## Abstract

A new technique capable of monitoring two-dimensional vibration fringe patterns in real-time is presented. It is based on optical reconstructing of time-averaged digital holograms. Recording of the holograms is realized in a quasi-Fourier off-axis setup whilst reconstructing in an extended Fourier transform setup capable of spatial filtering of the hologram reconstructions. The effectiveness of the proposed device, tested on a silicon wafer, was verified by performing operations such as monitoring of vibration modes, adjusting the hologram recording parameters, or searching for resonant frequencies. Additionally, mode beating between two distant vibration frequencies is observed and reported. The proposed technique is suitable for various scientific and industrial applications.

© 2006 Optical Society of America

## 1. Introduction

Vibrational properties of objects are of great interest in many applications. The asymmetries in modal structures, for instance, can provide information about the quality of the atomic force microscope sensors [1], or about the tuning possibilities of the percussion musical instruments [2]. Full-field vibration analysis is especially useful in industrial applications [3]. However, in various applications especially ones involving dynamic effects [4] it is desirable to monitor the vibration patterns in real-time.

For observing vibration patterns, the holography based techniques such as classical [5] and digital [4,6,7] time-averaged holography, holographic interferometry [8], and electron speckle pattern interferometry (ESPI) [9] have been frequently applied. Their advantages are usually connected to the property of mechanical non-contactness and the inherent ability of the full-field measurements. The main obstacles of these techniques to real-time monitoring are time-consuming operations like the wet processing, as in classical holography, or numerical processing, as in digital holography and ESPI. The use of the real-time holographic interferometry is possible, but it needs preparing the classical hologram before monitoring and is limited to observing the resonant modes only, since it provides extremely poor fringe contrast. However, the improved commercial systems using ESPI technique have been developed [10,11]. Other real-time systems for monitoring of vibrations use moire technique [12], speckle translation [13], Talbot interferometry [14], or laser vibrometers [15,16], but they observe the displacement of only one spot on the vibrating surface. The aim of this paper is to present a new monitoring technique that preserves the advantages of the holographic techniques and allows observation of high-contrast vibration fringe patterns in real-time. This paper for the first time demonstrates the instantaneous optical reconstruction of the digitally recorded time-averaged holograms, although optical reconstructing of digital holograms using various addressable spatial light modulators was discussed earlier [17].

The paper is organized as follows. Section 2 describes the proposed system and the constraints in numerical or optical reconstructing of digital holograms. The experimental device and the obtained results are presented in Sections 3 and 4, respectively.

## 2. Constraints in the reconstruction of digital holograms

#### 2.1. Two-stage digital holography system

Consider a system composed of two parts, part *A*, for digital recording of time-averaged holograms, and part *B*, for instantaneous optical reconstructing of the obtained holograms. Let *A* be a quasi-Fourier setup with a linearly polarized laser light as a light source and a CCD sensor as an array photo-detector. The input of *A* is an object wave front and a focused reference beam, while the output of *A* is a two-dimensional array of discrete hologram data. Let *B* be the extended Fourier transform (FT) setup [18] with a linearly polarized laser light as a light source, a liquid crystal display (LCD) as an input information carrier, and another CCD sensor as the output photo-detector. For vibration study, the input of *A* can be described as *U*(*x,y,t*) = δ(*x-X,y-Y*) + *s*(*x,y,t*), where *s*(*x,y,t*) is the object wave front and (*X,Y*) the location of the point source. Assuming that only harmonic out-of-plane vibrations of the object occur and that the illumination is nearly normal to the surface, then *s*(*x,y,t*) = *s*(*x,y*)exp[*i* (4*π*/λ)*h*(*x,y*)sin2*π*
*ft*], where *s*(*x,y*) is the static wave front, *h*(*x,y*) the vibration amplitude, λ the wavelength, and *f* the vibration frequency. The diffracted field at the hologram plane can be calculated using Fresnel approximation, *U*(ξ,η,*t*)∝ exp[*i*(π/*λd*)(ξ^{2} + η^{2})]*FT*{*U*(*x,y,t*)exp[*i*(π/λ*d*)(*x*
^{2} + *y*
^{2}]},where constant terms are omitted and *d* denotes the distance from the input plane. The exposure is obtained by integrating the instantaneous intensity over the time τ, *E*(ξ,η)= ${\int}_{0}^{\mathrm{\tau}}$|*U*(ξ,η,*t*)| *dt*, where τ ≫ 1/*f*.

The term of interest yielding one reconstructed image of the object is in the form:

where *J*
_{0} is the zero-order Bessel function of the first kind. This exposure is then captured by the CCD sensor, where the capturing can be described as smoothing of *E*(ξ,η) with the active zone of pixels, yielding

where (*m,n*) are integers, (∆ξ,∆η) the pixel pitches, (*α,β*) the fill factors, and (*M,N*) the numbers of pixels of the CCD sensor. The exposure term has now discrete values obtained by replacing the continuous variables (ξ,η) by (*m*∆ξ,*n*∆η).

#### 2.2. Numerical reconstruction of digital holograms

The obtained digital hologram can be reconstructed numerically by performing the inverse FT,

$${U}_{\mathit{num}}\left(x\prime ,y\prime \right)\propto \left(\left\{\delta \left(x\prime -X,y\prime -Y\right)\otimes s\left(x\prime ,y\prime \right){J}_{0}\left[\frac{4\pi}{\lambda}h\left(x\prime ,y\prime \right)\right]\right\}\mathrm{sin}c(\frac{\alpha \Delta \xi}{\mathit{\lambda d}}x\prime ,\frac{\beta \Delta \eta}{\mathit{\lambda d}}y\prime )\right)\otimes \mathit{comb}(\frac{\Delta \xi}{\mathit{\lambda d}}x\prime ,\frac{\Delta \eta}{\mathit{\lambda d}}y\prime )\otimes \mathrm{sin}c(\frac{M\Delta \xi}{\mathit{\lambda d}}x\prime ,\frac{N\Delta \eta}{\mathit{\lambda d}}y\prime )$$

and followed by squaring of the modulus: |*U*
_{num}(*x*′,*y*′)|^{2}.

#### 2.3. Optical reconstruction of digital holograms

For optical reconstruction, the discrete hologram values distributed along the LCD panel can be described as:

where (∆*ξ*′,∆η′) are the pixel pitches, (α′,β′) the fill factors, and (*M*′,*N*′) the numbers of pixels of the LCD panel. The FT of the Eq. (4),

is first spatial filtered in the first FT plane, then enlarged in accordance to the spatial resolution of the second CCD sensor, and finally captured by the second CCD sensor. There exists an expected degradation of the optically reconstructed patterns caused by changes of the starting continuous hologram information to a pixelated one (due to the first CCD sensor) and then to a scaled one with lower resolution (due to the LCD panel). At best, if the LCD panel and the first CCD sensor have similar parameters, a degradation of the optical reconstruction will be only due to the flatness imperfections of the panel [19]. Otherwise, the optical reconstructions will suffer from larger speckle noise than the numerical ones due to the number of pixels used in processing [20].

## 3. Experimental device

The experimental setup is schematically shown in Fig. 1. In *A*, a He-Ne laser (power = 35 mW, λ = 632.8 *nm*) and a digital CCD_{1} sensor (2048 × 2048 pixels, 7.4×7.4*μm*
^{2} each pixel) are used. A continuous variable beam splitter (VBS) served for adjusting the beam balance ratio and a loudspeaker (LS) served to excite the object externally. In *B*, another He-Ne laser (power = 10 mW) is used together with the LCD (VGA, pixel pitch = 40 × 40 μm^{2}, fill factor = 0.8, refreshing rate = 50 Hz) and an analog CCD_{2} sensor (752 × 582 pixels, 8.6 × 8.3 μm^{2} each pixel). The lenses (FTLs) serve to perform the FT operation, to allow the spatial filtering in the FT domain, and to enlarge the FT spectrum.

The object of study was a silicon wafer (diameter about 50 mm, thickness = 0.3 mm) clamped to a holder on one side and excited by a loudspeaker. The frequency and intensity of the loudspeaker sound was controlled by a PC program. For numerical reconstructing, a PC connected to the CCD_{1} sensor is used with Intel Pentium 4 (2.66 GHz CPU and 1 Gb RAM). Time needed to process digital holograms, defined mainly by the image loading and calculating the Fourier power spectrum, for 2048 × 2048 pixels images was approximately 10.5 seconds. Additional time is needed for post processing. On the other hand, time needed to access the optical reconstruction is 0.02 seconds which is due to the refreshing rate of the LCD, since the optical in-line processing is practically instantaneous. Thus, our experimental device enables monitoring of the vibration fringe patterns with a refreshing rate of 50 Hz.

## 4. Results

#### 4.1. Adjusting the hologram recording parameters

To record the best digital hologram in given experimental conditions, one of the practical problems is experimental adjusting of the hologram recording parameters. Two main parameters, namely the intensity ratio between the reference and object beams and the total intensity incident onto the CCD_{1} sensor, are optimized in a fast and easy way when observing the object reconstruction in real-time.

#### 4.2. Searching for the resonant vibration modes

The resonant frequencies of the wafer are found by using the sweep function, where searching within 1 KHz range of frequencies with 1 Hz resolution consumes 20 seconds of time. The obtained resonant patterns, shown in Fig. 2, demonstrate satisfactory fringe contrast. As evident from Fig. 2, a circular aperture is used to select only one object reconstruction and to remove the disturbances. For comparison, the corresponding numerical results are given in Fig. 3, where one object image is cut out from the reconstructed images. Basically, Figs. 2 and 3 demonstrate the same information. Thus, although with larger speckles, the optically obtained reconstructions are suitable for monitoring the vibration fringe patterns and locating any changes in these patterns in real-time. Note that optical reconstructions could be still improved by using some available better-quality spatial light modulator [21] instead of our LCD (as discussed in Section 2.3.).

#### 4.3. Mode beating

The beating occurs when two harmonic vibrations of different frequencies act along the same vibrating system simultaneously. These two harmonic motions form a superposed fringe pattern that can be static or dynamic depending on the mutual phase differencies. For a practical two-dimensional system the beat frequencies forming a dynamic fringe pattern are difficult to find, since the beating phenomenon depends on many experimental parameters. In the case of the silicon wafer such parameters are the sound intensities or the presure of the screws that fasten the wafer. To find beating, the wafer was stimulated to vibrate at two different frequencies among which one was fixed (220 Hz) and the other driven by the sweep function. By monitoring the time-averaged fringe pattern in real-time, finding the beat frequencies was a relatively easy task. Movies shown in Figs. 4–6 illustrate these results. Figure 4 shows that superposition of two resonant frequencies (220 Hz and 1980 Hz) result in a static vibration mode. Note that the second frequency is a harmonic of the first frequency. When the second frequency is shifted by the interval *∆f*, the beating occurs with the beat frequency defined by *∆f*. Figs. 5 and 6 show the beating with the interval *∆f* equal to 2 Hz and 4 Hz, respectively.

## 5. Conclusion

Although a number of optical methods are available for studying vibrations, new solutions and devices are of interest due to the versatility of practical vibration problems especially those involving dynamic effects. This paper presents an original device for the real-time monitoring of vibration fringe patterns. The device consists of two parts, one for recording the time-averaged digital holograms and the other for instantaneous optical reconstructing of the obtained holograms. Two optical architectures are used, namely, the quasi-Fourier off-axis for recording and the extended Fourier transform for reconstructing. In the latter part, the unwanted diffraction terms are optically filtered out before displaying the output image. Thus constructed device, where optical reconstructing showed to be three orders of magnitude faster than corresponding numerical, preserving at the same time the output information as well, demonstrated very useful features. The real-time operations such as monitoring the vibration modes, or adjusting the hologram recording parameters, or searching for the resonant frequencies are easily achieved. Furthermore, mode beating between two different frequencies was observed and detected. To conclude, the fastness combined with the effectiveness and robustness of the proposed device make it suitable for various scientific and industrial applications.

## Acknowledgments

This work was supported by the Croatian Ministry of Science, Education and Sports.

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