Principles of laser

1. Principles of laser 席鹏 北京大学工学院 生物医学工程系

2. Outline • Principle of laser • Laser modes • Features of laser

3. Energy levels and transition粒子的能级与辐射跃迁 • The ground state corresponds to having the atom's single electron in the lowest possible orbit. An excited state is any state with energy greater than the ground state. • By giving the atom additional energy (for example, by the absorption of a photon of an appropriate energy), the electron is able to move into an excited state (one with one or more quantum numbers greater than the minimum possible). If the photon has too much energy, the electron will cease to be bound to the atom, and the atom will become ionised.

4. Decay, Lifetime • The lifetime of a system in an excited state is usually short: spontaneous or induced emission of a quantum of energy usually occurs shortly (ps level) after the system is promoted to the excited state, returning the system to a state with lower energy (a less excited state or the ground state). • This return to a lower energy level is often loosely described as decay and is the inverse of excitation. • Long-lived excited states are often called metastable state (ns level). • Question: where is the energy?

5. Section 1 一．粒子的能级与辐射跃迁 1．粒子的能级 　组成物质的原子、分子等粒子总是处于一定的能态或能级，能量最低的能态称为基态，其它能量较高的状态称为激发态。基态是最稳定的状态，通常多数粒子处在基态上，当一粒子获得一定的能量跃迁到某一激发态时，它在激发态上停留的时间一般很短，其平均寿命大约在10-7秒量级。有些粒子的某些激发态寿命较长，平均寿命大约可达10-2秒量级，这样的激发态称为亚稳态。

6. Atom absorbts energy from photon 激发态excited state 基态ground state

7. Units • 1 • 10-3 : milli- • 10-6 : micro- (μ) • 10-9 : nano- (n) • 10-12 : pico- (p) • 10-15 : femto- (f) • 10-18 : atto- (a)

8. 2．辐射跃迁的三种基本过程 　光与物质的相互作用有三种基本过程，这就是光的吸收、自发辐射和受激辐射。 1)吸收：假设E1、E2为某个粒子的两个能级，如图所示。一个处在较低能级E1上的粒子，吸收一个能量hn= E2-E1的光子，跃迁到较高的能级E2上，这一过程称为光的吸收。

9. Emissionof light 激发态 基态

10. Feature The propagation direction, and initial phase are random in spontaneous emission. 1) Spontaneous emission is the process by which a light source in an excited state undergoes a transition to the ground state and emits a photon. It plays an essential role in many phenomena in nature and forms the basis of many applications, such as fluorescent tubes, older television screens (cathode ray tubes), plasma display panels, lasers (for startup - normal continuous operation works by stimulated emission instead) and light emitting diodes.

11. 特点 自发辐射发出的光子的传播方向、初相位彼此无关。 1)自发辐射：处在高能级上的粒子可以自发地辐射光子而跃迁到较低的能级，这种过程称为自发辐射。如图所示，自发辐射光子的能量hn= E2-E1。

12. 特点 A photon created in this manner has the same phase, frequency, polarization, and direction of travel as the photons of the incident wave. 2) Stimulated emission is the process by which an atomic electron (or an excited molecular state) interacting with an electromagnetic wave of a certain frequency may drop to a lower energy level, transferring its energy to that field.

13. 特点 受激辐射发出的光子的频率、传播方向、初相位与诱发光子相同。 2)受激辐射：处在高能级E2上的粒子，在一个外来的能量为hn= E2-E1的光子的诱发下，跃迁到能量较低的能级E1，同时释放出一个与诱发光子完全相同的光子的过程 称为受激辐射。

14. Principle of laser • 受激辐射过程(Stimulated Emission Radiation, SER) 激发态：寿命10-7s 亚稳态:寿命10-3s 基态

15. 二．粒子数按能级分布 1.Boltzmann distribution In thermal equilibrium state, the atom number for each energy level obeys Boltzmann distribution Where niin the number of atomic energy levels by Ei, giis the level degeneracyof Ei, and k is the Boltzmann constant. According to the Boltzmann distribution, under thermal equilibrium conditions, the number of particles on a higher energy level is always less than the number of particles on the lower level. For example: according to the Boltzmann distribution , the 3s excited state of neon atom versus the ground state at room temperature (T = 300K), the ratio of particles number of two energy level is N2/N1=e-653 <<1 This indicates that, under conditions of thermal equilibrium, the vast majority of the particles are at the ground state. The higher the energy level , the less the particle number.

16. 二．粒子数按能级分布 1.玻尔兹曼分布 在热平衡状态下，原子数按能级分布服从玻尔兹曼定律 其中ni为处在Ei能级的原子数，gi为Ei能级的简并度，k为玻尔兹曼常数。 根据玻尔兹曼分布，在热平衡条件下，处在高能级上的粒子的数目总是少于低能级上的粒子的数目。例如：根据玻尔兹曼分布计算得知，氖原子3s激发态与基态在常温下(T=300K)，两能级的粒子数之比为 N2/N1=e-653 <<1 　　这说明，在热平衡条件下，绝大多数的粒子都处在基态，能级能量越高，粒子数越少。

17. 激光产生基本原理 • 光放大(Light Amplification, LA) • 光放大的条件：粒子数反转 (population inversion)

18. 一象棋盘大米的故事 • 传说国际象棋是舍罕王的宰相西萨.班.达依尔发明的。他把这个有趣的娱乐品进贡给国王。舍罕王对于这一奇妙的发明异常喜爱，决定让宰相自己要求得到什么赏赐。 西萨并没有要求任何金银财宝，他只是指着面前的棋盘奏道：“陛下，就请您赏给我一些麦子吧，它们只要这样放在棋盘里就行了：第一个格里放一颗，第二个格里放两颗，第三个格里放四颗，以后每一个格里都比前一个格里的麦粒增加一倍。圣明的王啊，只要把这样摆满棋盘上全部六十四格的麦粒都赏给你的仆人，他就心满意足了”，舍罕王听了，心中暗暗欣喜：“这个傻瓜的胃口实在不算大啊”。他立即慷慨的应允道：”爱卿，你当然会如愿以偿的！”但当记麦工作开始后不久，舍罕王便暗暗叫苦了，因为尽管第一袋麦子放满了将近二十个格子，可是接下去的麦粒数增长的竟是那样的快，国王很快意识到，即使把自己王国内的全部粮食都拿来，也兑现不了他许给宰相的诺言了！ 舍罕王由于失算而欠了西萨一大笔债，他为顾全面子而选择了什么样的善后措施我们已不得而知，但计算一下他的债务确是一件很有趣的事。 • 我们知道，这位聪明的宰相所要求的麦粒总数，实际上是等比数列:1,2,4,8... 的前六十四项和，即二的六十四次方减一，为一个二十位的大数：18,446,744,073,709,551,615。 这些麦粒究竟是多少呢？如果一升小麦按150，000粒计算，这大约是140万亿升小麦，按目前的平均产量计算，这竟然是全世界生产两千年的全部小麦！！

19. 经济小常识 • 假如你从现在开始每个月存200元到银行，银行每年发一次利息（本金的5%），这样的话40年后你有多少钱？ • 30万。 • 假如你从现在开始每个月投资200元到某股市，股市年回报率为10%，这样的话40年后你有多少钱？（依据：中国GDP增长约为8%）。 • 117万。

20. 与物理的联系 • 想要在股市赚钱的一个必要条件：年回报率>0。 • 形成激光的必要条件：粒子数反转 (增益系数>0)。

21. 激发态：Excited state寿命10-7s Mesostatic state亚稳态:寿命10-3s 基态 激光腔的构成Main components of Laser cavity • Gain medium：medium for population inversion • Pump: provide energy for population inversion • Mirror: provide oscillation • 增益介质：为粒子数反转提供媒介 • 泵浦源：为粒子数反转提供能量 • 反射镜：提供震荡

22. 1964 Nobel Prize in Physics

23. Mainmann and his Ruby laser On June 22 of 1960, Maiman sent a paper to the Physical Review Letters about his achievement, but received a letter of rejection stating that the publisher was no longer interested in maser related papers. Next, he sent a short version of his paper to the British journal "Nature". Consequently, the first scientific report about the first laser appeared on August 6, 1960 not in the USA but in Great Britain. The paper was titled "Stimulated Optical Radiation in Ruby" (Nature, 1960, v.187, P.493). 1927-2007

24. Lamp pumped laser灯泵浦的激光器结构图 Laser output Streak lamp Ruby Total reflection Partial reflection

25. Small-signal gain and gain saturation • When the laser intensity is very small initially, there are sufficient inversion particles to enhance the laser in the oscillator. In such a case, the gain coefficient is only related to the frequency of the light, irrespective of the light intensity. Such a gain is called small signal gain. • After a while when the laser is not so weak, therefore to the large amount of photons, the gain will decrease. Finally, the gain and loss will gradually reach a balance, such balance is called gain saturation.

26. 小信号增益和增益饱和 • 在振荡器内激光信号很小的时候，有足够的反转粒子支持信号得到增强，在这样的情况下，增益系数仅与光的频率有关，而与光强无关，这样的增益叫做小信号增益。 • 在经过一定增益后，光强对于工作介质所能提供的反转粒子数不再是一个很小的量，通过增益与损耗将逐渐达到一种平衡，这种平衡叫做增益饱和。

27. Theory of standing wave in laser • When light waves reflected back and forth in the cavity mirrors, the incident and the reflected waves interfere. To form a stable oscillation in the cavity requires that the lightwave due to the interference to be strengthened and that the light waves in the cavity from the Total phase shift of an integer multiple of 2. Therefore, only the specific frequencies of the light which satisfy the resonance condition can meet the oscillate condition: ， q为整数

28. 激光腔驻波原理 • 当光波在腔镜中来回反射时，入射波和反射波会发生干涉。为在腔内形成稳定的震荡，要求光波因干涉得到加强，即光波在腔内往返一周的总相移为2的整数倍。因此，只有特定频率的光才能满足谐振条件： ， q为整数

29. Mode of the laser (1): longitudinal mode Frequency Frequency spacing Realistic： Ideal：

30. 激光的模式（1）：纵模 频率 模式间隔 对于一般的增益介质： 理想状况：

31. Mode of the laser (1): longitudinal mode 频率 模式间隔 • The wave inside the laser has to be standing wave: • Different longitudinal mode pattern represents the monochromatics of a laser

32. 激光的模式（1）：纵模 频率 模式间隔 • 由电磁波理论可知，激光腔内的波必须是驻波，即满足条件 • 不同纵模模式代表了激光器的单色性

33. Mode of the laser (1): longitudinal mode • The longitudinal mode representatives the monochromatics of the laser • Homogeneously broadened medium, the mode of competition will bring a decline in the overall gain, making only one longitudinal mode oscillates - single longitudinal mode operation • Most of the media are non-homogeneously broadened medium, which requires other techniques to deliver single longitudinal mode operation. 增益阈值

34. 激光的模式（1）：纵模 • 纵模代表了激光的单色性 • 在均匀加宽工作介质中，模式竞争会带来整体增益下降，使得只有一个纵模震荡—单纵模运转 • 大部分介质为非均匀加宽介质，需要其他技术来实现单纵模运转 增益阈值

35. 激光的模式（1）：纵模 • In the inhomogeneous broadening gain medium, the laser radiation will cause "gain hole burning" effect • 在非均匀加宽增益介质中，由于激光的辐射会造成“增益烧孔”效应。

36. He-Nelaser application example • In a normal He-Ne laser, the central fluorescence frequency of the Ne atom(wavelength 632.8nm) is , its linewidth • Set q=1,to a cavity length L=30cm laser, • Therefore, we can have 3 longitudinal modes. There corresponding wavelengths are:

37. He-Ne激光器应用实例 • 在普通的He-Ne激光器中，Ne原子辉光放电荧光光谱中心频率 (波长632.8nm)，其线宽 • 设q=1,对于腔长L=30cm的激光器， • 因此，可出现3个纵模。其波长分别为：

38. Laser mode (2): transversal mode • When the laser oscillates in the resonant cavity, as it is reflected by the resonator back and forth, the mirror border diffraction effect after a certain number of times will cause the transverse field distribution of the light beam reaching a balance. This distribution is referred to as the transverse mode of the laser beam. A stabled cavity output: (square mirror) (round mirror) Where are the parameters of the cavity. ， L is the cavity length of the resonant cavity; R1, R2 is the radius of curvature of the two spherical mirrors; m and n are the ordinal number of the transverse modes; q is the ordinal number of the longitudinal mode; η is the refractive index for laser working medium.

39. 激光的模式（2）：横模 • 当激光在谐振腔内震荡的时候，由于谐振腔的来回反射，以及腔镜限制引起的衍射效应，在经过一定次数后，光束的横向场分布便趋于平衡。这一分布被称为激光的横模。 稳定腔的输出频率特性为： (方形镜) (圆形镜) 其中： 为腔参数； ， L为谐振腔的腔长；R1、R2为两个球面反射镜的曲率半径；m和n为横模序数；q为纵模序数；η为激光工作介质的折射率。

40. Laser mode (2): transversal mode Transverse mode intensity distribution of the squar mirror Transverse mode intensity distribution of the round mirror Except those with special needs, generally the fundamental transverse mode should be used, as it has the following characteristics: high brightness, small divergence angle, more uniform light intensity distribution in the radial cross section of the laser beam, equal phase at the cross section points, and best spatial coherence.

41. 激光的模式（2）：横模 圆形镜一些低阶模式的光强图样。 方形镜一些低阶模式的光强图样； 除有特殊需要外，一般都选择基横模输出，因为基横模有以下特点：亮度高 、发散角小 、在激光光束的横截面上径向光强分布较均匀 、横截面上各点的位相相同，空间相干性最好。

42. Gaussian intensity distribution • The general stability of the spherical cavity can be found in its sole corresponding equivalent confocal cavity. Stable cavity light field of a Gaussian beam. Gaussian beam along the z-axis is a variant of the center of curvature and radius of curvature of the spherical wave. At the waist and infinity the wavefrontis flat, in which the radius of curvature is infinite; in， （ is the confocal parameter), there exists the minimal curvature radius http://202.113.13.85/webclass/gdzjsh/experiments/notes/13.doc

43. Gaussian intensity distribution • 一般稳定球面腔都可以找到与其唯一对应的等价共焦腔。稳定腔的光场为高斯光束。 高斯光束沿z轴是一个变曲率中心和曲率半径的球面波。在束腰和无穷远处的波阵面是平面,曲率半径无限大;在 ， （ 为共焦参量), 曲率半径有最小值 http://202.113.13.85/webclass/gdzjsh/experiments/notes/13.doc

44. Gaussian beam Size of the beam W(z)changes with z with hyperbolically: z=0, W(0) reaches min, called waist, written as W0 The far-field divergence angle (full angle) of a confocal cavity fundamental mode Corresponding high-order modes divergence angle: (方镜) (圆镜)

45. 高斯光束 光斑尺寸W(z)随坐标z按双曲线规律变化: 其中z=0处的W(0)最小,称为束腰,记为W0; 共焦腔基模远场发散角（全角） 相应高阶横模的远场发散角 (方镜) (圆镜)

46. M2factot The beam diffraction magnification factor M2 is widely accepted as parameter to evaluate the quality of the laser beam in space. It is defined as: (square mirror) where , (round mirror). The fundamental mode Gaussian beam having the smallest M2 value (M2 = 1). he beam waist radius and the angle of divergence is also minimized, to achieve the diffraction limit. high-order, multi-mode Gaussian beam or other non-ideal light beam (such as wavefront aberration) M2 values ​​are greater than the the M2 values ​​can characterize the actual beam deviates from the extent of the diffraction limit, so called diffraction magnification factor M2 larger the value the faster the divergence of beam diffraction.

47. M2因子 目前国际上普遍将光束衍射倍率因子M2作为衡量激光光束空域质量的参量.其 定义为: (方镜), 其中, , (圆镜)。 由以上论述可知,基模高斯光束具有最小的M2值(M2=1),其光腰半径和发散角也最小,达到衍射极限高阶、多模高斯光束或其他非理想光束(如波前畸变)的M2值均大于1. M2值可以表征实际光束偏离衍射极限的程度,因此被称为衍射倍率因子. M2值越大,光束衍射发散越快。

48. 2 Filter M1 Laser CCD M3 1 M4 M2 1m Measure the divergence angle experimentally First adjust to guarentee that between 1 and 2, the optical path is greater than 8 meters (4 bounds). With the CCD the beam profile is measured at the position 1 (z1) and position 2 (Z2) to obtain the spot radius W (z1) and W (z). Using the formula The far-field divergence angle can be estimated.

49. 2 Filter M1 Laser CCD M3 1 M4 M2 1m 测量激光发散角实验 首先调整位置1和2之间的光程大于8米(4次往返),分别用CCD在位置1(z1)和位置2(z2)处测量出光斑半径W(z1)和W(z2),利用公式 可以估算出该模式的远场发散角。

50. Different laser types (by the gain medium) • Gas laser气体激光器：He-Ne, Ar+, CO2 • Solid-state laser固体激光器：Nd:YAG, Ti:Sapphire • Dye laser染料激光器 • Diode laser半导体激光器 • Fiber laser光纤激光器