A Combination of a 3 Step Temporal Phase Algorithm and a High Speed Interferometer System for Dynamic Profile Measurements

This work presents the application of a temporal phase algorithm on a high speed optical interferometry vibration study. The object is a rectangular plate clamped on its four sides. The dynamic event of a complete cycle of vibration is registered by means of a high speed electronic speckle pattern interferometry system (HS ESPI). The HS ESPI system contains a CMOS camera set at 4000 frames per second leaving the shutter widely open all the time. A group of intensity patterns of interference are recorded, different sets of fringe patterns along the cycle of vibrat ion come from the subtraction of the intensity patterns of interference and are processed by a three-frame temporal phase measurement method to recover the phase of deformat ion of the plate. The result is reported and is also compared with the mechanical phase described on the theory.


Introduction
Optical techniques for vibrations measurements have had a v ery imp o rtan t ro le in in d u s trial ev o lu tio n . Th e introduct ion of h igh speed cameras to very well kno wn systems like electron ic speckle interfero met ry and dig ital holography gives a big opportunity to study dynamic events with much more precision. High speed optical interfero metry is a powerfu l optical tool fo r measuring the phase difference between two events of an object under some mechan ical change [1,2]. Th ere are so me industrial and b io medical applicat ions that use optical interfero metry measurements for vibrat ion analysis, also for testing mechanical objects, and recently for measuring of non solid objects (elastics) [3]. Temporal phase measurement techniques have been used to detect the phase for the study of the co mplete deformation evolution in static and dynamic events. The optical set ups us ed in th ese tech n iq ues emp lo yed tilted mirro rs o r piezoelectric co mponents to retrieve the optical phase of the reference beam. In this work we present the use of a known temp o ral ph ase measu rement alg o rith m fo r temp o ral as ynch ronous d emod u lat ion app lied to a s equ ence o f Hig h -Speed Electron ic Speckle Pattern In terfero metry (HS-ESPI) interferograms fro m an optical set up that does not have tilting mirrors or piezoelectric co mponent, in this case, the optical phase is obtained because the retrieved optical phase comes from the object beam which is under continuous movement due to an external excitat ion. The interferograms' sequence comprises a whole cycle of the harmonic event taken fro m a clamped rectangular metal plate excited at the first resonant frequency. In order to obtain the phase from these interferograms' sequence, the algorithm only needs 3 interferograms taken during various cycles. The High-Speed system works at 4000 fps (frames per second) using a CMOS camera and a continuous wave (cw) Verdi laser at 532 n m. Simu lated phase data is compared to experimental results and plotted to show errors in the proposed method.
To determine phase, electronic and analytic techniques are used.. So me o f these algorith ms have worked with a set of three, four or five recorded fringe patterns using a phase shift of 90°, but also, there are algorith ms that are independent of the amount of phase shift [4][5][6][7][8][9][10][11][12]. Recent works have reported new methods and algorith ms that go fro m the study of a single, two or more steps to get a phase recovery; nevertheless, they have to cover some requirements to be applied in o rder to wo rk properly [13][14][15][16][17][18]. The applicat ion of these algorith ms requires of a phase change in the reference beam. In this work, the change of phase comes directly fro m the excited ob ject. The three-frame technique by Wyant et al, established in 1984[6], is applied to a selected group of fringe patterns coming fro m a HS-ESPI system, where a continuous wave laser illu mination and a high speed camera are used. Figure 1 shows an out-of-plane sensitive HS ESPI. The beam coming fro m a cw laser with 6-Watts maximu m and 532 nm, is divided into an object beam and a reference beam, IO and IR respectively, by a beam splitter. The object beam is projected over the target by means of a 10x microscope objective and the reflection of this is recombined by a second beam splitter. The reference beam is re-directed to the second beam splitter and then to the sensor of the camera. For this experiment the laser is set to a power of 5.5 Watts, the exposure time of the CM OS camera is set at 4000 frames per second and the shutter is all the time open. Optical set up for a high-speed ESPI system. Beam splitter 1 is a 70/30, beam splitter 2 is a 50/50, microscope objectives are 10x. The object to study is a 140x190x1 mm rectangular aluminum plate A rectangular alu minum plate clamped in its four sides that carries a uniformly distributed load is the object to be studied. The plate is excited by an external sine wave generator placed behind it and plugged to a conventional speaker to generate vibration all over the plate. The first vibration modal for this plate is reported at 320 Hz [24]. The evolution of the v ibration is recorded by a h igh-speed CMOS camera and having an image lens with a focal length of 75mm in order to get the whole object into the CM OS sensor.

Experimental Setup and Object Description
The mechanical propert ies of the plate are mass density According to several authors, the equation for the vibrating rectangular plate is given by [19][20][21][22][23].
( ) Where the deformat ion variable ( ) and again, applying the variables separation method ( ) y x F , can be written as: After a mathemat ical procedure, the expression that gives the solution on function of the dimensions of the plate, a for the length, b fo r the width and for each horizontal and vertical v ibration modal, w and z respectively is:  The deformed profile of the vibrating plate on its first natural mode of frequency fits a polyno mial function of degree 2 or 4.

Theoretical Description
In the high speed optical interfero metry technique, the addition of the reference beam R I and the object beam O I is written as where Ψ is the subtraction of the phase of the wavefront of the object beam O φ and the reference beam R φ .
Once the surface of the object is deformed, the phase of 1 I changes by φ ∆ , and thus the sum of two beams after deformation 2 I can be written as The subtraction of intensities of the input images 1 I and 2 I can be given by Eq. (8) represents a fringe pattern that can be numerically processed to obtain a phase map for measuring the deformation of the object on a specific t ime.
When a sinusoidal signal produces an oscillated vibrat ion on the surface of the object, the phase change φ ∆ can be expressed as where Λ is the wavelength of a laser, A is the amp litude and ω is the natural frequency.
Eq. (9) indicates that the vibration on the object's surface will change cyclically fro m a minimu m until a maximu m amp litude A. It is considered that in a HS-ESPI system there are more than two intensities that can be stored by a sensor while the object is under def1ormation, n   I  I  I  I ., . . , , , , then, the phase of the intensity patterns of interference will change from = ∆φ 0 until π 2 .
The number of intensity patterns of interference stored in a complete vib ration cycle will depend on the natural frequency, ω , and on the exposure time of the CMOS camera, τ .
In this case, the intensity patterns of interference stored by the camera can be written as Each of these subtractions represents a fringe pattern and each of these fringe patterns can be processed for ext racting the corresponding phase map along the deformation of the object for a specific t ime of the co mp lete cycle of vibration, n' goes fro m 2 to n-number of frames per second recorded by the high speed camera.
With this information, it is very possible to do the reconstruction of the deformat ion during a co mplete vibration cycle of the object.
One of the most important considerations of this technique is that the measurement can be started at any time; it is only needed to select an intensity pattern and make a subtraction of the fo llo wing consecutive intensity patterns to it. It is not need of any kind of synchronization or any kind of carrier to do the measurement.
Some of the phase shifting techniques requires a particular and exact amount of phase change between consecutive intensity measurements in order to apply specific algorithms for phase calculation.
Wyant [6] presented in 1984 a technique of phase measurement which depends of the amount of phase shift between consecutive measurements equal to α = π/2 (90°), yielding three equations, where the phase shift, α, is assumed to be linear. I 0 is the background intensity, γ is the fringe modulation, φ(x,y) is the phase distribution to be measured. Fro m these equations, the phase at each point is

Simulating results
Consider a full resonant vibration cycle at ω = 320Hz, with the CMOS camera working at an exposure time, τ = 4000 fps. The relat ion of intensity patterns of interference recorded by the camera gives a sample of m=12 fringe patterns along the whole vibrat ion event. The intensity patterns are separated from each other by a constant time.
To reach the 12 fringe pattern, we start fro m any intensity pattern of intensity at any time, called I 1 , and then the subsequence intensity is subtracted fro m it, then, the followed intensity pattern, I 2 is selected and the subsequence intensity is subtracted fro m it, same procedure keeps doing until cover the fu ll cycle of v ibration according to eq. (11d). Then a selected fringe pattern is chosen for all the combinations and is used to be applied on the Wyant's algorith m.
The fringe patterns obtained by the subsequent subtraction of the intensity patterns are shown in figure 3. The figure represents a complete cycle o f vibrat ing evolution where the maximu m deformation of the plate is located in the center of it. In accordance to eq. (1) the deformation of the p late must follow a polyno mial function of degrade 2 or 4.
According to Wyant, a single fringe pattern is selected and then shifted by a certain amount of phase in order to recover the phase of deformat ion. For this simulat ion the fringe pattern selected is the one inside the rectangle in figure 3 (b), Figure 4 shows the profile of the fringe patterns to be used in Wyant's algorithm.  Figure 6 shows the intensity patterns of interference recorded by the HS ESPI.

Experimental Result
The experimental fringe patterns coming by the HS-ESPI are shown in figure 7. Th is group of fringe patterns shows how the plate changes in a time interval of 0 to 250µs, 250µs to 500µs, 500µs to 750µs, 750µs to 1ms, 1ms to 1.25ms until covering the 3.125ms needed for a co mplete cycle of vibration.
It is observed that in each time interval, there is a set of fringe patterns with a visible change of the optical phase between them. Each t ime interval represents also a part of the mechanical phase evolution, in an instant of time and also along the time until 3.125ms that is the time needed to cover a full cycle of vib ration.
Taking advantage of using a high speed system, it is possible to choose the fringe pattern with the maximu m informat ion about the mechanical deformation of the plate due to the sinusoidal vibration. By simp le observation (that can be also done by simple co mputational software to be more precise) it is possible to select the fringe pattern coming fro m the subtraction |I 8 -I 9 | as the fringe pattern with mo re informat ion, fro m there, the selected fringe pattern to be used in Wyne's algorith m is |I 8 -I 9 | and the respective fringe patterns containing more information co ming fro m following cycles of vib ration, as it is shown in figure 8. It is important to remember that the selected fringe pattern can be anyone along the cycle and also, within all the posible fringe patterns coming fro m the HS ESPI system.    Figures 7 and 8 there is a noise that looks like a group of horizontal lines. This phenomenon is due to manufacturing problems specifically fo r this model camera. According to the manufacturer, this error was corrected for subsequent versions. The camera's version is the NAC's Memrecam fx 6000, however, this phenomenon doesn't affect the measurement because it is a constant noise and not a phase changing.
Applying eqn. (19), the resulted phase is shown in figure  9.  figure 11 is measured along the center of the p late and it is shown that the profile fits a quadratic polynomial function just as the theory indicates. The maximu m value of deformation shown in this graph could or couldn't represent the maximu m deformation that the plate suffers in reality; y-axis represents the amp litude value on radians of the deformation, x-axis is in mm. For th is experiment the adjustment of experimental results to a polynomial function of grade 4 is equal to 0.9953. It means that the experimental profile measured on this work fits the theory and also to the simulated profiles because it follows a polino mial function of grade 4. Th is adjustment indicates that the method works without the need of synchronization and without the need of mechanically tilt a component for shift ing an amount of phase in the experiment.
As it is shown, by means of HS-ESPI, the mechanical amp litude that is resulted of the processing of the fringe patterns represents relative values of the profile that has been measured.
Selecting a different vibration frequency involves a different exposure time on the camera. We work at 320Hz because it is the first vibration natural mode of the rectangular plate and the relation mentioned on the manuscript between the exposure time and the v ibration frequency give us enough fringe patterns that can be used in the proposed method and ensure that we can measure a complete vib ration cycle. The second and third vibration natural modes are 540 Hz and 740 Hz respectively as they were found in [22].
It is important to notice that this profile could or couldn't exhibit the maximu m deformation suffered by the plate. It is because the method employed in this work doesn´t need to be calibrated or synchronized.

Conclusions
A HS-ESPI system working with the Wyant's technique of phase shifting is a very fast way to measure maximu m mechanical d isplacement. The introduction of a high speed camera to an ESPI set up eliminates the need for tilted mirrors or piezoelectric devices for obtaining the phase and also eliminates the need of electronically synchronization.
High speed cameras and continuous wave lasers give a very strong way for measuring real time vibrat ions; it can be a very interesting solution to be applied in industry. The system records a sequence of 4000 fps, which can be use for applying some of the phase shifting techniques proposed fro m many authors and can be useful to characterize the complete vibrat ion evolution of a dynamic event. HS ESPI is a technique that allows a very fine exploration of mechanical displacements considering either static or dynamic events. The most important issue of the technique corresponds to a non contact and non invasive measurements.