Double Slit Interference

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Double Slit Interference
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Double Slit Interference
d sin? m ? , m0,1,2, bright fringe
for m1, need ? lt d d sin? (m1/2) ? ,
m0,1,2, dark fringe
Light gun
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Problem

• In a double slit expt the distance between slits
  is d5.0 mm and the distance to the screen is
  D1.0 m. There are two interference patterns on
  the screen one due to light with ?1480 nm and
  another due light with ?2600 nm. What is the
  separation between third order(m3) bright
  fringes of the two patterns?
• Note same d,D but different ? gt one pattern
  magnified
• d/D 5 x 10-3 gt sin? tan ? ?

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Solution

• In general d sin ? m? and y D tan?
• since ? ltlt1, ym D ?m m ? D/d
• for ?1480 nm, D1.0m, d5.0 mm y3 3(480x
  10-9)(1.0)/5.0 x 10-3) 2.88 x 10-4 m
• for ?2600 nm, D1.0m, d5.0 mm y3 3(600x
  10-9)(1.0)/5.0 x 10-3) 3.60 x 10-4 m
• hence difference is .072 mm

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Problem

• A thin flake of mica (n1.58) is used to cover
  one slit of a double slit arrangement. The
  central point on the screen is now occupied by
  what had been the seventh bright fringe (m7)
  before the mica was used. If ? 550 nm, what is
  the thickness of the mica?

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Solution

• With no mica seventh bright fringe corresponds to
  path difference dsin ? 7 ?
  
• with one slit covered, there is an additional
  path difference due to change of ? ?/n in the
  mica
• Let unknown thickness of mica be L
• path difference in mica compared to no mica is
  L/ ? - L/ ? ? nL -L 1.58L - L .58L
• to shift seventh bright fringe to center, this
  must correspond to 7 ? .58 L
• hence L 7 ? /(.58) 7(550)x10-9/.58 6.64?m

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Intensity of Interference Pattern

• Each slit sends out an electromagnetic wave with
  an electric field E(r,t)Emsin(kr-?t) where r is
  the distance from either slit to the point P on
  the screen
• the two waves are coherent
• net field at P is the superposition of two waves
  which have travelled different distances and
  hence have a phase difference

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Intensity of Interference Pattern

• net effect at point P is the superposition of two
  waves
• E(r,t) Emsin(kr1-?t) Emsin(kr2- ?t) gtsame
  wavelength and frequency but travel different
  distances r1 and r2 r1 ?L
• E(r,t) Emsin(kr1- ?t) Emsink(r1 ?L)- ?t
• Emsin(kr1- ?t) sin(kr1- ?t ?)
• where ?k ?L is phase shift due to path
  difference!
• sin(A)sin(B) 2 cos(A-B)/2sin(AB)/2
• E(r,t) 2Emcos(?) sin(kr1- ?t ?) ? ?/2

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?k ?L
??/2k ?L/2
?L d sin?
d
k 2?/?
??/2?d sin? /?
E(r,t) 2Emcos(?) sin(kr1-?t ?)
Amplitude 2Emcos(?)
Intensity? (amplitude)2
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Intensity of Interference Pattern

• Amplitude of electric field at P is
  2Emcos(?) 2Emcos(?d sin?/?)
• intensity of field ? the amplitude squared
• I I0 cos2 (?d sin?/?) where I0 4(Em)2
• I I0 when ? m? gt d sin?m?
• I 0 when ? (m1/2)? gt d sin?(m1/2)?
• I I0 cos2 (?) with ? ?d sin?/?

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Intensity of Double Slit
E E1 E2
I E2 E12 E22 2 E1 E2 I1 I2
interference lt vanishes if incoherent
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MC 41-12

• Three coherent equal intensity light rays arrive
  at P on a screen to produce an interference
  minimum of zero intensity.
• If any two of the rays are blocked, the intensity
  at is I1 . What is the intensity at P if only one
  is blocked?
• a) 0 b)I1/2 c) I1 d) 2I1 e) 4I1

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Solution
yy1sin(kx-?t?1) y2sin(kx-?t?2)
y1sin(kx-?t?3)

• Consider adding three vectors of equal length to
  get zero resultant vector
• choose ?10 , ?2 2?/3, ?3 4?/3
• add any two gt phase difference 2?/3
• amplitude 2y1cos(??/2)2y1cos(?/3)y1
• hence intensity is I1 gt c)