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Topic:Optical Interferometry, Holography & Laser Speckles (OHL) Preference: Oral
INTERFEROGRAM ANALYSIS USING HILBERT TRANSFORM
U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal
Applied Optics Laboratory, Department of Physics,
Indian Institute of Technology Madras, Chennai 600036, India.E-Mail ID:[email protected]
Abstract: It is often required to obtain quantitative phase information from a single interferogram. For
quantitative analysis a /2 phase shifted interferogram is required. Hilbert transform can generate a new signal
with a phase altered by /2 while the amplitude is left unchanged with respect to the original signal. Hence we
apply the Hilbert transform method for phase analysis of static and vibration speckle fringes is described. This
method is demonstrated for the measurement of out-of-plane deformation and out-of-plane vibration amplitude
using a microscopic TV holography system. The usefulness of the method is demonstrated with the examples of
static and vibration measurements on small scale samples.
1. INTRODUCTION
TV holography (TVH) or digital/electronic specklepattern interferometry (DSPI/ESPI) allows both the
static and dynamic deformation measurement on
rough surfaces with interferometric sensitivity [1-2].
In either case, the speckle fringe overlaid images of
the object under study are obtained, where the fringes
denote contours of constant surface deformation or
constant vibration amplitude. The subtraction speckle
correlation is applied to visualize static fringes and
the time average method is used for vibration fringes.
For Microsystems analysis, the TV holographic
arrangement uses a microscopic imaging system in
the setup [2,3]. The TV Holography uses thetemporal phase method for static phase evaluation
and the space phase modulation method for vibration
phase evaluation. The temporal and the phase
modulation methods require at least three phase
shifted frames for analysis [1,4-5].
In this paper we present a Hilbert transform (HT)
method for phase analysis of static and vibration
fringe analysis using a single frame. We have
proposed in this paper to correct the error introduced
in the calculation of the phase due to limits in the HT
function by using look up tables. Hilbert transform
method which has several advantages: (i) It has
simplicity in calculation algorithms, (ii) It has
relatively shorter calculation time, compared, to
Fourier transform (FT) method, (iii) it can be fully
automated unlike FT method. The proposed method
is demonstrated for the measurement of out-of-plane
deflection and the vibration amplitude at resonant
frequencies on a PZT cantilever.
2. MICROSCOPIC TV HOLOGRAPHIC
(MTVH) SYSTEM
The schematic of the MTVH system is shown in
Fig.1. The narrow beam from a 532 nm CW Nd:
YAG laser is divided into two beams using a beamsplitter (BS1). One beam is expanded using a spatial
filtering setup (SF) and collimated with a collimating
lens (CL) to act as an object beam to illuminate theobject via a mirror M, cube beam splitter (BS2). The
microscopic imaging system consists of a Thales-
Optem Zoom 125C long working distance
microscope (LDM) with extended zoom range and a
Sony 2/3 CCD camera (XC-ST70CE). The CCD is
interfaced to a PC with an NI1409 frame grabber
card. The reference beam also expanded using a
spatial filtering setup (SF) and collimated with the
support of a collimating lens (CL). A Function
Generator (FG) is used for harmonic excitation of the
object. The scattered object wave and the smooth
reference wave are combined coherently onto the
CCD plane. The PZTM in the setup is used to
visualize the time average fringe patterns.
Fig.1 Schematic of the MTVH system static and
vibration fringe analysis: SF, Spatial filter, BS, Beam
splitter, NDF, Neutral density filter, M, Mirror,
PZTM, Piezoelectric transducer mirror, A, Amplifier,
FG, Function generator, and DAQ, Digital to analog
converter card.
3. THEORY
3.1. Static Fringe Analysis
The intensity distribution of the speckle fields before
and after deformation of the object can be expressed
as [1]b O R o RI I I 2 I I cos( )= + + (1)
ICOP 2009-International Conference on Optics and Photonics
CSIO, Chandigarh, India, 30 Oct.-1 Nov. 2009
7/27/2019 B4.3_U. Paul Kumar (2)
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Topic:Optical Interferometry, Holography & Laser Speckles (OHL) Preference: Oral
a O R o R I I I 2 I I cos( )= + + + (2)
where IO, IR are the intensities of the scattered object
and the reference waves, is the random phase, is the fringe locus function. The subtraction of Eq.(1)
from Eq.(2) yields, the speckle correlation fringes
which can be expressed as
a b O R S I I C I I sin( / 2)sin( / 2)= = + (3)
The Eq(3) describes the modulation of the high
frequency noise (sin ( / by a low frequency
speckle interference pattern (sin ( / ) related to the
phase change . HT of a sinusoidal function gives a
cosinesoidal function with amplitude unchanged.
Using this characteristic, we can determine the phase
of the signal such as in Eq.(3) using the equation
[6,7]
2))+2)
Hi{S}' = (4)arctanS
The value of ' differs from the correct argument
of the sinusoidal function at the extreme ends due to
the limits of the HT function, that is
(5)' s = +
where sis the error introduced due to HT. The error
s can be calculated for any value of to create alook-up table. Such a lookup table is shown in Fig.2.
To generate this table we calculate the known phase
distribution using HT method. A plot of input and the
calculated phase should be a straight line. The error,which is departure from the straight line, is shown in
Fig.2(b). The corrected phase can be related to
out-of-plane deformation (w) for normal illumination
and observation condition as [1]
(6)4 /w =
where is the wavelength.
Fig.2. (a) The look-up graph for the error sbetween
and ',(b) Phase error due to the Hilbert
transformation (HT).
3.2. Vibration Fringe Analysis
The intensity distribution of a time average frame
obtained by sinusoidally exiting the object with a
frequency much higher than the video frame rate can
be expressed as [4,5]
avg O R o R oI I I 2 I I cos( )J ( )= + + (7)
whereJ0() is the zero order Bessel function of firstkind. Eq.(7) results in poor contrast fringes due to
high level of the DC term. The DC level can be
removed by subtracting the intensity from a 180ophase shift (contrast reversal) frame. The intensity
distribution of the contrast reversal time average
frame, is then expressed as [4,5,9]
R I Vcos( )J ( )o 0= (8)
The J0() fringe pattern from Eq.(8) corresponds tothe contours of vibration amplitudes of the object.
The phase of the signal shown in Eq.(8) can be
determined using the Eq. (4) assuming J0 as cosine
function as [6,7]
Hi{R}" = arctanR (9)
The value of '' differs from the correct argument
of the J0because of the difference between the cosine
and J0 functions and it can be written as
''v = + (10)
where v is the total error due to the differencebetween the J0 and cosine functions plus that due to
the limits of the HT function. The error v can be
calculated for any value of to create a look-up
graph as shown in Fig.3(a). Fig.3(a) shows the
relation between '' (the calculated phase using
Eq.(9)) and , the known input phase. The departure
from a straight line gives the corresponding error r.
The look-up graph is used to correct the calculated
phase. Phase error plot due to the assumption of
Bessel function as cosine function is shown in Fig.
3(b). The corrected phase can be related to
amplitude of object vibration A as
= (4 / )A (11)
Fig.3. (a) The look-up graph for the error between
and , (b) Phase error due to the assumption of
Bessel function as cosine function and due to the
limits of HT function.
ICOP 2009-International Conference on Optics and Photonics
CSIO, Chandigarh, India, 30 Oct.-1 Nov. 2009
7/27/2019 B4.3_U. Paul Kumar (2)
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Topic:Optical Interferometry, Holography & Laser Speckles (OHL) Preference: Oral
4. EXPERIMENTAL RESULTS
4.1. Measurement of the out-of-plane deflection
Experiments were carried out on a PZT cantilever
beam (8x4x0.75 mm3) for deflection characterization
by applying the DC voltage externally using the
microscopic TV holographic system show in Fig.1.The PZT cantilever beam is coated with a white
spray to make the specimen diffuse. The object is
focused onto the CCD with the help of white light
illumination by adjusting the fine focusing knob in
the long working distance microscope (LDM) zoom
module. We have used (1.0X) magnification by
adjusting the magnification control knob of the
LDM. Once the initial adjustments are made, the test
surface as well as the reference mirror is illuminated
with a collimated laser beam via the cubic beam
splitter. The scattered object wave and the smooth
reference wave as explained in Fig.1 are coherentlycombined at the CCD which is interfaced to the PC
with NI frame grabber card. We have incorporated a
LabVIEW based software program to visualize and
store the speckle correlation fringes in real-time by
subtracting the initially stored image from the live
image during the deformation as explained in Section
3.1.
We have stored different sets of deflection fringes at
different applied voltages. Fig. 4(a) shows such a
speckle correlation fringe pattern at 150V. Fig.4(b)
shows the noisy line scan profile along the central x-
axis of the fringe pattern shown in Fig.4(a). For
quantitative static fringe analysis, we have used a
Prior to performing HT, the pattern shown in the
Fig.4(a) is filtered using average filtering with 3x3
window and is shown in the Fig.4(c) whose intensity
distribution can be expressed as a cosine function.
Fig.4(d) shows the HT generated fringe pattern
whose intensity distribution can be expressed as a
sine function. The line scan profiles along central x-
axis of the fringe patterns shown in Fig.4(c) and 4(d)
are combined in Fig.4(e) where one can observe the
HT generated signal is shifted by /2 compared to the
original signal leaving the amplitude part almostunchanged. Thus we have two fringe patterns: (a)
original fringe pattern shown in Fig.4(c) (i.e cosine
part), and (b) HT generated fringe pattern shown in
Fig.4 (d) (i.e sine part). The wrapped phase map
obtained using the patterns shown in Fig.4(c) and
4(d) is shown in Fig. 4(f) using the Eq.(4). The phase
map is wrapped between and +due to arctangent
function and is not correct due the limits of HT. The
phase map is correct using the look up generated as
shown in Fig.2. The wrapped phase map is
unwrapped using multi-grid method [10] and scaled
by Eq. (6) to generate the out-of-plane deflection
profiles. Fig.4(g) shows a 3-D out-of-plane deflection
profile PZT cantilever beam at 150V.
Fig.4. Static fringe analysis on a PZT cantilever
beam: (a) speckle deflection fringes at 150V, (b)
central line scan profile of the pattern shown in
Fig.4(a), (c) filtered pattern, (d) HT generated
pattern, (e) central line scan profiles along x- axis of
Fig.4(c) and Fig. 4(d), (f) wrapped phase map, (g) 3-
D plot.
4.2. Measurement of the out-of-plane amplitudes
of vibration at resonant frequencies
A PZT cantilever beam used for static fringe analysis
in section 4.1 is excited sinusoidally at different
frequencies and amplitudes by an earphone
connected to an amplifier (A2) and function generator
(FG). For real-time visualization of the vibrationmode shapes at resonant frequencies, we have used
an automatic refreshing reference frame technique at
a video rate of 25 frames per second. For this,
software is developed in LabVIEW in such a way
that it will first store a time-average frame during the
object vibration. It will then shift the mirror PZTM
(Fig. 1) for phase shift and store the modified time
average second frame. The subtraction of the two
time average frames results in a vibration fringes at a
resonant frequency on the monitor. The resonance
frequencies and their mode shapes on the PZT
cantilever beam are observed. The fundamental mode
of PZT cantilever beam at 5.72 KHz resonant
frequency is shown in Fig.5(a). . It is to be noted
ICOP 2009-International Conference on Optics and Photonics
CSIO, Chandigarh, India, 30 Oct.-1 Nov. 2009
7/27/2019 B4.3_U. Paul Kumar (2)
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Topic:Optical Interferometry, Holography & Laser Speckles (OHL) Preference: Oral
from the fringe patterns that the bright regions on the
subtracted frame are the positions of zero
displacement, i.e. the nodal lines. Fig.5(b) shows the
noisy line scan profile along the central x-axis of the
fringe pattern. Fig.5(c) and 5(d) shows the filtered
and corresponding HT generated fringe patterns.
Fig.5(e) shows the original zero order Bessel signaland HT generated signal. The wrapped phase map
generated using the Eq.(9) associated with an error
due to the difference between the Jo and cosine
function and it is shown in Fig.5(f). The error
corrected phase map is obtained by using the look-up
table generated a shown in Fig.3. The evaluated 3-D
profile of vibration amplitude of the PZT cantilever
beam excited at the fundamental mode and scaled
using the Eq.(11) is shown in Fig.5(g).
Fig.5. A PZT cantilever beam vibrating atfundamental frequency 5.72 KHz. (a) time average
speckle correlation fringes, (b) filtered pattern, (c)
HT generated pattern, (d) phase map, (e) line scan
profiles along x- axis of Fig.5(c) and Fig.5(d), (f)
wrapped phase map, (g)3-D plot.
5. CONCLUSIONS
We have presented the Hilbert transform method for
both static and vibration fringe analysis using a
microscopic TV holographic system. The proposed
method uses only one frame for quantifying the
fringe data. However, the error introduced due to the
HT is corrected using the look-up table. The
usefulness of the method is demonstrated for the
measurement of out-of-plane deflection and vibration
amplitude on a PZT cantilever beam.
ACKNOWLEDGEMENT
This work is supported by Defense Research and
Development Organization (DRDO).
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ICOP 2009-International Conference on Optics and Photonics
CSIO, Chandigarh, India, 30 Oct.-1 Nov. 2009
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