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# 17 繞射

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17 繞射

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### 17-5 X-Ray Diffraction

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3

Diffraction Pattern from a single narrow slit.

### and the Wave Theory of Light

36- Central

maximum

Side or secondary maxima

Light

Fresnel Bright Spot.

Bright spot

Light These patterns

cannot be explained using geometrical optics (Ch. 34)!

(4)

(5)

### 17-2 Diffraction by a single slit

sin (1 minima)st

a   asin 2 (2 minima) nd

(6)

(7)

(8)

8 36-

sin sin

2 2

a

 

a

(9)

9 36-

(second minimum)

sin sin 2

4 2

a

 

a

### 

(minima-dark fringes)

sin , for 1, 2,3 a

m

m

(10)

(11)

Fig. 36-7 11

### Intensity in Single-Slit Diffraction, Qualitatively

36-

phase 2 path length

difference difference

     

    

     

2

xsin

###  

 

N=18 = 0 small 1st min. 1st side max.

(12)

12

36-

Fig. 36-9

1

sin 2

2 E

R

Em

R

1 1 2

2

m sin

E E

22

m sin 2

m m

I E

I I

I E

 

    

2

asin

### 

 

  

(13)

13

Here we will show that the intensity at the screen due to a single slit is:

Fig. 36-8 36-

Intensity in Single-Slit Diffraction, Quantitatively

m sin 2 (36-5)

I

I

### 

 

  

 

where 1 sin (36-6) 2

a

 

, for 1, 2,3

m m

### 

In Eq. 36-5, minima occur when:

sin , for 1, 2, 3

or sin , for 1, 2, 3 (minima-dark fringes)

m a m

a m m

###  

 

 

If we put this into Eq. 36-6 we find:

(14)

### Ex.17-2 36-2

1 , 1, 2,3, m 2 m

     

(15)

15

### 17-3 Diffraction by a Circular Aperture

36-

Distant point source, e,g., star

lens

Image is not a point, as expected from geometrical optics! Diffraction is

responsible for this image pattern

d

Light

a

Light

a

sin 1.22 (1st min.- circ. aperture) d

### 

sin (1st min.- single slit) a

(16)

16

R

36-

### Fig. 36-11

small

sin 1 1.22 1.22 (Rayleigh's criterion)

R

R d d

 

(17)

(18)

(19)

λ

(20)

20

### The telescopes on some commercial and military surveillance satellites

36-

D

L  R  122. d

 = 550 × 10–9 m.

(a) L = 400 × 103 m ， D = 0.85 m → d = 0.32 m.

(b) D = 0.10 m → d = 2.7 m.

Resolution of 85 cm and 10 cm respectively

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21

### Diffraction by a Double Slit

36-

Two vanishingly narrow slits a<<

Single slit a~

Two Single slits a~

m

cos2

### 

sin 2 (double slit)

I

I

 

  

d sin

a sin

(22)

μ

μ

λ

1 2

sin , 1

sin for 0,1, 2,

a m m

d m m

1 1

sin , 2

a m m

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23

### 17-4 Diffraction Gratings

36-

Fig. 36-18 Fig. 36-19

sin for 0,1, 2 (maxima-lines) d

m

m

Fig. 36-20

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24

36-

### Fig. 36-22

sin hw , sin hw hw Nd

 

### 

hw (half width of central line) Nd

###  

 

hw (half width of line at ) cos

Nd

 

Fig. 36-21

(25)

25

36-

Fig. 36-23

Fig. 36-24

(26)

26

36-

sin

d

m

### 

Differential of first equation (what change in angle

does a change in

wavelength produce?)

Angular position of maxima

cos

d

dmd

For small angles

cos

d

  m

and

d

 

d

 

cosm

d

###  

 

(dispersion defined) D

(dispersion of a grating) (36-30)

cos D m

d

(27)

27

36-

hw Nd cos

 

### 

Substituting for  in calculation on previous slide

Rayleigh's criterion for half- width to resolve two lines

hw

N m

  

  

R

Nm

### 

avg (resolving power defined) R

(resolving power of a grating) (36-32) RNm

cos

d

  m

(28)

28

36-

(29)

### Compact Disc

(30)

30

X-rays are electromagnetic radiation with wavelength ~1 Å

= 10-10 m (visible light ~5.5x10-7 m)

36-

Fig. 36-29

### X-ray generation

X-ray wavelengths too short to be

resolved by a standard optical grating

1 1 1 0.1 nm

sin sin 0.0019

3000 nm m

d

### 

 

(31)

31

d ~ 0.1 nm

→ three-dimensional diffraction grating

36-

Fig. 36-30

2 sind

m

### 

for m  0,1, 2 (Bragg's law)

(32)

32 36-

Fig. 36-31

2 0

5

0 0

5 4 or 0.2236

20

da daa

(33)

(34)

(35)

(36)

36

36-

(37)

### Structural Coloring by Diffraction

(38)

A light beam incident on a small circular aperture becomes diffracted and its light intensity pattern after passing through the aperture is a diffraction pattern with circular

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