Chapter 4Chapter 4 Optical ResonatorsOptical 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.

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 dissipated 沉迷于酒色的沉迷于酒色的 , , 消散消散的的

field energy stored by resonator

power dissipated by resonatorQ 4.0-1

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

( , ) sin sine z t E t kz 4.0-2

According to :

e e2

elelctric

Volum

2

0 0( , )

2

l T

electric

Ae z t dzdt

T

4.0-3

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:

Where is the resonator volume. Since the average magnetic energy stored in a resonator is equal to electric energy, the total stored energy is :

V lA

21

8electric E V 4.0-4

21

4electric E V 4.0-5

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

2

4

E VQ

P

and the peak field is given by

4QPE

V 4.0-6

Mode density of optical resonatorsMode 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.

We consider the simple transverse electromagnetic (TEWe consider the simple transverse electromagnetic (TEM) two-mirror resonator with a field distribution as givM) two-mirror resonator with a field distribution as given by Equation (4.0-2). The resonant frequencies are deen by Equation (4.0-2). The resonant frequencies are determined by requiring that the field vanish at z=0 and atermined by requiring that the field vanish at z=0 and at the location z=L of the second reflector. This happens t the location z=L of the second reflector. This happens when when

sin(ksin(kmml)=0 kl)=0 kmml=mπl=mπ m=1,2,3……m=1,2,3……Using Using , where n is the index of refraction, we o, where n is the index of refraction, we o

btain ωbtain ωmm=m(πc/nL) for the resonance frequencies corre=m(πc/nL) for the resonance frequencies corresponding to a frequency separation between adjacent sponding to a frequency separation between adjacent modes of modes of △△ω=πc/nL. If we, arbitrarily, choose the criterω=πc/nL. If we, arbitrarily, choose the criterion of sufficient mode spacing as ion of sufficient mode spacing as △△ω=ωω=ω ， ， we obtain Lwe obtain L=λ/2n, i.e., the linear dimension needs to be comparabl=λ/2n, i.e., the linear dimension needs to be comparable to the wavelength (in the medium).e to the wavelength (in the medium).

nc

wm

nc

k cm

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.

312 3( 10 1 )cm at m

( )L

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,

( )L

?between and

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)

(4.0-8a)

22 2 2 2x y zk k k k n

c

, ,x y z

r s tk k k

a b c

, ,x y z

r s tk k k

a b c

(4.0-8)

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)

3 3

modVabc V

2 ( , , )( , , )

v r s tk r s t n

c

Figure 4-1 k space description of modes. EveFigure 4-1 k space description of modes. Every positive triplet of integers r,s,t defines a ury positive triplet of integers r,s,t defines a unique mode. We can thus associate a primitinique mode. We can thus associate a primitive volume πve volume π33/abc in k space with each mode./abc in k space with each mode.

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)

33

3 2

1 48 3

( )6

kk V

N k

V

2 3 3( ) ( ) / 8 /p v dN v dv v n V c

3 3

3

4( )

3

v n VN v

C

where we multiplied the final result by 2 to where we multiplied the final result by 2 to account for the two independent orthogonaaccount for the two independent orthogonally polarized modes that are associated witlly polarized modes that are associated with each r,s,t triplet.h each r,s,t triplet. The number of modes that fall within the i The number of modes that fall within the interval dν centered on ν is thus nterval dν centered on ν is thus

where V is the volume of the resonator. For where V is the volume of the resonator. For the case of V=1 cmthe case of V=1 cm33,υ=3×10,υ=3×101414 Hz and dν=3 Hz and dν=3×10×101010, as an example (4.0-12) yields N~2×1, as an example (4.0-12) yields N~2×10099 modes. If the resonator were closed, all t modes. If the resonator were closed, all these modes would have similar values of Q. hese modes would have similar values of Q. This situation is to be avoided in the case of This situation is to be avoided in the case of lasers, since it will cause the atoms to emit lasers, since it will cause the atoms to emit power (thus causing oscillation) into a largpower (thus causing oscillation) into a large number of modes, which may differ in the number of modes, which may differ in their frequency as well as in their spatial chareir frequency as well as in their spatial characteristics acteristics

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

Problem: we have a resonator which volume equal

and (in atmosphere), please calculate the number of modes that produce within the interval centered on .

33cm

139 10 Hz 106 10d

d

6 108 54 10 1.36 10N

The Phone Call Goodbye The Phone Call Goodbye 道别道别电话电话

When I was small, my Great-Aunt Nony was still alive. When I was small, my Great-Aunt Nony was still alive. Her life had been a terrible ordeal (Her life had been a terrible ordeal ( 严酷的考验严酷的考验 ) for her ) for her as she had been plagued with (as she had been plagued with ( 用用 ......来烦扰人来烦扰人 )) lots of dilots of different types of cancer. One day, my parents took me to fferent types of cancer. One day, my parents took me to see her in the nursing home in which she lived. It was qusee her in the nursing home in which she lived. It was quite scary (ite scary ( 引起惊慌的引起惊慌的 ) as the cancer had started to sho) as the cancer had started to show and you could see tumors (w and you could see tumors ( 瘤瘤 ) on her face and arms. ) on her face and arms. She seemed to just be wasting away (She seemed to just be wasting away ( 日益消瘦日益消瘦 ), but my ), but my parents wanted me to see her before she died.parents wanted me to see her before she died.

A few days after this, my parents were talking in the kitcA few days after this, my parents were talking in the kitchen when I received a seemingly real phone call on my plhen when I received a seemingly real phone call on my plastic toy phone. It was Nony.astic toy phone. It was Nony.

"Hello. It's Nony, Carrie. Don't worry. Everything "Hello. It's Nony, Carrie. Don't worry. Everything will be alright. Tell your parents. Don't worry, evewill be alright. Tell your parents. Don't worry, everything's going to be fine."rything's going to be fine."

So I hung up and told my parents what Nony had So I hung up and told my parents what Nony had said, but they didn't believe me. They just thought said, but they didn't believe me. They just thought that she was playing on my mind when I was playithat she was playing on my mind when I was playing on my pretend (ng on my pretend ( 假装假装 , , 装扮装扮 )telephone.)telephone.

One hour later, we received a phone call from my One hour later, we received a phone call from my grandmother. She told us that Nony had died abougrandmother. She told us that Nony had died about an hour ago. t an hour ago.

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，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。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。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。”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。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。小时前去世了。

Thank youThank you Thank you for your answers areThank you for your answers are almost almost identical. identical. Thus, you have saved me much time in review your sThus, you have saved me much time in review your s

choolworks. Thanks again!choolworks. Thanks again! I’m deeply moved by your kindness. So If you have aI’m deeply moved by your kindness. So If you have a

ny trouble in your study, please don’t hesitate to let ny trouble in your study, please don’t hesitate to let me know, I will do my best to help you.me know, I will do my best to help you.

For example, If you, unfortunately, failed in the final For example, If you, unfortunately, failed in the final test, I will be glad to prepare the second time test for test, I will be glad to prepare the second time test for you.you.

Even if more unfortunaly, you failed again in the secEven if more unfortunaly, you failed again in the second time test, I still would be glad to prepare the thirond time test, I still would be glad to prepare the third time test for you, and so on.d time test for you, and so on.

But, I do be more happy to test just one time, Do you But, I do be more happy to test just one time, Do you think so?think so?

4.1 Fabry-Perot etalon (4.1 Fabry-Perot etalon ( 标准具标准具 ))

The Fabry-Perot etalonetalon or 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′.

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 cosnl

4.1-1

the vacuum

wavelength of

the incident

wave

the internal 内 在 的angle of incidence

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

' 'sin sinn CD n CD cos

lBC cos

lBC

D

E

F

Figure 4-2(b) Two successive Figure 4-2(b) Two successive reflections, Areflections, A11 and A and A22. Their path . Their path difference is given by difference is given by

'' sinsin nn CDnCDn '' sinsin

CEnADn '

cos2cos

2cos

l

BCAB

cos2cos

2cos

l

BCAB

cos4)(2

cos2coscos

2cos

nlnL

ll

lBCABL

cos4)(2

cos2coscos

2cos

nlnL

ll

lBCABL

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 inci

dent from n’ toward n, and r’ and t’ are the corresponding quantities for waves traveling from n toward n’.

' ' ' '3 21 2 3, ,i i

i i iB rA B tt r Ae B tt r Ae

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:

2 4 2{ ' ' (1 ' ' ...)}i i ir iA r tt r e r e r e A 4.1-2

2 4 2'{ ' ' ...}i it iA Att r r e r e 4.1-3

(1 )

1 e

1 e

i

r ii

t ii

eA A

RT

A AR

4.1-4

4.1-5

where we used the fact that r’=-r, the conservation-of-energy ( 能量守恒 ) relation that applies to lossless mirrors

2 ' 1r tt

at the same time, we define

2 2'R r r 'T ttR : 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.

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:

*i iA A

* 2

* 2 2

* 2

* 2 2

4 sin ( / 2)

(1 ) 4 sin ( / 2)

(1 )

(1 ) 4 sin ( / 2)

r r r

i i i

t t t

i i i

I A A R

I A A R R

I A A R

I A A R R

4.1-6

4.1-7

Consider the transmission characteristics of a Fabry-Perot etalon, according to 4.1-7, the transmission is unity whenever :

4 cos2

nl

For maximum transmission can be written as:

For a fixed and , 4.1-9 defines the unity transmission (resonance)( 共振 ) frequencies of the etalon.

l

2 cosm

cv m m any integer

nl

4.1-8

4.1-9

These are separated by the so-called These are separated by the so-called free spectral rangefree spectral range

1 (4.1 10)2 cosm m

cv v v

nl

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

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

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:

2

2

(1 )

(1 )t

i

I R A

I RA

4.1-11

An experimental transmission plot of a FaAn experimental transmission plot of a Fabry-Perot etalon is shown in Figure 4-4bry-Perot etalon is shown in Figure 4-4

Are you a normal personAre you a normal person ？？你是正常人吗？你是正常人吗？

During a visit to the mental asylum (During a visit to the mental asylum ( 精神病院精神病院 ), a visitor asked t), a visitor asked the director ..., "What is the criterion that defines a patient to he director ..., "What is the criterion that defines a patient to be institutionalized (be institutionalized ( 把把 ...... 送交专门机构治疗送交专门机构治疗 [[ 拘留拘留 ])?” ])?”

"Well..." said the director, "we fill up a bathtub, and we offer a "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 teaspoon, a teacup, and a bucket to the patient and ask him to empty the bathtub." "Oh, I understand," said the visitor. "A empty the bathtub." "Oh, I understand," said the visitor. "A normal person would choose the bucket as it is larger than the normal person would choose the bucket as it is larger than the spoon or the teacup." spoon or the teacup."

"Noooooooo!" answered the director. "A normal person would p"Noooooooo!" answered the director. "A normal person would pull the plug." ull 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？” 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，“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。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“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大。” “错了”，“院长回答”“正常人会把浴缸塞子拔掉”。掉”。

4.2 Fabry-Perot etalons as optical spectrum 4.2 Fabry-Perot etalons as optical spectrum analyzersanalyzers

According to 4.1-8, the maximum transmission of a Fabry-Perot etalon occurs when

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

0o d

dl

△v: the intermode frequency separation

2 cosnlm

4.2-1

( / 2 )

d dl

n

4.2-2

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 ).

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

/ 2dl n

the operation of a scanning Fabry-Perot etalon

collimate使平行；使准直

oscilloscope

示波器

Intensity versus (与 ···· 相对 ) frequency data obtained by analyzing the output of a multimode ( 多状态，多种方式 ) He-Ne l

aser oscillating near 632.8nm

The peaks shown correspond to longitudinal (纵向的 ) laser modes, which will be discussed in section 4-5.

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 : :

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

12

22(1 )R

22 1/ 212

sin2 4

Rm

R

Assume then:1/ 2( 2 )m

Defining the etalon finesse ( 精密度 ) as

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.

Then

1/ 2 1/ 2

12

2 cos 2 cos

c c Rv m

nl nl R

1/ 2 2 cos

v cv

F lF

1

RF

R

4.2-4

4.2-5

Shcoolwork:Shcoolwork: Study the part of 4.3 in page 132 by youStudy the part of 4.3 in page 132 by you

rself.rself.

quiet

Obituary Obituary 死亡讣告死亡讣告 The phone rang in the obituary department of the The phone rang in the obituary department of the

local newspaper. "local newspaper. "How much does it cost toHow much does it cost to have an have an obituary printed"? asked the woman. "It's five dollars obituary printed"? asked the woman. "It's five dollars a word, ma'am," the clerk replied politely. "Fine," a word, ma'am," the clerk replied politely. "Fine," said the woman after a moment. "Got a pencil?" "Yes said the woman after a moment. "Got a pencil?" "Yes ma'am." "Got some paper?"ma'am." "Got some paper?"

"Yes ma'am." "Okay, write this down: 'Cohen dead'." "Yes ma'am." "Okay, write this down: 'Cohen dead'." ""That's allThat's all?" asked the clerk ?" asked the clerk disbelievinglydisbelievingly. "That's . "That's it." "I'm sorry ma'am, I it." "I'm sorry ma'am, I shouldshould have told you - there's have told you - there's a five word minimum." "Yes, you should've," a five word minimum." "Yes, you should've," snapped the woman. Now let me think a minute... snapped the woman. Now let me think a minute... okay, got a pencil?" "Yes ma'am."okay, got a pencil?" "Yes ma'am."

"Got some paper?" "Yes, ma'am." "Okay, here goes: "Got some paper?" "Yes, ma'am." "Okay, here goes: 'Cohen dead. 'Cohen dead. Cadillac for SaleCadillac for Sale.'".'"

地方报社负责刊登死亡讣告的部门电话响了。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“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？”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。”“是的，夫人。”“纸呢？”“是的，夫人。”“好的，这样写：‘科恩去世了’”“就这些“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。”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“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，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。”“纸嗯…拿着笔呢吗？”“是的，夫人。”“纸呢？”“是的，夫人。”“好的，这样写：‘科呢？”“是的，夫人。”“好的，这样写：‘科恩去世了，出售一辆卡迪拉克轿车。’”恩去世了，出售一辆卡迪拉克轿车。’”

How did I do?How did I do? A A rookierookie police officer was out for his first ride in a police officer was out for his first ride in a cruisercruiser

with an experienced partner. with an experienced partner. A call came in telling them to disperse some people who were A call came in telling them to disperse some people who were

loiteringloitering. The officers drove to the street and observed a small . The officers drove to the street and observed a small crowd standing on a corner. The rookie rolled down his crowd standing on a corner. The rookie rolled down his window and said, "Let's get off the corner, people." A few window and said, "Let's get off the corner, people." A few glances, but no one moved, so he glances, but no one moved, so he barkedbarked again, "Let's get off again, "Let's get off that corner...NOW!" that corner...NOW!" IntimidatedIntimidated, the group of people began , the group of people began to leave, to leave, castingcasting puzzled stares in his direction. Proud of his puzzled stares in his direction. Proud of his first official act, the young policeman turned to his partner first official act, the young policeman turned to his partner and asked, "Well, how did I do?" and asked, "Well, how did I do?"

"Pretty good," "Pretty good," chuckled chuckled the the veteranveteran policemen, "especially policemen, "especially since this is a bus stop!" since this is a bus stop!"

新手 巡游

闲荡

咆哮恐吓投掷

吃吃的笑声 老兵 , 老手 ,

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，“尤其样？”“你做得很好，”老警察笑着说，“尤其是在公共汽车站。”是在公共汽车站。”

1 (4.1 10)2 cosm m

cv v v

nl

Hznl

c

etal

9105.12

Hznl

c

etal

9105.12

Consider the problem of designing a scanning Fabry-PConsider the problem of designing a scanning Fabry-Perot etalon to be used in studying the mode structure erot etalon to be used in studying the mode structure of a He-Ne laser with the following characteristics: lof a He-Ne laser with the following characteristics: llalaserser

=100 cm and the region of oscillation =Δυ=100 cm and the region of oscillation =Δυgaingain≈1.5÷×10≈1.5÷×109 9 Hz.Hz.

The free spectral range of the etalon (that is, its inteThe free spectral range of the etalon (that is, its intermode spacing) must exceed the spectral region of inrmode spacing) must exceed the spectral region of interest, so from (4.1-10) we obtainterest, so from (4.1-10) we obtain

Numerical Example: Design of a Fabry-Perot EtalonNumerical Example: Design of a Fabry-Perot Etalon

cmnletal 202 cmnletal 202 or

The separation between longitudinal modes of The separation between longitudinal modes of the laser oscillation is c/2nlthe laser oscillation is c/2nllaserlaser=1.5×10=1.5×1077 Hz can Hz can resolved. According to (here we assume n=1). resolved. According to (here we assume n=1). We choose the resolution of the etalon to be a We choose the resolution of the etalon to be a tenth of this value, so spectral details as tenth of this value, so spectral details as narrow as 1.5×10narrow as 1.5×1077 Hz can be resolved. Hz can be resolved. According to (4.2-6), this resolution can be According to (4.2-6), this resolution can be achieved if achieved if

HzFnl

cv

etal

72/1 105.1

2 Hz

Fnl

cv

etal

72/1 105.1

2 Or

)82.4(1022 3 cmFnletal )82.4(1022 3 cmFnletal

To satisfy condition (4.2-7), we choose To satisfy condition (4.2-7), we choose 2nl2nletaletal=20cm; thus (4.2-8) is satisfied when =20cm; thus (4.2-8) is satisfied when F ≥100 F ≥100 (4.2-9) A finesse of 100 requires, (4.2-9) A finesse of 100 requires, according to (4.2-5), a mirror reflectivity of according to (4.2-5), a mirror reflectivity of approximately 97 percent.approximately 97 percent. As a practical note we may add that the As a practical note we may add that the finesse, as defined by the first equally in (4.2-6), finesse, as defined by the first equally in (4.2-6), depends not only on R but also on the mirror depends not only on R but also on the mirror flatness and the beam angular spread. These flatness and the beam angular spread. These points are taken up in Problems 4-3 and 4-4.points are taken up in Problems 4-3 and 4-4.

1

RF

R

Another important mode of optical spectrum analAnother important mode of optical spectrum analysis performed with Fabry-Perot etalons involves tysis performed with Fabry-Perot etalons involves the fact that a noncollimated monochromatic beahe fact that a noncollimated monochromatic beam incident on the etalon will emerge simultaneoum incident on the etalon will emerge simultaneously, according to (4.1-8), along many directions θsly, according to (4.1-8), along many directions θ33, , which corresponds to the various orders m. If the which corresponds to the various orders m. If the output is then focused by a lens, each such directioutput is then focused by a lens, each such direction θ will give rise to circle in the focal plane on the on θ will give rise to circle in the focal plane on the lens, and, there, each frequency component preselens, and, there, each frequency component present in the beam leads to a family of circles. This mont in the beam leads to a family of circles. This mode of spectrum analysis is especially useful under de of spectrum analysis is especially useful under transient conditions where scanning etalons canntransient conditions where scanning etalons cannot be employed. Further discussion of this topic is ot be employed. Further discussion of this topic is included in Problem 4-6.included in Problem 4-6.

collimate 使平行：使平行；使准直Monochromatic [ 物 ] 单色的 , 单频的transient 瞬变的 , 短暂的

4.34.3 optical resonators with spherical optical resonators with spherical ((球形的球形的 ) mirrors) mirrors

In this section we study the properties of optical resonators formed by two opposing spherical mirrors. We will show that the field solutions inside the resonators are those of the propagating Gaussian beams, which were considered in chapter 2. It is, consequently, useful to start by reviewing the properties of the beams.

The field distribution corresponding to the (l,m) transverse mode ( 横模 ) is given by :

The sign of R(z) is take as positive when the center of curvature is to the left of the wavefront, and vice versa ( 反之亦然 ).

0

, 0

2 2 2 2

2

1/ 22

00 0

0

( ) 2 2( )

exp ( 1)( ) 2 ( )

( ) 1 ,

l m t m

z

x yE E H H

z z z

x y x yik ikz i l m

z R z

nzz z

z

r

4.3-1

4.3-2

2( ) tan 1 tan 1

o o

z zz

n z

The loci (轨迹， locus 的复数形式 ) of the points at which the beam intensity (watts per square meter) is a given fraction of its intensity on the axis are the hyperboloids (双曲面 ).

2 2 2. ( )x y const z

The hyperbolas generated by the intersection of these surfaces with planes that include the z axis are shown in Figure 4-7. These hyperbolas are normal to the phase fronts and thus correspond to the local direction of energy flow. The hyperboloid x2 ＋ y2 = ω2 (z) is, according to (4.3-1), the focus of the points where the exponential factor in the field amplitude is down to 1/e from its value on the axis. The quantity ω (z) is thus defined as the mode spot size at the plane z.

2

Hyperbolic 双曲线的

Figure 4-7 Hyperbolic curves corresponding to the local directions of propagation. The nearly spherical phase fronts represent possible positions for reflectors. Any two reflectors form a resonator with a transverse field distribution given by (4.3-1).

2

Given a beam of the type describe by 4.3-1, we can form an optical resonator merely by inserting at points z1 and z2 two reflectors with radii of curvature that match those of the propagating beam spherical phase fronts at these points. Since the surface are normal to the direction of energy propagation

Figure 4-7 Hyperbolic curves corresponding to the local directions of propagation. The nearly spherical phase fronts represent possible positions for reflectors. Any two reflectors form a resonator with a transverse field distribution given by (4.3-1).

2

the reflected beam retrace itself, thus, if the phase shift between the mirrors is some multiple (倍 ) of radians, a self-reproducing stable field configuration results.

2

No rush No rush 别着急别着急 AS STUDENTS in the college of veterinary medicine AS STUDENTS in the college of veterinary medicine

((兽医兽医 ) at Texas A & M University, we frequently tre) at Texas A & M University, we frequently treated the farm animals at the state prison. While awated the farm animals at the state prison. While awkwardly performing a medical procedure on an unrkwardly performing a medical procedure on an unruly horse, a classmate said to the prisoner who was uly horse, a classmate said to the prisoner who was holding the animal, "Sorry I'm taking so long." "No holding the animal, "Sorry I'm taking so long." "No problem," the prisoner replied, "I'm doing seven yeproblem," the prisoner replied, "I'm doing seven years.ars.

我在德克萨斯我在德克萨斯 A&MA&M大学兽医药学系学习的时候，同学们经大学兽医药学系学习的时候，同学们经常把动物带到州监狱里去治疗。有一次我们笨手笨脚的给一常把动物带到州监狱里去治疗。有一次我们笨手笨脚的给一匹烈马做完检查，我同学对一直按着这匹马的犯人说：“真匹烈马做完检查，我同学对一直按着这匹马的犯人说：“真是不好意思，我用了这么长时间。”“没关系，”他回答说，是不好意思，我用了这么长时间。”“没关系，”他回答说，“我已经作了七年这种事了。”“我已经作了七年这种事了。”

1. Optical resonator 1. Optical resonator algebra (algebra ( 代数学代数学 ))

As mentioned in the preceding paragraphs, we can form an optical resonator by using two reflectors, one at z1, and the other at z2 , chosen so that their radii of curvature are the same as those of the beam wave-fronts at the two locations. The propagating beam mode (4.3-1) is then reflected back and forth between the reflectors without a change in the transverse profile. The requisite radii of curvature are

2

1 11

ozR zz

2

2 22

ozR zz

from above, we can get:

For a given minimum spot size , we can use 4.3-6 to find the positions z1 and z2 at which to place mirrors with curvature R2 and R1 , respectively.

1/ 2( / )o oz n

2 211 1

14

2 2 o

Rz R z

2 222 2

14

2 2 o

Rz R z

(4.3-6)

2. The symmetrical mirror resonator2. The symmetrical mirror resonator

The special case of a resonator with symmetrically (about z=0) placed mirrors merits a few comments. The planar phase front at which the minimum spot size occurs is, by symmetry, at z=0. Putting R2=-R1=R in (4.3-7) gives:

2 (2 )

4o

R l lz

1/ 2 1/ 2 1/ 4 1/ 40

1( ) ( ) ( ) ( )

2oz l

Rn n l

Which yields the following expression for the spot size at the mirrors:

For and the beam spread inside the resonator is small.

1,2, oR l

21/ 2 1/ 4

1,2

2( ) [ ]2 ( / 2)

l R

n l R l

(4.3-10)

The value of R (for a given l ) for which the mirror spot size is a minimum, is readily found from 4.3-10 to be R=l. When this condition is fulfilled we have what is called a symmetrical confocal ( 共焦 ) resonator, since the two foci (焦距 ) ， occurring at a distance of R/2 from the mirrors, coincide. From 4.3-9, we obtain:

1/ 2( ) ( )2o conf

l

n

whereas from 4.3-10, we get:

1,2( ) ( ) 2conf o conf

so the beam spot size increase by between the center and the mirrors.

2

4.3-11

4.3-12

3. Design of a Symmetrical Resonator3. Design of a Symmetrical Resonator

Consider the problem of designing a symmetrical resonator for with a mirror separation . If we were to choose the confocal geometry with , the minimum spot size (at the resonator center) would be, from 4.3-11 and for n=1:

410 cm 2l m

2R l m

1/ 2( ) ( ) 0.05642o conf

lcm

n

the spot size at the mirrors would have the value:

1,2( ) ( ) 2 0.0798conf o conf cm

Assume next that a mirror spot size is desired . assuming , we get:

1,2 0.3cm

R l

1,2 1/ 4

1/ 2

0.3 2( )

0.056( )2

Rl ln

where :

R=400l=800m

so that the assumption is valid. The minimum beam spot size is found, through 4.3-2 and 4.3-8, to be

R lo

1,20.994 0.3o cm

thus, to increase the mirror spot size from its minimum (confocal) value of 0.0798cm to 0.3cm, we must use exceedingly plane mirrors (R=800m). This also shows that even small mirror curvature (that is, large R) give rise to “narrow” beams.

o

The numerical example we have worked out applies equally well to the case in which a plane mirror is placed at z=0. The beam pattern is equal to that existing in the corresponding half of the symmetric resonator in the example, so the spot size on the planar reflector is

Results:

Shcoolwork:Shcoolwork:Study section 4.4 by yourself.Study section 4.4 by yourself.

quiet

4.4 Mode stability criteria4.4 Mode stability criteria((模式稳定准则模式稳定准则 ))

The ability of an optical resonator to support low (diffraction) loss modes depends on the mirrors separation l and their radii of curvature R1 and R2. To illustrate (举例说明 ) this point, consider first the symmetric resonator with R1=R2=R . The ratio of the mirror spot size at a give l/R to its minimum confocal (l/R=1):

1/ 4

1,2

1,2

1

( ) ( / )[2 ( / )]conf l R l R

4.4-1

Figure 4-8 Ratio of beam spot size at the mirrors of a symmetrical resonator to its confocul (l/R=1) value.

Shcoolwork:Shcoolwork:Study section 4.5 by yourself.Study section 4.5 by yourself.

quiet

4.6 Resonance Frequencies (4.6 Resonance Frequencies ( 共振频率共振频率 )) of Optical Resonator of Optical Resonator

The resonance frequencies are determined by the condition that the complete round-trip phase delay of a resonator mode be some multiple of . This requirement is equivalent to that in microwave waveguide resonators where the resonator length must be equal to an integral number of half-guide wavelengths. This requirement makes it possible for a stable standing wave pattern to establish itself along the axis with a transverse field distribution equal to that of the propagation mode.

2

If we consider a spherical mirror resonator with mirrors at z1 and z2 , the resonance condition for the l,m mode can be written as:

1 12 1( 1)(tan tan )qo o

z zk d l m q

z z

in which d=z2-z1 is the resonator length. It follows that:

1q qk kd

or using , we get:2 nk

c

1 2q q

c

nd

for the intermode frequency spacing.

4.6-1

4.6-2

4.6-3

Shcoolwork:Shcoolwork: In the case of confocal resonator (R=In the case of confocal resonator (R=

d), please derive the d), please derive the △△v.v.

quiet

4.7 Losses in optical resonators4.7 Losses in optical resonators

An understanding of the mechanisms by which electro-magnetic energy is dissipated in optical resonators and the ability to control them are of major importance in understanding and operating a variety of optical devices. For historical reasons as well as for reasons of convenience, these losses are often characterized by a number of different parameters.

c

d

dt t

Where is the energy stored in the mode so that in a passive resonator

( ) (0)exp( / ) (0)exp( / )ct t t t Q

If the fractional (intensity) loss per pass is L and the length of the resonator is , the fractional loss per unit time is , therefore:

l

/cL nl

d cL

dt nl

The decay lifetime (photon lifetime) tc of a cavity mode is defined by means of the equation:

4.7-1

So:

c

nlt

cL

1 2lnL l R R

So that, we have:

1 2 1 2[ (1/ ) ln ] [ (1 )]c

n nlt

c l R R c l R R

2R for the case of a resonator with mirror’s reflectivities and and an average distributed loss constant , the average loss per pass is for small losses

1R

4.7-2

4.7-3

Where the approximate equality applies when R1R2=1 . The quality factor of the resonator is defined universally as:

/Q

P d dt

Where is the stored energy, is the resonant frequency, and is the power dissipated. By comparing 4.7-4 and 4.7-1 we obtain:

/P d dt

cQ t

4.7-4

4.7-5