moiré technique

Transcription

moiré technique

MOIRÉ TECHNIQUE
ENGINEERING APPLICATIONS OF LASERS
(EAL 604)
DR. JALA EL-AZAB
MOIRE
O
TECHNIQUE
C
QU

What is moiré?

Moiré techniques

Sinusoidal gratings

Two angularly displaced gratings

Measurement of in-plane deformation and strains

Measurement of out-of-plane deformations

Sh d moiré
Shadow
i é and
d contour
t

Vibration analysis

Reflection moiré

Triangulation
p 7,, “Optical
p
Metrology”,
gy , Kjell
j J. Gasvik,, John
Ref: Chapter
Wiley & Sons Ltd, England, 3rd ed. 2002.
What is Moiré?
The moiré effect can be observed in our everyday
surroundings.
What is Moiré?

The French term “moiré”
originates from a type of
textile traditionally of silk
textile,
silk,
with a grained or watered
pp
appearance.
Now moiré is generally used
for a fringe that is created by
superposition of two (or more)
patterns such as dot arrays
and
d grid
id lines.
li
The moiré effect is therefore
often
ft termed
t
d mechanical
h i l
interference.

The mathematical description of
moiré patterns resulting from the
superposition of sinusoidal
gratings is the same as for
i t f
interference
patterns
tt
fformed
db
by
electromagnetic waves.
Moiré techniques

Moiré can be obtained by the principle of pattern formation
of in-plane and out-of plane moirés.
I l
In-plane
moiré:
i é measurement off iin-plane
l
d
deformation
f
i and
d
strains.
O t f l
Out-of-plane
moiré:
i é measurementt off out-of
t f plane
l
deformations (Contouring).
SINUSOIDAL GRATINGS

Often, gratings applied in moire methods are transparencies
with transmittances given by a square-wave function.
I
Instead
d off square-wave ffunctions,
i
we describe
d
ib linear
li
gratings
i
by sinusoidal transmittances (reflectances) bearing in mind that
all types of periodic gratings can be described as a sum of
sinusoidal gratings.
A sinusoidal grating of constant frequency is given by
where p is the grating period and where 0 < a < ½.
½
SINUSOIDAL GRATINGS

If the grating is modulated, it will be given by
where ψ(x) is the modulation function and is equal to the
displacement of the grating lines from its original position
di id d b
divided
by th
the grating
ti period
i d and
d u(x)
( ) is
i th
the displacement.
di l
t
SINUSOIDAL GRATINGS

When the two gratings are laid in contact, the resulting
transmittance t becomes the product of the individual
transmittances
represent the original
gratings
the second grating
with doubled frequency
depends on the
modulation
f
function
i only
l
THE MOIRÉ
N
PATTERN
SINUSOIDAL GRATINGS

Another way of combining gratings is by addition (or
subtraction). This is achieved by e.g. imaging the two gratings
by double exposure onto the same negative
negative. By addition we
get
The moiré fringes are amplitude modulating The original grating
SINUSOIDAL GRATINGS

A maximum results in a bright fringe whenever
and minima (dark fringes) whenever

Both grating t1 and t2 could be phase-modulated by
modulation functions ψ1 and ψ 2 respectively. Then ψ(x) has to
be replaced by
SINUSOIDAL GRATINGS

In both multiplication and addition (subtraction), the grating
becomes demodulated using Heterodyne technique, thereby
getting a term depending solely on ψ(x),
(x) describing the moiré
fringes.
the relations between ψ(x) (and u(x)) and the measuring
parameters depend on the different applications.
TWO ANGULARLY DISPLACED GRATINGS

The mathematical description of this case is the same as for
two plane waves interfering under an angle α.
Wh two gratings
When
i
off transmittances
i
t1 and
d t2 are llaid
id in
i
contact, the resulting transmittance is not equal to the sum
t1 + t2, b
butt the
th product
d t t1 t2.
The result is, however, essentially the same, i.e. the gratings
form a moiré pattern with interfringe distance
This can be applied for measuring α by measurement of d.
MEASUREMENT OF IN-PLANE DEFORMATION
AND STRAINS
T obtain
To
bt i the
th moiré
i é pattern,
tt
one may apply
l one off severall methods
th d
1. Place the reference grating with transmittance t1 in contact with the model
grating with transmittance t2. The resulting intensity distribution then
b
becomes
proportionall to the
h product
d t1 · t2.
2. Image the reference grating t1 onto the model grating t2. The resulting
intensity then becomes proportional to the sum t1 + t2. This can also be
done by forming the reference grating by means of interference between
two plane coherent waves.
3. Image
g the model grating
g
g t2, and p
place the reference g
grating
g t1 in the
image plane. t1 then of course has to be scaled according to the image
magnification. The resulting intensity becomes proportional to t1 · t2.
4. Image the reference grating given by t1 onto a photographic film and
thereafter image the model grating given by t2 after deformation onto
another film. Then the two films are laid in contact. The result is t1 · t2.
5 Do the same as under (4) except that t1 and t2 are imaged onto the same
5.
negative by double exposure. The result is t1 + t2.
AND STRAINS

When multiplying the gratings, the resulting intensity distribution is
proportional to t1 · t2 which can be written
DC term

Modulation function
When adding the gratings, the intensity distribution becomes equal
to t1 + t2 which can be written
DC term
Amplitude
p
modulation of the
original reference grating
AND STRAINS

This distribution has a

This corresponds to a displacement equal to

The linear and shear strain can be obtained by orienting the
model grating and the reference grating along the y-axis, we
can in the same manner find the modulation function ψy(y)
and the displacement v(y) in the y-direction.
AND STRAINS

An example of such an
intensity distribution with the
corresponding displacement
and strain
MEASUREMENT OF OUT-OF-PLANE
DEFORMATIONS

IIn-plane
l
moiré
i é pattern
tt
iis generated
t db
by superposing
i the
th
reference and object gratings in the same plane.
In applying out-of-plane moiré method to the contour
mapping, the object grating formed across the object is
distorted in accordance with object profile.
This out-of-plane moiré method is also termed moiré
topography.
Th moiré
The
i é topography
t
h iis categorized
t
i d mainly
i l iinto
t
shadow moiré and projection moiré methods.
Shadow moiré method is known to be the first example
of applying the moiré phenomenon to three-dimensional
measurement.
SHADOW MOIRE AND CONTOUR

The point light source and the
detector (the aperture of detector
lens is assumed a point) are at a
distance l from the reference
grating
g
g surface and their
interseparation is s.
Period of the reference grating is
p (p << l and p << s).
Without loss of generality, we may
assume that a point O on the
object surface is in contact with the
grating.
ti

The grating lying over the object
surface is illuminated by the point
source and its shadow is projected
source,
onto the object.
The moire pattern observed from
the detector is the result of
superposition between the grating
elements contained in OB of the
reference grating and the elements
contained
i d iin OP off the
h objected
bj
d
grating, which is the shadow of
elements contained in OA of the
reference grating.

Assuming that OA and OB have i and j grating elements,
respectively, from geometry
where
n is the order of the moiré pattern
hn is the depth of nth-order moiré pattern
as measured from the reference grating

Since
and u(x)= AB
A bright fringe is obtained whenever ψ(x) = n, for n = 0,1,2,...,
which gives
where xP is x component of OP.
Substituting above equations in hn and
rearranging leads to

In the case of plane wave illumination and observation from infinity,
α and β will remain constant across the surface and hn describes a
contour map with a constant,
constant fixed contour interval.
interval
With the point source and the viewing point at finite distances, α
and β will vary across the surface resulting in a contour interval
which is dependent on the surface coordinates. This is of course an
unsatisfactory condition.

However, if the point source and the viewing point are placed at
equal heights above the surface and if the surface height variations
are negligible compared to l,l then tan α + tanβ will be constant
across the surface resulting in a constant contour interval.
This is a good solution,
solution especially for large surface areas which are
impossible to cover with a plane wave because of the limited
aperture of the collimating lens.

If the surface height variations are large compared to the grating
period, diffraction effects will occur, prohibiting a mere shadow of
the grating to be cast on the surface.
surface
Shadow moiré is therefore best suited for rather coarse
measurements on large surfaces.
surfaces It is relatively simple to apply and
the necessary equipment is quite inexpensive.
It is a valuable tool in experimental mechanics and for measuring
and controlling shapes.

As the square wave grating that shadow moiré deals with. In
Fourier mathematics, it is known that all types of periodic
functions (including square wave function) can be described as
a sum of simple sinusoidal functions.
In shadow moiré,
moiré the amplitude transmittance of the square
wave grating is considered as that of a sinusoidal grating.
Then the resulting intensity at the point P is proportional to the
product TA(xA, y) · TB(xB, y):

It is seen that

The normalized intensity is given by

The last term is solely dependent on height and is the contour
term.
The other three cosine terms representing the reference
grating, although height-dependent, are also dependent on x
(location) and, hence, do not represent contours.

The patterns corresponding to these three terms can obscure the
contours.
Intensity distribution along cross-sectional line clearly shows the
disturbance of the reference grating itself.
To remove these unwanted patterns, Takasaki proposed, to translate
the grating in azimuth during exposure.

The resultant intensity is proportional to
where
a ((=K)) is the intensityy bias
b (=K/2) is the amplitude
φP is the phase related to the temporal phase shift of this cosine
variation

The unwanted noise patterns are mixed out due to the averaging
effect of the reference grating.
Vibration Analysis

Assume that a point on the surface executes harmonic
out-of-plane vibrations given by
where z0 is the equilibrium
position, a is the amplitude and
h frequency.
f
ω the
The intensity distribution of the projected pattern
becomes
Vibration Analysis

The expression can be written as
where

By photographing this pattern with an exposure
ti muchh longer
time
l
than
th the
th vibration
ib ti period
i d T , th
the
resulting transmittance t of the film becomes
proportional
ti l tto I (x,
( t) averaged
d over th
the vibration
ib ti
period.
Vibration Analysis

This is analogous to time-average holography and
gives for the transmittance
where J0 is the zeroth-order Bessel function
Vibration Analysis
The Bessel function amplitude modulates the fringe
pattern on the static surface given by z0.
Vibration Analysis

The light fringes occur when
and dark fringes occur when

The first dark fringe of this pattern thus corresponds
t an amplitude
lit d all equall tto
to
which is a figure representing the sensitivity of the
method.
REFLECTION MOIRE

The reflection moiré method can be applied on the
objects with mirror-like surfaces.
The smoothness of the surface S makes it possible to
image the mirror image of the grating G by means of
the lens L.
REFLECTION MOIRE

A grating can be placed in the image plane of L or
the mirror image of G can be photographed before
and after the deformation of S.
The result is a moiré pattern defining the derivative
of the height profile, i.e. the slope of the deformation.
TRIANGULATION

Shadow
Sh
d moiré
i é and
d projected
j t d
fringes are techniques based on the
triangulation principle.
A laser beam is incident on a
g surface under
diffuselyy scattering
an angle θ1.
The resulting
g light
g spot
p on the
surface is imaged by a lens onto a
detector D.
The optical axis of the lens makes
an angle θ2 to the surface normal.
Assume that the object moves a
distance s normal to its surface.
TRIANGULATION

The corresponding
Th
di movementt
of the imaged spot on the
detector is given by
where m is the transversal
magnification of the lens.
The detector is positionsensing,
g, i.e. it g
gives
v an output
p
voltage proportional to the
distance of the light spot from
the centre of the detector.
TRIANGULATION

It is
i th
the centroid
t id off th
the light
li ht spott that
th t is
i sensed
d and
d thus
th the
th
position measurement is independent of the spot diameter
as long as it is inside the detector area.
Therefore sharp focusing is not critical.
The position of an unexpanded laser beam directly incident
on such a detector can be determined to an accuracy of less
than 1μm.
μ
the movement s can be magnified by the lens, thereby
increasing the sensitivity.
However, the size of the light spot will also be magnified,
and this must always lie inside the detector area to avoid
measurement errors, thus limiting the usable magnification.