Chapter 2 the propagation of rays and beams

• Introduction: In this chapter , we will take up the subject

of optical ray propagation through a variety of optical media . These include homogeneous ( 均匀的 ) and isotropic ( 各向同性的 ) materials, for example, thin lenses( 薄透镜 ) ， dielectric interfaces ( 电介质界面 ),and curved mirrors ( 曲面镜 )

• Since a ray is, by definition, normal to the optical wavefront, an understanding of the ray behavior makes it possible to trace the evolution of optical wave when they are passing through various optical elements.

• We find that the passage of a ray(or its reflection) through these elements can be described by simple 2*2 matrices.

• Furthermore, these matrices will be found to describe the propagation of spherical waves and of Gaussian beams such as those characteristic of the output of lasers. Gaussian bean propagation is analyzed in second half of the chapter

2.1 Lens waveguide

ri r’i

ro r’o

r’if

A paraxial( 近轴的 ) ray passing through a thin lens of focal length ( 焦距 ) f is shown as:

Figure 2-1 Deflection of a ray by a thin lens

out in

out in in

r = r

r' = r' -

r /f

(2.1

-1)

'out out(r , r )

'in in(r , r )

Taking the cylindrical ( 圆柱的 ) axis of symmetry as z, denoting the ray distance from the axis by r and its slope( 斜率 ) dr/dz as r′, we can relate the output ray to the input ray by means of

Where the first term follows from the definition of a thin lens and the second can be derived from a consideration of the behavior of the undeflected( 不偏斜的 )central ray with a slope equal to .'

inr

' 'o i ir f r f r

ri r’i

ro r’o

r’if

-r’of

光轴

光线 1,r’ 为正

光线２ ,r’ 为负Focal

plane

f' ' i

o i

rr r

f

r(z)

r’(z)r(z)=

Representing a ray at any position z as a column matrix ,

From above , we can rewrite 2.1-1 using the rules for matrix multiplication as

Where f>0 for a converging( 会聚的 )lens and is negative( 负数 )for a diverging( 使（如光线）发散 ,使偏离 )one.

' '

1 0

1/ 1out in

out in

r r

fr r

2.1-2

Example 1:

1. As shown by figure 2.2, the propagation of a ray through a straight( 直的 ) section of a homogeneous( 均匀的 ) medium of length d followed by a thin lens of focal length f , this corresponds to propagation between planes a and b in figure 2.2.

a b d

f

Figure 2.2

Letter to God• A little boy needed $50 very badly and prayed for weeks,

but nothing happened. Then he decided to write God a letter requesting the $50. When the post office received the letter to God, USA, they decided to send it to the President. The president was so amused( 开心的 )that he instructed( 指示 )his secretary( 秘书 )to send the boy a $5.00 bill. The president thought this would appear to be a lot of money to a little boy. The little boy was delighted( 欣喜的 )with the $5.00 bill and sat down to write a thank-you note to God, which read: Dear God: Thank you very much for sending the money. However, I noticed that for some reason you sent it through Washington, D.C., and, as usual( 惯例 ), those turkeys( 笨蛋，傻瓜 ) kept $45 in taxes.

Since the effect of the straight section is merely that of increasing r by dr′, we can relate the output b and input (at a ) rays by:

' '

'

1 0 1

1/ 1 0 1

1

1/ (1 / )

out in

out in

in

in

r rd

fr r

rd

f d f r

2.1-3

2. As shown by figure 2.3, consider the propagation of a ray through a biperiodic ( 两个的，两片的 ) lens systems made up of lenses of focal lengths f1 and f2 separated by d:

a b plane s+1

f1 f2f1 f1

f2

plane s

Figure 2.3

The matrix relating the ray parameters at output of a unit cell to those at the input is:

1

'1 1 2 2 11

1

1 1

1/ (1 / ) 1/ (1 / )

A

s s

ss

s

s

r rd d

f d f f d f rr

rB

C D r

＝2.1-4

Where A, B, C, D are the elements of the matrix resulting from multiplying the two square matrices.

In equation form, we can get:

2.1-5's+1 s s

s s s

r = Ar + Br'

r +1= Cr + Dr

2

2

1 1 2

1 1 2

1 /

(2-d/f )

-[1/ (1- / ) / ]

[ / (1 / )(1 / )]

A d f

B d

C f d f f

D d f d f d f

2.1-6

1

' '

s s+2 s+1

s s s

r = (r - Ar ) B

r +1= Cr + Dr

' 1s s+1 s r = (r - Ar )

B

s+2 s+1 s+1 s r - Ar = BCrs+ D(r - Ar )

1's+2 s+1 s s(r - Ar ) = Cr +Dr

B'

' 1ss+2 s+1 s

s s+1 s

r - Ar = BCr + BDr

r = (r - Ar )B

as a result:

s+2 s+1 sr -(A+ D)r +(AD - BC)r = 0 2.1-9

'1

1s s+2 s+1 r = (r - Ar )

B

From equation 2.1-5:

AD-BC=1

Because:

( ) (0)exp[ ]r z r i Gz

2 1

22 1 1 2

2 0

1( ) (1 / / / 2 )

2

s s sr br r

b A D d f d f d f f

2.1-10

And then 2.1-8 can be rewrited as :

'' 0r Gr

Then

We are thus led to try a solution in the form of

0isq

sr r e

when substituted in 2.1-9, leads to

2 2 1 0iq iqe be and therefore :

21iq ie b i b e

2.1-10

2.1-12

2.1-13

( 2) ( 1)0 0 02 0i s q i s q isqr e br e r e

1 2? ?s sr r ( 1) ( 2)

1 0 2 0i s q i s q

s sr r e r r e

So we have :

0is

sr r e

from above, we know that the general solution can be taken as a linear superposition of andexp( )is exp( )is

the condition for a stable( 稳定的 )ray is that be a real number the necessary and sufficient condition for to be real is that:

1b

2.1-13: coswhere b

Example 2: Identical( 同样的 )-lens waveguide

The simplest case of a lens waveguide is one in which f1=f2=f ; that is, all lenses are identical.

The analysis of this situation is considerably simpler than that used for a biperiodic lens sequence. The reason is that the periodic unit cell (the smallest part of the sequence that can, upon translation( 平移 ),recreate the whole sequence) contains a single lens only. The (A,B,C,D) matrix for the unit cell is given by the square matrix in (2.1-3). Following exactly the steps leading to (2.1-11) through (2.1-14), the stability condition becomes:

0 4d f 2.1-18

A 1

1/ (1 / )

B d

C D f d f

1 2 /( )

2 2

d fb A D

1b

the condition for a stable( 稳定的 )ray is that

2 /1

2

d f 2 /

1 12

d f

And the beam radius at the nth lens is :

max sin( )

cos (1 / 2 )nr r n

d f

2.1-19

Algebraic代数的Derivation由来 , 起源

The stability criteria( 标准 ) can be demonstrated experimentally by tracing( 追踪 )the behavior of a laser beam as it propagates down a sequence of lenses spaced uniformly. One can easily notice the rapid escape of the beam once condition (2.1-18) is violated( 违反或藐视 ).

2.2 propagation of rays between mirrors

formalism 形式主义，法则 bouncing 跳跃的 curved 成曲形radius 半径 curvature 曲率 fold 折叠

Another important application of the fonltalism just veloped concems the boancing of a ray betweent wocurved mirrors Sincethe reflection at a mirror with a radiusof curvature R is equivalent ． except for the folding of the path ． to passage through a lens with a focaIlength f=R/2 ． we Can use the fomlaIism of the preceding section to describe the propagation of a ray between two curved reflectors with radii of curvatare R1 and R2 ， which are separated by d.Let us consider the simple case of a ray mat is iniected into a symmetric two—mirror systerm as shown in Figure 2.3(a) ．Since the x and y coordinares of the ray are independent variables ． we can take them according to (2.1-19)in the foml of

Since the x and y coordinates of the ray are independent variables, so:

max

max

sin( )

sin( )n x

n y

x x n

y y n

2.2-1

Parameter 参数 , 参量 immediately 立即 , 马上locus 地点，所在地 , [ 数 ] 轨迹 ellipse [ 数 ] 椭圆 , 椭圆形

If in (2.2-1) satisfies the condition :

wher v and l are any two integers ,A ray will return to its starting point following v round trips and will thus continuously retrace the same pattern on the mirrors.

2 2l

Example: reentrant(Example: reentrant( 重新进入的；凹入的重新进入的；凹入的 ) rays: ) rays: 重返光线重返光线

2.2-2

• trip ( 短途 ) 旅行 , 往返 , 差错 ,

• symmetric 相称性的 , 均衡的

• Confocal [ 数 ] 共焦的

If we consider as an example the simple case of L=1,v=2,so that ， from(2.1-19)we obtain d=2f=R;that is,if the mirrors are separated by a distance equal to their radius of curvature R,the trapped ray will retrace its pattern after two round trips(v=2).This situation(d=R)is referred to as symmertic confocal,since the two mirrors have a common focal point f=R/2,it will be discussed in detail in Chapter4.The ray pattern corresponding to v=2 is illustrated in Figure 2-3(b)

2/

2.3 Rays in lenslike media

responsible 有责任的 , 可靠的 , 可依赖的 quadratic 二次的account for 说明 , 解决 , 得分

The basic physical property of lenses that is responsible for their focusing action is the fact that the optical path across them (where n is the index of refraction of the medium) is a quadratic function of the distance r from the z axis. Using ray optics,we account for thes fact by a change in the ray’s slope as in(2.1-1).This same property can be represented by relating the complex field amplitude of the incident optical field immediately to the right of an ideal thin lens to that immediately to the left by

(2.3-1)Where is the focal length and

dzzrn )( 、

),( yxR),( yxL

]2

exp[),(),(22

1 f

yxikyxEyxR

The effect of the lens, therefore, is to cause a phase shift :

which increase quadratically ( 二次的 ) with the distance from the axis.

We consider next the closely related case of a medium whose index of refraction( 折射率 ) n varies according to

2 2( ) / 2k x y f 2-3-1

2 220( , ) [1 ( )]

2

kn x y n x y

k 2-3-2

Where k2 is a constant.

Since the phase delay of a wave propagation through a section dz of a medium with an index of refraction n is

It follows directly that a thin slab （厚平板 , 厚片） of the medium described by (2.3-2) will act as a thin lens,

(2 / )odz n

Introducing a phase shift proportional to 2 2( )x y The behavior of a ray in this case is described by the differential equation that applied to ray propagation in an optically inhomogeneous ( 不均匀的 )medium:

( )d dr

n nds ds

2-3-3

where s is the distance along the ray measured from some fixed position on it and r is the position vector of the point at s.

For paraxial rays, we may replace d/ds by d/dz, then

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

or'0r

22

2( ) 0kd r

rdz k

2.3-4

From the equation, we know the ray oscillates ( 振荡 ) back

and forth across the axis., as shown in Figure 2-4. A section of the quadratic index medium acts as a lens

'2 20 0

2

'2 2 20 0

( ) cos sin

'( ) sin cos

k kkr z z r z r

k k k

k k kr z z r z r

k k k

2.3-5

( / 2) 2 ( / 2) 2

( / 2) 2 ( / 2) 2

( / 4) 2 ( / 4) 2

( / 4) 2 ( / 4)

cos sin sin sin(2 )2( , ) , / 2,

sin sin(2 ) sin cos2

cos sin sin sin(2 )4( , )

2sin sin(2 ) sin c

4

i i

i i

i i

i i

e e iW for quarter waveplate so

i e e

e e iW

i e e

2

2 2

2 2

os

cos( ) sin( ) cos cos( ) sin( ) sin sin sin(2 )4 4 4 4 4

sin sin(2 ) cos( ) sin( ) sin cos( ) sin( ) cos4 4 4 4 4

1 cos 2 sin(2 )2

sin(2 ) 1 cos 22

i i i

i i i

i i

i i

Answer for p35 1.8

/

1 cos 2 sin(2 ) 02

sin(2 ) 1 cos 2 12

sin(2 ) sin(2 )2 2

1 cos 2 cos 22 2 y

y

i iWV

i i

i iV

i i

/

imaginary

2 2imaginary

sin(2 ) ( cos(2 ))

sin(2 ) 1 cos(2 )

sin(2 ) 1:

cos(2 )

: 1 1

y

real

real

represents V by a complex number results in

complex number x yi i i

i

xdenote

y i

then x y

2.3 Rays in lenslike media

responsible 有责任的 , 可靠的 , 可依赖的 quadratic 二次的account for 说明 , 解决 , 得分

The basic physical property of lenses that is responsible for their focusing action is the fact that the optical path across them (where n is the index of refraction of the medium) is a quadratic function of the distance r from the z axis. Using ray optics,we account for thes fact by a change in the ray’s slope as in(2.1-1).This same property can be represented by relating the complex field amplitude of the incident optical field immediately to the right of an ideal thin lens to that immediately to the left by

(2.3-1)Where is the focal length and

dzzrn )( 、

),( yxR),( yxL

]2

exp[),(),(22

1 f

yxikyxEyxR

The effect of the lens, therefore, is to cause a phase shift :

which increase quadratically ( 二次的 ) with the distance from the axis.

We consider next the closely related case of a medium whose index of refraction( 折射率 ) n varies according to

2 2( ) / 2k x y f 2-3-1

2 220( , ) [1 ( )]

2

kn x y n x y

k 2-3-2

Where k2 is a constant.

Since the phase delay of a wave propagation through a section dz of a medium with an index of refraction n is

It follows directly that a thin slab （厚平板 , 厚片） of the medium described by (2.3-2) will act as a thin lens,

(2 / )odz n

Introducing a phase shift proportional to 2 2( )x y The behavior of a ray in this case is described by the differential equation that applied to ray propagation in an optically inhomogeneous ( 不均匀的 )medium:

( )d dr

n nds ds

2-3-3

where s is the distance along the ray measured from some fixed position on it and r is the position vector of the point at s.

For paraxial rays, we may replace d/ds by d/dz, then

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

or'0r

22

2( ) 0kd r

rdz k

2.3-4

From the equation, we know the ray oscillates ( 振荡 ) back

and forth across the axis., as shown in Figure 2-4. A section of the quadratic index medium acts as a lens

'2 20 0

2

'2 2 20 0

( ) cos sin

'( ) sin cos

k kkr z z r z r

k k k

k k kr z z r z r

k k k

2.3-5

• This can be proved by showing,using(2.3-5),that a family of

parallel rays entering at z=0 at different radii will converge upon

emerging at to a common focus at a distance

(2.3-6)

From the exit plane,The factor accounts for the refraction at

the boundary.assuming the medium at to possess an

index and a small angle of incidence.The denvation

of(2.3-6)is left as an exercise(Problem2.3)

converge 会聚 account for 说明 , 解决 , 得分boundary 边界 , 分界线 possess 占有 , 拥有

lz

l

k

k

k

k

nh 2

20

cot1

0nlz

1n

• Equations(2.3-5)apply to a focusing medium with .In a medium where that is,where the index increases with the distance from the axis—the solutions for and become

So that increases with distance and eventually escapes.A section of such a medium acts as a negative lens

eventually 最后 , 终于 escape 逃 , 逃亡

negative 否定 , 负数

02 k 02 k

zr zr '

'0

2

20

2 )(sincosh rzk

kth

k

krz

k

kzr

'0

20

22' )cosh()(sin)( rzk

krz

k

kth

k

kzr

)(zr

Physical situations giving rise to quadratic index variation include:

1.Propagation of laser beams with Gaussian-like intensity profile in a slightly absorbing medium.The absorption heating gives rise,because of the dependence of on the temperature T,to an index profile[9].If ,as is the case for most materials,the index is smallest on the axis where the absorption heating is highest.This corresponds to a in(2.3-2) and the beam spreads with the distance .If ,as in certain lead glasses[10],the beams are focused.

2.In optically pumped solid-state laser rods,the portion of the absorbed pump power that is not converted to laser radiation is conducted as heat to the rod surface.This heat conduction requires a temperature gradient in which T is maximum on axis.

Gaussian ( 德国数学家 ) 高斯的 profile 剖面 , 侧面 , 轮廓

absorbe 吸收 , 吸引 temperature 温度

spread 伸展 , 展开 , 传播 rod 杆 , 棒 require 需要 , 要求

0/ dTdn

02 k0/ dTdn

n

dielectric 电介质 , 绝缘体 sandwich 三明治 , vt. 夹入中间

injection 注射 , 注射剂 cladd 在金属外覆以另一种金属

sheath 护套 , 外壳 pipe 管 , 导管

The dependence of n on T then gives rise to a positive lens effect for and a negative lens for 3.Dielectric waveguides made by sandwiching a layer of index between two layers with index This situation will be discussed further in Chapter 7 in connection with injection lasers.4.Optical fibers produced by cladding a thin optical fiber(whose radius is comparable to ) of an index with a sheath of index .Such fibers are used as light pipes.

0/ dTdn0/ dTdn

1n12 nn

1n 12 nn

• Optical waveguides consisting of glasslike rods or filaments,with radii large compared to ,whose index decreases with increasing Such waveguides can be used for the simultaneous transmission of a number of laser beams,which are injected into the waveguide at different angles,It follows from(2.3-5)that the beams will emerge,each along a unique direction,and consequently can be easily separated.Furthermore,in view of its previously discussed lens properties,the waveguide can be to transmit optical image information in much the same way as images are transmitted by a multielement lens systern to the image plane of a camera[11].

filament 细丝 , 灯丝

image 图象 , 肖像

multielement 多元素 , 多元件

r

Marry in the heaven A young couple 夫妇 was on their way to get married when they had an accident and died. Now they were in front of St. Peter 圣徒彼得 and the young lady asked if they could get married. St. Peter told them, he would have to get back to them with an answer. Around 30 days later St. Peter returns and tells the couple that they can get married in heaven. The young lady then asks St. Peter, “If things just don't work out can we get a divorce?" St. Peter looks at her and replies, " Lady it took me 30 days to find a preacher up here do you really think I am going to find a lawyer?!!"

一对年轻的夫妇在去结婚的路上出了车祸，双双死去了。于是，他们来到了面前，妻子问是否她还可以和丈夫结婚，圣徒彼得告诉他们，关于这个问题他一有了结果就会回来找他们。差不多 30 天以后，圣徒彼得回来了，并且告诉他们可以在天堂结婚。妻子又问：“如果生活的不愉快，我们可不可以离婚呢？”圣徒彼得看着她，回答说：“夫人，我花了 30 天才找到个传教士，难道你真的希望我再去找个律师吗？”