Optics Communications 338 (2015) 253–256

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Optics Communications

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Fast Track Communications

Heterodyne imaging speckle interferometer

Shengjia Wang, Zhan Gao n, Ziang Feng, Xiaoqiong Zhang, Dong Yang, Hao YuanKey Laboratory of Luminescence and Optical Information of Ministry of Education, Beijing Jiaotong University, Beijing 100044, China

a r t i c l e i n f o

Article history:Received 28 July 2014Received in revised form13 September 2014Accepted 16 September 2014Available online 3 November 2014

Keywords:HeterodyneElectro-optical devicesSpeckle interferometryPhase measurementImaging

x.doi.org/10.1016/j.optcom.2014.09.07818/& Elsevier B.V. All rights reserved.

esponding author.ail address: zhangao@bjtu.edu.cn (Z. Gao).

a b s t r a c t

A heterodyne imaging speckle interferometer coupled with lithium niobate is developed for whole fielddynamic deformation imaging. In this device, the carrier frequency is introduced by the dual-transverselinear electro-optic effect. It is electrically controlled within a large range, which is twice the angularvelocity of the driving alternating electric fields. By setting the angular velocity, the carrier frequency canfit most of area-array detectors, making it feasible to achieve whole field real time imaging. By usingtemporal evolution of the light intensity in heterodyne interferometry, the temporal intensity analysismethod is employed to extract the deformation at each pixel dynamically. The principle and systemconfiguration are described. The preliminary experiment is conducted with a cantilever beam and theresults are compared with theoretical simulations to validate the proposed approach.

& Elsevier B.V. All rights reserved.

1. Introduction

Heterodyne speckle pattern interferometry (HSPI) is a well-established technique for measuring deformation. The main ad-vantages of HSPI include noise immunity, high sensitivity, possi-bility of real time, non-contact, and no surface preparation of thetarget object. Thus, HSPI is often applied for measurement inacademic and industrial research.

Various heterodyne devices have been reported to introducecarrier frequency in a heterodyne system. Among them, dual-fre-quency laser [1,2] and acousto-optic modulator [3,4] are com-monly used. The carrier frequencies generalized by the above twomethods are from several megahertz to tens of megahertz which isconsidered too high for the sampling frequency of area-array de-tector (such as CCD industrial camera). In HSPI imaging system,the point-sensing device with scanning technique is con-ventionally demanded to collect heterodyne signal [4,5]. However,such a design limits the development of the whole field and dy-namic imaging in real time. In 2000, Haible et al. [6] introduced afrequency shifter [7], which consists of a rolling half-wave plateand a stationary quarter-wave plate, and it is able to slow downthe carrier frequency which fits the sampling frequency of CCDcameras. In practice, due to the mechanical rotating of the half-wave plate, the upper limit of the carrier frequency is restrictedand the system is vulnerable.

In this paper, a heterodyne imaging speckle interferometerbased on electro-optic effect is presented. There are no movingparts in this system, in which the carrier frequency is electricallycontrolled and determined by the rotating frequency of the drivingelectric fields, ranging from several hertz to several gigahertz. Bysetting a proper frequency of the electric field, the carrier fre-quency can fit the area-array detector well. Whole field real timeimaging can be achieved by using a CCD camera. On the otherhand, temporal intensity analysis method is employed to realizereal time dynamic measurement. Preliminary experiment, whichaimed at measuring the whole field deformation of a cantileverbeam in real time, was carried out to demonstrate the feasibility ofthis device. The experiment and its principle will be detailedbelow.

2. Principle

As shown in Fig. 1, the frequency shifter [8] consists of a pieceof LiNbO3 crystal and a stationary quarter-wave plate (QWP). TheLiNbO3 crystal is driven by two orthogonal sinusoidal electricfields with phase delay of π/2. The voltage amplitude betweeneach electric field is equal to the crystal's half-wave voltage.

The incident light can be denoted as

⎡⎣⎢

⎤⎦⎥= ω−V e1

0,

(1)ii t

Fig. 1. The frequency shifter consists of a piece of LiNbO3 crystal and a stationary QWP. The incident linearly polarized light passes through the crystal, then a pair ofcircularly polarized light is obtained. By using the stationary QWP, the circularly polarized lights are converted into two orthogonal linearly polarized lights with frequencydifference of 2ωm.

S. Wang et al. / Optics Communications 338 (2015) 253–256254

where ω is the frequency of the incident light. Assume the in-cident linearly polarized light passes through the LiNbO3 along itsthree-fold axis (the z-axis direction). The effect of the frequencyshifter can be described in Jones matrix as

⎡⎣⎢

⎤⎦⎥

ω ω ω ωω ω ω ω

=− +

− − −T

i t i t sin t i t

sin t i t t i t2

cos sin cos

cos cos sin,

(2)fs

m m m m

m m m m

where ωm is the angular velocity of the alternating electric fields.After the light passes through the frequency shifter, the emergentlight can be expressed as

⎛⎝⎜

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎞⎠⎟= = +ω ω ω ω− + − −V T V i e e

12

10

01

.(3)

o fs ii t i t( ) ( )m m

It can be seen that the two items in Eq. (3) are a pair of linearlypolarized lights with frequency difference of 2ωm. The polarizationdirections of this pair of lights are perpendicular to each other asshown in Fig. 1.

The emergent light (represented by Eq. (3)) is then put into aMichelson interferometer. The configuration of our homemadedual-transverse electro-optic HSPI system is shown in Fig. 2. Ac-cording to the orthogonal polarization directions of the twocomponents represented by Eq. (3), a pair of polarizers is em-ployed to separate the up/down frequency shifted beams (The firstitem in Eq. (3) is the up frequency shifted beam and the seconditem is the down frequency shifted beam.). The two beams are

Fig. 2. System configuration.

reflected from the measuring arm and the reference arm respec-tively. A polarizer with allowed polarization direction of 45° isplaced at the front of the imaging system, so that the two beamscan generate speckle pattern. At the imaging plane, the speck-legram is captured by a CCD camera. The intensity of the speck-legram before the deformation is

Φ ω= + +I x y t I x y I x y x y t( , , ) ( , ) ( , ) cos [ ( , ) 2 ], (4)c m0 0

where I0(x,y) is the bias intensity of the specklegram, Ic(x,y) is theamplitude of the specklegram, and Ф0(x,y) is the initial phasewhich usually can be ignored in calculation. In Eq. (4), 2ωmt is thecarrier frequency. It is generated by the up/down frequency shiftedbeams.

During the measurement, a piezoelectric transducer (PZT) isplaced at the free end of a cantilever beam to induce continuouschange of bending curvature. The deformation of the cantileverbeam is along the z-axis. The optical path difference between thereference arm and the measuring arm is changed and the intensityof the specklegram becomes

⎡⎣ ⎤⎦Φ π π λ

=

+ + + Δ

I x y t I x y

I x y x y f t z x y t

( , , ) ( , )

( , ) cos ( , ) 2 4 ( , , )/ , (5)c

0

0 0

where Δz(x,y,t) is the out-of-plane deformation along the z-axis, λis the wavelength of the incident light, and f0¼ωm/π is the carrierfrequency. Hence, at the imaging plane, the CCD will receive afrequency-modulated signal with a carrier frequency of 2πf0t, fromwhich the deformation information can be detected.

To illustrate the temporal intensity analysis method, an arbi-trary point of the cantilever beam is chosen. During the de-formation, the specklegram intensity of this point will change andis recorded by the CCD camera. The temporal intensity I(x,y,t) isobtained. By using Euler's formula, Eq. (5) can be expanded as

π π= + + −⁎( ) ( )I t I c j f t c j f t( ) exp 2 exp 2 , (6)0 0 0

where n represents the complex conjugate and c can be expressedas

Φ π λ= + Δc I j z t12

exp { [ 4 ( )/ ]}. (7)c 0

To obtain the desired phase in the cosine function of Eq. (5),Fourier analysis of time series intensity is performed on Eq. (6).The frequency spectrum can be denoted as

λ λ= + Δ + * − Δ( ) ( )F I A C f z t C f z t( ) , / , / , (8)0 0

Fig. 4. The temporal frequency spectrum of one pixel in the time series speck-legram. 0 Hz is removed to show the spectrum more clearly. The red part is isolatedto obtain the desired phase. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

S. Wang et al. / Optics Communications 338 (2015) 253–256 255

where F(I) is the temporal frequency spectrum of Eq. (6); A is thespectrum of the bias intensity which is regarded as a zero-frequency component; C and Cn are conjugated and representthe spectrum of the second and the third items in Eq. (6)respectively; f0 shows the position of the carrier frequency in thefrequency spectrum and the parameter Δz/λt is the equivalentfrequency of the deformation, which determines the spectralbroadening around the carrier frequency and carries the deforma-tion information. Obviously the three items in Eq. (8) occupydifferent frequency bands. The Fourier transformed fringes arenow filtered in the frequency domain to isolate the function C(f0,Δz/λt). With the inverse Fourier transform, the second item in Eq.(6) is retrieved. The desired phase can be obtained as [9]

⎡⎣ ⎤⎦ Φ= +−F C I i tlog ( ) log [(1/2) ] ( ), (9)c1

where F�1 represents the inverse Fourier transform function andthe imaginary partФ(t) is the desired phase in the cosine functionof Eq. (5). The phase unwrapping technique is then used to obtainthe total phase. The carrier frequency 2πf0t is subtracted from thetotal phase according to the angular velocity of the driving electricfields. The phase induced by the deformation is obtained in timesequence.

3. Experiment and results

To demonstrate the performance of this heterodyne imagingspeckle interferometer, an experiment system was establishedfollowing the design of Fig. 2. The experiment setup is shown inFig. 3. The light source was a semiconductor laser, of which thecentral wavelength was 532 nm, and the maximum output powerwas 120 mW. The size of LiNbO3 was 5�5�30 mm3 and thelength of the 3-fold axis (z-axis direction) was 30 mm. Betweenevery two opposite side surfaces (5�30 mm2), two alternatingelectric fields were generated by the power supply with a fre-quency of 17 Hz and a phase delay of π/2. The carrier frequencywas 34 Hz according to Eqs. (3) and (4). The voltage amplitudebetween one pair of electrodes was 700 V. The directions of theQWP and all the polarizers were adjusted according to Fig. 2. Thespatial filter was composed of the pinhole and the 40� microobjective. The CCD ran in continuous mode and the exposure timewas 5 ms for each frame with a resolution of 26�250. The can-tilever beam was made of aluminum with the size of 100�30�2 mm3 (L�H�W). During the measurement, as the PZTcontinued to push the free end of the cantilever beam, the opticalpath of each point of the measured object changed and the dy-namic whole field deformation was obtained in this system.

Fig. 3. Experiment setup.

Within a measuring time of 10 s, 2000 frames were taken, andeach pixel had 2000 values of intensity in time sequence. An ar-bitrary pixel was selected (in this experiment, the pixel located in(16, 13)) to illustrate the signal processing procedure. The temporalfrequency of this pixel is shown in Fig. 4. The red part was isolatedwith a frequency domain filter and then inverse Fourier transformwas performed on it. By using Eq. (9), the desired phase (in thecosine function of Eq. (5)) was obtained. Replacing the parametersin this experiment where λ¼532 nm and f0¼34 Hz, the de-formation at the selected pixel was obtained. This procedure wasthen applied over the entire surface, and the dynamic whole fielddeformation imaging was achieved.

Fig. 5(a) shows 4 statuses of the deformation at different times.For a single status, to get a continuous surface, quadratic curvefitting was implemented on each horizontal line of pixels. Themeasured result at 9.500 s was compared with the theoretical si-mulation of the cantilever beam as shown in Fig. 5(c).

From Fig. 5(c), it can be seen that the measured deformationmatches the theoretical simulation well. At the right side of Fig. 5(c) which is close to the fixed end of the cantilever beam, the er-rors become large. The reason is that at the fixed end of the can-tilever beam, theoretically, there is no deformation, while inpractice, considering the method to fix the cantilever beam andthe way to load the stress, it is difficult to keep the fixed end of thecantilever no shift at all.

4. Conclusion

A heterodyne imaging speckle interferometer is presented inthis paper. Compared with conventional HSPI systems, the pro-posed system can obtain the real time image slices of the de-formed object in time domain. In addition, it has no scanningmechanism and moving parts. Dual-transverse linear electro-opticeffect is introduced to generalize the adjustable carrier frequency.In terms of the signal processing, the temporal intensity analysismethod is employed to extract the deformation dynamically.Preliminary experiment shows our homemade imaging system isfeasible.

Fig. 5. (a) The deformation of the cantilever beam at 4 different times. (b) A profile of (a) at the 13th Line of pixels. From the bottom to the top, successively, the times are5.000 s (black), 6.500 s (red), 8.000 s (green), and 9.500 s (blue). (c) The comparison of the measured deformation at 9.500 s (blue) and the theoretical simulation (red) of thecantilever beam. The errors are shown as green line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of thisarticle.)

S. Wang et al. / Optics Communications 338 (2015) 253–256256

Acknowledgment

The authors acknowledge the financial support from NationalNatural Science Foundation of China 51275033.

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