湖北大学：《大学物理》第十七章（17-7） 衍射光栅

[4.分别光挪( Diffraction Gratings.) “历史”的回顾: 例题.N个同方向、同频率的谐振动,振幅相等 相位依次相差α,求合振动的振幅与相位。 x =a cos ot M x=a cos( at+a x =acos(at+2a) xa =a cos(at +3a) 4 ●●●。● ●●● 20 c c X 01 Pⅹ

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。 x acost 1 = cos( ) x2 = a t + cos( 2 ) x3 = a t +  cos( 3 ) x4 = a t +  a1   a5   a4   a3   a2  ………. O X a1  a5 4  2 a3   a2  A  P Q M R C N a4  3

结论:-)当a2mnr 1:(m≠k)A=0 N 1)m=1.2.3.Ny-1 N+1.N+2.…2N-1 3N+1,3N+2,…4N-1 ●●●●●●●●●●●●0●●●0●●●●●●●●●●●●●●●。● A=0 2)在两个最大之间有N-1个最小值A=0 m=0.N2N,3N 相当于k=1,2,3 二)当a=2K时:(/=0±1±2±…) A=M 请大家记住这一结论!

结论: ( k) N m  A = 0 N m  2 一）当 = 时: N +1,N + 2, 2N −1 3N +1,3N + 2, 4N −1 ……………………………... A = 0 m = 0,N,2N,3N 1) m =1.2.3....N −1 2) 在两个最大之间有N-1个最小值A=0 相当于 k =1,2,3 请大家记住这一结论! A = Na 二)当=2K时: (k = 01 2)

An array ofn 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 a b— width of a opaque strip d--grating constant b d=a+b d:105~106m (1cm contains 100010000 slits

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）

Device and phenomenon装置和现象 S D 中央 L1、L2:lens透镜 明纹 A: granting光栅 Characteristics of fringes: E screen 屏幕 bright、 sharp scattered 条纹特点:亮、细、疏

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

剖面图: Y 光栅 透镜 lens granting 屏幕 screen

Y I 剖面图： 光 栅 granting 透 lens 镜 屏 幕 screen

Multiple rays interference Grating diffraction is the combinations of N parallel rays. Any two adjacent parallel rays have a phase difference of Ao The path difference bc between BC=(a+b)sin 0=d sin e Ap= Bd I rays from adjacent slits are

Multiple rays interference I BC = (a +b)sin  = d sin    a b d B C   a b d B C   a b d B C   a b d B C The path difference BC between rays from adjacent slits are:    BC  = 2 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 (主最大) BC 3级明纹 Chien atb △qp=2兀 sin e 2 F2级明纹 k=0±1.±2.±3 (a+b)sinO=kk=0±1.±2±3 1级明纹 P中央 Grating equation光栅方程 明纹 >-1级明纹 dt 10 栅钅b 2级明纹 B 屏幕 -3级明纹

1) Grating equation 光栅方程： principal maxima 主明纹 Y I 光 栅 透 镜 屏 幕      2 2 sin BC a + b  = = k = 01. 2.3... k = 01. 2.3...   a b d B C Grating equation 光栅方程 = 2k (a +b)sin  = k 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 (主最大) BO a+b 3级明纹 △q=2x=2兀 2m兀 2级明纹 暗纹 ≠k N 1级明纹 m=12N-1,N+1N+2.2N-1Ⅰ中央 3N+1.+3N+2 明纹 >-1级明纹 (a+b)sin 0=m a(minima) 2级明纹 There are N-1 minima between two principal 纹 maxima. When n is very big, it is a dark region

2) minima 暗纹 Y 光 栅 透 镜 屏 幕   a b d B C   (minima) N m (a + b)sin = 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） 3 1. 3 2.... 1.2... 1; 1. 2....2 1; + + + = − + + − N N m N N N N      2 2 sin BC a + b  = = 暗纹 N 2m = k N m  There are N-1 minima between two principal maxima. When N is very big, it is a dark region

Analysis of the intensity Only is there diffraction果只有衍射 Only is there interference如果只有干涉: There are diffraction andinterference干涉衍射均有: - 2 级4A 2 △ △A 5 2

I I I -5 -4 -2 -1 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 -2 -1 1 2 -2 2

3)Analysis of the pattern: A)ifn is fixed, the smaller grating constant is the more scattered fringes are The angular distance between two principal maxima (a+b)sin 0=kn sin 0 k+1 sin [(k+1)-kx] atb + 6, k+1-6≈ atb B)if d=atb is fixed, the bigger n is, the wider angular distance Is

3) Analysis of the pattern： A) if  is fixed，the smaller grating constant is， the more scattered fringes are. a b k k a b k k + + − = + + − =    [( 1) ] 1 sin sin 1 a b k k + + −    1  The angular distance between two principal maxima：  B）if d=a+b is fixed , the bigger  is, the wider angular distance is.(a +b)sin  = k