J. Phys. zyxwvutsrq A: Math. Gen. 14 (1991) 2485-2505. Printed in the UK zyxwv Theory of photothermal wave diffraction tomography via spatial Laplace spectral decomposition Andreas Mandelis Photoacaurtic and Photothermal Sciences Laboratory, Department of Mechanical Engineering and Ontario Laser and Lightwave Research Center, University of Toronto, Toronto, Canada M5S 1A4 Received 4 October 1990 Abstract. Laser-generated thermal wave diffraction theory is presented as a perturbative Born (or Rytav) approximation in a two-dimensional spatial domain for use with tomo- graphic image reconstruction methodologies. The ranges of validity of the pertinent two- dimensional spatial-frequencylthermal wavenumber domain complex plane contours are investigated in terms of the existence af inverse spatial Laplace transforms in the mean- square sense. The spectral decomposition of the Laplace transforms according to a Laplace diffraction theorem is shown to involve zyxwvu regular complex-valued propagation functions, which represent the two-dimensional Laplace transform afa scattering abject alongsemicir- culm arcs comprising the objcct’s thermal wavenumber domain. A discussion ofthe complex thermal-wave spatial frequency domain content is also presented, with a view to tomo- graphic recovery of the scattering abject field. 1. Introduction The use of optically excited thermal waves in condensed phase materials as probes of subsurface features or defects has dramatically increased in recent years [I]. Most applications to imaging typically involve two-dimensional scanning with a laser beam, and photothermal detection of projection images by a spatially integrating sensor, such as a piezoelectric transducer [2] or a pyroelectric transducer [3] placed below the sample under investigation. Other popular schemes involve local recording of projec- tional photothermal images including microphonic photoacoustic microscopy [4] and photothermal beam deflection imaging zyxwv [5]. Total (or partial) spatial integration of the thermal-wave field is to be understood as a surface integral limited either by the active detector area (the case of back-surface piezoelectric or pyroelectric detection), or by the magnitude of the thermal diffusion length in the neighbourhood of the optically excited surface (the case of microphonic detection zyxwv [6]). Spatial integration may further take the form of a line integral, such as the case of photothermal beam deflection, or Mirage effect, imaging along the length of the intersection between a broad pump source and a probe laser beam. All the above photothermal scanning and detection modes using a single exciting laser beam are incapable of providing depth imaging of sample cross-sections involving the subsurface location of defects. More sophisticated imaging methods based on thermal-wave interference from two coherently [7,8] or anticoherently intensity-modulated laser beams have succeeded in providing estimates of the depth of a defectlscatterer. These methods depend on the resolution of an infrared blackbody radiometric emission detector [7], or a thin metallic pin capacitively 0305.4470/91/1124S5+2l$03.50 Q 1991 IOP Publishing Ltd 2485