04/19/23 Lecture VI 1

Physics 123, Spring 2006

Wave nature of lightthin films, diffraction

04/19/23 Lecture VI 2

Intensity in Young’s experiment• E=E1 +E2

• E=E0 (sin(t)+sin(t+))

2

cos2

sin2sinsinBABA

BA

2cos

2sin2 0

tEE

2EI =0 =0: amplitude

E(=0)=2E0

• I(=0)=4E02

• Amplitude E()=2E0cos(/2)

• I()=4E02 cos2(/2)

04/19/23 Lecture VI 3

Intensity in Young’s experiment• I(=0)=4E0

2

• I()=4E02 cos2(/2)

y

L

d

I

I

22 cos

2cos

)0(

)(

yL

d

2

md

Ly

mmyL

d

3,2,1,0,• Bright when cos=1, or -1

04/19/23 Lecture VI 4

Young’s experiment r=700 nm b=400 nm• d=2000nm• L=20cm • First fringes (bright

spots) yr, yb-?• m=1:• y=L /d• yr=7cm• yb=4cm• Blue is closer to the

center than red

04/19/23 Lecture VI 5

Young’s experiment

• Two different -

• Distance between slits – d• Multiple slits (diffractive

grating)– same pattern, sharper lines

• Interference pattern depends on – Maxima:– d sin = m

– d sin = m

04/19/23 Lecture VI 6

Coherence

• Why do not we observe an interference pattern between two different light bulbs?

• These two sources of light are incoherent:

• What does it mean for two sources to be coherent?– Same (or close) frequency – Constant shift in phase (not necessarily zero)

04/19/23 Lecture VI 7

Light in a medium (refraction)• Huygens principle

– each point forces oscillations with frequency f

• f1=f2

• v1=c/n1

• v2=c/n2

• n11=n22

• E.g. go from air to medium n:

/n• n=/n

n1

n2

04/19/23 Lecture VI 8

Light in medium

n=2

=400nmx=600nm

n=n400/2=200nm

Destructiveinterference

+

+

-

-

1=k1x=(2/)x=2600/400=3

Extra phase =3

2=k2x=(2/)x=2600/200=60

0)2/3cos()2/cos(

21

EE

04/19/23 Lecture VI 9

Reflection of a transverse wave pulse

•Reflection from fixed end –inverted pulse

•Reflection from loose end – the pulse is not inverted.

04/19/23 Lecture VI 10

Reflection

• Reflect from medium with higher n2>n1 phase change by =

– + -

• Reflect from medium with lower n2<n1 no phase change =0

– + +

+

- +

+

04/19/23 Lecture VI 11

Soap film• Soap film, air on both sides

• Thickness t

• n(soap)=1.42

• n(soap)>n(air) Ray 1 at A

• n(air)<n(soap)Ray 2 at B

• Relative shift ‘

• Ray 2 travels ABC = extra 2t

• “kl2t/n= t/n

• Relative shift “-’= t/n• If m t/n=m or t(m=1)=n/2

– Rays 1 and 2 are out of phase

– Destructive interference

• If m t/n-=m or t(m=0)=n/4

– Rays 1 and 2 are in phase

– Constructive interference

12

film is violet 2t=400nm/2/n t=70nmfilm is red2t=700nm/2/nt=123nm Violet is thinner than red.

B

04/19/23 Lecture VI 12

Diffraction on a single slit

sin22)(zl

z

))((0 Im ztkxie

D

dzEdE

l

x

z

Slit size D, z=-D/2 to D/2Observe diffraction at angle Interference of wavescoming fromdz

04/19/23 Lecture VI 13

Diffraction on a single slit

dzeeD

Edzee

D

EdEE

zitkxizitkxi

sin

2)(0)()(0

sin2)(z

z

l

x

zIntegrate overdz

/sin

)/sinsin(sin

2

sin2)(

0

2/

2/

)(0

D

DeE

zie

Di

EE tkxi

D

D

tkxi e

04/19/23 Lecture VI 14

Diffraction

• Dark spot at

/sin

)/sinsin()(0 D

DeEE tkxi

2

2

0 )/sin(

)/sin(sin/

D

DII

Dm

mD

sin

/sin

1)/sin(

)/sin(sin/

2

2

00 lim

D

DII

• Except =0 – must be bright spot:

04/19/23 Lecture VI 15

Diffraction

• Single slit diffraction• Angular half width of the

first peak:

D

sin