Depth Profiling of Optical Absorption in Thin Films via the Mirage Effect and a New Inverse Scattering Theory. Part I: Principles and Methodology

Abstract

Impulse mirage effect spectroscopy is developed in this work as a nondestructive method for depth profiling the optical properties of samples which are nearly thermally homogeneous with depth. Both a theory and an experimental methodology are presented. An inverse scattering theory of the experimental photothermal deflection signal is derived, based on a previous theory of the impulse mirage effect, which takes into account the effect of Fresnel diffraction on the probe beam. To reconstruct the depth profile of heat source density generated by light absorption in an unknown sample, we have applied our inverse theory to the experimental impulse response, using a regularized minimum square error reconstruction algorithm based on our previously published expectation minimum principle. Because this reconstruction problem is ill posed, it was necessary to identify and compensate for all experimental bias errors significantly affecting the fidelity of the depth profiles. A procedure for obtaining the overall best-fit model of the depth profile given the minimum prior experimental information is presented. These procedures have produced an agreement between the experimental and theoretically predicted mirage effect response to within typical root-mean-square error levels of 0.5% or less.

A regularized solution with weighted Bregman distances for the inverse problem of photoacoustic spectroscopy

“…[21][22][23][24][25][26], and most of them use Tikhonov's regularization. Such approach allows the use of a priori information, when available.…”

Beam Deflection Photothermal Spectroscopy

“…p ½ 5 a 0 C . 21,22 The key geometric factor affecting depth resolution is the offset a 0 . 20 As the probe beam is aligned closer to the surface, the time lag in the deflection response becomes smaller, according to the approximate expression:…”

“…This is true for single-ended irradiation, An experimental means of determining a 0 involves placing an optically opaque, thin (<5 µm) marker layer on the sample surface near a region of interest for depth analysis. 21,22 Assuming thermal matching between fluid and sample, irradiation of the marker layer and reconstruction of p the estimate of a 0 returned by the marker. a 0 is further refined by fine tuning its value in the vicinity of the marker estimate, and observing where the data residual (fit of theory to experiment for the sample depth region) is minimized.…”

“…A diffraction theory reducible to the form of equation (14), provides an alternative expression for G e x, x 0 , t , with an agreement <1% RMS of full scale between theory and experiment. 21,22 Finally, p C must be related back to b x , the quantity of spectroscopic interest. If the sample's total optical transmission, T 0 , is available, optical transmission as a function of depth is then given by equation (16) 21 :…”

“…Detection of the probe beam deflection is made using a bicell, or quadrant cell (for two-dimensional deflection signals). For theoretical interpretation of the measured impulse response, signal deconvolution of the instrumental components of v t is essential, especially at low frequency 20,21 where photothermal signals increase monotonically. Components such as preamps, modulators and other devices may cut off below a few hertz, severely affecting depth profile resolution in the sub 300-µm range for low diffusivity materials.…”