Phases and Density of States in a Generalized Su-Schrieffer-Heeger Model

ApoE—／—小鼠动脉粥样硬化模型的建立摘要：ApoE-/-基因敲除小鼠（ApoE-/-）经含有21%脂肪和0.15%胆固醇的高脂饲料喂食12周后进行各项血脂胆固醇水平检测，以及整体主动脉油红0染色与主动脉根部病理切片油红0染色等动脉粥样硬化病理分析。

结果显示经过高脂诱导的ApoE-/-小鼠的血浆总胆固醇和甘油三酯水平均比未经饮食诱导的ApoE-/-小鼠、经同样饮食处理的野生型小鼠以及未经处理的野生型小鼠均显著升高（P＜0.05）；低密度脂蛋白-胆固醇水平与野生型（正常饮食组和高脂组）相比升高了近3倍多；高脂诱导ApoE-/-小鼠的主动脉斑块面积占整体主动脉面积的65%，显著高于ApoE-/-小鼠的正常饮食组（21%）（P＜0.05），同时主动脉根部的血管壁明显增厚，管腔变窄。

实验结果表明通过高脂饲料饮食诱导，成功建立了动脉粥样硬化模型小鼠，可为下游的药物筛选、基因治疗以及动脉粥样硬化机理的体内研究提供理想的实验材料。

关键词：ApoE-/- 小鼠；动脉粥样硬化；胆固醇；血脂；主动脉中图分类号：R394 文献标识码：A 文章编号：1007-7847（2015）02-0141-04The Establishment of Atherosclerosis Model in ApoE-/- MiceOU Hai-long，ZHANG Li-lin，HE Xiao-lan，LI Hong-mei，LEI Ting-wen （Department of Biochemistry and Molecular Biology，Guiyang Medical University，Guiyang 550004，Guizhou，China）Abstract ：The ApoE gene knockout mice （ApoE-/-）were fed with a high-fat （HF）diet containing 21% fat and 0.15% cholesterol for 12 weeks to establish atherosclerosis model. After diet induction，the ApoE-/- mice were sacrificed and plasma total cholesterol （TC），triglyceride （TG），low -density lipoprotein -cholesterol （LDI-C）and high-density lipoprotein-cholesterol （HDL-C）levels were measured. The whole aortas and sequential sections of the aortic root were stained with oil red 0. Results showed that the plasma levels of TC and TG from HF diet induced ApoE-/- mice were both dramatically higher than normal diet（ND）-fed ApoE-/- mice as well as wild-type mice fed with normal diet or HF diet （P <0.05）. The contents of LDL-cholesterol in plasma were elevated by three-fold compared with wild-type （ND and HFD）. Atherosclerotic lesion sizes were significantly increased in whole aorta （65%）as compared with normal diet ApoE-/-mice （21%）. Similar result was obtained from cross -sections of the aortic root analysis. The results demonstrated that HF diet treatment greatly enhanced atherosclerosis development in ApoE-/-mice. The establishment of atherosclerosis model mice provides a valuable tool for drug screen，gene therapy and even in vivo mechanism analysis in atherosclerosis disease.Key words ：ApoE-/-mice；atherosclerosis；cholesterol；lipoprotein；aorta（Life Science Research，2015，19（2）：141 ?144）体内的脂类物质代谢异常时，多余的脂质沉积在血管壁上，并逐渐形成斑块，使血管内皮增厚、变硬，是引起动脉粥样硬化（atherosclerosis AS）的重要原因之一。

诚信应考,考试作弊将带来严重后果！大学课程考试试卷mutation, in which the mutated site changes back into the original wild-type sequence.9. Induced mutations : Mutations result from changes caused byenvironmental chemicals or physical factors.10. Gene expression is the production of proteins according to instructionsencoded in DNA. During gene expression, the information in DNA is transcribed into RNA, and the RNA message is translated into a string of amino acids.11. Loss-of-function mutation (or allele): DNA mutation that reduces orabolishes the activity of a gene; most (but not all) loss-of-function alleles are recessive .12. Mutation rate : The frequency with which a wild-type allele at a locuschanges into a mutant allele and is generally expressed as the number of mutations per biological unit, which may be mutations per cell division, per gamete, or per round of replication.13. cis-acting regulatory elements are regions of DNA sequence that lienearby on the same DNA molecule as the gene they control.14. Germ-line mutations*arise in cells that ultimately produce gametes*. Agerm-line mutation can be passed to future generations, producing individual organisms that carry the mutation in all their somatic and germ-line cells.15. Trans-acting factors. trans-acting genetic elements encode productscalled transcription factors that interact with cis-acting elements, either directly through DNA binding or indirectly through protein-protein interactions.二、[填空题] (每空2 分，共20 分)1. the gene map in a chromosome is changed from bog-rad-fox1-fox2-try-duf to bog-rad-fox2- fox1-try-duf: (Inversion)2. the gene map in a given chromosome is changed frombog-rad-fox1-fox2-try-duf to bog-rad-fox1-mel-qui-txu-sqm: (Inversion; translocation; deletion; insertion)3. an A–T base pair in the wild-type gene is changed to a G–C pair: ( transition; base substitution )4. an A–T base pair is changed to a T–A pair: (base substitution; transversion)5. the sequence AACGTTATCG is changed to AATGTTATCG: (transition; base substitution)6. the sequence AACGTCACACACACATCG is changed to AACGTCACATCG: (deletion; unequal crossing-over)7. the sequence AAGCTTATCG is changed to AAGCTATCG: (deletion)8. the sequence AAGCTTATCG is changed to AAGCTTTATCG: (insertion)9. the gene map in a given chromosome arm is changed frombog-rad-fox1-fox2-try-duf (where fox1 and fox2 are highly homologous, recently diverged genes) to bog-rad-fox1-fox3-fox2-try-duf (where fox3 is a new gene with one end similar to fox1 and the other similar to fox2 ): (insertion; unequal crossing-over)10. a C–G base pair in the wild-type gene is changed to a T–A pair: ( transition; base substitution )三、[简答题] (每题 10 分，共 20分)1. Describe the process of DNA replication machinery and DNA repairmachinery briefly,A chromosome contains double-stranded DNA, made from nucleotides with four different bases. Adenine always pairs with thymine, and guanine with cytosine. Together they form “base pairs”. The cell’s 46 chromosomes comprises approximately 6 billion base pairs.The DNA repair machinery1.Base excision repair2.Nucleotide excision repair3.Mismatch repairtake advantage of complementary base pairing, using the undamaged strand as a template to damaged DNA Some examples are base or nucleotide excision repair systems, and mismatch repair systems.2. What is promoter, please describe the three phases of transcription. Promoter: A regulatory region of DNA usually located upstream of a gene, providing a control point for regulated gene transcription.1.Initiation:①Initiation requires a special subunit of RNA polymerase—the sigma (σ) subunit—in addition to the two alpha (α), one beta (β), and one beta prime (β’) subunits that make up the core enzyme.2.Elongation:Elongation continues until the RNA polymerase encounters a signal in the RNA sequence that triggers termination.3. Termination•Termination signal ing part of transcription is started at the promoter and then ended at the termination signal.•Two types of termination signals in prokaryotes：Rho dependent and Rho independent.三、［简答题］（每题10分，共20分）1. What are transposable elements? How many and what types oftransposable elements there are in general? What is transposition?Write out the features of transposition in common.Transposable elements are DNA sequences capable of moving and are found in the genomes of all organisms.They often cause mutations, either by inserting into another gene and disrupting it or by promoting DNA rearrangements such as deletions, duplications, and inversions.Types of transposable elements:Simple structures, encompassing only those sequences necessary for their own transposition (movement),Complex structures and encode a number of functions not directly related to transposition.TranspositionTransposition is the movement of a transposable element from one location to another.Several features of transposition in common:⑴Staaaered breaks are made in the target;(2)The transposable element is joined to single-stranded ends of the target DNA;⑶DNA is replicated at the single-strand gaps.一．1.功能增益等位基因（或突变）：是增强基因功能或赋予基因产物新活性的罕见突变。

a r X iv:c ond-ma t/121v2[c ond-m at.s upr-con]14Jan2SUPERCONDUCTIVITY IN A MESOSCOPIC DOUBLE SQUARE LOOP:EFFECT OF IMPERFECTIONS V.M.Fomin ∗,J.T.Devreese ♯Theoretische Fysica van de Vaste Stof,Universiteit Antwerpen (U.I.A.),Universiteitsplein 1,B-2610Antwerpen,Belgi¨e V.Bruyndoncx,V.V.Moshchalkov Laboratorium voor Vaste-Stoffysica en Magnetisme,Katholieke Universiteit Leuven,Celestijnenlaan 200D,B-3001Leuven,Belgi¨e (February 1,2008)Abstract We have generalized the network approach to include the effects of short-range imperfections in order to analyze recent experiments on mesoscopic superconducting double loops.The presence of weakly scattering imperfec-tions causes gaps in the phase boundary B (T )or Φ(T )for certain intervals of T ,which depend on the magnetic flux penetrating each loop.This is accom-panied by a critical temperature T c (Φ),showing a smooth transition between symmetric and antisymmetric states.When the scattering strength of imper-fections increases beyond a certain limit,gaps in the phase boundary T c (B )or T c (Φ)appear for values of magnetic flux lying in intervals around half-integer Φ0=hc/2e .The critical temperature corresponding to these values of mag-netic flux is determined mainly by imperfections in the central branch.The calculated phase boundary is in good agreement with experiment.Typeset using REVT E XEarly experiments1,2have revealed that the effect of nonmagnetic impurities on the transition temperatures of bulk superconductors is very small.The critical temperature, T c,changes by about1%for1%concentration of impurities.An interpretation of these observations wasfirst proposed by Anderson3.The only effect of the impurities is to change the energy of a free electron to eigenenergies determined by those impurities.This modifies the density of states in the integral equation for T c.Therefore,scattering by non-magnetic impurities only slightly changes T c in bulk superconductors4.Investigations extending An-derson’s work have been performed for multiband superconductors using the Abrikosov-Gor’kov approach5.An interesting opportunity to intensify the effect of the imperfections occurs in mesoscopic superconducting structures where the confined condensate is much more sensitive to the action of impurities than in bulk structures.Recently,the onset of the superconducting state has been studied6,7in different meso-scopic structures of Al,comprising lines,dots,loops,double loops,microladders etc.,with sizes smaller than the coherence lengthξ(T).Refs.8,9show that experimentally observed phase boundaries for square loops with two attached leads are in excellent agreement with calculations based on Ginzburg-Landau(GL)theory10,11.In this present letter,we analyze the case of a superconducting In this present letter, we analyze the case of a superconducting mesoscopic double square loop(see the inset to Fig.1a)where experiments6have revealed the phase boundary shown in ing the micronet approach12–14for a superconducting double loop,we obtain a phase boundary (cf.Ref.7)which consists of the intersecting parabolas shown in Fig.1b.One set(with minima at integer values of the magneticfluxΦthrough one loop in units of the magnetic flux quantumΦ0≡hc/2e:Φ/Φ0)depends on the magneticflux quanta penetrating each loop with L=0,1,2,....The other set(with minima at half-integer values ofΦ/Φ0) depends on odd numbers of magneticflux quanta penetrating the double loop as a whole with L=1/2,3/2,....Since the critical temperature corresponds to the lowest Landau level E LLL(Φ),the way the parabolas intersect means that the minimum energy encounters a shift from one branch to another at certain values of magneticflux.Moreover,since the derivative of the lowest Landau level with respect to magneticflux is proportional to the persistent current(cf.Ref.15Eq.(4.5))we arrive at the following paradox:At the intersections,the left and right derivatives of E LLL(Φ)are different,the persistent current has a discontinuity and its value is consequently indefinite.In order to resolve this paradox,an analogy between superconducting loops and semiconducting quantum rings can be exploited.In such rings, in the presence of impurities and an Aharonov-Bohm magneticfield,a crossing is known to change into an anti-crossing.In this case,the gaps between the different eigenenergies as a function of the magneticflux widen and hence the degeneracy of states,which contribute to the lowest level,is raised(see Ref.16).In this letter we demonstrate that this also applies to the superconducting mesoscopic double square loop.a)and without(panel b)a parabolic background(the dashed line in panel a)due to thefinite width of the loop.When transforming the phase boundary to the plane T c(Φ)/T c versusΦ/Φ0,the value Q=1302nm is taken in order to ensure that the maxima of T c(Φ)/T c are at integer values ofΦ/Φ0. Theoretical phase boundaries(solid lines in panel b)calculated for a perfect double loop with Q=1302nm andξ0=128nm.The observed phase boundary for a superconducting mesoscopic double square Al loop is given in Fig.1a.After subtraction of the parabolic background,related to thefinite width, w=130nm,of the stripes(dashed line in Fig.1a),at least12practically identical periods are seen.One such period is plotted in Fig.1b(dots).From the period of the oscillations of the phase boundary with respect to the magneticfield,∆B=1.24mT,an effective loop side length Q=1.3µm is obtained.This is close to the average loop size.For further experimental details,the reader is referred to Ref.6.In order to understand the observed “anti-crossing”of the different elements of the experimentally observed phase boundary and,in particular,the smooth shape of the T c(Φ)minima,we consider a superconducting mesoscopic double square loop with imperfections.These imperfections may be introduced during the fabrication of the mesoscopic structures.Most probably one of the sources of such imperfections is the inhomogeneity of the geometrical superconducting lines written by e-beam lithography.Within the framework of the GL approach,the presence of imperfections in a supercon-ducting structure may be modelled by a spatial inhomogeneity in the parameters a and b in the GL equation:1A(r) 2ψ(r)+a(r)ψ(r)+b(r)|ψ(r)|2ψ(r)=0.(1)cNear the phase boundary,where the order parameterψ(r)is small,the system is adequately described by the linearized GL equation:1A(r) 2ψ(r)+a(r)ψ(r)=0.(2)cMoreover,the magneticfield may be assumed to be equal to the applied magneticfield.The vector potential A of the uniformfield B e z is taken in the symmetric gauge.The presence of imperfections,localized in the loop around several points r s over a distance which is much smaller than the coherence length or the typical loop size(“short range imperfections”),can be modelled by the following function:a(r)=a+ s V sδ(r−r s),(3) where a is the GL coefficient of the substance,and the magnitudes V s are determined by specific characteristics of the imperfections.Eq.(2)then becomes similar to the Schr¨o dinger equation for a particle of mass m and charge2e in the potentialfield described by the scalar form s V sδ(r−r s)and by the vector potential A(r),the quantity−a playing the role of the energy.Short-range imperfections are assumed to be present in all three branches of the loop at the points characterized by the coordinates Q s with s=L,M,R as shown in Fig. 2. Furthermore,takingξ(T)≡ξ0/Φ0A x)2+(i∇y+2π(1−T/T c)is temperaturedependent.FIG.2.A model configuration of imperfections(open circles)in the double loop.We have solved Eq.(4)following the micronet approach.A new conceptual feature compared with Refs.12–14is the use of additional nodal points at the positions of the imperfections.The presence of these imperfections implies an additional condition,which can be derived in the following way.In the vicinity of the point with y=Q s,Eq.(4)takes the form(i∇y+2πdy y=Q s+ǫ−dΨcos(ϕ)=cos(L)+12D M −D L D RD M+N iLD R −1D2L−12=0(8)withN1s=cos3Q+˜V s sin(2Q−Q s)cos(Q+Q s),N2s=cos3Q+˜V s cos(2Q−Q s)sin(Q+Q s), D s=sin3Q+˜V s sin(2Q−Q s)sin(Q+Q s);N1M=cos Q+˜V M sin(Q−Q M)cos(Q M),N2M=cos Q+˜V M cos(Q−Q M)sin(Q+Q M), D M=sin Q+˜V M sin(2Q−Q M)sin(Q+Q M),(9)where s=L,R.In Eq.(8),ϕ=2πΦ/Φ0withΦ,the magneticflux through each of the loops.Here,we recall that lengths in the above equations are expressed in units ofξ(T)and are therefore functions of the temperature.Consequently,the secular equation establishes a relation between the magneticfluxΦand the temperature T.The phase boundaries obtained by solving Eq.(8),with an imperfection in only one branch of the double loop,are shown in Figs.3to5(L in Figs.3and4,M in Fig.5).FIG.3.Theoretical phase boundaries for a double loop with an imperfection in one side branch at Q L=0.5Q for Q=1302nm andξ0=128nm.(a)The dimensionless magnitude C L=0and 0.01corresponds to dotted and solid lines,respectively.(b)The dimensionless magnitude C L= 0,0.5,1.0corresponds to dotted,solid and dashed lines.The valuesΦ1andΦ2are discussed in the text.As is seen in Fig.3a,at small values C s,an“energy”gap forms between solu-tions corresponding to different numbers of magneticflux quanta penetrating each loop (L=0,1/2,1,...).The resultant phase boundary indicates that a continuous change takes place from a symmetric superconducting order parameter at integer values ofΦ/Φ0to an antisymmetric state at half-integer values of the relative magneticflux.It is also worthy of note that the presence of imperfections slightly diminishes the critical temperature of the double loop at zero magneticfield.When increasing C s above a certain limit,the pattern of phase boundaries changes dramatically.Gaps appear in certainflux intervals(“flux gaps”)around half-integerΦ/Φ0 values.This behavior is illustrated by the curves in Fig.3b.For a given T<T c,a superconducting state exists whenΦranges from zero to a certain value ofΦ1,after which the sample turns into the normal state.With a further increase of magneticflux along a horizontal straight line shown in Fig.3b,the sample remains in the normal state until a value ofΦ2is reached,at which the sample becomes again superconducting.This demonstrates a re-entrant behavior as a function offield.[When approaching in Fig.3b the points where the phase boundaryΦ(T)has its extrema and,consequently,the derivatives∂T/∂Φwould diverge,the superconducting state apparently becomes unstable.]It should be noted that the existence of such a regime,where the system is normal in a certainflux interval,was reported by de Gennes12for a single superconducting ring with a lateral arm.The trend of lowering of the critical temperature of a double loop at zero magneticfield with increasing C s is clearly seen by comparison of the curves in Fig.3b which refer to different values of V L.FIG.4.Theoretical phase boundaries(solid lines)for a double loop with an imperfection in one side branch at Q L=0.5Q for Q=1302nm andξ0=128nm.The dimensionless magnitude is C L=0.02.Results obtained without and with renormalization of temperature are given in panels a and b,respectively.Experimental data are shown with points.In regions adjacent to integer values ofΦ/Φ0,a good agreement with experiment of the calculated phase boundary T c(Φ),is achieved for the zero-temperature coherence lengthξ0 =128nm.This value is in accordance with previous estimates(see Ref.17).The resulting set of phase boundaries is shown in Fig.4a.For comparison with the experimental data,it is necessary to renormalize the temperature scale,taking as a unit temperature the specific critical value of a loop with imperfections(see Fig.4b).In Fig.5,a plot of the phase boundary is shown for a double loop with an imperfectionin the middle branch(M in Fig.2).It is clear that by increasing V M,one shifts a minimum of the1−T c(Φ)/T c curve(using a renormalized temperature),at half-integer values ofΦ/Φ0, to higher values,without modifying those parts of the phase boundary close to the integer values ofΦ/Φ0.FIG.5.Theoretical phase boundaries for a double loop with an imperfection in the middle branch at Q M=0.1Q for Q=1302nm andξ0=128nm.The dimensionless magnitude C M= 0.11,0.125corresponds to dashed and solid lines.Results are obtained with renormalization of temperature.Experimental data are shown with points.The best agreement between the calculated phase boundary and experimental data is achieved for a configuration where imperfections are present on all three branches of the double loop(L,M,R in Fig.2).The corresponding curves are shown in Fig.6.The main observation which follows from thesefigures is that imperfections in the central branch play a decisive role in determining the critical temperature of a double loop at the half-integer values ofΦ/Φ0.FIG.6.Theoretical with imperfections in all three branches at Q L=Q R=Q M=0.5Q for Q=1302nm andξ0=128nm.The dimensionless magnitudes are:C L=−C R=0.017and C M=0.090.Results are obtained with renormalization of temperature.Experimental data are shown with points.In conclusion,we have generalized the network approach to include the effects of short-range imperfections.The presence of weakly scattering imperfections leads to the formation of gaps between solutions corresponding to integer and half-integer numbers of magnetic flux quanta penetrating each loop.The phase boundary is characterized by a smooth tran-sition between symmetric and antisymmetric states.For imperfections with relatively large magnitudes,gaps in the phase boundary T c(B)or T c(Φ)appear when the magneticflux lies in intervals around half-integer values.The critical temperature at the half-integer values of the relative magneticflux has been shown to be determined mainly by imperfections in the central branch.The calculated phase boundary for a mesoscopic double square loop is in good agreement with experiment.Acknowledgments.-We thank V.N.Gladilin for fruitful interactions.This work has been supported by the Interuniversitaire Attractiepolen—Belgische Staat,Diensten van de Eerste Minister—Wetenschappelijke,technische en culturele Aangelegenheden;the F.W.O.-V.projects Nos.G.0287.95,G.0232.96,G.0306.00W.O.G.WO.025.99N(Belgium),and the ESF Programme VORTEX.REFERENCES∗Also at:Technische Universiteit Eindhoven,P.O.Box513,5600MB Eindhoven,The Netherlands.Permanent address:Department of Theoretical Physics,State University of 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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];。

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