International Journal of Thermophysics

, Volume 28, Issue 1, pp 60–82 | Cite as

“Thermal Mirror” Method for Measuring Physical Properties of Multilayered Coatings


In this study theoretical principles underlying the photothermal displacement (“thermal mirror”) method for measuring physical properties of opaque multilayered and functionally graded coatings with low thermal conductivity are analyzed. In this method, the specimen is locally heated by a power laser beam, and a two-dimensional transient temperature field is formed in a specimen. The physical basis for the photothermal displacement method is the non-stationary buckling and displacement of an irradiated surface due to a non-uniform thermal expansion. The surface is monitored by a low-power probe beam of a second laser, which is reflected from the specimen, i.e., the system operates as a convex “thermal mirror.” The photoinduced displacement varies with time, and the probe beam is reflected at a different angle depending on the slope of the displacement. The deflection angle is measured as a function of time by a position sensor, and the results of these measurements are compared with the theoretical dependence of the deflection angle on time and physical properties of a coating. This dependence was determined analytically from the solution of the two-dimensional thermal elasticity problem. It is shown that for the specimen composed of a substrate and a coating it is feasible to determine the properties of the coating, e.g., the thermal diffusivity and coefficient of linear thermal expansion provided that the analogous properties of the substrate are previously measured or otherwise known.


laser heating multilayer coating surface displacement thermal elasticity 



specific heat


substrate thickness


total laser power absorbed by a specimen


Fourier number


shear module


laser beam intensity absorbed by a coating

Ji (x )

ith order Bessel function of the first kind


thermal diffusivity


radial coordinate


radius of a laser beam

p, s

parameters of Hankel (p) and Laplace (s) transforms with respect to r and t, respectively






Laplace–Hankel transform of temperature

u, w

radial and axial components of displacement


volume content (m: metal, c: ceramics)


axial coordinate


coefficient of linear thermal expansion


coating thickness


laser beam deflection angle


thermal conductivity


Hankel transform of temperature


Poisson’s ratio


thermal stress

\({\sigma_{rr}, \sigma_{\Psi\Psi}}\)

radial and tangential stresses, respectively


duration of a laser pulse





i =  1,...,n

number of a layer in a coating



 + , −

upper and lower surfaces of a layer in a coating, respectively


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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Pearlstone Center for Aeronautical Engineering StudiesBen-Gurion University of the NegevBeer-ShevaIsrael

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