1. Introduction

Electronic Speckle Pattern interferometry (ESPI) is a whole field optical interferometric technique appropriate for studies of objects with optically rough surfaces. The method is based on interference of the scattered light from the specimen with a reference speckle pattern, and the resultant pattern shows also a randomly distributed intensity. The changes in the object state, for instance, deformation and displacement, are proportional to the phase changes within each speckle [1–2]. The recording media in ESPI is a CCD camera, for which the method has found many applications, especially in areas where a real-time 2D picture of a given process is required. For quantitative data analysis, various phase analyzing methods are applied.

In optical interferometry, there are numerous well-established methods for quantitative phase analysis [1–4]. In general, these techniques use temporal or space phase modulation. In the former group of methods, for instance, phase shifting interferometry is a well-established technique. Since the algorithms require the object to be stable for at least three frames, while the phase modulation is introduced, they are usually suitable for the study of static or quasi-static events, and some of them are also well suited in ESPI. The latter group of methods, which are appropriate for dynamic studies, require either a complicated and expensive optical set-up, or the calculation procedure requires a two-dimensional Fourier transform, which makes the methods difficult to apply in practice. Recently, we have developed a phase analysis technique for dynamic ESPI [5] that does not require phase modulation. The method utilizes both the subtraction and addition of correlation fringes; however, it requires a special procedure for phase unwrapping, for example, skeletonizing the phase image, prior to unwrapping, in the presence of speckle noise. In the case of correlation fringe density comparable with the resolution of the CCD camera, the skeletonizing becomes particularly difficult.

In the past several years, there have been active research in developing phase analysis methods that could be appropriate for dynamic phenomena [6–11]. The main idea in these methods is to consider the processing of the interference signal in the temporal domain, instead of the spatial domain, which can improve the measurement range, and allow making the calculation automatic.

In this paper, we propose a method for phase analysis of ESPI measurements and for obtaining the 3-D picture of the deformation field, in which a temporal Hilbert transform (HT) method is applied for phase analysis. The HT method has several advantages. It can be applied to a wide range of interferometric measurements, it has a simplicity in the calculation algorithms and it has relatively shorter calculation time, compared, for example, to Fourier transform method. In addition, the application of HT can be made fully automatic. Another advantage is the high spatial resolution of the phase map, up to the same order of the speckle size. The HT method has also found applications in Low Coherence Optical Tomography, Doppler velocimetry and other fields of science [12–16]. We present two experiments using a Fizeau type interferometer [17] with a scattering plate for producing the reference speckle pattern.

2. Data-analysis method

In general, interference speckle images obtained in dynamic ESPI (DESPI) can formally be represented by

where, I ₀(x,y,t _i) and I_m(x,y,t_i) are the bias and the modulation intensities, respectively. ϕ(x,y,t_i)=θ(x,y,t_i)+φ(x,y,t_i) is the signal phase, where θ(x,y,t_i) is the random phase of the speckle field, and φ(x,y,t_i) is the phase introduced by the deformation of the object under study, which usually varies slowly in the space and the time domains. t_i is the time the i-th frame of speckle interference pattern is taken. We can treat each point in space domain having an independent time signal of interference intensity. To obtain the phase, the temporal history of the development of the interference signal at each point in the image is processed, applying signal-processing technique. One advantage of working in time domain in case of ESPI is the significantly low presence of noise. Some other advantages are significantly simpler algorithms for one-dimensional signals, and the straightforwardness in the unwrapping procedure. In addition, since the interference signal that we are using is three dimensional, we can receive the full spatio-temporal information of a given process.

In the time domain, the interference signal is a one dimensional cosine function. The task to carry out is to determine the phase at each time. In signal theory, Denis Gabor was the first to introduce the notion of analytic signal [16,18–19]. According to it, with each real wave function u(t), we may associate a complex wave function ψ(t)=u(t)+iv(t) where the imaginary part is the HT of the real signal. The HT of u(t) is defined by [16]

Hi{u(t)} is a linear functional of u(t). In fact, it is obtained from u(t) by convolution with (-πt)^-1 expressed by

Hilbert transform is equivalent to the filtering, in which the amplitudes of the spectral components are left unchanged, except that their phases are altered by π/2, positively or negatively according to the sign of t. The HT of the signal f(ϕ(t))=cos(ϕ(t)) gives sin(ϕ(t)). Using this characteristic, we can determine the phase of the signal through the equation given by

For more details on properties of HT, reader can refer to [16,18–20].

A typical temporal signal observed at a certain point in the interference pattern of ESPI is shown in Fig. 1. As it can be seen in Fig.1, the variation of the bias intensity is considerably large as well as the variation of the modulation intensity. To minimize the influence of the bias intensity on the calculation process for the phase, prior to performing the Hilbert transform, we removed the bias intensity using an averaging filter. The details about the averaging procedures and consideration for the averaging filter size were already presented in [5]. In general, the size of the averaging filter is chosen in a way so that it includes at least 3 periods of the cosine function with maximum error of 5%. In addition, the bias intensity should not change considerably within the filter window. The estimation of the window size was performed considering the deformation speed and the camera acquisition rate.

 

Fig. 1. Temporal history of the interference signal at single point

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In order to calculate the Hilbert transform, we used MATLAB Signal processing toolbox [21–22] for filter design. In the first stage, a three-dimensional matrix is created by successively taking frames of interference pattern. In the second stage, the bias intensity is estimated and removed form the interference signal for each pixel. Then the HT is calculated in temporal domain and the wrapped phase value is determined. The unwrapping procedure in time domain does not suffer of discontinuities problems, typical for spatial phase unwrapping problems [23]. The speckle phase is removed from the phase values by subtracting a reference phase value, which can be expressed by

where p represents the reference frame. The speckle phase value will change for a long term of experiments. For this reason, the reference value is renewed after a certain interval. The unwrapped phase values are recorded as two-dimensional images, which represent the space development of the deformation. Because some pixels have low modulation intensity, the phase in these pixels could not be determined correctly. These pixels give spiky noise in the final results. To remove this noise, a median filter is applied in space domain.

3. Experimental results and discussion

To demonstrate the validity of the current method based on HT in DESPI, two different experiments were carried out. The schematic of the optical set up is given in Fig. 2. It is a Fizeau type interferometer with scattering plate for reference filed. Its almost common path arrangement of the interferometer permits a stable observation. The light source was a laser diode with the output power of 50 mW and the wavelength λ=658 nm. The working region of the injection current of the laser was chosen over a linearly varying range of the wavelength of the emitted light with no apparent mode hop. The light from the laser is collimated with lenses L1 and L2 to illuminate the object normally with a parallel beam. Part of the illumination beam is scattered from the plate, which was placed in front of the object at a distance of 55 mm, and the transmitted light is scattered by the object. The two scattered lights are superposed and interfere in the observation plane where the CCD camera was positioned. The ground glass that was used in the experiment is a very weak scatterer in order to obtain an almost equal intensity ratio between the two interfering beams. In fact, one can observe the object through the scattering plate, although its image is slightly blurred. The sensitivity vector lies in a plane perpendicular to the x-y plane. The amount of deformation in z-direction per unit fringe spacing, λ/(cosα+1), was estimated to be 330 nm with observation angle to the normal of the specimen surface α=6°, and the interference speckle patterns were taken continuously with a constant acquisition rate.

 

Fig. 2. Out-of plane sensitive optical set-up for dynamic deformation simulations. L1, L2 and L3 - lenses, α-observation angle

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The specimens used in the two experiments are illustrated in Fig. 3. In the first experiment, we prepared a plastic plate, with a cutting in the middle of the specimen, as is shown in Fig. 3(a). The red dot indicates the position where the concentration load was applied to the object using PZT. The purpose of the experiment was to examine the validity of HT methods in distinguishing defects in objects. The width and the height of the specimen were 120 mm and 60 mm, respectively, and the thickness was 3 mm. The observation area indicated by the blue line was 50 mm by 55 mm. To increase the amount of light scattered in the direction of observation, the surface of the specimen was coated with white paint. The plate was deformed besides the cutting by PZT with a maximum displacement 15 µm. The PZT was driven with a signal of a saw tooth shape function at a frequency 0.8 Hz. A high-speed CCD camera (Photoron FASTCAM-PCI) with pixel size 7.4 µm was used. The acquisition rate of the camera was set to 250 fps. The diaphragm of camera lens was adjusted to f#11 so that the speckle size over the observation plane becomes approximately 10 µm. The window size of the filter to perform the HT spans over 35 frames along the time axis. For the bias intensity, a simple temporal averaging filter was applied with the size of 35 frames.

 

Fig. 3. Schematic of the front of the objects used in the experiments

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Fig. 4. Experiments of the plastic plate with the cutting in the middle of the specimen

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After the two-dimensional phase map was obtained, it was filtered using spatial Median filter of size [5×5]. The results from the calculations are shown in Fig. 4. The area of the cutting can clearly be distinguished in Fig. 4. There is a sharp jump in the deformation field, which indicates the place of the cutting in the specimen.

For the second experiment, in order to distinguish sign in the deformation field, the injection current of the laser diode was modulated with a saw-tooth signal to produce 2π phase change at a given interval of time. The range in the change of the injection current was estimated by measuring the laser characteristics. The offset of the injection current was set at 700 mA with modulation amplitude 5.6 mA and a modulation frequency 2 Hz. For the object, we used a plastic plate, as shown in Fig. 3 (b), which was deformed by the sinusoidal driving signal for the PZT, with amplitude of approximately 1.5 µm peak-to-peak. The loading point was at the upper right hand side.

In this experiment, a Sony XC-55 CCD camera, with pixel size 7.4 µm, was used. The acquisition rate was set to 30 fps. The f# of the lens was 5.6, and the speckle size was estimated to be 5.5 µm. The HT filter size was 35, and the averaging filter size was 51. After the unwrapping procedure, the modulation phase was removed by using linear approximation. The results of the spatio-temporal distribution at one cross-section of the image, as indicated with a solid line in Fig. 3(b), are given in Fig. 5(a). Figure 5(b) gives an animation that displays the temporal development of the two-dimensional deformation field. Figure 6 gives cross-section of the spatio-temporal distribution at the points indicated by A, B, C, and D on Fig. 3(b), which are separated by 10 mm along the solid line. Black line corresponds to the deformation at the place closest to the loading point, i.e., the point indicated by A, and the blue line corresponds to that at the place furthest from the loading point, i.e., the point indicated by D.

4. Conclusions

The method proposed in this paper uses the notion of the analytic signal and Hilbert transform to calculate the phase. The data processing is performed in the temporal domain, considering the temporal history of the interference signal at every single pixel. This results in a relatively high spatial resolution of the phase map, up to the same order of the speckle size. In contrast, obtaining the phase based on fringe analysis [5] or spatial carrier method [3] is limited by the smallest fringe spacing, which is usually several times larger than the speckle size. In addition, the phase method can be made fully automatic and does not require human interaction. The final results give a temporal development of the two-dimensional deformation field. To reduce the influence of the fluctuations of bias intensity on the calculated phase, it was removed prior to performing the HT. The proposed method for analysis of the phase of dynamic ESPI was examined in two different experiments, i.e., defects distinction, and determination with the sign change in the deformation field. The dynamic range of measurements is increased from several tens of nanometers to tens of micrometers, which makes the method very attractive for dynamic measurement.

 

Fig. 5 (a) Spatio-temporal distribution obtained from the experiment with two directional deformation, (b) (2.5 MB) Movie of the temporal distribution of the two-dimensional deformation field

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Fig. 6 Cross-sections of the deformation at different points in space

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