1.

Introduction

There are several optical noninvasive methods currently used in different fields of knowledge to study object displacements and deformations. For instance, automotive and aerodynamics industries, as well as medical and biological fields use, on a daily basis, optical noninvasive methods for remote inspection, design processes, measurement of mechanical parameters (such as strain/stress), and eye correction, among many other applications. Particularly, there are two such methods that have been applied extensively and have proved their effectiveness as problem solvers: electronic speckle pattern interferometry (ESPI) and digital holography (DH). For instance, the latter has been used to measure deformations on biological tissues from a pig and on a human hand,¹ or for deformation analysis and shape measurement.² A variation of the former based on the speckle pattern contrast has been used to acquire optical tomography images for the detection of inhomogeneities.³ Recently, DH and other techniques have been applied to measure depth position of inhomogeneities,⁴ and the detection and analysis of defects.^{5, 6}

In what follows, the research done on 3-D pulsed digital holography (3-D PDH) for the detection and depth quantification of inhomogeneities within semisolid objects, also known as phantoms, is presented. The object is opaque and the inhomogeneity is placed well inside the material so that it is not visible and does not cause a dent on the object surface. Previous reported research work⁷ used an out-of-plane sensitive PDH system to detect surface microdisplacements on phantoms. The optical setup for this out-of-plane sensitivity measures the three orthogonal components of surface displacement, with the largest component being out of the plane of the object surface, e.g., the $z$ component of a flat surface. The other two components, $x$ , $y$ , are there, however small, and cannot be separated from the former, i.e., the three displacement components are mixed. The inhomogeneities consisted in human breast tumors. Sinusoidal sound waves were used to scan frequencies from $20\mathrm{Hz}$ to $1\mathrm{KHz}$ to find the gel resonant modes response. At this frequency range, two resonant modes were found, the first at $44\mathrm{Hz}$ and the second at $810\mathrm{Hz}$ . The latter frequency was chosen for the experiments, since it produced the largest surface displacement, with an input sound power of approximately $661\mathrm{mW}$ , equivalent to a pressure of $2.3\times {10}^5\mathrm{Pa}$ . The digital hologram taken for the inhomogeneity is compared with the digital hologram taken from a phantom that does not contain an inhomogeneity. The result shows as a set of nonconcentric rings on the phantom surface. However, this is an out-of-plane sensitive system that does not render precise and reliable information regarding the actual inhomogeneity depth position, thus a 3-D PDH system was devised for its quantification.

3-D PDH employs a setup where the phantom is illuminated from three different directions.⁸ This is necessary to have from each illumination direction an equation containing the mixed surface displacement components $x$ , $y$ , and $z$ , thus having three unknowns with the corresponding three equations needed to solve the problem. The usual pulsed digital hologram subtraction is performed separately: a digital hologram pair from each illumination direction. Each pair is formed with the holograms taken before (first laser pulse), e.g., at the peak of the vibration cycle, and after surface displacement (second pulse), $14\mathrm{ms}$ after the first, giving as a result a phase map directly related with the direction and magnitude of the displacement. The second pulse is fired as needed according to the mechanical wave propagating through the gel and that reaches its surface. Moving on with the process, each phase map is unwrapped to get the surface displacement data, vector $\vec{\mathit{d}}$ in 3-D, via the following equation,

Fig. 1

The corresponding unity vectors for each of the different object illuminating source positions are shown, together with phantom dimensions and depth location of the inhomogeneities.

The characteristic unwrapped phase map for a phantom with and without an inhomogeneity is such that the concentric rings obtained for the latter are destroyed for the former, and this may be immediately noticed by simple observation of the phase map. The quantitative evaluation of such unwrapped phase maps through Eq. 1 gives the actual depth position of the inhomogeneity, a unique feature of 3-D PDH. The observed surface displacements are all in the region of micrometers.

2.

Experimental Method

Three hologram pairs were obtained sequentially for five locations of the inhomogeneity: 10, 14, 16, 18 and $28\mathrm{mm}$ below the gel surface and to the outer edge of the inhomogeneity. The inhomogeneities used were tumor tissues of approximately $1\mathrm{cm}$ in diameter.

The optical arrangement is shown in Fig. 2 , where a high resolution CCD is used, with $1024\times 1280\mathrm{pixels}$ at 12 bits. A beamsplitter divides the pulsed laser beam, $\mathrm{Nd}:\mathrm{YAG}$ at $532\mathrm{nm}$ , 15-ns pulse width, and $6\mathrm{m}$ of coherence length, in two: a reference and object beam. By using a rotating mirror (RM), the latter is directed sequentially, i.e., for each hologram pair, to mirrors ${\mathrm{M}}_1$ , ${\mathrm{M}}_2$ , and ${\mathrm{M}}_3$ , to illuminate the object from three different positions. The reference beam is conveyed to the CCD sensor via a single-mode optical fiber. Care was taken to match the optical path lengths by pairs to be within the laser coherence length. This arrangement is now sensitive for each object illumination direction to all $x$ , $y$ , and $z$ axes. An electronic circuit was designed to synchronize and control the sinusoidal sound waves, the firing of the laser pulse, and the CCD camera trigger in such a way that the first laser pulse was fired at the peak of the sound vibration cycle, i.e., the camera shutter opens a few microseconds before this, and closes a few microseconds after the laser pulse was fired, acquiring the first digital hologram. A second laser pulse comes $14\mathrm{ms}$ after the first one, so the second digital hologram is obtained. As is commonly done in digital holography, these two holograms are subtracted after the usual Fourier algorithm routine is employed, obtaining a phase map for an illumination direction. This procedure is repeated for the other two illumination directions. Thus, three phase maps ${\phi }_{1\text{-}3}$ are obtained containing the direction and magnitude of the surface displacement $\vec{\mathit{d}}$ in 3-D.

Fig. 2

The experimental setup to measure 3-D microdisplacements. RM is a rotating mirror that serves to direct the laser beam for the three illumination directions, ${\mathrm{M}}_{1,2,3}$ and M are mirrors, NL are negative lenses, BS is a beamsplitter, BC is a beam combiner, LS is a loud speaker, OF is a single mode optical fiber, L is a positive lens, A is an aperture, and T is the tumor.

The phantom is a semisphere with an 8.4 cm diameter and 4 cm height, subject to sinusoidal sound waves at $810\mathrm{Hz}$ by means of an off-the-shelf speaker. The phantom center is located at the origin of the Cartesian coordinate system. The $z$ axis data are used to evaluate the inhomogeneity depth position.

To validate and make sure that the results were repeatable, several holograms were taken for each of the phantoms at the positions given before. Since with 3-D PDH it is possible to separate individually displacement components along each axis, it is then possible to calculate normal and tangential vectors to the surface. With this in mind, it is now possible to probe that particular line that passes through the largest surface displacement. This procedure is repeated with all the available data previously taken for each object and its depth position, calibrating the 3-D PDH system in this manner.

3.

Results

Several experiments were performed to verify the repeatability of the technique. Figure 3 shows three unwrapped phase maps for each illumination direction, chosen at random from many taken. It is seen that the phase pattern changes according to the illumination direction, and is clearly nonsymmetric, taken for a tumor inside the phantom. These data are used in Eq. 1 to evaluate the object surface displacement along the $x$ , $y$ , and $z$ axes. Figure 4a shows the result in 3-D, for all three displacement components, for a 1 cm-diam tumor that gives a total phantom surface displacement of $3\mathrm{\mu }\mathrm{m}$ . Figure 4b corresponds to a phantom without any inhomogeneity and any sound present: the very small surface displacement was due to mechanical disturbances on the optics table, a feature that shows how the surface deforms under noncontrolled arbitrary conditions. The images in this figure may be compared, and by simple observation, or by looking at the color scale at its bottom, it would be possible to decide which sample has an inhomogeneity in it and which one does not.

Fig. 3

(a), (b), and (c) Unwrapped phase maps for each illumination direction.

Fig. 4

From Eq. 1 and all phase maps from Fig. 3: (a) for a tumor and (b) free-standing phantom.

Data, converted to a $78\times 25$ pixel size, along the $z$ axis were used to find the depth location of the inhomogeneity. Figure 5 shows images chosen at random for an inhomogeneity placed at the same depth. Several lines were scanned around the area where the inhomogeneity is located on the image, and the one corresponding to the largest surface displacement is chosen, shown in white. Vertical scans were also performed to corroborate this line data and position. The procedure was repeated for all acquired image data and for an inhomogeneity 1.2 cm diameter located $18\mathrm{mm}$ below the surface. It is pointed out here that to calibrate the system, all of the previous was done for all cited locations.

Fig. 5

Image samples chosen at random. Approximate surface displacement: (a) $-1.08\mathrm{\mu }\mathrm{m}$ , and (b) and (c) $-1.13\mathrm{\mu }\mathrm{m}$ .

Figure 6 shows the depth location, along the $z$ axis, of the inhomogeneity with respect to the surface. The maximum surface displacement for this tumor size and position is about $1\mathrm{\mu }\mathrm{m}$ . Blue, pink, and brown lines correspond to Figs. 5a, 5b, 5c, respectively. The actual value to determine the depth position is averaged, taking into consideration the system 0.297-mm/pixel resolution for 25 data points/line and a phantom container diameter of $84\mathrm{mm}$ . The same procedure was repeated for all images at the locations given.

Fig. 6

Depth profile for the inhomogeneity, taken from Fig. 5.

Figure 7 shows the displacement results for tumor tissue with a 1-cm diameter placed at about $27\mathrm{mm}$ below the phantom surface. The maximum surface displacement, averaged, is about $0.5\mathrm{\mu }\mathrm{m}$ . Figure 8 shows the error found between the optical measurements and the minimum square fit to the data, which shows a straight line that serves to predict the inhomogeneity depth location. The horizontal axis shows the real, physically measured depth of the inhomogeneity immersed in the gel, while the measurements on the vertical axis are displacement averages obtained with the optical noninvasive method dealt with here. The line scans indicate that the displacement measurements performed decrease as the depth of the inhomogeneity increases. The optical measured values are in very good agreement with the actual depth position, within an error of about 0.03 %. The plot serves as a calibration chart, and thus can be used to predict from the optical data the depth of an inhomogeneity immersed in the phantom.

Fig. 7

Displacements for the depth profile.

Fig. 8

Experimental results fitted to a straight line for calibration purposes.

4.

Concluding Remarks

Within the scope of this research, it is shown that 3-D pulsed digital holography is an optical noninvasive method capable of measuring with great accuracy the depth position of human tumor tissue, breast tumors, immersed in gel phantoms. Separation of the three displacement components, $x$ , $y$ , and $z$ , from unwrapped phase maps quantifies their depth. The technique was tested for different inhomogeneity known positions up to depths around $28\mathrm{mm}$ below the gel surface. The results show that the optical data can be related to the depth location via a straight line fitted through the experimental results. Future research will be oriented toward the application of the method in live animals to later transfer the gained knowledge to live human tissue. The pulsed laser used delivers pulses of $15\mathrm{ns}$ width at $20\mathrm{mJ}$ of energy for the experiments conducted here. Since the illuminating object beam is expanded, the live tissue will be exposed to an average power of $0.639\mathrm{\mu }\mathrm{W}∕{\mathrm{cm}}^2$ , which is well within the limits of human tissue exposure to this type of laser. Furthermore, the sound power that may be used in live tissues will be around that used here, which is also well below the audible decibel limits. It is envisaged that problems will arise in the placement of the human breast in the optical setup, and the finding of the optimum resonant mode frequency directly related to the type of tissue properties, like its density and fat.