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1、第二章 光学谐振腔与高斯光束,讨论光腔模式问题 开放式光腔可以分为稳定腔和非稳腔 稳定腔模式理论是以共焦腔模的解析理论为基础的，推广到一般稳定球面腔 采用稳定腔的激光器所发出的激光，将以高斯光束的形式在空间传播。研究高斯光束在空间的传播规律以及光学系统对高斯光束的变换规律 稳定腔不适用于某些高功率激光器，非稳腔却能同时满足高输出功率和良好光束质量这两个要求,2.1 概述,谐振腔的作用 无源谐振腔 理论依据 开放式光腔 开腔的分类 无源谐振腔的模式 光腔的损耗,Laser cavities differ in several significant ways from the closed mi
2、crowave cavities that are commonly treated in electromagnetic theory textbooks. Optical resonators first of all usually have open sides, and hence always have diffraction losses because of energy leaking out the sides of the resonator to infinity. Optical resonators are also usually described in sca
3、lar or quasi plane-wave terms, with emphasis on the diffraction effects at apertures and mirrors edged, rather than in vector terms with emphasis on matching boundary conditions. The distinction between “longitudinal” and “transverse” modes in the resonator is also much sharper in optical than in mi
4、crowave resonators.,一 谐振腔的作用,提供轴向光波模的反馈 控制腔内振荡光束的特性 （直接控制光束的横向分布特性、光斑大小、谐振频率及光束发散角等）,二 无源谐振腔,不考虑腔内激活介质的影响 无源腔模式可以作为具有激活介质腔（有源腔）的激光模式的良好近似 激活介质的作用主要是补充腔内电磁场在振荡过程中的能量损耗，使之满足阈值条件；激活介质对场的空间分布和振荡频率的影响是次要的，不会使模式发生本质的变化,三 采用的理论,几何光学理论-推导腔的稳定性条件（不能得到腔的衍射损耗） 衍射光学理论-深入了解模式特性,四 开放式光腔,激光器中使用的谐振腔通常是开放式的，即侧面没有光学边
5、界（理想化的处理方法），称为开式光学谐振腔，简称开腔。 气体激光器是采用开腔的典型例子。 对固体激光器，棒的直径远大于激光波长，棒的长度远小于腔长，可认为是开腔。 半导体激光器是使用介质腔的典型例子。,五 开腔的分类,根据光束几何逸出损耗的高低，分为稳定腔、非稳腔和临界腔。 稳定腔：旁轴（傍轴）光线在腔内多次往返而不逸出腔外，具有较低的几何损耗 非稳腔：傍轴光线在腔内经过少数几次往返就逸出腔外，具有较高的几何损耗 临界腔：性质介于稳定腔和非稳腔之间，只有少数特定光线能在腔内往返传播,Optical resonators can usually be divided into either “g
6、eometrically stable” or “geometrically unstable” categories (where these terms refer to ray stability within the resonators, and have nothing to do with whether or not the laser is or is not stable against laser oscillation),六 无源谐振腔的模式,模的概念 纵模和谐振频率 激光的横模,模的概念 腔与模的一般联系,通常将光学谐振腔内可能存在的电磁波的本征态称为腔的模式。 腔的
7、模式也就是腔内可区分的光子的状态。同一模式内的光子，具有完全相同的状态（如频率、偏振等）。 腔内电磁场的本征态由麦克斯韦方程组及腔的边界条件决定。一旦给定了腔的具体结构，则其中振荡模的特征也就随之确定下来。这就是腔与模的一般联系。,目的：弄清楚激光模式的基本特征及其与腔的结构之间的具体依赖关系。 模的基本特征：每一个模的电磁场分布，特别是在腔的横截面内的场分布；模的谐振频率；每一个模在腔内往返一次经受的相对功率损耗；与每一个模相对应的激光束的发散角。 只要知道了腔的参数，就可以唯一地确定模的基本特征。,纵模和谐振频率,利用均匀平面波模型讨论开腔中傍轴传播模式的谐振条件 考察均匀平面波在腔中沿轴
8、线方向往返传播的情形。当波在腔镜上反射时，入射波和反射波将会发生干涉，多次往复反射时就会发生多光束干涉。为了能在腔内形成稳定振荡，要求波能因干涉而得到加强。,发生相长干涉的条件是：波从某一点出发，经腔内往返一周再回到原来位置时，应与初始出发波同相（即相差是2的整数倍）。,F-P腔的谐振频率是分立的。,腔的光学长度应为半波长的整数倍-驻波条件 腔内光强沿z轴的分布不是均匀的，而是强弱相间地分布着。光强最强的明亮区，称为波腹；最弱的黑暗区，称为波节。 通常将由整数q所表征的腔内纵向光场的分布称为腔的纵模，不同的q相应于不同的纵模，或相应于驻波场波腹的个数。 纵模间隔与q无关，腔的纵模在频率尺度上是
9、等间隔排列的。,The oscillation frequency is determined by the requirement that phase delay per round trip be some integer, say q, of 2. The integer q corresponds to the number of maxima of the standing wave interference pattern between the two reflectors.,光腔的损耗 (losses in optical resonators),An understandi
10、ng of the mechanisms by which electromagnetic energy is dissipated in optical resonators and the ability to control them are of major importance in understanding and operating a variety of optical devices. For historical reasons as well as for reasons of convenience, these losses are often character
11、ized by a number of different parameters. This book uses the concepts of loss per pass, photon lifetimes, and quality factor Q to describe losses in resonators.,损耗类型(loss mechanisms ) 几何偏折损耗 衍射损耗(Diffraction losses) 腔镜反射不完全引起的损耗(loss resulting from nonperfect reflection) 固有损耗(absorption and scatteri
12、ng in the laser medium),几何偏折损耗 光线在腔内往返传播时，从腔的侧面偏折逸出的损耗。 取决于腔的类型和几何尺寸 几何损耗的高低依模式的不同而异，高阶横模损耗大于低阶横模损耗 是非稳腔的主要损耗,衍射损耗 腔镜具有有限大小的孔径，光波在镜面上发生衍射时形成的损耗 与腔的菲涅尔数（ ）有关，N愈大，损耗愈小 与腔的几何参数有关 与横模阶次有关(the higher the transverse mode indices m,n, the greater the loss),腔镜反射不完全引起的损耗：反射镜的吸收、散射和透射损耗。(Reflection loss is un
13、avoidable, since without some transmission no power output is possible. In addition, no mirror is ideal; and even mirrors are made to yield the highest possible reflectivities, some residual absorption and scattering reduce the reflectivity to somewhat less than 100 percent ),固有损耗：材料中的非激活吸收、散射、腔内插入物
14、所引起的损耗。(Transitions from some of the atomic levels, which are populated in the process of pumping, to higher lying levels constitute a loss mechanism in optical resonators when they are used as laser oscillators. Scattering from inhomogeneities and imperfections is especially serious in solid-state
15、laser media.),这后两种损耗称为非选择损耗，通常情况下它们对各个模式大体一样。 几何偏折损耗和衍射损耗称为选择损耗，不同模式的几何偏折损耗和衍射损耗各不相同。,损耗参数(loss per pass, photon lifetimes, and quality factor Q),1平均单程损耗因子 如果初始光强为I0，在无源腔内往返一周后光强衰减到I1，则 如果各种因素引起的单程损耗因子用i来表示，则总的单程损耗是=i 。,例：由腔镜反射不完全所引起的损耗,以r1和r2分别表示腔的两个镜面的反射率（功率反射系数），则初始光强为I0的光在腔内往返一周经两个镜面反射后，其强度I1应为
16、按的定义，对由腔镜反射不完全所引入的损耗r应有 由此得到,2光子在腔内的平均寿命R和线宽c,初始光强为I0，在腔内往返m次后,取t=0时刻的光强为I0,R称为腔的时间常数。由于腔内存在损耗，光场不再为简谐振动，而是振幅随时间指数衰减的阻尼振荡，其强度按频率的分布有一宽度 C=1/(2R) (full width at the half-power points),腔的时间常数等于光子在腔内的平均寿命,设t时刻腔内光子数密度为N(t)，N0表示t=0时刻光子数密度,3无源谐振腔的品质因数Q,：储存在腔内的总能量； P：单位时间内损耗的能量,腔的品质因数表示光腔的储能与损耗的特征。Q值大，表示光腔
17、的储能好，损耗小，腔内光子寿命长。,Optical resonators are used primarily in order to build up large field intensities with moderated power inputs. A universal measure of this property is the quality factor Q of the resonator. Q is defined by the relation,平均单程损耗因子 光子在腔内的平均寿命R和模式线宽c 无源谐振腔的品质因数Q 四者之间的关系：,损耗举例 由镜反射不完全所引
18、起的损耗 腔镜倾斜时的几何损耗 平行平面腔的调整精度要求极高 衍射损耗,N愈大，损耗愈小,2.2 共轴球面腔的稳定性,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：光线出射点、出射方向在腔轴线的上方时， r、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
19、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。,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
20、 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
21、 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, c
22、urved 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 understa
23、nding 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
24、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 tot
25、al 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
26、面上出发，向M2方向行进 傍轴光线在腔内完成一次往返总的变化矩阵为,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,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.,共轴球面腔的稳定性条件 (mode sta
27、bility 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,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、
28、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, 凹-凸稳定腔,(d) 平-凹稳定腔,(e) 半共焦腔,临界腔,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(symmetric concentric) 虚共心腔 R1、R2异号,(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 (stability diagram
29、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。,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.,