2.4 wave equation in quadratic index media the most widely encountered beam in quantum electronics...
TRANSCRIPT
2.4 wave equation in quadratic 2.4 wave equation in quadratic index mediaindex media
The most widely encountered beam in The most widely encountered beam in quantum electronics is one where quantum electronics is one where the intensity distribution at planes the intensity distribution at planes normal to the propagation direction normal to the propagation direction is Gaussian. To derive its is Gaussian. To derive its characteristics we start with the characteristics we start with the Maxwell’s equations in an isotropic Maxwell’s equations in an isotropic charge-free medium.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
Et
EE
12
22
Et
EE
12
22
t
EH
t
EH
t
HE
t
HE
0 E 0 E
33This neglect is justified if the fractional chanThis neglect is justified if the fractional change of ge of εε in one optical wavelength is << 1. in one optical wavelength is << 1.
quantity 量 , 数量
where
44If K is complex (for example, KIf K is complex (for example, Krr+iK+iKii), then a ), then a
traveling electromagnetic plane wave has the traveling electromagnetic plane wave has the form of:form of:
exp exp
exp exp
i r
i r
i t kz k z i t k z
k z i t k z
thus allowing for a possible dependence of ε on thus allowing for a possible dependence of ε on position r. We have also taken k as a complex position r. We have also taken k as a complex number to allow for the possibility of losses number to allow for the possibility of losses (σ>0) or gain (σ<0) in the medium(σ>0) or gain (σ<0) in the medium
)(
1)()( 22 rirrk
)(
1)()( 22 rirrk
We limit our derivation to the case in which k2We limit our derivation to the case in which k2(r) is given by(r) is given by
where, according to (2.4-4)where, according to (2.4-4) so that k2 is some constant characteristic of the so that k2 is some constant characteristic of the
medium. Furthermore, we assume a solution medium. Furthermore, we assume a solution whose transverse dependence is on r= whose transverse dependence is on r= only, so that in (2.4-3) we replace by only, so that in (2.4-3) we replace by
transverse 横向的 , 横断的
22
22 ),,( rkkkzrk 22
22 ),,( rkkkzrk 22 yxr 22 yxr 2.4-5
)0(
)0(1)0()0( 222
ikk
)0(
)0(1)0()0( 222
ikk
22 yx 22 yx 22
2
2
22
2
2
2
2
222 11
zzrrrzt 2
2
22
2
2
2
2
222 11
zzrrrzt 2.4-6
The kind of propagation we are considering is that The kind of propagation we are considering is that of a nearly plane wave in which the flow of enerof a nearly plane wave in which the flow of energy is predominantly along a single( for example, gy is predominantly along a single( for example, z) direction so that we may limit our derivation z) direction so that we may limit our derivation to a single transverse field component E. Taking to a single transverse field component E. Taking E as E as
We obtain from (2.4-3) and (2.4-5) in a few simple sWe obtain from (2.4-3) and (2.4-5) in a few simple steps, teps,
where and where we assume that the variation where and where we assume that the variation is slow enough that is slow enough that
predominantly 主要的 , 突出的 , 有影响的transverse 横向的
ikzezyxE ),,( ikzezyxE ),,( 2.4-7
02 2'2 kkikl 02 2'2 kkikl 2.4-8
Where and where we assume that the variation is slow enough that
z /' z /' 2''' kk 2''' kk
Next we take in the form of Next we take in the form of
that ,when substituted into (2.4-8) and after that ,when substituted into (2.4-8) and after using (2.4-6),givesusing (2.4-6),gives
If (2.4-10) is to hold for all r, the coefficients If (2.4-10) is to hold for all r, the coefficients of the different powers of r must be equal of the different powers of r must be equal to zero. This leads to to zero. This leads to
The wave equation (2.4-3) is thus reduced to The wave equation (2.4-3) is thus reduced to (2.4-11).(2.4-11).
coefficient [ 数 ] 系数
Mental deficiencyMental deficiency 智力缺陷智力缺陷 "Would you mind telling me, Doctor," Bob aske"Would you mind telling me, Doctor," Bob aske
d ...d ... "how you detect a mental deficiency in somebo"how you detect a mental deficiency in somebo
dy who appears completely normal?" dy who appears completely normal?" "Nothing is easier," he replied. "Nothing is easier," he replied. "You ask him a simple question which everyone "You ask him a simple question which everyone
should answer with no trouble. should answer with no trouble. If he hesitates, that puts you on the track.“If he hesitates, that puts you on the track.“ " Well, What sort of question?" " Well, What sort of question?" "Well, you might ask him, 'Captain Cook made t"Well, you might ask him, 'Captain Cook made t
hree trips around the world and died during onhree trips around the world and died during one of them. Which one?' e of them. Which one?'
Bob thought for a moment, and then said with a Bob thought for a moment, and then said with a nervousnervous 紧张的紧张的 laugh, "You wouldn't happen to laugh, "You wouldn't happen to have another example would you? I must confehave another example would you? I must confess(ss( 坦白坦白 )I don't know much about history.")I don't know much about history."
““ 医生，你能不能告诉我，”鲍勃问，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？”缺陷的呢？” ““ 再没有比这容易的了，”医生回答，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。”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？”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？’”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。”问题吗？坦率地说，我对历史了解的不是很多。”
高斯光束的基本性质及特征参数高斯光束的基本性质及特征参数
高斯光束的参数特征
高斯光束在自由空间的传播规律
基模高斯光束
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：
2 2iωt
2
γ γ zi k z arctan e
2Rz fω z e000
EE r,z,t e
ω z
式中： E0 为常数，其余符号的意义为
2k
2 2 2r x y
2
0 1z
zf
2
( )f
R z zz
20
0f Z
基模高斯光束的束腰半径
高斯光束的共焦参数
与传播轴线相交于 Z 点的高斯光束等相位面的曲率半径
与传播轴线相交于 Z点高斯光束等相位面上的光斑半径
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。
2
0 1z
zf
2
2 ( )
r
w ze
可见，光斑半径随着坐标 Z 按双曲线的规律扩展，即
2 2
2 20
1z z
f
如图 1-7 所示。
在 Z=0 处， ω （ z ） =ω0 达到极小值，称为束腰半径。
(2) 基模高斯光束场的相位因子
2
00 , arctan2
zr z k z
R z f
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。
R(z) 随 Z 变化规律为：
2 2
21
f fR z z z
z z
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 。
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。
(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/e
2 定义为 z→∞ 时，强度为中心的 1/e2 点所夹角的全宽度，即
21/0
2 ( )limez
z
z
高斯光束的发散度由束腰半径 ω0 决定。
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。
2.5 Gaussian beams in a homogeneous2.5 Gaussian beams in a homogeneous 均匀均匀 mmediumedium
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
22
1 1( ) 0
kd
q dz q k
In a homogeneous medium the quadratic coefficient( 系数 ) k2 of the equation above is zero, so that:
1 1( ) 0
d
q dz q
2.5-1
2.5-2
How to solve it? Have a try! please
2 2 2
0
1 1 10 1 0 1
dq dq dq
q q dz q dz dz
q Z q
/
o
i iP
q z q
Where q0 is an arbitrary integration constant. From (2.4-11) and (2.5-4) we have
( ) ln( )
ln
( )
( )
( ) ln ln
( ) ln 1
o
o
o o
o o
o
o
p z i z q c
and ch
d z qdP idp i
dz z q z q
p z i z q i q
zp
oose q
z iq
c i
then:
( ) ln(1 )o
zP z i
q
where q0 is an arbitrary( 任意的 ) constant. combining above discussion, we obtain:
2exp{ [ ln(1 ) ]}2( )o o
z ki i r
q q z
2.5-7
2.5-6
in which
2o
o
nq i
2 n
k
Let us consider, one a time, the two factors in 2.5-7, the first one becomes
2exp[ ln(1 )]
o
zi
n
vacuum ( 真空 )wave length
2.5-8
2.5-9
imaginary 虚数的 confine 限制 , 禁闭
substitution 代替 , 代入法 exponent 指数 , 解释者 , 典型
The choice of imaginary will be found to lead to physically meaningful waves whose energy density is confined near the z axis. With this last substitution, let us consider, one at a time, the two factors in (2.5-7). The first one becomes
where we used .Substituting (2.5-8) in the second term of (2.5-7) and separating the exponent into its real and imaginary parts, we obtain
Define the important following parameters:2
2 2 2 22 2
2 22
2
2
2
( ) [1 ( ) ] (1 )
[1 ( ) ] (1 )
( ) tan 1 tan 1
o oo o
o o
oo
o o
z zz
n z
n zR z z
z z
n z zz z
n z
2-5-11
2-5-12
Combine equations above in 2.5-4 , and obtain:Combine equations above in 2.5-4 , and obtain:
20
0
200 2
( , , ) exp ( )( ) 2 ( )
1exp ( )
( ) ( ) 2 ( )
2 /
krE x y z E i kz z i
w z q z
ikE i kz z r
w z z R z
k n
2.5-14
0
: the distance r at which the field amplitude is down by a factor 1/e compare to its valve on the axis; 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.
( )z
That is the basic result. we 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 .
From (2.5-14) the parameter w(z), which evolves aFrom (2.5-14) the parameter w(z), which evolves according to (2.5-11), is the distance r at which thccording to (2.5-11), is the distance r at which the field amplitude is down by a factor 1/e compare field amplitude is down by a factor 1/e compared to its value on the axis. We will consequently ed to its value on the axis. We will consequently refer to it as the beam spot size. The parameter refer to it as the beam spot size. The parameter is the minimum spot size. It is the beam spot sizis the minimum spot size. It is the beam spot size at the plane z=0. The parameter R in (2.5-14) is e at the plane z=0. The parameter R in (2.5-14) is the radius of curvature of the very nearly spherithe radius of curvature of the very nearly spherical wavefronts at z. We can verify this statemencal wavefronts at z. We can verify this statement by deriving the radius of curvature of the constt by deriving the radius of curvature of the constant phase surfaces (wavefronts) or ,more simplant phase surfaces (wavefronts) or ,more simply, by considering the form of a spherical wave ey, by considering the form of a spherical wave emitted by a point radiator placed at z=0. It is givmitted by a point radiator placed at z=0. It is given by en by
evolve ( 使 ) 发展 , ( 使 ) 进展
since z is equal to R, the radius of curvatsince z is equal to R, the radius of curvature of the spherical wave. Comparing ure of the spherical wave. Comparing (2.5-15) with (2.5-14), we identify R as t(2.5-15) with (2.5-14), we identify R as the radius of curvature of the Gaussian he radius of curvature of the Gaussian beam. The convention regarding the sibeam. The convention regarding the sign of R(z) is that it is negative if the cegn of R(z) is that it is negative if the center of curvature occurs at and vice venter of curvature occurs at and vice versa.rsa.convention 协定 , 习俗 , 惯例
The form of the fundamental Gaussian beam is, The form of the fundamental Gaussian beam is, according to (2.5-14),uniquely determined once according to (2.5-14),uniquely determined once its minimum spot size wits minimum spot size w0 0 and its location –that and its location –that is , the plane z=0– are specified. The spot size w is , the plane z=0– are specified. The spot size w and radius of curvature R at any plane z are and radius of curvature R at any plane z are then found from (2.5-11) and (2.5-12). Some of then found from (2.5-11) and (2.5-12). Some of these characteristics are displayed in Figure 2-these characteristics are displayed in Figure 2-5. The hyperbolas shown in this figure 5. The hyperbolas shown in this figure correspond to the ray direction and are correspond to the ray direction and are intersections of planes that include the z axis intersections of planes that include the z axis and the hyperboloids. and the hyperboloids.
uniquely 独特地 , 唯一地 hyperbola 双曲线hyperboloid [ 数 ] 双曲面
)(. 222 zconstyx )(. 222 zconstyx 2.5-16
These hyperbolas correspond to the These hyperbolas correspond to the local direction of energy local direction of energy propagation. The spherical surfaces propagation. The spherical surfaces shown have radii of curvature given shown have radii of curvature given by (2.5-12). For large z the by (2.5-12). For large z the hyperbolas xhyperbolas x22+y+y22=ω=ω22 are asymptotic are asymptotic to the cone to the cone
hyperboloid [ 数 ] 双曲面asymptotic 渐近线的 , 渐近的cone 锥形物 , 圆锥体
zn
yxr0
22
zn
yxr0
22
2.5-17
Some of these characteristics are displayed as following figure:
Propagation lines
+z
Z=0
Phase fronts
0beam
Whose half-apex angle, which we take as a Whose half-apex angle, which we take as a measure of the angular beam spread, ismeasure of the angular beam spread, is
This last result is a rigorous manifestation This last result is a rigorous manifestation of wave diffraction according to which a of wave diffraction according to which a wave that is confined in the transverse wave that is confined in the transverse direction to an aperture of radius wdirection to an aperture of radius w00 will will spread (diffract) in the far field (z>>π wspread (diffract) in the far field (z>>π w00 22n/λ) according to (2.5-18)n/λ) according to (2.5-18)
apex 顶点 尖端 rigorous 严格的 , 严厉的hyperboloid [ 数 ] 双曲面
nnbeam00
1tan
nnbeam00
1tan
for beam beam 2.5-18
2.6 Fundamental Gaussian beam in a lenslike 2.6 Fundamental Gaussian beam in a lenslike medium –the ABCD Law medium –the ABCD Law
We now return to the general case of a lenslWe now return to the general case of a lenslike medium so that kike medium so that k22 ≠0. The P and q functi ≠0. The P and q functions of (2.4-9) obey, according to (2.4-11)ons of (2.4-9) obey, according to (2.4-11)
011 2
2
k
k
qq0
11 2
2
k
k
qq q
ip '
q
ip '
2.6-1
2
)(2)(exp r
zq
kzPi
2
)(2)(exp r
zq
kzPi
If we introduce the functions defined If we introduce the functions defined by by
we obtain from (2.6-1)we obtain from (2.6-1)
turn to P:46 2.3-4s
s
q
'1s
s
q
'1 2.6-2
02'' k
kss 02'' k
kss
If at the input plane z=0 the ray has a radIf at the input plane z=0 the ray has a radius rius r00 and slope r and slope r00’, we can write the s’, we can write the solution of (2.3-4) directly as olution of (2.3-4) directly as
022
2
rk
k
dz
rd02
2
2
rk
k
dz
rd2.3-4
'0
2
20
2 sincos)( rzk
k
k
krz
k
kzr
'
02
20
2 sincos)( rzk
k
k
krz
k
kzr
'0
20
22
' cossin)( rzk
krz
k
k
k
kzr
'
02
02
2' cossin)( rz
k
krz
k
k
k
kzr
2.3-5
Using (2.6-3) in (2.6-4) and expressing the reUsing (2.6-3) in (2.6-4) and expressing the result in terms of an input value q0 gives thsult in terms of an input value q0 gives the following result for the complex beam re following result for the complex beam radius q(z) adius q(z)
The physical significance of q(z) in this case The physical significance of q(z) in this case can be extracted from (2.4-9). We expand tcan be extracted from (2.4-9). We expand the part ψ(r,z) that involves r. The result is he part ψ(r,z) that involves r. The result is
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
2.6-4
If we express the real and imaginary parts If we express the real and imaginary parts of q(z) by means of of q(z) by means of
We obtain We obtain
so that w(z) is the beam spot size and R itso that w(z) is the beam spot size and R its radius of curvature, as in the case of a s radius of curvature, as in the case of a homogeneous medium, which is descrihomogeneous medium, which is described by (2.5-14). For the special case of a bed by (2.5-14). For the special case of a homogeneous medium (khomogeneous medium (k22=0), (2.6-4) re=0), (2.6-4) reduces (2.5-4)duces (2.5-4)
)()(
1
)(
12 zn
izRzq
)()(
1
)(
12 zn
izRzq
2.6-5
)(2)(exp
2
2
2
zR
kri
z
r
)(2)(exp
2
2
2
zR
kri
z
r
＝ 1 ， for K2 ＝ 0 ＝ 1 ， for K2 ＝ 0
＝ 0 ， for K2 ＝ 0 ＝ 1 ， for K2 ＝ 0
2
2
2
2
22
0 02
sinsin
sinsinlim 1 lim 1
kk
kk
kk
x k
kk k
k
xt t
x kk
Transformation of the Gaussian Beam—the ABCD Transformation of the Gaussian Beam—the ABCD Law Law
We have derived above the transformation law of a We have derived above the transformation law of a Gaussian beam (2.6-4) propagation through a lenGaussian beam (2.6-4) propagation through a lenslike medium that is characterized by k2, we not slike medium that is characterized by k2, we not first by comparing (2.6-4) to Table 2-1(6) and to (2.first by comparing (2.6-4) to Table 2-1(6) and to (2.3-5) that the transformation can be described by 3-5) that the transformation can be described by
where A,B,C,D are the elements of the ray matrix twhere 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 hat relates the ray (r,r’) at a plane 2 to the ray at plane 1.It follows immediately that the propagatiplane 1.It follows immediately that the propagation through on through
DCq
BAqq
1
12 DCq
BAqq
1
12 2.6-6
2 2
2
2 2 2
0
0
cos sin
sin cos
k kkk k k
k k kk k kq
zqz
z z
Ray matrix a medium with a quadratic index profile:
21
where A,B,C,D are the elements of the ray matrix that relates 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 immethe ray (r,r’) at a plane 2 to the ray at plane 1.It follows immediately that the propagation through.diately that the propagation through.
DCq
BAqq
1
12 DCq
BAqq
1
12 2.6-6
12
1
12
2 11
0 1 1 11 0
111
1
Aq Bq
Cq Dq
qq q fq
ff
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)
fqq
111
12
fqq
111
12
2.6-7
21 1 1
2
22
2
2 2
2
22 2 2
21 1 1 1
22 1
22
1
2
1
1
2
1
1
1 1
1 1
1 1( ) ( ) ( )
1 1
1 1
1 1
1 11 1
1 11
1
iq R n
iq z R z n z
iq R n
andq f f
i iq R f
R f
R n n
in
iq R n
R
q R f
Consider next the propagation of a Gaussian beam thConsider next the propagation of a Gaussian beam through two lenslike media that are adjacent to each otrough two lenslike media that are adjacent to each other. The ray matrix describing the first one is her. The ray matrix describing the first one is (A(ATT,B,BTT,,CCTT,D,DTT)) while that of the second one is (A while that of the second one is (ATT,B,BTT,C,CTT,D,DTT) . T) . Taking the input beam parameter as q1 and the outpuaking the input beam parameter as q1 and the output beam parameter as q3, we have from (2.6-6) t beam parameter as q3, we have from (2.6-6)
Adjacent 邻近的 , 接近的
Output beam
12 3
012
l
DCq
BAqq
1
12 DCq
BAqq
1
12
for the beam parameter at the output of medium 1 and for the beam parameter at the output of medium 1 and
and after combining that last two equations,and after combining that last two equations,
where (Awhere (ATT,B,BTT,C,CTT,D,DTT) are the elements of the ray matrix r) are the elements of the ray matrix relating the output plane (3) to the input plane (1), thaelating the output plane (3) to the input plane (1), that is ,t is ,
TT
TT
DqC
BqAq
1
13
TT
TT
DqC
BqAq
1
13
222
2223 DqC
BqAq
222
2223 DqC
BqAq
It follows by induction that (2.6-9) applies It follows by induction that (2.6-9) applies to the propagation of a Gaussian beam tto the propagation of a Gaussian beam through any arbitrary number of lenslikhrough any arbitrary number of lenslike media and elements. The matrix (Ae media and elements. The matrix (ATT,B,BTT,C,CTT,D,DTT) is the ordered product of the m) is the ordered product of the matrices characterizing the individual meatrices characterizing the individual members of the chainmbers of the chaininduction 感应 , 感应现象 , 归纳
11
11
22
22
DC
BA
DC
BA
DC
BA
TT
TT 11
11
22
22
DC
BA
DC
BA
DC
BA
TT
TT 2.6-10
The great power of the ABCD law is that The great power of the ABCD law is that it enables us to trace the Gaussian beait enables us to trace the Gaussian beam parameter q(z) through a complicatm parameter q(z) through a complicated sequence of lenslike elements. The ed sequence of lenslike elements. The beam radius R(z) and spot size beam radius R(z) and spot size
ωω （（ zz ）） at any plane z can be recovereat any plane z can be recovered through the use of (2.6-5). The applicd through the use of (2.6-5). The application of this method will be made clear ation of this method will be made clear by the following example.by the following example.
Most wanted autographMost wanted autograph 签名签名 Our university newspaper runs a weekly qOur university newspaper runs a weekly q
uestion feature. Recently, the question wauestion feature. Recently, the question was: "Whose autograph would you most want s: "Whose autograph would you most want to have, and why?" As expected, most respto have, and why?" As expected, most responses mentioned music or sports stars, or onses mentioned music or sports stars, or politicians. The best response came from politicians. The best response came from a freshman, who said, "The person who sia freshman, who said, "The person who signs my diploma."gns my diploma."
我们大学的校报开办了一个每周一问的专栏。上我们大学的校报开办了一个每周一问的专栏。上周的问题是：“你最想要什么人的签名？为什周的问题是：“你最想要什么人的签名？为什么？”和预计的一样，大部分的回答都是歌星、么？”和预计的一样，大部分的回答都是歌星、体育明星或者政治家。但是，最优秀的答案来自体育明星或者政治家。但是，最优秀的答案来自一个一年级新生，他说：“在我毕业证上签字的一个一年级新生，他说：“在我毕业证上签字的那个人。”那个人。”
Example : Gaussian beam focusing
As an illustration of the application of the ABCD law, we consider the case of a Gaussian beam that is incident at its waist on a thin lens of focal length f , as shown:
1 2 3
012 Output beam
l
At the plane 1, we have so :01 1,R
2 21 1 01 01
1 1i i
q R n n
2.6-1
22 1 01
1 1 1 1i
q q f f n
So that: 2 2 2
a ibq
a b
The transformation can be discribed by:
2.6-2
2.6-3
in which:
201
1,a b
f n
2.6-4
At plane 3:
3 2 2 2 2 2
a ibq q l l
a b a b
from 2.6-5,we have:
23 3 3
1 1i
q R n
2.6-5
2.6-6
Since plane 3 is, according to the statement of the problem, to correspond to the output beam waist,
using this fact in the last equation leads to
3R
2 2 2 2011 ( / / )
a fl
a b f n
as the location of the new waist, and to
23 01 01
2 2 201 01 01
/ /
1 ( / ) 1 ( / )
w f n f z
w f n f z
2.6-7
2.6-8
for the output beam waist.
201
01
nz
The confocal （ [ 数 ] 共焦的） beam parameter
is, according to (2.5-11), the distance from the waist in which the input beam spot size increases by and is a convenient（便利的 , 方便的）measure of the convergence （会聚） of the input beam. The smaller Z01, the “stronger” the convergence.
2
2.7 A Gaussian beam in lens 2.7 A Gaussian beam in lens waveguidewaveguide
As another example of the application of the AAs another example of the application of the ABCD law, we consider the propagation of a GBCD law, we consider the propagation of a Gaussian beam through a sequence of thin lenaussian beam through a sequence of thin lenses, as shown in Figure 2-2. The matrix, relatses, as shown in Figure 2-2. The matrix, relating a ray in plane s+1 to the plane s=1 is ing a ray in plane s+1 to the plane s=1 is
where (A,B,C,D) is the matrix for propagation where (A,B,C,D) is the matrix for propagation through a single two-lens, unit cell (through a single two-lens, unit cell (△s=1) an△s=1) and is given by (2.1-6). We can use a well-known formd is given by (2.1-6). We can use a well-known formula for the sth ula for the sth
P ： 42
S
TT
TT
DC
BA
DC
BA
S
TT
TT
DC
BA
DC
BA 2.7-1
we can use a well-known formuwe can use a well-known formula for the sth power of a matrix with a unila for the sth power of a matrix with a unity determinant (unimodular) to obtainty determinant (unimodular) to obtain
unity 团结 , 联合 , 统一 , 一致
determinant 行列式 决定性的 Unimodular 【数】幺模的
sin
)1sin()sin(
ssAAT
sin
)1sin()sin(
ssAAT
sin
)sin(sBBT
sin
)sin(sBBT
sin
)sin(sCCT
sin
)sin(sCCT
sin
)1(sin)sin(
ssDDT
sin
)1(sin)sin(
ssDDT
The condition for the confinement of the The condition for the confinement of the Gaussian beam by the lens sequence is, Gaussian beam by the lens sequence is, from (2.7-4), that θbe real; otherwise, thfrom (2.7-4), that θbe real; otherwise, the sine functions will yield growing expoe sine functions will yield growing exponentials. From (2.7-3), this condition benentials. From (2.7-3), this condition becomes cosθ≤1,or comes cosθ≤1,or
That is, the same as condition (2.1-16) for That is, the same as condition (2.1-16) for stable-ray propagation.stable-ray propagation.
confinement (被 ) 限制 , (被 ) 禁闭
determinant 行列式 决定性的 Unimodular 【数】幺模的
12
12
1021
f
d
f
d1
21
210
21
f
d
f
d2.7-5
Open-book examOpen-book exam ON THE DAY of our final exam at my Community CON THE DAY of our final exam at my Community C
ollege in Santa Maria, Calif., we heard that the booollege in Santa Maria, Calif., we heard that the bookstore had changed its policy and would buy back kstore had changed its policy and would buy back our business-management textbooks. Before class, our business-management textbooks. Before class, several of us dashed over to the store and sold our several of us dashed over to the store and sold our books. We were seated and waiting for the test whebooks. We were seated and waiting for the test when our professor announced that considering the difn our professor announced that considering the difficulty of the final, it would be an open-book exam.ficulty of the final, it would be an open-book exam.
我在加利福尼亚的圣玛丽亚市一所社区大学读书。期末考我在加利福尼亚的圣玛丽亚市一所社区大学读书。期末考试那天，听说书店在回购我们的工商管理课本。考试前，试那天，听说书店在回购我们的工商管理课本。考试前，我们几个赶忙跑到书店把书卖了，随后，我们坐在教室里我们几个赶忙跑到书店把书卖了，随后，我们坐在教室里等着考试。这时候教授宣布：考虑到试题的难度，今天的等着考试。这时候教授宣布：考虑到试题的难度，今天的考试我们决定开卷。考试我们决定开卷。
As shown,
Please calculate the
1 , 632.8 , 100o cm nm z m
z
Z=0
o1
Schoolworks
Chapter 3 Propagation of optical beams in fibers
Introduction
Since the realization of low-loss fibers in 1970s, fibers become the most important medium for optics communication.
The technology of optical communication in fibers can be stated as that of feeding(饲养 , 吃 , 输送 ) optical pulses at a maximal rate into one end of a fiber and retrieving(恢复 ) them at the other end. The main goal of a communication system is to receive the pulses at the output end with minimal loss of energy, minimal spread, and minimal contamination(玷污 , 污染 ) by noise. For example, the silica([ 化 ]硅石 )glass fiber ,which has the character of low-loss propagation of confined optical modes, has become the most important transmission medium for long distance.
In this chapter, we will study : 1. the subject of optical guided modes in fibers
2. The problem of pulse spreading due to group velocity dispersion and various strategies of combatting it
3.1 wave equation in cylindrical 3.1 wave equation in cylindrical coordinates(coordinates( 柱坐标柱坐标 ))
In Chaoter 2 we have shown that optical waveguides with a quaIn Chaoter 2 we have shown that optical waveguides with a quadratic index profile (See Equation 2.9-1a) can support guided nodratic index profile (See Equation 2.9-1a) can support guided nondiffracting modes. The effect of diffraction spreading is counterndiffracting modes. The effect of diffraction spreading is counterbalanced by the lensing effort of the index profile of the guide.balanced by the lensing effort of the index profile of the guide.Commercial(商用的 ) silica-based optical fiber use a step index profile with a “high” index core and a “low” index cladding(覆层 ). These fibers form the backbone(脊椎 , 骨干 )of most modern communication systems, and the study of their modes of propagation is the subject at hand.
, , ,r z r zE E E H and H
Since the refractive(折射的 ) index profiles n(r) of most fibers are cylindrically symmetric (柱对称性的 ), it is convenient to use the cylindrical coordinate system. The field components are , the wave equation 2.4-3 assumes its simple form only for the Cartesian(笛卡儿 ) components of the field vectors(矢量 ) 。
n(r)n1
n2
2a
2b
Figure 3-1 Structure and index profile of a step-index circular waveguide
Since the unit vector ar and are not constant vectors, the wave equation involving the transverse( 横向的 ) components are very complicated. The wave equation for the z component of the field vectors, however, remains simple
a
2 2 0Z
Z
Ek
H
3.1-1
, , ,r z r zE E E H and H
The problems of wave propagation in a cylindrical structure are usually approached by solving for Ez and Hz first and then expressing in terms of Ez and Hz
Since we are concerned with the propagation along the waveguide, we assume:
( , ) ( , )exp[ ( )]
( , ) ( , )
t ri t z
t r
E r E
H r H
exp[ ( )]i t z every component of the field vector assume the same z-and t-dependence of .
3.1-2
Maxwell’s curl equations are now written in terms of the cylindrical components and are given by :
3.1-3
3.1-4
From above equation, we get
3.1-5
3.1-6
From the results, we know that these relations show it is sufficient to determine Ez and Hz in order to specify uniquely the wave solution. The remaining components can be calculated from (3.1-5) and (3.1-6)
With the assumed z-dependence of (3.1-With the assumed z-dependence of (3.1-2), the wave equation (3.1-1) becomes2), the wave equation (3.1-1) becomes
This equation is separable, and the solutThis equation is separable, and the solution takes the formion takes the form
where l=0,1,2,3….., so that E and H, are where l=0,1,2,3….., so that E and H, are single-valued functions of . Thensingle-valued functions of . Then
z
z
H
E
rrrr22
2
2
22
2 11
z
z
H
E
rrrr22
2
2
22
2 11
3.1-7
)exp()( ilrH
E
z
z
)exp()( ilr
H
E
z
z
3.1-8
(3.1-7) becomes(3.1-7) becomes
where where Equation (3.1-9) is the Bessel differentEquation (3.1-9) is the Bessel different
ial equation, and the solutions are callial equation, and the solutions are called Bessel functions of order l. If ed Bessel functions of order l. If , the general solution of , the general solution of
(3.1-9) is (3.1-9) is
01
2
222
2
2
r
lk
rrr0
12
222
2
2
r
lk
rrr3.1-9
zzHE zzHE
022 k 022 k
)()()( 21 hrYchrJcr ll )()()( 21 hrYchrJcr ll 3.1-10
where , and are constants, awhere , and are constants, and , are Bessel functions of the firsnd , are Bessel functions of the first and second kind, respectively, of order t and second kind, respectively, of order l. If , the general solution of (3.1-l. If , the general solution of (3.1-9) is 9) is
where , and are constants, where , and are constants, and are the modified Bessel functions oand are the modified Bessel functions of the first and second kind ,respectively ,f the first and second kind ,respectively ,of order l.of order l.
222 kh 222 kh 1c1c 2c2c lJ lJ lYlY
022 k 022 k
)()()( 21 qrKcqrIcr ll )()()( 21 qrKcqrIcr ll 3.1-11
222 kq 222 kq 1c1c 2c2c lI lI lK lK
To proceed with our solution, we need To proceed with our solution, we need the asymptotic forms of these the asymptotic forms of these functions for small and large functions for small and large arguments. Only leading terms will arguments. Only leading terms will be given for simplicity.be given for simplicity.
For x<<1:For x<<1:
asymptotic [ 数 ] 渐近线的 , 渐近的
t
l
x
nxJ
2
1)(
t
l
x
nxJ
2
1)(
5772.0
2ln
2)(
xxYo
5772.0
2ln
2)(
xxYo
Motivation Motivation 动机动机MY ENGLISH PROFESSOR once launched into a lecture MY ENGLISH PROFESSOR once launched into a lecture
on "motivation." "What pushes you ahead?" he asked.on "motivation." "What pushes you ahead?" he asked. "What is it that makes you go to school each day? What "What is it that makes you go to school each day? What driving force makes you strive to accomplish?" Turning driving force makes you strive to accomplish?" Turning suddenly to one young woman, he demanded: "What masuddenly to one young woman, he demanded: "What makes you get out of bed in the morning?" The student replkes you get out of bed in the morning?" The student replied: "My mother."ied: "My mother."
我们英文课的教授有一次在课上讲“动机”。“是我们英文课的教授有一次在课上讲“动机”。“是什么推动你在人生的路上向前走？”他问道，“是什么推动你在人生的路上向前走？”他问道，“是什么让你每天上学来？又是什么驱使你追求成什么让你每天上学来？又是什么驱使你追求成功？”冲着一个女学生，他问：“是什么让你早晨功？”冲着一个女学生，他问：“是什么让你早晨从床上爬起来的呢？”学生答道：“我妈妈。”从床上爬起来的呢？”学生答道：“我妈妈。”
The geometry of the step-index circular wavegThe geometry of the step-index circular waveguide is shown in Figure 3-1. It consists of a cuide is shown in Figure 3-1. It consists of a core of refractive index and radius a, and a ore of refractive index and radius a, and a cladding of refractive index and radius b. Thcladding of refractive index and radius b. The radius b of the cladding is usually chosen te radius b of the cladding is usually chosen to be large enough so that the field of confineo be large enough so that the field of confined modes is virtually zero at r=b. In the calculd modes is virtually zero at r=b. In the calculation below we will put b= ; this is a legitiation below we will put b= ; this is a legitimate assumption in most waveguides, as far mate assumption in most waveguides, as far as confined modes are concerned.as confined modes are concerned.
3.2 the step-index circular waveguide3.2 the step-index circular waveguide
legitimate 合法的 , 合理的
1n1n 2n2n
n(r)n1
n2
2a
2b
Figure 3-1 Structure and index profile of a step-index circular waveguide
The radial dependence of the fields EThe radial dependence of the fields Ezz. and H. and Hzz. ,is . ,is given by(3.1-10) or (3.1-11) depending on the siggiven by(3.1-10) or (3.1-11) depending on the sign of .For confined propagation , must n of .For confined propagation , must be larger than (i.e. ). This ebe larger than (i.e. ). This ensures that the wave is evanescent in the claddinsures that the wave is evanescent in the cladding region, r>a. The solution is thus given by(3.1-ng region, r>a. The solution is thus given by(3.1-11) with c11) with c11=0 .This is evident from the asymptoti=0 .This is evident from the asymptotic behavior for large r given by (3.1-13). The evanc behavior for large r given by (3.1-13). The evanescent decay of the field also ensures that the poescent decay of the field also ensures that the power flow is along the direction of the z axis, i.e., wer flow is along the direction of the z axis, i.e., no radial power flow exists. Thus the fields of a cno radial power flow exists. Thus the fields of a confined mode in the cladding (r>a) are given byonfined mode in the cladding (r>a) are given by
evanescent 渐消失的 , 易消散的
22 k 22 k
lcn 2 lcn 2 lcnkn 202 lcnkn 202
where C and D are two arbitrary constawhere C and D are two arbitrary constants, and q is given by nts, and q is given by
For the fields in the core , r<a, we must consider the For the fields in the core , r<a, we must consider the behavior of the fields as ,According to (3.1-1behavior of the fields as ,According to (3.1-12), Y2), Yll and K and Kll are divergent as .Since the fields are divergent as .Since the fields must remain finite at r=0, the proper choice for the must remain finite at r=0, the proper choice for the fields in the core (r<a) is (3.1-10) with cfields in the core (r<a) is (3.1-10) with c22=0. This bec=0. This becomes evident only when matching, at the interface omes evident only when matching, at the interface r=a, the tangential components of the field vectors r=a, the tangential components of the field vectors E and H in the core with the cladding field componeE and H in the core with the cladding field components derived from (3.2-1); we are unable to accomplisnts derived from (3.2-1); we are unable to accomplish this if the radial dependence of the core fields is gh this if the radial dependence of the core fields is given by Iiven by Ill. Thus the propagation constant must b. Thus the propagation constant must be less than , and the core fields are given bye less than , and the core fields are given by
divergent 分歧的tangential 切线的
0r 0r0r 0r
01kn 01kn
In the field expressions (3.2-1) and (3.2-3), we havIn the field expressions (3.2-1) and (3.2-3), we have taken a “+” sign in front of in the expone taken a “+” sign in front of in the exponents. A negative sign would yield a set of in depeents. A negative sign would yield a set of in dependent solutions, but with the same radial depenndent solutions, but with the same radial dependence. Physically, l plays a role similar to the qudence. Physically, l plays a role similar to the quantum number describing the z component of tantum number describing the z component of the orbital angular momentum of an electron in he orbital angular momentum of an electron in a cylindrically symmetric potential field. Thus, ia cylindrically symmetric potential field. Thus, if the positive sign in front of f the positive sign in front of
corresponds to a clockwise “circulation” of phcorresponds to a clockwise “circulation” of photons about the z axis, the negativeotons about the z axis, the negativeorbital 轨道的 ,potential 潜在的 , 可能的 , 势的 , 位的
ll
ll
sign would corresponds to a clockwise “cirsign would corresponds to a clockwise “circulation” of photons around the axis. Sincculation” of photons around the axis. Since the fiber itself does not possess any prefee the fiber itself does not possess any preferred sense of rotation, these two states are rred sense of rotation, these two states are degenerate.degenerate.
Equations (3.2-1) and (3.2-3) together reqEquations (3.2-1) and (3.2-3) together require that and , which translates uire that and , which translates to to
which can be regarded as a necessary conditwhich can be regarded as a necessary condition for confined modes to exist. This is ion for confined modes to exist. This is
preferred 首选的
02 h 02 h 02 q 02 q
0201 knkn 0201 knkn 3.2-5
which can be regarded as a necessary which can be regarded as a necessary condition for confined modes to exist. This is condition for confined modes to exist. This is identical to the condition discussed in Section identical to the condition discussed in Section 13.1 for the slab dielectric waveguide and 13.1 for the slab dielectric waveguide and can be expected on intuitive grounds from can be expected on intuitive grounds from our discussions of total internal reflection at a our discussions of total internal reflection at a dielectric interface.dielectric interface.
Using (3.2-) and (3.2-3) in conjunction Using (3.2-) and (3.2-3) in conjunction with (3.1-5) and (3.1-6), we can calculate all with (3.1-5) and (3.1-6), we can calculate all the field components in both the cladding and the field components in both the cladding and the core regions. The resultthe core regions. The resultslab 厚平板 , 厚片
The result isThe result is
The result isThe result is
Core ( r<a):Core ( r<a):
Nontrivial 非平凡的transcendental 超越的
The quantity B/A is of particular interest The quantity B/A is of particular interest because it is measure of the relative abecause it is measure of the relative amount of Emount of Ez z and Hand Hzz in a mode (i.c., B/A in a mode (i.c., B/A== HHzz// EEzz ). Note that E ). Note that Ez z and Hand Hzz are out of are out of phase by phase by
New business was opening New business was opening 开业大吉开业大吉
A new business was opening ... and one of the owner's friends wanA new business was opening ... and one of the owner's friends wanted to send him flowers for the occasion. They arrived at the new ted to send him flowers for the occasion. They arrived at the new business site and the owner read the card,.... "Rest in Peace." The business site and the owner read the card,.... "Rest in Peace." The owner was angry and called the florist to complain. After he had towner was angry and called the florist to complain. After he had told the florist of the obvious mistake and how angry he was, the flold the florist of the obvious mistake and how angry he was, the florist replied, "Sir, I'm really sorry for the mistake, but rather thaorist replied, "Sir, I'm really sorry for the mistake, but rather than getting angry, you should imagine this: somewhere, there is a fun getting angry, you should imagine this: somewhere, there is a funeral taking place today, and they have flowers with a note saying,neral taking place today, and they have flowers with a note saying, ... 'Congratulations on your new location!'" ... 'Congratulations on your new location!'"
新公司开业了，开业典礼上，经理的一个朋友送他一个花篮。经新公司开业了，开业典礼上，经理的一个朋友送他一个花篮。经理高声朗读着花篮上的贺卡：“安息吧。”经理生气极了，打电理高声朗读着花篮上的贺卡：“安息吧。”经理生气极了，打电话找来卖花的人要质问他是怎么回事。花店老板来了，看到这个话找来卖花的人要质问他是怎么回事。花店老板来了，看到这个明显的错误和经理气急败坏的样子，他说：“我真得很抱歉。但明显的错误和经理气急败坏的样子，他说：“我真得很抱歉。但是与其这么生气，你倒不如这样想：有另外一个地方，今天要举是与其这么生气，你倒不如这样想：有另外一个地方，今天要举办一个葬礼，他们将会收到一个花篮，留言条上写着‘恭喜你有办一个葬礼，他们将会收到一个花篮，留言条上写着‘恭喜你有了新的归属！’”了新的归属！’”
3.2 the step-index circular waveguide3.2 the step-index circular waveguide
n(r)n1
n2
2a
2b
Figure 3-1 Structure and index profile of a step-index circular waveguide
nn11: : core of refractive index ;
n2: cladding of refractive index
a, b: radius. the radius b of the cladding is usually chosen to be large enough so that the field of confined modes is virtually zero at r=b. in the calculation below we will put b=∞ ; this is a legitimate assumption in most waveguides, as far as confined modes are concerned.
The radial dependence of the fields Ez and Hz depending on the sign of .
2 2k
For confined propagation, this ensures that the wave is evanescent(逐渐消失的 ) in the cladding region, r>a. This is evident from the asymptotic( 渐进的 ) behavior for large r given by 3.1-13, the evanescent decay of the field also ensures that the power flow is along the direction of the z axis. i.e., no radial power flow exists.
2 /n c
For the field in the core, r<a, we must consider the behavior of the fields as r → o. And 1 0n k
Schoolwork:
1. Derive the equation 3.1-3 – 3.1-6
2. Translate section 3.2 into Chinese
?
?
??
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.
Example: One-Dimensional Example: One-Dimensional ResonatorResonator
sin( ) 0
1,2,3.......m mk l k l m
m
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)
(4.0-8)
( , , ) sin sin sinx y zE x y z k x k y k z
, ,x y z
r s tk k k
a b c
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 desription of modes. Every positive triplet of integers r,s,l defines a unique mode. We can thus associate a primitive volume π3/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
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 for her as she had beeHer life had been a terrible ordeal for her as she had been plagued with lots of different types of cancer. One day, n plagued with lots of different types of cancer. One day, my parents took me to see her in the nursing home in whmy parents took me to see her in the nursing home in which she lived. It was quite scary as the cancer had started ich she lived. It was quite scary as the cancer had started to show and you could see tumors on her face and arms. to show and you could see tumors on her face and arms. She seemed to just be wasting away, but my parents wanShe seemed to just be wasting away, but my parents wanted me to see her before she died.ted 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 telephone.ng on my pretend 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。
2.10 propagation in media with a quadratic gain pro2.10 propagation in media with a quadratic gain profilefile
In many laser media the gain is a strong fuction of pIn many laser media the gain is a strong fuction of position. This variation can be due to a variety of cauosition. This variation can be due to a variety of causes, among them: (1) the radial distribution of energses, among them: (1) the radial distribution of energetic electrons in the plasma region of gas lasers etic electrons in the plasma region of gas lasers ［［ 1199］］ , (2) the variation of pumping intensity in solid , (2) the variation of pumping intensity in solid state lasers, and (3) the dependence of the degree of state lasers, and (3) the dependence of the degree of gain saturation on the radial position in the beam.gain saturation on the radial position in the beam.
function 功能 , 作用 , 职责 , 典礼 , 仪式 , [ 数 ]函数
among them: 这些因素包括：
radial 光线的 , 光线状的 , 放射状的 , 半径的
energetic electron ：精力充沛的电子？高能电子
dependence 依靠 , 依赖 , 相关
We can account for an optical medium with quadratiWe can account for an optical medium with quadratic gain (or loss) vatiation by taking the complex proc gain (or loss) vatiation by taking the complex propagation constant k(r) in (2.4-5) as pagation constant k(r) in (2.4-5) as
where the plus (minus) sign applies to the case of gaiwhere the plus (minus) sign applies to the case of gai
n (loss). Assuming kn (loss). Assuming k22rr22<< k in (2.4-5), we have k<< k in (2.4-5), we have k22=la=la22.. Using this value in (2.4-11) to obtain the steady-stat Using this value in (2.4-11) to obtain the steady-state((1/q)’=0) solution of the complex beam radius yie((1/q)’=0) solution of the complex beam radius yieldselds66
account for 做出解释，提出理由the complex propagation constant 复数传播常数gain （ loss ） 增益（损耗）the steady－ state solution 稳态解
)2
1()( 2
20 raaikrk )2
1()( 2
20 raaikrk 2.10-1
k
iai
k
ki
q221
k
iai
k
ki
q221
2.10-2
The steady-state beam radius and spot sizThe steady-state beam radius and spot size are obtained from (2.6-5) and (2.10-2) e are obtained from (2.6-5) and (2.10-2)
We thus find that the steady-state solutioWe thus find that the steady-state solution corresponds to a beam with a constann corresponds to a beam with a constant spot size but with a finite radius of curt spot size but with a finite radius of curvature.vature.
2
2 2na
2
2 2na
2
2a
nR
2
2a
nR
2.10-3
The general (non-steady-state) beThe general (non-steady-state) behavior of the Gaussian beam in a quadhavior of the Gaussian beam in a quadratic gain medium is described by (2.ratic gain medium is described by (2.6-4), where k6-4), where k22=la=la22.. Experimental data showing a decr Experimental data showing a decrease of the beam spot size with increaease of the beam spot size with increasing gain parameter a2 in agreement sing gain parameter a2 in agreement with (2.10-3) are shown in Figure 2-9.with (2.10-3) are shown in Figure 2-9.
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 modFigure 4-2(a) Multiple reflections model for analyzing the Fabry-Perot etalon el for analyzing the Fabry-Perot etalon
' 'sin sinn CD n CD
' 'sin sinn CD n CD
' 'sin sinn CD n CD
D
' 'sin sinn n
E
F
'n AD n CE
' 'sin sinn CD n CD
cos 2
cos 2cos
AB BC
l
cos
lBC
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 free These are separated by the so-called free spectral rangespectral 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 etaTheoretical transmission plots of a Fabry-Perot etalon are shown in Figure 4-3. The maximum transmislon 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.
If we allow for the existence of losses in the elaton 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 Fabry-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, a visitor asked the director ...,During a visit to the mental asylum, a visitor asked the director ..., "What is the criterion that defines a patient to be institutionalize "What is the criterion that defines a patient to be institutionalized?" "Well..." said the director, "we fill up a bathtub, and we offer d?" "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 ea teaspoon, a teacup, and a bucket to the patient and ask him to empty the bathtub." "Oh, I understand," said the visitor. "A normmpty the bathtub." "Oh, I understand," said the visitor. "A normal person would choose the bucket as it is larger than the spoon or al person would choose the bucket as it is larger than the spoon or the teacup." "Noooooooo!" answered the director. "A normal perthe teacup." "Noooooooo!" answered the director. "A normal person would pull the plug." son 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，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”。 把浴缸塞子拔掉”。
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
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
2
2 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
1 (4.1 10)2 cosm m
cv v v
nl
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=100cm and the region of oscillation= ν△ gain≈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
Hznl
c
etal
9105.12
Hznl
c
etal
9105.12
cmnletal 202 cmnletal 202 or 4.2-7
The separation between longitudinal modes of the laser oscillation is c/2nllaser = 1.5×108 Hz (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 1.5×107Hz can be resolved. According to (4.2-6), this resolution can be achieved if
△υ1/2=C/2nletalF≤ 1.5×107 Hz or
2nletalF≥2×103cm
1
RF
R
As a practical note we may add that the finess, as defined by the first equality 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.
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
Another important mode of optical spectrum analysAnother important mode of optical spectrum analysis performed with Fabry-Perot etalons involves the fis performed with Fabry-Perot etalons involves the fact that a noncollimated monochromatic beam inciact that a noncollimated monochromatic beam incident on the etalon will emerge simultaneously, accodent on the etalon will emerge simultaneously, according to (4.1-8), along many directions θrding to (4.1-8), along many directions θ33, which cor, which corresponds to the various order m. If the output is theresponds to the various order m. If the output is then focused by a lens, each such directionθ will give rin focused by a lens, each such directionθ will give rise to a circle in the focal plane on the lens, and, therse to a circle in the focal plane on the lens, and, therefore, each frequency component present in the beefore, each frequency component present in the beam leads to a family of circles. This mode of spectruam leads to a family of circles. This mode of spectrum analysis is especially useful under transient condm analysis is especially useful under transient conditions where scanning etalons cannot be employed. itions where scanning etalons cannot be employed. Further discussion of this topic is include in ProbleFurther discussion of this topic is include in Problem 4-6.m 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 ).
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 locus 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.
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)
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
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
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)\
No rush No rush 别着急别着急
AS STUDENTS in the college of veterinary medicine at AS STUDENTS in the college of veterinary medicine at Texas A & M University, we frequently treated the farTexas A & M University, we frequently treated the farm animals at the state prison. While awkwardly perfom animals at the state prison. While awkwardly performing a medical procedure on an unruly horse, a clasrming a medical procedure on an unruly horse, a classmate said to the prisoner who was holding the animasmate said to the prisoner who was holding the animal, "Sorry I'm taking so long." "No problem," the prisonl, "Sorry I'm taking so long." "No problem," the prisoner replied, "I'm doing seven years.er replied, "I'm doing seven years.
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。”
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。’”
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.
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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.
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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
Shcoolwork:Shcoolwork: please summarize the most common please summarize the most common
loss mechanisms in optical resnoator aloss mechanisms in optical resnoator and write them in both English and Chind write them in both English and Chinese.nese.
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photomultiplier
photodiodephotodiode
Avalanche photodiode