3D Measurements and Characterizations of Refractive Index Distributions of Volume Holographic Gratings Using Low‐Coherence Holotomography

Holographic Optical Elements (HOEs) employ refractive index (RI) modulation to manipulate light, establishing their widespread application in augmented and virtual reality technologies. Despite significant strides in HOEs' fabrication, a comprehensive 3D examination and measurement of these elements remain largely unexplored. This study targets this void by conducting an in‐depth exploration of the 3D RI distributions of volume holographic gratings (VHGs), a quintessential representative of HOEs. Direct tomographic measurements of the 3D RI distributions within millimeter‐scale VHGs are performed at a spatial resolution of 156 nm × 156 nm × 1.2 µm and an RI precision of 2×10−4. This level of precision facilitates a systematic high‐resolution characterization of VHGs. The findings present a substantial evolution from traditional measurement techniques and promise to considerably advance the evaluation of intricate HOE structures and refine the corresponding fabrication processes.


Introduction
Holographic Optical Elements (HOEs) are specialized diffractive components that utilize the principles of holography to encode the interference pattern of two or more coherent light beams into refractive index (RI) modulations within a material.Over DOI: 10.1002/adom.202302048 the years, HOEs have garnered significant interest due to their multifaceted advantages such as cost-effectiveness, superior beam quality, excellent resolution, and ease of customization and fabrication. [1]These attributes have facilitated their widespread adoption across diverse sectors, rendering them integral to advanced optical systems.The unparalleled wavefront manipulation potential of HOEs allows their application in imaging, [2] display, [3] data storage, [4] solar concentrators for photovoltaic cells, [5] and prominently, in emerging extended reality (XR) industry comprising of augmented reality (AR), virtual reality (VR), and mixed reality (MR) technologies. [6]mong the various types of HOEs, volume holographic gratings (VHGs) exhibit exceptional precision in light property manipulation, superseding thin gratings.VHGs display notable angular selectivity, significantly amplified diffraction efficiency, and superior beam properties. [7]A critical parameter of a grating is its diffraction efficiency, [8] and a volume holographic gratings' diffraction efficiency can theoretically attain up to 100%, which greatly surpasses the maximum 33.8% and 40.5% efficiencies achievable by thin sinusoidal and binary phase gratings. [9]These salient features position VHGs as vital tools in imaging, [10] biophotonics, [11] and display technologies. [12]espite remarkable progress in the materials and techniques for recording gratings since the advent of VHGs, [13] methodologies for investigating and characterizing VHG and HOE remain largely untapped. [14]There is significant interest in understanding the relationship between diffraction efficiency and various factors such as groove count, RI, and relief depth. [8,15]However, no prior research has directly measured and characterized the 3D RI distributions of VHGs.Common practice involves measuring the diffraction efficiency (Figure 1a,b) without detailed information about the RI or grating formation thickness, or calculating the RI modulation of VHG based on angular selectivity and thickness, using the coupled wave theory. [16]Such measurements often assume uniform grating formation across the sample.
Most previous studies are limited to 2D grating profiling, with phase contrast microscopy offering only qualitative evaluation. [17]uantitative Phase Imaging (QPI) techniques like off-axis digital holography or interferometric microscopy can measure the quantitative optical phase delay of a coherent beam, [18] but they fail to separate the RI information and thickness profile from the measured phase delay.Ellipsometry can provide data about the thickness and RI of HOEs with prior knowledge, [19] but it does not deliver a tomographic reconstruction of the RI distributions within the HOEs.
Holotomography (HT), a 3D QPI technique, can reconstruct the 3D RI distribution of a sample from multiple 2D optical measurements with varying modulations. [20]With its capacity to non-invasively visualize high-resolution 3D RI distribution of transparent volumetric samples, [20b] HT has found broad applications in fields such as cell biology, [21] flow cytometry, [22] biophysics, [23] microbiology, [24] and preclinical studies. [25]n this study, we present a systematic approach to measuring 3D RI tomograms of volume holographic gratings and their characterization using low-coherence HT (Figure 1c).We demonstrate the acquisition of high-resolution RI tomograms within VHGs with a spatial resolution of 156 nm × 156 nm × 1.2 μm and an RI precision of 2×10 −4 .The RI distributions within the VHGs, along with defect locations, are precisely delineated.Additionally, we compute the RI modulation depth at various positions to quantitatively assess grating formation strength across the entire sample.Furthermore, we demonstrate the capability of detecting voids and debris in VHGs and wide-field RI imaging by stitching multiple fields of view, thereby enabling assessments within millimeter-scale VHGs.

Refractive Index Tomogram of a Volume Grating
To measure the refractive index tomogram of VHGs, we employed a low-coherence holotomography system (Figure 2a, See Methods).HT is one of 3D quantitative phase imaging techniques, and serves as an optical counterpart to X-ray computed tomography (CT), whereby multiple 2D images of a sample, captured with varied illumination modulations, are used to reconstruct a 3D RI distribution of the sample by inversely solving the wave equation.Unlike HT techniques based on a coherent light source such as optical diffraction tomography, [20c,26] lowcoherence HT utilizes temporally low-coherence light, for example, light-emitting diodes. [27]The utilized low-coherence system uses a LED with a center wavelength of 449 nm and is founded on self-interference, eliminating the need for a reference arm.Consequently, the low-coherence HT mitigates issues related to speckle noise and exhibits high stability.
To reconstruct the 3D RI distribution of a VHG, multipleintensity images were captured for each illumination pattern (Figure 2b).This procedure was repeated at several axial positions.The RI tomogram of the VHG was then reconstructed from the measured raw intensity images (Figure 2c).The reconstructed RI tomogram clearly displays a highly periodic distribution of RI values along the x-axis.The measured RI range of the VHG spans from 1.533 to 1.543, which aligns with the expected values.The period and axial formation of the volumetric gratings can be directly assessed from this reconstructed RI tomogram.

Assessments of Diffraction Efficiencies
For a holistic evaluation of VHGs, we fabricated and analyzed two distinct VHG samples using the methodology described, differentiated by their thickness and molecular weight.Sample 1 was designed using PMMA with a molecular weight of 120k and a thickness of 44 ± 5 μm, while Sample 2 was constructed using PMMA with a higher molecular weight of 350k and a thickness of 55 ± 5 μm.The grating period for both samples was set at 1 μm.
The measured RI tomograms of Samples 1 and 2, as shown in Figure 3, affirm the achieved grating period of 1 μm in both instances.Sample 1 demonstrates a volume grating formation with a thickness of 41 μm, extending from z = 8 μm to 49 μm, while Sample 2 showcases a comparable structure with a thickness of 47 μm, spanning from z = 12 μm to 59 μm.
The average modulation depth Δn of Sample 1 and Sample 2 were ascertained to be 0.0046 ± 0.0016 and Δn = 0.0047 ± 0.0041, respectively.The capability of our method to perform volumetric RI imaging allows us to deeply probe RI distribution at different axial positions within VHGs.For instance, the RI distributions of Sample 1 at unique depths are depicted in the subfigures of Figure 3a.The distributions at z = 53, 44, and 29 μm are respectively highlighted in orange, yellow, and blue boxes.Also provided are histograms detailing the RI distributions at varying z positions with a 6 μm interval.These results imply that RI modulations were executed non-uniformly across these axial positions in terms of both the modulation depth Δn and the distribution of RI values.
The RI modulation depth is the peak-to-valley value obtained from the sinusoidal fitting performed in each x-direction.The gray lines represent the index modulation at different y-positions every 1.1 μm and the red line in the graph is the average value of gray lines.The red line shows that Sample 2 has a higher maxi-mum RI modulation depth of 0.0088 compared to 0.0063 in Sample 1.However, the thickness of the area where the RI modulation exceeds 0.002 is 19.2 μm for Sample 2, which is roughly half of the 37.2 μm thickness of Sample 1. Consequently, the two samples display diffraction efficiencies of 22.7% for Sample 1 and 29.6% for Sample 2. This result is consistent with our expectation that the diffraction efficiency of the thick phase grating depends on both the thickness d and RI modulation depth n 1 .This is expressed by the equation DE = sin 2 Φ where DE represents diffraction efficiency and Φ = n 1 d/(2cos) [9,16b] when the beam is illuminated at Bragg angle  with the same wavelength  used during recording.The RI modulation depth calculated by coupled wave theory is 0.004 for both samples, which is in line with the average RI modulation depths of 0.0046 and 0.0047 observed in our experiment.
Three distinct regions are discernible within the samples: a region with nearly zero index modulation, presumed to be either air or the glass substrate, and regions with weak and strong index modulation.Three possible explanations can be surmised for this tripartite division; 1) Despite the very low permeability and transmittance of oxygen in PMMA, the surface's radical reaction can be influenced by atmospheric oxygen.This interference could have subjected the radical polymerizations in this region to oxygen's radical scavenging effect, [28] compromising the grating's quality.2) The monomer's diffusion rate could fluctuate depending on the solvent content.Although we placed the samples in a vacuum for 24 h to allow the solvent of the ascasted photopolymer to be fully evaporated, the remanent solvent could persist within the photopolymer.Therefore, the relatively dry surface region might exhibit subpar holographic recording performance due to deficient monomer diffusions. [29]After the holographic recording, any residual solvent may also cause the VHG's deformation as demonstrated in Figure 3. 3) Radical polymerization can cause substantial volume shrinkage of the photopolymer film, which could in turn lead to VHG deformation and a decrease in recording performance. [30]While the bottom region may resist volume shrinkage through its tight bond with the substrate, the upper region is directly affected by volume shrinkage.

Axial Profiles of Modulation Depths of VHGs
Exploring the impact of PMMA molecular weights on the formation of volume holographic grating structures, we analyzed the 3D RI distribution of Sample 2, which has a higher PMMA molecular weight (350k) compared to Sample 1 (120k).The results, as depicted in Figure 3b, showcase Sample 2′s formation of a volume grating with a thickness of 47 μm and a modulation depth Δn = 0.0047 ± 0.0041.Notably, Sample 2 presents two distinct domains: the top layer (z = 34 μm to 59 μm) with a shallow modulation Δn = 0.0011 ± 0.0005, and the bottom layer (z = 12 μm to 33 μm) with a deep modulation Δn = 0.0063 ± 0.0024.These variations in modulation depths across these domains can also be discerned from the lateral RI distributions (refer to insets, Figure 3b) and the histogram of RI values.
The ability to control VHG's optical properties on-demand, via the synthesis and modulation of photopolymers, is crucial for custom fabrication of VHGs for diverse applications.Recent research suggests that the use of photopolymers with smaller molecular weights promotes sufficient diffusion during holographic recording, thus enhancing the RI modulation of VHGs. [31]Our results, demonstrated in Figure 3a,b, affirm this proposition when considering the distributions and values of modulated RI in the upper layers of fabricated VHGs.However, our findings also reveal significant variability in RI modulation in thicker VHGs along axial positions.In the case of Sample 1, the VHG's top surface manifests the greatest RI modulation, which diminishes with depth until z = 35 μm, and subsequently increases toward the VHG's base.
The upper domain of Sample 2 (z = 34 μm to 59 μm) exhibits weaker RI modulation compared to Sample 1, aligning with our expectations.However, the lower domain of VHG in Sample 2 presents strong RI modulations, surpassing the RI modulation in Sample 1. Fascinatingly, there exists a clear demarcation (z = 34 μm) between these domains, indicating varying optical properties.

Detection of Micrometer-Sized Defects in VHGs
By leveraging the ability of holotomography to map 3D RI distributions of a sample with remarkable spatial resolution and RI precision, our current approach offers significant potential for detecting and analyzing micrometer-scale anomalies within VHGs, especially those associated with abnormal RI values.
For example, as depicted in Figure 3a, we can identify void structures displaying significantly lower RI values, as highlighted by the black arrows.The morphological characteristics of these structures -their spherical shape and below-average RI values -suggest they likely represent air bubbles unintentionally introduced during the VHG fabrication process.In contrast, Figure 3b reveals a structure ≈2 μm in size showing high RI values (n = 1.547), as pointed out by the white arrow.The properties of this structure suggest it may be particulate matter, such as dust or debris.
The advanced capabilities of HT to plot the volumetric distribution of RI values provide an invaluable tool for identifying and distinguishing these unwanted inconsistencies, such as voids or debris, within the VHG structure.This feature, together with an accurate evaluation of the optical properties of VHGs, offers significant potential for future applications in fabrication processes within the XR industry.

Extended Field-Of-View in Holotomography
Our method can deliver an extensive field of view.We carried out measurements over a field of view of 1 mm × 1 mm on a photopolymer sample (as detailed in Section 2.2, Sample 1), achieved by stitching together 7×7 RI tomogram measurements using an embedded function of the HT-X1 platform, with consistent hardware usage as outlined in the preceding section.Using an x-y motorized translation stage, we obtained raw bright field images, with a single shot reconstruction field of view of 164.6 μm × 164.6 μm.The resulting lateral and axial resolutions post-reconstruction were 156 nm and 1.1 μm respectively.
As illustrated in Figure 4a, across the entire x-direction, 1000 grooves of the grating are contained, along with observable defects and cracks.Wide-field imaging has a distinctive advantage in evaluating mesoscopic and macroscopic structures of HOEs, and in identifying sparse defects.Figure 4b magnifies the area outlined by the red box in Figure 4a, offering a detailed examination of this section.The distinctive groove pattern of the phase grating within the photopolymer is visible in the 3D representation.
Figure 4c presents a more detailed view of the area demarcated by the blue box in Figure 4b, showing the RI modulation of the sample is pronounced from z = 7 to 39 μm and gradually fades from z = 39 to 51 μm.To ascertain structural variations by location, we delineate the cross-section of the 3D RI from different areas within the same sample.The regions shown in Figure 4d are marked with black in Figure 4a, and the two different regions demonstrate a similar configuration.Although the RI modulations are comparable, the modulation at position 2 is marginally higher than at position 1.Through our RI tomogram and analysis of RI modulation depth, we confirmed that the grating depth remained consistent across different locations in this specimen.

Discussion
The 3D refractive index tomograms captured in this study provide comprehensive insights into VHGs.The precise and quantitative measurements of RI distributions deliver an unprecedented and detailed understanding of VHGs' optical properties.This research presents the tomographic RI measurements of VHGs with varying PMMA molecular weights.Moreover, we illustrate how our method can detect impurities, such as voids or debris, by evaluating the 3D RI tomograms of VHGs.This feature holds significant potential for implementation in quality inspection procedures within VHG manufacturing facilities.
An intriguing finding from our work is the variability in the periodic RI distributions of VHGs along the axial direction.Such phenomena have largely remained unexplored, as existing methods do not facilitate tomographic measurements of refractive index.The results depicted in Figure 3 are complex and extend beyond the simplicity of the diffusion theory. [31]The non-uniform formation of holographic grating structures along the axial direction introduces several uncharted mechanisms for photopolymer rearrangements.These could include the effects of light diffraction on the consequent layer due to the early formation of holographic grating structures, as well as transient dynamics associated with the diffusion and photopolymerization of PMMA molecules, among others.
The experimentally measured 3D RI distributions of VHGs not only offer critical insights for assessing fabricated VHGs, but also pave the way for unprecedented comprehensive numerical analysis.For instance, one can accurately compute the light propagation through VHGs by solving the wave equation, using scattering potential data from the measured 3D RI tomograms.This enables numerical determination of light diffraction patterns through the VHGs for any arbitrary wavefront.Furthermore, conditions involved in image formation, as well as potential aberration issues, can be systematically explored using the measured 3D RI tomograms of VHGs.
Our current approach has several technical limitations.One significant challenge is the slow acquisition and reconstruction time.It takes 6.5 seconds to capture raw data-multiple 2D intensity images for refractive index reconstruction-to cover a 3D volume of 165 μm × 165 μm × 70 μm.Furthermore, it necessitates a 55-second reconstruction process to calculate the 3D RI distribution of a sample from these captured multiple 2D images.However, optimizing the mechanical movements of the sample stage and employing multiple GPU processing approaches can expedite the speed of acquisition and reconstruction. [32]Machine learning algorithms can also be considered to expedite the process of reconstruction and regularization. [33]other limitation is the inability of our method to provide a distinct boundary between a sample and the media when the boundary aligns with the optical axis.This is because the holotomography system we utilized is based on transmission geometry and cannot capture reflection signals from surfaces normal to the optical axis.To overcome this limitation, one can collect reflection signals by integrating another imaging system and synthesizing both transmission and reflection signals during the reconstruction process.
17a,34] Our method can also measure multilayered HOEs or HOEs with layers possessing disordered refractive index distributions. [35]By adjusting the wavelengths of the illumination source, it is possible to measure the 3D RI tomograms of a VHG as a function of wavelengths.This adjustment potentially expands the applicability of our method to multicolor VHG structures. [36]The concept underlying our approach can be further extended to measure dielectric tensor tomography, thereby enabling the full optical characterization of VHGs by considering the polarization of light and the orientation of molecules within the VHGs. [37]

Conclusion
In this study, we present the direct experimental measurements of 3D RI distribution in VHGs with exceptional spatial resolution, leveraging low-coherence HT.RI distributions within VHGs, ranging in thickness from 40 to 50 μm, were ascertained with lateral and axial resolutions of 156 nm and 1.1 μm, respectively, and a RI precision of 2×10 −4 .We demonstrated a 1 mm × 1 mm field of view by stitching multiple refractive index tomograms.
In conclusion, this work reveals the substantial potential of HT in the study and fabrication of VHGs and HOEs.Leveraging its precision, non-invasiveness, and versatility, HT provides comprehensive 3D RI measurements, offering insights that could revolutionize the field.This technique promises to be a transformative tool in holography, with implications far beyond our current applications.As we continue to refine this approach, we anticipate exciting developments in the optical characterization and manipulation of holographic elements.

Experimental Section
Fabrication of Photopolymer Sample: The volume holographic gratings were fabricated utilizing a photopolymer.The initial photopolymer solution comprised 19 wt.% of linear polymethyl methacrylate (PMMA) average molecular weight of 3 50 000 Da (350k) and 1 20 000 Da (120k), 19 wt.% of acrylate monomer (BzMA, benzyl methacrylate), 2 wt.% of photo-initiator (Irgacure-784), and 60 wt.% of anisole.This photopolymer solution was subsequently coated onto a glass slide using a doctor blade.To minimize void formation in the specimen, the solvent was evaporated in a vacuum chamber for 24 h.Following sample fabrication, the sample's thickness was determined using a profilometer (DektakXT, Bruker).
The HOE was inscribed in the photopolymer through the modulation of illumination light intensity, achieved by the interference of two laser beams (i.e., intensity interference pattern, IIP).The RI of the photopolymer fluctuated with the beam intensity; it escalated where the incident beam intensity was high and diminished where the intensity was low.The RIs of benzyl methacrylate (BzMA) and the PMMA matrix, which functioned as the acrylate monomer and binder material respectively, were 1.57 and 1.5 (at a measurement wavelength of 449 nm). [38]Upon light illumination, the photo-initiator provoked the radical polymerization of the BzMA monomer, thereby reducing the monomer's concentration. [39]Consequently, BzMA molecules in low-light intensity areas diffused toward areas of lower concentration with higher light intensity. [40]As the RI of BzMA was greater than that of PMMA, regions where BzMA was concentrated exhibited a higher RI than the sample's average RI of 1.538, while areas with lower BzMA concentration displayed a value below 1.538.This resulted in the formation of a VHG within the photopolymer.In the experiment, two 532 nm laser beams, split via a 50:50 beam splitter, were directed onto the sample at angles of +15.5°and −15.5°(i.e., with a 1 μm period), while maintaining a constant intensity of 2 mW cm −2 for both beams and a holographic recording time of 10 min, respectively.The study employed this symmetric mixing of two beams as a means to achieve maximal contrast in intensities across the sinusoidally modulated dark and bright regions of the intensity interference pattern (IIP).
Measurement of the Diffraction Efficiency: The real-time measurement of the diffraction efficiency (DE) of the evolving volume holographic gratings (VHGs) followed a standard volume holographic recording and characterization protocol, as previously reported in the literature. [41]During the recording of the VHGs, one of the two recording beams that were symmetrically mixed was temporarily blocked for a brief duration (0.1 s) with intervals of 10 s.Subsequently, the intensities of both the first-orderdiffracted and the zeroth-order-transmissive beams were simultaneously measured for 0.1 s.In this real-time measurement of the DE, there was a consideration that using a single detecting laser beam with the same wavelength as that used for VHG recording could potentially dilute the refractive index (RI) modulation, as uniform photopolymerization across the grating vector of the intensity interference pattern (IIP) can occur.However, it is important to note that the time for this uniform polymerization is quite short, and its effect on the evolution of RI modulation was found to be negligible.Furthermore, the ability to perfectly match the Bragg angle of the evolving VHGs was achieved, especially through the utilization of in situ real-time recording and characterization schemes within the optical setup.Following the 10-minute VHG recording, the DE reached saturation, confirming sufficient photopolymerization selectively within the bright regions of the IIP.To ensure the complete consumption of monomers, an overall curing and fixing process involving flood exposure to ultraviolet light was performed.
3D Refractive Index Measurement: To measure the RI tomogram, a lowcoherence holotomography setup (HT-X1, Tomocube) was employed.This system used a light-emitting diode with a center wavelength of 449 nm as the illumination source.The light field of the illuminations was modulated using a digital mirror device with a 7.6 μm micromirror pitch (DMD, DLP4500, Texas Instrument).Multiple patterned illuminations were directed onto the sample plane using a condenser lens (f = 180 mm, numerical aperture (NA) = 0.72).The corresponding transmitted beams were collected using an objective lens (NA = 0.95) and a tube lens to form a 4-f telescopic imaging system and were then recorded by a CMOS camera (FS-U3-28S5, FLIR).Multiple-intensity images were captured with optimized illumination patterns for each axial position.The measurements were carried out along the axial direction to reconstruct the 3D RI distribution.The tomogram offered lateral and axial resolutions of 156 nm and 1.2 μm, respectively.The RI precision of the used system was 2×10 −4 , which was characterized by measuring the standard deviation of the reconstructed RI tomogram of a sample with a homogeneous distribution of RI values.Refer to Supporting Information for further details and principles of the system.

Figure 1 .
Figure 1.Analysis of volume holographic gratings.a) Photograph of a photopolymer VHG sample.b) The diffraction pattern of the VHG.A 638 nm laser (Cobolt 06-MLD 638 nm, Cobolt AB) is directed onto the grating, and the beam is projected toward the target after 5 cm of propagation.c) Conceptual diagram illustrating the measurement and reconstruction process of 3D RI tomogram of a VHG.The 3D RI map reconstruction involves recording the image penetrated by the illumination under four distinct illumination patterns, with each pattern involving a shift in the axial focal plane.

Figure 2 .
Figure 2. Process of 3D RI reconstruction.a) Schematic representation of the low-coherence HT setup.b) Raw intensity images captured under each illumination pattern (four quadrants) along axial positions.c) 3D RI distribution reconstructed from the raw images in (b).x-z and x--y cross-sectional images at various z positions are shown.

Figure 3 .
Figure 3. RI Tomograms and Modulation Depth Analysis.a,b) Provide cross-sectional visualizations of the 3D RI distribution along the x-y and x-z planes, as demarcated by the dashed lines.The RI modulation depth is denoted by Δn.The series of gray lines displayed in the Δn plots represent the variation of the refractive index at intervals of 1.1 μm.Meanwhile, the red line serves to illustrate the averaged value derived from these gray lines.Scale bar: 5 μm.For visualization purposes, the scales of the RI tomograms in the z-direction are halved compared to the x and y directions.

Figure 4 .
Figure 4. Wide-field RI image.a) 7×7 images are stitched to construct a 1 mm × 1 mm wide-field image.b) Crop image of the area marked with the red box in (a).c) Crop image of the area marked with the blue box in (b).d) Cross-section images of the area marked with black boxes in (a).