VIBRATION PHASE BASED ORDERING OF VIBRATION PATTERNS
ACQUIRED WITH A SHEARING SPECKLE INTERFEROMETER AND
PULSED ILLUMINATION
Peter A.A.M. Somers and Nandini Bhattacharya
Optics Research Group, Delft University of Technology
Lorentzweg 1, NL-2628 CJ Delft, the Netherlands
[email protected];
[email protected]
ABSTRACT
A method for both temporal and spatial characterization of harmonic vibrations is presented. The method is based on
simultaneous acquisition of phase-stepped speckle interference patterns along with measurement of vibration phase for each
vibration state. An unsynchronized free running pulse laser is used for illumination yielding speckle interference patterns in
random vibration phase order. Two π/2 phase-stepped speckle interference patterns are acquired simultaneously for each
recorded vibration state. The data-set is sorted using the vibration phase as sorting key. Subsequently the sorted speckle
interferograms are processed to obtain wrapped phase difference maps or accumulated unwrapped phase distributions.
Introduction
Harmonic vibrations of diffusely reflecting objects can be visualized using interferometric methods such as holography or
speckle interferometry [1, 2]. The vibrating object can be illuminated by a continuous wave laser or a pulse laser. When using
a continuous wave laser and exposing the holographic plate or CCD camera for a period that is relatively long compared to the
vibration period time average interferograms can be obtained, showing the vibration pattern. When a pulse laser is used two
exposures with appropriate timing can be made to obtain correlation fringes by subtracting the two speckle interference
patterns. A particular implementation with limited requirements on synchronization is based on the use of a double pulse laser,
and the spatial carrier phase-shift method to obtain phase [3]. Phase information can also be obtained by applying phase
stepping for measured vibration states, imposing more extensive requirements on synchronization of vibration, illumination,
camera exposure and phase stepping.
For quantitative evaluation of the vibration pattern for a particular vibration phase using a continuous wave laser, the camera
shutter must be synchronized to the vibration phase. In addition, phase stepping must also be synchronized to both camera
exposure and vibration phase. Generally several vibration periods are required to obtain the number of phase-stepped
interference patterns that are necessary for the chosen algorithm for the calculation of phase. When full temporal
characterization of the vibration pattern is required a series of measurements has to be taken with appropriate timing in order
to obtain data for a sufficient number of different vibration phase values. For this purpose a stroboscopic approach can be
used, either employing synchronized pulsed illumination or synchronized camera exposure [4, 5]. In most cases the object
vibration is considered to be the master process: the other processes are synchronized to the vibration frequency.
Consequently, stroboscopic measurements using a pulse laser require the laser pulses to be coupled with the vibration
process. However, not all pulse lasers that provide sufficient power can be easily synchronized to external events.
An alternative method for stroboscopic measurements that use a synchronized pulse laser is proposed. Instead of triggering
the pulse laser by pulses that are derived from the vibration process the laser is free running, acting as the master process.
Simultaneous acquisition of interference patterns and vibration phase
For our experiments a free running flash lamp pumped Nd:YAG pulse laser was available that could not be triggered
externally, but did provide access to synchronization pulses. Using those, a set-up has been developed in which the pulsed
illumination was the master process. Image acquisition was controlled by synchronization pulses that were provided by the
pulse laser.
The interferometer we have used to measure vibration patterns records two π/2 phase-stepped speckle interference patterns
for one particular vibration state simultaneously [6, 7]. When two of these phase-stepped speckle interferogram pairs are
taken, one for a reference vibration state and the other for a current one, four phase-stepped speckle interference patterns are
available, from which phase difference ∆ϕ between the two states of the object at times t and t + ∆t can be calculated, using
the 2-bucket algorithm [7, 8] given in Eq. (1)
 I 0(t ) − Iπ / 2(t + ∆t ) 
∆ϕ = π / 2 + 2 arctan 

 Iπ / 2(t ) − I 0(t + ∆t ) 
(1)
where I0 and Iπ/2 are the intensities corresponding with phase steps 0 and π/2.
There is no need to acquire the second pair during the same vibration period if displacement is periodic, which was assumed
to be the case for our experiments. The second pair can be acquired several vibration periods later and can still be combined
with the first pair if relative displacement is within a certain range. Available power per pulse and pulse duration can be chosen
such that the effective integration time of the camera is short enough to freeze the vibration pattern, even at high vibration
frequencies.
Vibration phase can be recorded also, simultaneously with speckle data for each state. For that purpose a counting board and
some custom electronics for pulse shaping were used to implement measuring of elapsed time between two types of input
pulses: vibration sync and laser sync pulses. The two pulse series, originating from the two independent sources, the vibration
set-up and the pulse laser were combined by an OR circuit to produce a combined series of pulses, basically at the vibration
frequency, including sparse laser sync pulses at random positions between two vibration sync pulses (Figure 1). The set-up is
shown in Figure 2. The counter is read out and restarted on every incoming pulse, which can be either a next vibration sync
pulse or a laser sync pulse. Elapsed time now represents either the vibration period or the vibration phase. Laser sync pulses
can be identified by also observing the next read-out: in that case the sum of the elapsed times belonging to a valid vibration
phase measurement and the next one is equal to one full vibration period (Figure 3). A valid vibration phase measurement can
now be assigned to a simultaneously acquired pair of phase-stepped speckle interference patterns.
Vibration period
Vibration phase
1.5
1
0.5
Vibration
amplitude
0
0
2
4
6
8
10
12
14
16
18
20
-0.5
-1
-1.5
Time
Vibration pulse
Laser pulse
Figure 1. Synchronization pulses from two independent sources: vibration set-up and pulse laser
After having acquired a given number of phase-stepped speckle interference pattern pairs, along with their respective vibration
phase, the latter can be used to sort the speckle interference patterns. Speckle patterns can be now acquired in random order
using a free running pulse laser as the illumination source. Sorting the speckle interference patterns using the vibration phase
as the sorting key results in a series of speckle pattern pairs that can be processed as if they were acquired in the sorted
order. There are some limitations to the laser repetition rate in order to achieve full coverage of the vibration period: the
sampling and vibration frequencies should be asynchronous.
CCD
Camera trigger
Laser pulses
OR
Counter
PC
Vibration pulses
20 MHz
clock
Figure 2. Overview of measurement system for simultaneous acquisition of vibration patterns and vibration phase
20000
15000
10000
Vibration period
(counts)
5000
0
1
3
5
7
9
11
13
15
17
19
Measurement #
Figure 3. Identification of a vibration phase measurement. A vibration phase measurement is characterized by a time shorter
than the vibration period and verified by also observing the next measurement. The sum of the two measurements is equal to
the vibration period
Vibration frequency is considered to be an independent variable while sampling frequency is governed by pulse repetition rate,
which is also independent in our present set-up. Although the laser could not be synchronized as a slave process, it was
possible to change its pulse repetition rate in order to avoid the vibration frequency to be a multiple of the sampling frequency,
a condition that also has to be met for stroboscopic systems. With proper settings an even distribution of samples over the
vibration period can be obtained. In practice, where sampling and vibration frequencies differed by one or two orders, it
appeared to be very hard to achieve settings for which an unacceptable distribution of samples over the vibration period was
obtained. Figures 4 and 5 show some examples for possible distributions of samples over the vibration period. Figure 4
presents an unacceptable distribution obtained for a setting for the laser pulse repetition rate close to 10 Hz and a vibration
frequency of 900 Hz. Figure 5 shows an acceptable distribution for a slightly changed value of the pulse repetition rate. Both
examples were obtained during testing of the set-up where laser sync pulses were replaced by a pulse generator and vibration
sync pulses were obtained from the vibration set-up.
If a sufficient number of interference patterns are acquired, phase difference between two states can be kept within the range
of -π to +π. In that case temporal phase unwrapping [9] can be applied, which does not rely on evenly distributed samples.
After sorting the series of speckle interference patterns using the vibration phase as the sorting key, the data can be
processed sequentially to produce phase difference maps that might be wrapped and unwrapped accumulated phase
difference distributions for all values of the vibration phase.
0
2π
Figure 4. Distribution of 50 vibration phase values when vibration frequency is close to a multiple of laser repetition rate
0
2π
Figure 5. Distribution of 50 vibration phase values when vibration frequency is not related to laser repetition rate
Experimental
Some experiments have been carried out on a vibrating Aluminum plate of 240x240 mm, with a thickness 0.5 mm. The plate
was elastically supported by four springs, attached to the corner of the plate (Figure 6). The specimen was excited by a V201
shaker, manufactured LDS, which was driven by a 100 W PA audio amplifier. The signal source was a Hewlett Packard
33120A arbitrary waveform generator, providing a sine wave of 900 Hz, which was one of the resonance frequencies of the
plate.
Figure 6. Experimental set-up
The specimen was illuminated by a divergent beam, provided by a system 2000 pulsed Nd:YAG laser, manufactured by JK,
delivering approximately 30 mJ at a repetition rate of 10 Hz. Pulse duration was 15 ns approximately. The laser provided two
synchronization pulses, one synchronized with the Pockel’s cell, the other was a pre-trigger signal about 100 µsec earlier. The
first pulse was used to read out and restart a general-purpose counter, available on a National instruments PCI-6713 DAQcard, that was used to measure vibration period or phase, the last one to trigger the CCD camera. Another trigger signal,
derived from the waveform generator was OR-combined with the laser sync pulses, and the combined pulse series fed to the
vibration period/phase counter. The counter was driven by a 20 MHz clock, yielding around 18,000 pulses during one vibration
period.
Vibration patterns were acquired in random order and sorted using vibration phase as the sorting key. After sorting, the
interference patterns were processed in sorted order using temporal phase unwrapping [9], allowing accumulation of phase
differences to obtain unwrapped phase data at any instance of the vibration phase.
Results
Figures 7-9 show phase differences between a series of intermediate vibration states and the reference state, which was close
to the static state, where vibration amplitude was zero. Figure 7 shows the results for an intermediate vibration state during the
first half of the vibration period. Figure 8 shows a result for a vibration state halfway the vibration period. In figure 9 a result for
a vibration state during the second half of the vibration is presented. For most of the measured intermediate states the
displacement related phase differences were higher phase than +/-π, resulting in wrapping. Since the interferometer is of the
shearing type spatial derivatives of the out of plane displacement are shown. Shear is in vertical direction. Figures 7 and 9
clearly show the reversal of phase for first and second halves of the vibration.
Figure 7. Intermediate vibration state during the first half of the vibration period
Figure 8. Intermediate vibration state halfway the vibration period
Figure 9. Intermediate vibration state during the second half of the vibration period
When a sufficient number of samples per period are taken, phase difference between samples can be smaller than +/- π,
avoiding wrapping and allowing phase differences to be accumulated over all or a selection of measurement intervals.
Following this procedure, the results for a selection of measurement intervals during the positive and the negative part of the
vibration period have been accumulated. Figure 10 shows the accumulated phase difference distribution for the positive
portion of the vibration; figure 11 shows the results for the negative part. Again the reversal of phase for first and second
halves of the vibration is clearly shown.
Figure 10. Accumulated phase difference distributions during the first half of the vibration period
Figure 11. Accumulated phase difference distributions during the second half of the vibration period
Conclusions
A set-up for performing phase-stepped measurements to measure dynamic phase changes using a free-running pulsed laser
has been established. Two phase-stepped speckle interference patterns are acquired simultaneously for each recorded
vibration state, in random vibration phase order and sorted using the vibration phase as sorting key. The set-up is capable of
characterizing periodic dynamic events both spatially and temporally.
Acknowledgments
This research was supported by the Technology Foundation STW, applied science division of NWO and the technology
program of the Ministry of Economic Affairs. We thank Günther Ouwendijk, Polytechnic of Rijswijk, for his contribution to this
work.
References
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6.
7.
8.
9.
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Somers, P.A.A.M. and Bhattacharya, N., “Polarization plane rotator used as a phase stepping device in a 2-channel
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W. Osten, C. Gorecki and E. Novak, SPIE, Bellingham, 664-673 (2005).
Somers, P.A.A.M. and van Brug, H., “A single camera, dual image real-time-phase-stepped shearing speckle
interferometer”, Proceedings of Fringe 2001, edited by W. Osten and W. Jüptner, Elsevier, Paris, 573-580, (2001).
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