# Interference in Thin Films

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```Chapter 24
The Wave Nature
of Light
AP Physics B
Mrs. Wallace
Wave Optics
Sections
24-01 Conditions for Interference
24-02
24-03
24-04
24-06
Young’s Double-Slit Experiment
Change of Phase Due to Reflection
Interference in Thin Films
Diffraction
24-07 Single-Slit Diffraction
24-08 The Diffraction Grating
Wave Optics 24 (02 of 41)
Conditions for Interference
For observable interference effects, sources must be
1. Coherent
Waves must maintain
a constant phase
relationship
2. Monochromatic
Waves must have
the same wavelength
Wave Optics 24 (03 of 41)
Chapter 24: Wave Optics
Two light sources are said to be coherent if they
(A) are of the same frequency.
(B) are of the same frequency, and maintain a constant
phase difference.
(C) are of the same amplitude, and maintain a constant
phase difference.
(D) are of the same frequency and amplitude.
Wave Optics 24 (04 of 41)
Young’s Double-Slit Experiment
If light is a wave, interference effects will be seen, where one
part of wavefront can interact with another part.
One way to study this is to do a double-slit experiment:
Wave Optics 24 (05 of 41)
Young’s Double-Slit Experiment
Wave Optics 24 (06 of 41)
Young’s Double-Slit Experiment
The interference occurs because each point on the screen is not
the same distance from both slits. Depending on the path length
difference, the wave can interfere constructively (bright spot) or
destructively (dark spot).
Wave Optics 24 (07 of 41)
Chapter 24: Wave Optics
What principle is responsible for alternating light and
dark bands when light passes through two or more
narrow slits?
(A) refraction
(B) polarization
(C) dispersion
(D) interference
Wave Optics 24 (08 of 41)
Young’s Double-Slit Experiment
We can use geometry to find the conditions for
constructive and destructive interference:
Bright
(constructive interference)
d sin θ  mλ ,
m  0, 1, 2, ....
Dark
(destructive interference)


d sin θ  m  1 λ ,
2
m  0, 1, 2, ....
Wave Optics 24 (09 of 41)
Young’s Double-Slit Experiment
PS2 - PS1 = d sin(q)
P
S1
q
d
Constructive Interference
S2
q
d sin(θ)  mλ
m  0,  1,  2, . . .
Destructive Interference
1

d sin(θ)   m   λ
2

m  0,  1,  2, . . .
screen
Wave Optics 24 (10 of 41)
Young’s Double-Slit Experiment
sin(θ)  tan(θ) 
Bright
y
L
y
S1
d
q
Constructive Interference
S2
L
d sinθ   mλ
y (bright ) 
mLλ
d
m  0,  1,  2,
Δy 
Lλ
d
screen
Wave Optics 24 (11 of 41)
Young’s Double-Slit Experiment
Dark
y
sin(θ)  tan(θ) 
L
y
S1
d
q
Destructive Interference
S2
1

d sin(θ)   m  
2

m  0,  1,  2, . . .
L
1

m


 Lλ
2
y dark   
d
Lλ
Δy 
d
screen
Wave Optics 24 (12 of 41)
Chapter 24: Wave Optics
In a Young's double slit experiment, if the separation
between the slits decreases, what happens to the distance
between the interference fringes?
(A) It decreases.
(B) It increases.
Lλ
Δy 
d
(C) It remains the same.
(D) There is not enough information to determine.
Wave Optics 24 (13 of 41)
Chapter 24: Wave Optics
At the first minima on either side of the central bright
spot in a double-slit experiment, light from each opening
arrives
(A) in phase.
(B) 90&deg; out of phase.
(C) 180&deg; out of phase.
(D) none of the given answers
Wave Optics 24 (14 of 41)
Chapter 24: Wave Optics
At the second maxima on either side of the central bright
spot in a double-slit experiment, light from
(A) each opening travels the same distance.
(B) one opening travels twice as far as light from the other
opening.
(C) one opening travels one wavelength of light farther than
light from the other opening.
(D) one opening travels two wavelengths of light farther than
light from the other opening.
Wave Optics 24 (15 of 41)
Chapter 24: Wave Optics
What principle is responsible for light spreading as it
passes through a narrow slit?
(A) refraction
(B) polarization
(C) diffraction
(D) interference
Wave Optics 24 (28 of 41)
Diffraction
Wave Optics 24 (27 of 41)
Single-Slit Diffraction
The resulting pattern of light and dark stripes is called a
diffraction pattern.
This pattern arises because different points along a slit create
wavelets that interfere with each other just as a double slit would.
The minima of the single-slit diffraction pattern occur when
D sin θ  mλ
m  1, 2, 3, ....
Wave Optics 24 (29 of 41)
Single-Slit Diffraction
q
d
L
y2
sin θ  2λ
y1
sin θ  λ
0
sin θ  0
 y1
sin θ   λ
 y2
sin θ   2λ
d
d
d
d
Wave Optics 24 (30 of 41)
Chapter 24: Wave Optics
A person gazes at a very distant light source. If she now
holds up two fingers, with a very small gap between them,
and looks at the light source, she will see
(A) the same thing as without the fingers, but dimmer.
(B) a series of bright spots.
(C) a sequence of closely spaced bright lines.
(D) a hazy band of light varying from red at one side to
blue or violet at the other.
Wave Optics 24 (31 of 41)
Single-Slit Diffraction
A single slit of width 0.30 mm is
illuminated with 650 nm light. An
interference pattern is formed on a
screen 2.5 m away. Find the distance
from the first dark band to the central
bright band.
λ
sin θ  
d
λ y

d L
y
tanθ  
L
Lλ
y
d
sinθ   tanθ 
y
d

2.5 m 6.5x10 7 m

3.00 x 10 4 m
y  0.0054 m
Wave Optics 24 (32 of 41)
Single-Slit Diffraction
A single slit of width 0.14 mm is illuminated by monochromatic
light and interference bands are observed on a screen 2.0 m away.
If the second dark band is 16 mm from the central bright band,
what is the wavelength of the light?
D sin θ  mλ
m2
2λ
 sin θ 
D
y
tan θ   sin θ
L
y
yD
λ
2L
L
q
D
y 2λ


L D
0.016 m0.00014 m 

22.0 m 
λ  5.6 x 10 7 m  560 nm
Wave Optics 24 (33 of 41)
The Diffraction Grating
A diffraction grating consists of a large number of equally
spaced narrow slits or lines. A transmission grating has slits,
while a reflection grating has lines that reflect light.
Two slits
The more lines or
slits there are, the
narrower the peaks.
Six slits
Wave Optics 24 (34 of 41)
Chapter 24: Wave Optics
Consider two diffraction gratings with the same slit separation,
one grating has 3 slits and the other 4 slits. If both gratings are
illuminated with a beam of the same monochromatic light, make
a statement concerning the separation between the orders.
(A) The grating with 3 slits produces the greater separation
between orders.
(B) The grating with 4 slits produces the greater separation
between orders.
(C) Both gratings produce the same separation between orders.
(D) Both gratings produce the same separation between orders,
but the orders are better defined with the 4-slit grating.
Wave Optics 24 (35 of 41)
The Diffraction Grating
The maxima of the diffraction pattern are defined by
d sin θ  mλ ,
m  0, 1, 2, 3, ......
Wave Optics 24 (36 of 41)
The Diffraction Grating (Problem)
A 3500-line/cm grating produces a third-order fringe at
a 28.0o angle. What is the wavelength of light being used?
d sin θ  mλ
λ
d sin θ
m
1
λ
350,000

sin 28.0o 
lines
m
3
 4.47 x 10 7 m
 447 nm
Wave Optics 24 (37 of 41)
The Diffraction Grating (Problem)
A diffraction grating has 6.0 x 105 lines/m. Find the angular
spread in second-order spectrum between red light of wavelength
7.0 x 107 m and blue light of wavelength 4.5 x 107 m.
B
qB
qR
R

Dq
d sin θ  mλ
m=2


 2 7.0 x 10  7 m 
  57.1o
θ R  sin 1 
1




5
 6.0 x 10 line/m 



 2 4.5 x 10  7 m 
  32.7o
θB  sin 1 
1




5
 6.0 x 10 line/m 

Δθ  57.1o  32.7o 

24.4o
Wave Optics 24 (39 of 41)
The Diffraction Grating
The Spectrometer and Spectroscopy
A spectrometer makes accurate measurements of wavelengths
using a diffraction grating or prism.
Wave Optics 24 (39 of 41)
The Diffraction Grating
The Spectrometer and Spectroscopy
The wavelength can be determined to high accuracy by
measuring the angle at which the light is diffracted.
λ
d
sin θ
m
Atoms and molecules can be identified when they are in a
thin gas through their characteristic emission lines.
Wave Optics 24 (40 of 41)
Change of Phase Due to Reflection
A reflected light wave undergoes a phase change of 180o.
Wave Optics 24 (16 of 41)
Change of Phase Due to Reflection
A reflected light wave undergoes no phase change.
Wave Optics 24 (17 of 41)
Chapter 24: Wave Optics
When a beam of light, which is traveling in glass,
strikes an air boundary, there is
(A) a 90&deg; phase change in the reflected beam.
(B) a 180&deg; phase change in the reflected beam.
(C) a 45&deg; phase change in the reflected beam.
(D) no phase change in the reflected beam.
Wave Optics 24 (18 of 41)
Interference in Thin Films
Another way path lengths can differ, and waves interfere, is
if they travel through different media.
If there is a very thin film of material – a few wavelengths
thick – light will reflect from both the bottom and the top of
the layer, causing interference.
This can be seen in soap bubbles and oil slicks, for example.
air
1λ
2
glass
No
phase
change
phase
change
air
Wave Optics 24 (19 of 41)
Interference in Thin Films
If
t
t  1 λ  Dark (minima)
2
t  1 λ  Bright (maxima)
4
Wave Optics 24 (20 of 41)
Interference in Thin Films
a
b
t
n


The phase displacement of (b)
relative to (a) is
1 λ  Due to reflection
2
2tn  Due to path difference


When 12   2tn is 12  , 32  , 52  , ... m  12 


When 12   2tn is  , 2 , 3 , ... m 
There is darkness
There is brightness
Wave Optics 24 (21 of 41)
Interference in Thin Films (Problem)
A tanker has dumped a large quantity of kerosene into the gulf
creating a large slick on the top of the water. You are looking
straight down from an airplane onto a region of the slick where
its thickness is t = 460 nm.
Which wavelength of visible light is reflected the greatest?
a
2tn'  mλ
b
air
n = 1.0
kerosene
n’ = 1.2
water
n’’ = 1.3
t
2tn'
λ
m
2460 nm  1.2
λ
m
m  1  λ  1104 nm
m  2  λ  552 nm
m  3  λ  368 nm
Wave Optics 24 (22 of 41)
Interference in Thin Films
A tanker has dumped a large quantity of kerosene into the gulf
creating a large slick on the top of the water. You are a scubadiver directly under the slick where its thickness is t = 460 nm.
Which wavelength of visible light is the transmitted intensity
the greatest?
1 λ  2tn'  mλ
2
n = 1.0
air
kerosene
n’ = 1.2
water
n’’ = 1.3
a
b
2tn'
λ
1
m

t
2
2460 nm 1.2
λ
m1
m  1  λ  2208 nm
m  2  λ  736 nm
m  3  λ  441 nm
m  4  λ  315 nm
2
Wave Optics 24 (23 of 41)
Interference in Thin Films
One can also create a thin film of air by creating a wedgeshaped gap between two pieces of glass.
Wave Optics 24 (24 of 41)
Interference in Thin Films
1

1
Dark  2   2t   m   
2

m=1
m=0
D
A
R
K
t 0
B
R
I
G
H
T
D
A
R
K
λ
t
2
m=2
B
R
I
G
H
T
D
A
R
K
tλ
Wave Optics 24 (25 of 41)
Interference in Thin Films
Problem Solving: Interference
Interference occurs when two or more waves arrive
simultaneously at the same point in space.
Constructive interference occurs when the waves are in phase.
Destructive interference occurs when the waves are out of phase.
An extra half-wavelength shift occurs when light
reflects from a medium with higher refractive index.
Wave Optics 24 (26 of 41)
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