Mapping active strain using terahertz metamaterial laminates

Visually opaque high-performance structural composite materials, including glass and carbon fiber-reinforced polymer matrix materials and high-temperature thermoplastics used in the aerospace, automotive, and defense industries, have been widely adopted because of their strength, durability, and reduced weight and cost, providing significant advantages over metallic alternatives.¹ However, a major limitation of many composite materials is that their failure can be sudden and catastrophic, and the precipitating strain-induced fatigue can be difficult to detect. Several techniques have been developed to monitor the structural health of composites, including embedded sensors, x-ray imaging, laser surface mapping, and other long wavelength imaging approaches.^2–7 Because many of these techniques fail to satisfy simultaneous constraints of high sensitivity, reduced complexity, and cost effectiveness, significant interest remains for innovative techniques that can image the spatial distribution of strain in composite materials.

Metamaterials (MMs) operating in the terahertz range have been actively explored over the last decade.^8–15 Because many composite materials are relatively transparent in this spectral range, terahertz MMs have been explored as possible embedded strain sensors.¹⁶ The resonance feature of terahertz MMs may be spectroscopically monitored in real-time to detect changes in strain, but the strain-induced frequency shifts have been quite small and require an unchanging, unstrained reference spectrum for comparison.^17–19 A more sensitive spatial mapping of the strain history of stretchable polydimethylsiloxane (PDMS)-based MM composites was recently achieved using aluminum bowtie antennas with a narrow strain-sensitive break-junction.²⁰ When the applied stress exceeded a predetermined damage threshold, the broken junctions permanently recorded these critical strain events through the disappearance of a terahertz polarimetric signature resonance arising from the intact asymmetric meta-atoms, thereby removing the a priori requirement for an unchanging, unstrained reference. A spatial distribution of the sample strain history was then obtained by ellipsometric mapping at the MM resonant frequency, but the signature was irreversible and only recorded where local strain exceeded that threshold.

Inspired by this reference-free technique for mapping strain history using polarization-dependent signals,^19–21 we introduce a reference-free polarimetric technique for sensing current strain in composite materials, featuring an embedded passive terahertz MM laminate composed of symmetric cross-based antennas with small (∼5 μm) gaps between adjacent meta-atoms. We show that for a simple one dimensional stretch, the polarization-dependent reversible MM response can be utilized in a self-referential manner to probe the currently active strain state reliably and quantitatively. This polarimetric technique opens up a route to facile multi-dimensional strain imaging of opaque, composite materials without the impractical a priori requirement of an unchanging, unstrained reference spectral map.

The meta-atom design consists of a fourfold rotationally symmetric aluminum Jerusalem cross structure, illustrated in Fig. 1. Cross-based metamaterials have been utilized previously to realize flexible devices capable of sensing strain associated with both uniaxial stretching^22,23 and bend curvature,^17,24 even along dual axes.¹⁸ To demonstrate strain-dependent phenomena in MM devices, the host dielectric material is typically chosen to be an elastomer such as PDMS, whose stiffness may be adjusted by controlling the degree of polymer cross-linking during preparation.^25,26 PDMS is partially transparent in the terahertz spectral range²⁷ and thus serves both as a suitable material for exploring strain-sensing concepts and as a surrogate for realistic stiffer and lossier composites.²⁸ 

Figure 1(a) schematically illustrates the interior of the MM layered structure. The structure consists of a square array of Jerusalem crosses, embedded in a 2 mm thick PDMS host material. The dimensions of the aluminum meta-atoms were chosen to give MM resonances in the transparency window of PDMS in the low terahertz range (⁠$<$1.0 THz). The MM unit cell is shown in Fig. 1(b), with periodicity a = 110 μm, cross length L = 85 μm, width = 10 μm, and crossbar length c = 78 μm. The gap between the crossbars of adjacent meta-atoms is 5 μm, and the metal thickness is 200 nm.

Figure 1(c) shows the simulated transmission response (black curve) of the MM composite. Design and simulation of the MM response were carried out using full-wave electromagnetic (EM) simulation software (CST Microwave Studio, frequency domain solver). The PDMS substrate was modeled with a permittivity of 2.38 and a loss tan δ = 0.044, representing a 2 mm thick deformable lossy host material similar to many modern fiber-reinforced polymer matrix composites. The MM layer consists of 200 nm thick aluminum crosses, simulated with periodic boundary conditions to represent a 2D lattice of meta-atoms, with a unit cell size of 110 μm.

Metamaterial resonances are dependent on the geometry of sub-wavelength meta-atoms that comprise the unit cell. They can be modeled by an equivalent circuit with the resonance frequency determined by various inductances and capacitances within the resonator elements.²⁹ The inductance originates from the “unwound” length of the metallic antennas, while the dominant capacitance comes from the electric fields concentrated at the edges of the unit cell between adjacent crosses.¹⁷ When embedded within a stretchable substrate such as PDMS and subjected to external stresses, deformations of the local geometry lead to small shifts in the MM resonance structure.

Previous studies have shown that small linear shifts in, for example, a resonance frequency can be linked directly to the applied stress and commensurate strain for PDMS-based metamaterials in the GHz and THz range.^{18,19,21,22,30} For the Jerusalem cross structures, this shift in the resonance occurs primarily through the modification of the gap capacitance between adjacent meta-atoms. These previous studies demonstrated predicted resonance behavior that can be used to measure the current strain in a composite material. However, because the resonance is much broader than the strain-induced shift, reliance on measuring very small frequency or amplitude shifts is quite prone to errors and would be quite difficult to map spatially when applied in a realistic setting. Perhaps of even greater concern is the requirement that the unstrained reference spectrum be known a priori, pixel-by-pixel, for the entire structure and that it remain unchanged under all environmental conditions other than strain. Any drift in the reference spectrum could lead to incorrectly ascribing a frequency shift to strain.

When considering the case of a uniform stretch along one axis, instead of measuring subtle spectroscopic shifts, one can take advantage of the contrasting behavior of the MM resonance for different polarizations of the incident probing beam. Figure 1(c) shows the simulated terahertz transmission spectrum for orthogonal polarizations, where T_s and T_p denote an electric field polarized along (s, blue curve) and orthogonal (p, red curve) to the direction of the stretch, respectively. For an unstretched MM device, the fourfold symmetry of the Jerusalem cross structure gives an identical transmission response for the two polarizations [black curve in Fig. 1(c)].

Upon applying stress, however, the resonant behavior of the strained MM becomes polarization-dependent in a manner that can only be caused by strain. Since the PDMS material has a Poisson’s ratio of 0.4–0.5,^31,32 as the PDMS is stretched horizontally, it will contract vertically. Thus, shifts in the MM resonance associated with applied stress will occur in opposite directions for s- and p-polarization, as shown in Fig. 1(c), for a simulated stretch of 2% (simulated spectra for other stretches are provided in Fig. S1 of the supplementary material). In this simulation, the dimensions of the metal meta-atoms do not change with strain. Instead, the effects of strain are manifested as small decreases or increases in the vertical and horizontal size of the unit cell by an amount determined by the local strain and the Poisson ratio of PDMS. This approximation is appropriate, given the stiffness of the meta-atoms and the microscopic confirmation that almost all the stretching occurs between meta-atoms or in the metal-free region around the periphery.

As the MM is stretched horizontally, the gap between adjacent crosses increases, and thus, the effective capacitance decreases, blueshifting the resonance for s-polarization [blue curve in Fig. 1(c)]. Conversely, as the PDMS contracts vertically, the gaps shrink, and the resonance for p-polarization accordingly redshifts (red curve). These reference-free, differential polarimetric shifts represent the principal innovation of this technique.

In addition, the PDMS material will slightly thin in the beam propagation direction as it is stretched, increasing transmission and shifting the Fabry–Perot-like fringes caused by interference of radiation reflected between the front and back PDMS surfaces. These fringes can subtly change the overall amplitude of the resonance peak, depending on both the applied stress and measured polarization, as seen in Fig. 1(c).

When designing metamaterials, a high Q-factor and very sharp resonances are often desirable so that small shifts are easily measurable.¹⁹ However, to reflect the effect of losses in practical composite materials, we have intentionally chosen a meta-atom design that gives a significantly broadened resonant feature so that the resonance is spread over a relatively wide frequency range and reference-dependent shifts would be harder to measure. For this more realistic case, instead of measuring frequency shifts, consider measuring the ratio of polarized terahertz transmission T_s/T_p as a function of strain. As shown below, this reference-free ratio reveals the active strain state of the sample more clearly than by simply measuring shifts of broad features such as those in Fig. 1(c). Figure 1(d) shows the combined effect across the range of the MM resonance, featuring spectrally separated regions on either side of the resonance. This distinctive polarimetric signature, where the terahertz transmission ratio is first above then below unity, is quite sensitive to small applied stresses. Note the small range of frequencies near the resonance peak where the effects of the strain are the largest. By contrast, the transmission is insensitive to polarization and strain for frequencies farther away, so these may serve as self-referential isosbestic points. This distinctive signature, composed of the polarimetric transmission ratio calibrated by another frequency such as an isosbestic point, is more robust in a practical sense because it does not rely on measuring subtle frequency shifts of a broad resonance relative to a presumptive unchanging, unstrained reference.

Thus, Fig. 1(d) reveals that the active strain profile across the sample may be obtained easily and in a self-referential manner. By measuring only the s- and p-polarized transmission not at the resonance frequency (0.42 THz) but at the peak value of T_S/T_P (∼0.39 THz) and at a second calibration frequency [either a transmission minimum (e.g., 0.48 THz) or a suitably selected isosbestic point (e.g., 0.65 THz)], a monotonically increasing quantitative measure of current local strain may be obtained. These relevant frequencies are indicated by gray dotted lines in Fig. 1. The inset in Fig. 1(d) shows, for example, the strain dependence of the transmission ratio at its peak value near ω = 0.39 THz, normalized by the calibration signal at ω = 0.48 THz. This normalized polarimetric ratio, defined as $η≡TS/TP(ω=0.39)TS/TP(ω=0.48)$⁠, provides a one-to-one mapping in response to the currently applied stress and can be used to quantify the resulting strain without having to measure the entire broadband terahertz response. The result is a rapid and more robust method for spatially mapping the local strain fields across the entire sample compared to techniques that rely on measuring small frequency or amplitude shifts relative to a previously measured unstrained reference. Furthermore, this strain visualization method uses reversible MMs and measures the active strain state, compared to, for example, MMs based on bowtie antennas that use break-junctions to record the strain history permanently.²⁰ 

The passive MM strain sensor was fabricated by standard photolithography, following a process flow similar to that detailed previously.²⁰ Figure 2(a) shows a photograph of the fabricated composite MM structure, and Fig. 2(b) shows a magnified photomicrograph of four unit cells. The PDMS substrate used in this work was prepared using a cross-linker ratio of 10:1. One notable difference compared to the previous work involves a change in the technique of transferring the patterned MM layer from the silicon wafer to the thick PDMS material after the photolithography step. Previously, an adhesion layer composed of uncured PDMS was used to adhere the MM to the surface of a 2 mm thick previously cured PDMS substrate, resulting in a ring of excess PDMS that produced an unintentional strain barrier.²⁰ In the present work, the MM layer is instead transferred to a partially cured half-filled mold of thick (∼1 mm) PDMS. The thin MM peel is made to be in contact with the partially cured PDMS and allowed to flatten from the center outward to prevent the formation of trapped air bubbles between the PDMS and MM. The rest of the mold is then filled with liquid PDMS, and the whole sample is further cured to produce a 2 mm thick PDMS substrate with an MM layer embedded at the center of the PDMS. Rippling of the deposited aluminum can be seen in the microscope image due to the thermal coefficient of expansion mismatch between the metal and polymer during cooling after metal evaporation. This rippling was observed previously and does not influence the electrical length of the meta-atom response.²⁰ 

Experimental transmission measurements were performed using a terahertz time domain spectrometer (THz-TDS) and confirmed using a terahertz spectroscopic ellipsometer.^20,33 The experimental transmission spectra in Fig. 2(c), taken with both terahertz time domain (red curve) and continuous wave ellipsometry (blue curve) techniques, show the MM resonance of interest at 0.416 THz. Due to the thickness of the lossy substrate, the simulated transmitted intensity through PDMS is very low above 1 THz, so we ignore the higher order, higher loss resonance near 1.4 THz. The sample was placed in a stretching apparatus and mounted on a pair of linear translation stages, enabling the sample to be raster scanned in 2D across the beam so that the complete terahertz response could be recorded at every pixel. Thus, each measurement consists of a hyperspectral image obtained in an 80 × 80 mm² area defined by the yellow dashed box in Fig. 2(a), with the MM region at the center of the sample.

All strain mapping measurements were performed using the THz-TDS system, whose beam waist was roughly 4–5 mm at the MM layer. The stage was scanned in 2 mm steps (with 2 mm oversampling), resulting in a 41 × 41 pixel² hyperspectral terahertz image (1661 total spectra) for each measurement. The THz-TDS parameters were chosen with a spectral resolution of 5 GHz and acquisition time of 10 s/pixel, giving a total measurement time of roughly 9 h for each polarization. To obtain transmission measurements at two orthogonal polarizations in the TDS instrument, the terahertz emitter and receiver antennas were rotated 90°, with the sample and stretching apparatus remaining fixed. Although the THz-TDS photoconductive antennas predominantly generate and detect linearly polarized radiation, we included an additional pair of wire grid polarizers (PureWavePolarizers) with a >30 dB extinction ratio to ensure s- or p-polarization. The scan area was kept the same for each measurement, but images for each polarization were slightly offset because of the antenna rotation, so this was corrected during post-processing.

The detection limit is set by the source power and receiver sensitivity, providing at least 40 dB of dynamic range over the spectral region of interest. Although increased signal averaging could increase the dynamic range, at the cost of increased acquisition time, we have shown above that only two frequencies at two polarizations are necessary to obtain a quantitative strain map. A more practical measurement—terahertz polarimetry at those two frequencies rather than a full hyperspectral measurement—could reduce the total imaging time by more than a factor of ten without sacrificing sensitivity. Real-time imaging will require a polarization-sensitive terahertz focal plane array of detectors.

Terahertz transmission spectra, measured through the center of the sample and normalized to the signal through an open channel, are shown in Fig. 2(c), along with the spectrum of the bare PDMS substrate (light gray curve). Note the good agreement with the salient features predicted by simulation (dotted curve), including the frequencies of the MM resonances, the low transmitted signal above 1 THz due to the lossy PDMS substrate, and the Fabry–Perot oscillations from internal reflections between the front and back surfaces of the sample. The spectra obtained from the terahertz ellipsometry are self-referenced and agree very well with both the THz-TDS data and with the simulated MM response.

We are ultimately interested in obtaining a quantitative 2D strain map across the sample. Since a full terahertz spectrum was collected at each pixel, the spatial maps of the unstretched sample may be examined at different frequency slices. After ascertaining where the MM region is located within the scanned area, the spatial variations in the transmitted signal, caused by a combination of intrinsic and extrinsic local strains, may be observed. Figure 3 shows a stack of 2D images, including a cropped square of the sample photograph indicating the scanned region (top), as well as normalized terahertz transmission images for s- (left) and p-polarization (right) at 0.2, 0.42, and 0.65 THz [indicated by the colored arrows in Fig. 2(c)]. The MM region is most evident in the image at 0.42 THz, where the low signal in the central region is due to the MM resonance. The edges of the sample, which do not contain the MM, exhibit higher transmission. The signal is more uniform off resonance at both 0.2 and 0.65 THz, while still showing variations in transmission, particularly at 0.2 THz, that are likely associated with beam diffraction or scattering due to non-uniformity in the fabricated sample (of course, diffraction is greater at higher frequencies, but so is the attenuation by the PDMS, so the greater diffraction is not observable). The images for s- and p-polarized transmission on resonance (ω = 0.42 THz) were used to identify the MM region, which was necessary to obtain a properly normalized hyperspectral image for the transmission ratio for the two polarizations T_s(ω)/T_p(ω) after correcting for the spatial offset.

Figure 4 shows measured 2D strain maps for the full 80 × 80 mm² area encompassing the reversible Jerusalem cross MMs. Maps are obtained by normalizing the T_s/T_p image measured at ω = 0.39 THz by the strain-insensitive calibration T_s/T_p image measured at ω = 0.48 THz, as described earlier. This is the desired reference-free map of the current strain distribution, for which different colors are used to aid visualization of the different amounts of strain quantitatively measured. For the unstretched sample [Fig. 4(a)], there is little strain throughout the measurement area. As the applied global stretch in the sample increases [Figs. 4(b)–4(d)], the local strain in the MM regions increases but by an amount significantly smaller than that applied globally. For the 6% global stretch, the quantitative measure of local stress in the MM region indicates a local strain approaching only 1% near the center. Thus, the majority of the stretching in the soft PDMS occurs in the unmeasured periphery of the MM region, while only a portion of the global strain produced local changes in spacing between meta-atoms. Unlike PDMS, which distributes strain more globally across the material, stiffer composite materials will exhibit stronger local effects.

To verify how our passive MM sensor responds locally and quantitatively to applied stress, Fig. 4(e) shows representative strain-dependent spectra for several locations indicated by the symbols triangle, circle, and X in Fig. 4(a) (complete strain- and polarization-dependent spectra for these and other example pixels are shown in Fig. S2 of the supplementary material). The left and center panels in Fig. 4(e) plot typical spectra for regions that were under minimal local strain (triangle and circle symbols, respectively) prior to the application of external stress. Notice that those regions quantitatively brighten with increasing strain by an amount measured by the distinctive signatures predicted, as shown in Fig. 1(d). An uncertainty or shift of 1% in monitored frequency, comparable to the spectral resolution of the THz-TDS spectrometer at the MM resonance frequency, produces an uncertainty of 1% in η and a commensurate error of only 0.5% in the extracted strain value.

This strain visualization is fully reversible, provided that there are no plastic deformations of the host material. In our experiments, we observed that after stretching the sample beyond 6% global strain, the PDMS did not subsequently relax fully, indicating that plastic deformation occurred and prevented the gaps between the meta-atom crosses from returning to their original unstretched distances. Consequently, even after measuring the fully relaxed sample, the strain profile resembles an image between 2% and 4% (not shown).

Interestingly, Figs. 4(b)–4(d) reveal that several regions near the perimeter of the MM area remained dark even as the sample was increasingly stretched. Because those dark regions are observed with no stretch, they cannot be caused by unevenly applied strain. Instead, we hypothesize that this may be due to fabrication imperfections: the edges of the sample were not perfectly flat when the MM peel was transferred to the PDMS mold during fabrication. Some waviness is evident from visual inspection of the sample [Fig. 2(a)]. As such, the MM in these regions is not planar, and some strain may be “frozen-in” during curing of the PDMS, producing built-in strain orthogonal to the direction of applied stretching in our experiment. As evidence of this built-in strain, consider the spectra [right panel in Fig. 4(e)] in one of the dark regions (denoted by an “X”). The transmission maxima/minima are inverted, consistent with the hypothesis that these regions were pre-strained in the orthogonal direction during sample fabrication. Notice that the effect of the global stretch is to oppose and relax this built-in strain, as seen by the green curve.

Although only a uniform one-dimensional stretch was applied in this experiment, a three-dimensional response is observed as the orthogonal axes thin. More generally, this technique may be used to map complex, non-uniformly applied strain fields, and this approach can be extended to include other strain axes through inclusion of multiple MM resonances. A similar modeling analysis to link applied stretches to polarization-dependent spectral signatures may enable full vector strain field mapping. Furthermore, the effect is fully reversible if the sample does not undergo plastic deformation since it relies on a simple mechanism for MM resonance tuning, compared to previous strain mapping paradigms that rely on a destructive signal.²⁰ The latter approach is ideal for recording strain history and critical stress events, while the current design is more advantageous when spatially mapping the current strain profile is desired. Compared to previous methods that monitor peak frequencies or raw amplitudes as a function of applied stress, the approach presented here allows the current strain state to be measured directly, without requiring a universal unstrained reference.

A reference-free polarimetric technique is reported for the visualization of active strain fields in composite materials using embedded reversible terahertz metamaterials. By acquiring the polarization-dependent terahertz transmission through a symmetric Jerusalem cross-based MM embedded in a stretchable host, a unique quantitative spectroscopic signature derived from the MM resonance enables the current local strain to be measured quantitatively. This concept of MM laminates for mapping strain may be tailored for any desired spectral range, and the resulting passive layer can be embedded in a variety of composite materials that are transparent in the spectral band of interest. Thus, strain mapping using terahertz metamaterials offers a facile, practical way to monitor the current strain state within a wide variety of visually opaque high-performance structural composites such as high-temperature thermoplastics used in the aerospace, automotive, and defense industries.

See the supplementary material for simulated transmission spectra for other stretches between 0% and 2%, as well as experimental strain- and polarization-dependent transmission spectra for several individual pixels in the metamaterial composite.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.