16 Interference – (干涉)

What produces the blue-green of a Morpho’s wing?

How do colorshifting inks shift colors?

airbrush

16 Interference – (干涉)

Huygens’ principle 惠更斯原理

All points on a wavefront

serve as point sources of

spherical secondary wavelets.

After a time t, the new

position of the wavefront will be that of a surface tangent to these secondary wavelets .

Fig. 35-2

子波

3 35-

Law of Refraction from Huygens’ principle

35- 4

Index of Refraction: c n  v

1 2 1 1

1 2 2 2

ec hg

t t v

v v v

  

     

1 1

2 2

sin (for triangle hce) sin (for triangle hcg)

hc hc

 

 





1 1 1

2 2 2

sin sin

v v

 

 ^  ^

1 2

1 2

and

c c

n n

v v

 

1 1 2

2 2 1

sin sin

c n n c n n



 ^{ } Law of Refraction: n₁sin₁  n₂ sin₂

5

Wavelength and Index of Refraction坡常與折射率

35-

n

n n

v v

c c n

    

 ^{    }

n

n

v c n c

f f

  n 

   

The frequency of light in a medium is the same as it is in vacuum

6

Phase Difference相差

35-

Fig. 35-4

Since wavelengths in n1 and n2 are different, the two beams may no longer be in phase

1

1 1

1 1

Number of wavelengths in :

n

L L Ln

n N ^  ^  n ^ 

2

2 2

2 2

Number of wavelengths in :

n

L L Ln

n N ^  ^  n ^ 

 

2 2

2 1 2 1 2 1

Assuming : Ln Ln L

n n N N n n

  

     

2 1 1/2 wavelength destructive interference

N  N  

Ex.13-1 35-1

wavelength 550.0 nm n

=1.600 and

L = 2.600 m

Young’s Experiment

9

Coherence

35-

Two sources to produce an interference that is stable over time, if their light has a phase relationship that does not change with time: E(t)=E₀cos(

w

f

Coherent sources: Phase

f

constant. When waves from coherent sources meet, stable interference can occur － laser light (produced by

cooperative behavior of atoms)

Incoherent sources:

f

10

Fig. 35-13

Intensity and phase

35-

   

 

0 0

1

0 0 2

sin sin ?

2 cos 2 cos

E t E t E t

E E E

w w f

 f

   

 

  f 

2 2 2 1

0 2

4 cos

E  E f

2

2 1 2 1

2 0 2

2

0 0

4 cos 4 cos

I E

I I

I  E  f   f

 

phase path length difference difference

2

phase 2 path length difference difference

2 d sin

 





f  



   

   

   

   

    

   



Eq. 35-22

Eq. 35-23 Phasor diagram

E₁ E₂

11

Intensity in Double-Slit Interference

35-

 

1 0 sin and 2 0 sin

E  E wt E  E w ft 

2 1

0 2

4 cos

I  I f 2

d sin

f  

 

   

1 1 1

2 2 2

minima when: f  m  d sin  m  for m  0,1, 2, (minima)

1 2

maxima when: for 0,1, 2, 2 2 sin

sin for 0,1, 2, (maxima)

m m m d

d m m

f  f   

  

    

  

12

Intensity in Double-Slit Interference

35-

Fig. 35-12

avg 2 0

I  I

Ex.13-2 35-2

wavelength 600 nm n

=1.5 and

m = 1 → m = 0

Interference from Thin Films

15

Reflection Phase Shifts

35-

Fig. 35-16 n₁ n₁ > n₂ n₂

n₁ n₁ < n₂ n₂

Reflection Reflection Phase Shift Off lower index 0

Off higher index 0.5 wavelength

16

Phase Difference in Thin-Film Interference

35-

Fig. 35-17

Three effects can contribute to the phase difference between r₁ and r₂.

1. Differences in reflection conditions 2. Difference in path length traveled.

3. Differences in the media in which the waves travel. One must use the wavelength in each medium ( / n), to calculate the phase.

2

 ₀

17

Equations for Thin-Film Interference

35- 2

odd number odd number

2 wavelength = (in-phase waves)

2 2 ⁿ

L   

½ wavelength phase difference to difference in reflection of r₁ and r₂

2L  integer wavelength = integer _n2 (out-of-phase waves)

2

2

n n

  





2

2L m for m 0,1, 2, (maxima-- bright film in air) n

   

2

2L m for m 0,1, 2, (minima-- dark film in air) n

  

18

Color Shifting by Paper Currencies,paints and Morpho Butterflies

35-

weak mirror

looking directly down ： red or red-yellow tilting ：green

better mirror soap film

大 藍 魔 爾 蝴 蝶

雙狹縫干涉之強度

Ex.13-3 35-3 Brightest reflected light from a water film

thickness 320 nm n

=1.33

m = 0, 1700 nm, infrared

m = 1, 567 nm, yellow-green m = 2, 340 nm, ultraviolet

Ex.13-4 35-4 anti-reflection

coating

Ex.13-5 35-5 thin air wedge

24

Fig. 35-23

Michelson Interferometer

35-

1 2

2 2 (interferometer)

L d d

  

1

2 (slab of material of thickness placed in front of )

Lm L

L M

 

25

Determining Material thickness L

35-

= 2 (number of wavelengths

in same thickness of air)

a

N L



2 2

= = (number of wavelengths in slab of material)

m

m

L Ln

N  

2 2 2

 

- = = n-1 (difference in wavelengths

for paths with and without thin slab)

m a

Ln L L N N

 ^  