共轴球面腔的稳定性条件

2.2 共轴球面腔的稳定性条件共轴球面腔的稳定性条件光线传输矩阵(optical ray matrices or ABCD matrices)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、临界腔稳区图一一 光线传输矩阵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腔内任一傍轴光线在某一给定的横截面内都可以由可以由两个坐标参数两个坐标参数来表征：光线离轴线的来表征：光线离轴线的距离距离r、光线与轴线的夹角、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。用一个列矩阵列矩阵描述任一光线的坐标，用一个二二阶方阵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变换矩阵。Ray optics-by which we mean the geometrical laws for optical ray propagation,without including diffraction-is a topic that is not only important in its own right,but also very useful in understanding the full diffractive propagation of light waves in optical resonators and beams.Ray matrices or paraxial ray optics provide a general way of expressing the elementary lens laws of geometrical optics,or of spherical-wave optics,leaving out higher-order aberrations,in a form that many people find clearer and more convenient.Ray optics and geometrical optics in fact contain exactly the same physical content,expressed in different fashion.Ray matrices or“ABCD matrices”are widely used to describe the propagation of geometrical optical rays through paraxial optical elements,such lenses,curved mirrors,and“ducts”.These ray matrices also turn out to be very useful for describing a large number of other optical beam and resonator problems,including even problems that involve the diffractive nature of light.Since a ray is,by definition,normal to the optical wavefront,an understanding of the ray behavior makes it possible to trace the evolution of optical waves when they are passing through various optical elements.We find that the passage of a ray(or its reflection)through these elements can be described by simple 2x2 matrices.Furthermore,these matrices will be found to describe the propagation of spherical waves and of Gaussian beams such as those which are characteristics of the output of lasers.Ray propagation through cascaded elements:A single 4-element ray matrix equal to the ordinary matrix product of the individual ray matrices can thus describe the total or overall ray propagation through a complicated sequence of cascaded optical elements.Note,however,that the matrices must be arranged in inverse order from the order in which the ray physically encounters the corresponding elements.二二 腔内光线往返传播的矩阵表示腔内光线往返传播的矩阵表示由曲率半径为由曲率半径为R1和和R2的两个球面镜的两个球面镜M1和和M2组组成的共轴球面腔，腔长为成的共轴球面腔，腔长为L，开始时光线从，开始时光线从M1面上出发，向面上出发，向M2方向行进方向行进当凹面镜向着腔内时，当凹面镜向着腔内时，R取正值；当凸面镜向取正值；当凸面镜向着腔内时，着腔内时，R取负值取负值光线从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，总的坐标变换坐标变换为傍轴光线在腔内完成一次往返总的变换矩阵变换矩阵为The sign of R is the same as that of the focal length of the equivalent.This makes R1(or R2)positive when the center of curvature of mirror 1(or 2)is in the direction of mirror 2(or 1),and negative otherwise.三三 共轴球面腔的稳定性条件共轴球面腔的稳定性条件(mode stability criteria)傍轴光线能在腔内往返任意多次而不横向逸傍轴光线能在腔内往返任意多次而不横向逸出腔外，要求出腔外，要求n次往返变换矩阵次往返变换矩阵Tn的各个元的各个元素素An、Bn、Cn、Dn对任意对任意n值均保持有限值均保持有限引入引入g参数参数，可写成简单共轴球面腔共轴球面腔的往返矩阵以及共轴球面腔的往返矩阵以及n次往返矩阵均次往返矩阵均与光线的初始坐标无关，可以描述任意傍与光线的初始坐标无关，可以描述任意傍轴光线在腔内往返传播的行为。轴光线在腔内往返传播的行为。随着光线在腔内的初始出发位置及往返一随着光线在腔内的初始出发位置及往返一次的行进次序的不同，矩阵次的行进次序的不同，矩阵T各元素的具体各元素的具体表达式也将各不相同。表达式也将各不相同。可以证明，(A+D)/2对于一定几何结构的球对于一定几何结构的球面腔是一个不变量，与光线的初始坐标、面腔是一个不变量，与光线的初始坐标、出发位置及往返一次的顺序都无关出发位置及往返一次的顺序都无关。稳定腔非稳腔临界腔或或The ability of an optical resonator to support low(diffraction)loss modes depend on the mirrors separation L and their radii of curvature R1 and R2.四四 常见的几种稳定腔、非稳腔、临界腔常见的几种稳定腔、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（L=R/2）a对称共焦腔(confocal)R1=R2=La平行平面腔(plane-parallel)R1=R2=a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(symmetric concentric)虚共心腔 R1、R2异号临界腔临界腔(a)对称共焦腔(b)平行平面腔(c)实共心腔(d)对称共心腔(e)虚共心腔五五 稳区图稳区图(stability diagram of optical resonator)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。共轴球面腔的稳定性条件共轴球面腔的稳定性条件改写为：改写为：From this diagram,for example,it can be seen that the symmetric concentric(R1=R2=L/2),confocal(R1=R2=L),and the plane-parallel(R1=R2=)resonator are all on the verge of instability and thus may become extremely lossy by small deviations of the parameters in the direction of instability.小结：可以证明，可以证明，(A+D)/2对于一定几何结构的球面对于一定几何结构的球面腔是一个不变量，与光线的初始坐标、出发腔是一个不变量，与光线的初始坐标、出发位置位置(如在腔面上或在腔内任何其他点如在腔面上或在腔内任何其他点)及往及往返一次的顺序都元关。返一次的顺序都元关。对于复杂开腔，稳定性条件为：对简单共轴球面腔，稳定对简单共轴球面腔，稳定性条件为：性条件为：end