Double Slit Interference

1
Double Slit Interference
2
Intensity of Double Slit
E E1 E2
I E2 E12 E22 2 E1 E2 I1 I2
interference lt vanishes if incoherent
3
Refraction

• In general v ?f and ? changes if v does
• in vacuum c ?f
• in a medium c/n ?nf
• hence ?n ?/n which is less than ?
• consider two light waves which are in phase in
  air (n1) and each passes through a thickness L
  of different material

upper wave has ?2 ?/n2
lower wave has ?1 ?/n1
4
Refraction

• Wave 2 has N2 L/?2 (L/?)n2 wavelengths
  in block
• Wave 1 has N1 L/?1 (L/?)n1 wavelengths
  in block
• hence N2 -N1 (L/?)(n2 -n1)
• phase change of wave 1 is k1x-?t (2?/?1)L
  - ?t
• phase change of wave 2 is k2x-?t (2?/?2)L
  - ?t
• phase difference (2?L/?)(n2 -n1) 2?(N2
  -N1 )

5
Refraction

• Emerging waves are out of phase
• interfere constructively if phase difference is
  2? x integer
• (2?L/?)(n2 -n1) 2?m
• hence L m ?/(n2 -n1)

6
Problem

• Which pulse travels through the plastic in less
  time?

7
Solution

• td/v
• pulse 2 tt1t2t3t4
• v1c/1.55, v2c/1.70, v3c/1.60, v4c/1.45
• t(L/c)( 1.551.701.601.45)6.30(L/c)
• pulse 1 t t1t2t3
• v1c/1.59, v2c/1.65, v3c/1.50
• t(L/c)(2 x 1.59 1.65 1.50)6.33(L/c)
• pulse 2 takes least time

8
Phase Change due to Reflection

• Soap films, oil slicks show interference effects
  of light reflected from the top and bottom
  surfaces

Why are there different colours?
Why does top portion of film appear dark?

• When a wave moves from one medium to another
  there is a phase shift of ? if it moves more
  slowly in the second medium and zero if it moves
  more quickly

9
Fixed End
Phase change of ?
10
Free End
No phase change
11

• What is phase difference between rays 1 and 2 ?
• ray 2 travels further gt phase difference due to
  path difference
• phase difference due to extra thickness is
  (2?/?)(2t)
• but ? is the wavelength in the water medium!
  ? ?/n
• ray 1 is reflected from a medium with slower
  speed
• ray 2 is reflected from a medium with higher
  speed
• extra phase difference of ? due to reflection of
  ray 1
• total phase difference ? ? (2?n/?)(2t)

12

• Both rays reflected from media in which wave
  moves more slowly
• phase difference only due to path difference
• ? (2?nwater/?)(2t)
• if ? 2?m, then constructive interference
• if ? ? (2m-1), then destructive interference
• ? (2?nwater/?)(2t) ? (2m-1), i.e. t
  (2m-1)(?/4nwater)
• non-reflecting glass uses this principle

13
The diameters of fine wires can be accurately
measured using interference patterns. Two
optically flat pieces of glass of length L are
arranged with the wire between them as shown
above. The setup is illuminated by monochromatic
light, and the resulting interference fringes are
detected. Suppose L 20 cm and yellow sodium
light (?? 590 nm) is used for illumination. If 19
bright fringes are seen along this 20-cm
distance, what are the limits on the diameter of
the wire? Hint The nineteenth fringe might not
be right at the end, but you do not see a
twentieth fringe at all.

• 1. Find d for the 19th and 20th bright fringe
  
• path difference? black ray travels extra distance
  2d in air gtphase diff (2?/?)(2d)
  note n1!
• black ray has extra phase difference of ? due to
  reflection
• bright fringe when ? (2?/?)(2d)?2?m gt
  d(m-1/2)(?/2)
• 2. Give the limits on d
• d19 (19 1/2) ? /2 5457 nm d20 5753
  nm
• hence 5.46 µm lt d lt 5.75 µm

14
Newtons Rings
Light reflected from curved lens interferes with
lift reflected from plate bright ring ???
(2?/?)(2d)2m? 2d(m-1/2) ? max