Lock-in thermography diffusion length

Lock-in (modulated) thermography heats the surface with a sinusoidal source and extracts the amplitude and phase of the surface temperature at the excitation frequency. The heat entering the part propagates as a heavily damped thermal wave whose amplitude decays by 1/e over the thermal diffusion length µ = √(α/(π·f)). That single expression sets the whole technique: depth is selected by choosing the modulation frequency, low frequencies reaching deep and high frequencies confining the interrogation to the near surface.

Because µ goes as the inverse square root of frequency, depth is expensive. Halving the frequency buys only a 41% increase in reach, and each halving doubles the acquisition time since several full cycles must be averaged. The amplitude image is usable to roughly one diffusion length; the phase image reaches deeper, conventionally about 1.5 times µ, and it has the decisive practical advantage of being largely immune to non-uniform heating, surface emissivity variation and reflections — the reasons phase images are what most lock-in procedures actually evaluate.

The thermal wavelength λ = 2πµ is the distance over which the thermal wave completes one cycle of phase. It is worth quoting alongside the diffusion length because depth is often estimated from the measured phase shift as a fraction of that wavelength, and because a defect deeper than about half a wavelength produces a phase change too small to separate from noise.

Run the calculation both ways. Given the frequency, it tells you how deep you are looking; given the deepest feature of interest, it gives the frequency to set. Then confirm on a reference standard of the same material and thickness containing known defects at known depths — the 1.5 µ phase-depth factor is a widely used approximation, not a physical constant, and the factor that applies to your material and defect type should be established by measurement.

µ = √(α / (π · f))
thermal wavelength λ = 2π · µ
probing depth (amplitude) ≈ µ
probing depth (phase) ≈ 1.5 · µ
f required for depth d:  f = α / (π · (d / 1.5)²)