Frequency Multiplexed Photothermal Tomography

• FM-PCT is a thermophotonic modality that uses broadband pulsed excitation and multiplexed correlation demodulation to generate 3D depth-resolved damage maps.
• It combines the area coverage and speed of infrared thermography with enhanced depth sensitivity through matched filtering and virtual-wave inversion.
• Integrating thermal diffusivity calibration and infrared super-resolution, FM-PCT enables precise subsurface inspection and defect quantification in composite materials.

Searching arXiv for the papers on arXiv and closely related FM-PCT work. Frequency Multiplexed Photothermal Correlation Tomography (FM-PCT) is a thermophotonic tomographic modality for depth-resolved subsurface inspection that combines infrared thermography (IRT), pulsed optical excitation, and correlation-based multi-frequency demodulation to reconstruct three-dimensional damage signatures from a single thermogram sequence. In the reported composite-inspection setting, it is designed to combine the area coverage and speed of standard IRT, the depth sensitivity of photothermal coherence tomography (PCT), and the efficiency advantages of frequency multiplexing (Zhu et al., 13 Sep 2025). A later generalized formulation places FM-PCT within a broader arbitrary-excitation photothermal tomography framework by linking diffusion fields to a virtual-wave representation through a Fredholm integral mapping, thereby supplying a physically grounded route from multiplexed thermal measurements to wave-like depth localization (Zhu et al., 4 May 2026).

1. Position within photothermal nondestructive evaluation

FM-PCT was introduced in contrast to several established photothermal methods. Pulsed Phase Thermography (PPT) applies a broadband optical pulse and computes phase and amplitude in the Fourier domain at individual frequencies to enhance defect contrast, but it typically yields two-dimensional contrast maps without a direct, validated depth inversion and suffers when frame rate constrains the usable frequency band (Zhu et al., 13 Sep 2025). Lock-in Thermography (LIT) uses single-frequency sinusoidal excitation and synchronous demodulation for in-phase and quadrature imaging and depth selectivity via frequency choice; however, it requires sequential acquisitions per frequency, which is slow and limited by the camera frame rate and mechanical constraints (Zhu et al., 13 Sep 2025). Standard PCT applies SVD/PCA to the temporal sequence to enhance contrast and denoise, but does not directly resolve depth nor provide physics-based inversion to damage metrics, whereas eTC-PCT uses chirped excitations and matched filtering to recover depth, but chirp acquisition is time-consuming and the imaging area is limited (Zhu et al., 13 Sep 2025).

FM-PCT extends PCT by introducing frequency multiplexing in post-processing. A single broadband pulsed excitation produces a thermal relaxation response $T(x,y,t)$, and reference sinusoids at multiple frequencies are synthesized from the pulse via FFT and used concurrently for correlation demodulation in in-phase and quadrature channels (Zhu et al., 13 Sep 2025). This improves acquisition efficiency versus sequential LIT, enables depth-selective tomographic slicing despite limited IR camera frame rate, and uses correlation matched filtering to increase SNR and suppress nonstationary noise (Zhu et al., 13 Sep 2025).

The generalized virtual-wave theory published subsequently treats arbitrary excitation, including pulsed, harmonic, and chirped waveforms, as instances of a unified diffusion-to-wave transformation (Zhu et al., 4 May 2026). Within that formulation, FM-PCT is described as using a superposition of orthogonal tones or coded sequences with correlation or lock-in demodulation to recover per-frequency amplitude and phase fields in one acquisition, making FM-PCT a special case of arbitrary excitation under a common inverse framework (Zhu et al., 4 May 2026). This suggests that FM-PCT can be interpreted both as a pragmatic correlation-tomography method and as part of a more formal wave-based inversion program.

2. Physical basis and depth selectivity

In the composite-imaging formulation, heat conduction under photothermal excitation is modeled, under an isotropic approximation, by

$$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$

For a semi-infinite medium with surface optical heating and lateral diffusion, the complex diffusion-wave Green’s function in the frequency domain along depth $z$ is given as

$$G(z,\omega) \propto \exp\!\left[-(1+i)\frac{z}{\mu(\omega)}\right],$$

with thermal diffusion length

$$\mu(\omega)=\sqrt{\frac{2\alpha}{\omega}}=\sqrt{\frac{\alpha}{\pi f}}.$$

Accordingly, amplitude decays as $e^{-z/\mu}$ and phase advances approximately as $\phi \approx z/\mu$ (Zhu et al., 13 Sep 2025). Low frequencies therefore probe deeper, while high frequencies emphasize shallow layers (Zhu et al., 13 Sep 2025).

The same diffusion-length scaling is reiterated in the generalized virtual-wave framework for harmonic excitation, where $| \hat{T}|$ decays as $\exp(-z/\delta)$ and the phase lag accrues as $\phi \approx z/\delta$, with $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$0 or $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$1 (Zhu et al., 4 May 2026). This continuity is important because FM-PCT’s multiplexing strategy exploits precisely the fact that multiple diffusion lengths can be sampled simultaneously.

For pulsed excitation and a reflector or absorber at depth $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$2, the frequency-domain solution in the one-dimensional diffusion model with lateral spatial frequency components $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$3 is reported as

$$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$4

with inversion to the spatial domain

$$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$5

These expressions formalize the dependence of thermal response on depth, diffusion length, and lateral spatial content (Zhu et al., 13 Sep 2025).

The later generalized framework introduces a distinct but related representation through a virtual wave field $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$6 governed by

$$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$7

and establishes the causal Fredholm mapping

$$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$8

with

$$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$9

The Heaviside factors enforce unilateral time support, and the Gaussian factor encodes thermodynamic irreversibility (Zhu et al., 4 May 2026). In that representation, depth follows wave time-of-flight through $z$0 for reflections (Zhu et al., 4 May 2026). The contrast with FM-PCT’s practical empirical depth mapping is significant: FM-PCT relies on diffusion-wave scaling calibrated by thermal diffusivity, whereas generalized virtual-wave reconstruction seeks a wave-like linear depth axis.

3. Excitation, demodulation, and tomographic reconstruction

In the reported FM-PCT implementation, the physical excitation is a Xenon flash pulse of approximately $z$1 in reflection or transmission mode. The pulse is treated as broadband, and reference sinusoids are synthesized for demodulation (Zhu et al., 13 Sep 2025). The in-phase and quadrature references are

$z$2

with frequencies

$z$3

or truncated bands depending on the experiment (Zhu et al., 13 Sep 2025). For conceptual clarity, a simultaneous multi-tone excitation can be written as

$z$4

although in FM-PCT the effective multi-frequency content comes from the single broadband pulse and is multiplexed through correlation to many references in a single acquisition (Zhu et al., 13 Sep 2025).

FM-PCT computes matched-filtered cross-correlations against the in-phase and quadrature references at multiple frequencies. In the frequency domain, with $z$5,

$z$6

$z$7

from which the amplitude and phase of the cross-correlation at frequency $z$8 are

$z$9

An equivalent time-domain lock-in form is

$$G(z,\omega) \propto \exp\!\left[-(1+i)\frac{z}{\mu(\omega)}\right],$$0

$$G(z,\omega) \propto \exp\!\left[-(1+i)\frac{z}{\mu(\omega)}\right],$$1

with

$$G(z,\omega) \propto \exp\!\left[-(1+i)\frac{z}{\mu(\omega)}\right],$$2

These relations explicitly connect FM-PCT to lock-in demodulation while preserving multiplexed acquisition (Zhu et al., 13 Sep 2025).

Tomographic slicing is obtained by time-gated truncation and multi-frequency stacking. The $$G(z,\omega) \propto \exp\!\left[-(1+i)\frac{z}{\mu(\omega)}\right],$$3 and $$G(z,\omega) \propto \exp\!\left[-(1+i)\frac{z}{\mu(\omega)}\right],$$4 maps are truncated at consecutive time intervals to suppress late-time lateral diffusion and noise, then grouped by frequency into depth slices (Zhu et al., 13 Sep 2025). The practical depth mapping used in the composite experiments is

$$G(z,\omega) \propto \exp\!\left[-(1+i)\frac{z}{\mu(\omega)}\right],$$5

where $$G(z,\omega) \propto \exp\!\left[-(1+i)\frac{z}{\mu(\omega)}\right],$$6 is measured by the partial time method and $$G(z,\omega) \propto \exp\!\left[-(1+i)\frac{z}{\mu(\omega)}\right],$$7 is an empirical constant (Zhu et al., 13 Sep 2025). The paper also notes the semi-infinite approximation $$G(z,\omega) \propto \exp\!\left[-(1+i)\frac{z}{\mu(\omega)}\right],$$8, hence $$G(z,\omega) \propto \exp\!\left[-(1+i)\frac{z}{\mu(\omega)}\right],$$9, but states that in practice the calibrated $$\mu(\omega)=\sqrt{\frac{2\alpha}{\omega}}=\sqrt{\frac{\alpha}{\pi f}}.$$0 mapping is used (Zhu et al., 13 Sep 2025).

The generalized virtual-wave framework formulates multiplexing somewhat differently. For multiplexed tones $$\mu(\omega)=\sqrt{\frac{2\alpha}{\omega}}=\sqrt{\frac{\alpha}{\pi f}}.$$1, the in-phase and quadrature components are

$$\mu(\omega)=\sqrt{\frac{2\alpha}{\omega}}=\sqrt{\frac{\alpha}{\pi f}}.$$2

and the amplitude and phase are

$$\mu(\omega)=\sqrt{\frac{2\alpha}{\omega}}=\sqrt{\frac{\alpha}{\pi f}}.$$3

Because the Fredholm mapping is linear, superposed multi-frequency inputs produce superposed thermal responses that decompose cleanly under lock-in or correlation (Zhu et al., 4 May 2026). The paper states that one may either reconstruct the time-domain virtual-wave field directly from $$\mu(\omega)=\sqrt{\frac{2\alpha}{\omega}}=\sqrt{\frac{\alpha}{\pi f}}.$$4 through Fredholm inversion, or apply frequency-domain continuation component-wise (Zhu et al., 4 May 2026). A plausible implication is that FM-PCT demodulation and virtual-wave inversion are mathematically compatible layers rather than mutually exclusive alternatives.

4. Instrumentation, specimens, and calibration procedures

The reported FM-PCT experiments used a FLIR X8501sc infrared camera with a $$\mu(\omega)=\sqrt{\frac{2\alpha}{\omega}}=\sqrt{\frac{\alpha}{\pi f}}.$$5–$$\mu(\omega)=\sqrt{\frac{2\alpha}{\omega}}=\sqrt{\frac{\alpha}{\pi f}}.$$6 InSb detector, NETD $$\mu(\omega)=\sqrt{\frac{2\alpha}{\omega}}=\sqrt{\frac{\alpha}{\pi f}}.$$7, and $$\mu(\omega)=\sqrt{\frac{2\alpha}{\omega}}=\sqrt{\frac{\alpha}{\pi f}}.$$8 pixels, in both reflection and transmission modes (Zhu et al., 13 Sep 2025). Photothermal excitation was provided by two Balcar Xenon flash lamps with $$\mu(\omega)=\sqrt{\frac{2\alpha}{\omega}}=\sqrt{\frac{\alpha}{\pi f}}.$$9 energy each and pulse width of approximately $e^{-z/\mu}$0; baffles were used to uniformize illumination (Zhu et al., 13 Sep 2025).

Three composite systems were examined: interlayer-toughened epoxy-based CFRP, PEEK/CF hybrid thermoplastic composites, and PA12/CF FDM-printed thermoplastic composites (Zhu et al., 13 Sep 2025). The Neat/CF and PA6.6/CF validation specimens were fabricated as 12-ply twill $e^{-z/\mu}$1 prepreg stacks using VTM260 epoxy, with PA6.6 veils inserted at 11 interlaminar interfaces for the toughened case; specimens were $e^{-z/\mu}$2 and impacted at $e^{-z/\mu}$3 and $e^{-z/\mu}$4 (Zhu et al., 13 Sep 2025). Their thermal diffusivities used for depth mapping were $e^{-z/\mu}$5 for Neat/CF and $e^{-z/\mu}$6 for PA6.6/CF (Zhu et al., 13 Sep 2025).

The PEEK/CF specimens consisted of a base layer with short CF and 12 upper layers with continuous CF in a cross-ply $e^{-z/\mu}$7 configuration, consolidated by hot pressing at $e^{-z/\mu}$8 and manufactured by 9T Labs (Zhu et al., 13 Sep 2025). These specimens were also $e^{-z/\mu}$9, impacted at $\phi \approx z/\mu$0 and $\phi \approx z/\mu$1, and tested at room temperature and $\phi \approx z/\mu$2 with an impact mass of $\phi \approx z/\mu$3 after 1 h preconditioning at target temperature (Zhu et al., 13 Sep 2025). The PA12/CF specimens were fabricated on a Stratasys Fortus 450mc with $\phi \approx z/\mu$4 slice height, 16 cross-ply layers, and no post-processing, and were tested at $\phi \approx z/\mu$5, $\phi \approx z/\mu$6, and $\phi \approx z/\mu$7 (Zhu et al., 13 Sep 2025).

Impact testing employed a CEAST/Instron 9340 drop-weight tester with a $\phi \approx z/\mu$8 hemispherical tip, total falling mass $\phi \approx z/\mu$9 for group 1 samples, and a circular unsupported area of diameter $| \hat{T}|$0 (Zhu et al., 13 Sep 2025). Temperature-controlled tests at $| \hat{T}|$1 were done after 1 h conditioning (Zhu et al., 13 Sep 2025).

Calibration of thermal diffusivity was central to FM-PCT depth mapping. Transmission-mode flash sequences were analyzed by the partial time method to estimate $| \hat{T}|$2 in sound and damaged regions, with multiple theoretical formulations averaged (Zhu et al., 13 Sep 2025). For PEEK/CF, the reported values were approximately $| \hat{T}|$3 in sound regions and $| \hat{T}|$4 in damaged regions at $| \hat{T}|$5 and room temperature, versus $| \hat{T}|$6 and $| \hat{T}|$7 at $| \hat{T}|$8 and $| \hat{T}|$9; at $\exp(-z/\delta)$0 the sound and damaged values were $\exp(-z/\delta)$1 and $\exp(-z/\delta)$2 at room temperature, and $\exp(-z/\delta)$3 and $\exp(-z/\delta)$4 at $\exp(-z/\delta)$5 (Zhu et al., 13 Sep 2025).

The reported interpretation is that low temperatures increase matrix brittleness and fiber/matrix CTE mismatch, promoting microcracks and interfacial debonding under impact (Zhu et al., 13 Sep 2025). The paper further uses a Maxwell–Eucken model for effective conductivity,

$\exp(-z/\delta)$6

and notes that with $\exp(-z/\delta)$7 for air-filled cracks, increasing crack or delamination volume fraction $\exp(-z/\delta)$8 reduces $\exp(-z/\delta)$9 and thus $\phi \approx z/\delta$0 (Zhu et al., 13 Sep 2025). This provides the physical rationale for integrating diffusivity calibration into depth reconstruction.

5. Experimental results and validation against micro-CT

The reported validation against X-ray micro-CT was carried out on Neat/CF and PA6.6/CF using acquisition parameters $\phi \approx z/\delta$1 and $\phi \approx z/\delta$2 frames, with frequency band $\phi \approx z/\delta$3 (Zhu et al., 13 Sep 2025). The reconstructed depth ranges were $\phi \approx z/\delta$4–$\phi \approx z/\delta$5 for Neat/CF and $\phi \approx z/\delta$6–$\phi \approx z/\delta$7 for PA6.6/CF (Zhu et al., 13 Sep 2025). For Neat/CF under $\phi \approx z/\delta$8, FM-PCT showed two fiber fractures on the front surface and a rear-surface crack, with features corresponding well to micro-CT $\phi \approx z/\delta$9–$$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$00 and $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$01–$$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$02 planes (Zhu et al., 13 Sep 2025). For PA6.6/CF under $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$03, FM-PCT revealed two prominent fiber fractures on the front and three on the rear, whereas micro-CT showed only one shallow crack; the authors interpret this as superior sensitivity to subsurface fiber breakage and possible interlaminar defects (Zhu et al., 13 Sep 2025). At $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$04, FM-PCT captured a deep central crack in Neat/CF consistent with micro-CT and no delamination, while in PA6.6/CF it detected two front-surface delaminations consistent with micro-CT $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$05–$$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$06 views and a larger surface crack than in Neat/CF (Zhu et al., 13 Sep 2025).

For PEEK/CF at room and low temperature, acquisition used $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$07 and $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$08 frames with $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$09 (Zhu et al., 13 Sep 2025). The reconstructed depth ranges were $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$10–$$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$11 for PEEK/CF/5 J at room temperature and $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$12–$$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$13 at $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$14, while the $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$15 tests similarly covered approximately $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$16–$$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$17 at room temperature and $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$18–$$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$19 at $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$20 (Zhu et al., 13 Sep 2025). Quantitatively, the front-side crack at $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$21 measured approximately $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$22 at room temperature versus $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$23 at $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$24; on the back side, the crack measured approximately $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$25 at room temperature, whereas at $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$26 two interlayer delaminations of approximately $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$27 plus fiber fractures were observed (Zhu et al., 13 Sep 2025). For $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$28, the $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$29-direction slices showed a crack of approximately $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$30 at room temperature versus $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$31 at $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$32, and at low temperature matrix fractures were visible in $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$33-direction slices together with delaminations (Zhu et al., 13 Sep 2025).

For PA12/CF at room temperature, no visible damage was found at $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$34; at $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$35 a front-side dent and back-side crack were detected; and at $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$36 two orthogonal cracks on the back side and four deep cracks were identified, with no delamination observed even at $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$37 (Zhu et al., 13 Sep 2025). The paper explicitly notes this as a distinct damage mode relative to PEEK/CF and PA6.6/CF (Zhu et al., 13 Sep 2025).

The generalized virtual-wave study does not reproduce the same specimen set, but it reports experimental validation on CFRP laminates with a Teflon insert under pulse, lock-in, and chirp-pulsed excitation using a 940 nm laser and a cooled MWIR camera at $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$38 (Zhu et al., 4 May 2026). Its generalized virtual-wave reconstruction converts blurred thermal responses into wave-like echoes with clear reflections from subsurface interfaces and provides linear depth mapping $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$39 (Zhu et al., 4 May 2026). Reported contrast-to-noise ratio improvements were substantial in the time domain, with raw CNR values of 1.17, 0.03, and 4.20 for pulse, lock-in, and chirp respectively, versus GVWR values of 34.87, 2.50, and 10651.57; in the frequency domain, raw values of 4.30, 3.29, and 16.85 became 12.88, 2.70, and 2.96 after GVWR (Zhu et al., 4 May 2026). While these results pertain to generalized virtual-wave reconstruction rather than FM-PCT alone, they are relevant because the same paper states that combining FM-PCT’s multiplexed acquisition with GVWR’s diffusion-to-wave inversion yields robust, fast, and quantitatively interpretable photothermal tomography under realistic industrial conditions (Zhu et al., 4 May 2026).

6. Super-resolution integration, limitations, and future directions

The composite-damage study integrates a transfer learning-based infrared super-resolution generative adversarial network, IR-SRGAN, to mitigate the lateral diffusion and pixel-limited resolution that restrict IRT-based sizing accuracy (Zhu et al., 13 Sep 2025). The generator consists of an initial convolution, 23 Residual-in-Residual Dense Blocks, three convolutional layers, and an upsampling module, while the discriminator is U-Net-based and provides both global realism and local structural fidelity through encoder–decoder paths with pixel-wise classification (Zhu et al., 13 Sep 2025). The training dataset was compiled from three infrared NDT studies and contained 165 HR and 165 LR thermograms, with LR images created by pixel removal; the discriminator was frozen after visible-spectrum pretraining, and fine-tuning was performed on the upsampling module and the last two convolutional blocks of the generator (Zhu et al., 13 Sep 2025).

In application to FM-PCT, IR-SRGAN was used to enhance the spatial fidelity of thermograms and tomograms prior to damage quantification, with the explicit caution that temporal coherence and amplitude/phase statistics must be preserved so that depth fidelity is maintained (Zhu et al., 13 Sep 2025). Qualitatively, compared with cubic interpolation, IR-SRGAN recovered fine weave texture and defect edges without blurring; compared with ESRGAN, it reduced hallucinated textures and brightness inconsistencies arising from domain mismatch, and across Neat/CF/15J, PA6.6/CF/15J, PEEK/CF/15J, PEEK/CF/15J at $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$40, and PA12/CF/40J it produced cleaner crack and delamination delineation and restored fabric patterns important for accurate size measurements (Zhu et al., 13 Sep 2025).

Several limitations are stated explicitly. The usable frequency band is bounded by camera frame rate and SNR; extremely high frequencies are noise-prone, whereas very low frequencies require long acquisition windows (Zhu et al., 13 Sep 2025). Lateral thermal diffusion blurs features, especially at later times and lower frequencies, and CFRP anisotropy can bias depth estimates, though calibration via measured $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$41 and consistent viewing geometry mitigates errors (Zhu et al., 13 Sep 2025). Surface emissivity variations, nonuniform heating, and deep-slice SNR degradation remain practical concerns, addressed in part by baffles, normalization, correlation demodulation, and the cooled camera’s NETD $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$42 (Zhu et al., 13 Sep 2025). The generalized virtual-wave paper adds that heterogeneity and anisotropy break homogeneous-$$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$43 assumptions, finite detector bandwidth impacts high-frequency recovery, spectral leakage and intermodulation require careful tone placement and window design, and robust regularization is essential because the inverse problem is ill-conditioned (Zhu et al., 4 May 2026).

Future directions are formulated along both empirical and model-based lines. The composite study lists optimized multiplexing schedules tied to measured $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$44 and defect depth range, compressive sensing and coded excitation to widen the effective frequency band under frame-rate limits, anisotropy-aware model-based deconvolution and multi-angle FM-PCT, expanded domain-specific IR-SRGAN training data, physics-aware losses, cross-validation against micro-CT to quantify PSNR/SSIM and depth accuracy gains, and joint inversion combining multi-frequency amplitude/phase with $$\rho c \frac{\partial T}{\partial t} = k \nabla^2 T + Q(x,y,z,t), \qquad \alpha = \frac{k}{\rho c}.$$45 maps via regularized least squares (Zhu et al., 13 Sep 2025). The generalized framework similarly recommends ADMM for sparse virtual-wave responses and truncated SVD for harmonic or modulated excitation, emphasizing that the Fredholm inversion is per-pixel and parallelizable and that real-time FM-PCT with virtual-wave reconstruction over modest ROIs is feasible with modern GPU hardware (Zhu et al., 4 May 2026).

A recurrent misconception in photothermal imaging is that defect contrast enhancement alone constitutes tomography. The FM-PCT results explicitly distinguish themselves from PPT and standard PCT on the grounds that those methods enhance two-dimensional damage contrast but lack direct three-dimensional depth profiling, whereas FM-PCT provides volumetric tomograms enabling measurement of crack length, delamination layer, and matrix fracture morphology with validated depth slices (Zhu et al., 13 Sep 2025). The generalized virtual-wave framework pushes that distinction further by arguing that diffusion-based depth surrogates can be replaced by wave time-of-flight in a virtual medium, yielding continuous, sharply bounded delamination profiles and linear depth mapping (Zhu et al., 4 May 2026). This suggests an emerging conceptual division within the field between calibrated diffusion-slice tomography and explicit diffusion-to-wave inversion, with FM-PCT occupying a central position because it is compatible with both paradigms.