1. Introduction

The two-dimensional spatial Fourier transform method (designated as 2DFTM henceforth) [1] is a widely used technique for retrieving the phase of speckle interferograms with spatial carrier. In short, it yields the optical phase of an interferogram through the calculation of a direct 2D Fourier transform, the selection of one interferential term of the spectrum by means of a filter, and an inverse 2D Fourier transform [Figs. 1(a) to 1(d)]. The information of interest is often the phase change between two states, so the aforesaid three steps are usually applied to pairs of interferograms whose optical phases are subtracted to yield an optical phase-change map [2], often called simply “phase map”. The usual assumption when using this method is that the Fourier spectrum of the interferograms has a trimodal distribution, i. e., the interferential terms are completely separated from the low-frequency terms related to the intensity of the object and reference beams. When this assumption does not apply, we have spectra of the type shown in Fig. 2 , which is an enlarged version of Fig. 1(b). The four spectral terms can be distinguished: the term related to the intensity of the reference beam is the narrow bright region around the origin. The faint halo centered in the image is related to the intensity of the object beam. When the reference beam is made to diverge from a point on the aperture plane –a usual configuration in image-plane digital holography– the conjugate interferential terms are focused images of the aperture symmetrically shifted from the origin by the spatial carrier. In this example, these terms partially overlap the spectrum of the object beam. A filter window applied to any of the side lobes, which carry the phase of interest, will also capture a part of the central lobe, thus inducing an error in the resultant phase. Whilst total spectral separation could be achieved by reducing the size of the aperture until there is no overlap with the central term, it would be at the cost of reducing the amount of light reaching the sensor and the spatial resolution of the resultant phase map, which may not be acceptable in some cases. The alternative of changing the shape of the aperture (for example, a narrow rectangular diaphragm [2]) is not an option when working with commercial camera objectives. This led us to develop a method that removes the aforesaid phase error (related to the overlap of the side lobes with the central lobe in the 2D spectrum) whilst keeping the spatial resolution and light collecting capability of large apertures (which would yield a large overlap of the side lobes with the central lobe in the 2D spectrum).

 

Fig. 1 Summary of the 2DFTM method–subfigures (a) to (d)– and the 3DFTM method –subfigures (e) to (h). (a) Single interferogram. (b) Modulus of the 2D Fourier transform of the data and bandpass filter. (c) Filtered spectrum. (d) Complex-valued map containing the optical phase of the interferogram. (e) Sequence of interferograms. (f) Modulus of the 3D Fourier transform of the data. A 3D bandpass filter is also shown. (g) Filtered spectrum. (h) Sequence of complex-valued maps containing the optical phases of the interferograms.

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Fig. 2 Modulus of the 2D Fourier transform of a speckle interferogram with spatial carrier.

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This seemingly contradictory situation is possible by changing from a 2D to a 3D spectrum. Instead of processing one interferogram with spatial carrier, as in the 2DFTM, we record a sequence of interferograms, where the phase of interest may be either constant or a function of time. Each interferogram has a spatial carrier and a temporal carrier is introduced between succesive acquisitions by modulating the reference beam. This stack of spatiotemporal data is then processed jointly with a 3D Fourier transform. The spectral terms that would overlap in a 2D spectrum are separated in the resultant 3D spectrum because the faint halo in Fig. 2 is transformed into a thin disk spanning a narrow range of temporal frequencies around the origin, and the interferential terms are separated from one another and from the central disk by the spatial and temporal carriers respectively [see Fig. 1(f)]. After the application of a 3D filter and an inverse 3D Fourier transform, the final result is a sequence of complex-valued maps that contain the optical phases of the interferograms. This procedure, designated as 3DFTM henceforth, follows the same rationale that the authors used some years ago to develop a technique [3] to obtain the mechanical complex amplitude of ultrasonic surface acoustic waves from a sequence of displacement maps, and was also applied some time later by Abdul-Rahman et al. [4] in the field of fringe projection profilometry.

The structure of the paper is the following: the theoretical development of the proposed method is presented in section 2. To corroborate the theory, some experimental measurements of the amplitude of surface acoustic waves in plates were carried out with a double-pulsed TV holography setup. Section 3 describes this setup and the experimental procedures. The results are presented in section 4 and discussed in section 5. A comparison with results obtained when individual interferograms are processed with the regular 2DFTM is also provided in section 4. Finally, the main conclusions of this work are summarized in section 6.

2. Theory

The general expression of a speckle interferogram with spatial carrier, recorded with single-pulse illumination, for a given instant t_n is

N interferograms I_n with n=0,…, N–1 are acquired, and an additional phase α_n is introduced in each interferogram. The increment of α_n from one interferogram to the next is the phase step α and is the same for all the interferograms, i. e. α_{n + 1}–α_n=α. The additional phase α_n can be seen as a temporal carrier of the form 2πf _{c t}t_n, with a frequency

The modulus of the resultant 3D spectrum is of the form shown in Fig. 1(f). A 3D bandpass filter, in the general case, selects one of the side lobes and the inverse Fourier transform of the filtered data gives a set of complex-valued phase maps

The quantity of interest is usually the optical phase-change ΔΦ between two states of the object, recorded in two interferograms i and j. ΔΦ can be calculated in a single step [5]

3. Experimental

3.1. Set-up

The experimental set-up and some details of the procedure employed to record the sequence of interferograms are shown in Fig. 3 . A frequency-doubled, injection-seeded, Nd:YAG pulsed laser (Spectron SL404 T) with two independent cavities running at 25 Hz delivered pairs of pulses (p^a, p^b) with λ=532 nm [Fig. 3(c)]. A modified Mach-Zehnder interferometer was used to form the interferograms that were recorded with a CCD camera (PCO Double Shutter). An aluminium slab of dimensions 400×130×30 mm³ was insonified with short bursts of ultrasonic surface acoustic waves of central frequency 1.000 MHz, which induced a nanometric displacement of the observed surface. The interferometer was sensitive to the out-of-plane component of this displacement, represented by u₃ in Fig. 3(d).

 

Fig. 3 (a) Experimental set-up; (b) phase modulation of the reference beam; (c) laser pulses; (d) out-of plane displacement u₃ of the surface due to the ultrasonic wave; (e) and (f) several interferograms from sequences I^b and I^a respectively.

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A sequence of N=64 pairs of interferograms (I^a, I^b) with 12 bits per pixel was recorded. I^a_n and I^b_n denote individual interferograms located at position n in the sequence, with n=0…N–1. Interferograms I^a_n [Fig. 3(f)] captured a reference state whereas interferograms I^b_n [Fig. 3(e)] captured a second state of the surface. A spatial carrier was introduced in each interferogram by tilting the output of the reference beam slightly off the optical axis. A phase-stepper constructed by wrapping the optical fiber around a cylindric piezoelectric transducer was used to modulate the phase of the reference beam and introduce a phase step α=π/2 between consecutive pairs of interferograms [Fig. 3(b)]. The recording speed was limited by the transfer rate from the camera to the computer to approximately three pairs of interferograms per second. The total recording time was ~21 s. The emission of the laser pulses, the excitation of the ultrasonic wave and the trigger of the camera were synchronized with a delay generator (DG535 of SRS) and dedicated ad hoc electronics. The delay between the excitation of the wave and the emission of the pulses is usually changed between pairs of interferograms, so that the wave is captured at different positions on the plate and a synthetic movie of its propagation can be composed. However, changing this delay takes a significant time compared to the recording and transfer of the interferograms to the computer. Due to instabilities of the laser source (see section 3.2), we decided to keep this delay constant to have the highest possible acquisition rate, so the wave was captured at the same position on the plate in all the images.

3.2. 3D Fourier transform processing

A sequence of interferograms is actually a discrete 3D set of experimental data of P×Q×N sampled points

The central temporal frequency of the interference lobes appears at two symmetric planes ±n' determined by the temporal carrier 2πf _{c t}t. From Eq. (2) with α=π/2, the temporal carrier is f _{c t}=1/(4Δt), and from Eq. (9), n' is

To verify this point, the 3D Fourier transform of 8 consecutive interferograms I^a_n, with n=15,…,22, was calculated. The resultant 3D spectrum consists of eight complex-valued planes whose moduli are shown in Fig. 4(a) . Though the spectral content of each lobe is spread along the temporal frequency axis, it is strongest at plane n'=8/4=2, in agreement with Eq. (10). In our opinion, the main cause of this spread was the fluctuation of the phase of the laser pulses, in addition to the relatively low acquisition rate available with our camera. This fluctuation added random phase jumps to the π/2 phase step, or, in other words, it broadened the spectrum of the temporal carrier (ideally a δ function, see Eq. (5)).

 

Fig. 4 (a) Modulus of the 3DFTM of a sequence of eight interferograms. Black and white represent zero and the maximum modulus respectively. (b) Average modulus calculated in the region delimited by the dotted rectangle in map n'=2. (c) Result obtained for a sequence of sixteen interferograms.

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To compare quantitatively the spectral content in different planes, the average modulus was computed for each map in the region bound by the dotted rectangle shown in Fig. 4(a), map n'=2, and plotted versus n' in Fig. 4(b). The relatively high value at n'=0 comes essentially from the spectrum of the object beam that enters the region. Figure 4(c) shows the result obtained for a sequence of sixteen I^a_n consecutive interferograms from the same experiment, with n=15,…,30. The maximum average modulus appears at n'=16/4=4, as expected, but the spread of the spectral content is more severe than before, since the appearance of strong random phase changes is more likely as the number n of interferograms increases. These sixteen pairs of interferograms (I^a_n, I^b_n), with n=15,…,30, were used to obtain the results presented in the following sections.

The 3DFTM was first applied to interferograms I^a_n and then to I^b_n, and is graphically described in Fig. 5 . The dimensions of the 3D filter, shown in Fig. 5(b), were 231×231 pixels in f_{p '} and f_{q '}; in f_{n '} it was a band-stop filter that removed the plane n' = 0, where the spectrum of the object beam was confined. Two sequences of 16 complex-valued maps (I'^a_, I'^b), containing the optical phase of the interferograms, were thus obtained [Fig. 5(c)]. The calculations were carried out with a 3D implementation of the fast Fourier transform algorithm [6, ch.12.4].

 

Fig. 5 Summary of the 3DFTM. Only 5 maps out of 16 are used to illustrate the different stages. (a) Sequences of temporally phase-shifted interferograms. (b) Modulus of the 3D Fourier transform (FT (3D)) of the sequences and the 3D filter used in the processing. (c) Sequence of optical phases Φ^a and Φ^b, obtained as the argument of the inverse 3D Fourier transform (FT⁻¹ (3D)) of the filtered data (d) Sequence of optical phase-change maps calculated as indicated in Eq. (11). * stands for complex conjugation.

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3.3. Optical phase-change maps ΔΦ_n

From each pair (I'^a_n, I'^b_n) obtained with the 3DFTM an optical phase-change map

As described in Figs. 1(a) to 1(d), the 2DFTM —which would be equivalent to the 3DFTM if the 3D filter were applied to all the temporal frequency planes, including n'=0— was applied individually to the sixteen interferograms of both sequences I^a and I^b. Another two sequences of 16 complex-valued maps (I'^a_, I'^b) were thus obtained. These maps were processed as indicated in Eq. (11) to yield another sequence of 16 optical phase-change maps ΔΦ_n, with n=15,…,30.

3.4. Maps of mechanical complex amplitude

From each ΔΦ_n obtained with the 2DFTM and the 3DFTM, a complex optical phase-change map $\mathrm{\Delta }{\widehat{\mathrm{\Phi }}}_n(x,t)$ that is proportional to the mechanical complex amplitude of the wavetrain was calculated. The details of the method are given in [7] and are summarized in Fig. 6 . The modulus of the complex optical phase-change is mod[$\mathrm{\Delta }{\widehat{\mathrm{\Phi }}}_n(x,t)$]=(4π/λ)u _{3m, n}(x,t), where u _{3m, n}(x,t) is the amplitude of the out of plane displacement of the surface at map n due to the ultrasonic acoustic wave (i.e., the acoustic amplitude). Several optical phase-change maps ΔΦ_n and acoustic amplitude maps u _{3m, n} yielded by the 3DFTM and the 2DFTM methods are compared in the following section.

 

Fig. 6 Calculation of a complex optical phase-change map $\mathrm{\Delta }{\widehat{\mathrm{\Phi }}}_n$ from an optical phase-change map ΔΦ_n. (a) ΔΦ₂₃ obtained by applying Eq. (11) to the pair of complex-valued maps (I'^a ₂₃,I'^b ₂₃). (b) Modulus of the 2D Fourier transform of (a). The filter size is 25×25 pixel. An inverse 2D Fourier transform yields a complex optical phase-change map $\mathrm{\Delta }{\widehat{\mathrm{\Phi }}}_n$. (c) Acoustic amplitude u _{3m, n}=[λ/(4π)]mod($\mathrm{\Delta }{\widehat{\mathrm{\Phi }}}_n$). The average value of u _{3m, n} is computed in the region A₁ defined by the dotted rectangle. Black and white represent, respectively (a) –π and π (b) minimum and maximum values of the modulus (c) minimum and maximum values of u _{3m, n}.

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4. Results

Figure 7(a) and 7(b) show the optical phase-change maps at n=20, obtained with the 2DFTM and the 3DFTM respectively. Figure 7(c) and 7(d) show the corresponding results for n=24. The wavetrain can be seen in all maps, though (b) and (d) are less noisy. A noise rejection filter and contrast enhancement were applied to all the images to improve their visualization.

 

Fig. 7 ΔΦ₂₀ obtained with (a) 2DFTM and (b) 3DFTM. ΔΦ₂₄ obtained with (c) 2DFTM and (d) 3DFTM.

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Figure 8 shows a 3D representation of the acoustic amplitude u _{3m, n}=[λ/(4π)]mod($\mathrm{\Delta }{\widehat{\mathrm{\Phi }}}_n$), expressed in nanometers, for n=17, 20 and 24, arranged in columns (a), (b) and (c), respectively. The results for 2DFTM and 3DFTM are in rows (i) and (ii) respectively.

 

Fig. 8 Acoustic amplitude u _{3m, n} obtained with (i) 2DFTM and (ii) 3DFTM. Columns (a), (b) and (c) correspond to maps at n=17, n=20 and n=24, respectively.

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5. Discussion

We observe that the improvement obtained with the 3DFTM is apparent. To compare quantitatively the performance of both methods, the average of the acoustic amplitude u _3m was calculated, for each of the 16 mechanical amplitude maps, in a rectangular area where the wave is not present (region A₁ in Fig. 6), considering that noise in a map should translate into a high value of the average acoustic amplitude in such a region. The results, plotted versus n in Fig. 9 , show that this is indeed the case. The average acoustic amplitude is significantly higher for the 2DFTM than for the 3DFTM, which is in agreement with the visual impression given by the 3D representations in Fig. 8.

 

Fig. 9 Average of the acoustic amplitude u _{3m, n} in region A₁ (see Fig. 6), calculated for a sequence of 16 maps with the 2DFTM (in black) and 3DFTM (in grey).

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It could be argued that spectral separation of the interferential lobes and the central disk could be achieved with a temporal carrier alone and a regular temporal phase-shifting algorithm [8] or a pixel-wise temporal Fourier transform [9]. However, these techniques could not ensure spectral separation in the case of interferential lobes with broadband spectral content, as in the situation illustrated by our experiments [see Fig. 4(a)]. If the spatial carrier were not present, the side lobes would overlap along the temporal frequency axis, preventing the retrieval of the optical phase.

The most probable cause of the spectral broadening of the temporal carrier, which spreads the spectral content of the interferential lobes along the temporal frequency axis f_t, is the random fluctuation of the phase of the laser pulses. A slight variation of the intensity of the interferograms along the sequence was discarded as a possible cause: the intensities were equalized before applying the 3DFTM and, apart from a scale factor, the resultant spectrum was almost identical to the one obtained with non-equalized images. The dynamics of the object and an imperfect phase shifting were also discarded, because the spectral broadening was present even with the object at rest and the phase stepper turned off. However, since the spectrum of the object beam was thoroughly confined to plane n'=0, it was still feasible to remove it by means of an appropriate filter.

A consequence of this spectral broadening is that a small portion of the spectral content of the interferential terms falls into plane n'=0 and is removed in the filtering process, which introduces a phase error in the resultant optical phase maps (the residual lobes at plane n'=0 are barely observable in Fig. 4(a) because their modulus is negligible compared to that of the spectrum of the object beam). This error could be avoided by reducing the phase fluctuation of the laser pulses, which would translate into a temporal carrier with a narrower spectrum [closer to the theoretical delta function in Eq. (5)] that would reduce the spread of the interferential lobes along the f_t axis. The experimental 3D spectrum in Fig. 4(a) would thus be closer to the theoretical one in Fig. 1(f). This phase fluctuation could be minimized by using a more stable laser, combined if possible with a faster camera. A faster acquisition would also relax the requirements of isolation from the environmental perturbations typically associated with the recording of a long sequence of consecutive images.

The proposed 3DFTM is a combination of Fourier-based spatial and temporal phase retrieval methods, and it provides capabilities that cannot be achieved separately by spatial or temporal methods alone. When spatial phase evaluation with the 2DFTM is used, the size of the aperture has to be reduced if phase errors due to the overlap of the interferential terms with the central term are to be removed, thus reducing the illumination of the sensor and the spatial resolution of the phase map. The 3DFTM avoids this phase error whilst allowing the use of larger apertures, whose size would then be limited by the size of the speckle (that has to be resolved by the sensor) and the requisite of total spatial separation of the conjugate interferential terms. Each side lobe could then cover a quarter of the size of the sensor (for circular or polygonal apertures) or half the sensor (for rectangular apertures). Temporal phase evaluation with temporal phase shifting or 1D temporal Fourier transform methods would allow spectral separation with maximum light collection capability and spatial resolution, but these methods fail if the spectral content of the interferential lobes is broadband. In comparison, the 3DFTM has less light collection capability and spatial resolution, but can work regardless of the temporal bandwidth of the interferential lobes. In experiments where the recording of sequences of interferograms has interest per se, (e.g., in the study of time varying phenomena) its implementation requires only an additional effort of phase modulation

6. Conclusions

An acquisition and processing method for sequences of speckle interferograms with temporal and spatial carriers, based on the 3D Fourier transform of the acquired data, is presented. Its aim is to separate the spectra of the interferential terms from the spectrum of the object beam and thus remove the noise that affects phase maps when the 2D spatial spectra of these terms overlap.

A sequence of temporally phase-shifted interferograms with spatial carrier is recorded with a digital camera, and a 3D Fourier transform is applied to this set of spatiotemporal data. In the resultant 3D spectrum, the term related to the object beam spans a wide range of spatial frequencies but is essentially confined to low temporal frequencies. The temporal carrier shifts the central temporal frequency of the interferential terms to high temporal frequencies, thus eliminating or minimizing their overlap with the low frequency terms. The role of the spatial carrier is crucial since it prevents a possible overlap of conjugate interferential lobes with broadband temporal spectral content. A 3D filter rejects the low temporal frequencies and selects one of the spatial side lobes, and a sequence of phase maps is obtained after the application of a 3D inverse Fourier transform. Experimental results of the measurement of the mechanical amplitude of guided acoustic waves in a metallic sample, obtained with a double-pulsed TV holography setup, are presented as a proof of principle of the method.

An apparent improvement is achieved with this new 3DFTM method compared to the regular spatial-carrier 2DFTM even in presence of phase instability of the laser.