物理科学与技术学院

Chapter 5

Polarization

5.1 polarized light and natural light 、 degree of polarization 　　　　　　　　　　　　　　　　　　　　　　　　　　　偏振光与自然光、偏振度5.2 polarization by reflection and refraction 、 Brewster’s law 由反射和折射获得偏振光　布儒斯特定律5.3 double refraction in uniaxial crystal 单轴晶体的双折射现象 5.4 Huygen’s explanation of double refraction by wavefront 用波面的概念解释双折射现象5.5 polarization prism and analyser Malus law 偏振棱镜和偏振片 马吕斯定律5.6 elliptically and circularly polarized light wave plate 椭圆偏振光与圆偏振光 波片

5.8 detection of the state of polarization 偏振态的实验检验5.9 interference of polarized light 偏振光的干涉

5.10 photoelastic effect and electro-optical effect 光弹性效应和电光效应5.11 rotation of the plane of linearly polarized light 线偏振光沿晶体光轴传播时振动面的旋转

5.1 polarized light and natural light 、degree of polarization

transverse wave 横波 —the light is propagated as transverse.

longitudinal wave 纵波

波的振动方向对传播方向具有对称性。

波的振动方向对传播方向没有对称性 偏振。

5.1.1 polarization and polarized lightaccording to frequency:

monochromatic 单色光

nonmonochromatic 复色光{E

在垂直于光传播方向的平面内光矢量的振动状态。

—the light is propagated as longitudinal.

electric vector

optical vector光矢量

state of polarization 偏振态— state of the vibration on the propagation of light.

Pre

fac

e

Monochromatic light is used to discuss the state of polarization

natural lightunpolarized

自然光

circularly polarized light圆偏振光

elliptically polarized light椭圆偏振光

right handed 右旋

sort of state of polarization 偏振态的分类

partial polarized light 部分偏振光

polarized light完全偏振光

linearly polarized light平面偏振光 ( 线偏振光 )

left handed 左旋

right handed 右旋

left handed 左旋

5.1.2 natural light and polarized light 自然光与偏振光

E ·

light travels towards to you

EEy

Ex

y

x

sinEE y

1. linearly polarized light 平面偏振光 ( 线偏振光 )

cosEEx

· · · ··

krtAE cos

resolve into two component

vibrations are perpendicular to plane of paper vibrations are parallel to plane of paper

2.natural light 自然光

yx EE xyx IIII 2

resolvent

a natural beam can be resolved into two incoherent components ,which are perpendicular to each other with equal magnitude

·· ·

incoherent

5.1.3 partially polarized light and degree of polarization

polarized+unpolarized

最常讨论的部分偏振光可看成是自然光和线偏振光的混合，

resolve

incoherent

·· ·· ·· ··

天空的散射光和水面的反射光就是这种部分偏振光，它可以分解如下：

degree of polarization 偏振度

minmax

minmax

IIII

Pdegree of

polarization

unpolarized P = 0

polarized P = 1

partially polarized 0<P<1

5.2.1 Brewster’s angle 布儒斯特角

221

iiwhen rp=0 polarized, have no parallel component

write as i10 i10——Brewster’s angle

22101 sinsin inin 102101 cossin inin

1

210tan

nn

i

unpolarizedpartial polarized

5.2.2 linearly polarized light by refraction 　由折射获得偏振光 transmitted light is polarized ,the vibration being in plane of incidence

Laser-tube cavity激光器谐振腔

Brewster window 布儒斯特窗

a pile of glass plates 玻璃堆

5.2 polarization by reflection and refraction 、 Brewster’s law

unpolarized beam is not incident at Brewster’s angle incident with ：

21

21

21

21

21

21

21

21'

cos

cos

cos

cos

sin

sin

tan

tan

1

1

ii

iir

ii

ii

ii

ii

ii

ii

A

Ar

s

p

pp

1

21

21

ii

ii

cos

cos

''

11 spsp AArr

11 sp AA

Reflected and refracted natural light are partially polarized light 自然光反射和折射后成为部分偏振光

·

···

· · ·n1

n2

i1 i1'

i2

·· S

P

5.2.3 explanation of polarization by reflection and refraction

书 P.206 例 5-1自然光以入射角 i 投射到水面上，使发射光成为完全偏振光。现有一块玻璃浸于水中，如光从玻璃面反射也为完全偏振光，试求水面与玻璃面之间的夹角。解：

33.1

1tan 1

水

空气

n

ni 0265361 i

9010 ii

65921112 ii

33.1

5.1tan 2

水

玻

n

ni 6162482 i

5.3 double refraction in uniaxial crystal

5.3.1 ordinary ray and extraordinary ray

2.ordinary ray and extraordinary ray寻常 (o) 光和非寻常 (e) 光

a ray of light is refracted by a crystal, it gives two refracted rays自然光入射到各向异性介质中，折射光分成两束的现象。

1. double refraction in1

n2

ro

reanisotropic

unpolarized

o e

o ornin sinsin 21 O ray is at plane of incidence

{

e E ray is not always at plane of incidence

.constsin

sin

er

i

{

obey laws of refraction 遵从折射定律

usually,not obey laws ofrefraction 一般不遵从折射定律

· · · calcite 方解石o

ee

o· · ·rotate calcite round the incident ray as an axis

calcite方解石

偏振片

vibrations of two refracted rays are perpendicular to each other双折射的两束光振动方向相互垂直

象折射现双折射现calcite

Phenomenon of double refraction

光光双折射

纸面纸面

光光双折射

纸面纸面

光光双折射

纸面纸面

光光双折射

纸面纸面

光光

paperpaper

image by o ray

image by E ray

when calcite is rotated,image of o ray remains stationary o 光的像不动，

image of e ray rotates with the rotation of crystal 光的像围绕 o 光的像旋转

calcite

5.3.2 optic axis 光轴

a direction, not a line ,along which the double refraction is a

bsent 特定的方向，晶体中光沿这个方向传播时，不发生双折射现象。

Crystallised CaCO3(Iceland spar)方解石晶体（冰洲石）

A

B

102° 78°

1. optic axis 光轴

propagation along optic axis n()=no

propagation perpendicular to optic axis n()=ne

{optic axis

Extraordinary ray

uniaxial crystal 单轴晶体： calcite 方解石、 quartz 石英

biaxial crystal 双轴晶体：云毋、 sulfur 硫磺、 topaz 黄玉{5.3.3 principal section 主截面

晶体光轴与晶体表面法线构成的平面叫主截面

a plane , which contains the optic axis and thenormal of the crystal face, is called principal section

A

B

102° 78°

optic axis

normal

for every point there are three principal section

note the difference from the definition in Chinese textbook

P.207

o rayoptic axis

principal plane of

o ray

· · ··e ray

optic axis

principal plane of e ray

o 光的振动方向垂直于垂直于 o 光的主平面

e 光的振动方向平行于平行于 e 光的主平面 usually, vibrations of o ray and e ray are not perpendicular to each other

vibration of o ray is perpendicular to its principal plane

vibration of e ray is parallel to its principal plane

principal plane 主平面a plane , which contains the optic axis and the rayof light, is called principal plane

principal plane of the ordinary ray

principal plane of the extraordinary ray

optic axis in the plane of incidencethe principal planes of o ray and e ray coincidevibrations of o ray and e ray are perpendicular to each other

commentscomments ： ： general, the angle between two principal planes is general, the angle between two principal planes is very small, the vibrations of o ray and e ray are almost normalvery small, the vibrations of o ray and e ray are almost normal

5.3.4 relative intensity of o ray and e ray o 光与 e 光的相对光强

a. incidence of natural light

optic axis

··

·

·

·

e

calcite

·

o

I

2,

222 I

AI

A eo

2I

II eo

special case: normal incidence

special case:

principal section

cos

sin

AA

AA

e

o

2tane

o

I

I

b.incidence of linearly polarized light

Ae

Ao θ

rotating the crystal round incident ray introduces the rotation of principal section θ changes

θ =0 ， I0=0, Ie=Iθ =90 ， I0=I ,Ie=0

in crystal the wavefront of o ray is spherical o 光波面是球面

in crystal the wavefront of e ray is an ellipsoid of revolution e 光波面是旋转椭球面

x

z

x

z

rrn

zen

xon o

o nc

v

ee n

cv

rn varies with the different vibration direction velocity of light varies

n

cv

uniaxial :symmetry of revolution around optic axis

vibration is normal to z:

vibration is parallel to z:

nc

v 'other direction vibration n between on en~

varies with the direction in crystal

5.4 Huygen’s explanation of double refraction by wavefront

Opic axis

principal plane----contains the optic axis and the ray of light

o ray----vibration is normal to principal plane

at any direction,vibration is normal to optic axis

oo n

cv wavefront is spherical——

e ray----vibration is parallel to principal plane

vibration is normal to z

ee n

cv vibration is parallel to z

oo n

cv

wavefront is an ellipsoid of revolution——

propagation along zpropagation along z

propagation is propagation is normal to znormal to z

o ray has same velocity at any direction

e ray has different velocity at different direction

positive crystal 正晶体negative crystal 负晶体

ve<vo

ve>vo

quartz 石英calcite 方解石{

5.4.2propagation direction of o ray and e ray in uniaxial crystal

a. optic axis in the plane of incidence, oblique incidence

r0

i····

voΔt

veΔt

cΔto

oo

nci

r

vsin

sinA···

·B

D

e o re

··

optic axis

··

b. optic axis is normal to the surface, normal incidence

c. optic axis is parallel to the surface, normal incidence

e e·· ··· ·

光轴 晶体

··

o o

·· ··· ··

···

no doublerefractionoptic axis

no separation

same velocity

double refraction happens

no separation but different velocity

d. optic axis is normal to the plane of incidence,normal incidence

· ·optic axis • • •

··

••

••

o oe e

double refraction happens

same as “c ”case

optic axis • • • o

e

••

e. optic axis is normal to the plane of incidence, oblique incidence

5.5 polarization prism and analyser Malus law

1 .Nicol prism 尼科耳棱镜

•• ••

••

no=1.658ne=1.486nb=1.550e

2.Wollaston prism 渥拉斯顿棱镜

• •• •

Canada balsam加拿大树胶

• • • •••

•e

o

tansin2 1eo nn

e

o

5.5.1 polarization prism 偏振棱镜

• •• •

• •

• •• •

• • • •••

•e

o

glass

examples:

（ 1 ）

（ 2 ）

Calcite(negative crystal)

no=1.658ne=1.486

no=1.658ne=1.486nglass=1.5

calcite

• •••

•e

o

5.5.2 analyser (polarization plate)

5.5.3 Malus law 马吕斯定律

P2

A1

P2

A 2= A1cos

Malus law 马吕斯定律 (1809)

2 2 1 2 cosAI , I

21

02

1 1 AI

202 cos

2I

I

—— extinguishintensity vanishes 消光

2/0 0max2 III ，

02 2 I，

linear IP1

plane of transmission

natural light I0

· ··

例 5-2 假定在两个静止的、理想的、正交的起偏器之间有另一个理想的起偏器以角速度 w 旋转，试证透射光的强度满足下列关系式： tI

I 4cos181

★ 例 5-3 设一水晶棱镜的顶角 A 为 60 度，光轴与棱镜主截面垂直，钠光以最小偏向角的方向在棱镜中折射，用焦距为0.5m 的透镜聚焦， no=1.54425,ne=1.5536, 试求 o 光和 e 光两谱线的间隔为多少？

5.6 elliptically and circularly polarized light wave plate

circularly polarized 圆偏振光—— tip of the vector draws a circle

elliptically polarized 椭圆偏振—— tip of the vector draws a ellipse{face to the ray

clockwise——right handed 右旋

counter-clockwise——left handed 左旋{

5.6.1 description of elliptically and circularly polarized light 圆偏振光和椭圆偏振光的描述

b.theory

a.definition

elliptically polarized light consists of two plane polarized rayswith same frequency, orthogonal vibrations, traveling along same direction

椭圆偏振光可通过两列频率相同，振动方向相互垂直， 沿同一方向传播的线偏振光叠加得到。

kztAE

kztAE

yy

xx

cos

cosamplitude

22

2

2

2

sincos2y

y

x

x

y

y

x

x

A

E

AE

A

E

AE

=0 ，

= /2

= 其他值

y

y

x

x

A

E

AE

12

2

2

2

y

y

x

x

A

E

AEdiscussion

linear polarized

AxAy elliptically polarized

Ax=Ay circularly polarized

oblique elliptically polarized

{

圆偏振光可通过两频率相同，振动方向相互垂直，且振幅相同， 位相差 (2k+1)/2, 沿同一方向传播的线偏振光叠加得到。

circularly polarized light consists of two plane polarized rayswith same frequency, orthogonal vibrations of equal amplitude, phase difference of (2k+1)/2 ,traveling along same direction

5.6.2 wave plate 波（晶）片wave plate——a plate of doubly refracting uniaxial crystal,which optic axis is parallel to the surface. o ray and e ray travel along same direction with different velocities

d—thickness of plate

nod——optical path of o ray

ned——optical path of e ray

= （ no- ne ） dphase difference:

¼ 波片 dnn eo 2

122

12

k

k

½ 波片

principal section

optical path difference:

quarter wave plate

half wave plate(半波片 )

incident linearly polarized light makes an angle of with optic axispassing a half wave plate ， the emergent light is still a linearly polarized light,which direction of polarization is rotated by 2

b.plane polarized light is incident

intensity ratio of o ray and e ray : sin2/cos2

phase difference ： dnn eo 2

e 光

o 光

a. natural light is incident

two components of incident light have unfixed phase difference o and e rays have unfixed phase difference emergent light is natural light

for half wave plate:

A

Ao

Ae

Ao'

2optic axis

y

d

xA

Ao

Ae

linear

when the ray have traveled through the thickness d in crystal, a phase difference is introducted between o 、 e rays:

A

Ao

Ae

optic axis P

elliptically

general case----- product an elliptically polarized light

d optic axis

wave plate

5.6.3 production of elliptically and circularly polarized light 圆偏振光和椭圆偏振光的产生

sinAAo cosAAe

2

dnn eo

a. linearly polarized light passes through a wave plate

and），（ eo AA 4

2

3,

2

b. linearly polarized light passes through a quarter wave plate

4

dnn oe

4

circularly polarized light

2,0 plane polarized light

thickness of the wave plate2

240

，， elliptically polarized light(normal)

product an circularly polarized light

special case -----

c. production of elliptically and circularly polarized light from natural light

incidence of natural lighttransmitted rays have no fixed phase difference-----natural light

at first, the natural light must pass a polarizer to become a linearly polarized light

Pnatural light I0

· ··

optic axis

quarter wave plate when =45°

circularly polarized

usually, elliptically polarized light

例 5-4 一束圆偏振光，（ 1 ）垂直入射到 1/4 波片上，求透射光的偏振态；（ 2 ）垂直入射到 1/6 波片上，求透射光的偏振态；

例 5-5 一单色自然光通过尼科尔棱镜 N1 ， N2 和晶片 C ，其次序如图示， N1 的主截面水平， C 对应于这波长的 1/4 波片，其主截面与竖直方向成 30 度，试问：（ 1 ）在 N1 和 C 之间， C 和 N2 之间，以及从 N2 透射出来的光各是什么性质的光？（ 2 ）若入射光的强度为 I0 ，则上述各部分的光的强度各是多少？ 注 : 此例最好放到偏振光干涉之后讲

Calculate the thickness of (i) a quarter wave plate and (ii) a half wave plate given that

nmnn oe ,544.1,553.1

5.7 detection of the state of polarization 偏振态的实验检验

5.7.1 Compensator

Babinet’s Compensator

Soleil’s Compensator

· ·· ·

· ·· ·

5.7.2 detection of the state of polarization

incident light

incident on a rotating polarizer

intensity changes and vanishes

intensity does not change

intensity changes but never vanishes

plane polarized either circularly polarized or unpolarized

either ellipticallypolarized or partially polarized

result result result

case 2 case 3case 1

case 2: original beam

incident on a rotating polarizer

intensity changes and vanishes

intensity does not vanish

plane polarized

either circularlypolarized or unpolarized

result result

incident on a quarter wave plate

unpolarized

unpolarized

circularly polarized

case 3: original beam

incident on a rotating polarizer

intensity changes and vanishes

intensity does not vanich

plane polarized

either elliptically polarized or partially polarized

result result

incident on a quarter wave plate,whose optic axis is parallel to max

partially polarized(linearly polarize +natural)

partially polarized

circularly polarized

▲ 如何区分由自然光和椭圆偏振光组成的 部分偏振光与由自然光和线偏振光组成 的部分偏振光？

思考

▲ 如何区分由自然光和圆偏振光组成的部分 偏振光与自然光？

5.8 interference of polarized light 偏振光的干涉

5.8.1 set up and intensity distribution 偏振光干涉装置和强度分布

natural light

polarizerP1

d

wave plate C

optic axis

A1o

A2o

P2

P1 CA1 A1e

A2e

after P1 ：

polarizerP2

interference

sin11 AA o cos11 AA e

sinsinsin 112 AAA oo

coscoscos 112 AAA ee

after P2 ：

Interference intensity ：

2'

sin2sin2sincos

'coscossincossin2

coscossinsin

'cos2

2221

21

2221

2221

2222

22

A

A

AA

AAAAI eoeo

dnn eo 2

phase difference ：

' =——P1 ， P2 are at same side of optic axis

+ ——P1 ， P2 are at opposite side of optic axis{

—destructive interference

if d is various ， there are equal thickness fringes in screen

),2,1(,2

122

knn

kdk

oe

oe nn

kdk

)12(

P1 ， P2 are at opposite side of optic axis:

),2,1(,2

122

knn

kdk

oe

,...)2,1,0(,)12(

knn

kdk

oe

—constructive interference

P1 ， P2 are at same side of optic axis:

—constructive interference

—destructive interference

especially ：

（ 1 ） P1 // P2 ，

（ 2 ） P1 P2 ，

2

2sin2sin1 222

1//

AI

2sin2sin

2sin2sin2sin

2221

22221

A

AI

P1 ， P2 are at opposite side of optic axis

21// AII

incidence of monochromatic light 单色光入射fixed A1

2,

dnn eo 2

phase difference ：

equal thickness: I is related to

unequal thickness: I is related to and d{intensity I is only related to ，

5.8.2 incidence of white light白光入射 is related tocolor fringes

（显）色偏振

if the thickness of wave plate is uniform, the screen presents coloured ------complementary colour 互补色 of destructive interference at a wavelength. It is called chromatic polarization

end