Chapter 2 the propagation of rays and beams

2. Chapter 2 the propagation of rays and beams • Introduction: In this chapter , we will take up the subject of optical ray propagation through a variety of optical media . These includehomogeneous(均匀的) and isotropic(各向同性的) materials,for example,thin lenses(薄透镜)，dielectric interfaces (电介质界面),and curved mirrors (曲面镜)

3. 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 wave 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 2*2 matrices. • Furthermore, these matrices will be found to describe the propagation of spherical waves and of Gaussian beams such as those characteristic of the output of lasers. Gaussian bean propagation is analyzed in second half of the chapter

4. ri r’i ro r’o r’if 2.1 Lens waveguide A paraxial(近轴的) ray passing through a thin lens of focal length (焦距) f is shown as: Figure 2-1 Deflection of a ray by a thin lens

5. Taking the cylindrical (圆柱的) axis of symmetry as z, denoting the ray distance from the axis by r and its slope(斜率) dr/dz as r′, we can relate the output ray to the input ray by means of Where the first term follows from the definition of a thin lens and the second can be derived from a consideration of the behavior of the undeflected(不偏斜的)central ray with a slope equal to .

6. 光轴 光线1,r’为正 光线２,r’为负 Focal plane ri r’i ro r’o -r’of r’if f

7. r(z) r’(z) r(z)= Representing a ray at any position z as a column matrix ,

8. 2.1-2 From above, we can rewrite 2.1-1 using the rules for matrix multiplication as Where f>0 for a converging(会聚的)lens and is negative(负数)for a diverging(使（如光线）发散,使偏离)one.

9. f Figure 2.2 a d b Example 1: 1. As shown by figure 2.2, the propagation of a ray through a straight(直的) section of a homogeneous(均匀的) medium of length d followed by a thin lens of focal length f , this corresponds to propagation between planes a and b in figure 2.2.

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11. Since the effect of the straight section is merely that of increasing r by dr′, we can relate the output b and input (at a ) rays by: 2.1-3

12. f1 f2 f1 f2 f1 b a plane s+1 plane s Figure 2.3 2. As shown by figure 2.3, consider the propagation of a ray through a biperiodic (两个的，两片的) lens systems made up of lenses of focal lengths f1 and f2 separated by d:

13. 2.1-4 The matrix relating the ray parameters at output of a unit cell to those at the input is: Where A, B, C, D are the elements of the matrix resulting from multiplying the two square matrices.

14. In equation form, we can get: 2.1-5 2.1-6

15. 2.1-9 as a result:

16. From equation 2.1-5: AD-BC=1 And then 2.1-8 can be rewrited as : 2.1-10 Because: Then

17. We are thus led to try a solution in the form of 2.1-10 when substituted in 2.1-9, leads to 2.1-12 and therefore : 2.1-13

18. 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: So we have : 2.1-13

19. 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

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

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

22. 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).

23. 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

24. 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

25. 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.

26. 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

27. 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

28. 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.

29. 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.

30. 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

31. Answer for p35 1.8

33. 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

34. 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.

35. 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.

36. 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

38. 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

39. 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

40. 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

41. 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

42. 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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