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Interference of light waves
Conditions for interference
Young’s double-slit experiment
  Intensity distribution of double-slit interference
pattern
  Change of phase due to reflection
  Interference in thin films


Young’s double-slit
experiment
Condition for interference
The sources must be coherent, that is, they
must maintain a constant phase difference
with respect to each other.
  The sources should be monochromatic, that
is, they have the same wavelength.

A common method to produce two coherent
light sources is to use one monochromatic
source to illuminate a barrier containing two
small openings.
Double-slit interference
Path difference:
δ = r2 − r1 = d sin θ
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Constructive interference:
δ = d sin θ = mλ
m = 0,±1,±2,...
Destructive interference:
1
δ = d sin θ = (m + )λ
2
m = 0,±1,±2,...
y = L tan θ ≈ L sin θ
λL
λL
1
m; ydark =
(m + )
d
d
2
ybright =
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Example 37.1 Measuring the
wavelength of a light source
A viewing screen is separated from a double-slit source by 1.2 m.
The distance between the two slits is 0.030 mm. The second-order
bright fringe is 4.5 cm from the center line. (a) Determine the
wavelength of the light, (b) Calculate the distance between adjacent
bright fringes.
λL
λL
ym =
m
y2 =
2
d
d
y2 d (3.0 ×10 −5 m)(4.5 ×10 −2 m)
=
= 5.6 ×10 −7 m = 560 nm
2L
2(1.2m)
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λL (5.6 ×10 −7 m)(1.2m)
ym+1 − ym =
=
= 2.2 ×10 −2 m
d
(3.0 ×10 −5 m)
λ=
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Intensity distribution
E1 = Emax sin ωt, E2 = Emax sin(ωt + φ )
φ=
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2 πδ 2 π
=
d sin θ
λ
λ
EP = E1 + E2 = Emax [sin ωt + sin(ωt + φ )]
A+ B
A−B
)cos(
)
2
2
φ
φ
EP = 2Emax cos( )sin(ωt + )
2
2
φ
2
I ∝< EP2 >= 2Emax
cos2 ( )
2
2 φ
I = I max cos ( )
2
π
πd
I = I max cos2 ( d sin θ ) = I max cos2 ( y)
λ
λL
sin A + sin B = 2 sin(
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Multiple-slit interference
patterns
I max ∝ N 2
Change of phase due to
reflection
An electromagnetic wave
undergoes a phase change of
180o upon reflection from a
medium that has a higher
index of refraction than the
one in which the wave is
traveling.
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Interference in thin films
Rules for thin film interference
Identify the thin film causing the interference
Phase relationship between the reflected wave
at the upper surface and the one at the lower
surface
  Phase difference caused by path difference
and phase change that may occur upon
reflection (180o change when rays travel from
lower to higher index of refraction, otherwise no
change)
  Constructive if the effective path difference is
mλ and destructive if the path difference is (m
+1/2)mλ

For destructive interference:
2t = mλn
m = 0,±1,±2,...
For constructive interference
1
2t = (m + )λn
2
m = 0,±1,±2,...
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where
λn =
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λ
n
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Newton’s Rings
r ≈ mλR / n
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Example 37.2 Nonreflective
coating for solar cells
Determine the minimum film
thickness that produces the least
reflection at a wave length of 550
nm.
•  The reflective light is a
minimum when two rays meet
the destructive interference
•  180o phase change for both
rays upon reflection
•  Destructive when 2nt=λ/2, t= λ/4n = 94.8 nm
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Example 37.3
Young’s double-slit experiment is performed with 589-nm light and a
distance of 2.00 m between the slits and the screen. The tenth
interference minimum is observed 7.26 mm from the central
maximum. Determine the spacing of the slits. Example 37.5
Interference effects are produced at point P on a screen as a result of
direct rays from a 500-nm source and reflected rays from the mirror, as
shown in the figure below. Assume the source is 100 m to the left of the
screen and 1.00 cm above the mirror. Find the distance y to the first
dark band above the mirror. Example 37.4
Two narrow parallel slits separated by 0.850 mm are illuminated by
600-nm light, and the viewing screen is 2.80 m away from the slits. (a)
What is the phase difference between the two interfering waves on a
screen at a point 2.50 mm from the central bright fringe? (b) What is
the ratio of the intensity at this point to the intensity at the center of a
bright fringe? Example 37.6
A thin film of oil (n = 1.25) is located on a smooth wet pavement.
When viewed perpendicular to the pavement, the film reflects most
strongly red light at 640 nm and reflects no blue light at 512 nm. How
thick is the oil film? 4