engineering physics

Topic 1

Interference and Diffraction

Spring 2013 1 ENGG 2520 Engineering Physics II

(Halliday/Resnick/Walker Ch.35, 36)

Spring 2013 ENGG 2520 Engineering Physics II 2

Interference and Diffraction

• Introduction to Interference

• Young’s Interference Experiment

• Intensity of Interference Pattern

• Thin Film Interference

• Optical Interferometers

• Introduction to Diffraction

• Single Slit and Double Slit Diffractions

• Diffraction Grating

• Diffraction Resolution Limit


Interference in Nature

butterfly

peacock

soap film and bubble beetle

CD/DVD


What is Interference? Interference: a physical phenomenon in which two or more waves superimpose

to form a resultant wave. It usually refers to the interaction of waves that are

coherent/correlated with each other. Optical interference is the interference

between light waves and is applied in many branches of science and engineering.

The resultant wave is obtained by the superposition principle. Its amplitude

depends on the phase difference between the interacting waves,

In-phase interference → amplitude doubling

Out-of-phase interference → amplitude cancelling

The blue colour on the top surface of a

Morpho butterfly wing is due to optical

interference. It shifts in color as you

change your viewing angle.


Young’s Interference Experiment

Thomas Young

(1773-1829)

Young’s interference experiment (Young’s double slit interference expt): ~1801

Significance: proved that light is a wave, contradicted Isaac Newton’s view

Formation of wave interference pattern

Optical interference

fringes

http://en.wikipedia.org/wiki/File:Thomas_Young_(scientist).jpg


Optical Path Difference

(1.1) sindL

parallel and rays

,For

21

rr

dD


Constructive and Destructive

Interferences

Bright fringes (constructive interference, maxima): L must be either zero or an

integer number of wavelengths. Appear at angles satisfying

Dark fringes (destructive interference, minima): L must be an odd multiple of

half a wavelength. Appear at angles satisfying

(1.2) 2,... 1, 0,,sin mmd

(1.3) 2,... 1, 0,,2

1sin

mmd


Fringe Spacing

Consider near pattern center, i.e. small

(1.4) tansinD

y

Spacing between 2 bright fringes:

(1.5)

1

sinsin 1

1

d

D

d

mD

d

mD

DD

yy

mm

mm

Conclusion: The spacing between

neighboring bright fringes remains

unchanged when and d are small.


Example: Measuring Plastic Thickness

Plastic block inserted → 1 (first order) bright fringe

moves to center for = 600nm

Given that a plastic

block of refractive

index 1.5 is placed on

the upper slit,

determine the plastic

thickness.

Let L be the plastic block thickness, the path difference at

the center bright fringe is and is equal to L x (1.5-1)

m102.1

m106005.0

6

9

L

L


Wave Coherence For the interference pattern to

appear on the screen, the light

waves reaching any point on the

screen must have a phase

difference that does not vary in

time. When the phase difference

remains constant, the light from

slits S1 and S2 are said to be

completely coherent.

If the phase difference constantly

changes in time, the light is said

to be incoherent.

Incandescent light – incoherent light source

Laser – highly coherent light source

Sunlight – partially coherent, i.e., phase difference is constant only if the

points under consideration are very close


Intensity of Interference Pattern

)71(sin

)61(sin

0

0

.tωEE

.t ωEE

2

1

Consider electric field components of the light waves at point P on the screen:

It can be shown (proof to follow) that the intensity (power/area) at P is

Constant phase difference → coherent

(1.9) sin2

where(1.8) 2

cos4 2

0

dII

Total energy is

conserved


Proof:

Recall

2sin

2cos2sinsin

BABABA

Hence, the electric field at point P is

(1.10) 2

sin2

cos2sinsin 00021

tEtEtEEEE

Constant amplitude

Thus, the electric field amplitude is changed by a factor of .2

cos2

Intensity (field amplitude)2 → changed by (1.8) 2

cos4 2

0

II.

2cos4 2

sin

2d Phase difference

Path difference (1.1)

(1.9) sin2

d

Intensity of Interference Pattern


Thin Film Interference

Ray representation of incident, transmitted, and reflected light waves.

Phase difference of the reflected waves is determined by:

• phase shift by reflection

• different physical path lengths

• different RI different optical path lengths


Phase shift on normal reflection (RI = refractive index):

Incident from high RI to low RI (e.g from water/glass to air) → no phase shift

Incident from low RI to high RI (e.g from air to water/glass) → phase shift

Phase Shift by Reflection

http://www.schoolphysics.co.uk

i.e., /2


Maxima and Minima Air Water Air

/2 reflection shift no phase shift

Assume normal incidence,

(1.11) 2,... 1, 0,,2

12 2

mmnL

Destructive interference (Minima): (1.12) 2,... 1, 0,,2 2 mmnL

Constructive interference (Maxima):


Different soap film thicknesses → maxima at different wavelengths

Soap Film and Bubble

Dark region: L<< (about to burst)

Path difference: negligible

Phase difference dominated by

phase shift in reflection =

m =1

m =2

m =3


Thin Film Coating Thin film coatings are widely used in glasses, camera lens, optical

instruments, semiconductor optical devices, fiber devices, etc. to

improve their performances through reduction or enhancement of

optical reflections/transmission of specific wavelengths.

CUHK Photonic Packaging Lab

Coating Machine


Example: Antireflection Coating A glass lens (RI=1.50) is coated with MgF2 (RI=1.38) to reduce optical

reflections. Determine the minimum coating thickness required to eliminate

reflections at the center of the visible spectrum, i.e. =550 nm. Assume

incident light is perpendicular to the lens surface.

Reflection is minimized if we

choose coating thickness L s.t.

waves reflected from the interfaces

are exactly out of phase.

2,... 1, 0,,2

12 2

mmnL

For minimum thickness, m = 0

nm 6.9938.14

nm 550

4 2

n

L


Optical Interferometer Interferometer: an optical instrument/device that measures lengths or

changes in length (or RI) with great accuracy by means of interference fringes.

Common types of Interferometer:

mirror Beam splitter

Mach-Zehnder Interferometer

Sagnac Interferometer

Michelson Interferometer

Fabry-Perot Interferometer


The Nobel Prize in Physics 1907 was awarded to Albert A. Michelson

"for his optical precision instruments and the spectroscopic and

metrological investigations carried out with their aid". He was the first

American to receive the Nobel Prize in Sciences.

Michelson Interferometer

http://en.bj-force.net/

MIPAS Interferometer Assembly (Michelson

Interferometer for Passive Atmospheric Sounding) https://earth.esa.int/web/guest/missions/esa-operational-eo-

missions/envisat/instruments/mipas

http://en.wikipedia.org/wiki/File:Albert_Abraham_Michelson2.jpg


)13.1( 12

nL

For each path change of (phase change = 2), the pattern is shifted by one

fringe. Thus, by counting the number of fringes through which the material causes

the pattern to shift, the thickness L can be determined in terms of .

Animation on shifting of interference fringes http://skullsinthestars.com/2008/10/16/fabry-perot-and-their-wonderful-

interferometer-1897-1899/

Michelson Interferometer Path difference of the interfering branches is 2d2-2d1.

When one of the mirrors is moved by /2, the path

difference is changed by and the interference pattern

will be shifted by one fringe.

Similarly, insertion of a transparent material of

thickness L and RI n on one of the paths will cause a

phase change equal to


Diffraction

vertical slit

razor blade

sunlight diffraction

diffraction photography

circular disk square aperature


Basics of Diffraction Light diffraction: light passes through a narrow slit (dimension ~

wavelength) or an edge interferes with itself and produces an interference

pattern called the diffraction pattern. It is a wave nature of light.

Interference pattern from two-

slit diffraction http://en.wikipedia.org/wiki/Diffraction

Water wave encounters a barrier with an

opening of dimensions similar to the

wavelength, the part of the wave that passes

through the opening will flare (spread) out—will

diffract—into the region beyond the barrier. The

flaring is consistent with the spreading of

wavelets according to Huygens principle.

http://en.wikipedia.org/wiki/File:Doubleslit.gif


Dependence on Slit Width

The narrower the slit, the stronger the diffraction.

Diffraction limits geometrical optics (ray optics) since a narrow slit will

cause the light to spread. Geometrical optics holds only when slits or

apertures have dimensions much greater than the wavelength of light.

In projecting an image of tiny features on a photo film, the image will

become blurred through optical diffraction.


Single Slit Diffraction: Locating Minima

First, if we divide the slit into two zones of equal widths a/2, and then

consider a light ray r1 from the top point of the top zone and a light ray r2

from the top point of the bottom zone. For destructive interference at P1,

.2/sin2/ a

)14.1( sin aHence, the first minimum is located at

For D>>a, rays r1 and r2

can be treated as parallel

and inclined at angle to

the central axis.

This pair of rays r1 and

r2 cancel each other at

P1. Similarly, we can

pair up other rays in

the two zones. This path length

difference shifts one

wave from the other,

which determines the

interference.


Single Slit Diffraction: Locating Minima After locating the first dark fringes, one can find the second dark fringes above and

below the central axis. The slit is now divided into four zones of equal widths a/4.

2sin i.e. ,2/sin4/ aaThe second minimum is located at

In general, the minima (dark fringes) are located at

)15.1( ,...3,2,1for ,sin mma

Spring 2013 ENGG 2520 Engineering Physics II

The slit is divided into 18

zones. Resultant

amplitudes E are shown for

the central maximum and

for different points on the

screen.

(b) Light waves have a small

phase difference and add to give

a smaller amplitude

(c) With a sufficiently large phase

difference, the waves can add

together to give zero amplitude.

(a) Light waves from the zones are

in phase and add constructively to

give the maximum amplitude.

(d) With even larger phase

difference, the waves add to

give a small net amplitude.

Single Slit Diffraction: Intensity Pattern

27


Double Slit Diffraction For slit width a << → central

maximum of the diffraction

pattern of each slit covers the

entire screen. Interference leads

to fringes of same intensities.

If a << is NOT satisfied → each slit

produces its own diffraction pattern

described on p.27 with the first

minima close to the center.

Hence, interference of light from the

2 slits produces fringes of different

intensities. The diffraction pattern in

(b) sets an envelope for the intensity

plot in (a) to produce plot (c).

Double slit

Single slit

Spring 2013 ENGG 2520 Engineering Physics II

29

Diffraction Gratings Diffraction gratings contain a large number of slits, often called rulings. They

can be made on a opaque surface with narrow parallel grooves arranged like

the slits, forming a reflection grating instead of a transmission grating.

The fine rulings of ~0.5 micron width

on a CD function as a reflection grating

Interference fringe width is inversely proportional to the number

N of rulings for a given ruling separation d and wavelength.

Hence, a diffraction grating with large N can be used to

determine the wavelength of a monochromatic light. In contrast,

the bright fringes from double slit interference are broad and

different wavelengths will overlap too much to be distinguished.

)16.1( ... 2, 1, 0,for ,sin mmd

Constructive interference (maxima) occurs at

Visible emission lines of Cadmium

resolved with a grating spectroscope


Diffraction by Circular Aperture A tiny circular aperture such as a circular lens can result in an optical

diffraction pattern.

Diffraction pattern of a circular

aperture showing the central

maximum and the interference

fringes. For an aperture of diameter

d, the first minimum is located at an

angle given by

)17.1( 22.1sind

Compared to (1.14), the factor 1.22

here is caused by the circular shape

of the aperture.

In photography, a larger aperture (smaller f-number) gives a better resolved

image due to weaker diffraction effect. You may take a look on photos

taken with different apertures in the following link: http://www.luminous-landscape.com/tutorials/understanding-series/u-diffraction.shtml


Resolvability Consider the lens image of two distant point objects (e.g. stars) with a small

angular separation. They cannot be resolved (distinguished) if their diffraction

patterns overlap. Two objects are barely resolvable when their angular separation

is given by That is, the central maximum of one source

coincides with the first minimum of the other source in the diffraction pattern.

This is called the Rayleigh’s criterion.

.22.1sind

RR

Indistinguishable marginally distinguishable clearly distinguishable


Diffraction Limit in Optical Imaging

Diffraction Limit: a fundamental limit to the

resolution of an imaging system, such as

camera, telescope, and microscope. The

resolution is proportional to the size of the

objective lens, and inversely proportional to the

wavelength. The resolution limit is roughly

equal to half the wavelength in modern optics.

E.g, for green light at 0.5 mm, the resolution limit

is ~0.25 mm. To increase the resolution, shorter

wavelength such as ultraviolet or X-ray can be

used for the imaging. Alternatively, instead of

using optical imaging, one can use electronic

imaging. Electron exhibits wave-particle duality

with a wavelength (de Broglie wavelength)

inversely proportional to its momentum, and

can reach ~0.01 nm at 10 keV energy.

Scanning electron microscope

(SEM), CUHK

10 mm

Photo taken with a SEM showing

nano-features fabricated on silicon at

CUHK (Courtesy of Prof. H. K. Tsang)

http://www.google.com/imgres?imgurl=http://www.phy.cuhk.edu.hk/centrallaboratory/s360/sem1.jpg&imgrefurl=http://www.phy.cuhk.edu.hk/centrallaboratory/s360/s360.html&usg=__VkzhIfliF2rBVNh1zu6u1rEHSBE=&h=1330&w=1662&sz=675&hl=en&start=1&zoom=1&tbnid=4rItO0LrCHeEFM:&tbnh=120&tbnw=150&ei=maTmUKrFLcrNkgW0-IAw&prev=/search?q=cuhk+scanning+electron+microscope&um=1&hl=en&rls=com.microsoft:en-us&tbm=isch&um=1&itbs=1

engineering physics

Documents

dark fringes

phase difference

high ri

low ri

central axis

light waves

resultant

constructive