Chapter 25 Wave Optics – Lecture 19

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 frm a Single

Slit

25.7 Diffraction Gratings

25.8 Optical Resolution and Rayleigh Criterion

Wave Optics vs. Geometric Optics

• When discussing image characteristics over

distances much greater than the wavelength,

geometric optics is extremely accurate

• When dealing with sizes comparable to or smaller

than the wavelength, wave optics is required

• Examples include interference effects and propagation

through small openings

Introduction

Interference

• One property unique to waves is interference

• Interference is a wave phenomenon

• Interference of sound waves can be produced by two speakers

Section 25.1

Constructive Interference

• Require the phase difernece of 0,

2, 4 .... or a path difference x given by

+

=

0, 1, 2,...x m m m m

Destructive Interference

• Here the phase difference between the

two waves is radians, 180, or /2

1

0, 1, 2,...2

x m m m m

+

=

Summary - Interference

• If light waves are traveling from some point, then the

phase difference can be related to the path difference between

the two waves

• The criterion for constructive interference is given by a path

difference x given by

• Destructive interference will take place if the path difference x

is a half wavelength plus an integer times the wavelength

• The inter m is called the order of the fringe

0, 1, 2,...x m m m m

1

0, 1, 2,...2

x m m m m

Summary- Interference Conditions

• For constructive interference, Δx = m λ

• For destructive interference, Δx = (m + ½) λ

• m is an integer in both cases

• If the interference is constructive, the light

intensity at the detector is large

• Called a bright fringe

• If the interference is destructive, the light

intensity at the detector is zero

• Called a dark fringe

Section 25.2

Interference

• Sunlight is composed of light containing a broad range

of frequencies and corresponding wavelengths

• We often see different colors

separated out of sunlight by

refraction in rainbows

• We also sometimes see various

colors from sunlight due to

constructive and destructive

interference phenomena on the

surface of DVD’s or CD’s

or in thin layers of oil or water

Michelson Interferometer

• The Michelson

interferometer is based

on the interference of

reflected waves

• Two reflecting mirrors

are mounted at right

angles

• A third mirror is partially

reflecting

• Called a beam splitter

Section 25.2

Michelson Interferometer, cont.

• The incident light hits the beam splitter and is divided

into two waves

• The waves reflect from the mirrors at the top and right

and recombine at the beam splitter

• After being reflected again from the

beam splitter, portions of the waves

combine at the detector

• The only difference between the two

waves is that they travel different

distances between their respective

mirrors and the beam splitter

• The path length difference is

ΔL = 2L2 – 2L1 Section 25.2

Michelson Interferometer, final

• The path length

difference is related to the

wavelength of the light

• If N is an integer, the two

waves are in phase and

produce constructive

interference

• If N is a half-integer the

waves will produce

destructive interference

LN

D=

l

Section 25.2

Measuring Length with a Michelson

Interferometer • Use the light from a laser and adjust the mirror to give

constructive interference

• This corresponds to one of the bright fringes

• The mirror is then moved, changing the path length

• The intensity changes from high to zero and back to high,

every time the path length changes by one wavelength

• If the mirror moves through N bright fringes, the distance

d traveled by the mirror is (for round trip) or N

d =2

l

Section 25.2

• The accuracy of the measurement depends on the

accuracy with which the wavelength is known

• Many laboratories use helium-neon lasers to make very

precise length measurements

d N2

He Ne nm632.99139822

• When a light wave

passes from one

medium to another, the

waves must stay in

phase at the interface

• The frequency must be

the same on both sides

of the interface

Section 25.3

Light Traveling in an Optical Medium (1)

Light Traveling in an Optical Medium (2)

• We have seen that the wavelength of light changes when traveling in an optical medium with index of refraction greater than one

• Taking with case 1 as a vacuum and case 2 as a medium with index of refraction n, we have found out that we can write

• Remembering that v = f we can write the frequency fn of light traveling in a medium as

• So the frequency does not change !

n

v

c n

/ ( / )n

n

v cf v v c f

1 2

1 2

tv v

nf f

Light Traveling in an Optical Medium (3)

• So the frequency of light traveling in an optical medium with n >

1 is the same as the frequency of that light traveling in vacuum

• We perceive color by frequency rather than wavelength

• Thus placing an object under water does not change our

perception of the color of the object

• Easy to demonstrate: take a colored object and put it in a jar of

water. Water has index of refraction n = 1.33. The object

appears to have the same color under water as in air

Phase Change and Reflection

• When a light wave reflects from a surface it may be

inverted

• Inversion corresponds to a phase change of 180°

• There is a phase change whenever the index of

refraction on the incident side is less than the index

of refraction of the opposite side

• If the index of refraction is larger on the incident side

the reflected ray is not inverted and there is no

phase change

Section 25.3

Phase Change and Reflection, Diagram

Section 25.3

Phase Changes in a Thin Film

• The total phase change in a thin film must be

accounted for

• The phase difference due to the extra distance

traveled by the ray

• Any phase change due to reflection

• For a soap film on glass, nair < nfilm < nglass

• There are phase changes for both reflections at the

soap-film interfaces

• The reflections at both the top and bottom surfaces

undergo a 180° phase change no phase change !

• The nature of the interference is determine only by

the extra path length

Section 25.3

Thin-Film Interference (1)

• Assume a thin soap film rests on a flat glass

surface (Fig. A and Fig. B)

• The upper surface of the soap film is similar to

the beam splitter in the interferometer

• It reflects part of the incoming light and allows the

rest to be transmitted into the soap layer after

refraction at the air-soap interface Section 25.3

Thin-Film Interference (2)

• The transmitted ray is partially reflected at the bottom

surface

• The two outgoing rays meet the conditions for interference

• They travel through different regions

• One travels the extra distance through the soap film

• They recombine when they leave the film

• They are coherent because they originated from the same

source and initial ray Section 25.3

Thin-Film Interference (3), nair < nfilm > nair

• Assume the soap bubble is

surrounded by air

• There is a phase change at the top

of the bubble

• There is no phase change at the

bottom of the bubble

• Since only one wave undergoes a

phase change, the interference

conditions are

film

film

m

d constructive interferencen

md destructive interference

n

æ ö+ç ÷

è ø=

=

1

22

2

l

l

Section 25.3

Thin-Film Interference (4), White Light

• Each color can interfere

constructively, but at

different angles

• Blue will interfere

constructively at a

different angle than

red

• When you look at the

soap film the white light

illuminates the film over

a range of angles

Section 25.3

Thin Film Interference (5) • When light travels from an optical medium with an index of refraction n1 into

a second optical medium with index of refraction n2,

(1) The light can be transmitted through the boundary

no change of the phase of light

(2) The light can be reflected

• If n1 < n2, the phase of the reflected wave will be changed by or

• If n1 > n2 then there will be no phase change

/ 2

Oil floating on water: The color seen corresponds to the

wavelength of light that is interfering constructively

n2

n1

n1

n2

• Due to the phase change of 1800, the criterion for

constructive interference is given by

• The minimum thickness tmin that will produce contructive

interference corresponds to (for m=0)

• Note that this result applies only to the case where we

have a material with index of refraction n and air on both

sides, like a soap bubble

min4

airtn

1

2 0, 1, 2,...2

x m t m m m

min min

12 0 4 =

2 4 4

airx m t m t tn

Thin Film Interference (6), nair < nfilm > nair

• The criterion for destructive interference is given by

• The minimum thickness tmin that will produce destructive

interference corresponds to (for m=1)

• Note that this result applies only to the case where we

have a material with index of refraction n and air on both

sides, like a soap bubble

min2

airtn

2 , 1 0, 1, 2,...x m t m m m m

1 1 1 2

2 2 2 1

/

/

v c n n

v c n n

2 2, 1

air

air air

n

n n

1 2

1 2

tv v

min min2 1 2 = 2 2

airx m t m t tn

Thin Film Interference (7), nair < nfilm > nair

Thin-Film Interference, nair < nfilm > nair

• Equations are

• These equations apply whenever

nair < nfilm > n(substance below the film)

film

film

md constructive interference

n

m

d destructive interferencen

2

1

22

Section 25.3

Thin Film Interference (8) - Demo

• The reflected wave undergoes a phase shift

of half a wavelength when it is reflected

because nair < n

• The light that is transmitted has no phase

shift and continues to the back surface of the

film

• At the back surface, again part of the

wave is transmitted and part of the wave is

reflected

• The reflected light has no phase shift because n >

nair and travels back to the front surface of the film

• The transmitted light has traveled a longer distance

than the originally reflected light and has a phase

shift given by the path length difference that is

twice the film thickness t

soap bubble

Phase Changes in a Thin Film, nair < nfilm < nglass

• For a soap film on glass, nair < nfilm < nglass

• There are phase changes for both reflections at the

soap-film interfaces

• The reflections at both the top and bottom surfaces

undergo a 180° phase change no phase change

!

• For nair < nfilm < n(substance below the film)

Section 25.3

film

film

md constructive interference

n

m

d destructive interferencen

2

1

22

Antireflection Coatings (1)

• Nearly any flat piece of

glass may act like a

partially reflecting mirror

• To avoid reductions in

intensity due to this

reflection, antireflective

coatings may be used

• The coating makes a

lens appear slightly dark

in color when viewed in

reflected light

Section 25.3

Antireflective Coatings (2)

• Many coatings are

made from MgF2

• nMgF2 = 1.38

• There is a 180° phase

change at both

interfaces

• Destructive interference

occurs when

Section 25.3

MgF MgF

m

d dn n

2 2

1

22

4

• The light transmitted through the

coating has no phase change: When it

is reflected from the lens, it will

undergo a phase change

• the criterion for destructive interference

is

n=1

n=1.38 n=1.51

1

2 0, 1, 2,...2

air

coating

m t m m mn

min

4

air

coating

tn

• Light reflected at the surface of

the coating will undergo a phase

change of half a wavelength

because nair < ncoating

Antireflective Coatings (3) – more details

• The criterion for destructive interference is given by

• The minimum thickness tmin that will produce destructive

interference corresponds to (for m=0)

• Note that this result applies to the case where we have a

material with index of refraction n coated on a lens and

air on one side, like a coated lens.

min4

airtn

1

2 0, 1, 2,...2

x m t m m m

min min

12 0 4 =

2 4 4

airx m t m t tn

Antireflective Coatings (4) – more details

Summary - Thin-Film Interference

• For nair < nfilm < n(substance below the film)

• For nair < nfilm > n(substance below the film)

Section 25.3

film

film

md destructive interference

n

m

d costructive interferencen

2

1

22

film

film

md constructive interference

n

m

d destructive interferencen

2

1

22

Light Through a Single Slit

• Light passes through a slit or opening and then

illuminates a screen

• As the width of the slit becomes closer to the

wavelength of the light, the intensity pattern on the

screen and additional maxima become noticeable

Section 25.4

Single-Slit Diffraction

• Water wave example of

single-slit diffraction

• All types of waves

undergo single-slit

diffraction

• Water waves have a

wavelength easily

visible

• Diffraction is the

bending or spreading of

a wave when it passes

through an opening

Section 25.4

Huygen’s Principle

• It is useful to draw the

wave fronts and rays for

the incident and

diffracting waves

• Huygen’s Principle

can be stated as all

points on a wave front

can be thought of as

new sources of

spherical waves

Section 25.4