1
第十二章 光的衍射 （ Diffraction）
光的衍射现象,光波在空间传播遇到障碍时，其传播方向会偏离直线传播，弯入到障碍物的几何阴影中，并呈现光强的不均匀分布的现象。
2
When light passes through a narrow slit,
it spreads out more than what could be
accounted for by geometric construction,
This is an example of diffraction,
Diffraction can be defined as any
departure from the predictions of
geometric optics.
What is the Diffraction?
3
The phenomenon that the light waves
tend to bend around and become spread
out when they pass near a barrier is called
diffraction,
Diffraction of light occurs when a light
wave passes by a corner or through an
opening or slit that is physically the
approximate size of,or even smaller than
that light's wavelength.
Diffraction
4
Related Concept
The terms diffraction and scattering
are often used interchangeably and
are considered to be almost
synonymous,
Diffraction describes a specialized
case of light scattering in which an
object with regularly repeating
features (such as a diffraction grating)
produces an orderly diffraction of
light in a diffraction pattern,
5
In the real world most objects
are very complex in shape and
should be considered to be
composed of many individual
diffraction features that can
collectively produce a random
scattering of light.
6
衍射实验 (Diffraction experiment)：
S
K
光的衍射 是光的波动性的主要标志之一 。
Light source
Barrier Screen
7
衍射现象的分类
(Classification of light diffraction)：
根据光源、衍射物（衍射屏）和衍射场（观察屏）三者之间的位置确定
（ 1）夫琅和费衍射 (Fraunhofer diffraction)：
光源和衍射场都在衍射物无限远处的衍射。
（ 2）菲涅耳衍射（ Fresnel diffraction ）：
光源和衍射场或二者之一到衍射物的距离比较小时的衍射。
8
第一节 光波的标量衍射理论一、惠更斯－菲涅耳原理
1、惠更斯原理 (Huygens’ principle)：
（ 1）波阵面的形成，
（ 2）波面的传播方向。
图 12－ 3 光波通过圆孔的惠更斯作图法
S
D
D'
K
v
9
S
Z
P
r
R
Q
Z '
图 1 点光源 S对 P点的作用
2、惠更斯－菲涅耳原理波阵面外任一点光振动应该是波面上所有子波相干叠加的结果。
10
波阵面外任一点光振动应该是波面上所有子波 相干叠加 的结果。S
Z
P
r
R
Q
Z '
dr i k rECKPEd Q e x p~~
子波向 P点的 球面波公式子波法线方向的振幅子波振幅随?角的变化
R
i k R
AE
Q
ZZS
Q
e x p~
'
点产生的复振幅：任意上在波面光源
Qd
P
点 处 大 小 的 面 元对 点 的 贡 献 为,
11
当? = 0 时,K(?)=Max,p/2 时,K(?)=0.
若 S发出的光源振幅为 A（ 单位距离处）,整个波面’的贡献
dr i k rKi k RRCAPE ex pex p~
菲涅尔假设：
（实验证明是不对的）
d
r
i k rECKPEd
PdQ
Q
e x p~~
点的作用：对点处的面光源求解此公式主要问题,C,K(?)没有确切的表达式。
S
Z
P
r
R
Q
Z '
12
二、菲涅耳－基尔霍夫衍射公式（确定了 C,K(?))
基尔霍夫 (Kirchhoff) 从波动方程出发，用场论得出了比较严格的衍射公式。
其中，设定方向角
( n,l ) 和 ( n,r )
为?的法线 与 l
和 r 的夹角。
w'
w"
S
R
w
( n,l )
( n,r )
r
P
l
d
lnrn
r
i k r
l
i k l
i
APE
2
,c os,c ose xpe xp＝～
Q
13
。前于入射波表示子波的振动位相超?
p
90
2
1 ]e x p [ i
i
i
成反比。成正比，与波长子波的复振幅与
2
lnrn
K
,c o s,c o s
)(
d
lnrn
r
i k r
l
i k l
i
APE
2
,c os,c ose xpe xp＝～
14
c o s121K则
R
i k R
l
i k l )e x p ()e x p (?
c o s),c o s (
,1),c o s (
rn
ln
( n,l )
( n,r )
r
P
当光线接近于正入射时
15
将近似条件代入得到：
菲涅耳－基尔霍夫衍射近似公式
S
R
( n,l )
( n,r )
r
P
d
r
i k r
R
i k RAiPE c o s1e x pe x p
2
~
16
三、基尔霍夫衍射公式的近似
C
Q
P
EK
y
1
x
1
y
z
1
r
P
0
x
图 12－ 4 孔径?的衍射
1、傍轴近似（两点近似）
1 c o sc o s rn 1c o s121K(1)
(2)在振幅项中
1
11
zr?
dr i k rR i k RAiPE c o s1e x pe x p2~
17
(3)设定孔径函数
1111 dydxdyxE,,~
。之内，在＝之外它在 R i k RAyxEyxE )e x p (,~,~ 1111 0
图 12－ 4 孔径?的衍射
C
Q
P
EK
y
1
x
1
y
z
1
r
P
0
x
1111
1
e x p,,~ dydxi k ryxEAziyxE
进一步的计算需要将 exp( ikr )中的 r表示成 (x,y,z)的函数。
18
2.菲涅耳近似（对位相项的近似）
....
8
][
2
1)(
3
1
22
1
2
1
1
2
1
2
1
1
2
1
2
1
2
1
1
2
1
2
1
2
1
z
yyxx
z
yyxx
z
z
yyxx
zyyxxzr
C
Q
P
EK
y
1
x
1
y
z
1
r
P
0
x
1
2
1
2
1
1 2 z
yyxxzr
p
4
3
1
22
1
2
1
z
yyxx ][
近似条件：
级数展开
19
1
2
1
2
1
1 2 z
yyxxzr 称为菲涅耳近似。
112121
1
11
1 2
e x p,~,~
1
dydxyyxxzkiyxEzieyxE
i k z
C
Q
P
EK
y
1
x
1
y
z
1
r
P
0
x
得到菲涅耳衍射：
20
1
22
1
11
1 2 z
yx
z
yyxxzr ＋
3.夫琅合费近似继续展开
1
2
1
2
1
1
22
1
11
1 22 z
yx
z
yx
z
yyxxz ＋
1
2
1
2
1
1 2 z
yyxxzr
取上式前三项
1
1
22
1 2
zi
z
yxzik
yxE
)](e x p [
,~
1111
1
11 dydxyyxxz
kiyxE?
e x p,~
21
菲涅耳衍射和夫琅合费衍射的判别式；
p
1
m a x
2
1
2
1
2 z
yxk
或者
m a x
2
1
2
1
1
yxz (菲涅耳衍射 )
m a x
2
1
2
1
1
yxz
(夫琅合费衍射 )
菲涅耳衍射和夫琅和费衍射是两个经常应用的衍射计算。
22
一、惠更斯－菲涅耳原理
1、惠更斯原理
2、惠更斯－菲涅耳原理
dr i k rKi k RRCAPE ex pex p~
本课内容回顾
23
二、菲涅耳－基尔霍夫衍射公式
d
lnrn
r
i k r
l
i k l
i
APE
2
,c os,c ose xpe xp＝～
dr i k rR i k RAiPE
W
co sex pex p~ 12
精确计算：
近似计算 （设平面波入射,cos(n,l )=-1 ）
24
三、基尔霍夫衍射公式的近似
1111 dydxi k ryxEAziyxE e xp,,~
1、菲涅耳近似（对位相项的近似）
1
2
1
2
1
1 2 z
yyxxzr
112121
1
11 2 dydxyyxxz
kiyxE
zi
eyxE i k z
e xp,~,~
25
2、夫琅合费近似
1
22
1
11
1 2 z
yx
z
yyxxzr ＋
)](e xp [,~
1
22
1
1 2
1
z
yxzik
ziyxE
1111
1
11 dydxyyxxz
kiyxE?
e x p,~
26
Finished § 12－ 1
下一节
27
Definition
Fraunhofer diffraction refers to parallel,
collimated light (far-field diffraction),
In Fresnel diffraction,the light need not
be parallel (near-field diffraction).
Fresnel diffraction is more general; it
includes Fraunhofer diffraction as a
special case,But Fraunhofer
diffraction is so much easier to discuss
that it is customarily presented first.
28
Fraunhofer
Joseph von Fraunhofer (1787-1826),
German,After working for a while as a lens grinder
and apprentice optician,he became a partner in an
optical company that made precision theodolites,
professor at the University of Munich,and was
knighted by King Maximilian of Bavaria,In his short
life (died of tuberculosis at age 39),he produced
large-aperture telescope lenses,exceptionally well
corrected for spherical and chromatic aberration,
ruled precision gratings and discovered their use for
spectroscopy,and found that the spectrum of the sun
is crossed by dark lines since named Fraunhofer
lines.
29
Fresnel
Augustin Jean Fresnel (1788-1827),a
nineteenth century French physicist,He
studied mathematics,then civil engineering;
he went into optics later,He is best known
for the invention of unique compound lenses
designed to produce parallel beams of light,
which are still used widely in lighthouses,In
the field of optics,Fresnel derived formulas
to explain reflection,diffraction,interference,
refraction,double refraction,and the
polarization of light reflected from a
transparent substance.
30
Huygens’ principle
Huygens’ principle states that each
point on a wavefront may be
considered the origin of new,
secondary wavelets,these form
another wavefront,and so on,In
this way the wave moves forward.
31
Huygens
Christiaan Huygens (1629-1695) - Christiaan
Huygens was a brilliant Dutch mathematician,
physicist,and astronomer who lived during the
seventeenth century,a period sometimes
referred to as the Scientific Revolution,Huygens,
a particularly gifted scientist,is best known for his
work on the theories of centrifugal force,the
wave theory of light,and the pendulum clock,His
theories neatly explained the laws of refraction,
diffraction,interference,and reflection,and
Huygens went on to make major advances in the
theories concerning the phenomena of double
refraction (birefringence) and polarization of light.
32
级数展开公式
32
!3
21
!2
111 nnnnnnn
第十二章 光的衍射 （ Diffraction）
光的衍射现象,光波在空间传播遇到障碍时，其传播方向会偏离直线传播，弯入到障碍物的几何阴影中，并呈现光强的不均匀分布的现象。
2
When light passes through a narrow slit,
it spreads out more than what could be
accounted for by geometric construction,
This is an example of diffraction,
Diffraction can be defined as any
departure from the predictions of
geometric optics.
What is the Diffraction?
3
The phenomenon that the light waves
tend to bend around and become spread
out when they pass near a barrier is called
diffraction,
Diffraction of light occurs when a light
wave passes by a corner or through an
opening or slit that is physically the
approximate size of,or even smaller than
that light's wavelength.
Diffraction
4
Related Concept
The terms diffraction and scattering
are often used interchangeably and
are considered to be almost
synonymous,
Diffraction describes a specialized
case of light scattering in which an
object with regularly repeating
features (such as a diffraction grating)
produces an orderly diffraction of
light in a diffraction pattern,
5
In the real world most objects
are very complex in shape and
should be considered to be
composed of many individual
diffraction features that can
collectively produce a random
scattering of light.
6
衍射实验 (Diffraction experiment)：
S
K
光的衍射 是光的波动性的主要标志之一 。
Light source
Barrier Screen
7
衍射现象的分类
(Classification of light diffraction)：
根据光源、衍射物（衍射屏）和衍射场（观察屏）三者之间的位置确定
（ 1）夫琅和费衍射 (Fraunhofer diffraction)：
光源和衍射场都在衍射物无限远处的衍射。
（ 2）菲涅耳衍射（ Fresnel diffraction ）：
光源和衍射场或二者之一到衍射物的距离比较小时的衍射。
8
第一节 光波的标量衍射理论一、惠更斯－菲涅耳原理
1、惠更斯原理 (Huygens’ principle)：
（ 1）波阵面的形成，
（ 2）波面的传播方向。
图 12－ 3 光波通过圆孔的惠更斯作图法
S
D
D'
K
v
9
S
Z
P
r
R
Q
Z '
图 1 点光源 S对 P点的作用
2、惠更斯－菲涅耳原理波阵面外任一点光振动应该是波面上所有子波相干叠加的结果。
10
波阵面外任一点光振动应该是波面上所有子波 相干叠加 的结果。S
Z
P
r
R
Q
Z '
dr i k rECKPEd Q e x p~~
子波向 P点的 球面波公式子波法线方向的振幅子波振幅随?角的变化
R
i k R
AE
Q
ZZS
Q
e x p~
'
点产生的复振幅：任意上在波面光源
Qd
P
点 处 大 小 的 面 元对 点 的 贡 献 为,
11
当? = 0 时,K(?)=Max,p/2 时,K(?)=0.
若 S发出的光源振幅为 A（ 单位距离处）,整个波面’的贡献
dr i k rKi k RRCAPE ex pex p~
菲涅尔假设：
（实验证明是不对的）
d
r
i k rECKPEd
PdQ
Q
e x p~~
点的作用：对点处的面光源求解此公式主要问题,C,K(?)没有确切的表达式。
S
Z
P
r
R
Q
Z '
12
二、菲涅耳－基尔霍夫衍射公式（确定了 C,K(?))
基尔霍夫 (Kirchhoff) 从波动方程出发，用场论得出了比较严格的衍射公式。
其中，设定方向角
( n,l ) 和 ( n,r )
为?的法线 与 l
和 r 的夹角。
w'
w"
S
R
w
( n,l )
( n,r )
r
P
l
d
lnrn
r
i k r
l
i k l
i
APE
2
,c os,c ose xpe xp＝～
Q
13
。前于入射波表示子波的振动位相超?
p
90
2
1 ]e x p [ i
i
i
成反比。成正比，与波长子波的复振幅与
2
lnrn
K
,c o s,c o s
)(
d
lnrn
r
i k r
l
i k l
i
APE
2
,c os,c ose xpe xp＝～
14
c o s121K则
R
i k R
l
i k l )e x p ()e x p (?
c o s),c o s (
,1),c o s (
rn
ln
( n,l )
( n,r )
r
P
当光线接近于正入射时
15
将近似条件代入得到：
菲涅耳－基尔霍夫衍射近似公式
S
R
( n,l )
( n,r )
r
P
d
r
i k r
R
i k RAiPE c o s1e x pe x p
2
~
16
三、基尔霍夫衍射公式的近似
C
Q
P
EK
y
1
x
1
y
z
1
r
P
0
x
图 12－ 4 孔径?的衍射
1、傍轴近似（两点近似）
1 c o sc o s rn 1c o s121K(1)
(2)在振幅项中
1
11
zr?
dr i k rR i k RAiPE c o s1e x pe x p2~
17
(3)设定孔径函数
1111 dydxdyxE,,~
。之内，在＝之外它在 R i k RAyxEyxE )e x p (,~,~ 1111 0
图 12－ 4 孔径?的衍射
C
Q
P
EK
y
1
x
1
y
z
1
r
P
0
x
1111
1
e x p,,~ dydxi k ryxEAziyxE
进一步的计算需要将 exp( ikr )中的 r表示成 (x,y,z)的函数。
18
2.菲涅耳近似（对位相项的近似）
....
8
][
2
1)(
3
1
22
1
2
1
1
2
1
2
1
1
2
1
2
1
2
1
1
2
1
2
1
2
1
z
yyxx
z
yyxx
z
z
yyxx
zyyxxzr
C
Q
P
EK
y
1
x
1
y
z
1
r
P
0
x
1
2
1
2
1
1 2 z
yyxxzr
p
4
3
1
22
1
2
1
z
yyxx ][
近似条件：
级数展开
19
1
2
1
2
1
1 2 z
yyxxzr 称为菲涅耳近似。
112121
1
11
1 2
e x p,~,~
1
dydxyyxxzkiyxEzieyxE
i k z
C
Q
P
EK
y
1
x
1
y
z
1
r
P
0
x
得到菲涅耳衍射：
20
1
22
1
11
1 2 z
yx
z
yyxxzr ＋
3.夫琅合费近似继续展开
1
2
1
2
1
1
22
1
11
1 22 z
yx
z
yx
z
yyxxz ＋
1
2
1
2
1
1 2 z
yyxxzr
取上式前三项
1
1
22
1 2
zi
z
yxzik
yxE
)](e x p [
,~
1111
1
11 dydxyyxxz
kiyxE?
e x p,~
21
菲涅耳衍射和夫琅合费衍射的判别式；
p
1
m a x
2
1
2
1
2 z
yxk
或者
m a x
2
1
2
1
1
yxz (菲涅耳衍射 )
m a x
2
1
2
1
1
yxz
(夫琅合费衍射 )
菲涅耳衍射和夫琅和费衍射是两个经常应用的衍射计算。
22
一、惠更斯－菲涅耳原理
1、惠更斯原理
2、惠更斯－菲涅耳原理
dr i k rKi k RRCAPE ex pex p~
本课内容回顾
23
二、菲涅耳－基尔霍夫衍射公式
d
lnrn
r
i k r
l
i k l
i
APE
2
,c os,c ose xpe xp＝～
dr i k rR i k RAiPE
W
co sex pex p~ 12
精确计算：
近似计算 （设平面波入射,cos(n,l )=-1 ）
24
三、基尔霍夫衍射公式的近似
1111 dydxi k ryxEAziyxE e xp,,~
1、菲涅耳近似（对位相项的近似）
1
2
1
2
1
1 2 z
yyxxzr
112121
1
11 2 dydxyyxxz
kiyxE
zi
eyxE i k z
e xp,~,~
25
2、夫琅合费近似
1
22
1
11
1 2 z
yx
z
yyxxzr ＋
)](e xp [,~
1
22
1
1 2
1
z
yxzik
ziyxE
1111
1
11 dydxyyxxz
kiyxE?
e x p,~
26
Finished § 12－ 1
下一节
27
Definition
Fraunhofer diffraction refers to parallel,
collimated light (far-field diffraction),
In Fresnel diffraction,the light need not
be parallel (near-field diffraction).
Fresnel diffraction is more general; it
includes Fraunhofer diffraction as a
special case,But Fraunhofer
diffraction is so much easier to discuss
that it is customarily presented first.
28
Fraunhofer
Joseph von Fraunhofer (1787-1826),
German,After working for a while as a lens grinder
and apprentice optician,he became a partner in an
optical company that made precision theodolites,
professor at the University of Munich,and was
knighted by King Maximilian of Bavaria,In his short
life (died of tuberculosis at age 39),he produced
large-aperture telescope lenses,exceptionally well
corrected for spherical and chromatic aberration,
ruled precision gratings and discovered their use for
spectroscopy,and found that the spectrum of the sun
is crossed by dark lines since named Fraunhofer
lines.
29
Fresnel
Augustin Jean Fresnel (1788-1827),a
nineteenth century French physicist,He
studied mathematics,then civil engineering;
he went into optics later,He is best known
for the invention of unique compound lenses
designed to produce parallel beams of light,
which are still used widely in lighthouses,In
the field of optics,Fresnel derived formulas
to explain reflection,diffraction,interference,
refraction,double refraction,and the
polarization of light reflected from a
transparent substance.
30
Huygens’ principle
Huygens’ principle states that each
point on a wavefront may be
considered the origin of new,
secondary wavelets,these form
another wavefront,and so on,In
this way the wave moves forward.
31
Huygens
Christiaan Huygens (1629-1695) - Christiaan
Huygens was a brilliant Dutch mathematician,
physicist,and astronomer who lived during the
seventeenth century,a period sometimes
referred to as the Scientific Revolution,Huygens,
a particularly gifted scientist,is best known for his
work on the theories of centrifugal force,the
wave theory of light,and the pendulum clock,His
theories neatly explained the laws of refraction,
diffraction,interference,and reflection,and
Huygens went on to make major advances in the
theories concerning the phenomena of double
refraction (birefringence) and polarization of light.
32
级数展开公式
32
!3
21
!2
111 nnnnnnn
