Use of Kramers&#x02013;Kronig relation in phase retrieval calculation in X-ray spectro-ptychography

Makoto Hirose; Kei Shimomura; Nicolas Burdet; Yukio Takahashi

doi:10.1364/OE.25.008593

1. Introduction

Coherent diffraction imaging (CDI) [1, 2] is a method for reconstructing the complex-valued images of objects from the diffraction intensities by using iterative phasing methods. CDI provides information on both the phase shift and the absorption of incident beams due to the interaction with electrons in objects. It is therefore considered that CDI substitutes computational image reconstruction for an image-forming lens. In particular, CDI using X-rays, which was first demonstrated by Miao et al. [3], is a promising tool for the high-resolution observation of specimens beyond the resolution obtained by conventional X-ray microscopy with lenses. In the phasing method, the correct phase is associated with the diffraction pattern by iteratively enforcing consistency with the measurement using two sets of constraints in real and reciprocal space. Convergence of the phase retrieval calculation is a crucial aspect in both the reliability and practical use of CDI.

The original concept of CDI employs a plane-wave geometry, in which the support plays an important role in the real-space constraint in the phasing method [4–6]. Nowadays, improved methods [7–11] are used in the image reconstruction from diffraction patterns of various specimens. Plane-wave CDI, however, has a significant limitation in that the sample must be spatially isolated. In addition, it is difficult to reconstruct a complex-valued image without using an optimal support [12, 13]. Therefore, isolated weak-phase objects, such as cells [14] or nanoparticles [15], have often been selected as samples in the hard X-ray regime. A scanning type of CDI, also referred to as ptychography, is a breakthrough that overcomes these limitations and difficulties, in which an extended specimen is scanned by an overlapping incident beam with a known separation, the diffraction pattern is observed at each sample position, and then the complex transmission function of the object is reconstructed by phasing methods based on ptychographic iterative phase retrieval algorithms [16, 17]. In ptychography, translational diversity due to an overlapping incident beam on the sample enables us to reliably reconstruct a complex-valued image [18, 19]. Experimental difficulties encountered in the initially demonstrated ptychography have been substantially alleviated by the development of various algorithms that, for example, reconstruct of the incident illumination field as well as the sample image [20, 21], correct the positional errors of sample [22, 23], and perform the mixed-state reconstruction of the probe and/or sample [24]. X-ray ptychography is now one of the most promising methods of CDI using X-rays for observing a variety of samples in biology and materials science.

X-ray ptychography can also provide us with chemical information of a sample by using an X-ray absorption edge, i.e., multiple-energy ptychographic diffraction data are collected in the vicinity of the absorption edge, and then element-specific images are reconstructed, which is often referred to as X-ray spectro-ptychography. So far, the visualization of Au in Au/Ag nanoboxes [25], the identification of the oxidation state of gold nanoparticles [26], the discrimination between PMMA and SiO₂ beads [27], the classification of CoFe₂O₄ nanoparticles buried in a mouse fibroblast cell [28], and the chemical mapping of Li_xFePO₄ nanoplates [29,30] have been reported. An ultimate form of X-ray spectro-ptychography is to reconstruct X-ray absorption fine structure (XAFS) at the nanoscale. Recently, the XAFS of Fe₃O₄ nanoparticles [31] has been reconstructed by soft X-ray spectro-ptychography. Extending this approach to the hard X-ray region will enable us to visualize the chemical state of nanostructures buried within thick samples. However, a limitation of this method is the weak absorption of incident X-rays in the hard X-ray region.

To improve the convergence of the phase retrieval for complex-valued images in X-ray spectro-ptychography, we propose the addition of a constraint based on the Kramers–Kronig relation(KKR) [32] to phase retrieval algorithms. First, we perform a numerical simulation to evaluate its usefulness, and then we apply it to the reconstruction of near-edge XAFS spectra at the Mn K edge of a Mn₂O₃ film at SPring-8.

2. Principle of constraint using Kramers–Kronig relation

X-ray ptychography reconstructs complex-valued images of an object T (r, E) and probe P (r, E) as a function of a real-space two-dimensional vector r and the incident X-ray energy E. The sample is modeled by the multiplicative complex transmission function T (r, E) = exp [2πi (n (r, E) − 1) t (r)/λ], where λ is the wavelength, n (r, E) is the complex refractive index described as 1 − δ (r, E) + iβ (r, E), and t (r) is the local sample thickness. The amplitude and phase of T (r, E) are written as

| T (r, E) | = \exp (- \frac{2 π}{λ} β (r, E) t (r)),

\arg (T (r, E)) = - \frac{2 π}{λ} δ (r, E) t (r) .

Here, δ and β are respectively expressed as the real and imaginary parts of the anomalous term of the atomic scattering factors f′ and f″ as follows:

δ = \frac{r_{e}}{2 π} λ^{2} \sum_{i} n_{i} (Z_{i} + {f^{'}}_{i}),

β = - \frac{r_{e}}{2 π} λ^{2} \sum_{i} n_{i} {f^{″}}_{i},

where r_e is the Thomson scattering length, n_i is the atom number in the unit cell of the ith atom, and Z_i is the atomic number of the ith atom. Here, the f″ spectrum is related to the f′ spectrum in the KKR as

f^{'} (ω) = \frac{2}{π} P \int_{0}^{\infty} \frac{ω^{'} f^{″} (ω^{'})}{{ω^{'}}^{2} - ω^{2}} d ω^{'},

where

P \int_{0}^{\infty}

is the Cauchy principal value and ω is the angular frequency. Using Eqs.(1)–(5), the following equation is derived.

- \frac{2}{π} P \int_{0}^{\infty} \frac{E^{'} {\ln | T (r, E) | / λ}}{{E^{'}}^{2} - E^{2}} d E^{'} - \frac{\arg (T (r, E))}{λ} = r_{e} t (r) \sum_{i} n_{i} (r) Z_{i} = c o n s t .

Note that this equation is used as a constraint in phase retrieval algorithms for reconstructing complex-valued images from a multiple-energy ptychographic data set. In addition, this constraint is applicable even when the sample consists of several elements. This constraint is referred to as the KKR constraint in this paper.

The KKR constraint can be easily installed using well-established ptychographical iterative phase retrieval algorithms [20, 21]. Here, it is assumed that K energies around an absorption edge are used in the ptychography measurement. The coherent diffraction intensity I_j (q, E_k) at the jth scanning position and the kth X-ray energy is described as

I_{j} (q, E_{k}) = {| ℱ ψ_{j} (r, E_{k}) |}^{2},

where ψ_j (r, E_k) is the wavefront exiting from the sample surface and

ℱ

represents the Fourier transform operator. ψ_j (r, E_k) is expressed as follows:

ψ_{j} (r, E_{k}) = T (r - r_{j}, E_{k}) \times P (r, E_{k}),

where r_j is the lateral displacement.

The phase recovery starts from initial guesses of object functions T₀ (r, E₁),⋯, T₀ (r, E_K) and probe functions P₀ (r, E₁),⋯, P₀ (r, E_K). At the jth position and nth iteration, the current estimates of exit waves ψ_j,n (r, E₁),⋯, ψ_j,n (r, E_K) are propagated to the detector plane. Then, the amplitudes are replaced with the square root of the measured diffraction intensities $\sqrt{I_{j} (q, E_{1})}, \dots, \sqrt{I_{j} (q, E_{K})}$ , to enforce consistency with the measurement. The updated functions are back-propagated to the sample plane where the current guesses of the object and probe are updated using a real-space constraint that relies on the overlap of the illumination field on the extended sample. After imposing two sets of constraints at all scanning positions, the KKR constraint is applied to each pixel of the series of updated object functions T_n₊₁ (r, E₁),⋯, T_n₊₁ (r, E_K). Figure 1 shows a flowchart of the KKR constraint at the nth iteration. The differences (i.e., c₁ (r_i),⋯, c_K (r_i)) between ln (|T (r, E₁)|)/λ,⋯, ln (|T (r, E_K)|)/λ and arg (T (r, E₁)) / λ,⋯, arg (T (r, E_K))/λ are calculated at each pixel at all X-ray energies, and then their average values (i.e., c (r_i)) are calculated at each pixel. By this averaging process, the complex-valued object images are rotated to a common origin. After that, the phase values arg (T (r, E₁))/λ,⋯, arg (T (r, E_K))/λ are updated using c (r_i). Once the KKR constraint has been applied to all pixels of the current estimates of the object functions, the iteration returns to the conventional procedure of the phase retrieval calculation mentioned above.

Fig. 1 Flowchart of the KKR constraint at the nth iteration in ptychographical iterative phase retrieval algorithms.

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3. Simulation

The performance of the KKR constraint was evaluated by numerical simulation. The optical parameters were selected to simulate our recent experiments at SPring-8. The sample is the “Mandrill” image. It is composed of the element Mn and its thickness varies from 0 to 1 μm. There is a vacuum area for the amplitude reference in the center of the image.

Twenty incident X-ray energies were chosen between 6.527 keV and 6.627 keV, around the Mn K absorption edge of 6.539 keV. The energy step was 2 to 12 eV, the smallest gap being in the vicinity of the absorption edge. The incident X-rays were two-dimensionally focused to a 500 nm (FWHM) spot size. Ptychographic coherent diffraction patterns in a 7 × 7 overlapping field of view with a 400 nm step width were calculated at each X-ray energy. In the phase recovery, all initial guesses of the object functions were constant values and the probe functions were blurred images of a sinc function with a constant phase. To prevent the reconstruction of object images with amplitude offsets, the vacuum area in the object was used as the reference, where the complex transmission function was unity, i.e., |T| = 1, at all X-ray energies. Complex-valued images of both the sample and probe were reconstructed using an extended ptychographical iterative engine (ePIE) [21] with and without the KKR constraint. The iterative process for the reconstruction was continued for up to 7.5 × 10³ iterations. The KKR constraint was applied every 100 iterations, not each iteration. The low-frequency application of the KKR constraint suppress the degradation of the quantitativeness due to the finite energy range in Kramers-Kronig inversions.

Figures 2(a) and 2(b) show the reconstructed amplitude and phase images, respectively, without and with using the KKR constraint. The convergence was evaluated using the root mean square (RMS) error between the original sample image and reconstructed object image as follows:

E_{e r r o r} (n) = \frac{1}{K} \sum_{k = 1}^{k = K} \frac{Σ_{r} {| T (r, E_{k}) - γ (E_{k}) T_{n} (r, E_{k}) |}^{2}}{Σ_{r} {| T_{n} (r, E_{k}) |}^{2}},

where

γ (E_{k}) = \frac{Σ_{r} T (r, E_{k}) \cdot T_{n}^{*} (r, E_{k})}{Σ_{r} {| T_{n} (r, E_{k}) |}^{2}}

and * denotes the complex conjugate. Figure 2(c) shows the iteration number dependence of the RMS error of the object image without and with using the KKR constraint. The use of the KKR constraint improves the convergence. Figure 2(d) shows the original and reconstructed near-edge XAFS spectra without and with using the KKR constraint. KKR constraint provides us with more precise XAFS spectra.

Fig. 2 (a) Reconstructed amplitude image without (left) and with (right) the KKR constraint. (b) Reconstructed phase image without (left) and with (right) the KKR constraint. The scale bar is 1 μm and X-ray energy is 6.557keV. (c) Iteration number dependence of the RMS error of the object image with and without using the KKR constraint. (d) Original (top) and reconstructed near-edge XAFS spectra without (middle) and with (bottom) using the KKR constraint. The spectra were extracted from the 40 × 40 nm² region (2 × 2 pixels) indicated by the arrows in (a) and (b).

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4. Experiment

To perform a proof-of-principle experiment on the reconstruction of the XAFS by X-ray spectro-ptychography in the hard X-ray region, a nanostructured object composed of Mn₂O₃ was fabricated. A 600-nm-thick Mn₂O₃ layer was deposited on a 500-nm-thick Si₃N₄ membrane chip, and then both a pattern comprising the characters of “SPring-8” and a square hole were fabricated using a focused ion beam. Figure 3(a) shows a field-emission scanning electron microscopy (FE-SEM) image of the sample. Ptychographic diffraction data of the sample were collected at BL29XUL in SPring-8. Figure 3(b) shows a schematic drawing of the experimental setup. A monochromatic X-ray beam was generated by an in-vacuum undulator device and a Si(111) double-crystal monochromator. Fourteen X-ray energies were selected between 6.530 keV and 6.588 keV, around the K absorption edge of Mn. The incident X-rays were two-dimensionally focused to a ~500 nm (FWHM) spot size using a pair of Kirkpatrick-Baez (KB) mirrors. To produce a sufficient spatial coherence length on the sample plane, a secondary slit with an 8 (horizontal) × 17 (vertical) μm² opening was placed ∼48 m upstream of the KB mirrors. The sample was positioned at the focal plane and mounted on piezoelectric stages inside a high-vacuum chamber. A diffraction dataset was collected at each position using an in-vacuum front-illuminated CCD detector (Princeton Instruments PyLoN 1300) with a pixel size of 20 × 20 μm², placed 1219 mm downstream of the sample position. To measure the bright field and increase the effective dynamic range of the diffraction intensity, 10-μm-thick Ta with a size of 640 × 640 μm² was installed in front of the CCD detector as a semitransparent central stop [33]. The exposure time at each position was 2.8 s and it took ~1 h to collect 49 patterns at each X-ray energy, which includes the readout time from the detector and the time required for position correction. The illumination position was corrected at each scanning column by the drift compensation method [34]. This method allows us to radiate the focused X-ray beam more accurately than 10 nm which was smaller than the pixel size of the reconstructed image. The fluctuation of the incident X-ray energies was much less than 1 eV which was smaller than the minimum step of the present experiment. Under these experimental conditions, the dynamic range of the diffraction intensity was 1.0 photon/pixel: 1.9 × 10⁶ photons/pixel.

Fig. 3 (a) FE-SEM image of the test sample. The scale bar is 1 μm. (b) Experimental setup at BL29XUL in SPring-8. (c) Coherent diffraction pattern of the sample at 6.554 keV.

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Both the sample image and the probe were reconstructed by using ePIE with and without the KKR constraint. The initial inputs of the object function were constant values and the initial input of the probe was estimated by considering our experimental parameters. Figures 4(a) and 4(b) show the reconstructed amplitude and phase images at 6.542 keV and 6.554 keV, respectively, after 2×10³ iterations using the KKR constraint every 100 iterations. The reconstructions were in good agreement with the FE-SEM image of the sample. The amplitude image showed better contrast at 6.554 keV, which is above the Mn K absorption edge. The spatial resolution of the images was evaluated using the phase retrieval transfer function (PRTF) as shown in Fig. 4(c). The spatial resolution was determined to be 41.1 nm from the intersection with the 1/e threshold, corresponding to the size of two pixels. Figure 4(d) shows the energy dependence of the full period spatial resolution determined using PRTF. Recently, the improvement of the resolution at the absorption edge has been reported in soft X-ray realm [35]. In the present study, the significant improvement of the resolution was not observed due to the weak resonant scattering in hard X-ray realm.

Fig. 4 (a-b) Amplitude (left) and phase (right) images reconstructed from the ptychographic diffraction patterns at (a) 6.542 keV and (b) 6.554 keV. The scale bar is 1 μm. (c) PRTF curve of the image reconstructed at 6.564 keV. (d) Energy dependence of the full period spatial resolution determined using PRTF.

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Next, near-edge XAFS spectra were derived from the absorption images by calculating μt = −2 ln (|T|). Figures 5(b) and 5(c) show four near-edge XAFS and phase spectra without and with using the KKR constraint, respectively, which were extracted from the 40 × 40 nm² regions indicated by the arrows in Fig. 5(a). The top spectrum in Fig. 5(b) is the reference spectrum measured in the transmittance mode. Without using the KKR constraint, a strong subpeak appears in addition to the main peak. On the other hand, using the KKR constraint, the intensity of the subpeak decreases and the spectra become closer to the reference spectrum. The RMS error between the reference spectrum and average of the spectra obtained by X-ray spectro-ptychography was improved from 0.232 to 0.178 by using the KKR constraint. It is clear that the KKR constraint works well and provides us with more precise XAFS spectra.

Fig. 5 (a) Amplitude image reconstructed at 6.554 keV. The scale bar is 1 μm. (b) Near-edge XAFS spectrum of the sample measured in transmittance geometry using a focused X-ray beam (top). Near-edge XAFS spectra reconstructed by X-ray ptychography without (left) and with (right) using the KKR constraint. The spectra were extracted from the 40 × 40 nm² regions (2 × 2 pixels) indicated by the arrows in (a). (c) Near-edge phase spectra at the same positions without (left) and with (right) using the KKR constraint.

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5. Summary and conclusions

In this paper, to improve the convergence of image reconstruction in X-ray spectro-ptychography, we have proposed an iterative phase retrieval algorithm using a novel constraint based on the KKR originating from resonant X-ray scattering around the absorption edge. To assess the effectiveness of this constraint, we investigated its feasibility by numerical simulation. More precise images were obtained by using the KKR constraint as well as faster convergence. Then, we performed a proof-of-principle experiment on the reconstruction of the near-edge XAFS spectra by X-ray ptychography at SPring-8. The near-edge XAFS spectra at the Mn K edge of Mn₂O₃ films were successfully reconstructed with a spatial resolution of ~42 nm by using the KKR constraint.

When the range of the X-ray energy in X-ray spectro-ptychography is extended to much higher regions, an extended X-ray absorption fine structure (EXAFS) appears in the absorption images. The EXAFS enables us to directly determine the local atomic structure around a specific element. The signal of the EXAFS is extremely small compared with that of the near-edge XAFS, and the present reconstruction algorithm with the KKR constraint would be useful for reconstructing the EXAFS signal. In the near future, we believe that the present approach will provide a novel way to understand the relationship between the local atomic structure and the chemical state in unprecedentedly small amounts of materials.

Funding

Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant Nos. JP25709057, JP16K13725, and JP16H00889), the X-ray Free Electron Laser Priority Strategy Program of Ministry of Education, Culture, Sports, Science and Technology (MEXT), and SENTAN of Japan Science and Technology Agency (JST).

Acknowledgments

We thank M. Tada, T. Ishikawa, and K. Yamauchi for many stimulating discussions, as well as Y. Kohmura and A. Suzuki for help in the synchrotron experiment.

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Use of Kramers–Kronig relation in phase retrieval calculation in X-ray spectro-ptychography

Abstract

1. Introduction

2. Principle of constraint using Kramers–Kronig relation

3. Simulation

4. Experiment

5. Summary and conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (5)

Equations (9)

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