Abstract

Scientific and technological progress depend substantially on the ability to image on the nanoscale. In order to investigate complex, functional, nanoscopic structures like, e.g., semiconductor devices, multilayer optics, or stacks of 2D materials, the imaging techniques not only have to provide images but should also provide quantitative information. We report the material-specific characterization of nanoscopic buried structures with extreme ultraviolet coherence tomography. The method is demonstrated at a laser-driven broadband extreme ultraviolet radiation source, based on high-harmonic generation. We show that, besides nanoscopic axial resolution, the spectral reflectivity of all layers in a sample can be obtained using algorithmic phase reconstruction. This provides localized, spectroscopic, material-specific information of the sample. The method can be applied in, e.g., semiconductor production, lithographic mask inspection, or quality control of multilayer fabrication. Moreover, it paves the way for the investigation of ultrafast nanoscopic effects at functional buried interfaces.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

1. INTRODUCTION

Imaging techniques have always been an important catalyst for technological and scientific advances. Nowadays, the ongoing miniaturization of electronic devices and the investigation of physical properties of complex, heterogeneous structures down to the nanoscale call not only for very high resolution but also for the extraction of quantitative information [1]. Ideally, such quantitative imaging should be nondestructive and thus enable the investigation and monitoring of the characteristics of nanostructures in situ. The combination of these requirements remains a challenge for current imaging techniques. Electron microscopes, for example, provide sufficient resolution but typically suffer from low penetration depth, which implies that samples have to be sliced in order to obtain three-dimensional images [2].

Microscopy with extreme ultraviolet (XUV) light can also reach nanoscopic resolution due to the short wavelengths and, at the same time, provide higher penetration depths [3]. Conventional imaging in the XUV and soft x-ray spectral region can be implemented with the use of zone plates [4,5]. The challenges in manufacturing optical elements suited for the XUV and the involved costs lead to the development of a range of lenseless imaging techniques [610]. Due to core-level electronic resonances, XUV imaging enables element specificity. This has already been used in, e.g., chemically specific coherent diffraction imaging (CDI) of buried structures [11].

A novel XUV imaging technique is XUV coherence tomography (XCT) [12,13]. It is related to optical coherence tomography (OCT) [1416] and as such is a reflective, nondestructive, three-dimensional imaging method based on white-light interferometry. The short coherence length of broadband XUV enables nanometer-scale depth resolution. In contrast to other interferometric methods such as reflectometry with x-rays (XRR) [17,18] or extreme ultraviolet radiation (EUVR) [19,20], which are heavily model-based in most cases, XCT can provide direct imaging of 3D structures. Here we show that XCT can be extended to provide material-specific spectroscopic information in addition to structural information. Since XCT is performed in reflection geometry, it is also sensitive to interface properties such as roughness or impurities that are smaller than the resolution of XCT.

In the visible and infrared range, the extraction of spectroscopic information from OCT measurements has already been demonstrated [2123]. Due to the higher technical complexity of XCT in comparison to its optical predecessor, a sophisticated algorithmic treatment of the measured data including one-dimensional phase retrieval is necessary. Therefore, the extraction algorithms of spectroscopic data known from OCT cannot be directly applied to XCT. Here we show that by a rigorous investigation of the signal formation, suitable algorithms could be developed for XCT to nonetheless enable spectroscopic imaging.

We start with a brief review of the theory of XCT, not only for the sake of a self-contained presentation, but also to carve out the differences to the additional procedures required for depth-resolved spectroscopic and material-specific information. These are described subsequently. Following a brief description of the experiment, we present and discuss results obtained on two different silicon-based samples.

2. METHODS

A. XUV Coherence Tomography

XCT uses white-light interferometry to image a sample’s depth structure. As a consequence, the axial resolution only depends on the coherence length of the source. Lateral resolution is achieved by scanning the sample and is consequently limited by the numerical aperture of the XUV optics. For a Gaussian-shaped source spectrum the coherence length ${l_c}$, and thus the axial resolution, is given by

$${l_c} \approx \frac{{2\ln 2}}{\pi}\frac{{{\lambda ^2}}}{{\Delta \lambda}}.$$
Consequently, XUV radiation of short wavelength $\lambda$ allows axial resolutions of a few nanometers independent of the focusing geometry if the bandwidth $\Delta \lambda$ (FWHM) is sufficiently large [24].

Due to the generally high absorption cross sections of all materials in the XUV [25], the effective bandwidth is typically defined by transmission windows of the dominant sample material. An example of particularly high potential for applications is silicon between 30 and 100 eV, provided that the light source also supports this bandwidth. XCT in the silicon window has been performed at synchrotron sources reaching an axial resolution of 16 nm [26]. In silicon-based samples, the penetration depth allows the detection of layers up to a few micrometers below the surface. Later it was shown that XCT can also be performed on the laboratory scale using a laser-driven XUV source based on high-harmonic generation (HHG) [12]. In fact, HHG sources turned out to be a perfect match for XCT since they are intrinsically broadband.

When broadband XUV light is used, the implementation of an interferometer with a reference arm, as typical for the optical counterpart of XCT, is not straightforward [27]. Both the high absorption in the XUV and the short wavelength entail obstacles for the production of broadband beam splitters. Therefore, in XCT a common-path Fourier domain setup is used [28]. Here the sample acts as its own reference. Specifically, the radiation reflected at the sample’s surface interferes with the radiation reflected inside the sample and thus provides the desired depth information. A sketch of the principle of the common-path XCT setup is depicted in Fig. 1. Broadband XUV light is focused onto a layer-structured sample. Each transition from the ($j - 1$)th layer to the $ j $th layer forms an interface $ j $, which partially reflects the light into a spectrometer. There, the spectral interference between all reflected beams modulates the spectrum and carries the structural information.

 

Fig. 1. Sketch of the principle of the common-path Fourier-domain interferometer: XUV radiation is focused on the sample under an angle $\alpha$ to the surface. Sample features with effective reflectivity $r_j^{{\rm eff}}$ are buried in a distance ${z_j}$ below the surface. The different optical path lengths lead to phase shifts ${\Phi _j} = {\kappa _D}{z_j}$ in the reflected light. When using a broadband source, the frequency-dependent phase shifts lead to interferences, which are shown in the left part of the figure. The interferences with the surface can be used to compute the depth position of a buried structure.

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An accurate reconstruction of the depth structure requires the dispersion of the sample as well as the angle of incidence to be taken into account. Therefore, it is beneficial to conduct a coordinate transform from the frequency $\omega$ to ${\kappa _D}(\omega)$, which is the $ z $ component of the wavevector in the dominant sample material. It can be computed from the frequencies $\omega = 2\pi c/\lambda$, according to the dispersion relation

$${\kappa _D}(\omega) = {\rm Re}\left[{\frac{{2\omega}}{c}\sqrt {n_D^2(\omega) - \mathop {\cos}\nolimits^2 (\alpha)}} \right],$$
with the angle of incidence to the surface $\alpha$; the complex refractive index ${n_D}(\omega)$ of the dominant material, which in this work is silicon; and the real part Re[].

The scheme of XCT and the reconstruction can be described in a simple way by the frequency-dependent overall complex field reflectivity of the sample $r({\kappa _D})$, which includes phase information. The overall field reflectivity of the sample can be understood as the sum over independent reflections at every interface:

$$r({\kappa _D}) = \sum\limits_{j = 1}^N r_j^{{\rm eff}}({\kappa _D}){e^{i{\Phi _j}({\kappa _D})}}\quad {\rm with}\quad {\Phi _j}({\kappa _D}) = {\kappa _D}{z_j},$$
if multiple reflections are neglected. In the XUV this is reasonable, considering the high absorption and low reflectivities (see also Section 7 of Supplement 1). Each interface $ j $ contributes with a depth-dependent phase shift ${\Phi _j}({\kappa _D}) = {\kappa _D}{z_j}$ and an effective complex reflectivity $r_j^{{\rm eff}}({\kappa _D})$, which is the product of the reflectivity of the isolated interface and the transmission and absorption of the layers on top of it. The phase shift depends on the propagation inside the sample and thus carries the structural information, whereas the effective reflectivity contains material-specific signatures.

In XCT, only the structural information has been exploited so far. Here we combine this depth information with the spectrally resolved reflectivity of individual surfaces. The mathematics of the depth reconstruction will be briefly reviewed in the next paragraph. The extraction of the spectrally resolved, and thus material-specific, effective reflectivities—the object of this work—will be discussed in the sections starting with 2.B

The depth reconstruction, which will be described in the following, is shown together with an example in Fig. 2. The depth structure of the sample $\mathfrak{r}(z)$ is encoded in the phases ${\Phi _j}({\kappa _D})$, which are proportional to the depth of the interface ${z_j}$ with respect to the surface (${z_1} = 0\;{\rm nm} $). Thus, the inverse Fourier transform ${\cal F}{\cal T}_{{\kappa _D},z}^{- 1} \equiv \int {e^{- i{\kappa _D}z}} {\rm d}{\kappa _D}$ of Eq. (3) gives access to the depth structure. For the suppression of Fourier artifacts, $r({\kappa _D})$ is multiplied by an appropriate window function $W({\kappa _D})$ (see Section 1 in Supplement 1). Consequently, the reconstructed depth structure is given by

$$\begin{split}\mathfrak{r}(z)&=\mathcal{F}\mathcal{T}_{{{\kappa }_{D}},z}^{-1}\left[ W({{\kappa }_{D}})r({{\kappa }_{D}}) \right] \\ & =\sum\limits_{j=1}^{N}\{\mathcal{F}\mathcal{T}_{{{\kappa }_{D}},z}^{-1}\left[ W({{\kappa }_{D}}) \right]\circledast\\&\quad \mathcal{F}\mathcal{T}_{{{\kappa }_{D}},z}^{-1}\left[ r_{j}^{\rm eff}({{\kappa }_{D}}) \right]\circledast \delta \left( z-{{z}_{j}} \right)\}, \end{split}$$
where $\mathfrak{f}(z)\circledast \mathfrak{g}(z)=\int\mathfrak{f}({z}^\prime)\mathfrak{g}(z-{z}^\prime){\rm d}{z}^\prime$ is the convolution and $\delta (z)$ the delta function. The absolute value of Eq. (4), $|\mathfrak{r}(z)|$, exhibits maxima at the position of the interfaces. Thus, the depth structure is completely recovered. The interface’s signals are broadened by the inverse Fourier transform of the window function ${\cal F}{\cal T}_{{\kappa _D},z}^{- 1}[{W({\kappa _D})}]$ and the effective reflectivities ${\cal F}{\cal T}_{{\kappa _D},z}^{- 1}[{r_j^{{\rm eff}}({\kappa _D})}]$.
 

Fig. 2. Sketch of the reconstruction algorithm: 1. With the measured broadband intensity reflectivity of the sample $R(\omega)$, a phase retrieval algorithm is performed and yields the complex reflectivity $r(\omega)$ of the sample. Then the depth structure of the sample $\mathfrak{r}(z)$ and the reflectivities of the individual interfaces are obtained separately. 2. For the depth profile, the retrieved complex reflectivity is mapped into the ${\kappa _D}$ domain again and then Fourier transformed. As shown in Eq. (4), this yields the depth structure $\mathfrak{r}(z)$. 3. An inverse Fourier transform yields the signal in the time domain. The contribution from a single interface $\mathfrak{r}_{j}^{F}(t)$ is obtained by filtering. Nevertheless, the linear phase shift associated with the propagation in the sample and the corresponding modulations are not eliminated by the filtering, as can be seen in the transparent part of the example on the right. A zero shift in the time domain eliminates these effects. Accordingly, a Fourier transform yields the interface contribution $r_j^F(\omega)$ in the frequency domain, which can then be assigned to the corresponding depth. Steps 2 and 3 are shown for a simulated sample (Si|Ti|Si|Ti) in the photon energy range of 42–86 eV.

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However, a XCT measurement does not yield the field reflectivity $r({\kappa _D})$ directly. Rather, only the intensity reflectivity $R({\kappa _D}) = |r({\kappa _D}{)|^2}$ is obtained. For both the unambiguous depth reconstruction and the extraction of the spectral effective reflectivities, the phase information is required. Such a one-dimensional phase problem can indeed be solved by a specific iteration procedure [see Ref. [12], Supplement 1], and the field reflectivity can be recovered from an intensity measurement.

B. Material Specificity

With the reconstructed field reflectivity $r(\omega)$ of the sample, the effective spectral reflectivity $r_j^{{\rm eff}}(\omega)$ of individual sample interfaces, a fingerprint of the sample composition, becomes accessible. The field reflectivity is

$$r(\omega) = W(\omega)\sum\limits_{j = 1}^N r_j^{{\rm eff}}(\omega){e^{i{\Phi _j}(\omega)}}.$$
Consequently, the effective reflectivity of each interface $r_j^{{\rm eff}}(\omega)$ can be obtained by isolating the corresponding summand in Eq. (5). Note that the field reflectivity is now analyzed as a function of frequency $\omega$ or photon energy rather than of ${\kappa _D}$. The remapping into the ${k_D}$ domain is only necessary to obtain the correct depth structure. For the discussion of spectral effects it can be omitted, and the reconstruction can be performed without previous knowledge of the dominant material. ${\Phi _j}(\omega)$ is again the mainly depth-dependent phase shift for each interface. In contrast to the phase shift as a function of the wavevector in the dominant medium ${\Phi _j}({\kappa _D})$, dispersion-related effects are now also included. Multiple reflections are neglected again.

The isolation of a summand in Eq. (5) is done by a truncated Fourier transform, i.e., by filtering the spectrum in the time domain and a back transform to the frequency domain. For this purpose, the inverse Fourier transform of Eq. (5) is performed analogously to Eq. (4):

$$\begin{split}\mathfrak{r}(t)&=\mathcal{F}\mathcal{T}_{\omega ,t}^{-1}[ r(\omega ) ]=\frac{1}{4{{\pi }^{2}}}\sum\limits_{j=1}^{N}\underbrace{\left[\mathcal{F}\mathcal{T}_{\omega ,t}^{-1}\big[ r_{j}^{\rm eff}(\omega ) \big]\right.}_{\rm reflectivity}\circledast \\& \quad \underbrace{\delta \!\left(\! t-{{ \left.\!\frac{d{{\Phi }_{j}}}{d\omega } \right|}_{{{\omega }_{0}}}} \right)}_{\rm separation}\circledast \underbrace{\left.\mathcal{F}\mathcal{T}_{\omega ,t}^{-1}\left[ {{e}^{i{{\mathcal{O}}_{{{\omega }_{0}}}}(\omega )}} \right]\circledast \mathcal{F}\mathcal{T}_{\omega ,t}^{-1}[ W(\omega ) ]\vphantom{r_{j}^{\rm eff}(\omega )}\right]}_{\rm broadening}.\end{split}$$
Again $W(\omega)$ is an appropriate window function (see Section 1 in Supplement 1). For the phase we use an expansion ${\Phi _j}(\omega) = {{\frac{{d{\Phi _j}}}{{d\omega}}} |_{{\omega _0}}}\omega + {{\cal O}_{{\omega _0}}}(\omega)$ to distinguish between the linear phase shift ${{\frac{{d{\Phi _j}}}{{d\omega}}} |_{{\omega _0}}}$ and the higher-order dispersion terms ${{\cal O}_{{\omega _0}}}(\omega)$. Here ${\omega _0}$ is the central frequency.

A single interface is isolated by applying a filter function in the time domain ${F_j}(t)$ to Eq. (6) (in our case a rectangular function, see also Section 2 in Supplement 1):

$$\mathfrak{r}_{j}^{F}(t)={{F}_{j}}(t)\cdot \mathfrak{r}(t).$$
The filtering is possible, if the temporal separation of the reflections of two neighboring interfaces due to the linear phase shift is larger than the broadening effects. Broadening is caused by the higher-order dispersion terms ${e^{i{{\cal O}_{{\omega _0}}}(\omega)}}$, the spectral window $W(\omega)$, and the effective reflectivity $r_j^{{\rm eff}}(\omega)$ itself. If the broadening of two neighboring interface reflectivities is larger than the distance between the two, their reflections cannot be resolved individually. In this case, the superposition of the two interfaces $r_j^{{\rm eff}}$ and $r_{j + 1}^{{\rm eff}}$ in Eq. (5) needs to be treated as an effective single interface $r_{{j^\prime}}^{{\rm eff}}$. The effect of the linear phase ${{\frac{{d{\Phi _j}}}{{d\omega}}} |_{{\omega _0}}}(\omega)$ can be eliminated with an additional shift of the maximum of the filtered reflectivity in the time domain to $t = 0$, which eliminates the effect of the linear phase:
$$\begin{split}\mathfrak{r}_{j}^{F}(t)&=\frac{1}{4{{\pi }^{2}}}{{F}_{j}}(t)\cdot \left[\mathcal{F}\mathcal{T}_{\omega ,t}^{-1}\big[ r_{j}^{\rm eff}(\omega ) \big]\circledast \right.\\&\quad \left.\delta ( t )\circledast \mathcal{F}\mathcal{T}_{\omega ,t}^{-1}\big[ {{e}^{i{{\mathcal{O}}_{{{\omega }_{0}}}}(\omega )}} \big]\circledast \mathcal{F}\mathcal{T}_{\omega ,t}^{-1}[ W(\omega ) ]\vphantom{r_{j}^{\rm eff}(\omega )}\right].\end{split}$$

In the final step, a Fourier transform ${\cal F}{{\cal T}_{\omega ,t}} \equiv \frac{1}{{2\pi}}\int {e^{{it\omega}}} {\rm d}t$ of Eq. (8) into the frequency domain leads to the effective spectral field reflectivity of each interface

$$r_{j}^{F}(\omega )=\mathcal{F}{{\mathcal{T}}_{\omega ,t}}\left[ {{F}_{j}}(t) \right]\circledast \left[ r_{j}^{\rm eff}(\omega )\cdot {{e}^{i{{\mathcal{O}}_{{{\omega }_{0}}}}(\omega )}}\cdot W(\omega ) \right].$$
The effect of the spectral window $W(\omega)$ can be minimized by a simple division; see also Section 3 in Supplement 1. This results in the desired $r_j^{{\rm eff}}(\omega)$ multiplied with the higher-order dispersion terms ${e^{i{{\cal O}_{{\omega _0}}}(\omega)}}$ and convoluted with the Fourier transform of the filter function ${F_j}(t)$. The higher-order dispersion terms are only phase terms that do not affect the subsequent analysis of the absolute values. The width of the filter function ${F_j}(t)$ defines the spectral resolution of the retrieved reflectivity in the spectral domain. It should be chosen in such a way that the time-dependent reflectivity $\mathfrak{r}(t)$ is zero or close to zero at its boundaries. Otherwise, sidelobes will appear in the frequency domain. Since the broadening in the time domain differs between interfaces, the filter width also varies.

In summary, the reconstruction of the depth profile and the extraction of spectroscopic information can be combined to obtain material-specific depth imaging. The complete algorithm and a simulated example is shown in Fig. 2.

We want to point out that, in contrast to other approaches typically used in reflective techniques like XRR or EUVR, the presented algorithm is completely model free. The effective reflectivities are obtained directly from the measurement without previous knowledge of the sample. The same holds true for the reconstruction of the axial structure. However, by adding an appropriate model for the light–matter interaction in the sample, which is laid out in Section 4 of Supplement 1, further model-based algorithms, as they are common in XRR, can be applied in order to extract even more information such as layer thickness or interface roughness. Besides the direct reconstruction, we also present the results of such a model-based evaluation in the second part of Section 3.

C. Setup and Samples

The experiments were performed at a laser-based, lab-scale XUV imaging setup [29]. A sketch of the setup is displayed in Fig. 3.

 

Fig. 3. Experimental setup: The XUV radiation is produced by high-harmonic generation from an infrared driving laser pulse. The harmonics are focused onto the sample using a toroidal mirror. Thin metal filters are used to suppress remaining infrared light. The light from the sample is reflected into a spectrometer, which consists of a cylindrical mirror, a reflective grating, and a camera.

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The XUV radiation is produced by HHG in argon. In order to measure the spectral modulations induced by the sample with high fidelity, a continuous XUV spectrum is needed. It is in fact sufficient if the accumulated harmonic spectrum is continuous. Even for heavily modulated harmonic spectra, this can be achieved by superimposing harmonic spectra generated by scanning the wavelength of the driving laser. We use an optical parametric amplifier (OPA, TOPAS, Light conversion) to this end. It is pumped by femtosecond laser pulses from a Ti:sapphire laser system (35 fs, 800 nm, 1 kHz, 10 mJ) and produces short infrared pulses (50 fs, 2.5 mJ). Within one exposure, the central wavelength of the OPA is swept in the range of 1250–1330 nm such that the superposition of the resulting harmonic combs is quasi-continuous [30]. Thin metal foils are used to filter out the remaining infrared radiation from the produced XUV light. Spectra recorded with complementary filter configurations (aluminum and zirconium) can be combined to increase the spectral width and thus the depth resolution. Here a maximum spectral width from 38–86 eV (edge to edge) was achieved. This corresponds to a minimum coherence length of 15.1 nm (FWHM). The XUV radiation is focused onto the sample at an angle of 75° with respect to the surface by a toroidal mirror. Together with the coherence length this incidence angle supports an axial resolution of up to 15.7 nm (FWHM). The reflected spectrum is measured with a high-resolution XUV spectrometer ($\lambda /\Delta \lambda \approx 800$) [31]. The exposure times were 120 s with the aluminum filter and 300 s with the zirconium filter limited by the available photon flux. The spectrometer was calibrated using a high-harmonic spectrum reflected from a polished ${\rm TiO}_2$ crystal, which serves as an inert reference material. The same reference is used to record the source spectrum. Then the spectrum reflected from the sample can be normalized to the source spectrum, which yields the sample’s reflectivity. A detailed description can be found in Section 5 in Supplement 1.

 

Fig. 4. (a), (b) Measured intensity reflectivity $R$ of samples 1 and 2 is shown in the energy range of 42–81 eV. (c), (d) The reconstructed field reflectivity $r(\omega ,z)$ is shown for samples 1 and 2 as a false color plot. For this purpose, the algorithm depicted in Fig. 2 was performed and the effective reflectivity $r_j^{{\rm eff}}(\omega)$ of each structure assigned to the corresponding depth ${z_j}$. The integrated depth structure $\mathfrak{r}(z)$, as well as the sample design, is shown on the right of each plot.

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Two samples were investigated using the approach described in Section 2.B. A sketch of both samples can be found in Fig. 4 and Section 6 in Supplement 1. Polished titanium dioxide crystals were chosen as substrates. Both samples were produced by electron beam evaporation (EBE) [32]. Sample 1 consists of a thin titanium layer, a ${\rm SiO}_2$ layer, and another ${\rm SiO}_2$ layer on top of the sample. In between these nanoscale layers, silicon with thicknesses of approximately 160, 90, and 350 nm was deposited. This sample was produced “ex situ,” meaning that the deposition chamber was opened for preparing each subsequent deposition step. The ${\rm SiO}_2$ layers were produced by simply exposing the sample to air [33,34]. For the production of the buried ${\rm SiO}_2$ layer, which is in fact an excellent benchmark for the sensitivity of an XCT measurement, the exposure to air lasted 24 h. For sample 2, we minimized the effects of oxidation between subsequent deposition steps by “in situ” production. A thin silver layer was deposited on a 160 nm silicon layer and covered by a second, 350 nm thick silicon layer. The resulting surface was again covered by a natural ${\rm SiO}_2$ layer.

3. RESULTS

A. Direct Reconstruction

The samples were investigated using the XCT setup. The measured intensity reflectivity $R$ and the spatial and spectral reconstruction $r_j^{{\rm eff}}(\omega ,z)$ of the samples are presented in Fig. 4. The measured spectra are shown in parts (a) and (b), whereas the spatio-spectroscopic reconstruction as well as the design of the samples are depicted in parts (c) and (d).

Following the procedure described in Section 2.B and visualized in Fig. 2, the contribution of each buried structure is extracted and can thus be analyzed individually. All interfaces are clearly revealed. The axial resolution of the measurement is 27 nm (FWHM) limited by the bandwidth and the used spectral window function $W({\kappa _D})$. The thickness of the thin structures (Ti, Ag, and ${\rm SiO}_2$) is below the axial resolution of the measurement. Thus, the superposition of the reflectivities of the front and back sides of these structures is obtained (see also Section 2.B and Section 8 in Supplement 1).

The frequency dependent effective reflectivity, which carries the material-specific information, was reconstructed for all layers in both samples with a spectral resolution of 6.3–14.1 eV. The resolution differs between the layers due to the different filter widths in the time domain. It is limited by the noise level in the time domain, which prevents larger filter widths (see Section 10 in Supplement 1 for detailed information). This noise is mainly caused by temporal instabilities in the source spectrum. Theoretically, a spectral resolution of down to 1.1 eV would be possible for the investigated samples.

The retrieved reflectivity $r_j^{{\rm eff}}(\omega ,z)$, as a function of space and photon energy, enables the analysis of all interfaces individually. Even the very thin naturally grown oxide layer in sample 1 could be detected and spectrally analyzed, which demonstrates the sensitivity of the method. The reconstructed effective reflectivities for the surface oxide layers are similar for both samples, as expected. The effective reflectivity of the buried oxide layer is lower, due to the absorption in the layers covering the oxide layer. The signals of the titanium oxide substrates and the thin titanium layer show a strongly curved behavior. This effect can be attributed to the titanium M-edge and can be used for the identification and localization of titanium in the sample. It demonstrates the strong and specific material contrast of the method, which will be further validated by a comparison to simulated data as shown in Section 3.B. We want to emphasize that, up to this point, the depth-resolved spectral reflectivities are retrieved directly from the measurement and do not require neither any previous knowledge of the sample nor the use of a model.

B. Model-Based Verification and Evaluation

Nevertheless, to verify our method, the absolute values of the effective reflectivities of the buried structures were compared to simulated data according to a model based on the Fresnel equations (see Section 4 in Supplement 1). Dispersion values were taken from [35]. In addition to the refractive index change, the reflectivity of a buried interface is also strongly dependent on the interface roughness. In our model, this was accounted for by the use of the Nevot–Croce factor [36]. Since the thickness of the thin structures (${\rm SiO}_2$, Ag, Ti) is significantly below the axial resolution of the XCT measurement, the front and back sides of these structures are treated as a single interface. The agreement between model and experiment is quantified by the root-mean-square error:

$${\rm err} = \int \frac{{|{r_{{\exp }}}{|^2} - |{r_{{\rm mod}}} - {\Delta _r}{|^2}}}{{{{(|{r_{{\exp }}}| + |{r_{{\rm mod}}} - {\Delta _r}|)}^2}}} {\rm d}E,$$
with ${r_{{\exp }}}$ and ${r_{{\rm mod}}}$ being the measured and modeled reflectivities and ${\Delta _r}$ a constant offset. The comparison between experimental and modeled data is shown for each interface of the two samples in Fig. 5. The modeled data closely matches the experimental curves. A mainly constant offset ${\Delta _r}$ between the experiment and model has been observed. The dominant cause of this offset can be traced back to a drift in the XUV flux between the measurement of the actual and the reference sample. The offset is measured at the center of the measured spectrum (64 eV) and accounted for in the calculation of the error [Eq. (10)]. However, the curvature of the signal is only weakly influenced by this offset. Thus, it is well suited to distinguish different elements of the samples, if specific features like absorption edges are present. Another source of fluctuations in the reconstructed reflectivities lies in the phase retrieval algorithm and its random initial phase guess. We analyzed the repeatability of the numerical reconstruction in Section 9 of Supplement 1.
 

Fig. 5. Comparison between experimentally obtained and simulated data in the energy range from 38–86 eV: For both samples, the contribution of each axial structure is compared to the model (Section 4 in Supplement 1 and tabulated dispersion data from [35]). For the thin layers, the superposition of the front and back sides is obtained (see Section 8 in Supplement 1). At the edges of the energy window, Fourier transform artefacts dominate. For this reason, when calculating the error according to Eq. (10), the energy regions before and after the first, and last extremum or point of inflection, respectively, were neglected (gray areas). The calculated error and model parameters can be found in Table 1. While for the ${\rm SiO}_2$ and Ag structures the reconstruction is flat, the structures containing Ti show a strongly nonlinear behavior, originating from a Ti M-edge.

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Due to the limited width of the filters in the time domain and the nonzero signal at the edges, Fourier artefacts appear in the results at the edges of the spectral window. Obviously, these sidelobes need to be neglected for the evaluation of the results. For this reason, only the signal between the two outermost extrema of the spectral reflectivity was considered in Eq. (10), and if there were no extrema, the signal between the two outermost points of inflection was chosen. The neglected area is shown in gray in Fig. 5.

The reconstruction of the individual spectral reflectivities matches the model for all investigated structures. While in the case of the silicon oxide layers both experiment and reconstruction are flat, it is noteworthy that these thin, naturally grown layers can be clearly resolved and reproduced in the model. The curvature of the reconstructed effective reflectivities of the layers that contain titanium is reproduced in the model. For the calculation of the refractive index of the silver layer, a density of $5.5 \;{{{\rm g}/{\rm cm}}^3}$ was assumed, corresponding to 53% of the literature value. This is in line with island formation found in thin silver layers deposited by evaporation techniques [37].

So far, our approach has been model free, in the sense that we have used modeled data only for the verification of the results. However, the model can also be used to extract even more information from the measurement like the distance of unresolved interfaces and the interface roughness. Two unresolved interfaces contribute to the overall reflectivity as if they were a single interface with a reflectivity that depends on the reflectivity of the individual interfaces and the distance between them (also see Section 8 in Supplement 1). For example, the difference in the reflectivity of the buried ${\rm SiO}_2$ layer in comparison to the surface layers is also influenced by the different thickness of these layers and not only by the superficial layers. Consequently, this thickness is taken as a parameter in the simulation of the interfaces’ combined reflectivity. In the case of the titanium layer, the experimentally obtained distance between the front and back interfaces (7.9 nm) does not match the design value (5 nm). In order to test the accuracy of our method, the deposition occurred on multiple substrates simultaneously. After each deposition step, a sample was taken out of the chamber for additional measurements. This enabled an x-ray reflectometry (XRR) measurement with a commercial system (Bruker) and yielded a Ti layer thickness of 7.5 nm in agreement with the XCT measurement. In the same manner, the interface roughness was obtained by XRR measurements and compared to the model parameter of XCT.

The measured thicknesses and roughnesses of all interfaces are presented in Table 1, as well as the comparative XRR results.

Tables Icon

Table 1. Model Parameters for the Simulation of the Individual Structuresa

The substrate roughness was determined correctly. However, the effect of interface roughness is difficult to distinguish from the offset ${\Delta _r}$ introduced by laser fluctuations. Comparative results were only obtained for some layers, as the characterization of the thin oxide layers and the silver layer with XRR was not possible due to sample degradation.

4. DISCUSSION

We used the phase information gained by phase-retrieved XCT to extract absolute values for the spectral reflectivity of all layers. We show that by including phase information, the resulting complex sample reflectivity can be considered as the sum of reflections at individual interfaces. A filtered Fourier transform of the complex sample reflectivity yields the frequency-dependent reflectivity contribution of each buried interface. These are then assigned to the depth structure, which is routinely generated in XCT.

While the axial resolution depends on the width of the input spectrum, the spectral resolution is limited by the filter width in the filtered Fourier transform. It follows that it is ultimately limited by the sample itself, since the maximum filter width is given by the distance between adjacent interfaces. In the current experiment, noise further limits the filter width and therefore the spectral resolution. Nevertheless, it can be seen in Fig. 5 that all spectral features that can be expected to be present based on simulations are clearly resolved.

XCT delivers spatially resolved information on the material inside nanostructured samples in a model-free approach. Materials can be analyzed by identifying spectral features in the reflectivity of interfaces. Consequently, it is essential that XCT delivers the broadband sample response in the XUV. This sets it apart from other lenseless imaging approaches in the XUV, e.g., coherent diffraction imaging or zone-plate microscopy, where monochromatization remains a prerequisite in most cases. Typically, imaging with elemental specificity is restricted to a sample’s surface or requires destruction of the specimen [3840]. We show here that with XCT it is possible to overcome these limitations. Here the maximum information depth is limited by absorption of the sample and the spectrometer resolution.

The method can be supplemented with a model-based approach to gain additional information on layer roughness and thickness below the nominal resolution. Typically, model-based algorithms, e.g., in XRR or EUVR, rely on fitting the reflectivity of the whole system. In XCT, it is possible to compare the reflectivity of each sample interface to the model individually and independent of the reconstruction of the depth structure. In principle, similar Fourier-transform-based approaches can be performed for XRR data [41,42] as well as direct reconstructions of a sample’s density profile [43]. Nevertheless, it is important to note that scanning of the incidence angle at a fixed wavelength is not the same as a broadband measurement under a fixed angle. Only in the latter case, which is XCT, is a spectroscopic measurement of the sample performed. Furthermore, XCT is performed under a steeper incidence angle with respect to the surface in comparison to the standard in reflectometry methods. This allows for a better lateral resolution.

The approach presented here is a fairly simple way to determine the composition of nanostructured samples. Based on only one calibrated reflected spectrum and suitable algorithms, the depth-resolved spectral reflectivity of a sample is obtained nondestructively. Lateral imaging in XCT is currently achieved by lateral scanning of the sample. This results in a lateral resolution of a few tens of micrometers (µm), limited by the numerical aperture of the focussing optic. Since the lateral imaging is completely independent from the axial and spectral reconstruction, we restricted ourselves to one spatial dimension (depth) in this work to achieve a proof of principle.

The presented method enables the material-sensitive inspection of, e.g., multilayer optics, semiconductor devices, solar cells, or functional heterostructures, even without a priori knowledge of the sample. This makes it an ideal fit for the controlling and development of manufacturing processes, especially since it is also nondestructive. Defects and oxidation inside the samples can be studied as well as interface roughnesses. The method is currently optimized for the use of silicon-based samples. The discrimination of materials works best for elements with electronic transitions in the chosen photon energy range. For the silicon window, this includes Ti, W, Mo, and Al among various others. However, by using different wavelength ranges, other materials can be studied as well. Particularly interesting is the so-called “water window” (282–532 eV), where XCT has recently been demonstrated [13].

Furthermore, the ability of XCT to perform energy-resolved, three-dimensional imaging enables the spectroscopy of buried structures, giving access to electronic processes inside the samples. This enables the combination of spatial imaging with depth-resolved near-edge x-ray absorption fine structure (NEXAFS) features [44,45]. Typically NEXAFS is performed in transmission geometry, but it has already been shown that the same information can be extracted in reflection geometry [4648], especially under steep angles [49]. This is particularly promising in combination with the ultrafast temporal resolution, which can be achieved by the femtosecond and subfemtosecond duration of high harmonics [50,51]. Consequently, carrier transportation effects at interfaces [52] and, if polarization is controlled, nanomagnetic structures [53] could be studied with temporal, depth, and spectral resolution.

5. CONCLUSION

In this work, XUV coherence tomography has been used to identify the material of nanoscale structures inside of samples based on their spectral reflectivity. Due to the broadband interferometric approach, the XUV flux provided by lab-based sources, i.e., HHG, is efficiently used. The method was tested on two samples, and the measurements were validated against simulated data and x-ray reflectometry. It was demonstrated that XCT is a model-free and non-invasive alternative to existing approaches such as x-ray reflectometry. XCT is particularly well suited for layered structures because its depth resolution is independent of the numerical aperture of the used optics and on the scale of a few tens of nanometers. Possible applications include the investigation of defects in multilayer samples or the spectroscopy at buried interfaces, which could even be performed ultrafast by exploiting the temporal structure of the high harmonics in a pump-probe scheme.

Funding

Deutsche Forschungsgemeinschaft (PA 730/91); Helmholtz Association (Exzellenznetzwerk phase 2, Exzellenznetzwerk phase 3); Thüringer Aufbaubank (2015FGR0094, 2018FGR0080); Bundesministerium für Bildung und Forschung (BMBF); Volkswagen Foundation.

Acknowledgment

F. W. is part of the Max Planck School of Photonics supported by BMBF, the Max Planck Society, and the Fraunhofer Society. C. R. acknowledges support from the Volkswagen Foundation and from the LOEWE excellence initiative of the state of Hesse.

Disclosures

The authors declare no conflicts of interest.

Supplemental document

See Supplement 1 for supporting content.

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  27. S. De Rossi, D. Joyeux, P. Chavel, N. De Oliveira, M. Richard, C. Constancias, and J.-Y. Robic, “Probing multilayer stack reflectors by low coherence interferometry in extreme ultraviolet,” Appl. Opt. 47, 2109–2115 (2008).
    [Crossref]
  28. A. B. Vakhtin, D. J. Kane, W. R. Wood, and K. A. Peterson, “Common-path interferometer for frequency-domain optical coherence tomography,” Appl. Opt. 42, 6953–6958 (2003).
    [Crossref]
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    [Crossref]
  30. M. Wünsche, S. Fuchs, S. Aull, J. Nathanael, M. Möller, C. Rodel, and G. G. Paulus, “Quasi-supercontinuum source in the extreme ultraviolet using multiple frequency combs from high-harmonic generation,” Opt. Express 25, 6936–6944 (2017).
    [Crossref]
  31. M. Wünsche, S. Fuchs, T. Weber, J. Nathanael, J. J. Abel, J. Reinhard, F. Wiesner, U. Hübner, S. J. Skruszewicz, G. G. Paulus, and C. Rödel, “A high resolution extreme ultraviolet spectrometer system optimized for harmonic spectroscopy and XUV beam analysis,” Rev. Sci. Instrum. 90, 023108 (2019).
    [Crossref]
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    [Crossref]
  34. R. J. Archer, “Optical measurement of film growth on silicon and germanium surfaces in room air,” J. Electrochem. Soc. 104, 619–622 (1957).
    [Crossref]
  35. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection E= 50-30,000 eV, Z= 1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
    [Crossref]
  36. L. Névot and P. Croce, “Caractérisation des surfaces par réflexion rasante de rayons X. Application à l’étude du polissage de quelques verres silicates,” Rev. Phys. Appl. 15, 761–779 (1980).
    [Crossref]
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2018 (3)

P. Wachulak, A. Bartnik, and H. Fiedorowicz, “Optical coherence tomography (OCT) with 2 nm axial resolution using a compact laser plasma soft X-ray source,” Sci. Rep. 8, 8494 (2018).
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2017 (5)

O. Kfir, S. Zayko, C. Nolte, M. Sivis, M. Möller, B. Hebler, S. S. P. K. Arekapudi, D. Steil, S. Schäfer, M. Albrecht, O. Cohen, S. Mathias, and C. Ropers, “Nanoscale magnetic imaging using circularly polarized high-harmonic radiation,” Sci. Adv. 3, eaao4641 (2017).
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M. Wünsche, S. Fuchs, S. Aull, J. Nathanael, M. Möller, C. Rodel, and G. G. Paulus, “Quasi-supercontinuum source in the extreme ultraviolet using multiple frequency combs from high-harmonic generation,” Opt. Express 25, 6936–6944 (2017).
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2016 (2)

E. R. Shanblatt, C. L. Porter, D. F. Gardner, G. F. Mancini, R. M. Karl, M. D. Tanksalvala, C. S. Bevis, V. H. Vartanian, H. C. Kapteyn, and D. E. Adams, “Quantitative chemically specific coherent diffractive imaging of reactions at buried interfaces with few nanometer precision,” Nano Lett. 16, 5444–5450 (2016).
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S. Fuchs, C. Rodel, A. Blinne, U. Zastrau, M. Wunsche, V. Hilbert, L. Glaser, J. Viefhaus, E. Frumker, P. Corkum, E. Forster, and G. G. Paulus, “Nanometer resolution optical coherence tomography using broad bandwidth XUV and soft x-ray radiation,” Sci. Rep. 6, 20658 (2016).
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2014 (4)

S. Witte, V. T. Tenner, D. W. Noom, and K. S. Eikema, “Lensless diffractive imaging with ultra-broadband table-top sources: from infrared to extreme-ultraviolet wavelengths,” Light Sci. Appl. 3, e163 (2014).
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M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8, 178–186 (2014).
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2013 (1)

2012 (3)

S. Döring, F. Hertlein, A. Bayer, and K. Mann, “EUV reflectometry for thickness and density determination of thin film coatings,” Appl. Phys. A 107, 795–800 (2012).
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M. Banyay, L. Juschkin, E. Bersch, D. Franca, M. Liehr, and A. Diebold, “Cross characterization of ultrathin interlayers in HfO2 high-k stacks by angle resolved x-ray photoelectron spectroscopy, medium energy ion scattering, and grazing incidence extreme ultraviolet reflectometry,” J. Vac. Sci. Technol. A 30, 041506 (2012).
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S. Fuchs, A. Blinne, C. Rödel, U. Zastrau, V. Hilbert, M. Wünsche, J. Bierbach, E. Frumker, E. Förster, and G. G. Paulus, “Optical coherence tomography using broad-bandwidth XUV and soft X-ray radiation,” Appl. Phys. B 106, 789–795 (2012).
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2011 (1)

G. Sansone, L. Poletto, and M. Nisoli, “High-energy attosecond light sources,” Nat. Photonics 5, 655–663 (2011).
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2010 (2)

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2008 (4)

S. De Rossi, D. Joyeux, P. Chavel, N. De Oliveira, M. Richard, C. Constancias, and J.-Y. Robic, “Probing multilayer stack reflectors by low coherence interferometry in extreme ultraviolet,” Appl. Opt. 47, 2109–2115 (2008).
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S. Döring, F. Hertlein, A. Bayer, and K. Mann, “EUV reflectometry for thickness and density determination of thin film coatings,” Appl. Phys. A 107, 795–800 (2012).
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M. Banyay, L. Juschkin, E. Bersch, D. Franca, M. Liehr, and A. Diebold, “Cross characterization of ultrathin interlayers in HfO2 high-k stacks by angle resolved x-ray photoelectron spectroscopy, medium energy ion scattering, and grazing incidence extreme ultraviolet reflectometry,” J. Vac. Sci. Technol. A 30, 041506 (2012).
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S. Fuchs, A. Blinne, C. Rödel, U. Zastrau, V. Hilbert, M. Wünsche, J. Bierbach, E. Frumker, E. Förster, and G. G. Paulus, “Optical coherence tomography using broad-bandwidth XUV and soft X-ray radiation,” Appl. Phys. B 106, 789–795 (2012).
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M. Schultze, K. Ramasesha, C. Pemmaraju, S. Sato, D. Whitmore, A. Gandman, J. S. Prell, L. Borja, D. Prendergast, K. Yabana, D. M. Neumark, and S. R. Leone, “Attosecond band-gap dynamics in silicon,” Science 346, 1348–1352 (2014).
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Callcott, T. A.

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Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1. Sketch of the principle of the common-path Fourier-domain interferometer: XUV radiation is focused on the sample under an angle $\alpha$ to the surface. Sample features with effective reflectivity $r_j^{{\rm eff}}$ are buried in a distance ${z_j}$ below the surface. The different optical path lengths lead to phase shifts ${\Phi _j} = {\kappa _D}{z_j}$ in the reflected light. When using a broadband source, the frequency-dependent phase shifts lead to interferences, which are shown in the left part of the figure. The interferences with the surface can be used to compute the depth position of a buried structure.
Fig. 2.
Fig. 2. Sketch of the reconstruction algorithm: 1. With the measured broadband intensity reflectivity of the sample $R(\omega)$, a phase retrieval algorithm is performed and yields the complex reflectivity $r(\omega)$ of the sample. Then the depth structure of the sample $\mathfrak{r}(z)$ and the reflectivities of the individual interfaces are obtained separately. 2. For the depth profile, the retrieved complex reflectivity is mapped into the ${\kappa _D}$ domain again and then Fourier transformed. As shown in Eq. (4), this yields the depth structure $\mathfrak{r}(z)$. 3. An inverse Fourier transform yields the signal in the time domain. The contribution from a single interface $\mathfrak{r}_{j}^{F}(t)$ is obtained by filtering. Nevertheless, the linear phase shift associated with the propagation in the sample and the corresponding modulations are not eliminated by the filtering, as can be seen in the transparent part of the example on the right. A zero shift in the time domain eliminates these effects. Accordingly, a Fourier transform yields the interface contribution $r_j^F(\omega)$ in the frequency domain, which can then be assigned to the corresponding depth. Steps 2 and 3 are shown for a simulated sample (Si|Ti|Si|Ti) in the photon energy range of 42–86 eV.
Fig. 3.
Fig. 3. Experimental setup: The XUV radiation is produced by high-harmonic generation from an infrared driving laser pulse. The harmonics are focused onto the sample using a toroidal mirror. Thin metal filters are used to suppress remaining infrared light. The light from the sample is reflected into a spectrometer, which consists of a cylindrical mirror, a reflective grating, and a camera.
Fig. 4.
Fig. 4. (a), (b) Measured intensity reflectivity $R$ of samples 1 and 2 is shown in the energy range of 42–81 eV. (c), (d) The reconstructed field reflectivity $r(\omega ,z)$ is shown for samples 1 and 2 as a false color plot. For this purpose, the algorithm depicted in Fig. 2 was performed and the effective reflectivity $r_j^{{\rm eff}}(\omega)$ of each structure assigned to the corresponding depth ${z_j}$. The integrated depth structure $\mathfrak{r}(z)$, as well as the sample design, is shown on the right of each plot.
Fig. 5.
Fig. 5. Comparison between experimentally obtained and simulated data in the energy range from 38–86 eV: For both samples, the contribution of each axial structure is compared to the model (Section 4 in Supplement 1 and tabulated dispersion data from [35]). For the thin layers, the superposition of the front and back sides is obtained (see Section 8 in Supplement 1). At the edges of the energy window, Fourier transform artefacts dominate. For this reason, when calculating the error according to Eq. (10), the energy regions before and after the first, and last extremum or point of inflection, respectively, were neglected (gray areas). The calculated error and model parameters can be found in Table 1. While for the ${\rm SiO}_2$ and Ag structures the reconstruction is flat, the structures containing Ti show a strongly nonlinear behavior, originating from a Ti M-edge.

Tables (1)

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Table 1. Model Parameters for the Simulation of the Individual Structuresa

Equations (10)

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l c 2 ln 2 π λ 2 Δ λ .
κ D ( ω ) = R e [ 2 ω c n D 2 ( ω ) cos 2 ( α ) ] ,
r ( κ D ) = j = 1 N r j e f f ( κ D ) e i Φ j ( κ D ) w i t h Φ j ( κ D ) = κ D z j ,
r ( z ) = F T κ D , z 1 [ W ( κ D ) r ( κ D ) ] = j = 1 N { F T κ D , z 1 [ W ( κ D ) ] F T κ D , z 1 [ r j e f f ( κ D ) ] δ ( z z j ) } ,
r ( ω ) = W ( ω ) j = 1 N r j e f f ( ω ) e i Φ j ( ω ) .
r ( t ) = F T ω , t 1 [ r ( ω ) ] = 1 4 π 2 j = 1 N [ F T ω , t 1 [ r j e f f ( ω ) ] r e f l e c t i v i t y δ ( t d Φ j d ω | ω 0 ) s e p a r a t i o n F T ω , t 1 [ e i O ω 0 ( ω ) ] F T ω , t 1 [ W ( ω ) ] r j e f f ( ω ) ] b r o a d e n i n g .
r j F ( t ) = F j ( t ) r ( t ) .
r j F ( t ) = 1 4 π 2 F j ( t ) [ F T ω , t 1 [ r j e f f ( ω ) ] δ ( t ) F T ω , t 1 [ e i O ω 0 ( ω ) ] F T ω , t 1 [ W ( ω ) ] r j e f f ( ω ) ] .
r j F ( ω ) = F T ω , t [ F j ( t ) ] [ r j e f f ( ω ) e i O ω 0 ( ω ) W ( ω ) ] .
e r r = | r exp | 2 | r m o d Δ r | 2 ( | r exp | + | r m o d Δ r | ) 2 d E ,