lecture 4 cont’d - encsusers.encs.concordia.ca/~nrskumar/index_files/mech6491...lecture 4 cont’d...
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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
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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
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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
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MECH 691T Engineering Metrology
and Measurement Systems
Lecture 5
Instructor: N R Sivakumar
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Introduction
General Description
Coherence
Interference between 2 plane waves
– Laser Doppler velocimetry
Interference between spherical waves
Interferometry
– Wavefront Division
– Amplitude Division
Heterodyne Interferometry
Outline
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Waves have a wavelength
Waves have a frequency
Light as Waves
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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
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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
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t
+1
-1
t
+1
-1
t
+2
-2
+
Destructive Interference
Out of Phase
180 degrees
Superposition
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Wavefront Division
A) Fresnel Biprism
B) Lloyds Mirror
C) Michelsons Stellar Interferometer
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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