2.4 wave equation in quadratic index media

1. 2.4 wave equation in quadratic index media The most widely encountered beam in quantum electronics is one where the intensity distribution at planes normal to the propagation direction is Gaussian. To derive its characteristics we start with the Maxwell’s equations in an isotropic charge-free medium. distribution 分布状态 isotropic 同向性的 charge 负荷, 电荷, 费用 curl 卷成小圈 Taking the curl of the second of (2.4-1) and substituting the first results in

2. 3This neglect is justified if the fractional change of εin one optical wavelength is << 1. where quantity 量, 数量

3. thus allowing for a possible dependence of ε on position r. We have also taken k as a complex number to allow for the possibility of losses (σ>0) or gain (σ<0) in the medium • 4If K is complex (for example, Kr+iKi), then a traveling electromagnetic plane wave has the form of:

4. 2.4-5 We limit our derivation to the case in which k2(r) is given by where, according to (2.4-4) so that k2 is some constant characteristic of the medium. Furthermore, we assume a solution whose transverse dependence is on r= only, so that in (2.4-3) we replace by 2.4-6 transverse 横向的, 横断的

5. The kind of propagation we are considering is that of a nearly plane wave in which the flow of energy is predominantly along a single( for example, z) direction so that we may limit our derivation to a single transverse field component E. Taking E as We obtain from (2.4-3) and (2.4-5) in a few simple steps, where and where we assume that the variation is slow enough that 2.4-7 2.4-8 Where and where we assume that the variation is slow enough that predominantly 主要的, 突出的, 有影响的 transverse 横向的

6. coefficient [数]系数

7. Mental deficiency智力缺陷 • "Would you mind telling me, Doctor," Bob asked ... • "how you detect a mental deficiency in somebody who appears completely normal?" • "Nothing is easier," he replied. • "You ask him a simple question which everyone should answer with no trouble. • If he hesitates, that puts you on the track.“ • " Well, What sort of question?" • "Well, you might ask him, 'Captain Cook made three trips around the world and died during one of them. Which one?' • Bob thought for a moment, and then said with a nervous紧张的laugh, "You wouldn't happen to have another example would you? I must confess(坦白)I don't know much about history."

8. “医生，你能不能告诉我，”鲍勃问， • “对于一个看上去很正常的人，你是怎样判断出他有智力缺陷的呢？” • “再没有比这容易的了，”医生回答， • “问他一个简单的问题，简单到所有人都知道答案，如果他回答得不干脆，那你就知道是怎么回事了。” • “那要问什么样的问题呢？” • “嗯，你可以这样问，‘库克船长环球旅行了三次，但是在其中一次的途中他去世了，是哪一次呢？’” • 鲍勃想了一会儿，紧张的回答道，“你就不能问另外一个问题吗？坦率地说，我对历史了解的不是很多。”

9. 高斯光束的基本性质及特征参数 基模高斯光束 高斯光束在自由空间的传播规律 高斯光束的参数特征

10. 4、高斯光束 由激光器产生的激光束既不是上面讨论的均匀平面光波，也不是均匀球面光波，而是一种振幅和等相位面在变化的高斯球面光波，即高斯光束。 以基模TEM00高斯光束为例，表达式为：

11. 式中：E0为常数，其余符号的意义为 基模高斯光束的束腰半径 与传播轴线相交于Z点高斯光束等相位面上的光斑半径 与传播轴线相交于Z点的高斯光束等相位面的曲率半径 高斯光束的共焦参数

12. 高斯光束的基本特征： (1)基模高斯光束在横截面内的光电场振幅分布按照高斯函数的规律从中心（即传播轴线）向外平滑地下降，如图1-6所示。由中心振幅值下降到1/e点所对应的宽度，定义为光斑半径。

15. 可见，光斑半径随着坐标Z按双曲线的规律扩展，即可见，光斑半径随着坐标Z按双曲线的规律扩展，即 如图1-7所示。 在Z=0处，ω（z）=ω0达到极小值，称为束腰半径。

17. (2)基模高斯光束场的相位因子 决定了基模高斯光束的空间相移特性。 其中，kz描述了高斯光束的几何相移；arctan(z/f)描述了高斯光束在空间行进距离z处，相对于几何相移的附加相移；因子kr2/(2R(z))则表示与横向坐标r有关的相移，它表明高斯光束的等相位面是以R(z)为半径的球面。

18. R(z)随Z变化规律为： 结论： a)当Z=0时，R(z)→∞，表明束腰所在处的等相位面为平面。 b)当Z→±∞时，│R(z)│≈z→∞表明离束腰无限远处的等相位面亦为平面，且曲率中心就在束腰处； c)当z=±f时，│R(z)│=2f，达到极小值 。

19. d) 当0<z<f时，R(z)>2f，表明等相位面的曲率中心在（-∞，-f）区间上。 e)当z>f时，z< R(z)<z+f，表明等相位面的曲率中心在（-f,0）区间上。

20. (3)基模高斯光束既非平面波，又非均匀平面波，它的发散度采用场发散角表征。 远场发散角θ1/e2定义为z→∞时，强度为中心的1/e2点所夹角的全宽度，即 高斯光束的发散度由束腰半径ω0决定。

21. 综上所述，基模高斯光束在其传播轴线附近，可以看作是一种非均匀的球面波，其等相位面是曲率中心不断变化的球面，振幅和强度在模截面内保持高斯分布。综上所述，基模高斯光束在其传播轴线附近，可以看作是一种非均匀的球面波，其等相位面是曲率中心不断变化的球面，振幅和强度在模截面内保持高斯分布。

22. 2.5 Gaussian beams in a homogeneous均匀medium From 2.4, the kind of propagation is a nearly plane wave in which the flow of energy is predominantly along a single direction, so 2.5-1 In a homogeneous medium the quadratic coefficient(系数) k2 of the equation above is zero, so that: 2.5-2 How to solve it? Have a try! please

23. Where q0 is an arbitrary integration constant. From (2.4-11) and (2.5-4) we have

24. where q0 is an arbitrary(任意的) constant. combining above discussion, we obtain: 2.5-7 then: 2.5-6

25. 2.5-8 in which vacuum (真空)wave length Let us consider, one a time, the two factors in 2.5-7, the first one becomes 2.5-9

26. imaginary 虚数的 confine 限制, 禁闭 substitution 代替, 代入法 exponent 指数 ,解释者, 典型

28. 2-5-11 2-5-12 Define the important following parameters:

29. Combine equations above in 2.5-4 , and obtain: 2.5-14 : the distance r at which the field amplitude is down by a factor 1/e compare to its valve on the axis; : the minimum spot size, it is the spot size at the plane z=0; R : the radius of curvature of the very nearly spherical wavefronts at z. and we identify R as the radius of curvature of the Gaussian beam.

30. That is the basic result. we refer to it as the fundamental Gaussian-beam solution, since we have excluded the more complicated solutions by limiting ourselves to transverse dependence involving r=(x2+y2) only. The higher-order modes will be discussed separately. The form of the fundamental Gaussian beam is uniquely determined once its minimum spot size w0 and its location ---that is, the plane z=0--- are specified, the spot size w0 and radius of curvature R at any plane z are then found .

31. From (2.5-14) the parameter w(z) evolves according to (2.5-11), is the distance r at which the field amplitude is down by a factor 1/e compared to its value on the axis. We will consequently refer to it as the beam spot size. The parameter w0 is the minimum spot size. It is the beam spot size at the plane z=0. The parameter R IN (2.5-11) is the radius of curvature of the very nearly spherical wavefronts at z. We can verify this statement by deriving the radius of curvature of the constant phase surfaces (wavefronts) or , more simply, by considering the form of a spherical wave emitted by a point radiator placed at z=0. It is given by evolve (使)发展, (使)进展

32. Since z is equal to R, the radius of curvature of the spherical wave. Comparing (2.5-15) with (2.5-14), we identify R as the radius of curvature of the Gaussian beam. The convention regarding the sign of R(z) is that it is negative if the center of curvature occurs at z’>z and vice versa convention 协定, 习俗, 惯例

33. The form of the fundamental Gaussian beam is, according to (2.5-14),uniquely determined once its minimum spot size w0 and its location –that is , the plane z=0– are specified. The spot size w and radius of curvature R at any plane z are then found from (2.5-11) and (2.5-12). Some of these characteristics are displayed in Figure 2-5. The hyperbolas shown in this figure correspond to the ray direction and are intersections of planes that include the z axis and the hyperboloids. 2.5-16 uniquely 独特地,唯一地 hyperbola 双曲线 hyperboloid [数]双曲面

34. These hyperbolas correspond to the local direction of energy propagation. The spherical surfaces shown have radii of curvature given by (2.5-12). For large z the hyperbolas x2+y2=ω2 are asymptotic to the cone 2.5-17 hyperboloid [数]双曲面 asymptotic 渐近线的, 渐近的 cone 锥形物, 圆锥体

35. Phase fronts Z=0 +z Propagation lines Some of these characteristics are displayed as following figure:

36. ＝1，for K2＝0 ＝1，for K2＝0 ＝0，for K2＝0 ＝1，for K2＝0

37. Whose half-apex angle, which we take as a measure of the angular beam spread, is This last result is a rigorous manifestation of wave diffraction according to which a wave that is confined in the transverse direction to an aperture of radius w0 will spread (diffract) in the far field (z>>π w02n/λ) according to (2.5-18) for 2.5-18 apex 顶点 尖端rigorous 严格的, 严厉的 hyperboloid [数]双曲面

38. Happy Birthday!!!

39. 2.6 Fundamental Gaussian beam in a lenslike medium –the ABCD Law We now return to the general case of a lenslike medium so that k2 ≠0. The P and q functions of (2.4-9) obey, according to (2.4-11) 2.6-1

40. If we introduce the functions defined by we obtain from (2.6-1) 2.6-2 turn to P:46 2.3-4

41. 2.3-4 If at the input plane z=0 the ray has a radius r0 and slope r0’, we can write the solution of (2.3-4) directly as 2.3-5

43. Using (2.6-3) in (2.6-4) and expressing the result in terms of an input value q0 gives the following result for the complex beam radius q(z) The physical significance of q(z) in this case can be extracted from (2.4-9). We expand the part ψ(r,z) that involves r. The result is 2.6-4 Significance 意义, 重要性 Extract 拔出, 榨取, 开方, 求根, 摘录, 析取, 吸取 Expand 使膨胀, 详述, 扩张

44. If we express the real and imaginary parts of q(z) by means of We obtain so that w(z) is the beam spot size and R its radius of curvature, as in the case of a homogeneous medium, which is described by (2.5-14). For the special case of a homogeneous medium (k2=0), (2.6-4) reduces (2.5-4) 2.6-5

45. ＝1，for K2＝0 ＝1，for K2＝0 ＝0，for K2＝0 ＝1，for K2＝0

46. Transformation of the Gaussian Beam—the ABCD Law We have derived above the transformation law of a Gaussian beam (2.6-4) propagation through a lenslike medium that is characterized by k2, we not first by comparing (2.6-4) to Table 2-1(6) and to (2.3-5) that the transformation can be described by where A,B,C,D are the elements of the ray matrix that relates the ray (r,r’) at a plane 2 to the ray at plane 1.It follows immediately that the propagation through 2.6-6

47. Ray matrix a medium with a quadratic index profile:

48. 2.6-6 where A,B,C,D are the elements of the ray matrix that relates the ray (r,r’) at a plane 2 to the ray at plane 1.It follows immediately that the propagation through.

49. plane 2 to the ray at plane 1. It follows immediately that the propagation through, or reflection from, any of the elements shown in Table 2-1 also obeys (2.6-6), since these elements can all be viewed as special cases of a lenslike medium. For future reference we not that by applying (2.6-6) to a thin lens of focal length f we obtain from (2.6-6) and Tbale 2-1(2) 2.6-7