---
title: Frequency Multiplexed Photothermal Tomography
url: https://www.emergentmind.com/topics/frequency-multiplexed-photothermal-correlation-tomography-fm-pct
type: topic
---

# Frequency Multiplexed Photothermal Tomography

Searching arXiv for the provided papers 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 [2509.10894]. 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 [2605.03747].

## 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 [2509.10894]. 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 [2509.10894]. 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 [2509.10894].

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 [2509.10894]. 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 [2509.10894].

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 [2605.03747]. 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 [2605.03747]. 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 $\phi \approx z/\mu$ [2509.10894]. Low frequencies therefore probe deeper, while high frequencies emphasize shallow layers [2509.10894].

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 $\delta(\omega)=\sqrt{2\alpha/\omega}$ or $\mu(f)=\sqrt{\alpha/(\pi f)}$ [2605.03747]. 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 $d$, the frequency-domain solution in the one-dimensional diffusion model with lateral spatial frequency components $\kappa^2 \equiv u^2+v^2$ is reported as
$$
\hat{T}(\kappa,d,\omega)=k \kappa^2 + i \omega/\alpha
\times \hat{Q}(\omega)
\times \frac{1-\exp(-2d\sqrt{\kappa^2+i\omega/\alpha})}{1+\exp(-2d\sqrt{\kappa^2+i\omega/\alpha})},
$$
with inversion to the spatial domain
$$
\hat{T}(x,y,d,\omega)=\iint \hat{Q}(\omega)\,\frac{k}{\kappa^2+i\omega/\alpha}
\times \frac{1-\exp(-2d\sqrt{\kappa^2+i\omega/\alpha})}{1+\exp(-2d\sqrt{\kappa^2+i\omega/\alpha})}
\times e^{-i(ux+vy)}\,du\,dv.
$$
These expressions formalize the dependence of thermal response on depth, diffusion length, and lateral spatial content [2509.10894].

The later generalized framework introduces a distinct but related representation through a virtual wave field $T_{\mathrm{virt}}(\mathbf{x},t)$ governed by
$$
\left(\nabla^2-\frac{1}{c_v^2}\partial_t^2\right)T_{\mathrm{virt}}(\mathbf{x},t)
=
-\frac{1}{c_v^2}Q(\mathbf{x},t),
$$
and establishes the causal Fredholm mapping
$$
T(\mathbf{x},t)=\int_0^\infty T_{\mathrm{virt}}(\mathbf{x},t')\,K(t,t')\,dt',
$$
with
$$
K(t,t')=
\frac{c_v^2}{k}\,
\frac{t'}{2\sqrt{\pi}(\alpha t)^{3/2}}
\exp\!\left(-\frac{c_v^2 t'^2}{4\alpha t}\right)H(t)H(t').
$$
The Heaviside factors enforce unilateral time support, and the Gaussian factor encodes thermodynamic irreversibility [2605.03747]. In that representation, depth follows wave time-of-flight through $z \approx c_v t'/2$ for reflections [2605.03747]. 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 $2\,\mathrm{ms}$ in reflection or transmission mode. The pulse is treated as broadband, and reference sinusoids are synthesized for demodulation [2509.10894]. The in-phase and quadrature references are
$$
r_{0,n}(t)=\cos(2\pi f_n t), \qquad r_{90,n}(t)=\sin(2\pi f_n t),
$$
with frequencies
$$
f_n=\frac{n f_s}{N}, \qquad n=1,2,\ldots,\frac{N}{4},
$$
or truncated bands depending on the experiment [2509.10894]. For conceptual clarity, a simultaneous multi-tone excitation can be written as
$$
s(t)=\sum_{k=1}^{K} a_k \cos(2\pi f_k t+\phi_k),
$$
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 [2509.10894].

FM-PCT computes matched-filtered cross-correlations against the in-phase and quadrature references at multiple frequencies. In the frequency domain, with $\omega_n=2\pi f_n$,
$$
S_{0,n}=F^{-1}\{\hat{T}(x,y,\omega-\omega_n)+\hat{T}(x,y,\omega+\omega_n)\},
$$
$$
S_{90,n}=F^{-1}\{-i\hat{T}(x,y,\omega-\omega_n)+i\hat{T}(x,y,\omega+\omega_n)\},
$$
from which the amplitude and phase of the cross-correlation at frequency $f_n$ are
$$
A_{cc,n}(x,y)=\sqrt{S_{0,n}^2+S_{90,n}^2},
\qquad
\phi_{cc,n}(x,y)=\tan^{-1}\!\left(\frac{S_{90,n}}{S_{0,n}}\right).
$$
An equivalent time-domain lock-in form is
$$
C_{c,n}(x,y)=\int_0^T T(x,y,t)\cos(2\pi f_n t)\,dt,
$$
$$
C_{s,n}(x,y)=\int_0^T T(x,y,t)\sin(2\pi f_n t)\,dt,
$$
with
$$
A_{cc,n}=\sqrt{C_{c,n}^2+C_{s,n}^2},
\qquad
\phi_{cc,n}=\operatorname{atan2}(C_{s,n},C_{c,n}).
$$
These relations explicitly connect FM-PCT to lock-in demodulation while preserving multiplexed acquisition [2509.10894].

Tomographic slicing is obtained by time-gated truncation and multi-frequency stacking. The $A_{cc,n}$ and $\phi_{cc,n}$ maps are truncated at consecutive time intervals to suppress late-time lateral diffusion and noise, then grouped by frequency into depth slices [2509.10894]. The practical depth mapping used in the composite experiments is
$$
z(f)=\frac{1.8\,\alpha}{m f},
$$
where $\alpha$ is measured by the partial time method and $m$ is an empirical constant [2509.10894]. The paper also notes the semi-infinite approximation $\phi_n \approx z/\mu(\omega_n)$, hence $z_n \approx \mu(\omega_n)\phi_n$, but states that in practice the calibrated $z(f)$ mapping is used [2509.10894].

The generalized virtual-wave framework formulates multiplexing somewhat differently. For multiplexed tones $\omega_m$, the in-phase and quadrature components are
$$
X_m=\frac{2}{T}\int_0^T T(t)\cos(\omega_m t)\,dt,
\qquad
Y_m=\frac{2}{T}\int_0^T T(t)\sin(\omega_m t)\,dt,
$$
and the amplitude and phase are
$$
A_m=\sqrt{X_m^2+Y_m^2},
\qquad
\phi_m=\tan^{-1}(Y_m/X_m).
$$
Because the Fredholm mapping is linear, superposed multi-frequency inputs produce superposed thermal responses that decompose cleanly under lock-in or correlation [2605.03747]. The paper states that one may either reconstruct the time-domain virtual-wave field directly from $T(\mathbf{x},t)$ through Fredholm inversion, or apply frequency-domain continuation component-wise [2605.03747]. 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 $3$–$5\,\mu\mathrm{m}$ InSb detector, NETD $<20\,\mathrm{mK}$, and $1280\times1024$ pixels, in both reflection and transmission modes [2509.10894]. Photothermal excitation was provided by two Balcar Xenon flash lamps with $6.4\,\mathrm{kJ}$ energy each and pulse width of approximately $2\,\mathrm{ms}$; baffles were used to uniformize illumination [2509.10894].

Three composite systems were examined: interlayer-toughened epoxy-based CFRP, PEEK/CF hybrid thermoplastic composites, and PA12/CF FDM-printed thermoplastic composites [2509.10894]. The Neat/CF and PA6.6/CF validation specimens were fabricated as 12-ply twill $2\times2$ prepreg stacks using VTM260 epoxy, with PA6.6 veils inserted at 11 interlaminar interfaces for the toughened case; specimens were $70\times70\times3\,\mathrm{mm}$ and impacted at $7.5\,\mathrm{J}$ and $15\,\mathrm{J}$ [2509.10894]. Their thermal diffusivities used for depth mapping were $\alpha=4.90\times10^{-7}\,\mathrm{m}^2/\mathrm{s}$ for Neat/CF and $\alpha=4.53\times10^{-7}\,\mathrm{m}^2/\mathrm{s}$ for PA6.6/CF [2509.10894].

The PEEK/CF specimens consisted of a base layer with short CF and 12 upper layers with continuous CF in a cross-ply $(0^\circ/90^\circ)$ configuration, consolidated by hot pressing at $450^\circ\mathrm{C}$ and manufactured by 9T Labs [2509.10894]. These specimens were also $70\times70\times3\,\mathrm{mm}$, impacted at $5\,\mathrm{J}$ and $15\,\mathrm{J}$, and tested at room temperature and $-70^\circ\mathrm{C}$ with an impact mass of $3.055\,\mathrm{kg}$ after 1 h preconditioning at target temperature [2509.10894]. The PA12/CF specimens were fabricated on a Stratasys Fortus 450mc with $0.250\,\mathrm{mm}$ slice height, 16 cross-ply layers, and no post-processing, and were tested at $2.5\,\mathrm{J}$, $5\,\mathrm{J}$, and $40\,\mathrm{J}$ [2509.10894].

Impact testing employed a CEAST/Instron 9340 drop-weight tester with a $12.7\,\mathrm{mm}$ hemispherical tip, total falling mass $8.055\,\mathrm{kg}$ for group 1 samples, and a circular unsupported area of diameter $40\,\mathrm{mm}$ [2509.10894]. Temperature-controlled tests at $-70^\circ\mathrm{C}$ were done after 1 h conditioning [2509.10894].

Calibration of thermal diffusivity was central to FM-PCT depth mapping. Transmission-mode flash sequences were analyzed by the partial time method to estimate $\alpha$ in sound and damaged regions, with multiple theoretical formulations averaged [2509.10894]. For PEEK/CF, the reported values were approximately $9.42\times10^{-7}\,\mathrm{m}^2/\mathrm{s}$ in sound regions and $9.26\times10^{-7}\,\mathrm{m}^2/\mathrm{s}$ in damaged regions at $5\,\mathrm{J}$ and room temperature, versus $9.79\times10^{-7}\,\mathrm{m}^2/\mathrm{s}$ and $8.11\times10^{-7}\,\mathrm{m}^2/\mathrm{s}$ at $5\,\mathrm{J}$ and $-70^\circ\mathrm{C}$; at $15\,\mathrm{J}$ the sound and damaged values were $8.42\times10^{-7}\,\mathrm{m}^2/\mathrm{s}$ and $5.92\times10^{-7}\,\mathrm{m}^2/\mathrm{s}$ at room temperature, and $9.07\times10^{-7}\,\mathrm{m}^2/\mathrm{s}$ and $4.08\times10^{-7}\,\mathrm{m}^2/\mathrm{s}$ at $-70^\circ\mathrm{C}$ [2509.10894].

The reported interpretation is that low temperatures increase matrix brittleness and fiber/matrix CTE mismatch, promoting microcracks and interfacial debonding under impact [2509.10894]. The paper further uses a Maxwell–Eucken model for effective conductivity,
$$
k_{\mathrm{eff}}
=
k_m
\frac{2k_m+k_c-2\phi(k_m-k_c)}{2k_m+k_c+\phi(k_m-k_c)},
$$
and notes that with $k_c \approx 0$ for air-filled cracks, increasing crack or delamination volume fraction $\phi$ reduces $k_{\mathrm{eff}}$ and thus $\alpha=k_{\mathrm{eff}}/(\rho c)$ [2509.10894]. 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 $f_s=150\,\mathrm{Hz}$ and $N=749$ frames, with frequency band $f \in [f_s/N,\,100f_s/N]$ [2509.10894]. The reconstructed depth ranges were $0.16$–$1.59\,\mathrm{mm}$ for Neat/CF and $0.15$–$1.53\,\mathrm{mm}$ for PA6.6/CF [2509.10894]. For Neat/CF under $7.5\,\mathrm{J}$, FM-PCT showed two fiber fractures on the front surface and a rear-surface crack, with features corresponding well to micro-CT $z$–$x$ and $z$–$y$ planes [2509.10894]. For PA6.6/CF under $7.5\,\mathrm{J}$, 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 [2509.10894]. At $15\,\mathrm{J}$, 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 $z$–$x$ views and a larger surface crack than in Neat/CF [2509.10894].

For PEEK/CF at room and low temperature, acquisition used $f_s=70\,\mathrm{Hz}$ and $N=999$ frames with $f \in [f_s/N,\,55f_s/N]$ [2509.10894]. The reconstructed depth ranges were $0.50$–$3.00\,\mathrm{mm}$ for PEEK/CF/5 J at room temperature and $0.55$–$3.00\,\mathrm{mm}$ at $-70^\circ\mathrm{C}$, while the $15\,\mathrm{J}$ tests similarly covered approximately $0.53$–$3.00\,\mathrm{mm}$ at room temperature and $0.55$–$3.00\,\mathrm{mm}$ at $-70^\circ\mathrm{C}$ [2509.10894]. Quantitatively, the front-side crack at $5\,\mathrm{J}$ measured approximately $0.78\,\mathrm{mm}$ at room temperature versus $3.36\,\mathrm{mm}$ at $-70^\circ\mathrm{C}$; on the back side, the crack measured approximately $2.09\,\mathrm{mm}$ at room temperature, whereas at $-70^\circ\mathrm{C}$ two interlayer delaminations of approximately $2.87\,\mathrm{mm}$ plus fiber fractures were observed [2509.10894]. For $15\,\mathrm{J}$, the $x$-direction slices showed a crack of approximately $6.31\,\mathrm{mm}$ at room temperature versus $7.99\,\mathrm{mm}$ at $-70^\circ\mathrm{C}$, and at low temperature matrix fractures were visible in $z$-direction slices together with delaminations [2509.10894].

For PA12/CF at room temperature, no visible damage was found at $2.5\,\mathrm{J}$; at $5\,\mathrm{J}$ a front-side dent and back-side crack were detected; and at $40\,\mathrm{J}$ two orthogonal cracks on the back side and four deep cracks were identified, with no delamination observed even at $40\,\mathrm{J}$ [2509.10894]. The paper explicitly notes this as a distinct damage mode relative to PEEK/CF and PA6.6/CF [2509.10894].

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 $f_{\mathrm{cam}}=100\,\mathrm{Hz}$ [2605.03747]. Its generalized virtual-wave reconstruction converts blurred thermal responses into wave-like echoes with clear reflections from subsurface interfaces and provides linear depth mapping $z=c_v t'/2$ [2605.03747]. 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 [2605.03747]. 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 [2605.03747].

## 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 [2509.10894]. 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 [2509.10894]. 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 [2509.10894].

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 [2509.10894]. 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 $-70^\circ\mathrm{C}$, and PA12/CF/40J it produced cleaner crack and delamination delineation and restored fabric patterns important for accurate size measurements [2509.10894].

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 [2509.10894]. Lateral thermal diffusion blurs features, especially at later times and lower frequencies, and CFRP anisotropy can bias depth estimates, though calibration via measured $\alpha$ and consistent viewing geometry mitigates errors [2509.10894]. 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 $<20\,\mathrm{mK}$ [2509.10894]. The generalized virtual-wave paper adds that heterogeneity and anisotropy break homogeneous-$\alpha,k$ 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 [2605.03747].

Future directions are formulated along both empirical and model-based lines. The composite study lists optimized multiplexing schedules tied to measured $\alpha$ 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 $\alpha$ maps via regularized least squares [2509.10894]. 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 [2605.03747].

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 [2509.10894]. 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 [2605.03747]. 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.

Source: https://www.emergentmind.com/topics/frequency-multiplexed-photothermal-correlation-tomography-fm-pct