Lecture 4 Cont’d - Encsusers.encs.concordia.ca/~nrskumar/Index_files/Mech6491...Lecture 4 Cont’d Instructor: N R Sivakumar In 1669, Huygens studied light through a calcite crystal

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MECH 6491 Engineering Metrology

and Measurement Systems

Lecture 4 Cont’d

Instructor: N R Sivakumar

In 1669, Huygens studied light through a calcite crystal – observed two

rays (birefringence) .

Here, we are talking about separating out different parts of light when we

discuss polarization. Specifically, we are interested in the electric field of

the electromagnetic wave.

Light –Polarization

Electric field only going

up and down – say it is

linearly polarized.

Light can have other types of polarizations such as circularly polarized

or elliptically polarized. We will only look at linearly polarized light.

Net electric field is zero – Unpolarized light!

Light –Polarization

Vertical

Horizontal

)/sin( txAEy

Plane-polarized light

)/sin( txAEz

Right circular

Left circular

Circularly polarized light

)90/sin( txAEy

)/sin( txAEz

)90/sin( txAEy

)/sin( txAEz

Polarizers are made of long

chained molecules which absorb

light with electric fields

perpendicular to the axis.I. Polarizers-

2

2

cosoII

EI

Because

Malus’s

Law

Light –Polarization

I0 is the initial intensity,

and θi is the angle between the light's initial polarization direction and the axis of the polarizer.

II. Scattering -

Light –Polarization

III. Reflection -

1

2

21

2

221

tan

cossin

90

sinsin

n

n

nn

nn

p

pp

p

o

p

Brewster’s

Law

Light –Polarization

Spherical Mirrors

A spherical mirror is a mirror which has the shape of

a piece cut out of a spherical surface.

There are two types concave, and convex mirrors.

Concave mirrors magnify objects placed close to

them

shaving mirrors and makeup mirrors.

Convex mirrors have wider fields of view

passenger-side wing mirrors of cars

but objects which appear in them generally look

smaller (and, therefore, farther away) than they

actually are.

The normal to the mirror centre is called principal axis.

V, at which the principal axis touches the mirror surface is called the

vertex.

The point C, on the principal axis, equidistant from all points on the

reflecting surface of the mirror is called the centre of curvature.

C to V is called the radius of curvature of the mirror R

Spherical Mirrors - Concave

Rays parallel principal axis striking a

concave mirror, are reflected by the mirror

at F (between C and V) is focal point.

The distance along the principal axis from

the focus to the vertex is called the focal

length of the mirror, and is denoted f.

The definitions of the principal axis, C, R, V of a convex mirror are

same as that of concave mirror.

When parallel light-rays strike a convex mirror they are reflected

such that they appear to emanate from a single point F located

behind the mirror.

Spherical Mirrors - Convex

This point is called the virtual focus

of the mirror.

The focal length f of the mirror is

simply the distance between V and

F.

f is half of R

Graphical method - just four simple rules:

An incident ray // to principal axis is reflected through the focus F of the mirror

An incident ray which passes through the F of the mirror is reflected // to the

principal axis

An incident ray which passes through the C of the mirror is reflected back along

its own path (since it is normally incident on the mirror)

An incident ray which strikes the mirror at V is reflected such that its angle of

incidence wrt the principal axis is equal to its angle of reflection.

Image Formation - Concave

ST is the object at distance p from the mirror (P > f).

Consider 4 light rays from tip T to strike the mirror

1 is // to axis so reflects through point F. 2 is through point F

so reflects // to the axis. 3 is through C so it traces its path

back. 4 is towards V so it has same angles of incidence and

reflectance.

The point at which all these meet is where the object will be

S’T’ at distace Q (if seen further from q, the image is seen)

This is a real image

We only need 2 rays (minimum) to get the image position:

In this case, the object ST is located within the focal length of the mirror f

Consider 2 lines from tip T of the object

Line 2, passes through F and reflects // to the principal axis

Line 3 passes through C and is reflected along its path

If these two lines are connected, the image is formed on the other side of the

lens

Here the image is magnified, but not inverted like the previous case

Image Formation - Concave

There are no real light-rays behind the mirror

Image cannot be viewed by projecting onto a screen

This type of image is termed a virtual image

The difference between a real and virtual image is,

immediately after reflection from the mirror, light-rays

from object converge on a real image, but diverge

from a virtual image.

Graphical method - just four simple rules:

Incident ray // to principal axis is reflected as if it is from virtual focus F of mirror.

Incident ray directed towards the virtual focus F of the mirror is reflected // to

principal axis.

Incident ray directed towards C of the mirror is reflected back along its own path

(since it is normally incident on the mirror).

Incident ray which strikes the mirror at V is reflected such that its angle of

incidence wrt the principal axis is equal to its angle of reflection. .

Image Formation - Convex

ST is the object. Consider 2 light rays from tip T to

strike the mirror

1 is // to axis so appears to reflect through point F.

2 is through point C so so it traces its path back.

The point of intersection of these two lines is

where the image will be

Vitural or Real ????

Inverted and Magnified or otherwise ?????

If we do this by analytical method, we will get the following formula

Magnification M is given by the object and image distances

It is negative if the image is inverted and positive if it is not

For expression that relates the object and image distances to radius of

curvature

Image Formation - Concave

If object is far away (p = ) all the lines are //

and focus on the focal point F

The focal length is R/2 which can be combine

to give

For plane mirrors the radius of curvature R =

If R = , then f = ±R/2 =

because 1/f = 0

1/p + 1/q = 1/f = 0

q = -p (which means that it is a virtual image far behind

the mirror as much the object is in front)

Magnification is –q/p = 1

So plane mirror does not magnify or invert the image

Image Formation – plane mirror

Image Formation – plane mirror

Sign conventions may vary based

on different text books. So follow

consistently which ever method is

used

Image Formation – LensesFor thin lenses this

distance is taken as 0

Image Formation – Lenses A lens is a transparent medium bounded by two

curved surfaces (spherical or cylindrical)

Line passing normally through both bounding

surfaces of a lens is called the optic axis.

The point O on the optic axis midway between the

two bounding surfaces is called the optic centre.

There are 2 basic kinds: converging, diverging

Converging lens - brings all incident light-rays

parallel to its optic axis together at a point F, behind

the lens, called the focal point, or focus.

Diverging lens spreads out all incident light-rays parallel to its optic

axis so that they appear to diverge from a virtual focal point F in front

of the lens.

Front side is conventionally to be the side from which the light is

incident.

Image Formation – Lenses Relationship between object and image distances to

focal length is given by

Magnification of the lens is given by

Example (Object outside Focal Point)

Object distance S = 200mm Object height h = 1mm

Focal length of the lens f = 50mm

Find image distance S’ and Magnification m

Image Formation – Lenses Relationship between object and image distances to

focal length is given by

Magnification of the lens is given by

Example (Object inside Focal Point)

Object distance S = 30mm Object height h = 1mm

Focal length of the lens f = 50mm

Find image distance S’ and Magnification m

Image Formation – Lenses Relationship between object and image distances to

focal length is given by

Magnification of the lens is given by

Example (Object at Focal Point)

Object distance S = 30mm Object height h = 1mm

Focal length of the lens f = -50mm (diverging lens)

Find image distance S’ and Magnification m

F-Number and NA The calculations used to determine

lens dia are based on the concepts

of focal ratio (f-number) and

numerical aperture (NA).

The f-number is the ratio of the

lens focal length of the to its clear

aperture (effective diameter ).

The f-number defines the angle of the cone of

light leaving the lens which ultimately forms

the image.

The other term used commonly in defining

this cone angle is numerical aperture NA.

NA is the sine of the angle made by the

marginal ray with the optical axis. By using

simple trigonometry, it can be seen that

Different Lenses

Spherical aberration comes from the spherical

surface of a lens

The further away the rays from the lens center, the

bigger the error is

Common in single lenses.

The distance along the optical axis between the

closest and farthest focal points is called (LSA)

The height at which these rays is called (TSA)

TSA = LSA X tan u″

Spherical aberration is dependent on lens shape,

orientation and index of refraction of the lens

Aspherical lenses offer best solution, but difficult to

manufacture

So cemented doublets (+ve and –ve) are used to

eliminate this aberration

Spherical Aberration

26

When an off-axis object is focused by a spherical lens, the natural

asymmetry leads to astigmatism.

The system appears to have two different focal lengths. Saggital and

tangential planes

Between these conjugates, the image is either an elliptical or a circular

blur. Astigmatism is defined as the separation of these conjugates.

Astigmatism

The amount of astigmatism depends on lens

shape

27

Astigmatism

Chromatic Aberration

Material usually have

different refractive indices

for different wavelengths

nblue>nred

This is dispersion

blue refracts more than the

red, blue has a closer

focus

As in the case of spherical aberration, positive

and negative elements have opposite signs of

chromatic aberration.

By combining elements of nearly opposite

aberration to form a doublet, chromatic

aberration can be partially corrected

It is necessary to use two glasses with

different dispersion characteristics, so that the

weaker negative element can balance the

aberration of the stronger, positive element.

Achromatic Doublets

n1

n2

R1R2

R3

Achromatic doublet (achormat) is often used to compensate

for the chromatic aberration

the focuses for red and blue is the same if

0)11

)(()11

)((32

22

21

11 RR

nnRR

nn rbrb

Achromatic Doublets

Lens maker’s formula

31

MECH 691T Engineering Metrology

and Measurement Systems

Lecture 5

Instructor: N R Sivakumar

32

Introduction

General Description

Coherence

Interference between 2 plane waves

– Laser Doppler velocimetry

Interference between spherical waves

Interferometry

– Wavefront Division

– Amplitude Division

Heterodyne Interferometry

Outline

33

Waves have a wavelength

Waves have a frequency

Light as Waves

34

Thousand (103) oscillations/second - kilohertz (kHz)

Million (106) oscillations/second - megahertz (MHz)

Billion) (109) oscillations/second - gigahertz (GHz)

Thousand billion (1012) oscillations per second -

terahertz (THz)

Million billion) (1015) oscillations per second -

petahertz (PHz)

Frequency

35

The superposition principle for electromagnetic

waves implies that, two overlapping fields Ul and U2

add to give Ul + U2.

This is the basis for interference.

Introduction

36

Interference in water waves…

37

Overlapping Semicircles

38

Superposition

t

+1

-1

t

+1

-1

t

+2

-2

+

Constructive Interference

In Phase

39

t

+1

-1

t

+1

-1

t

+2

-2

+

Destructive Interference

Out of Phase

180 degrees

Superposition

40

Superposition

+

Different f

1) Constructive 2) Destructive 3) Neither

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5

-1

-0.5

0

0.5

1

1.5

41

Superposition

+

Different f

1) Constructive 2) Destructive 3) Neither

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5

-1

-0.5

0

0.5

1

1.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

42

Interference Requirements

Need two (or more) waves

Must have same frequency

Must be coherent (i.e. waves must have definite

phase relation)

43

General Description

Interference can occur when two or more waves overlap each

other in space. Assume that two waves described by

and overlap

The electromagnetic wave theory tells us that the resulting field

simply becomes the sum

The observable quantity is intensity (irradiance) I which is

Where ei = (cos + isin ) and

44

General Description

Resulting intensity is not just (I1 + I2).

When 2 waves interfere is called the interference

term

We also see that when then

and I reaches minima (cos 180°) which means destructive

interference

Similarly when then and I

reaches maxima (cos 0°) constructive interference

When 2 waves have equal intensity I1 = I2 = I0

45

Coherence

Detection of light is an averaging process in space and

time

We assume that ul and u2 to have the same single

frequency

Light wave with a single frequency must have an infinite

length

However sources emitting light of a single frequency do

not exist

46

Coherence

Here we see two successive wave

trains of the partial waves

The two wave trains have equal

amplitude and length Lc, with an abrupt,

arbitrary phase difference

a) shows the situation when the two

waves have traveled equal paths. We

see that although the phase of the

original wave fluctuates randomly, the

phase difference remains constant in

time

47

Coherence

In c) wave 2 has traveled Lc longer

than wave 1. The head of the wave

trains in wave 2 coincide with tail of

the corresponding wave trains in

partial wave 1.

Now the phase difference fluctuates

randomly as the successive wave

trains pass by

Here cos varies randomly between

+1 and -1 and for multiple trains it

becomes 0 (no interference) I = I1 + I2

48

Coherence

In b) wave 2 has traveled l longer than

wave 1 where 0<l<Lc.

For many wavetrains the phase

difference varies in time proportional

to

we still can observe an interference

pattern but with a reduced contrast

is the coherence length and

is the coherence time

For white light, the coherence length is

1 micron

49

Plane Wave Interference

When two plane wave interfere the

resultant fringe spacing is given by

50

Laser Doppler Velocimetry (LDV)

Method for measuring the velocity of moving objects

Based on the Doppler effect –

light changes its frequency (wavelength) when

detected by a stationary observer after being scattered

from a moving object

example - when the whistle from a train changes from

a high to a low tone as the train passes by

51

Laser Doppler Velocimetry (LDV)

Particle is moving in a test volume where

two plane waves interfere at an angle

These two waves will form interference

planes which are parallel to the bisector of a

and separated by a distance

As the particle moves through test volume, it will scatter light when it

passes a bright fringe and scatter no light while passing a dark fringe

The resulting light pulses can be recorded by a detector

The time lapse between pulses is td and frequency is fd = 1/td

52

Laser Doppler Velocimetry (LDV)

53

Laser Doppler Velocimetry (LDV)

If there are many particles of different Vs

many different frequencies can be recorded

on a frequency analyzer and the resulting

spectrum will tell how the particles are

distributed among the different velocities

This method does not distinguish between particles moving in opposite

directions

LDV can be applied for measurement of the velocity of moving

surfaces, turbulence in liquids and gases (where the liquid or gas

seeded with particles). Examples - of stream velocities around ship

propellers, velocity distributions of oil drops in IC engines

You know , the and

the f can be recorded.

V can be calculated

54

Figure shows the fringe pattern in xz-

plane when spherical waves from two

point sources P1 and P2 on the z-axis

interfere.

Fringe density increases as distance

between PI and P2 increases

Interference between other Waves

55

Interference between other Waves The intensity distribution in XY plane is

Where

This called circular zone pattern

56

Interference between other Waves

By measuring the distance between interference

fringes over selected planes in space, quantities

such as the angle and distance can be found.

One further step would be to apply for a wave reflected from a rough

surface

By observing the interference - can determine the surface topography

For smoother surfaces, however, such as optical components (lenses,

mirrors, etc.) where tolerances of the order of fractions of a wavelength

are to be measured, that kind of interferometry is quite common.

57

Interferometry

Light waves interfere only

if they are from the same

source (why???)

Most interferometers have the following elements

light source

element for splitting the light into two (or more) partial waves

different propagation paths where the partial waves undergo

different phase contributions

element for superposing the partial waves

detector for observation of the interference

58

Interferometry

Depending on how the light is split,

interferometers are commonly classified

Wavefront division interferometers

Amplitude division interferometers

59

Wavefront Division

Example of a wavefront dividing interferometer, (Thomas Young)

The incident wavefront is divided by passing through two small

holes at SI and S2 in a screen 1.

The emerging spherical wavefronts from SI and S2 will interfere,

and the pattern is observed on screen 2.

The path length differences of the light reaching an arbitrary point

x on S2 is found from Figure

When the distance D between screens is much greater than the

distance d between S1 and S2, we have a good approximation

60

m=0

m=1

m=1

m=2

m=2

D

y

Wavefront Division

61

ym mD

d

dsin

dsin m m = 0,1,2,3... Maximum

dsin (m 1

2) m = 0,1,2,3... Minimum

tan ym

D or ym Dtan Dsin

m ym +/-

0

1

2

3

0

D/d

2D/d

3D/d

Maximam ym +/-

0

1

2

3

D/2d

3D/2d

5D/2d

7D/2d

Minima

ym (m 1/2)D

d

Wavefront Division

62

13E Suppose that Young’s experiment is performed with blue-green

light of 500 nm. The slits are 1.2mm apart, and the viewing screen is

5.4 m from the slits. How far apart the bright fringes?

From the table on the previous slide we see that the separation

between bright fringes is

D /d

D /d (5.4m)(500109m) /0.0012m

0.00225m 2.25mm

Wavefront Division

63

Wavefront Division

A) Fresnel Biprism

B) Lloyds Mirror

C) Michelsons Stellar Interferometer

64

Amplitude Division

Example of a amplitude dividing interferometer, (Michelson)

Amplitude is divided by beamsplitter BS which partly reflects and

partly transmits

These divided light go to two mirrors M1 and M2 where they are

reflected back.

The reflected lights recombine to form interference on the

detector D

The path length can be varied by moving one of the mirrors or by

mounting that on movable object (movement of x give path

difference of 2x) and phase difference

65

Amplitude Division

As M2 moves the

displacement is measured by

counting the number of light

maxima registered by D

By counting the number of

maxima per unit time will give

the velocity of the object.

The intensity distribution is

given by

Ezekiel, Shaoul. RES.6-006 Video Demonstrations in Lasers and

Optics, Spring 2008. (Massachusetts Institute of Technology: MIT

OpenCourseWare), http://ocw.mit.edu (Accessed 15 May, 2012).

License: Creative Commons BY-NC-SA

66

Michelson Interferometer

Split a beam with a Half Mirror, the use

mirrors to recombine the two beams.

Mirror

Mirror

Half Mirror

Screen

Light

sourc

e

67

If the red beam goes the same length as the blue

beam, then the two beams will constructively

interfere and a bright spot will appear on screen.

Mirror

Mirror

Half Mirror

Screen

Light

sourc

e

Michelson Interferometer

68

If the blue beam goes a little extra distance, s,

the the screen will show a different interference

pattern.

Mirror

Mirror

Half Mirror

Screen

Light

sourc

e

s

Michelson Interferometer

69

If s = /4, then the interference pattern

changes from bright to dark.

Mirror

Mirror

Half Mirror

Screen

Light

sourc

e

s

Michelson Interferometer

70

If s = /2, then the interference pattern changes

from bright to dark back to bright (a fringe shift).

Mirror

Mirror

Half Mirror

Screen

Light

sourc

e

s

Michelson Interferometer

71

By counting the number of fringe shifts, we can

determine how far s is!

Mirror

Mirror

Half Mirror

Screen

Light

sourc

e

s

Michelson Interferometer

72

If we use the red laser (=632 nm), then each

fringe shift corresponds to a distance the mirror

moves of 316 nm (about 1/3 of a micron)!

Mirror

Mirror

Half Mirror

Screen

Light

sourc

e

s

Michelson Interferometer

73

Amplitude Division

Twyman Green Interferometer

Mach Zehnder Interferometer

74

Dual Frequency Interferometer

We stated that two waves of different frequencies do not produce

observable interference.

By combining two plane waves

The resultant intensity becomes

If the frequency difference VI - V2 is very small and constant, this

variation in I with time can be detected

This is utilized in the dual-frequency Michelson interferometer for

length measurement

Also called as Heterodyne interferometer

75

Dual Frequency Interferometer