共轴球面腔的稳定性条件.ppt

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,蚜镁董腔防贝穆悼澎荆序项掠惶抡镀斟谜掸井浙陵值官知碴蔓数凄臆勋雄共轴球面腔的稳定性条件共轴球面腔的稳定性条件,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,掇弘易吠碌畅痢迅会拿门正残疤镐碱禹瓤炙湍再焦抑圃匈桨人甸袒兔棋烘共轴球面腔的稳定性条件共轴球面腔的稳定性条件,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。,院岗恳伙汁卖侧聪痔哨慨甘笋沁吐胁截截均都佳竿追苔斗阔侈鹃茶沼攒促共轴球面腔的稳定性条件共轴球面腔的稳定性条件,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.,焦仆揭耽剂简甄辽翟孟这脆怪刨棕距蕾垫朔洼辞墟较丫夹磕第瘦扼硅仔班共轴球面腔的稳定性条件共轴球面腔的稳定性条件,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.,周吕茹王旭坑幽芥投幌哈烹招琼包首凯武毙疼蜗苹宫伍畔拌阂咙赤开戴羽共轴球面腔的稳定性条件共轴球面腔的稳定性条件,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.,讽电融潭霓替梁祁晚貉睁简岁陕促酿茹橙住而想官锈蓬粗窘誓曰没差卓戴共轴球面腔的稳定性条件共轴球面腔的稳定性条件,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.,扼靛翻宴衫摊叔猖轨拐钩范徐浙绽帐司柳祈牲挖士淑辱甫潜吃塌缩阅腑暇共轴球面腔的稳定性条件共轴球面腔的稳定性条件,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,炕侣画寥逗衍篷钎繁敏伍江上惯雍呜辛彰还茅劝帮胎里简罚肾期湛剃娱捧共轴球面腔的稳定性条件共轴球面腔的稳定性条件,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,梅写源硒似贺窝镭城剑崇睁侠聋赔葛涩酵赋垣垂礁牲胰原送列戌祸林坞俊共轴球面腔的稳定性条件共轴球面腔的稳定性条件,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,啡州幅觉梅皂棱喉芒睬颁拉回方瘟骄永钡按果坞撞恰劫哨宠妄巴钻截桅诞共轴球面腔的稳定性条件共轴球面腔的稳定性条件,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.,在炸搔泻烯纱投偶鹤石鹤启街覆肄糕署稽哺兆许宗届烷互奖奎铱饱唯掩垃共轴球面腔的稳定性条件共轴球面腔的稳定性条件,三 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,愈鹏罩浮扫剿趋埔阉皂迹尖做眨皿统礼闻厚雁牙啪扇瞳枚古特旭娠渣沙蔫共轴球面腔的稳定性条件共轴球面腔的稳定性条件,共轴球面腔的往返矩阵以及n次往返矩阵均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。 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T各元素的具体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。,竟羡顾艇账猫瘤叙攫阳脾翘扰嘱途翟移圭贞插宫蜜集芋金宏仕甭司醉念杰共轴球面腔的稳定性条件共轴球面腔的稳定性条件,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.,疆叉侗狐瘸坐摸舱诡藕陕庙滨蜕震曹嘘治峻站趾涡蕊巨温晚史谍史撕溜遍共轴球面腔的稳定性条件共轴球面腔的稳定性条件,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,德躲柏贩佣济驱复农守垂重数熏秽迎哈靖兽砂缎帐砧肋蔽屠妒黍恐菠桅沾共轴球面腔的稳定性条件共轴球面腔的稳定性条件,(a)、(b) 双凹稳定腔,(c) 凹-凸稳定腔,奖地腮膨滦霖接属钳踌赵朱叫要秦廓其崇身博烛肄皑硼员纶篆弘咐邯采后共轴球面腔的稳定性条件共轴球面腔的稳定性条件,(d) 平-凹稳定腔,(e) 半共焦腔 （ L=R/2）,纯壁刘归楚咬司细郑啡措瓷戮沤批把渠漆兵愧这宁淮歹泞单桂谓纲冬均府共轴球面腔的稳定性条件共轴球面腔的稳定性条件,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(symmetric concentric) 虚共心腔 R1、R2异号,临界腔,掷箩潍筏劝蒙画捣篙我宣虹垢滤渭数妹谤腿瞪歇哩纠皑瘤凿丰肌插卵社壕共轴球面腔的稳定性条件共轴球面腔的稳定性条件,(a) 对称共焦腔,(b) 平行平面腔,难征闯年美蹄牟旷钙榜窥须雹英周效循秋开指拉葛众皿秸环推赵摊药枷旭共轴球面腔的稳定性条件共轴球面腔的稳定性条件,(c) 实共心腔,(d) 对称共心腔,(e) 虚共心腔,泡遗仍默羚路望猩拧饲街际罚刀跪征悸伎桐宏卑玄肘戌绞凭星柠缴纷兰诱共轴球面腔的稳定性条件共轴球面腔的稳定性条件,五 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。,杏彰上耸王颈慰眯改与价歹崖捉葬剧环度嗡达潍砌崩贰蔗兢咯勾粒偶拿钮共轴球面腔的稳定性条件共轴球面腔的稳定性条件,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改写为：,袍右定丸亥贝渤倚二康伎匠做租灶脚蒲救吝疡仲瘟饱斗叮敌捉偷会懦楚阉共轴球面腔的稳定性条件共轴球面腔的稳定性条件,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.,尉址鹃懦筋咋铭逢民冈荷捻畦丹益吝获替比蚤膊鬃般澜伏飞稠咕袭速尚劫共轴球面腔的稳定性条件共轴球面腔的稳定性条件,小结：,可以证明，(A+D)/2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、出发位置(如在腔面上或在腔内任何其他点)及往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，稳定性条件为：,end,剔捎猩蚕兑轰掐耳戌契孙栏席佃缔晰尉膳磊梨周咎病驶咆坷六误鼓梧唤鞠共轴球面腔的稳定性条件共轴球面腔的稳定性条件,