深圳大学：《激光原理》课程教学课件（PPT讲稿）第二章 开放式光腔与高斯光束 第二部分 共轴球面腔的稳定性条件

S 2.2共轴球面腔的稳定性条件·光线传输矩阵(optical ray matrices or ABCDmatrices)·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

§2.2 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LTL =0

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       = 0 1 1 L TL

球面镜对傍轴光线的002变换矩阵为（R为球R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Ba线的变换矩阵0.0DC

        − =         − = 1 1 1 0 1 2 1 0 R f TR 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。 2 1 2 1 r r A B   C D       =             该矩阵称为光学系统对光 线的变换矩阵

Ray optics---by which we mean the geometricallaws for optical ray propagation, withoutincluding diffraction---is a topic that is not onlyimportant in its own right, but also very useful inunderstanding the full diffractive propagation oflight waves in optical resonators and beams.. Ray matrices or paraxial ray optics provide ageneral way of expressing the elementary lenslaws of geometrical optics, or of spherical-waveoptics, leaving out higher-order aberrations, in aform that many people find clearer and moreconvenient

• 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 containexactly the same physical content, expressed indifferent fashion· Ray matrices or “ABCD matrices"’ are widelyused to describe the propagation of geometricaloptical rays through paraxial optical elements,such lenses, curved mirrors, and “ducts. Theseray matrices also turn out to be very useful fordescribing a large number of other optical beamand resonator problems, including evenproblems that involve the diffractive nature oflight

• 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 opticalwavefront, an understanding of the ray behaviormakes it possible to trace the evolution of opticalwaves when they are passing through variousoptical elements. We find that the passage of aray (or its reflection) through these elements canbe described by simple 2x2 matricesFurthermore, these matrices will be found todescribe the propagation of spherical waves andof Gaussian beams such as those which arecharacteristics of the output of lasers

• 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 theordinary matrix product of the individual raymatrices can thus describe the total or overall raypropagation through a complicated sequence ofcascaded optical elements. Note, however, thatthe matrices must be arranged in inverse orderfrom the order in which the ray physicallyencounters the corresponding elements

• 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

二腔内光线往返传播的矩阵表示，由曲率半径为R,和R,的两个球面镜M,和M,组成的共轴球面腔，腔长为L，开始时光线从M面上出发，向M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M面上出发到达M面上时120

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 2 1 1 2 1 1 1 0 1 L r r r L T            = =                

当光线在曲率半径为R,的镜M,上反射时0r2r2132=T=R2,]0.22R2当光线再从镜M行进到镜M,面上时r3r4L-= TA9,.0然后又在M上发生反射0rs4427一Rg0404R

当光线在曲率半径为R2的镜M2上反射时 2 3 2 2 3 2 2 2 1 0 2 1 R r r r T R              = =         −           当光线再从镜M2行进到镜M1面上时 4 3 3 4 3 3 1 0 1 L r L r r T           = =               然后又在M1上发生反射 1 5 4 4 5 4 4 1 1 0 2 1 R r r r T R              = =         −          

傍轴光线在腔内完成一次往返，总的坐标变换为0122C0.09SR.R傍轴光线在腔内完成一次往返总的变换矩阵为LR1R20R

傍轴光线在腔内完成一次往返，总的坐标变换 为 TR TL TR TL L R L R C D A B T 1 2 1 2 0 1 1 1 2 1 0 0 1 1 1 2 1 0 =               −              − =      = 5 1 1 1 5 1 1 1 1 2 1 0 1 0 1 1 2 2 1 1 0 1 0 1 r L L A B r r r T C D R R                           = = =                   − −                       傍轴光线在腔内完成一次往返总的变换矩阵为