© The Author(s), published by EDP Sciences, 2024

1 Introduction

The see-through capability of an Augmented Reality (AR) device defines the purpose of the latter, which is to superimpose a virtual image on the surrounding environment of the user. Aiming for this feature, Holographic Optical Elements (HOEs) are one of the most promising solutions for the in-coupler and out-coupler elements, enabling environmental light to interact with the HOE and subsequently producing an image.

Using photopolymers as the recording material has recently been a matter of study within AR [1, 2], thanks to their tuneable capacities, high efficiency, high resolution and low cost. The optical property that undergoes a change under illumination in this kind of recording material is either the refractive index or the absorption coefficient, mainly because of the photopolymerization and diffusion processes that occur during the exposure and thanks to which the optical phase information from an optical element can be stored.

For what follows, the commercial photopolymer Bayfol HX200 from Covestro AG has been used. Among its many studied features [3], the easy processing, long-term stability, accuracy of grating reproduction and compatibility with standard industrial product-integration process highlight to our purpose. Moreover, as discussed in [4], the non-local diffusion model limits the variety of photopolymers that can be used for reflection holography, where this commercial product has been tested successfully. Particularly, it has been optimized in multiplexing procedures for reflection holograms of diffusing objects [5].

Regarding the core of this work, the majority of the recording holographic geometries, using prisms and microlens arrays, are based on reflection holograms [6], as they provide higher values of the Field of View (FOV), in terms of a wider angular tolerance [7], than the transmission ones and eliminate the stray light that produces ghost images, key factors among many others related to the design of see-through devices.

The use of prisms and index matching oil [8], accounting for the waveguide combiners, permits impinging angles very oblique inside the material inside the material to achieve diffraction angles higher than the critical angle inside the substrate. Even being readily achievable in a laboratory setting, it poses much larger challenges in mass production, mainly because of the need to avoid an air gap, and it is not suitable for any kind of non-contact master-copying automated production method.

Hence, the novelty of this work lies in presenting an examination of the recording conditions to design the appropriate gratings, extending the fundamentals in previous works [2, 9, 10], using a reflection scheme without the use of prisms, and the experimental evidence about the viability of this approach.

2 Design and theoretical background

Intensity holography can be used to produce the aforementioned HOEs. This technique is built upon the translation of an interference pattern of two waves, known as reference and object waves, into a change in the optical properties of the recording material due to illumination, whose fundamentals are described next.

Focusing the design into the sinusoidal pattern arising from the interference of two plane waves, its main parameters are closely related to the recording geometry, which is depicted in Figure 1. Denoting as ${\mathit{k}}_o$ and ${\mathit{k}}_r$ the wave vectors of the object and reference beams, the grating is described by its reciprocal lattice vector, $\mathit{K}$, as:$$\mathit{K}={\mathit{k}}_o-{\mathit{k}}_r.$$(1)

Figure 1 Ewald’s spheres for the selected geometry, using ${\lambda }_r=532$ nm for the recording process and ${\lambda }_c=473$ nm as the designed wavelength for the reconstruction stage. The recording structure (green) is built upon the recording angles in the material: $19.4^{\circ }$ and $35.1^{\circ }$. The equivalent assembly for the reconstruction step (blue) yields angles of $1.0^{\circ }$ and $55.5^{\circ }$. The average refractive index is $n_0=1.505$, so the critical angle is ${\theta }_c=41.6^{\circ }$.

Then, the hologram may act as a diffraction grating. In this simplest case, spatial dependence is induced in the refractive index $n$ of the material due to the periodicity of the interference pattern, described as:$$n\left(\mathit{r}\right)=n_0+n_1\cos \left(\mathit{K}\cdot \mathit{r}\right),$$(2)

where $n_0=1.505$ is the average refractive index of the photopolymer [11], $n_1$ is the refractive index modulation and $\mathit{r}$ is the position vector of a point in the holographic $z-x$ plane, as in Figure 1.

That being so, with $\mathrm{\Lambda }$ as the grating spacing and $\phi$ as the slant angle of the grating vector given by (1), the Bragg’s condition during the reconstruction step yields the angle of incidence ${\theta }_B^{\text{'}}$ for which the maximum diffraction occurs, given the reconstruction wavelength ${\lambda }_c$:$$\cos \left({\theta }_B^{\text{'}}-\phi \right)=\frac{{\lambda }_c}{2n_0\mathrm{\Lambda }}.$$(3)

Moreover, following Kogelnik’s notation [12], we can establish the angle of the diffracted beam, as its associated wave vector $\mathit{\sigma }$ is related to the transmitted reference reference wave vector $\mathit{\rho }$ and the grating vector within the Bragg condition through:$$\mathit{\sigma }=\mathit{\rho }-\mathit{K}.$$(4)

Given the detailed relations (1)–(4) between the recording geometry and the grating formation and diffraction properties, we can design a specific geometry with convenient values for the angle of the diffracted beam and the Bragg angle. Then, if we can impose that the former is greater than the critical angle of the substrate, the hologram may work as a holographic coupler [9]. This is why the concrete configuration depicted in Figure 1 has been adopted, using ${\lambda }_0=532$ nm as the recording wavelength and ${\theta }_r=30^{\circ }$ and ${\theta }_o=60^{\circ }$ as the recording angles in air, which converts to ${\theta }_r^{\text{'}}=19.4^{\circ }$ and ${\theta }_o^{\text{'}}=35.1^{\circ }$ in the material.

Specifically, using (1), the designed grating has a spatial frequency of $1/\mathrm{\Lambda }\sim 5600$ lines/mm and a slant angle of $\phi =-27.3^{\circ }$. Hence, values obtained from (3) and (4) with ${\lambda }_c=473$ nm show that the diffracted beam propagates under the Total Internal Reflection (TIR) principle, as Figure 1 illustrates, when a near-normal incidence beam impinges on the grating.

Using different wavelengths in the recording and reconstruction processes (${\lambda }_0\ne {\lambda }_c$) permits us to accomplish this condition without using prisms. Noteworthily for this case, the guided wavelength is smaller than the recording wavelength, contrary to the transmission geometries where a wavelength greater than the recording one can be guided.

This can be explained by a general fact: if a grating vector $\mathit{K}$ is recorded with a reference and an object beam and, then, the reconstruction is performed with a greater wavelength, the angle between the reference and diffracted beam becomes also higher. Specifically, if (1) and (4) are combined, then$$\frac{1}{{\lambda }_c}\sin \left|\frac{\widehat{\left(\mathit{\rho },\mathit{\sigma }\right)}}{2}\right|=\frac{1}{{\lambda }_o}\sin \left|\frac{\widehat{\left({\mathit{k}}_{\mathit{o}},{\mathit{k}}_{\mathit{r}}\right)}}{2}\right|.$$(5)

Therefore, to achieve TIR propagation in the reflection schemes, as in Figure 1, we need to reduce this angle as the wave vectors are located at different half z-planes; one in $z>0$, the other in $z<0$. For transmission geometries, where both wave vectors share the same region, the angle between them needs to be higher and, hence, the intended operating wavelength needs to be greater than the recording one.

The use of this kind of grating in a waveguide combiner can be straightforwardly assembled if two complementary couplers are placed on the same substrate. Hence, the coupled light from the first one begins to propagate under the TIR, until it reaches the second grating, where it is decoupled, as they both share the same Bragg angle. Moreover, if the multiplexing technique is incorporated into the design, one may build a RGB system.

3 Experimental methods

3.1 Holographic recording setup

Aiming to produce the gratings with the recording geometry described in the preceding section, a two-beam holographic set-up has been used, as represented in Figure 2. The beam from the Excelsior Continuous Wave Laser – 532 nm (Spectra-Physics, Santa Clara, CA, USA) is split into two secondary beams, whose diameter is increased to 1 cm after being spatially filtered and collimated. Also, its ratio can be monitored by the variable beam splitter BC. The mirror M4 ensures the optical path is the same for both arms, that ideally reach the sample with an intensity of ratio of $1$:$1$, inside the material, in order to enhance the fringe’s visibility and, thus, the refractive index modulation [13].

Figure 2 Experimental set-up. M: mirror, NDF: neutral density filter, BC: beam combiner BC: beam combiner, SF: spatial filter, L: lens, D: diaphragm. The photopolymer film is attached to a glass (both depicted in blue) that is placed upon a rotation stage to recombine the beams at the desired angles in the material.

In this particular case, we have set that $I_o=0.7$ mW/cm² and $I_r=0.35$ mW/cm², yielding an average exposure power of $0.5$ mW/cm² in the interference region, where both beams reach with the same intensity due to the correction with both the area’s projection and the Fresnel loss. In a previous work, we have proven that this range of intensities is optimal for different recording schemes [14].

3.2 Characterization of the gratings

In the next section, we will present experimental evidence about the see-through capability of the presented recording geometry. To do so, we need to provide a characterization of the gratings, ensuring that they have the desired properties.

A basic and essential property to study is the transmitted efficiency $\mathrm{TE}$: the ratio of the transmitted intensity from the grating ($I_t$) and from the cured material where there is no hologram ($I_0$)$$\mathrm{TE}=I_t/I_0,$$(6)

that can be ultimately converted to the relative diffraction efficiency $\eta$ in the volume hologram regime through $\eta =1-\mathrm{TE}$. For the following analysis, it is important to note that we work under s-polarization ($\perp$).

The first aspect to study is the optimal exposure energy that allows the hologram to be fully recorded in the material. As the Bayfol HX 200 has a broad absorption spectrum [15], real-time reconstruction is not possible. In this work, we have recorded samples with different exposure conditions, through the exposure time and the intensity ratio, in order to estimate the sensitivity and the dynamic range of the material. The set-up of Figure 2 enables to study the angular response in the first case, while a spectrophotometer (Jasco 650-V UV-Vis) is used in the second one. The information about the diffraction peaks in the spectral scan is then used to corroborate the grating’s geometry quantitatively, instead of the angular information where the grating is less selective [7], but also offers qualitative information about the hologram’s performance.

Additionally, for an estimation of the refractive index modulation $n_1$ from (2) and the optical thickness of the hologram $d$, Coupled-Wave (CW) theory can be used [12, 16]. Hence, the response of the grating (through its transmitted or diffracted efficiency) can be determined as a function of either the angle of incidence for a reconstruction wavelength ${\lambda }_c$ or the wavelength for a fixed angle of incidence $\theta$. Comparing the latter expressions with experimental data of TE as a function of the corresponding variable, $n_1$ and $d$ are determined.

Lastly, we validate both the Bragg condition near normal incidence, through the playback angle ${\theta }_p$ and the propagation of the diffracted beam through TIR if the intended ${\lambda }_c=473$ nm is used. In this case, we also use an Excelsior Continuous Wave Laser – 473 nm (Spectra-Physics, Santa Clara, CA, USA) as the laser source.

4 Results and discussion

4.1 Exposure conditions

As stated in Section 3, the optimal exposure range of the photopolymer is studied from the transmission response at Bragg incidence as a function of the exposure conditions. In this sense, the results are shown in Figure 3.

Figure 3 TE of different samples as a function of exposure time $t$, with intensities of $I_o=0.7$ mW/cm2 and $I_r=0.35$ mW/cm2, measured in air, and ${\lambda }_0={\lambda }_c=532$ nm.

The sensitivity of the material, in this particular scenario, is proven to be about $15$ mJ/cm² according to the mean power exposure during the recording process, which is the threshold to produce an efficient hologram with the presented recording geometry. Also, one may reason that the inherent uncertainty of the measuring process accounts for the repeatability of the same response through different samples, as seen from the higher exposure values in Figure 3.

4.2 Angular and spectral selectivity of the gratings

To illustrate the dispersion properties of the designed gratings, we characterize the transmission response of a probe hologram in the saturation regime of Figure 3 through either an angular or a spectral scan. First, experimental data of TE with ${\lambda }_c={\lambda }_0=532$ nm is measured as a function of the incident angle, plotted in Figure 4. Then, experimental data of TE is plotted in Figure 5 as a function of the recording wavelength for a certain angle of incidence $\theta$. Also, an estimation of the refractive index modulation $n_1$ and the thickness $d$ of the hologram is achieved with the angular scan in Figure 6, where ${\lambda }_0\ne {\lambda }_c=473$ nm.

Figure 4 Experimental values of TE as a function of the incident angle in air $\theta$ with ${\lambda }_c=532$ nm.

Figure 5 Experimental values of TE as a function of the reconstruction wavelength ${\lambda }_c$ with different angles of incidence $\theta =0^{\circ }$ (blue) and $\theta =30^{\circ }$ (green).

Figure 6 Experimental values of TE (points) as a function of the incident playback angle in air ${\theta }_p$ with ${\lambda }_c=473$ nm and the corresponding CW curve (solid).

At this point, the recording wavelength is used for the reconstruction, so the Bragg condition is fulfilled for incidence at the corresponding recording angles, $+30^{\circ }$ and $+60^{\circ }$, from Figure 1. It is stated earlier that the transmitted intensity from the hologram is compared with the equivalent intensity in the case where no grating is recorded, so TE not being $100\%$ for the complete angular range in Figure 4 indicates that there is an energy trade-off from the 0th-order to the diffracted ones. Both minimum occur at the designed recording angles, which provide the expected response.

As seen in Figure 5, one may recognize the main features of this grating: when impinging with one of the recording angles, the hologram has a diffraction maximum at the corresponding recording wavelength $532$ nm. Moreover, at normal incidence, there is a significant peak near $473$ nm, whose diffracted counterpart is intended to be guided along the glass substrate.

In detail, for this latter case, there is a $2\%$ discrepancy between the measured and the designed wavelength at normal incidence, that may be explained with a shift in the grating structure: $\mathrm{\Lambda }$ and $\phi$ may be directly computed considering the experimental values of both peaks (${\theta }_1=0^{\circ },{\lambda }_1=482.0$ nm), (${\theta }_2=30^{\circ },{\lambda }_2=532.0$ nm) and its uncertainties $\Delta \theta =1^{\circ },\Delta \lambda =0.2$ nm using (3), thanks to which it holds:$$\frac{\cos ({\theta }_1^{\text{'}}-\phi )}{{\lambda }_1}=\frac{\cos ({\theta }_2^{\text{'}}-\phi )}{{\lambda }_2}$$(7)

Then, with (7) and values from Figure 5, it yields a slant angle of $\phi =-25.8\pm 1^{\circ }$ and a grating period of $\mathrm{\Lambda }=177.9\pm 1.3$ nm; or, conversely, a spatial frequency of $5620\pm 40$ lines/mm. Recalling the purpose of this work, this HOE exhibits the expected features and the quantitative differences, which may be caused by small deviations during the recording process, are not significant to our cause as it is checked in Section 4.3.

Subsequently, we may proceed with the estimation of the thickness and the refractive index modulation. It is important to note that the whole information about the angular or spectral response is required to do so, while for the calculations with (7) only the peak values are necessary. In this case, the 473 nm-beam in Figure 2 is used, as it provides the intended operating wavelength.

Turning now to the experimental evidence, using ${\lambda }_c=473$ nm, best-fitting refractive index modulation of $n_1=0.027\pm 0.002$ and $d=14.3\pm 0.1$ μm are obtained, which minimize the RMS between the experimental and the theoretical values, as plotted in Figure 6, where the transmitted efficiency is measured using (6) as a function of the playback angle ${\theta }_p$.

There is a good agreement between the experimental data and the theoretical curve from the CW theory, where the refractive index modulation is consistent with other estimations [5].

It is remarkable that the fitted optical thickness is less than expected for the photopolymer Bayfol HX 200, which is typically 16 μm [11]. This fact may be due to the presence of a subsidiary grating that emerges from the interference between one of the different 2-beam combinations, if accounting for the reflected beams in the recording process, which is also supported by the observation that there is not 100% efficiency between the central lobe and the secondary peaks. These also may explain the transmitted component in the corresponding curve from Figure 5, which can be isolated in this latter case.

4.3 Experimental validation

As the final step to study the viability of the proposed geometry, we present here Figure 7 in which the sample shows the desired behavior that has just been mentioned: the reconstruction with ${\lambda }_c=473$ nm yields a diffracted beam that is guided through the glass due to TIR. Indeed, this complete and complement the information in Figures 5 and 6, which also confirmed this near-normal Bragg condition.

Figure 7 Reflections gratings, recorded with the geometry in Figure 1, acting as in- and out- couplers under blue-light reconstruction.

Hence, the grating acts as a reflection holographic coupler, whose area is the hologram’s size, about $1\times 1$ cm². The whole system, thus, performs as a waveguide combiner, as the schematic in Figure 7a depicts, with see-through capability where an image can be retrieved which do not affect the observer’s surrounding environment as in Figure 7b.

5 Conclusions

A new recording geometry has been proposed to produce holographic reflection gratings in photopolymers, devoted to working as couplers in a see-through device. The inspection of the theoretical background has led us to a specific configuration where prisms are not needed, thanks to the ratio of the recording and the reconstruction wavelength being different from unity.

The proper functioning of the designed gratings has been proven and validated in the lab. The sensitivity for the concrete geometry has been found to be around $15$ mJ/cm², using Bayfol HX 200 as the photopolymer film, for which the grating vector $\mathit{K}$ has been experimentally obtained using the Bragg wavelength for different angles where only subtle differences have been found. Also, the estimated values for the refractive index modulation and the optical thickness of the material hinted the presence of subsidiary gratings. As these geometries share the Bragg condition partially with each other, the main purpose of the hologram is not affected: we confirmed that the grating has a near-normal incidence Bragg condition if reconstructed with ${\lambda }_c=473$ nm and the corresponding diffracted beam propagates under TIR in the glass substrate.

The presented evidence allows us to look forward to further experimental investigation, either by using a different photopolymer (like the HPDLC, with tuneable capacities after the recording process [17]) or by studying the imaging properties of the system when two complementary gratings are placed in the same glass substrate, with both the in-coupler and the out-coupler as a whole.

Funding

This work was funded by the “Generalitat Valenciana” (Spain) (IDIFEDER/2021/014, cofunded by EU through FEDER Programme; PROMETEO/2021/006 and INVEST/2022/419 financed by Next Generation EU), “Ministerio de Ciencia e Innovación” (Spain) (PID2021-123124OB-I00).

Conflicts of interest

The authors declare that they have no competing interests to report.

Data availability statement

Data associated with this article can be provided through direct contact and under the authors’ permission.

Author contribution statement

J.J.S-V. performed the experiments, analysed the data and wrote the paper; J.C.B and J.C-M. implemented the computer code regarding the data analysis; G.N. assisted with the instrumentation and optimized the experiments; C.N. formulated the research goal, wrote, reviewed and edited the paper; J.F validated the overall reproducibility of the results and managed project administration; S.G. formulated and conceptualized the research goal, designed the experiments and wrote the paper; A.B. conceptualized and supervised the research activities and has been responsible of the acquisition of the financial support. Each author has actively contributed to this work and has reviewed and approved the final version of the manuscript.

References

All Figures

Figure 1 Ewald’s spheres for the selected geometry, using ${\lambda }_r=532$ nm for the recording process and ${\lambda }_c=473$ nm as the designed wavelength for the reconstruction stage. The recording structure (green) is built upon the recording angles in the material: $19.4^{\circ }$ and $35.1^{\circ }$. The equivalent assembly for the reconstruction step (blue) yields angles of $1.0^{\circ }$ and $55.5^{\circ }$. The average refractive index is $n_0=1.505$, so the critical angle is ${\theta }_c=41.6^{\circ }$.

In the text

	Figure 2 Experimental set-up. M: mirror, NDF: neutral density filter, BC: beam combiner BC: beam combiner, SF: spatial filter, L: lens, D: diaphragm. The photopolymer film is attached to a glass (both depicted in blue) that is placed upon a rotation stage to recombine the beams at the desired angles in the material.
In the text

	Figure 3 TE of different samples as a function of exposure time $t$, with intensities of $I_o=0.7$ mW/cm2 and $I_r=0.35$ mW/cm2, measured in air, and ${\lambda }_0={\lambda }_c=532$ nm.
In the text

	Figure 4 Experimental values of TE as a function of the incident angle in air $\theta$ with ${\lambda }_c=532$ nm.
In the text

	Figure 5 Experimental values of TE as a function of the reconstruction wavelength ${\lambda }_c$ with different angles of incidence $\theta =0^{\circ }$ (blue) and $\theta =30^{\circ }$ (green).
In the text

	Figure 6 Experimental values of TE (points) as a function of the incident playback angle in air ${\theta }_p$ with ${\lambda }_c=473$ nm and the corresponding CW curve (solid).
In the text

	Figure 7 Reflections gratings, recorded with the geometry in Figure 1, acting as in- and out- couplers under blue-light reconstruction.
In the text