共轴球面腔的稳定性条.ppt

2.2共轴球面腔的稳定性条件,光线传输矩阵(opticalraymatricesorABCDmatrices)腔内光线往返传播的矩阵表示共轴球面腔的稳定性条件常见的几种稳定腔、非稳腔、临界腔稳区图,一光线传输矩阵,腔内任一傍轴光线在某一给定的横截面内都可以由两个坐标参数来表征：光线离轴线的距离r、光线与轴线的夹角。规定：光线出射方向在腔轴线的上方时，为正；反之，为负。光线在自由空间行进距离L时所引起的坐标变换为,球面镜对傍轴光线的变换矩阵为（R为球面镜的曲率半径）,球面镜对傍轴光线的反射变换与焦距为f=R/2的薄透镜对同一光线的透射变换是等效的。,用一个列矩阵描述任一光线的坐标，用一个二阶方阵描述入射光线和出射光线的坐标变换。,该矩阵称为光学系统对光线的变换矩阵。,Rayoptics-bywhichwemeanthegeometricallawsforopticalraypropagation,withoutincludingdiffraction-isatopicthatisnotonlyimportantinitsownright,butalsoveryusefulinunderstandingthefulldiffractivepropagationoflightwavesinopticalresonatorsandbeams.Raymatricesorparaxialrayopticsprovideageneralwayofexpressingtheelementarylenslawsofgeometricaloptics,orofspherical-waveoptics,leavingouthigher-orderaberrations,inaformthatmanypeoplefindclearerandmoreconvenient.,Rayopticsandgeometricalopticsinfactcontainexactlythesamephysicalcontent,expressedindifferentfashion.Raymatricesor“ABCDmatrices”arewidelyusedtodescribethepropagationofgeometricalopticalraysthroughparaxialopticalelements,suchlenses,curvedmirrors,and“ducts”.Theseraymatricesalsoturnouttobeveryusefulfordescribingalargenumberofotheropticalbeamandresonatorproblems,includingevenproblemsthatinvolvethediffractivenatureoflight.,Sincearayis,bydefinition,normaltotheopticalwavefront,anunderstandingoftheraybehaviormakesitpossibletotracetheevolutionofopticalwaveswhentheyarepassingthroughvariousopticalelements.Wefindthatthepassageofaray(oritsreflection)throughtheseelementscanbedescribedbysimple2x2matrices.Furthermore,thesematriceswillbefoundtodescribethepropagationofsphericalwavesandofGaussianbeamssuchasthosewhicharecharacteristicsoftheoutputoflasers.,Raypropagationthroughcascadedelements:Asingle4-elementraymatrixequaltotheordinarymatrixproductoftheindividualraymatricescanthusdescribethetotaloroverallraypropagationthroughacomplicatedsequenceofcascadedopticalelements.Note,however,thatthematricesmustbearrangedininverseorderfromtheorderinwhichtherayphysicallyencountersthecorrespondingelements.,二腔内光线往返传播的矩阵表示,由曲率半径为R1和R2的两个球面镜M1和M2组成的共轴球面腔，腔长为L，开始时光线从M1面上出发，向M2方向行进,当凹面镜向着腔内时，R取正值；当凸面镜向着腔内时，R取负值,光线从M1面上出发到达M2面上时,当光线在曲率半径为R2的镜M2上反射时,当光线再从镜M2行进到镜M1面上时,然后又在M1上发生反射,傍轴光线在腔内完成一次往返，总的坐标变换为,傍轴光线在腔内完成一次往返总的变换矩阵为,ThesignofRisthesameasthatofthefocallengthoftheequivalent.ThismakesR1(orR2)positivewhenthecenterofcurvatureofmirror1(or2)isinthedirectionofmirror2(or1),andnegativeotherwise.,三共轴球面腔的稳定性条件(modestabilitycriteria),傍轴光线能在腔内往返任意多次而不横向逸出腔外，要求n次往返变换矩阵Tn的各个元素An、Bn、Cn、Dn对任意n值均保持有限,引入g参数，可写成,简单共轴,球面腔,稳定腔非稳腔临界腔,或,或,Theabilityofanopticalresonatortosupportlow(diffraction)lossmodesdependonthemirrorsseparationLandtheirradiiofcurvatureR1andR2.,四常见的几种稳定腔、非稳腔、临界腔,双凹稳定腔、非稳腔凹凸稳定腔、非稳腔平凹稳定腔、非稳腔（如果L=R/2，称为半共焦腔；如果L=R，称为半共心腔）双凸腔、平凸腔都是非稳腔,(a)、(b)双凹稳定腔,(c)凹-凸稳定腔,(d)平-凹稳定腔,(e)半共焦腔,对称共焦腔(confocal)R1=R2=L平行平面腔(plane-parallel)R1=R2=共心腔R1+R2=L实共心腔R1、R2均为正值，当R1=R2=L/2时，称为对称共心腔(symmetricconcentric)虚共心腔R1、R2异号,临界腔,(a)对称共焦腔,(b)平行平面腔,(c)实共心腔,(d)对称共心腔,(e)虚共心腔,五稳区图(stabilitydiagramofopticalresonator),任意一个球面腔唯一地对应于g1-g2平面上的一个点。由g1=0、g2=0和g1g2=1双曲线的两支围成的区域属于腔的稳定工作区域，其余的区域属于非稳区。如果满足g1=0、g2=0或g1g2=1，则是临界腔。,任意一个具有确定（R1、R2、L）值的球面腔唯一地对应于图中一个点，但反过来，图中每个点并不单值地代表某一具体尺寸的球面腔。对称共焦腔（本属于临界腔g1=0，g2=0），其中任意傍轴光线均可在腔内往返多次而不横向逸出，而且经两次往返即自行闭合。在这种意义上，共焦腔属于稳定腔之列。,Fromthisdiagram,forexample,itcanbeseenthatthesymmetricconcentric(R1=R2=L/2),confocal(R1=R2=L),andtheplane-parallel(R1=R2=)resonatorareallonthevergeofinstabilityandthusmaybecomeextremelyl