4,衍射光栅（ Diffraction Gratings）
,历史”的回顾,
X
例题, N个同方向、同频率的谐振动，振幅相等
相位依次相差 ?，求合振动的振幅与相位。
tax ?c o s1 ?
)c o s (2 ?? ?? tax
)2c o s (3 ?? ?? tax
)3c o s (4 ?? ?? tax
1a
?
?
5a
?
?
4a
?
?
3a
?
?
2a
?
………,
O
X
1a?
4? 5a?
2?
3a?
?
2a?
A?
P
Q
M
R
C
N? 4a? 3?
结论, )( k
N
m ? 0?A
N
m ?? 2? 时, 一）当
12,2,1 ??? NNN ?
14,23,13 ??? NNN ?……………………………..,
0?A
?NNNm 3,2,,0?
1.,,,3.2.1 ?? Nm1)
在两个最大之间有 N-1个最小值 A=0 2)
相当于 ?3,2,1?k
请大家记住这一结论 !
NaA ?
二 )当 ?=2K?时, )210( ?????k
a
b
An array of N parallel equidistant slits,
called a diffraction grating,
Gratings are made by ruling equally spaced parallel grooves
on a glass plate,using a diamond cutting point motion,
a — width of a
transparent slit
b — width of a
opaque strip
d — grating constant
d=a+b
d,10-5～ 10-6m
(1cm contains 1000 ～ 10000 slits）
S
Device and phenomenon 装置和现象
L1
E
A,granting 光栅
E,screen 屏幕
L1,L2, lens 透镜
d
中央
明纹
A f
Characteristics of fringes,
bright,sharp,scattered,
条纹特点：亮、细、疏,
L2
D
Y
I
剖面图,
光
栅
granting
透 lens
镜 屏 幕
screen
Multiple rays interference
I
?? s ins in)( dbaBC ???
?
?
a
b d B C
?
?
a
b d B C
?
?
a
b d B C
?
?
a
b d
The path difference BC between
rays from adjacent slits are,
?
?? BC2??
Grating diffraction is the combinations of N
parallel rays,Any two adjacent parallel rays
have a phase difference of ??,
1) Grating equation 光栅方程,
principal maxima 主明纹 Y
I
光
栅
透
镜
屏
幕
?
?
?
?
?? s in22 baBC ????
...3.2.10 ????k
...3.2.10 ????k
?
?
a
b d B C
Grating equation 光栅方程
?k2?
?? kba ?? s in)(
中央
明纹
1级明纹
2级明纹
3级明纹
-1级明纹
-2级明纹
-3级明纹
（主最大）
2) minima 暗纹 Y
光
栅
透
镜
屏
幕
?
?
a
b d B C (minima) ??
N
mba ?? s in)(
I 中央
明纹
1级明纹
2级明纹
3级明纹
-1级明纹
-2级明纹
-3级明纹
（主最大）
....23.13;12....2.1;1...2.1
???
?????
NN
NNNNm
?
?
?
?
?? s in22 baBC ????
暗纹
N
m ?2?
k
N
m ?
There are N-1 minima between two principal
maxima,When N is very big,it is a dark region,
I
I
I
1 2 4 5 -1 -2 -4 -5
Analysis of the intensity,
Only is there diffraction 如果只有衍射,
Only is there interference 如果只有干涉,
There are diffraction and interference干涉衍射均有,
缺
级
缺
级
-1 -2 1 2
-2 2
3) Analysis of the pattern,
A) if ? is fixed,the smaller grating constant is,
the more scattered fringes are,
ba
kk
bakk ?
???
?
??? ????? ])1[(1s ins in 1
bakk ?
??? ??? 1
The angular distance between two principal
maxima,
?
B） if d=a+b is fixed,the bigger ? is,the wider
angular distance is,
?? kba ?? s in)(
4) Lack of the order 缺级
Only is there diffraction 如果只有衍射,
I
I
1 2 4 5 -1 -2 -4 -5
Only is there interference 如果只有干涉,
There are diffraction and interference 干涉衍射均有,
缺
级
缺
级 -2 2
,
-1 -2 1 2
I
4) Lack of the order 缺级,
From grating equation( principal maxima),
)1(s in)( ??? kba ??,..3.2.10 ????k
)2('s in ??? ka ?,..3.2.1' ????k
k is the interference order
'k is the diffraction order
,
From minima condition of the single slit diffraction,
from（ 1） /（ 2）,
'k
k
a
ba ??
If θ conforms two conditions above,
)3('' ?
a
dk
a
bakk ???
...3.2.1' ????k
缺级公式,
)3('' ?
a
dk
a
bakk ???
缺级公式,
1 2 4 5 -1 -2 -4 -5
I
缺
级
缺
级 -2 2
example,A gratings,b=2a,find lack of the order：
'32'' k
a
aak
a
bakk ?????,..3.2.1' ????k
故,..9.6.3'3 ????? kk
...3.2.1' ????k
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1 2 4 5 -1 -2 -4 -5
缺
级
缺
级
N=3
2 4 5 -1 -4 -5
I
1 -2
缺
级
缺
级 N=4
I
1 2 4 5 -1 -2 -4 -5
缺
级
缺
级
N=5
I
1 2 4 5 -1 -2 -4 -5
缺
级
缺
级
N=2
3?
a
d
5） if white light falls on,colour spectra appear
from violet to red beside a central maximum,
非连续光谱
6） grating spectrographs
1级光谱 2级光谱 -1级光谱 -2级光谱 中央明纹
1级光谱 2级光谱 -1级光谱 -2级光谱 中央明纹
?? kba ?? s in)(
6） grating spectrographs grating
telescope
collimator
construction,
principl,
?? kba ?? s in)(
definition,resolving power of a grating,
Two light waves whose wavelengths(?1 ?2)are close;
? is the mean wavelength of the two light waves
12 ??? ???
?
?
?
?R
Here is the wave lengths difference
between them and ? is the mean wavelength of
the two spectrum lines that can barely be
recognized as separate by the grating
12 ??? ???
Resolving power
of a grating R ????R
The smaller Δλ/λis,the closer the lines can
be and still be resolved; hence the greater the
resolving power R of the grating
We can prove that the resolving power of a grating,
R=k N
N is the total number of rulings in the grating,
exp,a grating,width 5 cm,with 120 rulings
every cm,Suppose λ=6000A,k=1,find the
resolving power,
6 0 051 2 01 ????? kNR
The range of differentiable wavelength,
?
?
A
A
R
10
600
6000
????
?
?
summary,
??
N
mba ?? s in)( k
N
m ?
...3.2.10 ????k ?? kba ?? s in)(
minimum,
missing
order,adka bakk '' ???,..3.2.1' ????k
缺
级
缺
级
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1 2 4 5 -1 -2 -4 -5
main formulas Equation
maximum,