Correcting errors in the theory for mirage-effect measurements
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Correcting errors in the theory for mirage-effect measurements D. Josell, E. J. Gonzalez, and G. S. White Materials Science and Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899 (Received 10 April 1997; accepted 24 September 1997)
Errors are noted in two publications on the theory for Mirage-effect measurements of the thermal diffusivity of materials. The works include theory for interpreting Mirage experiments on homogeneous samples mounted on a support with ambient (typically air) above and theory for interpreting Mirage experiments on coated substrates with ambient both above and below.
In Mirage-effect experiments, a periodic, localized heating pulse is used to generate thermal waves in a sample. Deflection of a laser beam directed parallel to the sample surface by gradients in the refractive index (temperature) of the ambient (air) above the sample surface probes these thermal waves. The deflection of the probe beam as a function of probe beam position (see Fig. 1) thus provides an integral measurement along the probe beam path of the thermal fields in the sample. The basic theory for this phenomenon has been published.1,2 In this work we note mistakes in the solution published by Kuo et al.3 and utilized by Reyes et al.4 and the solution presented in the doctoral dissertation of Wei5 and utilized in a recent publication by Wei et al.6 For the sake of brevity, we simply state the results necessary to understand and to correct the errors, both of which are in the solution for the temperature field on the surface of the sample. An error that appears in both the analysis of Kuo et al.3 and Wei5 arises from sign errors in the heat flux boundary condition for the interface with the heat source. We therefore note that the heat flux boundary condition at the interface (z 0) for one-dimensional heat flow is given by k1
≠T1 ≠T2 2 k2 2Q ≠z ≠z
at z 0 ,
replace the three-dimensional radially symmetric, time dependent problem with the integral solution Z ` Tsx, z, td e2ivt cosskxdti sz, kd dk , (2) 0
where the subscript i is either g, s, or b for the gas (ambient), substrate, or backing, respectively (maintaining the notation of Kuo et al.). The expressions given for ti [Eqs. (A12) through (A14)] are tg sz, kd Askde2kg z ts sz, kd Bskdeks z 1 Cskde2ks z tb sz, kd Dskdekb sz2ad
The coefficient Askd determines tg , giving the temperature field on the sample surface through Eq. (2); the forms of the expressions kg skd, ks skd, and kb skd are unnecessary for the purposes of this paper. Kuo et al. state that Askd is evaluated by solving four equations derived from the continuity conditions on temperature and heat flux [including the unit heat source exps2ivtd at x y z 0 as per their equation (A7)] for the four unknowns A, B, C, and D. We give the four correct boundary conditions on the left and Kuo et al.’s
(1)
where the “1” subscript indicates quantities defined for z > 0, the “2” subscript indicates qua
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