17 繞射

17 繞射

Sections

17-1 Diffraction and the Wave Theory of Light 17-2 Diffraction by a single slit

17-3 Diffraction by a Circular Aperture 17-4 Diffraction Gratings

17-5 X-Ray Diffraction

3

Diffraction Pattern from a single narrow slit.

17-1 Diffraction

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)!

The Fresnel Bright Spot (1818)



Newton



corpuscle



Poisson/Arago



Fresnel



wave

17-2 Diffraction by a single slit

sin (1 minima)st

a   asin  2 (2 minima) ^nd

單

狹

縫

繞

射

之

強 度

 Double-slit diffraction (with interference)

 Single-slit diffraction

雙狹縫與單狹縫

8 36-

Diffraction by a Single Slit:

Locating the first minimum

sin sin

2 2

a





 

(first minimum)

9 36-

Diffraction by a Single Slit:

Locating the Minima

(second minimum)

sin sin 2

4 2

a









(minima-dark fringes)

sin , for 1, 2,3 a





Ex.17-1 36-1 Slit width

Fig. 36-7 11

Intensity in Single-Slit Diffraction, Qualitatively

36-

phase 2 path length

difference difference





     

    

    ^{ }

^

_ ^



^ 

 

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

12

Intensity and path length difference

36-

Fig. 36-9

1

sin 2

2 E

R





1 1 2

2

m sin

E_ E







 

 

m m

I E

I I

I E



 



 

    

 

2



 



 

  

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

 

I







 

  

 

where 1 sin (36-6) 2



  

 



, 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:

Ex.17-2 36-2

1 , 1, 2,3, m 2 m

  ^  ^ 

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

Rayleigh’s Criterion: two point sources are barely

resolvable if their angular separation θ

results in the central maximum of the diffraction pattern of one

source’s image is centered on the first minimum of the diffraction pattern of the other source’s image.

Resolvability

36-

Fig. 36- 11

small

sin 1 1.22 1.22 (Rayleigh's criterion)

R

R d d









Diffraction and Pointillism

Why do the colors in a pointillism

painting change with viewing distance?

Ex.17-3 36-3 pointillism

D = 2.0 mm

d = 1.5 mm

(diameter of

the pupil)

Ex.17-4 36-4

d = 32 mm f ^{= 24 cm}

λ

= 550 nm

(a) angular

separation (b) separation

in the focal

plane

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

21

Diffraction by a Double Slit

36-

Two vanishingly narrow slits a<<

Single slit a~

Two Single slits a~

 





I



 



 

  

d sin

  





a sin

  





Ex.17-5 36-5

d = 19.44

m a ^{= 4.050}

m

λ

= 405 nm

1 2

sin , 1

sin for 0,1, 2,

a m m

d m m

 

 

 

 

1 1

sin , 2

a   m  m 

23

17-4 Diffraction Gratings

36-

Fig. 36-18 Fig. 36-19

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





Fig. 36-20

24

Width of Lines

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

Separates different wavelengths (colors) of light into distinct diffraction lines

Grating Spectroscope

36-

Fig. 36-23

Fig. 36-24

26

Gratings: Dispersion

36-

sin

d





Differential of first equation (what change in angle

does a change in

wavelength produce?)

Angular position of maxima





d

 



For small angles





d

 



d













d



 

 



(dispersion defined) D 



 

 (dispersion of a grating) (36-30)

cos D m

d 



27

Gratings: Resolving Power

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









avg (resolving power defined) R 

 



(resolving power of a grating) (36-32) R  Nm





d

 



28

Dispersion and Resolving Power Compared

36-

Compact Disc

30

X-rays are electromagnetic radiation with wavelength ~1 Å

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

17-5 X-Ray Diffraction

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

d ~ 0.1 nm

→ three-dimensional diffraction grating

Diffraction of x-rays by crystal

36-

Fig. 36-30

2 sind





32 36-

Fig. 36-31

X-Ray Diffraction, cont’d

2 0

5

0 0

5 4 or 0.2236

20

d  a d  a  a

A Holograph

Viewing a holograph

全像術

36

Optically Variable Graphics

36-

Fig. 36-27

Structural Coloring by Diffraction