### 1. INTRODUCTION

### 2. EXPERIMENTAL PROCEDURES

*n*= 1 –

*δ*+

*iβ*, which consists of the real part

*δ*related to a nuclear potential and the imaginary part

*β*related to absorption and incoherent scattering. In the case of a magnetic material, the neutron interaction is additionally affected by the magnetic field of the material, and the refractive index of neutron becomes

*n*= 1 -

*δ*-

_{nuc}*δ*+

_{mag}*iβ*, where a nuclear contribution

*δ*and a magnetic contribution

_{nuc}*δ*, in the interaction of the magnetic material [28].

_{mag}^{3}) which are laser-irradiated in distances of 5 mm and 2 mm perpendicularly to the direction of domain walls, was observed by the neutron grating interferometer.

*G*

_{0}), a phase grating (

*G*

_{1}), and an analyzer grating (

*G*

_{2}), and they have a line structure of micrometer scale. The

*G*

_{0}has a neutron absorbing material such as a gadolinium (Gd) or gadolinium oxysulfide (Gadox). It is positioned in front of the neutron source and generates several neutron beams to achieve a spatial coherence for the initial condition of interference. The

*G*

_{1}has a neutron phase shifting material such as silicon. It induces a phase modulation of the neutron beam and creates the interference pattern, called a Talbot pattern, which is generated behind of

*G*

_{1}. The Talbot pattern is an intensity structure along the beam path, and it includes a periodic intensity structure, called a self-image, as to the detector plane at the Talbot distance. The

*G*

_{2}, another neutron absorbing grating, located at the Talbot distance in front of the detector creates a sufficiently wide periodic intensity structure, called a Moiré pattern, to resolve the self-image of several micrometers, allowing to be analyzed by the detector pixel of tens of micrometers.

*m,n*) along phase steps

*x*is expressed in a Fourier series by [28]

_{g}*A*is the amplitude coefficient,

_{i}*Φ*is the phase coefficient,

_{i}*k*= 2π

*p*is the wave number,

_{i}*p*is the period of

_{i}*G*

_{1}, and

*x*is the fractional interval of

_{g}*p*(i.e. the size of phase step). The sinusoidal intensity oscillation is characterized by coefficients of the offset

_{i}*A*

_{0}, the amplitude

*A*

_{1}, and the phase

*Φ*

_{1}, which can be extracted by Fourier analysis.

*V*is the visibility, and

*A*is the extracted Fourier coefficients of

_{i}*i*amplitude coefficient in Eq. (1). Here, we denote indices in Eq. (2) for without sample (reference) and with sample by superscripts

^{th}*r*and

*s*respectively. The visibility is the intensity modulation of the Moiré fringe, which is the ratio of 0

^{th}and 1

^{st}amplitude coefficients, and the DFI is the ratio of visibility without sample (

*V*) and with sample (V

^{r}^{s}).

^{st}wavelet filtering as show in Fig 2 (a). The wavelet filtering separates the structural information of the image in the low frequency, horizontal, vertical and diagonal detail bands corresponding to L1, H1, V1, and D1 respectively as shown in Fig 2 (b). In the experiment, the stripe artifact of the amplifiers is perpendicular to the domain wall, so the information of those structures is decomposed into vertical and horizontal detail bands respectively. Subsequently, the structural information of the stripe artifact is processed by Fourier filtering the frequency region for amplifier arrays, indicated by the vertical yellow line box in H1 in Fig 2 (c), by multiplication of a Gaussian window function. The former filtering processes are additionally conducted for the low frequency detail band of lower classes of wavelet filtering to obtain further improved image result within the final image that is not distorted. Figure 2 shows the combined wavelet-Fourier filtering of decomposition levels of two. The filtering parameters are the specific type of wavelet filtering, the standard deviation of Gaussian function, and the number of class levels which is the repetition of combined wavelet-Fourier filtering. In this work, the type of wavelet filtering was set Daubechies Level 1, the standard deviation of Gaussian function was 0.2, and the decomposition level was repeated 5 times.

*G*

_{1}produces the phase shift of π at the effective wavelength of 4.4 Å. The detector was an Andor sCMOS camera [31] with Nikon lens of 105 mm [31] with reproduction ratio of 1, and the image array was 2560 × 2160 pixels with effective pixel pitch of 6.5 μm. The exposure time for each image was 180 s, and 15 images were merged by median filter to reduce non-statistical noise. The scintillator was a Gadox screen of 20 μm in height. The phase stepping consisted of 8 steps of

*G*

_{0}driven in direction of

*x*-axis and the electrical steel sheets were positioned 15 mm from

*G*

_{1}.