Rydberg spectroscopy of Lyman polarons

We fabricated three classes of heterostructures—a WSe₂ ML, a 3R-stacked WSe₂ bilayer (BL), and a WSe₂/WS₂ (tungsten disulfide) heterobilayer (see Supplementary Note 1)—by mechanical exfoliation and all-dry viscoelastic stamping (see “Methods”). All samples were covered with a mechanically exfoliated gypsum layer and transferred onto diamond substrates. Figure 1b shows an exemplary optical micrograph of the WSe₂ ML/gypsum heterostructure, where strong photoluminescence can be observed from the WSe₂ ML, attesting to the radiative recombination of 1s A excitons (Fig. 1c). The MIR transmission spectrum of gypsum (Fig. 1d) features two absorption peaks caused by the vibrational $$ {\nu }_4$$ and $$ {\nu }_3$$ modes of the $$ {\mathrm{SO}}_4$$ tetrahedral groups at 78 and 138 meV, respectively (see “Methods”). These modes are spectrally close to the internal resonance between the orbital 1s and 2p states of excitons in WSe₂^21,22 and are, thus, ideal for exploring the polaron physics that arises from the proximity-induced exciton–phonon coupling at the van der Waals interface. If the coupling strength exceeds the linewidth of both modes one may even expect exciton–phonon hybridization as the excitonic Lyman transition is resonantly dressed by the spatially nearby phonon field (Fig. 1a). In this proximity-induced strong-coupling scenario, Lyman polarons would emerge as new eigenstates of mixed electronic and structural character.

In the experiment, we interrogate the actual spectrum of low-energy elementary excitations by a phase-locked MIR pulse. The transmitted waveform is electro-optically sampled at a variable delay time, t_pp, after resonant creation of 1s A excitons in the K valleys of WSe₂ by a 100 fs near-infrared pump pulse (see “Methods”). A Fourier transform combined with a Fresnel analysis directly reveals the full dielectric response of the nonequilibrium system (see “Methods”). The pump-induced change of the real part of the optical conductivity, $$ \mathrm{\Delta }{\sigma }_1$$, and of the dielectric function, $$ \mathrm{\Delta }{\epsilon }_1$$, describe the absorptive and inductive responses, respectively. The dielectric response of a photoexcited WSe₂ ML covered with hBN at t_pp = 0 ps (Fig. 2, gray spheres) is dominated by a maximum in $$ \mathrm{\Delta }{\sigma }_1$$ (Fig. 2a) and a corresponding zero crossing in $$ \mathrm{\Delta }{\epsilon }_1$$ at a photon energy of 143 meV (Fig. 2b). This resonance matches with the established internal 1s–2p Lyman transition in hBN-covered WSe₂ MLs²¹ and lies well below the E_1u phonon mode in hBN (~172 meV, see “Methods”).

Fig. 2

Pump-induced dielectric response of WSe₂/hBN and WSe₂/gypsum heterostructures.

a,b Pump-induced changes of the real part of the optical conductivity $$ \mathrm{\Delta }{\sigma }_1$$ (a) and the dielectric function $$ \mathrm{\Delta }{\epsilon }_1$$ (b) as a function of the probe photon energy for different heterostructures at t_pp = 0 ps following resonant femtosecond photogeneration of 1s A excitons. Gray spheres: photoinduced dielectric response of a WSe₂ ML/hBN heterostructure. Red spheres: photoinduced dielectric response of a WSe₂ ML/gypsum heterostructure. The data are vertically offset for clarity. The dashed lines are fits to the experimental data based on the theoretical model in Eq. (1), by setting V₁ and V₂ to zero. The arrows indicate the characteristic dip in $$ \mathrm{\Delta }{\sigma }_1$$ arising from the strong exciton–phonon coupling.

In marked contrast, $$ \mathrm{\Delta }{\sigma }_1$$ features a distinct mode splitting for the WSe₂/gypsum heterostructure (Fig. 2a, red spheres). The two peaks and corresponding dispersive sections in $$ \mathrm{\Delta }{\epsilon }_1$$ (Fig. 2b, red spheres) are separated by ~35 meV and straddle the internal 1s–2p Lyman resonance of the WSe₂/hBN heterostructure. Interestingly, each peak is much narrower than the bare 1s–2p transition in the WSe₂/hBN heterostructure. Since the background dielectric constants (neglecting phonons) of gypsum and hBN are similar, the bare 1s–2p Lyman resonance in a gypsum-covered WSe₂ ML is expected to appear at an energy close to 143 meV, which gives rise to only a small detuning (ΔE ≈ 5 meV) to the vibrational $$ {\nu }_3$$ mode in gypsum (138 meV). The prominent splitting of $$ \mathrm{\Delta }{\sigma }_1$$ of ~35 meV in the WSe₂/gypsum heterostructure clearly exceeds the detuning energy and, thus, implies that the two new resonances are indeed Lyman polarons caused by strong-coupling.

Interlayer exciton–phonon hybridization

Hybridization between the intra-excitonic resonance and a lattice phonon across the van der Waals interface should lead to a measurable anticrossing signature. To test this hypothesis, we perform similar experiments on the WSe₂ BL/gypsum heterostructure, where the intra-excitonic transition can be tuned through the phonon resonance. Strong interlayer orbital hybridization in the WSe₂ BL shifts the conduction band minimum from the K points to the Λ points, leading to the formation of K–Λ excitons ($$ {\mathrm{X}}^{\mathrm{K}-\mathrm{\Lambda }}$$) with wavefunctions delocalized over the top and bottom layer^22,23. Such interlayer orbital hybridization, which is also commonly observed in other 2D transition metal dichalcogenide (TMD) heterostructures^1–5,8,9,24, renders the internal 1s–2p Lyman transition more susceptible to many-body Coulomb renormalization than in a single ML. This offers a unique opportunity to tune the intra-excitonic resonance from 87 to 69 meV by merely increasing the excitation fluence from 5 to 36 µJ cm⁻² (see Supplementary Note 2).

Figure 3a displays the MIR response of the WSe₂ BL/gypsum heterostructure at t_pp = 3 ps and various excitation densities. Strikingly, we observe a distinct anticrossing near the 1s–2p Lyman transition of K–Λ excitons in the WSe₂ BL and the $$ {\nu }_4$$ mode of gypsum upon increasing the excitation density. This is unequivocal evidence of hybridization of exciton and phonon modes across the atomic interface. In addition, the absorption for all excitation densities exhibits a discernible shoulder at a photon energy of ~115 meV (Fig. 3a, red arrow), which is very close to the 1s–2p resonance of K–K excitons ($$ {\mathrm{X}}^{\mathrm{K}-\mathrm{K}}$$)²². Such a transition is indeed expected to occur at short delay times t_pp < 1 ps, when the bound electron–hole pairs are prepared in the K valleys through direct interband excitation. However, the subtle interplay between 2D confinement and interlayer orbital overlap in the BL gives rise to a complex energy landscape^1–5,8,9,24, where the lowest-energy exciton state is given by K–Λ species. Thus, sub-picosecond thermalization of the electron to Λ valleys via intervalley scattering^22,23,25 should render the 1s–2p transition of K–K excitons weak. Yet, we clearly observe its spectral signature during the entire lifetime (see Supplementary Note 3). In addition, a new absorption band appears above the $$ {\nu }_3$$ resonance of gypsum at an energy of ~150 meV (Fig. 3a, blue arrow). Its spectral position is nearly independent of the excitation density. We will show next that these surprising observations hallmark interlayer exciton–phonon hybridization involving as many as two phonon and two exciton resonances across the atomic interface, at once.

Fig. 3

Anticrossing in interlayer exciton–phonon quantum hybridization.

a Experimentally observed pump-induced changes of $$ \mathrm{\Delta }{\sigma }_1$$ (t_pp = 3 ps, T = 260 K) of a 3R-stacked WSe₂ bilayer covered with few-layer gypsum, for different excitation fluences Φ indicated on the right. When Φ is increased from 5 to 36 µJ cm⁻² (from bottom to top), many-body renormalization shifts the intra-excitonic resonance $$ E_{1s-2p}^{\mathrm{K}-\mathrm{\Lambda }}$$ from above to below the ν₄ phonon resonance, unveiling exciton–phonon anticrossing. The red dashed lines are guides to the eyes for the peak position of $$ ∣{\mathrm{\Psi }}_1⟩$$ and $$ ∣{\mathrm{\Psi }}_2⟩$$. The excellent agreement between theoretical simulation (black dashed lines) and experimental data (solid spheres) across all $$ E_{1s-2p}^{\mathrm{K}-\mathrm{\Lambda }}$$ confirms the interlayer exciton–phonon hybridization. b Two-dimensional plot of the simulated $$ \mathrm{\Delta }{\sigma }_1$$ spectra based on the Hamiltonian shown in Eq. (1) as a function of $$ E_{1s-2p}^{\mathrm{K}-\mathrm{\Lambda }}$$. The color scale, vertical axis, and horizontal axis represent $$ \mathrm{\Delta }{\sigma }_1$$, $$ E_{1s-2p}^{\mathrm{K}-\mathrm{\Lambda }}$$, and the probe photon energy, respectively. Dashed white lines indicate the energies of Lyman polaron eigenstates ($$ ∣{\mathrm{\Psi }}_n⟩$$, n = 1, 2, 3, 4). c Illustration of the effective coupling between different excitonic states (black lines) and phonon states (gray lines). When the zero-phonon 1s–2p transition energy of the K–Λ exciton state $$ ∣2p^{\mathrm{K}-\mathrm{\Lambda }},0{\nu }_4,1s^{\mathrm{K}-\mathrm{K}},0{\nu }_3⟩$$ is tuned across the $$ {\nu }_4$$ state $$ ∣1s^{\mathrm{K}-\mathrm{\Lambda }},1{\nu }_4,1s^{\mathrm{K}-\mathrm{K}},0{\nu }_3⟩$$ (denoted by the thick red arrow) anticrossing occurs. d, e Solid lines show the calculated energies of the Lyman polaron eigenstates (d) and projection (P_n) of the eigenstates to the bare $$ ∣2p^{\mathrm{K}-\mathrm{\Lambda }},0{\nu }_4,1s^{\mathrm{K}-\mathrm{K}},0{\nu }_3⟩$$ state (e) as a function of $$ E_{1s-2p}^{\mathrm{K}-\mathrm{\Lambda }}$$. The dashed lines in d show the energies of $$ ∣1s^{\mathrm{K}-\mathrm{\Lambda }},0{\nu }_4,2p^{\mathrm{K}-\mathrm{K}},0{\nu }_3⟩$$ (115 meV), $$ ∣1s^{\mathrm{K}-\mathrm{\Lambda }},1{\nu }_4,1s^{\mathrm{K}-\mathrm{K}},0{\nu }_3⟩$$ (78 meV) and $$ ∣1s^{\mathrm{K}-\mathrm{\Lambda }},0{\nu }_4,1s^{\mathrm{K}-\mathrm{K}},1{\nu }_3⟩$$ (138 meV). The symbols in d, e correspond to $$ ∣{\mathrm{\Psi }}_n⟩$$ and were obtained from fits to the measured $$ \mathrm{\Delta }{\sigma }_1$$ spectra.

The dominant anticrossing feature in Fig. 3a occurs at an energy close to the $$ {\nu }_4$$ mode of gypsum (78 meV) and the 1s–2p resonance of K–Λ excitons in WSe₂ (69–87 meV, depending on the excitation fluence), while additional optical transitions emerge at energies close to the $$ {\nu }_3$$ mode (138 meV) and the 1s–2p resonance of K–K excitons (115 meV). Therefore, we consider how the 1s–2p transition of K-Λ and K–K excitons (see Supplementary Note 4) hybridize with $$ {\nu }_3$$ and $$ {\nu }_4$$ phonons in gypsum. The electron–phonon interaction is commonly described by the Fröhlich Hamiltonian, which is linear in the phonon creation and annihilation operators and couples only states differing by one optical phonon²⁶. The energetically lowest excited states of the uncoupled system, in which only one of the two exciton species or one of the two phonons is excited, can be denoted as $$ ∣2p^{\mathrm{K}-\mathrm{\Lambda }},0{\nu }_4,1s^{\mathrm{K}-\mathrm{K}},0{\nu }_3⟩$$, $$ ∣1s^{\mathrm{K}-\mathrm{\Lambda }},1{\nu }_4,1s^{\mathrm{K}-\mathrm{K}},0{\nu }_3⟩$$, $$ ∣1s^{\mathrm{K}-\mathrm{\Lambda }},0{\nu }_4,2p^{\mathrm{K}-\mathrm{K}},0{\nu }_3⟩$$, $$ ∣1s^{\mathrm{K}-\mathrm{\Lambda }},0{\nu }_4,1s^{\mathrm{K}-\mathrm{K}},1{\nu }_3⟩$$. The coupling between different states is illustrated in Fig. 3c. Using these basis vectors, we derive an effective Hamiltonian

Here, $$ E_{1s-2p}^{\mathrm{K}-\mathrm{\Lambda }}$$ $$ \left(E_{1s-2p}^{\mathrm{K}-\mathrm{K}}\right)$$ and $$ E_{\mathrm{ph}}^{{\nu }_4}$$ $$ \left(E_{\mathrm{ph}}^{{\nu }_3}\right)$$ denote the 1s–2p resonance energy of K–Λ (K–K) excitons and the energy of the $$ {\nu }_4$$ ($$ {\nu }_3$$) mode, respectively, whereas $$ V_1$$, $$ V_2$$, $$ V_3$$, and $$ V_4$$ describe the exciton–phonon coupling constants (Fig. 3c, red arrows). At exciton densities for which $$ E_{1s-2p}^{\mathrm{K}-\mathrm{\Lambda }}$$ is tuned through $$ E_{\mathrm{ph}}^{{\nu }_4}$$, the Hamiltonian shows that the resonant exciton–phonon hybridization leads to an avoided crossing. Quantitative comparison between experiment and theory can be achieved by directly solving the effective Hamiltonian and yields four new hybrid states ($$ ∣{\mathrm{\Psi }}_n⟩$$, n = 1, 2, 3, 4) that consist of a superposition of the basis modes.

Figure 3b displays a 2D map of the simulated optical conductivity of the new polaron eigenstates $$ ∣{\mathrm{\Psi }}_n⟩$$ as a function of the probe energy ($$ ħ\omega $$) and the position of $$ E_{1s-2p}^{\mathrm{K}-\mathrm{\Lambda }}$$. Since the excitons in WSe₂ are largely thermalized as K–Λ species on a sub-picosecond scale, the oscillator strength of the resulting polarons observed thereafter depends on their projection $$ P_n=⟨{\mathrm{\Psi }}_n\mid 2p^{\mathrm{K}-\mathrm{\Lambda }},0{\nu }_4,1s^{\mathrm{K}-\mathrm{K}},0{\nu }_3⟩$$ onto the bare zero-phonon K–Λ exciton (see Supplementary Note 3). To validate our model, we fit the simulated optical conductivity to the experimental data (Fig. 3a). Again, we set $$ E_{\mathrm{ph}}^{{\nu }_4}$$ = 78 meV and $$ E_{\mathrm{ph}}^{{\nu }_3}$$ = 138 meV (see Fig. 1d), and $$ E_{1s-2p}^{\mathrm{K}-\mathrm{K}}$$ = 115 ± 5 meV (ref. ²²), while $$ E_{1s-2p}^{\mathrm{K}-\mathrm{\Lambda }}$$ and the oscillator strength of the 1s–2p transition of the K–Λ exciton serve as fit parameters. For oscillator strengths similar to published values in ref. ²², the numerical adaption yields excellent agreement between theory and experiment and reproduces all optical transitions (Fig. 3a) and the prominent anticrossing (Fig. 3d).

The coupling constants retrieved from fitting the model to the experimental data amount to $$ V_1\approx V_3=$$ 20 ± 2 meV and $$ V_2\approx V_4=$$ 31 ± 2 meV (see Supplementary Note 3), even exceeding values reported in quantum dots²⁷. This result is remarkable given that in our experiments strong-coupling is only achieved by proximity across the van der Waals interface. The ratio $$ \frac{V_2}{V_1}~\frac{V_4}{V_3}~\sqrt{2}$$ qualitatively reflects the relative dipole moments of the $$ {\nu }_4$$ and $$ {\nu }_3$$ modes (see Supplementary Note 3). Our analysis also allows us to assign the high-frequency features in Fig. 3a to $$ ∣{\mathrm{\Psi }}_3⟩$$ and $$ ∣{\mathrm{\Psi }}_4⟩$$. Even when the K–K excitons are weakly populated at t_pp = 3 ps and the $$ {\nu }_3$$ phonon resonance is far-detuned from the 1s–2p resonance of K–Λ excitons, the strong-coupling scenario allows for these Lyman polarons to emerge. In addition, by increasing the excitation density, many-body Coulomb correlations shift the bare 1s–2p resonance of K–Λ excitons and thereby modify the Lyman composition of $$ ∣{\mathrm{\Psi }}_n⟩$$, as shown in Fig. 3e. For example, for $$ E_{1s-2p}^{\mathrm{K}-\mathrm{\Lambda }}$$ = 86.5 meV, $$ ∣{\mathrm{\Psi }}_1⟩$$ consists of 40% (6%) 1s–2p Lyman transition of the K–Λ (K–K) exciton and 10% (44%) $$ {\nu }_3$$ ($$ {\nu }_4$$) phonon (see Supplementary Note 3).

Shaping the interlayer exciton–phonon coupling strength

The interlayer exciton–phonon hybridization can be custom-tailored by engineering the spatial overlap of exciton and phonon wavefunctions on the atomic scale. To demonstrate this possibility, we create spatially well-defined intra- ($$ {\mathrm{X}}^{\mathrm{intra}}$$) and interlayer exciton ($$ {\mathrm{X}}^{\mathrm{inter}}$$) phases by interfacing the WSe₂ ML with a WS₂ ML in a WSe₂/WS₂/gypsum heterostructure. Unlike in the WSe₂ BL/gypsum heterostructure, the intralayer excitons in WSe₂ are now spatially separated from gypsum by the WS₂ ML (Fig. 4a). Ultrafast charge separation at the interface between WSe₂ and WS₂ depletes the Lyman resonance of $$ {\mathrm{X}}^{\mathrm{intra}}$$, while the transition of $$ {\mathrm{X}}^{\mathrm{inter}}$$ emerges on the sub-picosecond timescale²¹. Figure 4b shows $$ \mathrm{\Delta }{\sigma }_1$$ of the WSe₂/WS₂/gypsum heterostructure at t_pp = 1 and 10 ps. The MIR response arising from exciton–phonon coupling is qualitatively similar to that observed in the WSe₂ BL/gypsum heterostructure. This is partly because both the inter- ($$ E_{1s-2p}^{\mathrm{inter}}$$ = 69 meV) and intralayer ($$ E_{1s-2p}^{\mathrm{intra}}$$ = 114 meV) 1s–2p resonance in the WSe₂/WS₂ heterostructure are similar to $$ E_{1s-2p}^{\mathrm{K}-\mathrm{\Lambda }}$$ (69–87 meV, depending on the excitation fluence) and $$ E_{1s-2p}^{\mathrm{K}-\mathrm{K}}$$ (115 meV) in the WSe₂ BL, respectively. However, a direct comparison between $$ \mathrm{\Delta }{\sigma }_1$$ of both systems reveals a strong enhancement of the oscillator strength of the $$ ∣{\mathrm{\Psi }}_4⟩$$ mode and larger splitting between $$ ∣{\mathrm{\Psi }}_1⟩$$and $$ ∣{\mathrm{\Psi }}_2⟩$$ in the WSe₂/WS₂/gypsum heterostructure (Fig. 4c). By fitting the experimental data with our coupling model, we found that the large oscillator strength of $$ ∣{\mathrm{\Psi }}_4⟩$$ arises from enhanced phonon coupling to $$ {\mathrm{X}}^{\mathrm{inter}}$$ of V₁ = 22 ± 2 meV and V₂ = 36 ± 2 meV, which may be related with the dipolar nature of interlayer excitons. Meanwhile, the interlayer exciton–phonon coupling strength, which results directly from $$ {\mathrm{X}}^{\mathrm{intra}}$$ confined in the WSe₂ layer, amounts to V₃ = 16 ± 2 meV and V₄ = 25 ± 2 meV (see Supplementary Note 5). The reduction of coupling strength, V₃ and V₄, by at least 20% compared to the WSe₂ BL/gypsum case can be attributed to the atomically small spatial separation of the WSe₂ ML from gypsum, which illustrates how exciton–phonon interaction could be fine-tuned in search for new phases of matter.

Fig. 4

MIR response of the WSe₂/WS₂/gypsum heterostructure and comparison to the WSe₂ BL/gypsum heterostructure.

a Schematic of the spatial distribution of the excitons in the WSe₂/WS₂/gypsum (left) and the WSe₂ BL/gypsum (right) heterostructure. The relative distance of intralayer excitons ($$ {\mathrm{X}}^{\mathrm{intra}}$$ in the WSe₂/WS₂ heterostructure and $$ {\mathrm{X}}^{\mathrm{K}-\mathrm{K}}$$ in the WSe₂ BL) to the vibrational modes in gypsum yields different coupling strength. b Pump-induced change of the real part of the optical conductivity Δσ₁ for different t_pp for a WSe₂/WS₂/gypsum heterostructure (T = 230 K). Green spheres: experimental data. Black dashed line: theoretical simulation. Gray dotted lines indicate the spectral positions of $$ ∣{\mathrm{\Psi }}_n⟩$$. c Comparison between Δσ₁ of the WSe₂/WS₂/gypsum (green) and the WSe₂ BL/gypsum (blue) heterostructure at t_pp = 10 ps. Spheres: experimental data. Shaded areas: theoretical simulation. Green and blue arrows indicate the positions of $$ ∣{\mathrm{\Psi }}_1⟩$$ and $$ ∣{\mathrm{\Psi }}_2⟩$$.

Sample preparation

All heterostructure compounds were exfoliated mechanically from a bulk single crystal using the viscoelastic transfer method³¹. We used gypsum and hBN as dielectric cover layers. The vibrational $$ {\nu }_4$$ and $$ {\nu }_3$$ modes of the $$ {\mathrm{SO}}_4$$ tetrahedral groups³² in gypsum (Fig. 1d) are close to the internal 1s–2p transition of excitons in WSe₂. In contrast, the prominent E_1u mode in hBN at 172 meV³³ is far-detuned from the internal 1s–2p exciton transition in the WSe₂ layer. The exfoliated gypsum, hBN, and TMD layers were inspected under an optical microscope and subsequently stacked on top of each other on a diamond substrate with a micro-positioning stage. To remove any adsorbates, the samples were annealed at a temperature of 150 °C and a pressure of 1 × 10⁻⁵ mbar for 5 h. The twist angle of the WSe₂ BL was ensured by the tear and stack method: Starting from an extremely large exfoliated monolayer, only half of it is transferred onto the substrate. Consequently, transferring the remaining part of the ML onto the diamond substrate yields a perfectly aligned WSe₂ BL.

Ultrafast pump-probe spectroscopy

Supplementary Figure 6a depicts a schematic of the experimental setup. A home-built Ti:sapphire laser amplifier with a repetition rate of 400 kHz delivers ultrashort 12-fs NIR pulses. The output of the beam is divided into three branches. A first part of the laser output is filtered by a bandpass filter with a center wavelength closed to the interband 1s A exciton transition in the WSe₂ layer, and a bandwidth of 9 nm, resulting in 100-fs pulses. Another part of the laser pulse generates single-cycle MIR probe pulses via optical rectification in a GaSe or an LGS crystal (NOX1). The probe pulse propagates through the sample after a variable delay time t_pp. The electric field waveform of the MIR transient and any changes induced by the nonequilibrium polarization of the sample are fully resolved by electro-optic sampling utilizing a second nonlinear crystal (NOX2) and subsequent analysis of the field-induced polarization rotation of the gate pulse. Supplementary Figure 6b shows a typical MIR probe transient as a function of the electro-optic sampling time t_eos. The MIR probe pulse is centered at a frequency of 32 THz with a full-width at half-maximum of 18 THz (Supplementary Fig. 6c, black curve) and a spectral phase that is nearly flat (Supplementary Fig. 6c, blue curve). Using serial lock-in detection, we simultaneously record the pump-induced change ΔE(t_eos) and a reference E_ref(t_eos) of the MIR electric field as function of t_eos.

Extracting the dielectric response function

To extract the pump-induced change of the dielectric function of our samples with ultrafast NIR pump-MIR probe spectroscopy, we use serial lock-in detection. Hereby, a first lock-in amplifier records the electro-optic signal of our MIR probe field. Due to the modulation of the optical pump, the transmitted MIR probe field varies by the pump-induced change $$ \mathrm{\Delta }E\left(t_{\mathrm{eos}},t_{\mathrm{pp}}\right)$$. This quantity is read out in a second lock-in amplifier at the modulation frequency of the pump. Simultaneously, the electro-optic signal is averaged in an analog low-pass to obtain a reference signal $$ E_{\mathrm{ref}}\left(t_{\mathrm{eos}}\right)=\frac{1}{2}\left(E_{\mathrm{ex}}\left(t_{\mathrm{eos}},t_{\mathrm{pp}}\right)+E_{\mathrm{eq}}\left(t_{\mathrm{eos}}\right)\right)$$, where $$ E_{\mathrm{eq}}\left(t_{\mathrm{eos}}\right)$$ is the signal after transmission through the sample in thermal equilibrium and $$ E_{\mathrm{ex}}\left(t_{\mathrm{eos}},t_{\mathrm{pp}}\right)=E_{\mathrm{eq}}\left(t_{\mathrm{eos}}\right)+\mathrm{\Delta }E\left(t_{\mathrm{eos}},t_{\mathrm{pp}}\right)$$ is the signal after transmission through the excited sample at $$ t_{\mathrm{pp}}$$. From these quantities $$ E_{\mathrm{eq}}\left(t_{\mathrm{eos}}\right)$$ and $$ E_{\mathrm{ex}}\left(t_{\mathrm{eos}}\right)$$ are directly extracted. Subsequently, a Fourier transform for a fixed $$ t_{\mathrm{pp}}$$ yields $$ E_{\mathrm{eq}}\left(\omega \right)$$ and $$ E_{\mathrm{ex}}\left(\omega ,t_{\mathrm{pp}}\right)$$, which in turn provides us with the complex-valued field transfer coefficient of our layered structure