Chapter 4 Optical Resonators

1. Chapter 4Optical Resonators Introduction: Optical resonators, like their low-frequency, radio-frequency无线电频率, and microwave counterparts,副本, 极相似的人或物, 配对物 are used primarily in order to build up large intensities with moderate 中等的, 适度的 power inputs. They consist in most cases of two, or more, curved mirrors that serve to “trap”, by repeated reflecticons and refocusing, an optical beam that thus becomes the mode of the resonator.

2. 4.0-1 A universal (普遍的, 全体的) measure of optical resonators’ property is the quality factor Q of the resonator, Q is defined by the relation : dissipated 沉迷于酒色的, 消散的

3. 4.0-2 4.0-3 Consider the case of a simple resonator formed by bouncing (使)反跳, 弹起 a plane TEM wave between two perfectly conducting 导体 planes of separation l so that the field inside is : the average electric energy stored in the resonator is

4. Where A is the cross-sectional(代表性的)area 横截面积, is the dielectric constant, and T is the period. Using 4.0-2 we obtain: 4.0-4 Where is the resonator volume. Since the average magnetic energy stored in a resonator is equal to electric energy, the total stored energy is : 4.0-5

5. Thus, recognizing （认可, 承认）that in steady state the input power is equal to the dissipated power, and designating the power input to the resonator by P, we obtain from 4.0-1 and the peak field is given by 4.0-6

6. Mode density of optical resonators The main challenge in the optical frequency regime is to build resonators that possess a very small number, ideally only one, high Q modes in a given spectral(光谱的) region. The reason is that for a resonator to fulfill this condition, its dimensions need to be of the order of the wavelength.

7. We consider the simple transverse electromagnetic (TEM) two-mirror resonator with a field distribution as given by Equation (4.0-2). The resonant frequencies are determined by requiring that the field vanish at z=0 and at the location z=L of the second reflector. This happens when sin(kml)=0 kml=mπ m=1,2,3…… Using , where n is the index of refraction, we obtain ωm=m(πc/nL) for the resonance frequencies corresponding to a frequency separation between adjacent modes of △ω=πc/nL. If we, arbitrarily, choose the criterion of sufficient mode spacing as △ω=ω， we obtain L=λ/2n, i.e., the linear dimension needs to be comparable to the wavelength (in the medium).

8. Mode control in the optical regime would thus seem to require that we construct resonators with volume(体积)-- . This is not easily achievable. An alternative is to build large resonators but to use a geometry that endows only a small fraction of these modes with low losses (a high Q). In our two-mirror example any mode that does not travel normally to the mirror will “walk off” after a few bounce and thus will possess a low Q factor.

9. We will show later that when the resonator contains an amplifying (inverted(倒转的)population) medium, oscillation(振动) will occur preferentially (优先)at high Q modes, so that the strategy of modal discrimination(识别)by controlling Q is sensible(明智的), we shall also find that further model discrimination is due to the fact that the atomic(原子的)medium is capable of amplifying radiation only within a limited frequency region so that modes outside this region, even if possessing high Q, do not oscillate. One question asked often is the following: given a large optical resonator, how many of its modes will have their resonant frequencies in a given frequency interval ,say,

10. To answer this problem, consider a large, perfectly reflecting(反射的) box resonator with sides,a , b, c along the x, y, z directions. Without going into modal details, it is sufficient for our purpose to take the amplitude field solution in the form : For the field to vanish at the boundaries, we thus need to satisfy : In the equation above, the triplet （三个一组）r, s, t is any integers ,and they define a mode. (4.0-7) (4.0-8) (4.0-8a)

11. We will restrict, without loss of generality, r,s,t to positive integers. It is convenient to describe the modal distribution in K space. Since each (positive) triplet r, s, t generate an independent mode, we can associate with each mode an elemental volume in K space. V: is the physical volume of the resonator. We recall that the length of the vector K satisfies equation 4.0-8, rewrite here as : (4.0-9) (4.0-10)

12. Figure 4-1 k space description of modes. Every positive triplet of integers r,s,t defines a unique mode. We can thus associate a primitive volume π3/abc in k space with each mode.

13. to find the total number of modes with K values between 0 and k, we divide the corresponding volume in K space by the volume per mode: We next use 4.0-10 to obtain the number of modes with resonant frequencies between 0 and v . The mode density, that is, the number of modes per unit near v in a resonator with volume V, is thus: (4.0-11)

14. where we multiplied the final result by 2 to account for the two independent orthogonally polarized modes that are associated with each r,s,t triplet. The number of modes that fall within the interval dν centered on ν is thus where V is the volume of the resonator. For the case of V=1 cm3,υ=3×1014 Hz and dν=3×1010, as an example (4.0-12) yields N~2×109 modes. If the resonator were closed, all these modes would have similar values of Q. This situation is to be avoided in the case of lasers, since it will cause the atoms to emit power (thus causing oscillation) into a large number of modes, which may differ in their frequency as well as in their spatial characteristics

15. This objection (缺陷) is overcome to a large extent by use of open resonators, which consist essentially(本质上)of a pair of opposing (对立的）flat or curved reflectors. In such resonators the energy of the vast majority of the modes does not travel at right angles (直角) to the mirrors and will thus be lost in essentially a single traversal. (横向)往返移动 These modes will consequently possess a very low Q. if the mirrors are curved, the few surviving (能继续存在的) modes will have their energy localized （停留）near the axis; thus the diffraction （衍射）loss caused by the open sides can be made small compared with other loss mechanisms （机构, 机制）such as mirror transmission.传输, 转播→镜面透射

16. Problem: we have a resonator which volume equal and (in atmosphere), please calculate the number of modes that produce within the interval centered on .

17. The Phone Call Goodbye 道别电话 • When I was small, my Great-Aunt Nony was still alive. Her life had been a terrible ordeal (严酷的考验) for her as she had been plagued with (用...来烦扰人)lots of different types of cancer. One day, my parents took me to see her in the nursing home in which she lived. It was quite scary (引起惊慌的) as the cancer had started to show and you could see tumors (瘤) on her face and arms. She seemed to just be wasting away (日益消瘦), but my parents wanted me to see her before she died. • A few days after this, my parents were talking in the kitchen when I received a seemingly real phone call on my plastic toy phone. It was Nony.

18. "Hello. It's Nony, Carrie. Don't worry. Everything will be alright. Tell your parents. Don't worry, everything's going to be fine." • So I hung up and told my parents what Nony had said, but they didn't believe me. They just thought that she was playing on my mind when I was playing on my pretend (假装, 装扮)telephone. • One hour later, we received a phone call from my grandmother. She told us that Nony had died about an hour ago.

19. 我的姑奶奶诺尼在我小的时候还在世了，但是她患有很多种的癌症，生命中的每时每刻都在遭受着病痛的折磨，苦不堪言。有一天，爸爸妈妈带我去疗养院看她，她的身体状况简直把我吓坏了，癌细胞已经扩散到了皮肤上，甚至脸上胳膊上都能看到肿瘤。她一天一天地消瘦下去，爸妈带我来就是想让我见她最后一面。我的姑奶奶诺尼在我小的时候还在世了，但是她患有很多种的癌症，生命中的每时每刻都在遭受着病痛的折磨，苦不堪言。有一天，爸爸妈妈带我去疗养院看她，她的身体状况简直把我吓坏了，癌细胞已经扩散到了皮肤上，甚至脸上胳膊上都能看到肿瘤。她一天一天地消瘦下去，爸妈带我来就是想让我见她最后一面。 • 几天以后的一天里，爸妈正在厨房说话，我的塑料玩具电话居然接到了一个真实的来电，是诺尼姑奶奶打来的。 • “你好，凯丽，我是诺尼。别担心，事情会好起来的。告诉你爸妈别担心，事情会好起来的。” • 我挂了电话，跟爸妈说了这一切。但是他们并不相信，他们以为是我玩电话玩具的时候突然想起她了。 • 一小时之后，我们接到了奶奶的电话，她告诉我们诺尼姑奶奶一小时前去世了。

21. Thank you • Thank you for your answers are almost identical. • Thus, you have saved me much time in review your schoolworks. Thanks again! • I’m deeply moved by your kindness. So If you have any trouble in your study, please don’t hesitate to let me know, I will do my best to help you. • For example, If you, unfortunately, failed in the final test, I will be glad to prepare the second time test for you. • Even if more unfortunaly, you failed again in the second time test, I still would be glad to prepare the third time test for you, and so on. • But, I do be more happy to test just one time, Do you think so?

22. 4.1 Fabry-Perot etalon (标准具) The Fabry-Perotetalonor interferometer, named after its inventors (Fabry (1867.6- 1945.12)是法国物理学家), can be considered as the archetype (原型) of the optical resonator. It consists of a plane-parallel plate of thickness l and index n that is immersed（浸入的）in a medium of index n′.

23. Let a plane wave be incident on the etalon（标准具）at an angle to the normal, as shown by figure 4-2. We can treat the problem of the transmission（and reflection） of the plane wave through the etalon by considering the infinite number of partial waves produced by reflections at the two end surfaces. The phase delay between two partial waves --- which is attributable（可归于...的）to one additional round trip--- is given, according to figure 4-2(a)， by 4.1-1 the internal内在的angle of incidence the vacuum wavelength of the incident wave

24. Figure 4-2(a) Multiple reflections model for analyzing the Fabry-Perot etalon

25. D F E Figure 4-2(b) Two successive reflections, A1 and A2. Their path difference is given by

26. If the complex amplitude of the incidence wave is taken as Ai， then the partial reflections, B1 ,B2 , and so forth(往前), are given by: as shown by figure 4-2. r: the reflection coefficient (反射系数) (radio of reflected to incident amplitude); t: is the transmission coefficient (传输系数) for waves incident from n’ toward n, and r’ and t’ are the corresponding quantities for waves traveling from n toward n’.

27. 4.1-2 The complex amplitude of the (total) reflected wave is given by : for the transmitted wave: For the complex amplitude of the total transmitted wave. We notice that the terms within the parentheses(圆括号) in 4.1-2 and 4.1-3 form an infinite geometric progression (无穷级数), adding them, we get: 4.1-3 4.1-4 4.1-5

28. where we used the fact that r’=-r, the conservation-of-energy (能量守恒) relation that applies to lossless mirrors at the same time, we define R : the fraction of the intensity reflected ; T: the transmitted at each interface and will be referred to in the following discussion as the mirrors’ reflectance and transmittance.

29. If the incident intensity (watts per square meter) is taken as , we obtain from 4.1-4 the following expression for the fraction of the incident intensity that is reflected . They are: 4.1-6 4.1-7

30. For a fixed and , 4.1-9 defines the unity transmission (resonance)(共振) frequencies of the etalon. Consider the transmission characteristics of a Fabry-Perot etalon, according to 4.1-7, the transmission is unity whenever : 4.1-8 For maximum transmission can be written as: 4.1-9

31. These are separated by the so-called free spectral range • Theoretical transmission plots of a Fabry-Perot etalon are shown in Figure 4-3. The maximum transmission is unity, as stated previously. The minimum transmission, on the other hand, approaches zero as R approaches unity.

32. Theoretical transmission plots of a Fabry-Perot etalon are shown in Figure 4-3. The maximum transmission is unity, as stated previously. The minimum transmission, on the other hand, approaches zero as R approaches unity.

33. If we allow for the existence of losses in the etalon medium, the peak transmission is less than unity. Taking the fractional intensity loss per pass as (1-A), the maximum transmission drops from unity to: 4.1-11

34. An experimental transmission plot of a Fabry-Perot etalon is shown in Figure 4-4

35. Are you a normal person？你是正常人吗？ During a visit to the mental asylum (精神病院), a visitor asked the director ..., "What is the criterion that defines a patient to be institutionalized (把...送交专门机构治疗[拘留])?” "Well..." said the director, "we fill up a bathtub, and we offer a teaspoon, a teacup, and a bucket to the patient and ask him to empty the bathtub." "Oh, I understand," said the visitor. "A normal person would choose the bucket as it is larger than the spoon or the teacup." "Noooooooo!" answered the director. "A normal person would pull the plug." 参观一所精神病院的时候一个参观者问院长，“你们是用什么标准来决定一个人是否应该被关进精神病院呢？” “呃… …”院长说，“是这样，我们先给一个浴缸放满水，然后我们给病人一个调茶匙，一个茶杯和一个水桶去把浴缸里面的水放清。” “噢，我明白了”， 参观者说。“一个正常人会选择水桶， 因为水桶比茶匙，茶杯的体积大。” “错了”，“院长回答”“正常人会把浴缸塞子拔掉”。

36. 4.2-1 Taking, for simplicity, the case of normal incidence ( ), we obtain the following expression for the change in the resonance frequency of a given transmission peak due to a length variation 4.2-2 4.2 Fabry-Perot etalons as optical spectrum analyzers According to 4.1-8, the maximum transmission of a Fabry-Perot etalon occurs when △v:the intermode frequency separation

37. According to 4.2-2, we can tune the peak transmission frequency of the etalon by △v by changing its length by half a wavelength. This property is utilized (利用) in operating the etalon as a scanning interferometer. The optical signal to be analyzed passes through the etalon as its length is being swept (扫描).

38. If the width of the transmission peaks is small compared to that of the spectral detail in the incident optical beam signal, the output of the etalon will constitute a replica (复制品) of the spectral profile of the signal. In this application it is important that the spectral width of the signal beam be smaller than the intermode spacing of the etalon so that the ambiguity (模糊) due to simultaneous (同时性的) transmission through more than one transmission peak is avoid. For the same reason the total length scan is limited to

39. the operation of a scanning Fabry-Perot etalon collimate oscilloscope 使平行； 使准直 示波器

40. Intensity versus (与····相对) frequency data obtained by analyzing the output of a multimode (多状态，多种方式) He-Ne laser oscillating near 632.8nm The peaks shown correspond to longitudinal (纵向的) laser modes, which will be discussed in section 4-5.

41. the value of corresponding to the two half-power points --- that is, the value of at which the denominator (分母) of 4.1-7 is equal to It is clear from the foregoing (前述的) that when operating as a spectrum analyzer the etalon resolution---- that is, its ability to distinguish details in the spectrum---is limited by the finite width of its transmission peaks. If we take, somewhat arbitrarily, the limiting resolution of the etalon as the separation between the two frequencies at which the transmission is down to half of its peak value, from 4.1-7 we obtain :

42. Assume then: 4.2-4 Defining the etalon finesse (精密度) as Then 4.2-5 F:the radio of the separation between peaks to the width of a transmission bandpass (通带). This ratio can be read directly from the transmission characteristics such as those of figure 4-4, for which we obtain F=26.

43. Shcoolwork: Study the part of 4.3 in page 132 by yourself. quiet

44. Obituary 死亡讣告 • The phone rang in the obituary department of the local newspaper. "How much does it cost to have an obituary printed"? asked the woman. "It's five dollars a word, ma'am," the clerk replied politely. "Fine," said the woman after a moment. "Got a pencil?" "Yes ma'am." "Got some paper?" • "Yes ma'am." "Okay, write this down: 'Cohen dead'." "That's all?" asked the clerk disbelievingly. "That's it." "I'm sorry ma'am, I should have told you - there's a five word minimum." "Yes, you should've," snapped the woman. Now let me think a minute... okay, got a pencil?" "Yes ma'am." • "Got some paper?" "Yes, ma'am." "Okay, here goes: 'Cohen dead. Cadillac for Sale.'"

45. 地方报社负责刊登死亡讣告的部门电话响了。“登一篇讣告多少钱？”一位女士问。“五美元一个字，太太。”书记员礼貌地回答。“好的，”女士沉默了一小会儿，“拿着笔呢吗？”“是的，夫人。”“纸呢？”“是的，夫人。”“好的，这样写：‘科恩去世了’”“就这些了？”书记员疑惑地问道。“对，就这些。”“很抱歉，夫人，我刚才没有告诉您，在我们这登讣告最少也得五个字。”“没错，你就应该告诉我，”女士有点生气了，“现在我得考虑一下，嗯…拿着笔呢吗？”“是的，夫人。”“纸呢？”“是的，夫人。”“好的，这样写：‘科恩去世了，出售一辆卡迪拉克轿车。’”地方报社负责刊登死亡讣告的部门电话响了。“登一篇讣告多少钱？”一位女士问。“五美元一个字，太太。”书记员礼貌地回答。“好的，”女士沉默了一小会儿，“拿着笔呢吗？”“是的，夫人。”“纸呢？”“是的，夫人。”“好的，这样写：‘科恩去世了’”“就这些了？”书记员疑惑地问道。“对，就这些。”“很抱歉，夫人，我刚才没有告诉您，在我们这登讣告最少也得五个字。”“没错，你就应该告诉我，”女士有点生气了，“现在我得考虑一下，嗯…拿着笔呢吗？”“是的，夫人。”“纸呢？”“是的，夫人。”“好的，这样写：‘科恩去世了，出售一辆卡迪拉克轿车。’”

46. How did I do? 新手 巡游 • A rookie police officer was out for his first ride in a cruiser with an experienced partner. • A call came in telling them to disperse some people who were loitering. The officers drove to the street and observed a small crowd standing on a corner. The rookie rolled down his window and said, "Let's get off the corner, people." A few glances, but no one moved, so he barked again, "Let's get off that corner...NOW!" Intimidated, the group of people began to leave, casting puzzled stares in his direction. Proud of his first official act, the young policeman turned to his partner and asked, "Well, how did I do?" • "Pretty good," chuckled the veteran policemen, "especially since this is a bus stop!" 闲荡 咆哮 恐吓 投掷 吃吃的笑声 老兵, 老手,

47. 一名新警察与老警察开着警车第一次出去巡逻。一名新警察与老警察开着警车第一次出去巡逻。 • 他们得到命令去疏散一群闲逛的人，于是他们开车去了那条街，看到路口站着一群人。新警察摇下窗户：“大家注意了，快离开这里。”人们看了他几眼，没理他。他喊起来：“离开这里，马上离开！”大家都不知道怎么回事，但是在他的威胁下还是离开了。新警察对他第一次执行公务的结果很满意，对老警察说：“我干得怎么样？”“你做得很好，”老警察笑着说，“尤其是在公共汽车站。”

48. Numerical Example: Design of a Fabry-Perot Etalon Consider the problem of designing a scanning Fabry-Perot etalon to be used in studying the mode structure of a He-Ne laser with the following characteristics: llaser =100 cm and the region of oscillation =Δυgain≈1.5÷×109 Hz. The free spectral range of the etalon (that is, its intermode spacing) must exceed the spectral region of interest, so from (4.1-10) we obtain or

49. The separation between longitudinal modes of the laser oscillation is c/2nllaser=1.5×107 Hz can resolved. According to (here we assume n=1). We choose the resolution of the etalon to be a tenth of this value, so spectral details as narrow as 1.5×107 Hz can be resolved. According to (4.2-6), this resolution can be achieved if Or

50. To satisfy condition (4.2-7), we choose 2nletal=20cm; thus (4.2-8) is satisfied when F ≥100 (4.2-9) A finesse of 100 requires, according to (4.2-5), a mirror reflectivity of approximately 97 percent. As a practical note we may add that the finesse, as defined by the first equally in (4.2-6), depends not only on R but also on the mirror flatness and the beam angular spread. These points are taken up in Problems 4-3 and 4-4.