# Optical coherence tomography using broad-bandwidth XUV and soft X-ray radiation

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## Abstract

We present a novel approach to extend optical coherence tomography (OCT) to the extreme ultraviolet (XUV) and soft X-ray (SXR) spectral range. With a simple setup based on Fourier-domain OCT and adapted for the application of XUV and SXR broadband radiation, cross-sectional images of semiconductors and organic samples becomes feasible with current synchrotron or laser-plasma sources. For this purpose, broadband XUV radiation is focused onto the sample surface, and the reflected spectrum is recorded by an XUV spectrometer. The proposed method has the particular advantage that the axial spatial resolution only depends on the spectral bandwidth. As a consequence, the theoretical resolution limit of XUV coherence tomography (XCT) is in the order of nanometers, e.g., 3 nm for wavelengths in the water window (280–530 eV). We proved the concept of XCT by calculating the reflectivity of one-dimensional silicon and boron carbide samples containing buried layers and found the expected properties with respect to resolution and penetration depth confirmed.

## Keywords

Optical Coherence Tomography Coherence Length Boron Carbide Optical Coherence Tomography Signal Optical Coherence Tomography Device## 1 Introduction

*l*

_{ c }which depends on the central wavelength

*λ*

_{0}and the spectral width (FWHM) \(\varDelta\lambda_{\text{FWHM}}\) of a light source with a, e.g., Gaussian shaped spectrum As a consequence, the axial resolution only depends on the spectral rather than geometrical properties of the radiation. This is particularly interesting for applications where a high numerical aperture is not feasible. OCT with broadband visible and near-infrared sources typically reach axial (depth) resolutions in the order of a few micrometers [1, 2, 3].

Within the last decade and in conjunction with the quickly developing sector of advanced material design, the scale length of interest has dropped from micrometers to a few nanometers. Both the semiconductor circuit industry, aiming at fast and power-saving solutions, as well as structural biology and environmental chemistry with their enormous interest in nanostructures, call for resolutions in the nanometer regime. The ideas presented here take advantage of the fact that the coherence length can be significantly reduced if broadband XUV and SXR radiation is used.

Microscopy using XUV and SXR radiation has regularly ineluctable practical restrictions imposed by the optics and sources available in this regime. Coherence tomography with short wavelengths can circumvent some of these limitations in principle. A major limitation of XUV radiation is the absorption within a few tens or hundreds of nanometers [4] depending on the actual composition of the material and the wavelength range. Consequently, XUV coherence tomography (XCT) can only display its full capabilities when used in the transmission windows of the sample materials. For instance, the silicon transmission window (30–99 eV) corresponds to a coherence length of about 12 nm assuming a rectangular spectrum and an absorption length of about 200 nm, thus suggesting applications for semiconductor inspection. In the water window at 280–530 eV as defined by the K absorption edges of carbon and oxygen, respectively, a coherence length as short as 3 nm can be achieved and highlights possible applications of XCT for life sciences.

## 2 Theory of XUV coherence tomography

*k*=2

*π*/

*λ*and the temporal domain frequency

*ω*a spectral component of the electric field of the light source can be written as Assuming a time-integrating detector at the output of an interferometer setup with an ideal achromatic beamsplitter with equal splitting ratio, the detector signal is given by the temporally averaged sum of all partial field components where

*z*

_{ R }is a delay adjusted by moving the reference arm,

*R*

_{ R }the wavelength-independent intensity reflectivity of the reference mirror, and

*R*

_{ Sn }and

*z*

_{ Sn }the wavelength-independent intensity reflectivity and depth of the

*n*th layer of the sample structured discretely for purposes of proof-of principle [5].

*k*

_{0}and width

*Δk*), this integration leads to the detector signal, which depends only on the reference arm length. Equation (5) describes a typical time-domain OCT signal as shown in Fig. 1. Only the second term depends on

*z*

_{ R }. The other terms produce a constant offset \(\mathcal{I}_{\text{DC}}\). For an observation of a modulation of the detector signal a difference of the optical path length between the reference arm and the sample arm smaller than the coherence length

*l*

_{ c }is required; see (1). The measured detector signal as a function of the delay gives the reflectivity distribution of the sample broadened by the coherence function

*Γ*that is linked to the spectral intensity \(\mathcal {S}(k)\) via the Wiener–Chinchin theorem. Thus, the coherence function is directly connected to the axial resolution. Another well-known OCT setup is the so-called Fourier-Domain OCT. Here, for a fixed reference arm length, the entire spectral dependence of the spectral interferogram (4) is measured with a spectrometer at the detector position. This procedure has the particular advantage that the entire information is obtained with a single measurement rather than performing mechanical depth scans necessary in time-domain OCT. The information about the sample structure can be extracted from a Fourier transform. Using (6), it can be seen that a Fourier transform of the spectral interferogram (4) that depends on \(k=2\pi n(\omega) /\lambda _{\text{vacuum}}\), has an intensity distribution depending on the depth only. In fact, this expression contains the entire information about the sample reflectivity and thus its structure. The discrete

*δ*-structure is convoluted with the coherence function

*Γ*(

*z*) that limits the resolution in a similar way as in time-domain OCT. The first line of (7) contains the DC-components of the signal, the second line cross-correlation terms, and the third line’s terms are called autocorrelation terms. The cross and autocorrelation terms each contain the information about the sample’s structure.

In most cases of Michelson-type OCT, the reflectivity of the reference mirror is much higher than the reflectivities of the sample layers. Therefore, the amplitudes of the autocorrelation terms are weak as compared to the cross-correlation terms. Hence, the autocorrelation terms are usually not evaluated and only the cross-correlation components are used to reconstruct the depth structure.

Unfortunately, the wave number *k*=*nω*/*c* cannot be calculated without uncertainties, due to the unknown refractive index *n*(*ω*,*z* _{ S }) of the sample. In the simplest case, one could approximate the sample as an ideal non-dispersive object and calculate the Fourier transform. However, accuracy and precision will decrease, if the media is dispersive. Therefore, the absolute values of the depths of the buried layers as computed by the Fourier frequencies have an uncertainty due to the inaccurate assumptions about the refractive index and the optical path lengths in the sample, respectively. In order to reduce these inaccuracies, we assume a dispersion given by the material that is dominant in the sample. In practice, the dominant substance is known for most nano-scale samples of interest. For the present case of XCT, knowledge of the dominant substance is a prerequisite for determining the transparency window anyhow.

## 3 Simulated reflected signals of a layer structure based on a matrix method algorithm

In order to test the ideas outlined above, a simulation of the reflectivity of a multilayer structure is essential. For this purpose, we developed a computational model based on the matrix method algorithm [9] and using the refractive index database from [4]. Molybdenum (Mo) and Lanthanum (La) layers buried under a silicon layer were investigated as a sample for proof-of-principle studies in the silicon transmission window. These materials have a well-pronounced absorption contrast with respect to the silicon substrate. Further, layer combinations of different materials, e.g., B_{4}C and SiO_{2} were investigated for the water window. The simulations of the sample reflectivity in both spectral ranges show strong modulations in the reflected spectrum, as predicted by the theory of coherence tomography.

_{4}C layer. As expected, the resolution is on the order of 3 nm; see Fig. 6. The reflectivity in the SXR range is much smaller as compared to the XUV regime used for the silicon window. In order to measure such a signal in a tolerable time, a source with a sufficient photon flux is required.

Let us perform a rough estimation to show that such an experiment is indeed possible: The simulations show a reflectivity of the used materials of about 10^{−2} in the XUV range and about 10^{−5} in the SXR range. The efficiency of a typical XUV spectrometer consisting of a toroidal mirror and a gold transmission grating [10] or a reflection grating [11] depends strongly on the photon energy and is on the order of 10^{−2} at 50 eV. A typical XUV CCD camera needs about 10^{4}–10^{5} photons for saturation. With a synchrotron undulator source, photon fluxes of about 10^{12} photons/s at 0.1% bandwidth are possible. This means that exposure times of a few microseconds in the XUV regime and some seconds in the SXR are required, proving the feasibility of the scheme. A proof-of-principle experiment was performed at synchrotron radiation sources at DESY (Deutsches Elektronen-Synchrotron) and BESSY (Berliner Elektronenspeicherring-Gesellschaft für Synchrotronstrahlung), which produce the required photon flux. The broad spectrum could be achieved by changing the undulator gap over time. The results will be published soon.

Furthermore, the broadband XUV radiation from laser plasmas could be a suitable source. Using a calibrated spectrometer, we observed photon fluxes at the entrance of the spectrometer of about 10^{11} photons per laser shot at 0.1% bandwidth [12] in experiments irradiating solid density targets with intensities of approximately 10^{19} W/cm^{2}. The corresponding spectrum is very broad and smooth and the source dimension is small, i.e., very suitable for XCT.

## 4 Conclusion

We have presented calculations for a proof-of-principle experiment of optical coherence tomography using broad bandwidth XUV and SXR radiation (XCT). Evidence has been given that synchrotron and laser plasma XUV and soft X-ray sources offer sufficient flux for XCT. This strongly suggests its application as a new non-invasive tomographic method to investigate nanometer-scale structures of layered systems and simple three-dimensional samples by lateral raster scanning. We emphasize in particular the opportunities offered by the silicon and water spectral windows. The former is highly relevant for a new non-destructive method of imaging semiconductor devices. The water window, on the other hand, is known for its significance in the life sciences. The remarkable contrast of carbonic and oxygenic materials in the SXR range promises high-quality images of biological samples. The axial resolution of the imaged structures is only affected by the dynamic range of the spectrometer’s detector and the width and wavelength of the used transmission.

## Notes

### Acknowledgements

This work was partially supported by Deutsche Forschungsgemeinschaft (project SFB/TR 18) and the German Federal Ministry for Education (BMBF) (project FSP 301-FLASH). C.R. acknowledges support from the Carl Zeiss Stiftung.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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