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So we have :

from above, we know that the general solution can be taken as a linear superposition of and. the condition for a stable( 稳定的 ) ray is that be a real number,? the necessary and sufficient condition for to be real is that:.

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So we have :

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  1. from above, we know that the general solution can be taken as a linear superposition of and the condition for a stable(稳定的)ray is that be a real number,? the necessary and sufficient condition for to be real is that: why the condition for a stable(稳定的)ray is that be a real number So we have : 2.1-13

  2. If is not a real number, generally, we can take Where x and y are all real numbers.

  3. In terms of the system parameters, we can use (2.1-11) to reexpress (2.1-15) ?

  4. Example 2: Identical(同样的)-lens waveguide The simplest case of a lens waveguide is one in which f1=f2=f ; that is, all lenses are identical. The analysis of this situation is considerably simpler than that used for a biperiodic lens sequence. The reason is that the periodic unit cell (the smallest part of the sequence that can, upon translation(平移),recreate the whole sequence) contains a single lens only. The (A,B,C,D) matrix for the unit cell is given by the square matrix in (2.1-3). Following exactly the steps leading to (2.1-11) through (2.1-14), the stability condition becomes: 2.1-18

  5. the condition for a stable(稳定的)ray is that

  6. And the beam radius at the nth lens is : Algebraic 代数的 Derivation 由来, 起源 2.1-19

  7. The stability criteria(标准) can be demonstrated experimentally by tracing(追踪)the behavior of a laser beam as it propagates down a sequence of lenses spaced uniformly. One can easily notice the rapid escape of the beam once condition (2.1-18) is violated(违反或藐视).

  8. Marry in the heaven A young couple夫妇was on their way to get married when they had an accident and died. Now they were in front of St. Peter圣徒彼得and the young lady asked if they could get married. St. Peter told them, he would have to get back to them with an answer. Around 30 days later St. Peter returns and tells the couple that they can get married in heaven. The young lady then asks St. Peter, “If things just don't work out can we get a divorce?" St. Peter looks at her and replies, " Lady it took me 30 days to find a preacher up here, do you really think I am going to find a lawyer?!!" 一对年轻的夫妇在去结婚的路上出了车祸，双双死去了。于是，他们来到了面前，妻子问是否她还可以和丈夫结婚，圣徒彼得告诉他们，关于这个问题他一有了结果就会回来找他们。差不多30天以后，圣徒彼得回来了，并且告诉他们可以在天堂结婚。妻子又问：“如果生活的不愉快，我们可不可以离婚呢？”圣徒彼得看着她，回答说：“夫人，我花了30天才找到个传教士，难道你真的希望我再去找个律师吗？”

  9. 2.2 propagation of rays between mirrors Another important application of the fonltalism just veloped concems the boancing of a ray betweent wocurved mirrors Sincethe reflection at a mirror with a radiusof curvature R is equivalent．except for the folding of the path．to passage through a lens with a focaIlength f=R/2．we Can use the fomlaIism of the preceding section to describe the propagation of a ray between two curved reflectors with radii of curvatare R1 and R2，which are separated by d.Let us consider the simple case of a ray mat is iniected into a symmetric two—mirror systerm as shown in Figure 2.3(a)．Since the x and y coordinares of the ray are independent variables．we can take them according to (2.1-19)in the foml of formalism 形式主义，法则 bouncing 跳跃的 curved 成曲形 radius 半径 curvature 曲率 fold 折叠

  10. Since the x and y coordinates of the ray are independent variables, so: 2.2-1 Parameter 参数, 参量 immediately 立即, 马上 locus 地点，所在地, [数]轨迹ellipse [数]椭圆, 椭圆形

  11. If in (2.2-1) satisfies the condition : Example: reentrant(重新进入的；凹入的) rays: 重返光线 2.2-2 wher v and l are any two integers ,A ray will return to its starting point following v round trips and will thus continuously retrace the same pattern on the mirrors.

  12. If we consider as an example the simple case of L=1,v=2,so that， from(2.1-19)we obtain d=2f=R;that is,if the mirrors are separated by a distance equal to their radius of curvature R,the trapped ray will retrace its pattern after two round trips(v=2).This situation(d=R)is referred to as symmertic confocal,since the two mirrors have a common focal point f=R/2,it will be discussed in detail in Chapter4.The ray pattern corresponding to v=2 is illustrated in Figure 2-3(b) • trip (短途)旅行, 往返, 差错, • symmetric 相称性的, 均衡的 • Confocal [数]共焦的

  13. Answer for p35 1.8

  14. 2.3 Rays in lenslike media The basic physical property of lenses that is responsible for their focusing action is the fact that the optical path across them (where n is the index of refraction of the medium) is a quadratic function of the distance r from the z axis. Using ray optics,we account for thes fact by a change in the ray’s slope as in(2.1-1).This same property can be represented by relating the complex field amplitude of the incident optical field immediately to the right of an ideal thin lens to that immediately to the left by (2.3-1) Where is the focal length and responsible 有责任的, 可靠的, 可依赖的 quadratic 二次的 account for 说明, 解决, 得分

  15. 2-3-1 2-3-2 The effect of the lens, therefore, is to cause a phase shift : which increase quadratically (二次的) with the distance from the axis. We consider next the closely related case of a medium whose index of refraction(折射率) n varies according to Where k2 is a constant.

  16. Introducing a phase shift proportional to 2-3-3 Since the phase delay of a wave propagation through a section dz of a medium with an index of refraction n is It follows directly that a thin slab（厚平板, 厚片） of the medium described by (2.3-2) will act as a thin lens, The behavior of a ray in this case is described by the differential equation that applied to ray propagation in an optically inhomogeneous (不均匀的)medium: where s is the distance along the ray measured from some fixed position on it and r is the position vector of the point at s.

  17. If at the input plane z=0 the ray has a radius and slope , we can write the solution of (2.3-4) directly as 2.3-4 2.3-5 For paraxial rays, we may replace d/ds by d/dz, then From the equation, we know the ray oscillates (振荡) back and forth across the axis., as shown in Figure 2-4. A section of the quadratic index medium acts as a lens

  18. This can be proved by showing,using(2.3-5),that a family of parallel rays entering at z=0 at different radii will converge upon emerging at to a common focus at a distance (2.3-6) From the exit plane,The factor accounts for the refraction at the boundary.assuming the medium at to possess an index and a small angle of incidence.The denvation of(2.3-6)is left as an exercise(Problem2.3) converge会聚 account for 说明, 解决, 得分 boundary 边界, 分界线 possess 占有, 拥有

  19. Equations(2.3-5)apply to a focusing medium with .In a medium where that is,where the index increases with the distance from the axis—the solutions for and become So that increases with distance and eventually escapes.A section of such a medium acts as a negative lens eventually 最后, 终于 escape 逃, 逃亡 negative 否定, 负数

  20. Physical situations giving rise to quadratic index variation include: 1.Propagation of laser beams with Gaussian-like intensity profile in a slightly absorbing medium.The absorption heating gives rise,because of the dependence of on the temperature T,to an index profile[9].If ,as is the case for most materials,the index is smallest on the axis where the absorption heating is highest.This corresponds to a in(2.3-2) and the beam spreads with the distance .If ,as in certain lead glasses[10],the beams are focused. 2.In optically pumped solid-state laser rods,the portion of the absorbed pump power that is not converted to laser radiation is conducted as heat to the rod surface.This heat conduction requires a temperature gradient in which T is maximum on axis. Gaussian (德国数学家)高斯的 profile 剖面, 侧面, 轮廓 absorbe 吸收, 吸引temperature 温度 spread 伸展, 展开, 传播 rod 杆, 棒 require需要, 要求

  21. The dependence of n on T then gives rise to a positive lens effect for and a negative lens for 3.Dielectric waveguides made by sandwiching a layer of index between two layers with index This situation will be discussed further in Chapter 7 in connection with injection lasers. 4.Optical fibers produced by cladding a thin optical fiber(whose radius is comparable to ) of an index with a sheath of index .Such fibers are used as light pipes. dielectric 电介质, 绝缘体sandwich 三明治, vt.夹入中间 injection 注射, 注射剂 cladd 在金属外覆以另一种金属 sheath 护套, 外壳pipe 管, 导管

  22. Optical waveguides consisting of glasslike rods or filaments,with radii large compared to ,whose index decreases with increasing Such waveguides can be used for the simultaneous transmission of a number of laser beams,which are injected into the waveguide at different angles,It follows from(2.3-5)that the beams will emerge,each along a unique direction,and consequently can be easily separated.Furthermore,in view of its previously discussed lens properties,the waveguide can be to transmit optical image information in much the same way as images are transmitted by a multielement lens systern to the image plane of a camera[11]. filament 细丝, 灯丝 image 图象, 肖像 multielement 多元素, 多元件

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