Wavelength Reusability in Wavelength-Routed Optical Networks

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Regularizing phase-based stereo

Regularizing Phase-Based StereoT.Fr¨o hlinghaus,J.M.BuhmannRheinische Friedrich–Wilhelms–Universit¨a tInstitut f¨u r Informatik IIID-53117Bonn,GermanyE-Mail:ft|jb@informatik.uni-bonn.deAbstractWavelet-based techniques proved to be a promising ap-proach for estimating the disparity between two stereo im-ages.The complex-valued Gabor f ilter responses reduce the ambiguity of raw image intensities,and their phase dif-ferences between left and right image provide a direct mea-sure for the disparities.Experience shows that such phase-based measurements are reliable near edges but yield poor results between them.To improve these phase-based disparity estimations,a regularization scheme is proposed which directly compares possible matching pairs.Unreliable regions are f illed by using a simple smoothness constraint.In the spirit of Markov random f ields,we propose a probabilistic lattice model which describes the complete disparity distribution instead of representing only a single conf iguration.Exper-imental results are presented for an artif icial image pair generated by computer graphics.1IntroductionReconstructing the three-dimensional environment from its two-dimensional stereo projections is essentially an as-signment problem:For each pixel in the left stereo imageits corresponding pixel in the right image has to be deter-mined,i.e.the pixel which is projected from the same point in3D.In this way,a disparity map is constructed which represents the spatial shift between both stereo images.The computation of a depth map from the disparities and the known camera parameters is now left as a geometric prob-lem[2].Instead of directly evaluating the raw intensities of stereo images,further transformation into low level feature maps may facilitate the search for corresponding points because they yield a better discrimination than simple pixel intensi-ties.Sanger[10]proposed the use of Gabor wavelets[5](1) with extension andﬁlter frequency for solving the as-signment problem.This product of a complex harmonic and a Gaussian envelope yields a joint spatial and frequency rep-resentation of an image[1]by convolving with the image intensities:(2)(3) Without searching for corresponding points,the Gaborﬁl-ter responses themselves give already an estimation of the disparities by exploiting the Fourier Shift Theorem(4) This theorem states that a spatial shift in the image domain effects a phase shift in Fourier domain.Since Gaborﬁl-ters produce a localized frequency description,the normal-ized phase differenceDeviating from usual conventions,we subtract from to obtain positive disparities.revealed several consistency measures–e.g.amplitude and singularities–for detecting unreliable disparity estimations, but none of them could really catch the whole spectrum of error sources.In accordance with these observations we classify phase-based stereo as an ill-posed problem in the Hadamard sense:A solution of the stereo problem does not exist forprojections of an object which is occluded from theother viewpoint.Gaborﬁltering reduces the ambiguity problem,butthe solution is not unique for pixels within a smoothregion without any additional structures.Depth discontinuities usually produce smallﬁlter re-sponse variations,thus the solution does not dependcontinuously on theﬁltered data.To make an ill-posed problem well-posed,regulariza-tion schemes impose additional constraints like piecewise smoothness[7,9].For our purposes we use a simple smoothness constraint toﬁll the unreliable regions within a sparse disparity map by extending the reliable estimations near edges.As computations of regularization algorithms on general purpose workstations are too time consuming for real-time applications,our grid-like model was suited to the locality constraints of analog VLSI.Also,the restriction to elemen-tary operations guided by physical considerations should fa-cilitate a later hardware realization.2A Probabilistic Lattice ModelDue to the hardware constraints listed above we restrict our depth map representation to lattice models which suit to the solid image structure.The input to such a lattice are alreadyﬁltered images while the regularization is realized by coupling neighboring sites.Typically,spatial coherency is introduced by stating a cost function likeFor simplicity we assume that the epipolar line constraint holds and we restrict our formulas to the one-dimensional case,even though the smooth-ing is a two-dimensional operation.proved to be a weak information source.More reliable re-sults were produced by directly searching for the most sim-ilar point(7) within a given range of disparities,respectively.Obvi-ously,the underlying data structures have no appropriate implementation in analog VLSI since this would demand for an online modiﬁcation of the internal wiring.To overcome the above restrictions we propose a proba-bilistic lattice model,which represents the discretized prob-ability distribution of every pixels disparity and thus allows us to model the uncertainty in our estimations.Since the lat-tice resolution limits the granularity of the disparity range, the joint matching measure(8) is used as a compatibility estimation for possible matching points,i.e.it states whether the discrete disparities approximate the real one with pixel accuracy.We use a Gaussian model(9)with residual disparities(11) which sum up to one for each pixel.Note that this nor-malized expressions resemble the deﬁnition of a Gibbs dis-tribution with the squared residual disparities as its energyor cost function[7].Regularized disparity probabilities are introduced which are composed of a mixture a smoothnessprocess and the interesting correspondence process,i.e.reg-ularization is directly applied to the probability distribu-tions:(12) These regularized disparity probabilities are represented by a mixture model,which weights the actual disparity distri-butions against the averages of their surrounding neighbors,respectively.The reader should note the differ-ence to standard MRF approaches which balance raw dis-parities and smoothness in a regularization cost function with corresponding Gibbs distribution for disparities.To-gether with the normalization above,we thus realize a cooperative stereo algorithm similar to[8].The weighting factors(13) depend on the localﬁlter amplitude as well as on a global control parameter,called temperature.A weight-ing schedule(14) with initial temperature,decay rate and a time variable gradually lowers the smoothing constraint until reaching an empirically choosen temperaturewhere noise and distortion effects become perceptible.Thus the originally coarse grained disparity map is continuously reﬁned to a more and more detailed representation.Final output of such a probabilistic lattice model are the expecta-tion values(15) of its discretized disparity distributions.Note that even though this is a continuous output,the internal lattice struc-ture bases completely on a discrete model.For evaluating the probabilistic lattice model we used a pair pf synthetical images with known disparities as ground truth.The computer graphics images“Corridor”(Fig.1) were originally generated from a ray traycing program sim-ulating two parallel aligned cameras with a base line of10 cm.The reciprocal of the rays distances between camera and object yielded the ground truth disparities of the com-puter graphics scenes.The errors at the lower left of the ﬁgures were computed as the difference between real and estimated disparity map,whereby‘white’encodes a correct result,and‘black’encodes an incorrect estimate with a dif-ference equal to the largest disparity for the actual image pair.For the“Corridor”images in Fig.1we used aﬁlter with wavelength and with standard deviation to compute the phases in.Since thisﬁlters is too wide for aﬁne grained edge detection,we additionallyused aﬁlter with and to compute for the left image the amplitudes used in equation.Tospeed up our computations the disparity range was suitedto the actual image pair,i.e.we used a lower limit of0and an upper limit of10.The annealing schedule for the stereo pair started with andﬁnally reachedin3200steps.The“Corridor”results in Fig.1demonstrate that ourprobabilistic lattice model can extract the rough structure ofa disparity map even though it does not reveal every detail.Its main problem areﬁne structures like the pictures on the wall,and depth discontinuities as produced by the objects on theﬂoor.Theﬁrst phenomenon can be explained by the fact that the left and right images are not simply shifted rela-tive to each other.Due to discretisation they probably sam-ple the scene at points with different intensities,which of course distorts theﬁlter response ofﬁne structured regions. The second phenomenon is caused by limitations of our reg-ularization scheme which yet does not take discontinuities into consideration.For comparision Fig.1also shows a dis-parity map based onﬁlter responses which were corrected my means of the known(!)occlusions and discontinuities, and thus our current research focuses on estimating occlu-sion probabilities.3Limitations of the Phase-Based ApproachAccording to the Fourier shift theorem,two Gaborﬁlter responses and should only differ in their phase by a value of.When convolved with a Gabor kernel of frequency,theﬁlter amplitude is expected to remain constant while the phase increases linearly with position:(16)This characteristics is actually observed for short distances along the images scanlines.Nevertheless,longer shifts expose signiﬁcant nonlinearities.An analysis of the convo-lution results(17)(18)reveals that even for constant disparities there are feature modiﬁcations beyond a simple phase shift.This long-range variation is explained by the Gaborﬁlters Gaussian weight being aligned to the actual position.In addition to theﬁl-ter supports spatial shift its single pixels are also weighteddistinctly,which justiﬁes our approach of using phase dif-ferences only for the small residual disparities.Due to the problems near depth discontinuities we have to question the basic assumption of feature-based stereo, i.e.do corresponding points always have the sameﬁlter re-sponse?In fact,this is not case!The main source for feature distortions are occlusions because their neighboring points use these regions forﬁltering even though they are not con-tained in the other image.Since Gabor kernels typically have a support larger than their wavelength,distortions due to occlusions generally have a wider range than just nearest neighbors.Now given a region without occlusions,viewpoint differ-ences may still violate the requirement of feature equality. Assuming that(19) holds,the leftﬁlter response(20) differs from the response(21) of the corresponding right point unless the disparity is constant.Further,small structures like the pictures in the “Corridor”images cause problems because their continu-ous projections contain high frequencies larger than the im-ages Nyquist frequency.A simple sampling of them leads to aliasing effects,thus the high frequencies are projected onto lower frequencies,which e.g.could be located in the Gaborﬁlters band spectrum.4ConclusionsOur experiments revealed that regularization canﬁll sparse disparity maps,while other phase-based algorithms are conﬁned to detect unreliable disparity estimations with-out correcting them.The results for the synthetical and for the natural stereo images demonstrate that our probabilistic lattice model perceives the rough structure of a scene,but it has problems with entailing the detailed structure of the single objects.The limitations of pure regularization and the feature distortions near occlusions and discontinuities suggest that further research should concentrate on methods for feature adaptation.Based on the known or estimated occlusion in-formation,one has to correct theﬁlter responses such that corresponding points again yield similar features.Our ex-periments with known occlusions(publication in prepara-tion)demonstrate that modeling occlusion and discontinu-ities promises a signiﬁcant step toward yielding a dense and reliable depth map.Apart from correcting this likelihood knowledge,incorporation of further a priori knowledge like edge continuity should also improve the disparity estima-tions.AcknowledgementsWe gratefully acknowledge the support by the computer graphics group of D.Fellner.This work was funded by the European Community under grant CORMORANT8503.References[1]J.Daugman.Uncertainty relation for resolution inspace,spatial frequency,and orientation optimized by two-dimensional visual corticalﬁlters.J.Opt.Soc.Am.A, 2:1160–1169,1985.[2]O.Faugeras.Three-Dimensional Computer Vision:A Geo-metric Viewpoint.MIT Press,1993.[3] D.Fleet and A.Jepson.Stability of phase information.IEEETransactions on Pattern Analysis and Machine Intelligence, PAMI-15(12):1253–1268,1993.[4] D.Fleet,A.Jepson,and M.Jenkin.Phase-based disparitymeasurement.CVGIP:Image Understanding,53:198–210, 1991.[5] D.Gabor.Theory of communication.Journal of the Institu-tion of Electrical Engineers,93:429–459,1946.[6]G.Granlund and H.Knutsson.Signal Processing for Com-puter Vision.Kluwer Academic Publishers,1995.[7]S.Li.Markov Random Field Modeling in Computer Vision.Springer,1995.[8] D.Marr and T.Poggio.A computational theory of humanstereo vision.Proc.R.Soc.Lond.B,204:301–328,1979. [9]T.Poggio,V.Torre,and putational vision andregularization theory.Nature,317:314–319,1985.[10]T.Sanger.Stereo disparity computation using gaborﬁlters.Biological Cybernetics,59:405–418,1988.Figure1.“Corridor”images(top),ground truth(middle left),computed disparity(middle right),error (bottom left),disparity based on adapted features(bottom right).。

Wavelength monitor

专利名称：Wavelength monitor发明人：Minoru Maeda申请号：US10051507申请日：20020117公开号：US06980297B2公开日：20051227专利内容由知识产权出版社提供专利附图：摘要：A wavelength monitor having a Michelson interferometer (or Mach-Zehnder)optical system of a spatial light type having optical input from a light source has aninterference pattern generating means which inclines the wavefronts of interfering beams of collimated light to generate an interference pattern in the light intensity distributionin an interference light beam planes a first slit and a second slit which are adjustable in position and provided in front of a first photo-detector and a second photo-detector, respectively, which receive split beams of interference light, and a signal processing means by which the changes in the intensity of light from the first photo-detector and the second photo-detector are counted and subjected to necessary arithmetic operations to output signals representing wavelength data for the input light.申请人：Minoru Maeda地址：Tokyo JP国籍：JP代理机构：Fish & Richardson P.C.更多信息请下载全文后查看。

透射电子显微镜ppt课件

Lenses
Curved glass or mirror for
Visible light
concavห้องสมุดไป่ตู้ convex
Concave surface of metal (e.g. satellite dish)
Radio waves
Concave mirror or Fresnel lens Heat
Wavelength - distance between peaks of the waveform
Blue light has a wavelength of 0.47 um Resolution max - 0.2 um (200 nm)
Cannot go beyond this even with better optics. Solution? Use illumination of shorter wavelength
Resolution - The point at which two or more objects can be distinguished as separate.
History:
First record of using glass lens for magnification was by an Arabian from what is now known as Iran, Alhazen, in the 10 and 11th century. He contradicted Ptolemy's and Euclid's theory of vision that objects are seen by rays of light emanating from the eyes; according to him the rays originate in the object of vision and not in the eye. Because of his extensive research on vision, he has been considered by many as the father of modern optics.

菲涅耳非相干关联全息图(综述)

Fresnel incoherent correlation hologram-a reviewInvited PaperJoseph Rosen,Barak Katz1,and Gary Brooker2∗∗1Department of Electrical and Computer Engineering,Ben-Gurion University of the Negev,P.O.Box653,Beer-Sheva84105,Israel2Johns Hopkins University Microscopy Center,Montgomery County Campus,Advanced Technology Laboratory, Whiting School of Engineering,9605Medical Center Drive Suite240,Rockville,MD20850,USA∗E-mail:rosen@ee.bgu.ac.il;∗∗e-mail:gbrooker@Received July17,2009Holographic imaging oﬀers a reliable and fast method to capture the complete three-dimensional(3D) information of the scene from a single perspective.We review our recently proposed single-channel optical system for generating digital Fresnel holograms of3D real-existing objects illuminated by incoherent light.In this motionless holographic technique,light is reﬂected,or emitted from a3D object,propagates througha spatial light modulator(SLM),and is recorded by a digital camera.The SLM is used as a beam-splitter of the single-channel incoherent interferometer,such that each spherical beam originated from each object point is split into two spherical beams with two diﬀerent curve radii.Incoherent sum of the entire interferences between all the couples of spherical beams creates the Fresnel hologram of the observed3D object.When this hologram is reconstructed in the computer,the3D properties of the object are revealed.OCIS codes:100.6640,210.4770,180.1790.doi:10.3788/COL20090712.0000.1.IntroductionHolography is an attractive imaging technique as it oﬀers the ability to view a complete three-dimensional (3D)volume from one image.However,holography is not widely applied to the regime of white-light imaging, because white-light is incoherent and creating holograms requires a coherent interferometer system.In this review, we describe our recently invented method of acquiring incoherent digital holograms.The term incoherent digi-tal hologram means that incoherent light beams reﬂected or emitted from real-existing objects interfere with each other.The resulting interferogram is recorded by a dig-ital camera and digitally processed to yield a hologram. This hologram is reconstructed in the computer so that 3D images appear on the computer screen.The oldest methods of recording incoherent holograms have made use of the property that every incoherent ob-ject is composed of many source points,each of which is self-spatial coherent and can create an interference pattern with light coming from the point’s mirrored image.Under this general principle,there are vari-ous types of holograms[1−8],including Fourier[2,6]and Fresnel holograms[3,4,8].The process of beam interfering demands high levels of light intensity,extreme stability of the optical setup,and a relatively narrow bandwidth light source.More recently,three groups of researchers have proposed computing holograms of3D incoherently illuminated objects from a set of images taken from diﬀer-ent points of view[9−12].This method,although it shows promising prospects,is relatively slow since it is based on capturing tens of scene images from diﬀerent view angles. Another method is called scanning holography[13−15],in which a pattern of Fresnel zone plates(FZPs)scans the object such that at each and every scanning position, the light intensity is integrated by a point detector.The overall process yields a Fresnel hologram obtained as a correlation between the object and FZP patterns.How-ever,the scanning process is relatively slow and is done by mechanical movements.A similar correlation is ac-tually also discussed in this review,however,unlike the case of scanning holography,our proposed system carries out a correlation without movement.2.General properties of Fresnel hologramsThis review concentrates on the technique of incoher-ent digital holography based on single-channel incoher-ent interferometers,which we have been involved in their development recently[16−19].The type of hologram dis-cussed here is the digital Fresnel hologram,which means that a hologram of a single point has the form of the well-known FZP.The axial location of the object point is encoded by the Fresnel number of the FZP,which is the technical term for the number of the FZP rings along the given radius.To understand the operation principle of any general Fresnel hologram,let us look on the diﬀerence between regular imaging and holographic systems.In classical imaging,image formation of objects at diﬀerent distances from the lens results in a sharp image at the image plane for objects at only one position from the lens,as shown in Fig.1(a).The other objects at diﬀerent distances from the lens are out of focus.A Fresnel holographic system,on the other hand,as depicted in Fig.1(b),1671-7694/2009/120xxx-08c 2009Chinese Optics Lettersprojects a set of rings known as the FZP onto the plane of the image for each and every point at every plane of the object being viewed.The depth of the points is en-coded by the density of the rings such that points which are closer to the system project less dense rings than distant points.Because of this encoding method,the 3D information in the volume being imaged is recorded into the recording medium.Thus once the patterns are decoded,each plane in the image space reconstructed from a Fresnel hologram is in focus at a diﬀerent axial distance.The encoding is accomplished by the presence of a holographic system in the image path.At this point it should be noted that this graphical description of pro-jecting FZPs by every object point actually expresses the mathematical two-dimensional (2D)correlation (or convolution)between the object function and the FZP.In other words,the methods of creating Fresnel holo-grams are diﬀerent from each other by the way they spatially correlate the FZP with the scene.Another is-sue to note is that the correlation should be done with a FZP that is somehow “sensitive”to the axial locations of the object points.Otherwise,these locations are not encoded into the hologram.The system described in this review satisﬁes the condition that the FZP is depen-dent on the axial distance of each and every objectpoint.parison between the Fresnel holography principle and conventional imaging.(a)Conventional imaging system;(b)fresnel holographysystem.Fig.2.Schematic of FINCH recorder [16].BS:beam splitter;L is a spherical lens with focal length f =25cm;∆λindicates a chromatic ﬁlter with a bandwidth of ∆λ=60nm.This means that indeed points,which are closer to the system,project FZP with less cycles per radial length than distant points,and by this condition the holograms can actually image the 3D scene properly.The FZP is a sum of at least three main functions,i.e.,a constant bias,a quadratic phase function and its complex conjugate.The object function is actually corre-lated with all these three functions.However,the useful information,with which the holographic imaging is real-ized,is the correlation with just one of the two quadratic phase functions.The correlation with the other quadratic phase function induces the well-known twin image.This means that the detected signal in the holographic system contains three superposed correlation functions,whereas only one of them is the required correlation between the object and the quadratic phase function.Therefore,the digital processing of the detected image should contain the ability to eliminate the two unnecessary terms.To summarize,the deﬁnition of Fresnel hologram is any hologram that contains at least a correlation (or convolu-tion)between an object function and a quadratic phase function.Moreover,the quadratic phase function must be parameterized according to the axial distance of the object points from the detection plane.In other words,the number of cycles per radial distance of each quadratic phase function in the correlation is dependent on the z distance of each object point.In the case that the object is illuminated by a coherent wave,this correlation is the complex amplitude of the electromagnetic ﬁeld directly obtained,under the paraxial approximation [20],by a free propagation from the object to the detection plane.How-ever,we deal here with incoherent illumination,for which an alternative method to the free propagation should be applied.In fact,in this review we describe such method to get the desired correlation with the quadratic phase function,and this method indeed operates under inco-herent illumination.The discussed incoherent digital hologram is dubbed Fresnel incoherent correlation hologram (FINCH)[16−18].The FINCH is actually based on a single-channel on-axis incoherent interferometer.Like any Fresnel holography,in the FINCH the object is correlated with a FZP,but the correlation is carried out without any movement and without multiplexing the image of the scene.Section 3reviews the latest developments of the FINCH in the ﬁeld of color holography,microscopy,and imaging with a synthetic aperture.3.Fresnel incoherent correlation holographyIn this section we describe the FINCH –a method of recording digital Fresnel holograms under incoher-ent illumination.Various aspects of the FINCH have been described in Refs.[16-19],including FINCH of re-ﬂected white light [16],FINCH of ﬂuorescence objects [17],a FINCH-based holographic ﬂuorescence microscope [18],and a hologram recorder in a mode of a synthetic aperture [19].We brieﬂy review these works in the current section.Generally,in the FINCH system the reﬂected incoher-ent light from a 3D object propagates through a spatial light modulator (SLM)and is recorded by a digital cam-era.One of the FINCH systems [16]is shown in Fig.2.White-light source illuminates a 3D scene,and the reﬂected light from the objects is captured by a charge-coupled device (CCD)camera after passing through a lens L and the SLM.In general,we regard the system as an incoherent interferometer,where the grating displayed on the SLM is considered as a beam splitter.As is com-mon in such cases,we analyze the system by following its response to an input object of a single inﬁnitesimal point.Knowing the system’s point spread function (PSF)en-ables one to realize the system operation for any general object.Analysis of a beam originated from a narrow-band inﬁnitesimal point source is done by using Fresnel diﬀraction theory [20],since such a source is coherent by deﬁnition.A Fresnel hologram of a point object is obtained when the two interfering beams are two spherical beams with diﬀerent curvatures.Such a goal is achieved if the SLM’s reﬂection function is a sum of,for instance,constant and quadratic phase functions.When a plane wave hits the SLM,the constant term represents the reﬂected plane wave,and the quadratic phase term is responsible for the reﬂected spherical wave.A point source located at some distance from a spher-ical positive lens induces on the lens plane a diverging spherical wave.This wave is split by the SLM into two diﬀerent spherical waves which propagate toward the CCD at some distance from the SLM.Consequently,in the CCD plane,the intensity of the recorded hologram is a sum of three terms:two complex-conjugated quadratic phase functions and a constant term.This result is the PSF of the holographic recording system.For a general 3D object illuminated by a narrowband incoherent illumination,the intensity of the recorded hologram is an integral of the entire PSFs,over all object intensity points.Besides a constant term,thehologramFig.3.(a)Phase distribution of the reﬂection masks dis-played on the SLM,with θ=0◦,(b)θ=120◦,(c)θ=240◦.(d)Enlarged portion of (a)indicating that half (randomly chosen)of the SLM’s pixels modulate light with a constant phase.(e)Magnitude and (f)phase of the ﬁnal on-axis digi-tal hologram.(g)Reconstruction of the hologram of the three characters at the best focus distance of ‘O’.(h)Same recon-struction at the best focus distance of ‘S’,and (i)of ‘A’[16].expression contains two terms of correlation between an object and a quadratic phase,z -dependent,function.In order to remain with a single correlation term out of the three terms,we follow the usual procedure of on-axis digital holography [14,16−19].Three holograms of the same object are recorded with diﬀerent phase con-stants.The ﬁnal hologram is a superposition of the three holograms containing only the desired correlation between the object function and a single z -dependent quadratic phase.A 3D image of the object can be re-constructed from the hologram by calculating theFresnelFig.4.Schematics of the FINCH color recorder [17].L 1,L 2,L 3are spherical lenses and F 1,F 2are chromaticﬁlters.Fig.5.(a)Magnitude and (b)phase of the complex Fres-nel hologram of the dice.Digital reconstruction of the non-ﬂuorescence hologram:(c)at the face of the red dots on the die,and (d)at the face of the green dots on the die.(e)Magnitude and (f)phase of the complex Fresnel hologram of the red dots.Digital reconstruction of the red ﬂuorescence hologram:(g)at the face of the red dots on the die,and (h)at the face of the green dots on the die.(i)Magnitude and (j)phase of the complex Fresnel hologram of the green dots.Digital reconstruction of the green ﬂuorescence hologram:(k)at the face of the red dots on the die,and (l)at the face of the green dots on the position of (c),(g),(k)and that of (d),(h),(l)are depicted in (m)and (n),respectively [17].Fig.6.FINCHSCOPE schematic in uprightﬂuorescence microscope[18].propagation formula.The system shown in Fig.2has been used to record the three holograms[16].The SLM has been phase-only, and as so,the desired sum of two phase functions(which is no longer a pure phase)cannot be directly displayed on this SLM.To overcome this obstacle,the quadratic phase function has been displayed randomly on only half of the SLM pixels,and the constant phase has been displayed on the other half.The randomness in distributing the two phase functions has been required because organized non-random structure produces unnecessary diﬀraction orders,therefore,results in lower interference eﬃciency. The pixels are divided equally,half to each diﬀractive element,to create two wavefronts with equal energy.By this method,the SLM function becomes a good approx-imation to the sum of two phase functions.The phase distributions of the three reﬂection masks displayed on the SLM,with phase constants of0◦,120◦and240◦,are shown in Figs.3(a),(b)and(c),respectively.Three white-on-black characters i th the same size of 2×2(mm)were located at the vicinity of rear focal point of the lens.‘O’was at z=–24mm,‘S’was at z=–48 mm,and‘A’was at z=–72mm.These characters were illuminated by a mercury arc lamp.The three holo-grams,each for a diﬀerent phase constant of the SLM, were recorded by a CCD camera and processed by a computer.Theﬁnal hologram was calculated accord-ing to the superposition formula[14]and its magnitude and phase distributions are depicted in Figs.3(e)and (f),respectively.The hologram was reconstructed in the computer by calculating the Fresnel propagation toward various z propagation distances.Three diﬀerent recon-struction planes are shown in Figs.3(g),(h),and(i).In each plane,a diﬀerent character is in focus as is indeed expected from a holographic reconstruction of an object with a volume.In Ref.[17],the FINCH has been capable to record multicolor digital holograms from objects emittingﬂuo-rescent light.Theﬂuorescent light,speciﬁc to the emis-sion wavelength of variousﬂuorescent dyes after excita-tion of3D objects,was recorded on a digital monochrome camera after reﬂection from the SLM.For each wave-length ofﬂuorescent emission,the camera sequentially records three holograms reﬂected from the SLM,each with a diﬀerent phase factor of the SLM’s function.The three holograms are again superposed in the computer to create a complex-valued Fresnel hologram of eachﬂu-orescent emission without the twin image problem.The holograms for eachﬂuorescent color are further combined in a computer to produce a multicoloredﬂuorescence hologram and3D color image.An experiment showing the recording of a colorﬂuo-rescence hologram was carried out[17]on the system in Fig. 4.The phase constants of0◦,120◦,and240◦were introduced into the three quadratic phase functions.The magnitude and phase of theﬁnal complex hologram,su-perposed from theﬁrst three holograms,are shown in Figs.5(a)and(b),respectively.The reconstruction from theﬁnal hologram was calculated by using the Fresnel propagation formula[20].The results are shown at the plane of the front face of the front die(Fig.5(c))and the plane of the front face of the rear die(Fig.5(d)).Note that in each plane a diﬀerent die face is in focus as is indeed expected from a holographic reconstruction of an object with a volume.The second three holograms were recorded via a redﬁlter in the emissionﬁlter slider F2 which passed614–640nmﬂuorescent light wavelengths with a peak wavelength of626nm and a full-width at half-maximum,of11nm(FWHM).The magnitude and phase of theﬁnal complex hologram,superposed from the‘red’set,are shown in Figs.5(e)and(f),respectively. The reconstruction results from thisﬁnal hologram are shown in Figs.5(g)and(h)at the same planes as those in Figs.5(c)and(d),respectively.Finally,an additional set of three holograms was recorded with a greenﬁlter in emissionﬁlter slider F2,which passed500–532nmﬂuo-rescent light wavelengths with a peak wavelength of516 nm and a FWHM of16nm.The magnitude and phase of theﬁnal complex hologram,superposed from the‘green’set,are shown in Figs.5(i)and(j),respectively.The reconstruction results from thisﬁnal hologram are shown in Figs.5(k)and(l)at the same planes as those in Fig. 5(c)and(d),positions of Figs.5(c), (g),and(k)and Figs.5(d),(h),and(l)are depicted in Figs.5(m)and(n),respectively.Note that all the colors in Fig.5(colorful online)are pseudo-colors.These last results yield a complete color3D holographic image of the object including the red and greenﬂuorescence. While the optical arrangement in this demonstration has not been optimized for maximum resolution,it is im-portant to recognize that even with this simple optical arrangement,the resolution is good enough to image the ﬂuorescent emissions with goodﬁdelity and to obtain good reﬂected light images of the dice.Furthermore, in the reﬂected light images in Figs.5(c)and(m),the system has been able to detect a specular reﬂection of the illumination from the edge of the front dice. Another system to be reviewed here is theﬁrst demon-stration of a motionless microscopy system(FINCH-SCOPE)based upon the FINCH and its use in record-ing high-resolution3Dﬂuorescent images of biological specimens[18].By using high numerical aperture(NA) lenses,a SLM,a CCD camera,and some simpleﬁlters, FINCHSCOPE enables the acquisition of3D microscopic images without the need for scanning.A schematic diagram of the FINCHSCOPE for an upright microscope equipped with an arc lamp sourceFig.7.FINCHSCOPE holography of polychromatic beads.(a)Magnitude of the complex hologram 6-µm beads.Images reconstructed from the hologram at z distances of (b)34µm,(c)36µm,and (d)84µm.Line intensity proﬁles between the beads are shown at the bottom of panels (b)–(d).(e)Line intensity proﬁles along the z axis for the lower bead from reconstructed sections of a single hologram (line 1)and from a wideﬁeld stack of the same bead (28sections,line 2).Beads (6µm)excited at 640,555,and 488nm with holograms reconstructed (f)–(h)at plane (b)and (j)–(l)at plane (d).(i)and (m)are the combined RGB images for planes (b)and (d),respectively.(n)–(r)Beads (0.5µm)imaged with a 1.4-NA oil immersion objective:(n)holographic camera image;(o)magnitude of the complex hologram;(p)–(r)reconstructed image at planes 6,15,and 20µm.Scale bars indicate image size [18].Fig.8.FINCHSCOPE ﬂuorescence sections of pollen grains and Convallaria rhizom .The arrows point to the structures in the images that are in focus at various image planes.(b)–(e)Sections reconstructed from a hologram of mixed pollen grains.(g)–(j)Sections reconstructed from a hologram of Convallaria rhizom .(a),(f)Magnitudes of the complex holograms from which the respective image planes are reconstructed.Scale bars indicate image size [18].is shown in Fig. 6.The beam of light that emerges from an inﬁnity-corrected microscope objective trans-forms each point of the object being viewed into a plane wave,thus satisfying the ﬁrst requirement of FINCH [16].A SLM and a digital camera replace the tube lens,reﬂec-tive mirror,and other transfer optics normally present in microscopes.Because no tube lens is required,inﬁnity-corrected objectives from any manufacturer can be used.A ﬁlter wheel was used to select excitation wavelengths from a mercury arc lamp,and the dichroic mirror holder and the emission ﬁlter in the microscope were used to direct light to and from the specimen through an inﬁnity-corrected objective.The ability of the FINCHSCOPE to resolve multicolor ﬂuorescent samples was evaluated by ﬁrst imaging poly-chromatic ﬂuorescent beads.A ﬂuorescence bead slidewith the beads separated on two separate planes was con-structed.FocalCheck polychromatic beads(6µm)were used to coat one side of a glass microscope slide and a glass coverslip.These two surfaces were juxtaposed and held together at a distance from one another of∼50µm with optical cement.The beads were sequentially excited at488-,555-,and640-nm center wavelengths(10–30nm bandwidths)with emissions recorded at515–535,585–615,and660–720nm,respectively.Figures7(a)–(d) show reconstructed image planes from6µm beads ex-cited at640nm and imaged on the FINCHSCOPE with a Zeiss PlanApo20×,0.75NA objective.Figure7(a) shows the magnitude of the complex hologram,which contains all the information about the location and in-tensity of each bead at every plane in theﬁeld.The Fresnel reconstruction from this hologram was selected to yield49planes of the image,2-µm apart.Two beads are shown in Fig.7(b)with only the lower bead exactly in focus.Figure7(c)is2µm into theﬁeld in the z-direction,and the upper bead is now in focus,with the lower bead slightly out of focus.The focal diﬀerence is conﬁrmed by the line proﬁle drawn between the beads, showing an inversion of intensity for these two beads be-tween the planes.There is another bead between these two beads,but it does not appear in Figs.7(b)or(c) (or in the intensity proﬁle),because it is48µm from the upper bead;it instead appears in Fig.7(d)(and in the line proﬁle),which is24sections away from the section in Fig.7(c).Notice that the beads in Figs.7(b)and(c)are no longer visible in Fig.7(d).In the complex hologram in Fig.7(a),the small circles encode the close beads and the larger circles encode the distant central bead. Figure7(e)shows that the z-resolution of the lower bead in Fig.7(b),reconstructed from sections created from a single hologram(curve1),is at least comparable to data from a wideﬁeld stack of28sections(obtained by moving the microscope objective in the z-direction)of the same ﬁeld(curve2).The co-localization of theﬂuorescence emission was conﬁrmed at all excitation wavelengths and at extreme z limits,as shown in Figs.7(f)–(m)for the 6-µm beads at the planes shown in Figs.7(b)((f)–(i)) and(d)((j)–(m)).In Figs.7(n)–(r),0.5-µm beads imaged with a Zeiss PlanApo×631.4NA oil-immersion objective are shown.Figure7(n)presents one of the holo-grams captured by the camera and Fig.7(o)shows the magnitude of the complex hologram.Figures7(p)–(r) show diﬀerent planes(6,15,and20µm,respectively)in the bead specimen after reconstruction from the complex hologram of image slices in0.5-µm steps.Arrows show the diﬀerent beads visualized in diﬀerent z image planes. The computer reconstruction along the z-axis of a group ofﬂuorescently labeled pollen grains is shown in Figs. 8(b)–(e).As is expected from a holographic reconstruc-tion of a3D object with volume,any number of planes can be reconstructed.In this example,a diﬀerent pollen grain was in focus in each transverse plane reconstructed from the complex hologram whose magnitude is shown in Fig.8(a).In Figs.8(b)–(e),the values of z are8,13, 20,and24µm,respectively.A similar experiment was performed with the autoﬂuorescent Convallaria rhizom and the results are shown in Figs.8(g)–(j)at planes6, 8,11,and12µm.The most recent development in FINCH is a new lens-less incoherent holographic system operating in a syn-thetic aperture mode[19].Synthetic aperture is a well-known super-resolution technique which extends the res-olution capabilities of an imaging system beyond thetheoretical Rayleigh limit dictated by the system’s ac-tual ing this technique,several patternsacquired by an aperture-limited system,from variouslocations,are tiled together to one large pattern whichcould be captured only by a virtual system equippedwith a much wider synthetic aperture.The use of optical holography for synthetic apertureis usually restricted to coherent imaging[21−23].There-fore,the use of this technique is limited only to thoseapplications in which the observed targets can be illu-minated by a laser.Synthetic aperture carried out by acombination of several oﬀ-axis incoherent holograms inscanning holographic microscopy has been demonstratedby Indebetouw et al[24].However,this method is limitedto microscopy only,and although it is a technique ofrecording incoherent holograms,a specimen should alsobe illuminated by an interference pattern between twolaser beams.Our new scheme of holographic imaging of incoher-ently illuminated objects is dubbing a synthetic aperturewith Fresnel elements(SAFE).This holographic lens-less system contains only a SLM and a digital camera.SAFE has an extended synthetic aperture in order toimprove the transverse and axial resolutions beyond theclassic limitations.The term synthetic aperture,in thepresent context,means time(or space)multiplexing ofseveral Fresnel holographic elements captured from vari-ous viewpoints by a system with a limited real aperture.The synthetic aperture is implemented by shifting theSLM-camera set,located across theﬁeld of view,be-tween several viewpoints.At each viewpoint,a diﬀerentmask is displayed on the SLM,and a single element ofthe Fresnel hologram is recorded(Fig.9).The variouselements,each of which is recorded by the real aperturesystem during the capturing time,are tiled together sothat theﬁnal mosaic hologram is eﬀectively consideredas being captured from a single synthetic aperture,whichis much wider than the actual aperture.An example of such a system with the synthetic aper-ture three times wider than the actual aperture can beseen in Fig.9.For simplicity of the demonstration,the synthetic aperture was implemented only along thehorizontal axis.In principle,this concept can be gen-eralized for both axes and for any ratio of synthetic toactual apertures.Imaging with the synthetic apertureis necessary for the cases where the angular spectrumof the light emitted from the observed object is widerthan the NA of a given imaging system.In the SAFEshown in Fig.9,the SLM and the digital camera movein front of the object.The complete Fresnel hologramof the object,located at some distance from the SLM,isa mosaic of three holographic elements,each of which isrecorded from a diﬀerent position by the system with thereal aperture of the size A x×A y.The complete hologram tiled from the three holographic Fresnel elements has thesynthetic aperture of the size3(·A x×A y)which is three times larger than the real aperture at the horizontal axis.The method to eliminate the twin image and the biasterm is the same as that has been used before[14,16−18];。

CS_2宽带光限幅和非线性吸收特性研究_柳永亮

#光物理#C S2宽带光限幅和非线性吸收特性研究*柳永亮1,2,刘智波1,张冰1,田建国1**,臧维平1,周文远1,宋峰1,张春平1(1.南开大学泰达应用物理学院,弱光非线性光子学材料先进技术及制备教育部重点实验室,天津300457;2.德州学院物理系,山东德州253023)摘要:利用ns脉冲激光,研究了CS2在420~470nm波长范围内的非线性吸收和光限幅特性。

实验结果表明,CS2对420~450nm波长的脉冲激光具有大的非线性吸收和优良的光限幅性能,随着波长的增加,其非线性吸收和光限幅性能逐渐减小;CS2良好的宽带光限幅性能源于其大的非线性吸收,其非线性吸收机制为双光子吸收(TPA)以及由TPA诱导的激发态吸收(ESA)。

关键词:激发态吸收(ESA);反饱和吸收;光限幅;双光子吸收(TPA)中图分类号:O437文献标识码:A文章编号:1005-0086(2007)12-1492-04Broad band O ptical Lim itin g and N onlinear Optical Ab sorp tion Properties of C arbon D-isu lfideLIU Yong-liang1,2,LIU Zh-i bo1,ZHANG Bing1,TIAN Jian-guo1**,ZANG We-i ping1,ZH OU Wen-yuan1,SONG Feng1,ZH ANG Chun-ping1(1.T he Key Laborator y of Advanced Technique and Fabrication fo r Weak-light Nonlinear Photonics M aterials,Ministry of Education and T EDA A pplied Physical School,N ankai University,T i anji n300457,China;2.Departmentof Physics,Dezhou University,Dezhou253023,China)A bs tra ct:T he nonlinear absorption and optical limiti ng properties of CS2were studied in the wavelength region of420~470nm with nanosecond pulses.Resuls show that CS2has large nonlinear absorpti on and good opti cal limiting properties i n thewavelength of420-450nm.With the increasing of wavelength,the nonli near absorption and optical li mi ting effect of CS2de-crease gradually.T his strong and broad band opti cal limiting p erformance can be attributed to its large nonli near absorptionwi th combi nation of two-photon absorption(T PA)and the excited state absorpti on(ESA)induced by T PA.Key words:excited state absorption(ESA);reverse saturable absorpti on;optical li miting;two-photon absorption(TPA)1引言光限制器由于具有保护人眼或光学敏感元件免受高功率强激光损伤的功能,近20年得到了广泛的研究。

Wavelength conversion element, wavelength conversi

专利名称：Wavelength conversion element, wavelengthconversion method, phase matchingmethod, and light source device发明人：Muramatsu, Kenichi,Kurimura, Sunao申请号：EP13167494.7申请日：20080417公开号：EP2634625A1公开日：20130904专利内容由知识产权出版社提供专利附图：摘要：A wavelength conversion element is provided as one including amonocrystalline nonlinear optical crystal. The nonlinear optical crystal has: a plurality offirst regions having a polarity direction along a predetermined direction; a plurality of second regions having a polarity direction opposite to the predetermined direction; an entrance face into which a fundamental incident wave having a wavelength λ and a frequency ω is incident in a direction substantially perpendicular to the predetermined direction; and an exit face from which a second harmonic with a frequency 2ω generated in the crystal emerges. The plurality of first and second regions are formed as alternately arranged in a period substantially equal to d expressed by a predetermined expression, between the entrance face and the exit face.申请人：Nikon Corporation,National Institute for Materials Science地址：12-1, Yurakucho 1-chome Chiyoda-ku Tokyo 100-8331 JP,2-1, Sengen 1-chome Tsukuba-shi, Ibaraki 305-0047 JP国籍：JP,JP代理机构：Viering, Jentschura & Partner更多信息请下载全文后查看。

纹理物体缺陷的视觉检测算法研究--优秀毕业论文

摘要
在竞争激烈的工业自动化生产过程中，机器视觉对产品质量的把关起着举足轻重的作用，机器视觉在缺陷检测技术方面的应用也逐渐普遍起来。与常规的检测技术相比，自动化的视觉检测系统更加经济、快捷、高效与安全。纹理物体在工业生产中广泛存在，像用于半导体装配和封装底板和发光二极管，现代化电子系统中的印制电路板，以及纺织行业中的布匹和织物等都可认为是含有纹理特征的物体。本论文主要致力于纹理物体的缺陷检测技术研究，为纹理物体的自动化检测提供高效而可靠的检测算法。纹理是描述图像内容的重要特征，纹理分析也已经被成功的应用与纹理分割和纹理分类当中。本研究提出了一种基于纹理分析技术和参考比较方式的缺陷检测算法。这种算法能容忍物体变形引起的图像配准误差，对纹理的影响也具有鲁棒性。本算法旨在为检测出的缺陷区域提供丰富而重要的物理意义，如缺陷区域的大小、形状、亮度对比度及空间分布等。同时，在参考图像可行的情况下，本算法可用于同质纹理物体和非同质纹理物体的检测，对非纹理物体的检测也可取得不错的效果。在整个检测过程中，我们采用了可调控金字塔的纹理分析和重构技术。与传统的小波纹理分析技术不同，我们在小波域中加入处理物体变形和纹理影响的容忍度控制算法，来实现容忍物体变形和对纹理影响鲁棒的目的。最后可调控金字塔的重构保证了缺陷区域物理意义恢复的准确性。实验阶段，我们检测了一系列具有实际应用价值的图像。实验结果表明本文提出的纹理物体缺陷检测算法具有高效性和易于实现性。关键字: 缺陷检测；纹理；物体变形；可调控金字塔；重构
Keywords: defect detection, texture, object distortion, steerable pyramid, reconstruction
II

An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geo

a rXiv:g r-qc/3124v56Mar28An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry F.Finster,N.Kamran ∗,J.Smoller †,and S.-T.Yau ‡June 2004Abstract We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon.We derive an integral representation which expresses the solution as a superposition of solutions of the radial and angular ODEs which arise in the separation of variables.In particular,we prove completeness of the solutions of the separated ODEs.This integral representation is a suitable starting point for a detailed analysis of the long-time dynamics of scalar waves in the Kerr geometry.Contents 1Introduction 22Preliminaries 73Spectral Properties of the Hamiltonian in a Finite Box 114Resolvent Estimates 165Separation of the Resolvent216WKB Estimates257Contour Deformations32References 401IntroductionIn a recent paper[8],the long-term behavior of Dirac spinorﬁelds in the Kerr-Newman geometry,which describes a charged rotating black hole in equilibrium,was investigated. It was shown that solutions of the Dirac equation for Cauchy data in L2outside the event horizon and bounded near the event horizon,decay in L∞loc as t→∞.In this paper,we turn our attention to the scalar wave equation in the Kerr geometry.Our main result is to derive an integral representation for the propagator,similar to the one obtained for the Dirac equation in[8].In our next paper[9],we will use this integral representation to analyze the long-time dynamics and the decay of solutions in L∞loc.The analysis of the wave equation is quite diﬀerent from that for the Dirac equation. The main diﬃculty is that,in contrast to the Dirac equation,there is no conserved density for the scalar wave equation which is positive everywhere outside the event horizon.This is due to the fact that the charge density,which was positive for the Dirac equation, is not positive for the wave equation.The other conserved density,the energy density, is non-positive either:it is in general negative inside the ergosphere,a region outside the event horizon in which the Killing vector corresponding to time translations becomes space-like.For these reasons,it is not possible to introduce a positive scalar product which is conserved in time.In more technical terms,we are faced with the diﬃculty that it is impossible to represent the Hamiltonian(i.e.the operator generating time translations) as a selfadjoint operator on a Hilbert space.We remark that the existence of the ergosphere is a direct consequence of the fact that the Kerr black hole has angular momentum[4].Thus the ergosphere vanishes in the spherically symmetric limit.This simpliﬁes the analysis considerably.A number of important contributions have been made to the rigorous study of the scalar wave equation in black hole geometries.The current last word on the stability of spherical black holes under scalar wave perturbations is the paper by Kay and Wald[14], who proved using energy estimates together with a reﬂection argument that all solutions of the wave equation in the Schwarzschild geometry are bounded in L∞.More recently, Klainerman,Machedon,and Stalker[15]proved decay in L∞loc of spherically symmetric solutions.These papers use the spherical symmetry of the Schwarzschild metric in an essential way.Whiting[21]proved the absence of exponentially growing modes for the Teukolsky equation with general spin s=0,1selfadjoint,and has a spectral decomposition involving aﬁnite set of complex spectral points,which appear in complex conjugate pairs,together with a discrete spectrum of real eigenvalues.We write the projectors onto the invariant subspaces as contour integrals of the resolvent.In order to obtain estimates for the resolvent,it is useful to consider the Hamiltonian as a non-selfadjoint operator on a Hilbert space.This procedure also works in the original inﬁnite volume setting,and we derive operator estimates which compare the resolvent inﬁnite volume to that in inﬁnite ing these estimates,we can represent the spectral projector corresponding to the non-real spectrum as integrals over contours which are not closed and lie inside a region of of the form|Imω|<c(1+|Reω|)−1 around the real axis.At this point,we make use of the fact that the scalar wave equation in the Kerr geometry is separable into ordinary diﬀerential equations for the radial and angular parts[4].For the angular equation,we rely on the results of[10],where a spectral representation is obtained for the angular operator,and estimates for the eigenvalues and spectral projectors are derived.For the radial equation,we here derive rigorous estimates which are based on the semi-classical WKB ing these estimates,we can express the resolvent in terms of solutions of the ing furthermore Whiting’s result that the ODEs admit no normalizable solutions for complexω,we can deform the contours onto the real line.Thisﬁnally gives an integral representation for the propagator in terms of the solutions of the ODEs withωreal.To be more precise,recall that in Boyer-Lindquist coordinates(t,r,ϑ,ϕ)with r>0, 0≤ϑ≤π,0≤ϕ<2π,the Kerr metric takes the form[4,12]ds2=g jk dx j x k=∆∆+dϑ2 −sin2ϑM2−a2and r1=M+∂t,g ijξiξj=g tt=∆−a2sin2ϑU.(1.2)This shows thatξis space-like in the open region of space-time wherer2−2Mr+a2cos2ϑ<0,(1.3) the so-called ergosphere.It is a bounded region of space outside the event horizon,and intersects the event horizon at the polesϑ=0,π.The scalar wave equation in the Kerr geometry isΦ:=g ij∇i∇jΦ=1−g∂−g g ij∂∂r ∆∂∆ (r2+a2)∂∂ϕ2−∂∂cosϑ−1∂t+∂∂r∆∂∆((r2+a2)ω+ak)2(1.8) Aω,k=−∂∂cosϑ+1the wave equation(1.5)takes the form i ∂t Ψ=H Ψ,(1.12)where H is the HamiltonianH = 01αβ .(1.13)Here αand βare the diﬀerential operatorsα= (r 2+a 2)2∆−1∆−a 2sin 2ϑ −1 r 2+a 22πi k ∈Z e −ikϕ n ∈I N lim εց0 C ε−2π 2π0e ikϕΨ0(r,ϑ,ϕ)dϕ.We consider ωin the lower complex half plane {Im ω<0},and C εis a contour which joins the points ω=−∞with ω=∞and stays in an ε-neighborhood of the real line.A typical example is C ε={x −iεe −x 2:x ∈R }.2πi Ce −iωt (A −ω)−1dω,where A is a ﬁnite-dimensional matrix and C a contour which encloses the whole spectrum of A .For given ωand k ,the wave operator is a sum of a radial operator R ω,k and an angular operator A ω,k .As shown in [10],the angular operator has for ωnear the real line a purely discrete spectrum consisting of eigenvalues (λn )n ∈I N (see Lemma 2.1).The spectralU (dt −a sin 2ϑdϕ),(1.18)where q denotes the charge of the black hole,and the parameters M,a,q satisfy the inequality M 2>a 2+q 2.2PreliminariesIn this section we brieﬂy recall the variational formulation of the wave equation and the separation of variables.Furthermore,we bring the equation into aﬁrst-order Hamilto-nian form.Finally,we introduce and discuss scalar products which are needed for the construction of the propagator.The wave equation(1.5)is the Euler-Lagrange equation corresponding to the action S= ∞−∞dt ∞r1dr 1−1d(cosϑ) π0dϕL(Φ,∇Φ),(2.1) where the Lagrangian L is given byL=−∆|∂rΦ|2+1sin2ϑ (a sin2ϑ∂t+∂ϕ)Φ 2.(2.2) According to Noether’s theorem,symmetries of the Lagrangian give rise to conserved quantities.The symmetry under local gauge transformations yields that the vectorﬁeldJ k=−Im(2πQ,where Q is the charge densityQ=i∂L∆r2+a2 −a2sin2ϑa sin2ϑ .Moreover,since the Kerr metric is stationary,the Lagrangian is invariant under time translations.The corresponding conserved quantity is the energy E,E[Φ]= ∞r1dr 1−1d(cosϑ) 2π0dϕ∂ΦtΦt−L= (r2+a2)2sin2ϑ−a2Our analysis is based on a few properties of the angular operator Aω,which we now state.For realω,the angular operator Aωclearly is formally selfadjoint on L2(S2). However,this is not suﬃcient for our purpose,because we need to consider the case that ωis complex.In this case,Aωis a non-selfadjoint operator.Nevertheless,we have the following spectral decomposition,which is proved in[10].Lemma2.1(angular spectral decomposition)For any given c>0,we deﬁne the open set U⊂C by the condition|Imω|<cAfter separation (1.6),thereducedwave equation takes the form−∂∂r −1∂cos ϑsin 2ϑ∂sin 2ϑ(aωsin 2ϑ+k )2 Φ=0.(2.10)Under the separation,the above expressions for the charge and energy densities becomeQ =|Φ|2(r 2+a 2)2r 2+a 2 −a 2sin 2ϑ Re ω+k ∆ |ω|2−a 2k 2a 2sin 4ϑ+∆|∂r Φ|2+sin 2ϑ|∂cos ϑΦ|2.(2.12)It is a subtle point to ﬁnd a scalar product <.,.>which is well-suited to the analysis of the wave equation.It is desirable to choose the scalar product such that the Hamiltonian H is Hermitian (i.e.formally selfadjoint)with respect to it.Since H is the inﬁnitesimal generator of time translations,H is Hermitian w.r.to <.,.>if and only if the inner product <Ψ,Ψ>is time independent for all solutions Ψ=(Φ,i∂t Φ)of the wave equation.This can for example be achieved by imposing that <Ψ,Ψ>should be equal to the energy E corresponding to Ψ.This leads us to introduce a scalar product by polarizing the formula for the energy,(2.4,2.5).We thus obtain the so-called energy scalar product<Ψ,Ψ′>= ∞r 1dr1−1d (cos ϑ) (r 2+a 2)2∂t Φ∂t Φ′+∆∂cos ϑΦ∂cos ϑΦ′+ 1∆∆−a 2sin 2ϑ ω)ΦΦω,λ+2ak r 2+a 2ΦΦω,λ.(2.14)In the special case Ψ=Ψ′,this reduces to<Ψω,λ,Ψω,λ>=2ω ∞r 1dr1−1d (cosϑ)|Φω,λ|2× (r 2+a 2)2∆−1 .(2.15)By construction,the Hamiltonian is Hermitian with respect to the energy scalar product.However,the energy scalar product is in general not positive deﬁnite.This is obviousin (2.13)because the factor (sin −2ϑ−a 2/∆)is negative inside the ergosphere.Likewise,the integrand in (2.15)can be negative because the factor ak in the second term in the brackets can have any sign.Apart from the energy,also the charge Q gives rise to a conserved scalar product.It is a natural idea to try to obtain a positive scalar product by taking a suitable linear combination of these two scalar products.Unfortunately,comparing (2.12)and (2.11)one sees that it is impossible to form a non-trivial linear combination of Q and E which is manifestly positive everywhere.One might argue that a suitable linear combination might nevertheless be positive because the positive term ∆|∂r Φ|2+sin 2ϑ|∂cos ϑΦ|2might compensate the negative terms.However,comparing (2.15)with (2.11),one sees that there is a simple relation between the energy scalar product and the charge,<Ψω,λ,Ψω,λ>=2ωQ [Ψω,λ],making it again impossible to form a linear combination such that the integrand of the corresponding scalar product is everywhere positive.Stephen Anco showed that it is indeed impossible to introduce a conserved density for the wave equation which gives rise to a positive deﬁnite scalar product [1].We conclude that if we want to consider H as a selfadjoint operator,the underlying scalar product will necessarily be indeﬁnite.But we can clearly consider H as a non-selfadjoint operator on a Hilbert space,and this point of view will indeed be useful for the estimates of Section 4.Our method for constructing a positive scalar product is to simply replace the negative term −a 2/∆in (2.13)by a positive term.More precisely,we introduce the scalar product (.,.)by(Ψ,Ψ′)= ∞r 1dr1−1d (cos ϑ) (r 2+a 2)2∂t Φ∂t Φ′+∆∂cos ϑΦ∂cos ϑΦ′+1∂ϕΦ∂ϕΦ′+(r 2+a 2)2ΦΦ′ .(2.16)We denote the corresponding Hilbert space by H and the norm by . .This norm dom-inates the energy scalar product in the sense that there is a constant c 1>0depending only on the geometry such that the “Schwarz-type”inequality|<Ψ,Ψ′>|≤c 1 Ψ Ψ′ (2.17)holds for all Ψ,Ψ′∈H .We ﬁnally bring the Hamiltonian and the above inner products into a more convenient form.First,we introduce the Regge-Wheeler variable u bydu∆,∂∆∂r 2+a 2(2.19)β=−2ak r 2+a 2 (2.20)δ=1r 2+a 2(2.21)as well as the operatorA=1∂u(r2+a2)∂r2+a2∆S2−a2k2 c≤ρbelow).An important special case of a Krein space is when K is positive except on a ﬁnite-dimensional subspace,i.e.κ:=dim K−<∞.(3.3) In this case the Krein space is called a Pontrjagin space of indexκ.Classical results ofPontrjagin(see[3,Thms7.2and7.3,p.200]and[16,p.11-12])yield that any selfadjointoperator A on a Pontrjagin space is deﬁnitizable,and that it has aκ-dimensional negative subspace which is A-invariant.We now explain how the abstract theory applies to the wave equation in the Kerr ge-ometry.In order to have a spectral theorem,the Hamiltonian must be deﬁnitizable.There is no reason why H should be deﬁnitizable on the whole space(r1,∞)×S2,and this leads us to consider the wave equation in“ﬁnite volume”[r L,r R]×S2with Dirichlet boundary conditions.Thus settingΨ=(Φ,iΦt)and regarding the two components(Ψ1,Ψ2)ofΨasindependent functions,we consider the vector space P rL,r R =(H1,2⊕L2)([r L,r R]×S2)with Dirichlet boundary conditionsΨ1(r L)=0=Ψ1(r R).(3.4) Our deﬁnition of H1,2([r L,r R]×S2)coincides with that of the space W1,2((r L,r R)×S2)in[11,Section7.5].Note that we only impose boundary conditions on theﬁrst componentΨ1ofΨ,which lies in H1,2.According to the trace theorem[7,Part II, Section5.5,Theorem1],the boundary values of a function in H1,2([r L,r R]×S2)are in L2(S2),and therefore we can impose Dirichlet boundary conditions.We endow this vector space with the inner product associated to the energy;i.e.in analogy to(2.13),<Ψ,Ψ′>= r R r L dr 1−1d(cosϑ) (r2+a2)2Ψ2Ψ′2+∆∂cosϑΨ1∂cosϑΨ′1+ 1∆∂rΦ∂rΦ′+sin2ϑsin2ϑ−a2ΦΦ′ .(3.6)Transforming to the variable u,(2.18),and using the representation(2.23),one sees that on the subspace C2([u L,u R]×S2)the inner product(3.6)can be written as<Φ,Φ′>=(Φ,AΦ′)L2([uL,u R]×S2,dµ)(3.7)with A according to (2.22).Here weset u L =u (r L ),u R =u (r R ),and dµis the mea-sure (2.26).A is a Schr¨o dinger operator with smooth potential on a compact domain.Standard elliptic results [20,Proposition 2.1and the remark before Proposition 2.7]yield that H is essentially selfadjoint in the Hilbert space H =L 2([u L ,u R ]×S 2,dµ).It has a purely discrete spectrum which is bounded from below and has no limit points.The corresponding eigenspaces are ﬁnite-dimensional,and the eigenfunctions are smooth.Let us analyze the kernel of A .Separating and using that the Laplacian on S 2has eigenvalues −l (l +1),l ∈N 0,A has a non-trivial kernel if and only if for some l ∈N 0,the solution of the ODE−∂∂u +∆r 2+a 2φ(u )=0(3.8)with boundary conditions φ(u R )=0and φ′(u R )=1vanishes at u =u L .Since this φhas at most a countable number of zeros on (−∞,u R ](note that φ(u )=0implies φ′(u )=0because otherwise φwould be trivial),φvanishes at u L only if u L ∈E l with E l countable.We conclude that there is a countable set E =∪l E l such that the kernel of A is trivial unless u L ∈E .Assume that u L /∈E .Then A has no kernel,and so we can decompose H into the positive and negative spectral subspaces,H =H +⊕H −.Clearly,H −is ﬁnite-dimensional.Since its vectors are smooth functions,we can consider H −as a subspace of P r L ,r R ,and according to (3.7)it is a negative subspace.Its orthogonal complement in P r L ,r R is contained in H +and is therefore positive.We conclude that P r L ,r R is positive excepton a ﬁnite-dimensional subspace.It remains to show that the topology induced by <.,.>is equivalent to the H 1,2-topology.Since on ﬁnite-dimensional spaces all norms are equivalent,it suﬃces to consider for any λ0>0the spectral subspace for λ≥λ0,denoted by H λ0.We choose λ0such that1−λ0≤V 0:=min [r L ,r R ]−a 2k 22 Ψ,A Ψ L 2(dµ)+λ02c Ψ 2H 1,2+V 0−12Ψ 2L 2(dµ)≥1We always choose r L and r R such that P r L ,r R is a Pontrjagin space and that our initialdata is supported in [r L ,r R ]×S 2.We now consider the Hamiltonian (1.13)on the Pontrjagin space P r L ,r R with domainC ∞([r L ,r R ]×S 2)2⊂P r L ,r R .For clarity,we shall often denote this operator by H r L ,r R .Lemma 3.2H r L ,r R has a selfadjoint extension in P r L ,r R .Proof.On the domain of H,the scalar product can be written in analogy to(2.23)as<Ψ,Ψ′>=(Ψ,SΨ′)L2([uL,u R]×S2,dµ),where the operator S acts on the two components ofΨas the matrixS= A001 ,(3.9)where A is again given by(2.22)and dµis the measure(2.26).As shown in Lemma3.1, S has a selfadjoint extension and is invertible.We introduce on C∞0((u L,u R)×S2)2theoperator B by B=|S|−12.The fact that H is symmetric in P rL,r R implies that Bis symmetric in L2([u L,u R]×S2,dµ).A short calculation shows thatB2= |A||A|−12A|A|+β2 .Treating the terms involvingβas a relatively compact perturbation,we readilyﬁnd that B2is selfadjoint on L2([u L,u R]×S2,dµ)with domain D(B2)=D(A)⊕D(A).Con-sequently,the spectral calculus gives us a selfadjoint extension of B with domain D(B)= D(A12).We extend H to the domain D(H):=|S|−12Ψ,B|S|12Φ,|S|12Ψ,S|S|−12Ψ,B˜Ψ)L2(dµ)=(S|S|−12Ψlies in the domain of B and that B|S|12Φ.This implies thatΨ∈D(H)and that HΨ=Φ.with deg p0≤κminimal.Furthermore,we let p be the real polynomial of degree≤2κdeﬁned by p=p0p0(H rL,r R )x,L−>=<x,p0(H rL,r R)L−>=0,(3.10)so thatim p(H rL,r R)⊂im4Resolvent EstimatesIn this section we consider the Hamiltonian H as a non-selfadjoint operator on the Hilbert space H with the scalar product(.,.)according to(2.16).We work either in inﬁnite volume with domain of deﬁnition D(H)=C∞0((r1,∞)×S2)2or in theﬁnite box r∈[r L,r R] with domain of deﬁnition given by the functions in C∞((r L,r R)×S2)2which satisfy the boundary conditions(3.4).Some estimates will hold in the same way inﬁnite and inﬁnite volume.Whenever this is not the case,we distinguish betweenﬁnite and inﬁnite volumewith the subscripts rL,r R and∞,respectively.We always consider aﬁxed k-mode.The next lemma shows that the operator H−ωis invertible if either|Imω|is large or|Imω|=0and|Reω|is large.The second case is more subtle,and we prove it using a spectral decomposition of the elliptic operator A which generates the energy scalar product.This lemma will be very useful in Section7,because it will make it possible to move the contour integrals so close to the real axis that the angular estimates of Lemma2.1 apply.By a slight abuse of notation we use the same notation for H and its closed extension.Lemma4.1There are constants c,K>0such that for allΨ∈D(H)andω∈C,(H−ω)Ψ ≥11+|Reω| Ψ .Proof.For every unit vectorΨ∈D(H),(H−ω)Ψ ≥|(Ψ,(H−ω)Ψ)|≥|Im(Ψ,(H−ω)Ψ)|≥|Imω|−1space L 2(dµ):=L 2(R ×S 2,dµ),with dµaccording to(2.26).Clearly,A is bounded from below,A ≥−c ,and thus σ(A )⊂[−c,∞).For given Λ≫1we let P 0and P Λbe the spectral projectors corresponding to the sets [−c,Λ2)and [Λ2,∞),respectively.We decompose a vector Ψ∈H in the form Ψ=Ψ0+ΨΛwith Ψ0= P 000P 0 Ψ,ΨΛ= P Λ00P ΛΨ.This decomposition is orthogonal w.r.to the energy scalar product,<ΨΛ,Ψ0>= ΨΛ, A 001Ψ0 L 2(dµ)=0.However,our decomposition is not orthogonal w.r.to the scalar product (.,.),because (ΨΛ,Ψ0)= ΨΛ, A +δ001 Ψ0 L 2(dµ)= ΨΛ, δ000Ψ0 L 2(dµ).But at least we obtain the following inequality,|(ΨΛ,Ψ0)|≤c Ψ0 Ψ1Λ L 2(dµ),(4.2)where Ψ1Λdenotes the ﬁrst component of ΨΛ.Using that Ψ1Λ 2L 2(dµ)=<Ψ1Λ,A −1Ψ1Λ>≤1ΛΨ0 ΨΛ .Choosing Λsuﬃciently large,we obtain Ψ 2= ΨΛ 2+2Re (ΨΛ,Ψ0)+ Ψ0 2≤4( ΨΛ + Ψ0 )2and thusΨ ≤2( ΨΛ + Ψ0 ).(4.3)Furthermore,we can arrange by choosing Λsuﬃciently large that <ΨΛ,ΨΛ>= ΨΛ, A 001ΨΛ L 2(dµ)≥12 ΨΛ 2.Next we estimate the inner products <ΨΛ,H Ψ0>,(Ψ0,H ΨΛ)and (Ψ0,H Ψ0).The calculations <ΨΛ,H Ψ0>= ΨΛ, 0A A β Ψ0 L 2(dµ)= ΨΛ, 000β Ψ0 L 2(dµ)(Ψ0,H ΨΛ)= Ψ0, 0A +δA β ΨΛ L 2(dµ)= Ψ0, 0δ0βΨΛ L 2(dµ)|(Ψ0,H Ψ0)|= Ψ0, 0A +δA βΨ0 L 2(dµ) ≤c Ψ0 L 2(dµ)+2 A Ψ10 L 2(dµ) Ψ20 L 2(dµ) A Ψ10 2L 2(dµ)= Ψ0, A 2000 Ψ0 L 2(dµ)≤Λ2 Ψ0, A 001 Ψ0 L 2(dµ)=Λ2 Ψ0 2give us the bounds|<ΨΛ,HΨ0>|≤c ΨΛ Ψ0|(Ψ0,HΨΛ)|≤c Ψ0 ΨΛ|(Ψ0,HΨ0)|≤(c+2Λ) Ψ0 2.Using the above inequalities,we can estimate the inner product<ΨΛ,(H−ω)Ψ>by |<ΨΛ,(H−ω)Ψ>|≥|<ΨΛ,(H−ω)ΨΛ>|−|<ΨΛ,(H−ω)Ψ0>|≥|Imω|c ΨΛ − Ψ0 .(4.4) Next we estimate the inner product(Ψ0,(H−ω)Ψ),|(Ψ0,(H−ω)Ψ)|≥|(Ψ0,(H−ω)Ψ0)|−|(Ψ0,(H−ω)ΨΛ)|≥(|ω|−c−2Λ) Ψ0 2−c 1+|ω|Λ ΨΛ .(4.5) ChoosingΛ=(|ω|−c)/4and increasing c,the inequalities(4.4)and(4.5)give for suﬃ-ciently large|ω|the bounds(H−ω)Ψ ≥|Imω|2 Ψ0 −c ΨΛ .Multiplying the second inequality by4/|ω|and adding theﬁrst inequality,we conclude that2 (H−ω)Ψ ≥ |Imω||ω| ΨΛ + Ψ0 .The result now follows from(4.3).1+|Reω| (4.6) with K as in Lemma4.1.Corollary4.2Ifω∈Ω,the operator H−ωis invertible.The corresponding resolventS(ω):=(H−ω)−1satisﬁes the boundS(ω) ≤cUsing(4.6)in(4.7),we immediately get the boundS(ω) ≤c(1+|Reω|).(4.9) Since S(ω)is a bounded operator,its domain of deﬁnition can clearly be chosen to be the whole Hilbert space.We shall assume until the end of this section thatω∈Ω.The next lemma gives detailed estimates for the diﬀerence of the resolvents S rL,r R andS∞inﬁnite and inﬁnite volume,respectively.By Qλ(ω)we denote a given projector onto an invariant subspace of the angular operator Aωcorresponding to the spectral parameter λof dimension at most N(see Lemma2.1for details).Lemma4.3For everyΨ∈C∞0((r L,r R)×S2)2and every p∈N,there is a constant C=C(Ψ,p)(independent ofω)such that|<Ψ,[S r L,r R(ω)−S∞(ω)]Ψ>|≤C|Imω|.(4.10) Furthermore,for everyΨ∈C∞0((r L,r R)×S2)2and every p∈N and q≥N,there is a constant C=C(Ψ,p,q)(independent ofωandλ)such that|<Ψ,Qλ[S r L,r R(ω)−S∞(ω)]Ψ>|≤C|Imω| Qλ .(4.11) Proof.By deﬁnition of the resolvent,(H−ω)S(ω)Ψ=Ψ.This relation holds both in ﬁnite and in inﬁnite volume,and thus((H−ω)[S rL,r R(ω)−S∞(ω)]Ψ)(r,ϑ)=0if r L≤r≤r R.Iterating this identity and using the fact that H and S commute,we see that on[r L,r R]×S2,ωp+1[S rL,r R (ω)−S∞(ω)]Ψ=[S rL,r R(ω)−S∞(ω)]H p+1Ψ.(4.12)Combining this identity with the Schwarz-type inequality (2.17),we obtain|<Ψ,[S r L ,r R (ω)−S ∞(ω)]Ψ>|≤c 1 S r L ,r R (ω)−S ∞(ω) Ψ 2|ωp +1||<Ψ,[S r L ,r R (ω)−S ∞(ω)]Ψ>|≤ <Ψ,[S r L ,r R (ω)−S ∞(ω)]H p +1Ψ> ≤c 1 S r L ,r R (ω)−S ∞(ω) Ψ H p +1Ψ .Since Ψis smooth and has compact support,H p +1Ψalso has these properties.Theestimate (4.9)gives (4.10).In order to prove (4.11),we ﬁrst combine (4.12)with (2.17)to obtain(1+|ω|p +1)|<Ψ,Q λ(S r L ,r R −S ∞)Ψ>|≤c 1 Q λ S r L ,r R −S ∞ Ψ Ψ + H p +1Ψ .(4.13)Since q is at least as large as the dimension of the invariant subspace corresponding to λ,(A ω−λ)q Q λ=0.Therefore,for every Ψ′∈C ∞0((r L ,r R)×S 2)2,0=<Ψ,(A ω−λ)q Q λΨ′>=<(A ∗ω−λ)q and using (2.17),we obtain|λ|q <Ψ,Q λΨ′> ≤q l =1c l |λ|q −l (A ∗ω)l Ψ χ[r L ,r R ]Q λΨ′ with combinatorial factors c l (here χ[r L ,r R ]is the operator of multiplication by the char-acteristic function).Since the angular operator A ∗ωis according to (1.9)a polynomial inωof degree two,the function (A ∗ω)l Ψis also polynomial in ω,i.e.(A ∗ω)l Ψ=2l p =0ωp Ψp ,where the functions Ψp are composed of Ψand its angular derivatives,as well as the coeﬃcient functions of A ∗ω.This gives the estimate(A ∗ω)l Ψ ≤2l p =0|ω|p Ψp ≤c (1+|ω|2l )with a constant c which depends only on Ψand l .We thus obtain|λ|q<Ψ,Q λΨ′> ≤q l =1c l (Ψ)|λ|q −l (1+|ω|2l ) χ[r L ,r R ]Q λΨ′ .Young’s inequality allows us to compensate the lower powers of λ,|λ|q <Ψ,Q λΨ′> ≤c (q,Ψ)(1+|ω|2q ) χ[r L ,r R ]Q λΨ′ .We now choose Ψ′equal to the left side of (4.12)with p =0and p =r and take the sum of the resulting inequalities.Applying again the Schwarz inequality,we obtain|λ|q (1+|ω|r )|<Ψ,Q λ(S r L ,r R −S ∞)Ψ>|≤c (1+|ω|2q ) Q λ S r L ,r R −S ∞ ( Ψ + H r Ψ ).By choosing r suﬃciently large,we can compensate the factor (1+|ω|2q )on the right.More precisely,|λ|q (1+|ω|p +1)|<Ψ,Q λ(S r L ,r R −S ∞)Ψ>|≤c ′ Q λ S r L ,r R −S ∞ Ψ + H p +2q +1Ψ .Adding this inequality to (4.13)and substituting the estimate (4.9)gives (4.11).5Separation of the ResolventIn this section weﬁxω∈σ(H),so that the resolvent S=(H−ω)−1exists.As in the previous section,we assume that Qλis a given projector onto aﬁnite-dimensional invariant subspace of the angular operator Aωcorresponding to the spectral parameterλ.Our goal is to represent the operator product QλS in terms of the solutions of the radial ODE.According to(1.10)and(1.8),the radial ODE is−∂∂r−(r2+a2)2r2+a2 2+λ R(r)=0,(5.1) whereλis the separation constant.We can assume that k≥0because otherwise we reverse the sign ofω.We again work in the“tortoise variable”u,(2.18),and setφ(r)=r2+a2∂∂u+ ω+ak(r2+a2)2 φr2+a2=0.(5.3)Using that(r2+a2)∂2=−12∂∂u(r2+a2)1∂u2+V(u) φ(u)=0(5.4)with the potentialV(u)=− ω+ak(r2+a2)2+1r2+a2∂2uLemma 5.1The functions (u,u ′):=1∂u 2+V (u )s (u,u ′)=δ(u −u ′).Proof.By deﬁnition of the distributional derivative,∞−∞η(u )(−∂2u +V )s (u,u ′)du =∞−∞(−∂2u +V )η(u ) s (u,u ′)du for every test function η∈C ∞0(R ).It is obvious from its deﬁnition that the function s (.,u ′)is smooth except at the point u =u ′,where its ﬁrst derivative has a discontinuity.Thus after splitting up the integral,we can integrate by parts twice to obtain∞−∞(−∂2u +V )η(u ) s (u,u ′)du=u ′−∞η(u )(−∂2u +V )s (u,u ′)du +lim u րu′η(u )∂u s (u,u ′)+∞u ′η(u )(−∂2u+V )s (u,u ′)du −lim u ցu′η(u )∂u s (u,u ′).Since for u =u ′,s is a solution of (5.4),the obtained integrals puting thelimits with (5.7),we get∞−∞(−∂2u +V )η(u ) s (u,u ′)du = lim u րu′−lim u ցu′ η(u )∂u s (u,u ′)=1In what follows we also regard s (u,u ′)as the integral kernel of a corresponding operators ,i.e.(sφ)(u ):=du ′s (u,u ′)φ(u ′)du ′.If Q λprojects onto an eigenspace of A ω,we see from (1.10),(1.7),and (5.2)that(r 2+a 2)−12Q λ(ϑ,ϑ′)δ(u −u ′).(5.8)Loosely speaking,this relation means that the operator product Q λs is an angular modeof the Green’s function of the wave equation.Unfortunately,Q λmight project onto an invariant subspace of A ωwhich is not an eigenspace.In this case,the angular operator has on the invariant subspace the “Jordan decomposition”A ωQ λ=(λ+N )Q λ(5.9)with N =N (ω,λ)a nilpotent operator.Lemma 5.3extends (5.8)to this more general case.In preparation,we need to consider powers of the operator s .Lemma5.2For every l∈N0,the operator s l is well-deﬁned.Its kernel(s l)(u,u′)has regularity C2l−2.Proof.Writing out the operator products with the integral kernel,one sees that the operator s l is obtained from s by iterated convolutions,s p+1(u,u′)= s(u,u′′)s p(u′′,u′)du′′.(5.10)In theﬁnite box,these convolution integrals are allﬁnite because s(u,u′)is continuous and the integration range is compact.In inﬁnite volume,the function s(u,u′)decays exponentially as u,u′→±∞(see Corollary6.4),and so the integrals in(5.10)are again ﬁnite.Hence s l is well-deﬁned.Let us analyze the regularity of the integral kernel of s l.By deﬁnition,s(u,u′)is continuous,and(5.10)immediately shows that the same is true for s p(u,u′).Diﬀeren-tiating through(5.10)and applying Lemma5.1,one sees that s p satisﬁes for p>1the distributional equation−∂2Lemma5.3For givenλ∈σ(Aω)we let g be the operatorg=∞l=0(−N)l s l+1,(5.11)where N is the nilpotent matrix in the Jordan decomposition(5.9).Then(r2+a2)−12Qλ(ϑ,ϑ′)δ(u−u′).(5.12) Note that if Qλprojects onto an eigenspace,N vanishes and thus g=s.Furthermore, since N is nilpotent,the series in(5.11)is actually aﬁnite sum.Thus in view of Lemma5.2, (5.11)is indeed well-deﬁned.Proof of Lemma5.3.Denoting the radial operator with integral kernelδ(u−u′)by11u, we can write the result of Lemma5.1in the compact form(−∂2u+V)s=11u.Hence on the invariant subspace,we can do a Neumann series calculation,(−∂2u+V)g=∞l=0(−N)l(−∂2u+V)s l+1=∞ k=0(−N)l s l=11u−N g,to obtain that(−∂2u+V+N)g=11u.According to(1.10),(1.7),and(5.2),this is equiv-alent to(5.12).。

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Wavelength Reusability inWavelength-Routed Optical NetworksXiong Yizhi, Zeng Qingji, Wu Kai and Cheng YangR&D Center for Broadband Optical Networking TechnologyShanghai Jiao Tong University, Shanghai 200030, P.R.ChinaTel: +86-21-62932166Fax: +86-21-62820892E-mail: xyz70678@1. IntroductionRecently routing and wavelength assignment (RWA) problem in wavelength-routed optical networks (WRON's) has been studied extensively. Different optimum criterions are proposed to evaluate the performance of designed networks, such as carried traffic [1], wavelength requirement [2,3], fiber requirement [4], and OXC system scale [5]. Because wavelength is a very limited network resource, most of the research works focus on the wavelength requirements. However, for the non-uniform traffic pattern,the wavelength requirement N λ will increase with the traffic loads increased, it is difficult to evaluate the network performance by using wavelength requirement. In this paper, we will take wavelength reusability ρ as a network performance criterion. By using a novel heuristic RWA algorithm, we evaluate the wavelength reusability of several real planned transport networks. In addition, a large number of randomly generated WRONs are analyzed in order to quantify the relationship between wavelength reusability and network topology parameters, such as average nodal degree, etc.2. RWA Heuristic AlgorithmWe model the WRON's as an undirected graph G. Each vertex in the graph corresponds to a routing node (e.g. OXCs) and each edge to a link. For a Given WRON's with N nodes and L links, the average nodal degree is N L 2¥ave =. The network physical connectivity α is defined in [2], as 1)1(2¢−=−=N N N L ave δ. We define !i, k to be the set consisting of the K-shortest possible paths connecting the node-pair i. Here the shortest path length is measured by the number of physical links or hops. The traffic demand is T=(t i ), t i denote the number of connections demanded by s-d pair i.In this summary, the heuristic algorithms base on a shortest path routing and longer path wavelength assignment first policy. The accuracy and efficiency of this algorithm have been confirmed in [6]. It can apply to the designing of large-scale networks. In the case of virtual wavelength path (VWP) routing scheme, the VWP-Routing design algorithm is performed as follows:(1) All the node-pairs are ranked in decreasing manner according to their shortest path length firstly,then assign the shortest path route to each node-pair sequentially.(2) Usually the result achieved in step 1 is not optimal in minimizing wavelength requirement, so we reroute these previous assigned paths. Each established lightpath between node-pair is re-considered randomly. Each lightpath will be re-assigned a new route with the alternative shortest paths or longer paths in !i, k (k=3). If and only if the congestion of the most load link in the re-assigned path is lower than the congestion of the most loaded link in the previously assigned path. The re-assigned path is adopted.This process is repeated until no further improvement is achieved.Wavelength path (WP) routing scheme can be regarded as the special case of VWP-Routing where each VWP is assigned the same wavelength for all the links along the path route. The path routes selection of WP-Routing scheme is the same as the case of VWP-Routing scheme. The wavelength assignment algorithm is described as follows. The longest path route is assigned wavelength first, and then followed by those requiring shorter path length. If two or more paths have the same length, thealgorithm randomly selects the one to be assigned wavelength first. If all paths have been assigned a wavelength, the wavelength assignment is halted. The highest assigned wavelength-number determines the network wavelength requirement N λ of WP-Routing scheme.According to the obtained N λ, wavelength reusability ρ can be express as ∑=ii N t λ/². It denotes thetimes that the same wavelength used in the network.3. Numerical Simulations3.1 Real Planned NetworksWe take several real planned networks as examples. These networks include EuroCore, EON, UKnet and NSFnet [2]. The main topology parameters are summarized in Table 1. They are the minimum and maximum nodal degree (δmin , δmax ), average nodal degree δave , physical connectivity α, and nodal degree standard deviation σ. We define β as the average traffic load per node, that is ∑=ii N t /£. The trafficmatrix is generated randomly. The result of wavelength reusability in this paper is average over 50different traffic matrixes.Table 1The main topology parameters for real planned networks topologies NetworkN L (δmin , δmax )δave ασEuroCore1125(4,5) 4.540.450.498EON2039(2,7) 3.90.2 1.72UKnet2139(2,7) 3.710.19 1.137NSFnet1421(2,4)30.230.535For these real planned network. Fig.1 shows the wavelength reusability ρ for various value of β in the case of the VWP-Routing. It is easy to found that the wavelength reusability of these real networks will reach margin by increasing the traffic loads. For a network with given topology, the maximum wavelength reusability ρmax is limited. In contrast to the wavelength reusability, wavelength requirement will always increase with the traffic. So wavelength requirement is not an effective criterion to evaluate the network performance. It is interesting to see that in Fig.1, the network with large average nodal degree has a higher maximum wavelength reusability. For the case of WP-Routing, we can obtain the same results. The difference of wavelength reusability between VWP-Routing case and WP-Routing case is not more than 3%. Due to the limited space, the results of WP-Routing case are not presented.3.2 Random NetworksWe will analyze the randomly generated networks with size of 11, 14, 20 and 21 nodes. They are generated according to the approach proposed in [2]. The average nodal degree δave range from 2.0 to 5.0,the minimum and maximum nodal degree (δmin , δmax ) is set to [2, δave +2σ]. The chosen of standard deviation σ depends on N and α. For a given (N, δave ), networks can have different physical topologies. In order to derive general results, we will analyze 100 different topologies for each given (N, δave ).In Fig.2, the mean value of maximum wavelength reusability ρmax is plotted versus the average nodal degree. The mean value of ρmax increases with the δave increased. It is worth noting that the ρmax of the networks with δave =2.0 is slightly larger than that of the networks with δave a little greater than 2.0. It is because the networks with δave =2.0 correspond to the ring networks, the nodal degree standard deviation σ of ring network is zero, and δmin is equal to δmax . In addition, it can be seen that the ρmax of EuroCore is almost the same with the average ρmax , NSFnet is better than the average ρmax . Whereas the ρmax of EON and UKnet is lower than the average ρmax , because they have a high value of the node degree standard deviation, it will lead to the network topology cannot match the traffic load perfectly.Fig. 1. Wavelength reusability ρ for various valueFig. 2. The mean maximum wavelength reusability of β in the case of the VWP-Routing versus average nodal degree for the VWP-Routing case4. ConclusionsWavelength reusability is an efficient criterion to evaluate the network performance in wavelength-routed optical networks. Our study shows that the wavelength reusability will reach limit with traffic loads increasing. The maximum wavelength reusability ρmax is limited for a network with given topology parameters. The maximum wavelength reusability mainly depends on the average nodal degree δave .5. References[1] N. Wauters and P. Demeester, "Design of the optical path layer in multiwavelength cross-connected networks", IEEE J. Select.Areas Commun., Vol.14, No.5, pp.881-892, June. 1996[2] S.Baroni and P. Bayvel, "Wavelength requirements in arbitrarily connected wavelength-routed optical networks", Journal of lightwave technology, Vol.15, No.2, pp.242-251, Feb. 1997[3] N. Nagatsu, Y. Hamazumi, and K. I. Sato, " Optical path accommodation designs applicable to large scale networks", IEICE Trans. on Commun., Vol.E78-B, No.4, pp.597-607, April 1995[4] N.Nagatsu, S. Okamoto, and K.I.Sato, "Large scale photonic transport network design based on optical paths", in Proc.GLOBECOM'96, pp. 321-327, 1996[5] N.Nagatsu, S. Okamoto, and K.I.Sato, "Optical path cross-connect system scale evaluation using path accommodation design for restricted wavelength multiplexing", IEEE J.Select. Areas Commun., Vol.14, No.5, pp.893-902, June, 1996[6] Xiong YiZhi, Zeng Qingji, Cheng Yang and Zhou Yuxun, "New Heuristic Algorithms for Routing and Wavelength Assignment in Large Wavelength-Routed Optical Networks", Acta Photonica Sinica, Vol.28, No.6, pp.508-513, June, 1999。