Chapter 25 Wave Optics – Lecture 20

25.1 Coherence and Conditions for Interference

25.2 The Michelson Interferometer

25.3 Thin-Film Interference

25.4 Light through a Single-Slit: Qualitative Behavior

25.5 Double-Slit Interference: Young’s Experiment

25.6 Single-Slit Diffraction: Interference of Light from a Single

Slit

25.7 Diffraction Gratings

25.8 Optical Resolution and Rayleigh Criterion

Young’s Double-Slit Experiment

• Light is incident onto two narrow

slits and after passing through

them strikes a screen

• When the two slits are both very

narrow, each slit acts as a

simple point source of outgoing

new spherical waves

Section 25.5

• The light intensity on the screen shows an interference

pattern showing conclusively that light is a wave

• Experiment was first carried out by Thomas Young

around 1800

Young’s Experiment, final

• Assume the slits are very narrow

• According the Huygen’s principle,

each slit acts as a simple source

with circular wave fronts as

viewed from above

• The light intensity on the screen

alternates between bright and

dark as you move along the

screen

• These areas correspond to

regions of constructive

interference and destructive

interference Section 25.5

Double Slit Analysis

• Need to determine the

path length between

each slit and the screen

• Assume W is very large

• If the slits are separated

by a distance d, then

the difference in length

between the paths of

the two rays is

ΔL = d sin θ

Section 25.5

Double Slit Analysis, cont.

• If ΔL is equal to an integral number of complete

wavelengths, then the waves will be in phase when

they strike the screen

• The interference will be constructive

• The light intensity will be large (bright fringes)

d sin θ = m λ m = 0, ±1, ±2, …

• If ΔL is equal to a half number of complete

wavelengths, then the waves will not be in phase

when they strike the screen

• The interference will be destructive

• The light intensity will be zero (dark fringes)

d sin θ = (m + ½) λ m = 0, ±1, ±2, …

Section 25.5

ΔL = d sin θ

Double-Slit Intensity Pattern

• The angle θ varies as

you move along the

screen

• Each bright fringe

corresponds to a

different value of m

• Negative values of m

indicate that the path to

those points on the

screen from the lower

slit is shorter than the

path from the upper slit

Section 25.5

Spacing Between Slits

• Notation:

• d is the distance between the slits

• W is the distance between the slits and the screen

• h is the distance between the adjacent bright fringes

• For m = 1,

• Since the angle is very small,

• sin θ ~ θ and θ ~ λ/d

• Between m = 0 and m = 1,

h = W tan θ

• For small angles, tan θ ~ θ and sin θ ~ θ

• Using the approximations, h = W θ = W λ / d

Section 25.5

sind

Interference with Monochromatic Light

• The conditions for interference state that the

interfering waves must have the same frequency

• This means they must have the same wavelength

• Light with a single frequency is called

monochromatic (one color)

• Light sources with a variety of wavelengths are

generally not useful for double-slit interference

experiments

• The bright and dark fringes may overlap or the total

pattern may be a “washed out” sum of bright and dark

regions

• No bright or dark fringes will be visible

Section 25.5

Single-Slit Interference

• Slits may be narrow

enough to exhibit

diffraction but not so

narrow that they can be

treated as a single point

source of waves

• Assume the single slit

has a width, w

• Light is diffracted as it

passes through the slit

and then propagates to

the screen

Section 25.6

Single-Slit Analysis

• The key to the calculation of

where the fringes occur is

Huygen’s principle

• All points across the slit act

as wave sources

• These different waves

interfere at the screen

• For analysis, divide the slit

into two parts

• Detailed analysis is given

in the supplemental slides

Section 25.6

Single-Slit Fringe Locations

• Destructive interference

will produce a dark fringe

• Detailed analysis is

given in the

supplemental slides

Section 25.6

• Conditions for destructive interference are

w sin θ = ±m λ m = 1, 2, 3, …

• The negative sign will correspond to a fringe below the

center of the screen

Single-Slit – Central Maximum

• The full angular width of the central bright fringe is

θ = 2 λ / w (from w sin θ = ±m λ m = 1, 2, 3, … )

• If the slit is much wider than the wavelength, the light

beam essentially passes straight through the slit with

almost no effect from diffraction

Section 25.6

• Conditions for destructive

interference are

w sin θ = ±m λ

m = 1, 2, 3, …

• Detailed analysis is

given in the

supplemental slides

Double-Slit Interference with Wide Slits

• When the slits of a

double-slit experiment

are not extremely

narrow, the single-slit

diffraction pattern

produced by each sit is

combined with the

sinusoidal double-slit

interference pattern

• A full calculation of the

intensity pattern is very

complicated (see the

attached supplemental

ppt slides)

Section 25.6

Diffraction Grating

• An arrangement of

many slits is called a

diffraction grating

• Assumptions

• The slits are narrow

• Each one produces a

single outgoing wave

• The screen is very far

away

Section 25.7

Diffraction Grating, cont.

• Since the screen is far away, the rays striking the

screen are approximately parallel

• All make an angle θ with the horizontal axis

• If the slit-to-slit spacing is d, then the path length

difference for the rays from two adjacent slits is

ΔL = d sin θ

• If ΔL is equal to an integral number of wavelengths,

constructive interference occurs

• For a bright fringe, d sin θ = m m = 0, ±1, ±2,

…

Section 25.7

Diffraction Grating, final

• The condition for bright

fringes from a diffraction

grating is identical to the

condition for constructive

interference from a

double slit

• The overall intensity

pattern depends on the

number of slits

• The larger the number of

slits, the narrower the

peaks

Section 25.7

Grating and Color Separation

• A diffraction grating will produce an intensity pattern on the screen for each color

• The different colors will have different angles and different places on the screen

• Diffraction gratings are widely used to analyze the colors in a beam of light

Section 25.7

Diffraction and CDs

• Light reflected from the

arcs in a CD acts as

sources of Huygens

waves

• The reflected waves

exhibit constructive

interference at certain

angles

• Light reflected from a

CD has the colors

“separated”

Section 25.7

Optical Resolution

• For a circular opening of

diameter D, the angle

between the central

bright maximum and the

first minimum is

• The circular geometry

leads to the additional

numerical factor of 1.22

Section 25.8

1.22

D

Telescope Example

Section 25.8

• If the two sources are sufficiently far apart, they can

be seen as two separate diffraction spots (A)

• Two objects will be resolved when viewed through an

opening of diameter D if the light rays from the two

objects are separated by an angle at least as large

as θmin

min

1.22'Rayleigh s Criterion

D

Diffraction by a Circular Aperture (1)

telescope example

Diffraction by a Circular Aperture (2)

• The diffraction pattern consists of a bright circular

region and concentric rings of bright anddark fringes.

• The first minimum for the

diffraction pattern of a circular

aperture of diameter D is located by sin 1.22D

geometric factor

1 1.22 1.22sinR

D D

sinsin

2 2

ax

a

For a single slit

diffraction, the first

dark fringe occurs at

Rayleigh’s criterion

Diffraction by a Circular Aperture (3)

2 c

1.22cD

1 1.22 1.22sinR

D D

c 2 c

c

Example: Hubble Space Telescope (1)

Example: Hubble Space Telescope (2) • The diameter of the Hubble Space

Telescope is 2.4 m

Question: What is the minimum angular

resolution of the Hubble Space

Telescope?

Answer:

Using Rayleigh’s Criterion with green light of wavelength 550 nm we get

which corresponds to the angle subtended by a dime located 64 km away

91 71.22 550 10 m

sin 1.22 2.8 102.4 m

Rd

Demo - Diffraction Gratings

• Because diffraction gratings produce widely spaced narrow maxima, they can be used to determine the wavelength of monochromatic light by rearranging

• Monochromatic light incident on a diffraction grating will produce lines on a screen at widely separated angles

sin 0, 1, 2,...d

mm

sin , ( 0, 1, )d m m

Demo

1sin 0, 1, 2,...m

md

Supplemental Slides

Single Slit Diffraction: First Dark Fringe (1)

• We start with light emitted from the top edge of the slit and from the center of the slit as shown

• To analyze the path difference we show an expanded version of our figure to the right

• Here we assume that the point P on the screen is far enough away that the rays r1 and r2 are parallel and make an angle with the central axis

C

Single Slit Diffraction: First Dark Fringe (2)

• We assume that the point P on the screen is far

enough away that the rays r1 and r2 are parallel and

make an angle with thecentral axis

• Therefore the path length difference

for these two rays is

• The criterion for the first dark fringe is

• Although we chose one ray originating from the top edge of the

slit and one from the middle of the slit to locate the first dark

fringe, we could have used any two rays that originated a/2

apart inside the slit

sinsin or

/ 2 2

x ax

a

sinsin

2 2

ax or a

Single Slit Diffraction: Second Dark Fringe (1)

• Now let’s consider four rays instead of two

• Here we choose a ray from the top edge of the slit and three more rays originating from points spaced a/4 apart

Single Slit Diffraction: Second Dark Fringe (2)

• The path length difference between the pairs of

rays (r1,r2), (r2,r3), (r3,r4) is given by

• The dark fringe is given by

sinsin or

/ 4 4

x ax

a

sin or sin 2

4 2

ax a

This equation describes the second dark fringe

At this point we could take six pairs and eight pairs and describe the third and fourth dark fringes, etc.

The final result is

sin or sin 2

4 2

aa

sin 1,2,3,...a m m

Single Slit Intensity

• If the screen is placed a sufficiently large distance from the slits, the angle will be small and we can make the approximation

• We can express the position of the dark fringes as

• The intensity of light passing through a single slit can be calculated but we will not present the derivation here

• The intensity I relative to Imax that we would get if there were no slit is

• We can see that this expression for the intensity will be zero for sin() = 0, which means = m for m = 1, 2, 3, … consistent with

sin or 1,2,3,...ay m L

a m m y mL a

2

max

sin where sin

aI I

sin tan /y L

sin 1,2,3,...a m m

Single Slit Intensity: Dark Fringes

• We can write

which gives the same result for the diffraction minima as our

Huygens’ construction

• If the screen is placed a sufficiently large distance from the slits,

we can make the small angle approximation

and write

sin sin or sin ( 1,2,3,...)a a

m a m m

ay

L

2

max

sinI I

,L

y ma

or 1,2,3,...

ay m Lm y m

L a

Example - Single Slit Intensity Distribution

• If we take L = 2.0 m, a = 5.010-6 m = 5.0x 103 nm, and = 550

nm, we get the following intensity distribution

• Note that a >

1,2,3,...m L

y ma

2

max

sin sin

aI I

ay

L

,L

y ma

Review - Fringes on a Distant Screen from Double Slits

• The distance y of the bright fringes (constructive

interferences) from the central maximum

along the screen

• The distance y of the dark fringes (destructive interferences)

from the central maximum along the screen as

• In addition to the interference pattern described above, there will be also

the diffraction pattern originating from each of double slits

0, 1, 2,...m L

y m m md

1

0, 1, 2,...2

Ly m m m m

d

Double Slits - Interference + Diffraction (1)

Double Slit (slit width a<< Single Slit (slit width a~

Double Slit (slit width a~

Interference Difraction

0, 1, 2,...m L

y m m md

1

0, 1, 2,...2

Ly m m m m

d

2

max4 cosdy

I IL

d is the distance between two slits

Pattern for single slit diffraction, a ~ or a >

Pattern for double slit interference, a <<

Double Slits - Interference + Diffraction (2)

Intensity pattern for a double slit including interference and diffraction assuming L = 2.0 m, a = 5.010-6m = 5.0x103 nm, d = 1.010-5 m, and = 550 nm d is two slits spacing

Double Slits - Interference + Diffraction (3) (a ~

• With diffraction effects the intensity of the interference pattern

from double slits is given by

• If the screen is placed a sufficiently large distance from the slits

then we can write

2

2

max

sincos sin sin

a dI I

and ay dy

L L

Real Life Double Slits Diffraction Pattern

Diffraction minima