found in these comparisons within thestatisticalfluctuations inherent in the Monte Carlomethod.For example,with a slab of nonabsorbing Rayleighscatterers,of a thickness of1mean free path͑mfp͒oran optical depth of1,above a perfectly absorbingsurface,the Stokes vector of returning light was cal-culated for different zenith and azimuthal angles,␪and␾,respectively.Figure1shows the results forunpolarized light entering normal to the slab,i.e.,␪0ϭ0°.These graphs compare the Stokes parameters with data from Coulson et al.,34results from the AD code of Evans et al.,35and the Monte Carlo simula-tion.Since the latter two Stokes parameters are zero,they are not shown.Our simulation ran for 160million entry rays.The error on the Monte Carlo results is estimated from the standard devia-tion of the data when they are subdivided into16 subsets.͓Coulson’s data are renormalized by1͞␲, and his Q parameter is multiplied byϪ1to reconcile the different definitions of the Stokes vector.͔Another example is the above situation with an entry angle of cos␪0ϭ0.9,with Mishchenko’s data,36 and with the AD code used as benchmarks.Figure 2shows the good agreement of the simulation with other published results.Thefinal Stokes parameter V is always zero for this situation and is not shown; the third Stokes parameter U is also zero for␾ϭ0°or␾ϭ180°and is not shown.͓Mishchenko’s data36 are renormalized by1͑͞␲cos␪0͒,and his U parame-ter is multiplied byϪ1to reconcile the different def-initions of the Stokes vector.͔4934APPLIED OPTICS͞Vol.39,No.27͞20September2000Finally,a slab of nonabsorbing spherical Mie scat-terers,with a relative refractive index of 1.2and specified by the size parameter ka ϭ5.0,was used.Because of a restriction in the AD code,only the first two columns of the Mueller matrix of light returning from the slab could be found.Nevertheless,the sim-ulation was executed with an entry angle of ␪0ϭ40°and 64million rays,and the results are shown in Fig.3for a 10-mfp-thick slab.The typical errors in the Monte Carlo results are of the order of 5ϫ10Ϫ3down to 4ϫ10Ϫ4,depending on the Mueller matrix ele-ments M ij under examination.The M 41element is statistically consistent with the AD result,although it is not very useful.To obtain greater precision,the number of rays would need to be increased a hun-dredfold to reduce the standard deviation by a factor of 10.Because the current code takes ϳ7h to run in total,it is currently impractical to increase the rays by that factor.Since the code ran on a computer equipped with an Intel Pentium II 350MHz,with faster computers,in the near future,it will become more realistic to attempt such a validation of the M 41element.Meanwhile,a form of variance reduction could be implemented to subdue or to suppress sta-tistical noise should this element be required.The optical arrangement of the simulation to as-sess image subtraction is shown in Fig.4.Hence-forth all distances are quoted in terms oftheFig.1.Plots of the first two Stokes parameters against the cosine of the zenith angle from published data ͑labeled Coulson ͒,AD code,and the simulation ͑MC ͒.The upper graph represents I ,the lower,Q ,for a plane-parallel slab of thickness 1mfp with Rayleighscatterers.Fig.2.Plots of the first three Stokes parameters against the cosine of the zenith angle from published data ͑labeled Mish ͒,AD code,and the simulation ͑MC ͒.The upper graphs represents I ;the middle graphs,Q ;and the lower graph,U .The columns,from left to right,show the results at azimuthal angles of 0°,90°,and 180°.20September 2000͞Vol.39,No.27͞APPLIED OPTICS4935scattering mfp.Images are formed by a lens of focal length and diameter both equal to 0.1mfp.The il-lumination may be either collimated ͑dashed arrows ͒or off axis ͑solid arrows ͒.The object is modeled as a randomly depolarizing diffuse reflector with a reflec-tion coefficient of unity and zero in alternate quad-rants.We modeled the depolarization caused by the sur-face by generating random numbers for the Stokesparameters subject to the constraint Q 2ϩU 2ϩV 2ϭ1.This is equivalent to inducing a random rotation in ͑Q ,U ,V ͒and can be represented by a Mueller matrix M of the following block-diagonal formM (␣,␤,␥)ϭͫ103T03R (␣,␤,␥)ͬ,where R is a Euler matrix of dimensions 3ϫ3,03is a zero three-column vector,the superscript T denotes a matrix transpose,and ͑␣,␤,␥͒is the set of Euler angles.This is a pure Mueller matrix 39͑i.e.,it has a corresponding Jones matrix ͒.The average of this matrix over all allowable Euler angles is͗M ͘ϭ΄1000000000000000΅,that is,a completely depolarizing Mueller matrix that justifies the technique used to model depolarization.This model is used because underwater painted sur-faces act principally as volume scatterers,owing to index matching with the binder,which results in de-polarization from multiple scattering beneath the painted surface.404.Simulation ResultsFor purposes of comparison the first three columns of Fig.5show the results of a simulation toillustrateFig.3.Surface plots of the first two columns of the Mueller matrix.The set of plots in the two leftmost columns represents results from the AD code,and simulated data are shown in the two rightmost columns.The axis that runs from 0°to 90°is the zenith angle ␪,and the axis that ranges from 0°to 180°is the azimuthal angle ␾.Fig.4.Simulation geometry with an f ͞1focusing arrangement.4936APPLIED OPTICS ͞Vol.39,No.27͞20September 2000Fig.6.Results of applying polarization enhancement,at a target depth of z ϭ1.5mfp.Each image represents a linear combination of the opposite-polarized image and the copolarized image—the combinations are fractions,ranging from 0%to 20%in intervals of 2%,of the copolarized image subtracted from the opposite-polarizedone.Fig.5.Summary of the images from an object behind a medium of Mie scatterers ͑ka ϭ0.01͒illuminated with a collimated source and imaged with a simulated f ͞1optical system.Each column represents a different imaging technique:The first is the unfiltered intensity,the second is the copolarized output,the third is the opposite-polarized output,the fourth is the optimal linear combination of the opposite-polarized image and the copolarized image,the fifth is the optimal combination of the unfiltered and defocused images,and the final column is the optimal unsharp masked image.The rows,from top to bottom,represent different object depths z ϭ0.5,1.0,1.5,2.0,2.5,3.0,3.5mfp.20September 2000͞Vol.39,No.27͞APPLIED OPTICS4937the effectiveness of the simple polarization technique for collimated illumination through media compris-ing particles with ka ϭ0.01,with circularly polarized light.Each image shown is composed of 60ϫ60pixels.Each column shows the effect of increasing object depth or equivalently of increasing turbidity.The first three columns of this figure are similar to Fig.2͑d ͒of Ref.15,but more simulated photons have been used here,giving less statistical noise.It may be seen that the images formed in the opposite-polarized state ͑third column ͒show better contrast and that the visibility depth is extended beyond that for viewing in either the total intensity ͑first column ͒or the copolarized light ͑second column ͒,albeit to a limited extent.It may also be observed that the form of the copolarized image has a scattered compo-nent similar to the scattered component in the opposite-polarized images.An illustration of applying the subtraction tech-nique to the images in the third row ͑at z ϭ1.5mfp ͒of Fig.5is given in Fig.6.The images shown are the opposite-polarized image minus the given fractions of the copolarized image.If the subtraction results in a negative value,this has been set to zero,which shows as black in the images.It is clear that the image contrast and clarity are significantly improved by the subtraction process.The variation in the im-age contrast with subtraction fraction is shown in Fig.7.Image contrast is calculated over the central 64pixels and is defined here as ͑S t ϪS b ͒͑͞S t max ͒,where S t is the sum of the intensity from light re-turning in the reflecting quadrants,S b is that from light returning in the absorbing quadrants,and S t max is the product of the maximum intensity value and the number of pixels within the reflecting quadrants.This definition is used because it measures not only the variation between the reflecting and the absorb-ing quadrants but also the uniformity within the re-flecting quadrants.For this case the optimum subtraction fraction is seen to be ϳ13%,although contrast in excess of 80%is achieved for subtraction fractions from 12%to 16%,indicating that the selec-tion of the optimum value is not critical.The fourth column of Fig.5shows the subtracted image for a range of object depths,when the optimum subtraction fraction is used at each depth.When we compare these images with the total intensity images ͑first column ͒and the simple polarization method ͑third column ͒,it is evident that the application of the subtraction procedure has significantly improved the visibility depth by ϳ1mfp.It should also be noted that the subtraction fractions used at each depth ͑20%,14%,14%,12%,12%,10%,and 10%,respec-tively,with increasing depth ͒are similar.This sug-gests that for any given conditions a single subtraction fraction could be applied obviatingtheFig.7.Image contrast as a function of the fraction of the copo-larized image subtracted from the opposite-polarized image.The curve corresponds to the polarization-enhanced images for a target depth of z ϭ1.5mfp shown in Fig.6.Fig.8.Results of applying defocusing enhancement,at a target depth of z ϭ1.5mfp.Each image represents a linear combination of the focused and the defocused images—the combinations are fractions,ranging from 0%to 50%in intervals of 5%,of the defocused images subtracted from the focused one.The defocused image is formed closer to the lens than the focused image ͑at v Ϫ␦f ,where v is the distance of lens to focused image,f is the focal length,and ␦ϭ0.50͒.Fig.9.Image contrast as a function of the fraction of defocused image subtracted from the focused image.The different curves correspond to different defocus distances.The curve for ␦ϭ0.5corresponds to that used in Fig.8.4938APPLIED OPTICS ͞Vol.39,No.27͞20September 2000necessity to optimize it precisely for each pair of im-ages.Having demonstrated the potential of the polarization-subtraction procedure,we find it inter-esting to compare its performance with other subtracted-image enhancement methods.We ex-amine two techniques,both variants of unsharp masking.In the first technique 41an estimate of the form of the glare contribution to an image is formed by means of recording a defocused image.The en-hancement is achieved by subtracting a fraction of the defocused image from the in-focus image.The basis for this is that true image detail is rapidly blurred by defocus,whereas the glare,which arises from many depths,is not so strongly changed by de-focus.The second unsharp masking approach 42is that sometimes applied in microscopy;the method is to subtract a low-pass-filtered version of the in-focus image.An illustration of applying the defocus-subtraction technique to the total intensity image in the third row ͑at z ϭ1.5mfp ͒of Fig.5is given in Fig.8.The images shown are the in-focus image from which the given percentages of the defocused image,for a defo-cus distance of half of the focal length of the lens,have been subtracted.As above,if the subtraction results in a negative value,this has been set to zero.It is clear,for this amount of defocus and with a suitable amount of subtraction,that the image con-trast and clarity are improved compared with the intensity image shown in the first column.The vari-ation in image contrast,as defined above,with defo-cus image subtraction fraction,is shown in Fig.9for the defocus value used in Fig.8along with two other values.It may be noted that the maximum contrast value is similar for all three defocus values,indicat-ing that further enhancement by use of other values is unlikely.The peak value of the enhanced contrast is half of that achieved by the polarization-subtraction technique.The fifth column of Fig.5shows the subtracted image for a range of object depths,when the optimal defocus subtraction fraction is used at each depth.When we compare these images with those previ-ously described,it is evident that the defocus method does give some improvement compared with the orig-inal intensity images,but the image contrast and clarity are inferior to those obtained with the polar-ization approach.An illustration of applying the unsharp masking subtraction technique to the total intensity image in the third row ͑at z ϭ1.5mfp ͒of Fig.5is given in Fig.10.The images shown are the total intensity image minus the given percentages of a filtered version of this image.The filter applied in this case was of the form H ͑f ͒ϭexp ͑Ϫf 2͞w 2͒,with w ϭ0.032pixel Ϫ1.For this filter width it may be noted that the image contrast is improved with a suitable amount of sub-traction.The change in image contrast with sub-traction fraction is shown in Fig.11for the filter width used in Fig.10and for two other widths.It can be seen that the maximum contrast value is not strongly dependent on the filter width used,and it may be noted that the peak enhanced contrast value is ϳ86%of that achieved with the polarization-subtraction method.The sixth column of Fig.5shows the subtracted image for a range of object depths,when the optimal filter width and subtraction fraction are used at each depth.When we compare these images with the other columns,it is clear that the unsharpmaskingFig.10.Results of applying unsharp masking enhancement,at a target depth of z ϭ1.5mfp.Each image is a linear combination of the original image and a Gaussian-filtered image.The filter width is w ϭ0.032pixel Ϫ1.From left to right the fraction of the filtered image subtracted from the original image varies from 94%to 130%in steps of4%.Fig.11.Image contrast as a function of the fraction of filtered image subtracted from the original image.The different curves correspond to different Gaussian-filter widths.The curve for w ϭ0.032pixel Ϫ1corresponds to that used in Fig.10.20September 2000͞Vol.39,No.27͞APPLIED OPTICS4939method gives a marked contrast improvement com-pared with the intensity images in thefirst column. The unsharp masking images are also superior to the defocus subtraction images;but it is clear that they show less image contrast and less visibility depth than the images provided by the polarization ap-proach.5.ConclusionsA method for further enhancing the image contrast in active-illumination imaging through turbid media has been described.The method uses polarized il-lumination,and images are recorded in both the orig-inal and the opposite,or orthogonal,polarization states.The method involves the subtraction of a given fraction of the copolarized image from the opposite-polarized puter simulation re-sults have shown that the method is effective in ex-tending the visibility depth beyond that achieved with the simple polarization technique byϳ1mfp. The effectiveness of the polarization-subtraction ap-proach has been compared with that of two previ-ously proposed subtraction methods and has been found to be superior to both.The major drawback of the polarization-subtraction approach is that its im-plementation would necessitate the use of image cap-ture and processing,whereas the basic polarization method may be implemented by simple optics and could be incorporated into a direct view system. This study was supported by the UK Defense Eval-uation and Research Agency,Malvern.We are grateful to D.L.Jordan,G.D.Lewis,E.Jakeman, and P.J.Roberts for access to experimental data and for a number of useful discussions.References1.E.P.Zege,A.P.Ivanov,and I.L.Katsev,Image Transferthrough a Scattering Medium͑Springer-Verlag,Berlin,1991͒.2.P.Naulleau and D.Dilworth,“Motion-resolved imaging ofmoving objects embedded within scattering media by the use of time-gated speckle analysis,”Appl.Opt.35,5251–5257͑1996͒.3.B.Chance and R.R.Alfano,eds.,Optical Tomography,PhotonMigration,and Spectroscopy of Tissue And Model Media:The-ory,Human Studies,and Instrumentation,Proc.SPIE2389͑1995͒.4.G.E.Anderson,F.Liu,and R.R.Alfano,“Microscope imagingthrough highly scattering media,”Opt.Lett.19,981–983͑1994͒.5.S.R.Arridge and J.C.Hebden,“Optical imaging in medicineII:modelling and reconstruction techniques,”Phys.Med.Biol.42,841–853͑1997͒.6.S.Fantini,S.A.Walker,M.A.Franceschini,M.Kaschke,P.M.Schlag,and K.T.Moesta,“Assessment of the size,position, and optical properties of breast tumors in vivo by noninvasive optical methods,“Appl.Opt.37,1982–1989͑1998͒.7.S.L.Jacques and K.Lee,“Imaging tissues with a polarizedlight video camera,”in1999International Conference on Bio-medical Optics,Q.Luo,B.Chance,L.V.Wang,and S.L.Jacques,eds.,Proc.SPIE3863,68–74͑1999͒;S.L.Jacques, J.R.Roman,and K.Lee,“Imaging superficial tissues with polarized light,”Lasers Surg.Med.26,119–129͑2000͒.8.J.S.Jaffe,K.D.Moore,D.Zawada,B.L.Ochoa,and E.Zege,“Underwater optical imaging:new hardware and software,”Sea Technol.39,70–74͑1998͒.9.R.X.Li,H.H.Li,W.H.Zou,R.G.Smith,T.A.Smith,and T.A.Curran,“Quantitative photogrammatic analysis of digital un-derwater video imagery,”IEEE J.Ocean.Eng.22,364–375͑1997͒.10.S.M.Christie and F.Kvasnik,“Contrast enhancement of un-derwater images with coherent optical image processors,”Appl.Opt.35,817–825͑1996͒.11.X.M.Lui,J.L.He,X.H.Sun,and M.D.Zhang,“Instrumentfor collimating and expanding Gaussian beams for underwater laser imaging systems,”Opt.Eng.37,2467–2471͑1998͒. 12.G.D.Gilbert and J.C.Pernicka,“Improvement of underwatervisibility by reduction of backscatter with a circular polariza-tion technique,”Appl.Opt.6,741–746͑1967͒.13.J.Cariou,B.Le Jeune,J.Lotrian,and Y.Guern,“Polarizationeffects of seawater and underwater targets,”Appl.Opt.29, 1689–1695͑1990͒.14.G.R.Fournier,D.Bonnier,J.L.Forand,and P.W.Pace,“Range-gated underwater laser imaging system,”Opt.Eng.32,2185–2190͑1993͒.15.P.C.Y.Chang,J.G.Walker,K.I.Hopcraft,B.Ablitt,and E.Jakeman,“Polarization discrimination for active imaging in scattering media,”mun.159,1–6͑1999͒.16.K.Turpin,J.G.Walker,P.C.Y.Chang,K.I.Hopcraft,B.Ablitt,and E.Jakeman,“The influence of particle size in active polarization imaging in scattering media,”mun.168, 325–335͑1999͒.17.G.N.Plass and G.W.Kattawar,“Monte Carlo calculations oflight scattering from clouds,”Appl.Opt.7,415–419͑1968͒.18.G.N.Plass and G.W.Kattawar,“Reflection of light pulsesfrom clouds,”Appl.Opt.10,2304–2310͑1971͒;E.A.Bucher,“Computer simulation of light pulse propagation for commu-nication through thick clouds,”Appl.Opt.12,2391–2400͑1973͒.19.B.Maheu,J.-P.Briton,and G.Gouesbet,“Four-flux model anda Monte Carlo code:comparisons between two simple,com-plementary tools for multiple scattering calculations,”Appl.Opt.28,22–24͑1989͒;J.-P.Briton,B.Maheu,G.Gre´han,andG.Gouesbet,“Monte Carlo simulation of multiple scattering inarbitrary3-D geometry,”Part.Part.Syst.Charact.9,52–58͑1992͒.20.P.Bruscaglioni,P.Donelli,A.Ismaelli,and G.Zaccanti,“Anumerical procedure for calculating the effect of a turbid me-dium on the MTF of an optical system,”J.Mod.Opt.38, 129–142͑1991͒;P.Donelli,P.Bruscaglioni,A.Ismaelli,and G.Zaccanti,“Experimental validation of a Monte Carlo procedure for the evaluation of the effect of a turbid medium on the point spread function of an optical system,”J.Mod.Opt.38,2189–2201͑1991͒.21.G.W.Kattawar and G.N.Plass,“Radiance and polarization ofmultiple scattered light from haze and clouds,”Appl.Opt.7, 1519–1527͑1968͒;G.W.Kattawar and G.N.Plass,“Degree and direction of polarization of multiple scattered light. 1.Homogeneous cloud layers,”Appl.Opt.11,2851–2865͑1972͒.22.T.Aruga and T.Igarashi,“Narrow beam light transfer in smallparticles:image blurring and depolarization,”Appl.Opt.20, 2698–2705͑1981͒;erratum,20,3831͑1981͒.23.J.M.Schmitt,A.H.Gandjbakhche,and R.F.Bonner,“Use ofpolarized light to discriminate short-path photons in a multi-ply scattering medium,”Appl.Opt.31,6535–6546͑1992͒. 24.P.Bruscaglioni,G.Zaccanti,and Q.Wei,“Transmission of apulsed polarized light beam through thick turbid media:nu-merical results,”Appl.Opt.32,6142–6150͑1993͒.25.D.Bicout,C.Brosseau,A.S.Martinez,and J.M.Schmitt,“Depolarization of multiply scattered waves by spherical dif-fusers:influence of the size parameter,”Phys.Rev.E49, 1767–1770͑1994͒;A.S.Martinez and R.Maynard,“Faraday effect and multiple scattering of light,”Phys.Rev.B50,3714–3732͑1994͒.4940APPLIED OPTICS͞Vol.39,No.27͞20September2000。

Fundamentals of Control Engineering Ch8 Frequency Response MethodsHaoping WANG E-mail: Hp.wang@Automation SchoolNanjing University of Science and Technology Fundamentals of Control Engineering -hp.wang@j g y gyFrequency Response Methodsq y pp q y pBasic concept of frequency responseFrequency response plotsDrawing the Bode diagramPerformance specification in the frequency domainFundamentals of Control Engineering -hp.wang@The frequency response of a system is defined as the steady-state response of the system to a sinusoidal input signal . The resulting output signal for a linear system, is y p g g p g y ,also a sinusoidal in the steady state; it differs from the input waveform only in amplitude and phase angle .G(s)−0()sin r t R t ω=0()sin()y t t Y ωφ=+H()The frequency response of a system is different from the sinusoidal input signal H(s)e eque cy espo se o a syste s d e e t o t e s uso da put s g a only in the magnitude and the phase , which are functions of the frequency of the sinusoidal input signal.Fundamentals of Control Engineering -hp.wang@T he advantage of the frequency response method is the ready availability of sinusoid test signals for various ranges ofavailability of sinusoid test signals for various ranges offrequencies and amplitudes.It provides good designs in the face of uncertainty in the plant It provides good designs in the face of uncertainty in the plant modelp g p pExperimental information can be used for design purposes.No intermediate processing of the data (such as finding poles and zeros) is required to arrive at the system model.The basic disadvantage of the frequency response method for analysis and design is the indirect link between the frequency and analysis and design is the indirect link between the frequency and the time domain.Fundamentals of Control Engineering -hp.wang@Consider a stable time-invariant linear system with distinct poles asR ()()()s m s T s ==T ω=∠Fundamentals of Control Engineering -hp.wang@ ()j φFrequency Response Methodsq y pp q y pBasic concept of frequency responseFrequency response plotsDrawing the Bode diagramPerformance specification in the frequency domainFundamentals of Control Engineering -hp.wang@z Transfer function and Laplace transformY()sFundamentals of Control Engineering -hp.wang@z Frequency characteristic, Transfer function and differential equation are equivalent in representation of systemequivalent in representation of systemFundamentals of Control Engineering -hp.wang@Polar plot: the polar plot of a sinusoidal transfer function G(jw)is a plot of |G(jw)|, the magnitude of G(jw)versus (), the phase angle of G(jw ) on |(j )|g (j )(ϕ)p g (j )polar coordinates as w is varied from zero to infinity. The polar plot is often called the Nyquist plot .G G s R jX ωωω==+()()()()s j j j ω=Fundamentals of Control Engineering -hp.wang@Exe 1. Frequency response of an RC filter: A simple RC filter is shown asthe referred transfer function is , And the 2()()()V s G s V s RCs ==Fundamentals of Control Engineering -hp.wang@Solution 1: then the polar plot (or Nyquist plot) can be obtained from the relation1/)j ωωωω−+1(()()()G j R j jX j ω==plane with real-axis R(w)and the imaginary axis X(w)as shown belowFundamentals of Control Engineering -hp.wang@plane with real axis R(w) and the imaginary axis X(w) as shown below.Solution 1:Fig. the polar plot of the RC filter system Fundamentals of Control Engineering -hp.wang@g t e po a p ot o t e C te systeNote that)()()()()G j G j R j jX j ωωωω−==−H th it d d th h l b ittHence the magnitude and the phase angle may be written as (12()KG j ω=)24211()tan ()j ωωτφω−+=−−Fundamentals of Control Engineering -hp.wang@ωτSolution 2:K()122421()1G j ωωωτ−=+()tan ()j φωωτ=−−Fundamentals of Control Engineering -hp.wang@Blode Diagram Plot: A sinusoidal transfer function may be represented by two separate plots, one giving the magnitude in decibel versus frequenc y, and the other p p ,g g g q y,==−Fundamentals of Control Engineering -hp.wang@()220log[()]20log 10log ()1()1G j ωωτωτ+⎜⎟+⎝⎠1/22120log[()]20log 10log ()1)1G j ωωτ⎛⎞==−+⎟⎠Solution 3:()2(ωτ⎜+⎝=−Fundamentals of Control Engineering -hp.wang@()tan ()j φωωτSolution 3:Fig 3Bode diagram for 1==Fundamentals of Control Engineering -hp.wang@Fig 3. Bode diagram for()1G j RCj ωτωτ+Blode Diagram Plot: a linear scale of frequency is not the most convenient or judicious choice, and we consider the use of a logarithmic scale of frequency. The j ,g q ythat for the magnitude curve is a straight line with respect to that for the magnitude curve is a straight line with respect to.1/ωτ ωslope of the asymptote line for p y pthis 1-order transfer function is −20 dB/decade.Fundamentals of Control Engineering -hp.wang@Blode Diagram Plot:Decade: An interval of two frequencies with a ratio equal to 10 is called a decade ,q q ,Fundamentals of Control Engineering -hp.wang@1ωτConsider a general form of transfer function asNote that the transfer function contains Q zeros, N poles at the origin, M real poles, and R pairs of complex conjugate poles.The logarithmic magnitude isFundamentals of Control Engineering -hp.wang@Assuming , the phase given by (When K b < 0, ±180 should be added.)0b k ≥()φωThe results shows that the logarithmic magnitude and the phase angle is g g p g Complex conjugated poles (or zeros)ξFundamentals of Control Engineering -hp.wang@ . Complex conjugated poles (or zeros)221()n ns ξωω++Consider four different kinds of factors as follows:Constant gain : The contribution to the logarithmic gain of the constant gain is a constant given by90 90 (ωω°==°−Fundamentals of Control Engineering -hp.wang@ ()())(φφConsider four different kinds of factors as follows:Multiple Poles at origin: the logarithmic magnitude M lti l P l t i i th l ith i it dFundamentals of Control Engineering -hp.wang@The Bode plot of 1/s N and s N are symmetrical with respect to the frequency axis, is shownfrequency axis is shownFig 5: Bode diagram for (jw)±N.Fundamentals of Control Engineering -hp.wang@(3) Poles (or zeros) on the real axis:Fundamentals of Control Engineering -hp.wang@Fundamentals of Control Engineering -hp.wang@ Fig 6When , the magnitude and phase may be approximated as ⎧1/ωτ≤()1d th h l t −Fundamentals of Control Engineering -hp.wang@and the phase angle 1()tan φωωτ=+11j ωτ+The Bode plots of and are symmetrical with respect to the frequency axis.1j ωτ+q y−Fundamentals of Control Engineering -hp.wang@ 1uWhen .u<<1, the magnitude and phase may be approximated as )20l [10l 0dB (222220log[()]10log (1)40 dBG j u u ωξ=−−+≈Fundamentals of Control Engineering -hp.wang@Fig 7 Bode diagram for1221()jjξωω−⎡⎤++⎢⎥ωω⎣⎦Fundamentals of Control Engineering -hp.wang@The frequency at which the magnitude becomes the maximum is called the resonant frequency and denoted by ωq y y rFundamentals of Control Engineering -hp.wang@Fig 11:The maximum value of theFig 11: The maximum value of thefrequency response, andthe resonant frequencyVersus .Versus .ξFundamentals of Control Engineering -hp.wang@Frequency Response Methodsq y pp q y pBasic concept of frequency responseFrequency response plotsDrawing the Bode diagramPerformance specification in the frequency domainFundamentals of Control Engineering -hp.wang@g gDrawing the Bode diagram Drawing Bode diagram (p. 545):gmagnitude.Fundamentals of Control Engineering -hp.wang@Drawing the Bode diagram g gFundamentals of Control Engineering -hp.wang@Drawing the Bode diagram g gFundamentals of Control Engineering -hp.wang@g gDrawing the Bode diagramFundamentals of Control Engineering -hp.wang@Drawing the Bode diagramg gFundamentals of Control Engineering -hp.wang@Frequency Response Methodsq y pp q y pBasic concept of frequency responseFrequency response plotsDrawing the Bode diagramPerformance specification in the frequency domainFundamentals of Control Engineering -hp.wang@Performance specification in the frequency domainThe bandwidth of a system with sinusoidal TF T is the frequency, ,B ω1()(0)2B T j T j ω=Fundamentals of Control Engineering -hp.wang@ Fig: Magnitude characteristic.The bandwidth of the closed-loop control system is an excellent measurement of the range of the fidelity of the response of the system.of the range of the of the response of the systemFig: Magnitude characteristic.Fundamentals of Control Engineering -hp.wang@The frequency response the step response of the two systems are shown in the following Fig. (a) and (b), respectively.Fig . Response of two first-order systems.For a typical second-order system2ωT s=()nB3,4Fig: Magnitude characteristic.Fundamentals of Control Engineering -hp.wang@Fig : Response of two second-order systems.Both system has the overshoot of 15%, butT4(s) has a peak time of 0.12 secondscompared to 0.36 seconds for T3(s).Note also that the settling time for T 4(s)isN l h h li i f()i0.37 seconds, while it is 0.9 seconds forT3(s)Fundamentals of Control Engineering -hp.wang@(i) The magnitude of the resonant peak is indicative of the relativeSatisfactory transient performance is usually obtained if P M ωstability . Satisfactory transient performance is usually obtained if which corresponds to an effective damping ratio of . In general a large value ofcorresponds to a large overshoot in the transient response to the step input.03dB P M ω≤≤0.40.7ξ≤≤P M ωresponse to the step input.(ii) The magnitude of the resonant frequency is indicative of the speed of thetransient response. The larger the value of , the faster the time response is.r ωp g ,p (iii) A large bandwidth corresponds to a small rise time, or fast response. r ωωRoughly speaking, we can say that the bandwidth is proportional to the speed of response.B Fundamentals of Control Engineering -hp.wang@SummaryB i t f fBasic concept of frequency responseF lFrequency response plotsi h d diDrawing the Bode diagramPerformance specification in the frequency domaind iFundamentals of Control Engineering -hp.wang@Assignment•E8.1•E8.5E86•E8.6Fundamentals of Control Engineering -hp.wang@。

点和线结合自然界英语The Dance of Dots and Lines in Nature.In the vast canvas of nature, dots and lines are not just abstract concepts but powerful narrators of the universe's stories. They are the silent language of the natural world, telling tales of growth, transformation, and harmony.Dots, in nature, are often the manifestation of seeds, stars, or the smallest of particles. They are the genesis of everything that grows and evolves. Seeds, for instance, are tiny dots of potential, carrying within them the blueprint for a future plant or tree. They are the beginning of a journey that will unfold into branches, leaves, and flowers, creating a vibrant ecosystem. Similarly, stars in the night sky are like dots of light, burning millions of miles away, illuminating our world with their distant glow.Lines, on the other hand, are the threads that connect and guide. They are the paths of rivers, the trajectories of birds in flight, the contours of mountains, and the veins of leaves. Lines define shape, direction, and flow. They are the architects of nature's design, creating patterns and symmetries that are both aesthetically pleasing and functionally essential.When dots and lines combine in nature, they create a dance of life that is both beautiful and profound. Seeds, scattered like dots across the landscape, sprout and grow, tracing lines of growth and expansion. Trees, with their branches extending like lines into the sky, support dots of leaves that in turn capture the sun's energy, fueling the cycle of life.In the ocean, coral reefs are built by the collective efforts of millions of tiny coral polyps. These polyps,like dots, form a dense matrix that creates a habitat for a vast array of marine life. The lines created by their interconnected branches provide structure and shelter, creating a three-dimensional city under the sea.The dance of dots and lines is also seen in the migrations of animals. Birds, following the lines of their migratory paths, form dots in the sky as they fly in formation. These migrations are not just about survival but also about the perpetuation of species, as generations of birds follow the same lines, connecting the dots of their ancestors' journeys.Moreover, the patterns created by dots and lines in nature are often a testament to the laws of physics and mathematics. The spiral patterns seen in many natural objects, from nautilus shells to galaxy arms, are examples of the mathematical principles at work in the universe. These patterns, although abstract, are the visible manifestation of the underlying order and complexity of the natural world.The beauty of the dance of dots and lines in nature is that it is both constant and ever-changing. Seeds may sprout into trees that live for centuries, while the lines of rivers erode and reshape the landscape over time. Starsin the sky may burn for millions of years, while the migration patterns of animals adapt to changing environments. This dance is a continuous dialogue between stability and flux, between the timeless laws of nature and the ever-evolving world we inhabit.In conclusion, the combination of dots and lines in nature is not just a visual treat but a profound expression of the universe's complexity and harmony. It is a reminder that even the smallest of beginnings can lead to grand endings, and that the invisible threads of connection and guidance are often more powerful than we can imagine. As we observe and appreciate the dance of dots and lines in nature, we are reminded of our own place in this vast and wonderful world.。