Fourier-Laplace Spectral Theory for Generalized Non-Steady Thermal Fields Predicts How Long it takes for Photothermal Multifrequency Systems to Reach Steady State


A thermal wave is a critically-damped temperature oscillation that, in a homogeneous material, decays exponentially with distance from the heated surface. Over the past decades, thermal wave theory has evolved mainly along the lines of the various experimental modalities developed by research groups around the world, with the primary motivator being the application of thermal waves to photoacoustic spectroscopy and related photothermal detection techniques. From the onset of the modern-day renaissance of the field of condensed matter photo-acoustics in the 1970s, two major signal-generation and detection modalities emerged as discrete branches along with the associated theoretical analyses: frequency domain and time domain. Looking at the history of the (photo)thermal and photoacoustic signal-generation and evolution over the past half century, a modality intermediate between the time and frequency domains was pioneered in the 1980s by Professor Andreas Mandelis’s group at the Center of Advanced Diffusion-Wave and Photoacoustic Technologies (CADIPT), University of Toronto, in Canada. This modality was based on linear frequency modulation (LFM) chirps, correlation, and spectral analysis. To date, this modality evolved into the concept of the thermal-wave radar and related approaches, resulting in the recent rapid growth of two dynamic infrared (photo)thermal imaging methods using pulse compression and match filtering: The Thermal-Wave Radar (TWR) and Truncated-Correlation Photothermal Coherence Tomography (TC-PCT).

Over and above, several theoretical attempts have been made to quantify time-dependent LFM-chirp-based thermal wave signals resulting in approximate approaches to the associated generalized boundary-value problem developed as phenomenological extensions of the conventional single-frequency thermal-wave formalisms; unfortunately, a rigorous approach to the generalized time- and multifrequency modulation problem is still lacking. On this account, Professor Mandelis and Dr. Xinxin Guo developed the generalized theory of hybrid time-frequency domain thermal-wave fields with arbitrary temporal excitation sources, and investigated applications to purely time-transient and frequency-domain case studies, while paying special attention to LFM excitation. Their work is currently published in the research journal, Physical Review Applied.

The spectral theory of thermal fields subject to arbitrary boundary surface flux conditions was developed with the goal of establishing a universal approach to non-steady-state (photo)thermal responses of solids under transient or modulated thermal excitation through a combined Fourier-Laplace formalism.

The spectral theory was applied to LFM chirp thermal excitation waveforms associated with thermal-wave radar imaging. Further, the spectral theory was used to define the thermal diffusion length (TDL) associated with steady-state LFM thermal-wave fields in terms of linear superpositions of partial thermal waves. This mathematical approach turned out to be more general and impactful than the generalized theory of periodic hybrid thermal-wave fields as it sets the conditions for attainment of the dynamic steady state in terms of numbers of excitation waveform repetition periods. The authors showed that for single-frequency and multifrequency modulated thermal-wave fields, the Fourier-Laplace formalism could provide a passage to the modulated steady state that is theoretically attained only in the limit of infinite repetition periods of the excitation waveform, consistent with a purely Fourier domain spectral approach. The potential applications of the approach to other wavefields beyond thermal waves are obvious.

In summary, the study introduced a combined Fourier-Laplace formalism based on the frequency spectrum of thermal fields generated in solids by arbitrary time-dependent boundary surface flux conditions as a universal approach to defining generalized non-steady-state transient (thermal) responses and modulated multifrequency thermal waves. Overall, good agreement with previous relevant studies was reported and also experimentally validated. In a statement to Advances in Engineering, Professor Andreas Mandelis highlighted that an important practical outcome of the generalized theory was the emergence of a quantitative methodology on how photothermal and photoacoustic experimentalists can determine the minimum number of arbitrary excitation waveform repetition periods required before their experimental system can perform accurate (time-invariant) measurements of material properties, optical, thermophysical and otherwise. Without the application of this criterion, instrumentation operation under non-steady-state conditions (even for lock-in detection) was found to lead to imprecise measurements with errors larger than 10 – 14% of actual property values.

Fourier-Laplace Spectral Theory for Generalized Non-Steady Thermal Fields Predicts How Long it takes for Photothermal Multifrequency Systems to Reach Steady State - Advances in Engineering
(Top) Screen shot of a glassy carbon sample measured with linear frequency modulation (LFM chirp) (f1 = 0.1 Hz, f2 = 10 Hz) and 30-s chirp repetition period. The yellow curve is the incident heat flux waveform from a function generator. The green curve is the photothermal LFM response. (Bottom) Time dependence of the real part of the fundamental thermal wavelet component n = 1 of the spectral superposition representing the theoretical photothermal LFM response. The vertical line indicates attainment of steady state after 3500 s.

About the author

Dr. Xinxin Guo is a Senior Research Associate at the Center for Advanced Diffusion-Wave and Photoacoustic Technologies (CADIPT), Department of Mechanical and Industrial Engineering, University of Toronto, Canada. She has been working in the field of photothermal radiometry and its application in biomedical diagnostics. Her research interests are focused on the development of non-invasive glucose biosensor, alcohol biosensor and cannabis biosensor.


About the author

Andreas Mandelis is a Full Professor of Mechanical and Industrial Engineering; Electrical and Computer Engineering; and the Institute of Biomaterials and Biomedical Engineering, University of Toronto. He is the Canada Research Chair in Diffusion-Wave and Photoacoustic Sciences and Technologies and Director of the Center for Advanced Diffusion-Wave and Photoacoustic Technologies (CADIPT) at the University of Toronto.

He is also the President and CTO of Diffusion-Wave Diagnostic Technologies, Inc., Toronto, ON, and the CTO of Quantum Dental Technologies, Inc., Toronto, ON, Canada. He received his BS degree (Magna cum Laude) in physics from Yale University, and MA, MSE, and Ph.D. degrees from the Applied Physics and Materials Laboratory, Princeton University. He is the author and co-author of 430+ scientific papers in refereed journals and 190+ scientific and technical proceedings papers. He has been Editor-in-Chief of the Springer International Journal of Thermophysics (2014-19) and Topical Editor of the OSA Journal Optics Letters (2012-18). Currently he is an Associate Editor of the AIP Journals Review of Scientific Instruments, Journal of Applied Physics and member of the editorial board of the SPIE Journal of Biomedical Optics. He is Contributing Editor of the AIP flagship magazine Physics Today. He has several inventions, 39 patents and patents pending in the areas of photothermal tomographic imaging, signal processing and measurement, hydrogen sensors, dental laser diagnostics (biothermophotonics), several semiconductor non-destructive diagnostic technologies and laser biophotoacoustic and biothermophotonic tissue imaging. He holds the Canada Research Chair (Tier 1) in Diffusion-Wave and Photoacoustic Sciences and Technologies at the University of Toronto. He is also a National 1000-Talents Professor at the University of Electronic Science and Technology of China in Chengdu.

Professor Mandelis has received numerous national and international prizes and awards including the APS Keithley Award in Instrumentation Science, the Discovery Award in Science and Engineering (the Ontario Premier’s Innovation Award), the ASME 2009 Yeram Touloukian Award (and Medal) in Thermophysics, the Senior Prize of the International Photoacoustic and Photothermal Association, the Canadian Association of Physicists (CAP) Medal for Outstanding Achievement in Industrial and Applied Physics and the CAP-INO Medal for Outstanding Achievement in Applied Photonics. In 2014 he was elected Killam laureate, recipient of the Killam Prize in Engineering, one of Canada’s highest academic prizes awarded annually by the Governor General of Canada. In 2017 he was the recipient of the Inaugural Canadian NDT Research Award, Canadian Institute for Non-Destructive Evaluation (CINDE).


Andreas Mandelis, Xinxin Guo. Fourier-Laplace Spectral Theory for Non-Steady-State Thermal Fields with Applications to Problems in Steady-State Photothermal Linear Frequency Modulation. Physical Review Applied; volume 14, 024058 (2020).

Go To Physical Review Applied

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