For both light and X-rays, Lambert-Beer’s law

holds, where L, α, and I are the specimen length and absorption coefficient, and radiation incident and transmitted intensities, respectively. If we divide the absorption distribution along the path into small cells, the total absorption along the line is the sum of the absorption coefficients. This is the basic principle of X-ray CT.

In X-ray CT [7], the distribution of the X-ray absorption coefficient is related to a measured quantity, called the “forward projection” (also called the “Radon transform” [7,13]). This is defined as the sum of absorption coefficients along the propagation line. It is a function of the inclination angle θ and scanning step X along an axis inclined from the X-axis, as shown in Figure 1.

The 1D Fourier transform of the forward projection is defined as

where the parameter w is the 1D Fourier transform of the forward scanned step X. The distribution of the X-ray absorption coefficient is reconstructed from

This process, which recovers the original distribution, is called the “inverse Radon transform” [7,24].

In photoacoustic spectroscopy, the PA signal in opaque solids (including metals) is generally proportional to the product of the absorption coefficient α of the specimen (assuming that the surface defect is an equivalent surface absorber) and laser intensity I0, and inversely proportional to the modulation frequency f. Then, the obtained signal for the PAT, in which the laser LFB is focused on the specimen surface and the PA signal is forward-projected over the specimen surface along a LFB, is

where summation is along the LFB.

In photoacoustic imaging, it is known that the obtained PA amplitude and phase images and are coupled together as a form of (6) [1,25], therefore the PA amplitude and phase images and obtained from the PAT measurement are related to the forward projection (2) as

There is no need to take the logarithm of the output X-ray intensity as in X-ray CT. The obtained set of PA amplitude and phase signals itself is a forward projection. The equivalence between 2D-PAT and X-ray CT is shown schematically in Figure 2.

In photoacoustic (PA) imaging of undersurface defects, periodic thermal diffusion in a solid should be considered. It is governed by the Green’s function Equation [1]

where the complex parameter σ is defined in terms of the thermal conductivity σ, density ρ, heat capacity c, and modulation frequency f as

where μ is the thermal diffusion length. The Green’s function in free space is the spherical wave with complex wave number σ

When a condenser microphone is used as the detector in undersurface imaging, it will collect all sounds generated over the whole specimen surface by the thermal wave diffracted at the undersurface defect. By the reciprocal property of the Green’s function

In photoacoustic (PA) imaging with a condenser microphone by a point-source laser beam is equivalent to single-point detection of the plane thermal wave generated from the whole surface that is diffracted from the undersurface defect [2].

Hence PA imaging with LFB excitation and condenser microphone detection is equivalent to collecting the PA signal along a laser beam line, which is generated by a diffracted plane thermal wave at the undersurface defect located at a depth δ. As a result, the detected PA signal generated by LFB laser irradiation is described by (see Equation (10) below):

where X and θ are the step position and inclination angle, respectively. In Equation (10), integration is performed along all points on the LFB. A plane wave diffraction formula [26] along all points on the LFB. A plane wave diffraction formula [26] is assumed in the derivation of (10).

We rewrite (10) as

The function is defined as the second term integral in the bracket in (10)

It represents the complex PA image, obtained using a point-focus laser beam. Its summation along a LFB is the PAT image.

Comparison of (11) with (5) for surface defects shows that the PAT image of the undersurface defect with a LFB is back-projected and reconstructed as a PA image corresponding to that obtained with a point focus beam using the same inverse-Radon transform algorithm [7].

In the non-destructive evaluation (NDE) of undersurface defects or specimens such as welded plates, PA measure ment by a laser LFB can provide macroscopic information on the  internal status of the specimen quicker than that by a PFB [10]. During rapid pattern-matching inspection by a LFB that scans in the horizontal and vertical only, the measurement time for 65 steps was 1 min, 22 s (82 s). In comparison, the time needed to acquire a PAT forward-projected image with 65 steps along 65 directions was 4 min, 37 s (277 s). Thus, about a three-fold more rapid inspection was achieved compared with the PAT inspection.

For comparison, the time required for the PAM measurement using a 65 × 65-step scan is 1 h, 16 min, 39 s (4599 s). The measurement time was thus about 56 times that needed for the PA LFB. Even with diagonal scans added, the time needed for 4 (vertical, horizontal and two diagonal) scans with 65 steps was 2 min, 49 s (169 s). The times needed for PAT and PAM measurements at different resolutions are summarized in Table 1.

In the PAT measurements for the simulated surface and undersurface specimens, the power and size of the laser beam on the specimen was 30 mW and 25 mm at 650 µm, respectively. The modulation frequencies for surface and undersurface defect measurements were 390 and 30 Hz, respectively. The measured area was 27 × 27, while the reconstructed area was 18 × 18 mm². The rotation and translation steps were 1.8˚ (π/100) and 270 µm (27 mm/100), respectively. The measurement time was 87 min. The laser beam was uniform to within 5%.

To compare the image resolution, a point-focus laser beam with a power and spot size of 80 mW and 30 µm, respectively, was used. The modulation frequencies for the surface and undersurface defect measurements were the same as for the PAT measurements. The resolution was 100 × 100 pixels, which was set equal to that of the PAT measurement.

The translation steps were 270 µm (27 mm/100) in both the x and y directions. The measurement time in this case was 135 min.