3.1.

Methods

Digital shearography,¹² or digital speckle shearing interferometry, is a full-field, noncontact coherent optical technique used to measure directly the displacement gradient (deformation/strain) across a surface by using a digital camera as the recording device. The key element in a shearing interferometry system is the shearing device, which causes the light scattered from two neighboring regions on a test surface to interfere at a point in the image plane of the camera.

The shearographic interferometer used in this work¹³ is shown in Fig. 2 . A frequency-doubled Nd:YAG laser ( $532\mathrm{nm}$ , $30\mathrm{mW}$ ) polarized at $45°$ with respect to the plane of the optical table was used as the light source. A speckle pattern is produced by reflection of the incident light from the test object’s surface. In order to produce well-developed speckle patterns in our experiments, a steel plate $(300\times 300\times 1{\mathrm{mm}}^3)$ was used as a test object. The interferograms (shearograms) recorded by the CCD camera result from the interference between the object speckle wavefront and the same speckle wavefront shifted laterally along the $x$ direction by the shearing element¹³ (shearing distance of $1.9\mathrm{mm}$ in the experiments). For every state of the test object (unloaded reference state and loaded states) in the experiments, a set of four shearograms, phase-shifted relative to each other, was recorded by varying the control voltage of the phase modulator [liquid crystal variable retarder (LCVR)]. Wrapped phase difference maps sensitive to the $x$ -component of the out-of-plane displacement gradient were obtained with a standard four-step algorithm¹³ applied separately to the two sets of phase-shifted shearograms acquired before and after loading, followed by phase map subtraction.

Fig. 2

Experimental setup for shearography. The black square corresponds to the imaged area on the steel plate.

In order to produce exactly the same load at the same point of the test object for each experiment, the plate was held horizontally by three steel posts screwed into the optical bench. A small hole was pierced at the center of mass of the isosceles triangle formed by the three posts in order to hold, in a reproducible way, a ball of known mass $\left(63.8\mathrm{g}\right)$ . The scattering phantom was located in front of the test surface.

3.2.

Results and Discussion

A $7.5\times 10{\mathrm{mm}}^2$ area of the steel plate roughly centered on the loading point was imaged by the shearographic interferometer. A qualitative evaluation of the influence of the scattering medium on the interferometer is given in Fig. 3 , which shows three wrapped phase maps obtained without the scattering cell as well as with the cell at two different concentrations of $611\text{-}\mathrm{nm}$ beads. All other experimental parameters were kept constant during the different experiments—only the CCD exposure time was increased to maintain a constant mean irradiance in the recorded shearograms. One fringe in the phase maps corresponds to a strain (displacement gradient) magnitude of ${10}^{-5}$ in the shearing direction $\left(x\right)$ . In observing the different phase maps in Fig. 3, it is apparent that a deterioration of the contrast occurs with increasing optical density.

Fig. 3

Wrapped differential phase maps ( $5\times 5$ median filtered) obtained with the shearography setup. (a) Without a scattering cell; (b) with a scattering cell: $\mathrm{OD}=0.144$ ; (c) with a scattering cell: $\mathrm{OD}=0.576$ .

A quantitative analysis of the phase map degradation as a function of optical density was undertaken using the contrast,¹⁴ $C$ , of the fringes, defined as follows:

Fig. 4

Steps involved in the determination of $⟨I_{\max }⟩$ and $⟨I_{\min }⟩$ from the shearography phase maps.

Shearographic fringe contrast versus optical density for $611\text{-}\mathrm{nm}$ silica sphere scattering phantoms is shown in Fig. 5 for various concentrations. The figure shows that the fringe contrast decreases with increasing optical density of the scattering cells, as expected from the images in Fig. 3. The decrease in fringe contrast is attributable to progressively greater loss of spatial coherence due to the Brownian motion of the particles in the scattering cells. It has long been known that speckle noise can be reduced in holographic images by placing a moving diffuser in the light path.¹⁵ Since spatial coherence is critical in shearography, the effect of deteriorating spatial coherence is clearly seen in the experimental results.

Fig. 5

Contrast of the shearographic fringes versus optical density for different scattering phantoms composed of silica spheres $(d=611\mathrm{nm})$ in suspension in distilled water.

Note that from Rayleigh and Mie theory, it is expected that experiments with bead diameters of $250\mathrm{nm}$ and $400\mathrm{nm}$ should yield similar results and tendencies but with slightly different numerical values. Because of the unavoidable fairly large uncertainty (10%) in determining the speckle fringe contrast, the expected small numerical differences in the data obtained with $250\text{-}\mathrm{nm}$ and $400\text{-}\mathrm{nm}$ beads could not be observed experimentally. We chose therefore to conduct the remainder of our contrast measurement experiments with $611\text{-}\mathrm{nm}$ beads only.

4.1.

Methods

Coherence-gated photorefractive holography³ is a full-field, noncontact coherent optical technique similar, in some respects, to OCT² (both make use of a short-coherence light source). However, photorefractive holography has two advantages over OCT: First, optical coherence tomography is generally a pointwise scanning technique, whereas photorefractive holography is full-field (an entire plane is imaged in a single acquisition). Second, since the photorefractive effect is only sensitive to the gradient of optical illumination, incoherent light (i.e., nonballistic photons) that is not involved in interference does not contribute to the photorefractive effect.¹⁶ Hence, the photorefractive medium naturally filters out background-scattered light that would otherwise decrease the signal-to-noise ratio.

The experimental setup for photorefractive holography is shown in Fig. 6 . Holograms are recorded in a strontium-barium-niobate (SBN) photorefractive crystal using a TM-polarized frequency-doubled Nd:YAG laser $\left(532\mathrm{nm}\right)$ . A beamsplitter (BS) separates the incident beam into two equal-intensity paths: the reflected “object” (signal) beam and the transmitted “reference” (pump) beam. Both beams interfere in the photorefractive crystal, which is positioned so that its crystal lattice orientation is appropriate for the photorefractive effect using TM-polarized light. Once a hologram is recorded in the crystal, the signal beam is switched off using a digitally controlled shutter and the hologram can then read out of the interferometer by the pump beam.

Fig. 6

Experimental setup for photorefractive holography.

Note that for the experiments presented here, a long coherence-length laser source was used in order to simplify the optical alignment. The optical sectioning properties of coherence-gated photorefractive holography afforded by a short coherence-length source were not required since the test object’s surface was opaque. To study the influence of a scattering medium located in front of the test object, holograms were recorded through the tissue phantoms discussed in Section 2.

4.2.

Results and Discussion

As with the shearographic interferometer, qualitative and quantitative evaluations of the influence of the scattering medium on the interferometer were carried out. Again, all the experimental parameters were kept constant in order to permit a comparison of the different recorded holograms. The hologram write time in the photorefractive crystal was kept constant at $45\mathrm{s}$ .

Pictures of holograms of the letter “S” (the Université de Sherbrooke logo etched in chromium on a glass substrate) are shown in Fig. 7 , obtained both without (a) and with [(b) and (c)] the scattering cell containing $611\text{-}\mathrm{nm}$ spheres at increasing concentrations. As opposed to shearography, insertion of the scattering cell in front of the object improved the image quality, as expected.¹⁵ Indeed, as the concentration of silica sphere diffusers increases, the hologram image quality is enhanced and noise is reduced due to a loss of spatial coherence and a corresponding decrease in speckle amplitude.

Fig. 7

Holograms obtained with the photorefractive holography setup (a) Without a scattering cell; (b) with a scattering cell: $\mathrm{OD}=0.288$ ; (c) with a scattering cell: $\mathrm{OD}=0.864$ .

As in Section 3.2, the contrast, $C$ , between abrupt dark-to-light transitions in the images [Eq. 3] was used to evaluate quantitatively the quality of the recorded holograms. Figure 8 confirms that the contrast in the holograms improves with increasing optical density of the scattering cell. This improvement in image contrast occurs (1) because of the decrease in laser speckle due to the loss of spatial coherence and (2) because coherent imaging through a liquid scattering medium produces a time-varying diffuse scattered background that averages to a uniform field¹⁷ and so is readily rejected by photorefractive effect.

Fig. 8

Hologram contrast versus optical density for different scattering phantoms composed of silica spheres $(d=611\mathrm{nm})$ in suspension in distilled water.

5.1.

Methods

The coupled setup,⁴ consisting of the photorefractive holography interferometer (top) followed by the shearographic interferometer (bottom), is shown in Fig. 9 . When the sample under study has an optically rough surface, the light field emerging from the holographic interferometer will be a speckle pattern. Therefore, this “holographic speckle field” can be used for speckle interferometry by the second stage, the digital speckle shearing interferometer. Note that the light emerging from the holographic interferometer is TM-polarized. A half-wave plate is therefore required between the holographic and shearing interferometers in order to rotate the polarization by $45°$ and thus inject equal amounts of TE- and TM-polarized light into the shearing interferometer.

Fig. 9

Schematic of the coupled experimental setup: photorefractive holography at the top and digital shearography at the bottom.

A first hologram of the object in a reference state is recorded, requiring $45\mathrm{s}$ to write the hologram into the SBN crystal. Next, the signal-beam shutter is switched off and the four shearograms required by the phase-shifting algorithm are recorded by the CCD. The acquisition of the four shearograms required $0.3\mathrm{s}$ in total, which is faster than the hologram erasure time. Under constant illumination by the probe beam, the hologram is completely erased in approximately one minute. Subsequently, the steel plate is centrally loaded and a second hologram is recorded in the SBN crystal, followed by the acquisition of the second set of four phase-shifted shearograms. Finally, the wrapped differential phase maps shown in Fig. 10 were obtained with the four-step algorithm and phase map subtraction.

Fig. 10

Three sheared holograms ( $5\times 5$ median filtered) obtained with the coupled system. (a) Without a scattering cell, hologram write time of $120\mathrm{s}$ ; (b) with a scattering cell $(\mathrm{OD}=0.288)$ , hologram write time of $120\mathrm{s}$ ; (c) with a scattering cell $(\mathrm{OD}=0.288)$ , hologram write time of $300\mathrm{s}$ .

5.2.

Results and Discussion

Here again, a qualitative and quantitative evaluation of the influence of the scattering medium on the coupled system was carried out. All the experimental parameters were kept constant during the different experiments.

Three differential phase maps were determined from the sheared holograms without the scattering cell [Fig. 10a] and with the cell containing $611\text{-}\mathrm{nm}$ spheres [Figs. 10b and 10c]. It can be seen by comparing Figs. 10a and 10b that, under the same experimental conditions, the phase map obtained through the scattering medium is degraded compared with that obtained without the scattering cell. Figs. 10b and 10c show the phase maps determined from the sheared holograms obtained with the same scattering cell but with different recording times for the holograms ( $120\mathrm{s}$ and $300\mathrm{s}$ , respectively). By comparing these figures it is apparent that the fringe contrast increases with the recording time. The contrast C [Eq. 3] shown in Fig. 11 confirms this behavior: the more diffusing the medium, the lower the contrast. On the other hand, the greater the recording time, the higher the contrast.

Fig. 11

Shearographic fringe contrast recorded with the coupled system versus optical density for different scattering phantoms composed of silica spheres $(d=611\mathrm{nm})$ in suspension in distilled water, as well as different hologram write times. Increased hologram write times can be seen to restore fringe contrast by restoring the spatial coherence of the input light to the shearographic interferometer.

This important result is attributable to the fact that, though the presence of the scattering cell reduces the spatial coherence of the light at “relatively fast” photorefractive crystal hologram recording times $\left(45\mathrm{s}\right)$ , thereby drastically reducing the quality of the shearograms, longer hologram recording times $\left(300\mathrm{s}\right)$ will restore spatial coherence because ballistic photons will eventually dominate over scattered light due to the properties of the photorefractive effect (sensitivity to the gradient of illumination only, time-averaged integration).