Thermodynamic Limits of Spatial Resolution in Active Thermography

Abstract

Thermal waves are caused by pure diffusion: their amplitude is decreased by more than a factor of 500 within a propagation distance of one wavelength. The diffusion equation, which describes the temperature as a function of space and time, is linear. For every linear equation the superposition principle is valid, which is known as Huygens principle for optical or mechanical wave fields. This limits the spatial resolution, like the Abbe diffraction limit in optics. The resolution is the minimal size of a structure which can be detected at a certain depth. If an embedded structure at a certain depth in a sample is suddenly heated, e.g., by eddy current or absorbed light, an image of the structure can be reconstructed from the measured temperature at the sample surface. To get the resolution the image reconstruction can be considered as the time reversal of the thermal wave. This inverse problem is ill-conditioned and therefore regularization methods have to be used taking additional assumptions like smoothness of the solutions into account. In the present work for the first time, methods of non-equilibrium statistical physics are used to solve this inverse problem without the need of such additional assumptions and without the necessity to choose a regularization parameter. For reconstructing such an embedded structure by thermal waves the resolution turns out to be proportional to the depth and inversely proportional to the natural logarithm of the signal-to-noise ratio. This result could be derived from the diffusion equation by using a delta-source at a certain depth and setting the entropy production caused by thermal diffusion equal to the information loss. No specific model about the stochastic process of the fluctuations and about the distribution densities around the mean values was necessary to get this result.

Three-dimensional thermographic imaging using a virtual wave concept

“…35,36 In this Fourier space, called "k-space" corresponding to x-space for the temporal frequency domain, the temperature evolution in time is a simple multiplication by expðÀk 2 atÞ, with a wave vector k 37 and thermal diffusivity a, which was assumed to be homogeneous in the sample (see Eq. (1)).…”

“…By applying the inverse Fourier transform, Eq. (4) up to the frequency x trunc it follows that the reconstructed temperature signal T rec for the initial time t ¼ 0 is proportional to 36 1 2p…”

“…We have shown before that the lost information, quantitatively described by the Kullback-Leibler divergence (see Appendix), is equal to the entropy production. 36 As known from thermodynamics, the direct forward process always shows an entropy increase and, therefore, the inverse problem has to be ill-posed. The lost information is just the produced entropy.…”

“…In this Appendix, the main results from our work in the past on the connection of entropy production, information loss, and time reversal of heat diffusion are shortly reviewed 17,[34][35][36] which are used in Section II B. In the past, rank deficient and discrete ill-posed problems like the inversion of Eq.…”

“…Nevertheless, for macroscopic systems, we could show the equality of entropy production and information loss from recent results of nonequilibrium statistical physics, where we only used the heat diffusion equation modeling the mean value evolution of the temperature without the necessity to describe the fluctuations. 36 The influence of fluctuations is implicitly taken into account by calculating the entropy production from the mean value equation as a simple linear differential equation, and no sophisticated stochastic process has to be used. As the mean entropy production during the process, which is always positive, is equal to the information loss, the reconstruction of an earlier state from a later state of the process has to be an ill-posed problem.…”