# Holographic nanodiamond–polymer composite grating with unprecedented slow-neutron refractive index modulation amplitude

Elhoucine Hadden, Martin Fally, Yuko Iso, Tobias Jenke, Jürgen Klepp, Atsushi Kume, Koichi Umemoto, Yasuo Tomita

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Elhoucine Hadden, Martin Fally, Yuko Iso, Tobias Jenke, Jürgen Klepp, et al., Holographic nanodiamond–polymer composite grating with unprecedented slow-neutron refractive index modulation amplitude. Applied Physics Letters, 2024, 124 (7), pp.071901. 10.1063/5.0186753. hal-05442000

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# Holographic nanodiamond–polymer composite grating with unprecedented slow-neutron refractive index modulation amplitude

Elhoucine Hadden ✉ ; Martin Fally ✉ ; Yuko Iso; Tobias Jenke ✉ ; Jürgen Klepp ✉ ; Atsushi Kume ✉ ; Koichi Umemoto; Yasuo Tomita ✉

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Appl. Phys. Lett. 124, 071901 (2024)

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# Holographic nanodiamond-polymer composite grating with unprecedented slow-neutron refractive index modulation amplitude

Cite as: Appl. Phys. Lett. **124**, 071901 (2024); doi: 10.1063/5.0186753

Submitted: 8 November 2023 · Accepted: 26 January 2024 ·

Published Online: 14 February 2024

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Elhoucine Hadden,<sup>1,2,a)</sup> Martin Fally,<sup>3</sup> Yuko Iso,<sup>4</sup> Tobias Jenke,<sup>2</sup> Jürgen Klepp,<sup>3</sup> Atsushi Kume,<sup>5</sup> Koichi Umemoto,<sup>5</sup> and Yasuo Tomita<sup>4</sup>

## AFFILIATIONS

## ABSTRACT

We demonstrate exceptionally high slow-neutron diffraction efficiency (≈70% of transmitted intensity diffracted to the first order) from a holographic nanodiamond-polymer composite grating of only tens of micrometers thickness at the average neutron wavelength of 5.3 nm. By meticulous choice of materials for extreme refractive index modulation in a thin grating structure, we overcome typical wavelength and angular selectivity issues usually encountered when pursuing high diffraction efficiencies with a thick grating. This achievement paves the way for the implementation of flux-efficient diffractive elements, well-suited for polychromatic beams, and weakly collimated slow-neutron beams.

© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0186753

Neutron optics—and the neutron scattering and spectroscopy methods into which it has evolved—are key techniques for fundamental physics as well as condensed-matter physics.<sup>1–7</sup> The ongoing deployment and construction of powerful neutron sources<sup>8</sup> show clear necessity for the concurrent development of neutron-optical techniques, alongside well-established methods. This enables the exploration of opportunities offered by the recently established (or foreseen) neutron sources’ flux gains. Next to provision of cold neutrons (CN, with a de Broglie wavelength $\lambda_N \ge 0.4$ nm<sup>3</sup>), sources providing very cold neutrons (VCN, $\lambda_N \ge 2$ nm) are foreseen for some of these neutron facilities.<sup>9,10</sup> There is room for flux-efficient, compact, versatile, and low-cost neutron-optical devices, complementing the ones currently in use for CN and VCN. To spark development of the former, we intend to put forward holographic gratings recorded in the nanodiamond (ND)-dispersed photopolymer as highly efficient neutron-optical elements in this wavelength range. In particular, such devices are decisive elements for next-generation grating interferometers for cold and very cold neutrons. Gratings will be employed in such instruments, where they serve as beam splitters and mirrors.<sup>7</sup>

For further development of CN/VCN optical devices, holographic PMMA binder-based photopolymer gratings of several millimeters thickness were proposed and demonstrated.<sup>11</sup> While they achieved high diffraction efficiency and were used in neutron interferometers,<sup>12,13</sup> this notably thick holographic gratings exhibit also undesirable properties: very high sensitivity to both angle and wavelength. Such properties typically pose challenges in angular adjustment and result in substantial intensity loss when aiming for efficient usage of weakly collimated and polychromatic beams. They limit access to the full wavelength spectrum of beamlines. To steer clear of these drawbacks, we developed non-binder-based thinner holographic gratings recorded in nanoparticle-polymer composites (NPCs) dispersed with inorganic (SiO<sub>2</sub> and ZrO<sub>2</sub>) nanoparticles.<sup>14–19</sup> Although these holographic NPC gratings (at typical film thickness between about 50–100 µm) show good diffraction performance, thinner (of the order of tens of micrometer) holographic NPC gratings giving high diffraction efficiency were desired to overcome excessive wavelength- and angle-selectivity. To this end, we improved holographic NPC gratings further by use of nanodiamonds as nanoparticles.<sup>20,21</sup> Diamond exhibits the

<footnote n="1"><sup>1</sup>Faculty of Physics and Vienna Doctoral School in Physics, University of Vienna, 1090 Vienna, Austria</footnote>

<footnote n="2"><sup>2</sup>Institut Laue-Langevin, 71 avenue des Martyrs, CS 20156, 38042 Grenoble Cedex 9, France</footnote>

<footnote n="3"><sup>3</sup>Faculty of Physics, University of Vienna, 1090 Vienna, Austria</footnote>

<footnote n="4"><sup>4</sup>Department of Engineering Science, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan</footnote>

<footnote n="5"><sup>5</sup>Central Research Center, Daicel Corp., 1239 Shinzaike, Aboshi, Himeji, Hyogo 671-1283, Japan</footnote>

<footnote><sup>a)</sup>Author to whom correspondence should be addressed: a12049205@unet.univie.ac.at</footnote>
best combination of high coherent scattering and low incoherent scattering as well as low absorption cross sections²² for neutrons, as compared with other materials. These properties provide excellent slow-neutron refractive index modulation amplitudes (i.e., higher diffraction efficiency) than formerly used NPC materials.¹⁶⁻¹⁸ However, diffraction efficiency from previously reported holographic nanodiamond-polymer composite (NDPC) gratings of around 30 µm thickness was found to be lower than practical target values (≥50 %).²⁰ This is because the refractive index modulation amplitude remains low due to the poor mutual diffusion of ND nanoparticles and the monomer during holographic recording.

In this Letter, we describe a holographic NDPC grating of 33 µm thickness, which exhibits high diffraction efficiency and notably low angle-selectivity for a broad wavelength distribution around an average neutron wavelength of 5.3 nm. We have tackled the above-mentioned issues through careful selection of the constituents in NDPC. We have accomplished high spatial modulation of ND across a relatively large area (with a diameter nearing a couple of centimeters) by holographic means to form a strongly modulated grating structure that efficiently diffracts imperfectly collimated and polychromatic CN/VCN beams. As a result, we observe exceptionally high and—at the same time—broad diffraction-efficiency peaks, i.e., high values of refractive-index modulation amplitude with low wavelength- and angle-selectivity. These properties demonstrate optimal flux usage characteristics upon diffraction, vital for advancing the development of slow-neutron optical devices.

In NDPC film sample preparation, synthesized ND sol in a solution of methyl isobutyl kitone (MIBK) was dispersed to a blend of two acrylate monomers, single functional acrylic acid tetrahydrofurfuryl ester (THFA, Tokyo Chemical Industry Co. Ltd., purity > 98.0%), and multifunctional monomer, ethoxylated dipentaerythritol hexaacrylate (A-DPH, Shin-Nakamura Chem. Co. Ltd., purity ≥ 99.0%), respectively. The concentration of ND after evaporation of MIBK was 19.6 vol. % in 55 vol. % of THFA and 25.4 vol. % of A-DPH. The synthesis method and physico-chemical characterization of ND were described in detail in Ref. 20. A-DPH possesses ethylene oxide chains and also lowers the viscosity of the monomer blend, by which it effectively facilitates cross-linking reactions and mutual diffusion of ND and monomer during holographic recording. A radical electron-donor/acceptor photoinitiator system consisting of Rose Bengal ester (Tokyo Chemical Industry Co. Ltd., RB) and N-phenyl glycine (Tokyo Chemical Industry Co. Ltd., NPG, purity > 97.0%) was used to provide the photosensitivity in the green. They were doped at their concentrations of 2.5 and 5.6 wt. %, respectively, with respect to A-DPH. All of them were used as received from manufacturers without further purification step. Note that the present composition of the monomer and the photoinitiator system was different from that used in Ref. 20 in order to further facilitate mutual diffusion of ND and monomer and increase the concentration of dispersed ND and the photosensitivity. The mixed syrup was kept in a vial at 60 °C in a vacuum chamber for three hours to remove MIBK, followed by casting it on a spacer-loaded glass substrate. Finally, it was covered with another glass plate to be used as an NDPC film sample.

In the holographic recording experiment, we used a two-beam interference setup to record unslanted and plane-wave transmission gratings in ND-dispersed NPC films at a grating period $Λ = 0.5$ µm at equal intensities from a diode-pumped frequency-doubled Nd:YVO₄

laser operating at a wavelength of 532 nm. A low-intensity He-Ne laser beam operating at a photo-insensitive wavelength of 633 nm was employed as a readout beam to monitor the buildup dynamics of the first order diffraction efficiency $η_1(t)$ from the transmission NDPC grating until reaching a state of saturation after the monotonic buildup growth (not shown here). Holographic assembly in NPC films was confirmed by TEM, electron probe microanalyzer (EPMA), Raman imaging, and optical measurements, as shown in Refs. 19 and 20. In this state, Bragg-angle detuning of the grating was inspected, allowing to estimate the grating's physical thickness $d$ as well as possible variations of the first-order (Bragg-matched) refractive index modulation amplitude $Δn_1(z)$ along the film thickness direction (z-direction). Figure 1 shows the Bragg-angle detuning curve of the saturated light diffraction efficiency $η_{sat}(Δθ_B)$ as a function of Bragg-angle detuning $Δθ_B$ inside the NDPC film sample. From an experimental point of view, we define the diffraction efficiency of the sth diffraction order as $η_s = I_s/I_{tot}$, where $I_s$ and $I_{tot}$ are the measured intensities of the sth order and the sum of all measured diffraction orders, respectively. In Fig. 1, only the first order diffraction efficiency is shown as light diffraction occurs clearly in the Bragg regime with only two coupled waves propagating at a given position $Δθ_B$ (zero order not shown here). For diffraction measurements with lasers, statistical uncertainty for data points is, in general, very small. Therefore, no error bars are shown here. It can be observed that the data exhibit non-zero sidelobe minima, indicating that the recorded grating is nonuniform along the thickness direction.²³ Collected data were evaluated by employing the standard theories for volume holographic gratings.²³,²⁴ The solid red line is a least-squares fit of Uchida's formula that considers exponential decay of the saturated refractive-index modulation amplitude $Δn_{sat}$ along the film thickness (z-direction) for a phase grating in the form of $Δn_{sat}(z) = Δn_{sat} \exp(-z/L)$ with the 1/e decay-length $L$. From the fit, it was found that the thickness $d = (33.0 ± 1.3)$ µm, while $L = (22.2 ± 0.3)$ µm. There are two potential origins of the grating decay, which lead to sidelobe suppression as seen in Fig. 1: the first can be attributed to non-negligible linear absorption²⁵ during recording at 532 nm, with the light intensity of a beam decreasing to 1/e of its surface value at $z=0$, on propagating a distance $1/α$ (also called skin

<figure><img src="image_11.png" /><figcaption>FIG. 1. Bragg-angle detuning curve of $η_{sat}$ probed at 633 nm.</figcaption></figure>
depth) within the material, $\alpha$ being the absorption coefficient. As shown below, this is primarily induced by the relatively high doping concentrations of the RB dye.²⁶ The second origin is related to holographic scattering in NPCs,²⁷ particularly for thick NPC gratings with high values of $\Delta n_{sat}$. In this latter case, a decrease in intensity-interference fringe contrast along the thickness direction takes place during recording. Indeed, the appearance of sidelobe suppression in a Bragg-angle detuning curve due to both origins was reported in silica NPC.²⁶ In our case, a decrease in the recording intensity is significant during propagation in the NDPC film and linear absorption seems predominant. In fact, the skin depth $1/\alpha = 31.1 \ \mu m$ (measured before recording, at 532 nm) is comparable to $d = (33.0 \pm 1.3) \ \mu m$. Furthermore, grating buildup mostly exhibited monotonic growth before saturation. As a result, the grating buildup time along the thickness direction (at $z > 0$) gets longer than that at $z=0$ as the grating buildup time is semi-linearly proportional to the recording intensity.²⁸ Therefore, $\Delta n_{sat}(z)$ tends to be lowered as $z$ becomes larger, creating a decayed grating along the thickness direction. Compared to previously published results,²¹ obtained from a sample composition that incorporates an additional photoinitiator along with A-DPH, the uniformity of the grating is considerably improved. The much less lifted sidelobe minima of the Bragg angle detuning curve (shown in Fig. 1) clearly demonstrate this improvement.

It was also estimated from the fitting procedure that $\Delta n_{sat}(z=0)$ was approximately $1.23 \times 10^{-2}$ and its mean value (calculated by averaging over the grating thickness) $\overline{\Delta n_{sat}} \approx 6.4 \times 10^{-3}$. When $\overline{\Delta n_{sat}}$ is compared to that of our previous holographic NDPC grating $(2.2 \times 10^{-3}$ at $\Lambda = 0.5 \ \mu m$),²⁰ an increase by a factor of 3 is noted. This improvement is achieved, thanks to enhanced mutual diffusion during the recording process. This is confirmed by examination of the spatial modulation amplitude of the ND's volume fraction $\Delta f$ formed by holographic recording.²⁰ For the first-order Bragg diffraction at 633 nm, it is $a_1 \Delta f = \overline{\Delta n_{sat}}/|n_{NP} - n_P|$, where $a_1$ is related to the first-order Fourier component of a periodically modulated refractive index and $n_{NP}(n_P)$ is the refractive index of ND (the formed host polymer), respectively.²⁰ We find that $a_1 \Delta f$ is 0.015 for the present holographic NDPC grating. This value is approximately three times larger than the previously reported one.

Neutron optics is theoretically based upon the one-particle Schrödinger equation (see, for instance, Refs. 1, 3, 5, and 7) which contains the optical potential $V_N$ describing the interaction of neutrons with the nuclei of a given material. The neutron-refractive index of the (mono-isotopic and spin independent, for simplicity) material is given by $n_0 = \sqrt{1 - V_N/E_0} \approx 1 - \lambda_N^2 b_c \rho/(2\pi)$. Here, $b_c$ is the bound coherent scattering length of the particular isotope, $\rho$ is the atomic number-density, and $E_0$ is the energy of the incident free neutron. A perfectly sinusoidal diffraction grating for neutrons can then be characterized by the refractive-index function $n(x) = n_0 + \Delta n_{sat,N}(z) \cos(2\pi x/\Lambda)$. The scattering length density $b_c \rho$ is, thus, periodically modulated with the grating period $\Lambda$ along the x-direction with amplitude $b_c \Delta \rho$ which translates to a refractive-index modulation amplitude $\Delta n_{sat,N} = \lambda_N^2 b_c \Delta \rho_1 / (2\pi)$. We set $\Delta n_{sat,N} = \Delta n_{sat,N}(z)$, since a decay of the NDPC grating along the z-direction may also be observed by neutron diffraction. It may be necessary to add more terms to the above-mentioned formula for $n(x)$ if a grating exhibits a non-sinusoidal profile shape, i.e., has a profile described by more than one Fourier component.

Neutron experiments were carried out at the beamline PF2/VCN of the Institut Laue-Langevin (ILL) in Grenoble, France.

The measurement principle is illustrated in Fig. 2. The grating was—similarly to the light diffraction measurements—mounted in transmission geometry. The diffraction efficiency $\eta$ vs angle of incidence $\theta$ of the recorded holographic NDPC grating was measured at tilt angle $\zeta = 0^\circ$. A VCN beam with an extremely broad spectrum in the range of about 2–10 nm is available at PF2/VCN.²⁹ A Ti/Ni neutron mirror is necessary to redirect the incident neutron beam to achieve sufficient collimation. The resulting wavelength distribution at the average wavelength of 5.3 nm is similar to the one depicted in Fig. 2(b) of Ref. 30. The divergence of the beam was about 2 mrad. A 2D detector with $2 \times 2 \ \text{mm}^2$ pixel size was used to detect the diffraction spots. When $\Lambda$ greatly exceeds the neutron wavelength $\lambda_N$, the Bragg angles for different diffraction orders, represented as $\theta_{B,s}$, can be approximated by $\theta_{B,s} \approx s \frac{\lambda_N}{2\Lambda}$, where $s$ takes values of 0, $\pm 1$, $\pm 2$, .... These angles are typically in a range of 10 mrad. Given the spatial resolution of the detector system, about 1.8 m distance had to be maintained between the grating and the 2D detector to be able to observe separated diffraction spots. At each angle $\theta$, the sum over all counts in each diffraction spot on the 2D-detector was calculated and the resulting intensities plugged into the above-mentioned definition for $\eta_s$.

From the angular dependence curve (Fig. 3), it can clearly be seen that neutron diffraction from our holographic NDPC grating occurs in the intermediate diffraction regime where multiple diffraction orders propagate at given $\theta$. The regime can also be inferred from the values of the so-called Klein–Cook parameter and the grating strength parameter.³² For neutrons, they can be written as $Q_N = 2\pi d \lambda_N / (\Lambda^2 \cos \zeta)$ and $\nu_N \approx \pi d \Delta n_{sat,N} / (\lambda_N \cos \zeta)$,²⁰ respectively, where we have used that $\cos \theta_B \approx 1(\theta_B = 0.3^\circ)$. For the parameter estimations corresponding to the data in Fig. 3, $Q_N \approx 4.2$ and $\nu_N \approx 0.7$. Hence, according to Ref. 32, the diffraction data should be analyzed using the rigorous coupled-wave analysis (RCWA).³³,³⁴ Since we observe only seven diffraction orders, a truncated first order multi-wave coupling theory³⁵ (taking into account seven waves and including also the decay length) instead of a full RCWA model was used to fit the neutron data. For transmission geometry at small diffraction angles and a neutron-refractive index close to unity, differences between the RCWA and the truncated seven-wave coupling theory are negligible for the relevant parameter range. To take into account the wavelength distribution of

<figure><img src="image_12.png" /><figcaption>FIG. 2. Schematic of the measurement setup for neutron diffraction by a holographic NDPC grating. Stepwise rotation through angles $\theta$ around the y axis leads to diffraction of the incident neutron beam measured by a 2D detector. The grating can be tilted by an angle $\zeta < 90^\circ$ to increase the thickness $d$ to an effective thickness $d_{eff} = d/\cos \zeta$ (see the text).</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>FIG. 3. Angular dependence of the diffraction efficiency $\eta_{\text{sat}}(\theta)$ for the holographic NDPC grating at an average neutron wavelength of about 5.3 nm (see the text) and $\zeta = 0^{\circ}$. Solid curves correspond to least squares fits of a multi-wave coupling theory to the data (see the text), taking into account the wavelength distribution by averaging. Data are made available by the ILL.$^{31}$ Error bars represent statistical uncertainties calculated by standard error propagation from Poissonian distribution of neutron counts.</figcaption></figure>

the incident beam, we numerically integrated the product of the theory with the exponentially modified-Gaussian-shaped distribution over the full wavelength range$^{30,35}$ for fitting.

From the fit, we obtained the parameter estimates to be $d_N \approx 32$ and $L_N \approx 42$ µm, where $d_N$ and $L_N$ are the grating thickness and the grating decay length, respectively, estimated by the neutron diffraction measurement. The parameter estimation for the grating thickness $d_N$ agrees well with the light diffraction result [$d = (33.0 \pm 1.3) \mu\text{m}$], which indicates that the transversal coherence properties of the neutron beam are sufficient for investigating our grating. However, the wide wavelength distribution may lead to the significant disagreement of the decay lengths $L_N$ with $L$ (as obtained from light diffraction) due to averaging over different wavelengths.

Via the multi-wave coupling analysis of the data, we were able to compute the amplitudes of the first three Fourier components of the coherent scattering length density,$^{36}$ yielding $b_c \Delta\rho_1 \approx 11.7$, $b_c \Delta\rho_2 \approx 5.2$, and $b_c \Delta\rho_3 \approx 2.1 \mu\text{m}^{-2}$. These are by far the highest ever observed values for holographic NPC structures.$^{14,15}$ The saturated, thickness-averaged, refractive-index modulation amplitude at the average wavelength of $\lambda_N = 5.3$ nm for neutrons can be calculated to be $\bar{\Delta}n_{\text{sat,N}} \approx 3.6 \times 10^{-5}$ (for the first order), which is four times higher than that of our previously reported holographic NDPC.$^{20}$ The corresponding value for the second order is $\bar{\Delta}n_{2,N} = -1.6 \times 10^{-5}$ and for the third order is $\bar{\Delta}n_{3,N} = 0.7 \times 10^{-5}$, where the minus sign is due to a relative phase shift of $\pi$ of the second order Fourier component with respect to the first.$^{36}$ Note that the latter values are almost one order of magnitude larger than the ones previously reported for holographic NDPC gratings.$^{20}$ The results presented set a record for diffraction gratings for long wavelength neutrons. Moreover, $|\bar{\Delta}n_{2,3:N}|$ of the second and third order Fourier components are of the same order of magnitude as $|\bar{\Delta}n_{\text{sat,N}}|$. Consequently, the refractive index profile of the grating deviates considerably from a pure sinusoidal shape.

To further explore and showcase the exceptional (neutron-) optical characteristics of our grating, we tilted the sample to an angle of $\zeta \approx 69^{\circ}$ (see Fig. 2) to achieve higher peak diffraction efficiency via the Pendellösung effect (see, for instance, Ref. 1) and repeated the experiment. At this tilt angle, the sample area is still of reasonable size for the neutron beam to pass (the diameter is about 0.4 cm at $\zeta \approx 69^{\circ}$) and the effective sample thickness is $d/\cos\zeta \approx 89 \mu\text{m}$. Here, from estimating $Q_N \approx 11.9$ and $\nu_N \approx 1.9$, the diffraction regime is again expected to be in the intermediate diffraction regime. The results are shown in Fig. 4. Note the high peak diffraction efficiency of about 70% and, importantly, the high diffraction efficiency of over 50% exhibited within a rather wide angular range of several tenths of a degree. While the high diffraction power of such gratings will enable us to build better diffractive optical elements for long wavelength neutrons, the width of the angular dependence curves of those elements will save neutron flux and considerably simplify adjustment of beamlines. It will, for instance, make interferometer experiments less sensitive to experimental disturbances, such as vibrations and slight temperature changes. Previously impossible neutron interferometry experiments$^7$ could become feasible deploying holographic NDPC gratings as diffractive beam splitters and mirrors for CN and VCN.

We have experimentally demonstrated exceptionally high diffraction efficiency for slow neutrons at an average neutron wavelength of 5.3 nm, combined with very low angular and wavelength selectivity. This was accomplished mainly due to a record for coherent scattering length density/refractive index modulation amplitudes. Such structures could also be used for neutron-diffractive optics other than simple uniform gratings. Fresnel lenses and chirped gratings—adopted to a particular wavelength distribution of the source—are conceivable. Furthermore, NDPC phase-gratings with low wavelength- and angular-selectivity will trigger more interferometer experiments with long wavelength neutrons (CN and VCN), as grating periods down to

<figure><img src="image_14.png" /><figcaption>FIG. 4. Angular dependence of the diffraction efficiency $\eta_{\text{sat}}(\theta)$ for the holographic NDPC grating at $\zeta \approx 69^{\circ}$, again at about 5.3 nm wavelength (see the text). Solid lines are drawn to guide the eye. Diffraction efficiency is observed to be over 50% for a wide angular range. Data are made available by the ILL.$^{31}$ Error bars represent statistical uncertainties calculated by standard error propagation from Poissonian distribution of neutron counts.</figcaption></figure>
300 nm and sufficiently large grating diameter of up to a couple of centimeters, at excellent homogeneity, have become state of the art.¹⁹ Finally, we expect that optimization of the RB dye concentration will provide less decaying grating profile along the thickness direction, giving further increase in $|\Delta n_{sat,N}|$ at light- and slow-neutron wavelengths.

We acknowledge the financial support by the Vienna Doctoral School in Physics (VDSP), in particular, a VDSP mobility fellowship provided to E.H. We thank Dr. Michael Jentschel for the support during experiments and Mr. Thomas Brenner for the technical assistance at the ILL. We wish to acknowledge the financial support of the Ministry of Education, Culture, Sports, Science and Technology of Japan (Project Nos. 15H03576 and 25-03052). This research was funded in part by the Austrian Research Promotion Agency (FFG), Quantum-Austria NextPi (Grant No. FO999896034). This research was funded in part by the Austrian Science Fund (FWF) (No. P-35597-N). For the purpose of open access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Elhoucine Hadden:** Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). **Martin Fally:** Conceptualization (lead); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (lead); Project administration (lead); Resources (equal); Supervision (lead); Validation (lead); Writing – review & editing (equal). **Yuko Iso:** Conceptualization (equal); Data curation (equal). **Tobias Jenke:** Data curation (equal); Resources (lead); Validation (equal); Writing – review & editing (equal). **Juergen Klepp:** Conceptualization (equal); Data curation (equal); Funding acquisition (equal); Methodology (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (equal); Writing – review & editing (lead). **Atsushi Kume:** Conceptualization (supporting); Resources (equal). **Koichi Umemoto:** Conceptualization (supporting); Resources (equal). **Yasuo Tomita:** Conceptualization (lead); Data curation (lead); Funding acquisition (equal); Methodology (equal); Resources (equal); Supervision (lead); Validation (lead); Writing – original draft (supporting); Writing – review & editing (lead).

## DATA AVAILABILITY

Raw data were generated at the ILL large scale facility.³¹ Derived data supporting the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

¹V. F. Sears, *Neutron Optics* (Oxford University Press, Oxford, UK, 1989).

²H. Abele, “The neutron. Its properties and basic interactions,” *Prog. Part. Nucl. Phys.* **60**, 1 (2008).

³B. T. M. Willis and C. J. Carlile, *Experimental Neutron Scattering* (Oxford University Press, Oxford, UK, 2009).

⁴A. Furrer, J. Mesot, and T. Strässle, *Neutron Scattering in Condensed Matter Physics* (World Scientific, Singapore, 2009).

⁵M. Utsuro and V. K. Ignatovich, *Handbook of Neutron Optics* (John Wiley and Sons, Inc., NJ, 2010).

⁶D. Dubbers and M. G. Schmidt, “The neutron and its role in cosmology and particle physics,” *Rev. Mod. Phys.* **83**, 1111 (2011).

⁷H. Rauch and S. A. Werner, *Neutron Interferometry: Lessons in Experimental Quantum Mechanics, Wave-Particle Duality, and Entanglement* (Oxford University Press, Oxford, UK, 2015).

⁸K. H. Andersen and J. G. Barker, “Promising times for neutron scattering,” *J. Appl. Crystalogr.* **47**, 1163 (2014).

⁹F. Mezei, “Very cold neutrons in condensed matter research,” *JNR. 24*, 205 (2023).

¹⁰H. Abele, A. Alekou, A. Algora, K. Andersen, S. Baßler, L. Barron-Pálos, J. Barrow, E. Baussan, P. Bentley, Z. Berezhiani, Y. Beßler, A. Bhattacharyya, A. Bianchi, J. Bijnens, C. Blanco, N. B. Kraljevic, M. Blennow, K. Bodek, M. Bogomilov, C. Bohm, B. Bolling, E. Bouquerel, G. Brooijmans, L. Broussard, O. Buchan, A. Burgman, H. Calén, C. Carlile, J. Cederkall, E. Chanel, P. Christiansen, V. Cirigliano, J. Collar, M. Collins, C. Crawford, E. C. Morales, P. Cupiał, L. D’Alessi, J. M. Damian, H. Danared, D. Dancila, J. de André, J. Delahaye, S. Degenkolb, D. Di Julio, M. Dracos, K. Dunne, I. Efthymiopoulos, T. Ekelöf, L. Eklund, M. Eshraqi, I. Esteban, G. Fanourakis, A. Farricker, E. Fernandez-Martinez, M. Ferreira, M. Fertl, P. Fierlinger, B. Folsom, A. Frank, A. Fratangelo, U. Friman-Gayer, T. Fukuda, H. Fynbo, A. G. Sosa, N. Gazis, B. Gålnander, T. Geralis, M. Ghosh, G. Gokbulut, J. Gomez-Cadenas, M. Gonzalez-Alonso, F. Gonzalez, L. Halić, C. Happe, P. Heil, A. Heinz, H. Herde, M. Holl, T. Jenke, M. Jenssen, E. Jericha, H. Johansson, R. Johansson, T. Johansson, Y. Kamyshkov, A. K. Topaksu, B. Kildetoft, K. Kirch, B. Kliček, E. Klinkby, R. Kolevatov, G. Konrad, M. Koziol, K. Krhač, A. Kupšć-Lćny, L. Larizgoitia, C. Lewis, M. Lindroos, E. Lychagin, E. Lytken, C. Maiano, P. Marciniewski, G. Markaj, B. Märkisch, C. Marrelli, C. Martins, B. Meirose, M. Mezzetto, N. Milas, D. Milstead, F. Monrabal, G. Muhrer, A. Nepomuceno, V. Nesvizhevsky, T. Nilsson, P. Novella, M. Oglakci, T. Ohlsson, M. Olvegård, A. Oskarsson, T. Ota, J. Park, D. Patrzalek, H. Perrey, M. Persoz, G. Petkov, F. Piegsa, C. Pistillo, P. Poussot, P. Privitera, B. Rataj, D. Ries, N. Rizzi, S. Rosauro-Alcaraz, D. Rozpedzik, D. Saiang, V. Santoro, U. Schmidt, H. Schober, I. Schulthess, S. Silverstein, A. Simón, H. Sina, J. Snamina, W. Snow, T. Soldner, G. Stavropoulos, M. Stipčević, B. Szybniński, A. Takibayev, Z. Tang, R. Tarakeshian, C. Laroine, J. Thorne, F. Terranova, J. Thomas, T. Tolba, P. Torres-Sánchez, E. Trachanas, R. Tsenov, U. Uggerhøj, G. Vankova-Kirilova, N. Vassilopoulos, R. Wagner, X. Wang, E. Wildner, M. Wolke, J. Wurtz, S. Yiu, S. Yoon, A. Young, L. Zanini, J. Zejma, D. Zerzion, O. Zimmer, O. Zormpa, and Y. Zou, “Particle physics at the European spallation source,” *Phys. Rep.* **1023**, 1–84 (2023).

¹¹R. A. Rupp, J. Hehmann, R. Matull, and K. Ibel, “Neutron diffraction from photoinduced gratings in a PMMA matrix,” *Phys. Rev. Lett.* **64**, 301 (1990).

¹²U. Schellhorn, R. A. Rupp, S. Breer, and R. P. May, “The first neutron interferometer built of holographic gratings,” *Physica B* **234–236**, 1068–1070 (1997).

¹³C. Pruner, M. Fally, R. A. Rupp, R. P. May, and J. Vollbrandt, “Interferometer for cold neutrons,” *Nucl. Instrum. Methods Phys. Res., Sect. A* **560**, 598–605 (2006).

¹⁴M. Fally, J. Klepp, Y. Tomita, T. Nakamura, C. Pruner, M. A. Ellabban, R. A. Rupp, M. Bichler, I. Drevenšek-Olenik, J. Kohlbrecher, H. Eckerlebe, H. Lemmel, and H. Rauch, “Neutron optical beam splitter from holographically structured nanoparticle-polymer composites,” *Phys. Rev. Lett.* **105**, 123904 (2010).

¹⁵J. Klepp, C. Pruner, Y. Tomita, C. Plonka-Spehr, P. Geltenbort, S. Ivanov, G. Manzin, K. H. Andersen, J. Kohlbrecher, M. A. Ellabban, and M. Fally, “Diffraction of slow neutrons by holographic SiO₂ nanoparticle-polymer composite gratings,” *Phys. Rev. A* **84**, 013621 (2011).

¹⁶J. Klepp, C. Pruner, Y. Tomita, K. Mitsube, P. Geltenbort, and M. Fally, “Mirrors for slow neutrons from holographic nanoparticle-polymer free-standing film-gratings,” *Appl. Phys. Lett.* **100**, 214104 (2012).
<footnote n="17">¹⁷J. Klepp, Y. Tomita, C. Pruner, J. Kohlbrecher, and M. Fally, “Three-port beam splitter for slow neutrons using holographic nanoparticle-polymer diffraction gratings,” *Appl. Phys. Lett.* **101**, 154104 (2012).</footnote>

<footnote n="18">¹⁸J. Klepp, C. Pruner, Y. Tomita, P. Geltenbort, I. Drevenšek-Olenik, S. Gyergyek, J. Kohlbrecher, and M. Fally, “Holographic gratings for slow-neutron optics,” *Materials* **5**, 2788 (2012).</footnote>

<footnote n="19">¹⁹Y. Tomita, E. Hata, K. Momose, S. Takayama, X. Liu, K. Chikama, J. Klepp, C. Pruner, and M. Fally, “Photopolymerizable nanocomposite photonic materials and their holographic applications in light and neutron optics,” *J. Mod. Opt.* **63**, S1–S31 (2016).</footnote>

<footnote n="20">²⁰Y. Tomita, A. Kageyama, Y. Iso, K. Umemoto, A. Kume, M. Liu, C. Pruner, T. Jenke, S. Roccia, P. Geltenbort, M. Fally, and J. Klepp, “Fabrication of nanodiamond-dispersed composite holographic gratings and their light and slow-neutron diffraction properties,” *Phys. Rev. Appl.* **14**, 044056 (2020).</footnote>

<footnote n="21">²¹E. Hadden, Y. Iso, A. Kume, K. Umemoto, T. Jenke, M. Fally, J. Klepp, and Y. Tomita, “Nanodiamond-based nanoparticle-polymer composite gratings with extremely large neutron refractive index modulation,” *Proc. SPIE* **12151**, 1215109 (2022).</footnote>

<footnote n="22">²²Neutron Data Booklet, edited by A.-J. Dianoux and G. Lander (Institut Laue-Langevin, Philadelphia, 2003).</footnote>

<footnote n="23">²³N. Uchida, “Calculation of diffraction efficiency in hologram gratings attenuated along the direction perpendicular to the grating vector,” *J. Opt. Soc. Am.* **63**, 280–287 (1973).</footnote>

<footnote n="24">²⁴H. Kogelnik, “Coupled wave theory for thick hologram gratings,” *Bell Syst. Tech. J.* **48**, 2909–2947 (1969).</footnote>

<footnote n="25">²⁵Y. Tomita and R. Kuze, “Apodization effect of a reflection photorefractive hologram with linear absorption,” *Opt. Commun.* **183**, 327–331 (2000).</footnote>

<footnote n="26">²⁶Y. Tomita and H. Nishibiraki, “Improvement of holographic recording sensitivities in the green in SiO₂ nanoparticle-dispersed methacrylate photopolymers doped with pyrromethene dyes,” *Appl. Phys. Lett.* **83**, 410–412 (2003).</footnote>

<footnote n="27">²⁷N. Suzuki and Y. Tomita, “Holographic scattering in SiO₂ nanoparticle-dispersed photopolymer films,” *Appl. Opt.* **46**, 6809–6814 (2007).</footnote>

<footnote n="28">²⁸Y. Tomita, N. Suzuki, K. Furushima, and Y. Endoh, “Volume holographic recording based on mass transport of nanoparticles doped in methacrylate photopolymers,” *Proc. SPIE* **5939**, 593909 (2005).</footnote>

<footnote n="29">²⁹T. Oda, M. Hino, M. Kitaguchi, H. Filter, P. Geltenbort, and Y. Kawabata, “Towards a high-resolution TOF-MIEZE spectrometer with very cold neutrons,” *Nucl. Instrum. Methods Phys. Res., Sect. A* **860**, 35–41 (2017).</footnote>

<footnote n="30">³⁰M. Blaickner, B. Demirel, I. Drevenšek-Olenik, M. Fally, P. Flauger, P. Geltenbort, Y. Hasegawa, R. Kurinjimala, M. Ličen, C. Pruner, S. Sponar, Y. Tomita, and J. Klepp, “Monte-Carlo simulation of neutron transmission through nanocomposite materials for neutron-optics applications,” *Nucl. Instrum. Methods Phys. Res., Sect. A* **916**, 154–157 (2019).</footnote>

<footnote n="31">³¹J. Klepp, M. Fally, E. Hadden, T. Jenke, and C. Pruner, “Neutron diffraction from commercially available Bayfol photopolymer film gratings,” [Institut Laue-Langevin (ILL), Grenoble, 2021].</footnote>

<footnote n="32">³²T. K. Gaylord and M. G. Moharam, “Thin and thick gratings: Terminology clarification,” *Appl. Opt.* **20**, 3271–3273 (1981).</footnote>

<footnote n="33">³³M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” *J. Opt. Soc. Am.* **71**, 811–818 (1981).</footnote>

<footnote n="34">³⁴M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” *J. Opt. Soc. Am. A* **12**, 1068–1076 (1995).</footnote>

<footnote n="35">³⁵J. Klepp, C. Pruner, Y. Tomita, P. Geltenbort, J. Kohlbrecher, and M. Fally, “Advancing data analysis for reflectivity measurements of holographic nanocomposite gratings,” *J. Phys.* **746**, 012022 (2016).</footnote>

<footnote n="36">³⁶M. Fally, Y. Tomita, A. Fimia, R. F. Madrigal, J. Guo, J. Kohlbrecher, and J. Klepp, “Experimental determination of nanocomposite grating structures by light- and neutron-diffraction in the multi-wave-coupling regime,” *Opt. Express* **29**, 16153–16163 (2021).</footnote>