Abstract
Scientists have continually tried to improve the spatial resolution of imaging ever since the invention of the optical microscope in around 1610 by Galileo1. Recently, a spatial resolution near λ/10 was achieved in a near-field scheme by using surface plasmon polaritons2,3. However, further improvement in this direction is hindered by the size of metallic nanostructures2. Here we show that atom-scale resolution is achievable in the extreme-ultraviolet region by using X-ray parametric down-conversion, which detaches the achievable resolution from the wavelength of the probe light. We visualize three-dimensionally the local optical response of diamond at wavelengths between 103 and 206 Å with a resolution as fine as 0.54 Å. This corresponds to a resolution from λ/190 to λ/380, an order of magnitude better than ever achieved. Although the present study focuses on the relatively high-energy optical regions, our method could be extended into the visible region using advanced X-ray sources4,5,6,7, and would open a new window into the optical properties of solids.
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The optical response is recognized as a powerful tool to investigate materials and as a useful property for widespread application in science and industry. In spite of its importance, our knowledge of the optical property is quite limited. At present, we can only measure the macroscopic optical response, and cannot see how electrons in materials respond to the light due to the limited spatial resolution. In other words, the charge response can be investigated only at the Γ point, the origin in the momentum space8. Such a situation contrasts with the case of the magnetic response, which is measurable over the whole Brillouin zone9. For example, inelastic neutron scattering indicates fluctuating stripes in high-temperature superconductors10,11. Microscopic structures, such as stripes and orbital order, are observed commonly in so-called strongly correlated electron systems10,12. If the optical probe had the atomic resolution to unveil the charge response with large momentum transfer, it could give direct and clear evidence about the charge dynamics of microscopic structures for deeper understanding of the physical properties11.
Our basic idea to realize super-resolution is that the linear optical susceptibility, χ(1)(r), is incorporated in the second-order X-ray nonlinear susceptibility, χ(2)(r). Here, χ(1)(r) has a microscopic structure on the atomic scale that determines the local optical response. We consider one of the second-order nonlinear processes13, X-ray parametric down-conversion (PDC) into the optical region, where an X-ray pump photon (labelled as p) decays spontaneously into an X-ray signal photon (s) and an optical idler photon (i). The origin of nonlinearity is considered to be the Doppler shift14, where the induced charge at the idler frequency scatters X-rays at a different frequency (Fig. 1a). We expect that this nonlinear process reveals the structure of induced charge, similar to X-ray structural analysis. Our picture is confirmed by calculation on the basis of early works14,15 as the following relation for isotropic systems:
Here, χQ(1)(ωi) and χQ(2)(ωp=ωs+ωi) are the Qth Fourier coefficients of χ(1)(r,ωi) and χ(2)(r,ωp=ωs+ωi), respectively, Qis the reciprocal lattice vector, θpsi is the polarization factor and the other symbols have their ordinary meanings. We note that our interpretation is more general than the early works, which take into account specific electronic states, such as bond charge14 or valence charge16, and ignore the core charge. We treat χQ(2)(ωp=ωs+ωi) instead of χ(2)(r,ωp=ωs+ωi) itself, because we observe X-ray PDC as nonlinear diffraction (Fig. 1b) and determine χQ(2)(ωp=ωs+ωi) experimentally17,18. The structure of local optical response to the idler light is to be reconstructed by the Fourier synthesis: . Now the diffraction limit is imposed at the X-ray pump wavelength, despite investigating the optical response at the idler frequency.
Figure 1c shows a typical set of rocking curves of the nonlinear diffraction measured with synthetic type IIa diamond19. The photon energies are ℏωi=100 eV, ℏωs=11.007 keV and ℏωp=11.107 keV. The rocking curve was measured with the first five Q for which the Bragg reflection is observable. Three more sets were measured for ℏωi=60, 80 and 120 eV at the same ℏωp. The estimation of χQ(1)(ωi) is not straightforward due to the characteristic asymmetric peaks of the Fano effect18,20,21,22. We determined the magnitude of χQ(2)(ωp=ωs+ωi) by analysing the Fano spectra22. Then, |χQ(2)(ωp=ωs+ωi)| was converted to |χQ(1)(ωi)| by equation (1) with the polarization factor, θpsi=sin2θB, for the present experimental set-up. Here, θB is the Bragg angle for Q at ℏωp. Note that there is no clear sign of X-ray PDC for Q=(2 2 2) (see Supplementary Information for discussion).
Now let us focus on the linear susceptibility at the idler energy (Fig. 2). To compare it at different photon energies, we normalize |χQ(1)(ωi)| to |χ0(1)(ωi)|. Here, the average linear susceptibility, χ0(1)(ωi), is calculated from the tabulated refractive index23. We corrected the local field effect on the macroscopic linear susceptibility using the Lorentz model8, although the amount of correction is a few per cent. The linear structure factor, namely the Fourier transform of charge density, of the core (valence) electrons, FQC (FQV), is plotted for comparison. We estimated FQV using the structure factor measured by X-ray powderdiffraction24 and calculated FQC with the XD2006 program25. The |Q| dependence of |χQ(1)(ωi)/χ0(1)(ωi)|shows rapid decay similar to that of |FQV|, indicating that χ(1)(r,ωi) is not localized, but extended. At the same time, we notice a longer tail of |χQ(1)(ωi)/χ0(1)(ωi)| at larger |Q|, which is due to a weak contribution from the localized core electrons. These observations indicate that the bond-charge model14 is not sufficient to describe X-PDC into the extreme-ultraviolet region. The strong ℏωi dependence of |χQ(1)(ωi)/χ0(1)(ωi)|, especially for the smaller |Q|, relates to a change in the structure of χ(1)(r,ωi).
Our key result shown in Fig. 3a–d represents the Fourier-synthesized microscopic structure of linear susceptibility, χ(1)(r,ωi), for the light with the idler energies. The resolution is estimated to be 0.61×2π/|Qmax|=0.54 Å (ref. 1), which is much shorter than the idler wavelengths (corresponding photon energies), ranging from 103 Å (120 eV) to 206 Å (60 eV). The resolution in the unit of λ reaches λ/380 for ℏωi=60 eV, which is the highest ever achieved. As shown in Fig. 3g, we succeed in reconstructing the three-dimensional structure of χ(1)(r,ωi) with a 0.54 Å resolution. It is the most interesting finding that small spherical regions around each atom respond in phase to the light, whereas disc-shaped regions, which respond in the opposite phase, have a stronger contribution and dominate the optical response. The weak structures in the outer region are artefacts of the Fourier synthesis with a limited value of χQ(1)(ωi). Note that χ(1)(r,ωi) is normalized to its negative average level, χ0(1)(ωi).
An important point to be noted in the Fourier synthesis is the phase problem due to the fact that we measured only the amplitude of χQ(1)(ωi). At present, we do not have any established procedure to recover the missing phase, and carry out an exhaustive search. We synthesize χ(1)(r,ωi) using all possible combinations of the phase, and find a structure compatible with physical pictures, such that χ(1)(r,ωi) should be small in the region where the electron density is low and the bonding sites should have a major contribution (see Supplementary Information for details). The number of combinations can be reduced greatly by the following considerations. As we investigate the optical response far from resonances, we regard χ(1)(r,ωi) as real. Then, χQ(1)(ωi) is real, because the diamond structure possesses a centre of inversion. We need to determine only the sign of χQ(1)(ωi). The symmetry of the diamond structure fixes the relative phase difference among different Q with the same magnitude, for example, χ111(1)(ωi)=−χ11−1(1)(ωi).
Before we discuss the microscopic structure of χ(1)(r,ωi), we review the macroscopic optical response of diamonds to make clear what we find. For simplicity, we regard the band structure of diamond as a three-level system, which consists of the 1s core level, the valence ‘level’ and the conduction ‘level’. The macroscopic optical response of this three-level system may be described as a sum of two Lorentz oscillators5:
Here, nc (nv) and ωc (ωv) are the number density and the resonance frequency of core (valence) electrons. We ignore the lifetime (damping factor) for simplicity, and assume ℏωc=289 eV (ref. 22) and ℏωv=12 eV (ref. 26). It is clear that the valence electron determines χ0(1)(ω) around ℏω=100 eV, because the denominator of the first term is much larger.
According to the macroscopic picture above, it may be thought that the microscopic structure of χ(1)(r,ωi) should be similar to the density distribution of valence electrons. We notice, however, a large difference between them. The valence-electron density (Fig. 3e) has considerable weight at the atomic site, which originates from the 2s orbital, whereas χ(1)(r,ωi) at the atomic site is smaller or has opposite sign to the rest (Fig. 3a–d).
The discrepancy between χ(1)(r,ωi) and the density distribution of valence electron is explained well by taking the contribution from the 1s core electrons into account. The radii of the s orbitals have a quadric dependence on the main quantum number, so the radius of the 1s orbital is four times smaller than that of the 2s orbital, which makes the density of the 1s core electrons 43 times higher (Fig. 3e,f). Such a high concentration compensates the larger denominator for the 1s core electrons in equation (2). As is clear from equation (2), the 1s core electron responds in phase to the light around 100 eV, whereas the 2s-like state respond in the opposite phase. The 1s core and the 2s-like states, which give comparable but opposite contributions to each other, nearly cancel out at the atomic site.
We make a rough estimate of χQ(1)(ωi) by replacing n in equation (2) with FQ/vc, where vc is the volume of the unit cell. The 111 structure factors for the 1s core electrons and the valence electrons are estimated to be F111C=−10.8 and F111V=−7.70, respectively24. We evaluate equation (2) at ℏω=100eV, and find χ111(1)(100 eV)=1.53×10−3 e.s.u., whereas our experimental result is (3.7±0.57)×10−3 e.s.u. The agreement between the calculation and our experimental estimation is considered to be satisfactory for the following reason. As for the average susceptibility, we can compare the calculation by equation (2) with that reported. The calculation gives χ0(1)(100 eV)=−3.66×10−3 e.s.u., whereas the reported value is −8.31×10−3 e.s.u. (ref. 23). The calculation on the basis of equation (2) gives the same orders of magnitude, and has a tendency to underestimate the susceptibility, which is attributed to the model being simpler than the real band structure.
Finally, we consider qualitatively the energy dependence of χ(1)(r,ωi). It may be thought naively that only the magnitude of χ(1)(r,ωi)changes in accordance with χ0(1)(ωi) and that the shape remains unchanged even when the energy of light changes. We find that the shape of χ(1)(r,ωi) remains basically the same; however, the contrast shows a strong energy dependence. The higher contrast at 120 eV (Fig. 3d) is considered to be due to the stronger resonance effect of 1s core electrons at higher photon energies.
In summary, we have demonstrated that we can make use of the spatial resolution of X-rays, keeping the probing wavelength in the optical region, and can visualize the local optical response to extreme-ultraviolet radiation with atomic resolution. The application of our method in a lower-energy region, such as the visible region, requires a deeper understanding of X-ray PDC and a more sophisticated procedure for the phase determination, because the optical response becomes more complicated than equation (2). We discuss briefly a straightforward application of X-ray PDC to the soft-X-ray region. Recently, soft-X-ray resonant scattering has been recognized as a powerful tool to investigate microscopically the strongly correlated system27. For example, doped holes in high-temperature superconductors are investigated directly using soft-X-ray resonant scattering at the K edge of oxygen28. However, the drawback of using soft X-rays is the limited spatial resolution, 22.8 Å for the oxygen case. Our method should remove this limitation, enlarging the potential of the soft-X-ray probe.
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Acknowledgements
We are grateful to T-H. Arima for discussions, and to M. Takata and Y. Tanaka for encouragement. This work was supported by the JST PRESTO program and in part by a grant-in-aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan (2340081).
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K.T. and T.I. designed the experiment. K.T. acquired the experimental data, and K.T., K.S. and E.N. carried out the data analysis. K.T. and K.S. wrote the manuscript. All authors discussed the results and commented on the manuscript.
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Tamasaku, K., Sawada, K., Nishibori, E. et al. Visualizing the local optical response to extreme-ultraviolet radiation with a resolution of λ/380. Nature Phys 7, 705–708 (2011). https://doi.org/10.1038/nphys2044
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DOI: https://doi.org/10.1038/nphys2044
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