B4.3_U. Paul Kumar (2)

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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

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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.



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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



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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).

REFERENCES

[1] P. K. Rastogi, Digital speckle patterninterferometry and related techniques, John

Wiley & Sons, New York, 2001.

[2] W. Osten, Ed., Optical inspection ofMicrosystems, CRC Press, Boca Raton, FL,

USA(2007).

[3] U. Paul Kumar, B. Bhaduri, N. KrishnaMohan, M. P. Kothiyal and A. K. AsundiMicroscopic TV holography for MEMS

deflection and 3-D surface profile

characterization, Opt. Las. Eng., 46,

687(2008).

[4] R.J. Pryputniewicz, and K.A. Stetson,Measurement of vibration pattern using

electro-optic holography,Proc. SPIE.1162,

456 (1989).

[5] U. Paul Kumar, Y. Kalyani, N. KrishnaMohan, M.P.Kothiyal, Time average TV

holography for vibration fringe

analysis,Appl. Opt., 48, 3094(2009).

[6] D.A. Zweg, R.E. Hufnagel, A Hilberttransform algorithm for fringe pattern

analysis, Proc. SPIE, 1332, 295 (1990).

[7] V.D. Madjarova, and H. Kadono, Dynamicelectronic speckle pattern interferometry

(DESPI) phase analyses with temporal

Hilbert transform, Opt.Exp., 11, 617(2003).

[8] Z.Wang, J.Zhang, Z.Zhao, New methods ofthe filtering the phase noise in the

interferometric SAR, IEEE,4, 2622(2004).

[9] L. X. Yang and A. L. Bhangaonka,Investigation of natural frequencies under

free-free conditions on objects by digitalholographic speckle pattern interferometry,

ISEM, Charlotte, USA, June 2-4, 2003.

[10] D.C. Ghiglia, and M.D. Pritt, Two-dimensional phase unwrapping: Theory,

algorithms and software, New York, John

Wiley and son, 1998.



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