1. Introduction

Since the birth of the Augmented Reality (AR) concept in early 1990s, AR has drawn tremendous attention in many fields [1–2]. However, the traditional diffractive optical elements used as the core of AR devices, like volume holographic grating and surface relief grating [3–5], still have many problems, including low diffraction efficiency, non-adjustable refractive index, small diffraction angle, and complex manufacturing process. These lead to low efficiency and a large volume of the entire device, seriously hindering their applications in high efficiency and comfortableness desired products, such as Head Mounted Display (HMD), smart glasses, etc. Compared with the traditional diffractive coupling component, the reflective polarized volume grating (RPVG) has high diffraction efficiency, larger diffraction angle, and polarization selectivity for the waveguide coupling element. Many works have shown that RPVGs are very suitable for the current AR near-eye display system [6–14].

To the best of our knowledge, the RPVG as one of the holographic waveguide display elements, has a two-dimensional anisotropic periodicity by periodically rotating the liquid crystal (LC) molecules in a two-dimensional direction, retaining the high-efficiency single-order diffraction characteristics, possessing a large angular bandwidth, and the polarization selectivity to the incident light [6,7]. Such excellent performance makes it a favorable competitor among the waveguide coupling components. For the holographic waveguide display system, it is not only necessary to investigate the optical characteristics of RPVG, but also to realize imaging simulation based on RPVG. Till now, the diffraction analysis methods for RPVG mainly focus on the rigorous coupled wave analysis (RCWA) and finite element methods (FEM) based on electromagnetic field simulation models [7,11,15–19]. Although, there are various analytical models for imaging simulation such as Zemax Opticstudio (Zemax), CODEV, and VirtualLab Fusion, few reports that so many imaging software is applied to simulate the RPVG. So combining these theories with practical imaging simulation tools would be very meaningful for this RPVG. Furthermore, imaging simulation can not only be used to simulate the imaging quality of the RPVG, but also be guiding for the entire holographic waveguide display device, so it is a very valuable work for the design of the waveguide display system utilized in the near-eye device.

In recent years, many meaningful researches have been carried out on the RPVG, and mainly focus on the diffraction characteristics through the RCWA and FEM [7–19]. Specifically, Weng et al. reported firstly the numerical analysis using COMSOL Multiphysics (COMSOL) software based on the FEM to explain the polarization selectivity and realized a unique 2D/3D display model with the RPVG used as the in-coupling and out-coupling gratings [7]. Lee et al. also chose the FEM to investigate the properties of RPVG via the COMSOL Multiphysics, and the simulated results agreed well with the experiment [11]. Xiong et al. developed the RCWA to research various liquid crystal polarization gratings [19]. Despite these excellent efforts, many practical RPVG samples with such high diffraction efficiency can be obtained, but the imaging simulation of the unique RPVG has not been realized as the coupling component. In our previous work, a ray-tracing model in Zemax for the imaging simulation of volume holographic grating was built based on coupled wave theory [20]. Nevertheless, commercial optical software such as Zemax only supports the simple grating structure and isotropic materials which is not applicable to the anisotropic RPVG, and there are few reports of applying such excellent RPVG for imaging simulation. Herein, how to solve it is therefore a must for imaging simulation of the RPVG in the holographic waveguide display system.

In this paper, we introduce the self-build RCWA model for RPVG. The polarization characteristics are studied thoroughly firstly based on the RCWA in detail. Then, the RPVG model as the in-coupling and out-coupling grating is integrated into the commercial optical software Zemax through the dynamic link library (DLL) to realize the imaging simulation of the holographic waveguide display system. This work provides an entirely analysis methodology from polarization behavior to the imaging simulation of the RPVG based on the RCWA.

2. Slanted structure of RPVG

In terms of structure and principle, this kind of RPVG can be regarded as a combination of Volume holograph grating (VHG) and Pancharatnam-Berry (PB) phase grating [6–8]. As shown in Fig. 1, there is the incident light beam along the direction of the y-axis. The same rotation pattern of the optical axis in the x direction as the PB phase grating, but the optical axis shows a helical structure on the y-axis because of the chiral-doped liquid crystal. Recently, many research groups reported the existence of slanted angle in this structure during the spin-coated produce, which the helix axis of the liquid crystal molecule is inclined to the substrate normal [13,14]. Such a unique slanted RPVG has the characteristics of the conventional gratings that can produce single-order diffraction with high diffraction efficiency when the Bragg condition is satisfied, and possesses polarization sensitivity characteristics because of helical structure of chiral liquid crystal.

 

Fig. 1. A schematic diagram of the proposed slanted RPVG.

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In Fig. 1, the yellow dashed line is similar to the Bragg period plane which realizes the Bragg diffraction. The angle between this Bragg plane and the x-axis is defined as φ. The Bragg period is expressed as the P_B. In principle, enough thickness of the RPVG is designed to realize the Bragg condition, single-order diffraction with high diffraction efficiency when the Bragg condition is met. When the incident angle is 0°, the Bragg condition of RPVG can be expressed as:

3. Polarization behavior analysis of RPVG

The rigorous coupled wave analysis (RCWA) is a relatively simple and direct method for investigating the electromagnetic fields of the periodic grating structure. By expanding the electromagnetic field in the refractive index modulation region according to the diffraction order, combining with Maxwell's equations and matching the boundary conditions, the diffraction characteristics of each diffraction order can be obtained, which is very suitable for analyzing the periodic diffraction structure. Especially for the realization of subsequent imaging simulation based on the RPVG, the diffraction parameters can be easily linked to optical software data interface for ray tracing, so as to realize imaging simulation. Based on many previous efforts [15–19], the RCWA program has been implemented for RPVG via using the Matlab in this work.

3.1 Response bandwidth of RPVG

The bandwidth response characteristics of RPVG are very important for holographic waveguide display system which is also affected by the angle of incidence. The simulated diffracted efficiency dependence of the spectral response of RPVG with incidence angle increasing from -80° to 80° is depicted in the Fig. 2. Noted that the center wavelength of RPVG, the slanted angle and grating thickness are set for 550nm, 30° and 3μm respectively in the RCWA model. On the whole, the efficiency distribution is a parabolic curve which is symmetrical about the tilt angle (30°). Specifically, when the incident angle meets the 0° and 60°, although the efficiency is comparatively similar, a wider spectral response bandwidth (40nm to 90nm) appears at the 60°. And as the incident angle is reduced to 30°, the phenomenon of corresponding wavelength red-shift appears (560nm to 620nm), which is mainly attributed to the special RPVG structure of the inclination. Simultaneously, in practical applications, the bandwidth of the light source and the total internal reflection conditions of the waveguide also need to be considered, so generally the angle of incidence will be controlled within a relatively small range. For instance, the OLED light source is chosen to simulate the real condition whose center wavelength is 532nm. As shown in the insert of Fig. 2, the response angular bandwidth of the incident light in the waveguide medium can reach approximately 10°. This means that employing a multiple layer gratings structure, a bigger field of view can be obtained by expanding the wavelength bandwidth.

 

Fig. 2. Simulated diffracted efficiency dependence of the spectral response of RPVG with incidence angle increasing from -80° to 80°. The inset shows the enlarged part and the OLED image source. The color bar is the diffracted efficiency.

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3.2 Polarization properties of RPVG

In order to minimize energy loss during propagation, it must guarantee that the light beam from the RPVG and subsequently into the waveguide is circularly polarized with the same chirality as the incident light. Thanks to the reliable RCWA model, the novel polarization behavior can be investigated systematically in this section. The typical performance of RPVG is its unique polarization sensitive to the polarization state of incident light beam. The polarized behavior of RPVG originates from the periodic rotation of the optical axis of the anisotropic liquid crystal material. Therefore, the different tilt angles of the RPVG will inevitably affect the polarization characteristics of RPVG. Since incident linearly polarized (LP) light can be regarded as the sum of left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) light, the diffraction efficiency of RPVG for LP light is about half of that of LCP or RCP light. This known conclusion from the following polarization analysis can be verified, thus showing the accuracy of our self-built RCWA model. Then the RCWA model is utilized to simulate the grating diffraction efficiency of different polarized incident light under various tilt angle. It is worth noting that the polarization state of the incident light is expressed by Jones matrix in local coordinates as:

It can be seen from the Fig. 3(a) that when the helical structure of the liquid crystal molecules is right-handed, the highest diffraction efficiency of RPVG corresponds to the polarization state of the incident light is RCP (ψ= -45°, Δφ= -90°). The lowest point of diffraction efficiency corresponds to LCP (ψ= -45°, Δφ= 90°) that means the diffraction efficiency is almost zero and no diffracted light. The diffraction efficiency of LP (ψ= 0°, -90°, and Δφ= 0°, 180°, -180°) is about 50%, which is consistent with previous assumption. Furthermore, the RPVG with tilted angle is more sensitive to the polarization state of the incident light than the planar one (φ =0°). Although there are the highest and lowest diffraction efficiency, the state of incident light deviates from the circularly polarized light, and which means the slanted structure has a certain effect on efficiency control during the experiment.

 

Fig. 3. The diffraction efficiency of RPVG with different incident polarization states at various slanted angles, (a) φ = 0°, (b) φ = 30°, and (c) φ = 60°. The incident polarization states are expressed by Δφ and ψ as defined in Eq. (4) and the color bar is the diffraction efficiency.

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Then, the angle and wavelength response characteristics at the maximum diffraction efficiency are investigated. It is shown in Fig. 4(a) that the highest efficiency point is no longer at normal incidence with the right-handed circularly polarized light incident, but the highest diffraction efficiency can be achieved at the incidence angle of -3.5°. The angular response curve also shows that when incident light corresponds to the highest efficiency polarization state, the highest efficiency is still obtained at normal incidence, and it is similar to the wavelength response curve. In order to obtain high-efficiency single-order diffraction efficiency, the polarization state of incident light should be strictly maintained during the experiment. These results also provide a design idea for the structure of the waveguide that the polarization incident should match the corresponding incident angle.

 

Fig. 4. (a) Angular response and (b) Wavelength response of the RPVG with different incident polarization state at φ=30°.

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Whether the diffracted light obtained after passing the in-coupling grating can still propagate in the waveguide while maintaining the original polarization characteristics is of importance to the waveguide imaging system. In order to figure out the polarization of diffraction light under the various state of the incident light, the Stokes parameter S₃ is applied for describing the polarization of diffracted light. For example, S₃= 1 means RCP light, S₃= -1 means LCP light. The polarization condition of light is closest to circularly polarized when the absolute value of S₃ is near to 1. The polarized state of the diffraction light is also investigated as the various incident polarized light under different slanted angle. As seen in the Fig. 5, the area enclosed by the three white dotted circles corresponds to the point of lowest diffraction efficiency in Fig. 3. These simulation results show that only the incident polarization state corresponding to the lowest diffraction efficiency will have a greater impact on the polarization state of the diffracted light, but there is almost no diffracted light at this condition (diffraction efficiency is near to zero). There is a conclusion that the polarization state of the diffracted light is almost irrelevant to the polarization state of the incident light. Therefore, when the Bragg condition is satisfied, the polarization state of the diffracted light has not been affected by the polarization state of the incident light, which means obtained diffracted light cross the in-coupling RPVG would maintain the circular polarized state propagating in the waveguide when the incidence light is the circular polarized.

 

Fig. 5. The stokes parameter S₃ with different incident polarization states is determined by different slanted angles, (a) φ = 0°, (b) φ = 30°, and (c) φ = 60°. The incident polarization states are expressed by Δφ and ψ as defined in Eq. (4) and the color bar is the stokes parameter S₃.

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4. Imaging simulation

4.1 Implementation and verification of the DLL

Based on the analysis of the polarization behavior of the RPVG in the previous section, this part focus on mainly realizing the application of the diffraction coupling component as the RPVG in the imaging simulation software ZEMAX. Herein, the available commercial software ZEMAX is applied to analyze the imaging simulation and the working modes of ZEMAX software is set for non-sequential mode, in which the light is no longer traced in the order of the surface, and the propagation direction of the light beam is completely determined by the incident direction of the light and the physical location and properties of the diffracted components (RPVG). In addition, the object shape and diffraction characteristics of diffractive elements (including various gratings and holographic lenses) in non-sequential mode are defined separately, that is to say, an external dynamic link library (DLL) program can be used to define the order, propagation direction, and polarization state of the diffracted light. The RCWA model is very suitable for interacting with the software via the DLL having the RPVG diffractive behavior.

The DLL with RPVG diffraction behavior is compiled by C++ that contains two functions, UserDiffraction and UserParamNames. ZEMAX transfers data array to these two functions which contains the normal of the surface of the object, the refractive index before and after the surface, the direction cosine of the incident light, the wavelength, the polarization state information, the coordinates on the surface, and the diffraction of the light beam, etc. The diffraction DLL file in non-sequential mode, the direction of the emitted light needs to be defined first. In order to provide complete light beam data for ray tracing, it is necessary to ensure the orthogonality of the wave vector in the electromagnetic field, that is, the electric displacement vector must be perpendicular to the light propagation vector. Therefore, the cosine directions on the three coordinate axes and the electric field complex amplitudes in the three directions in space need to be defined. The real and imaginary parts of the electric field complex amplitude in each direction are stored in one data, so there are nine variables. The advantage of this method is that the diffracted light energy is included in the electric field amplitude, and can be calculated by RCWA.

Since the ray tracing algorithm of ZEMAX treats the light as an independent light beam, when the total energy of the light source does not change, the more the number of light is set, the smaller the amplitude of the electric field transmitted to the DLL program. Therefore, when defining the polarization state and diffraction efficiency of the diffracted light beam, the normalized electric field amplitude calculated by the RCWA model cannot be directly transferred to ZEMAX. The diffraction electric field is described as below:

 

Fig. 6. (a) Non-sequence model schematic diagram, (b) Input interface of the reflection volume holographic grating.

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In order to check the accuracy of this diffraction element of the RPVG, which is set to be sensitive to RCP, and the polarization state of the light source is set to RCP at the normal incident. Then, the holographic lenses are selected as the geometric object of the RPVG grating in the non-sequential element editor. The energy of the light source is set to 1 lumen and the three blue objects are all photodetectors, which are used to detect the energy of the 0-th order reflection, 1-th order diffraction and 0-th order transmitted light of RPVG respectively. Finally, the actual diffraction effect of RPVG in Zemax can be obtained under making the above preparations.

After the ray tracing is completed in the Zemax, the energy measured by each photodetector is shown in the Figs. 7(a)–7(c). The energy of 1-th order diffracted light, 0-th order reflected light and 0-th order transmitted light are 0.99118 lumens, 1.0557E-3 lumens and 7.7601E-7 lumens, respectively. On the contrary, the polarization characteristics of the incident light source to LCP, the measured results are shown in the Figs. 7(d)–7(f) with other parameters unchanged. The energy of the 1-th order diffracted light, the 0-th order reflected light, and the 0-th order transmitted light are 9.1528E-5 lumens, 8.0277E-2 lumens and 0.91963 lumens respectively. From this verification, it is concluded that the RPVG diffraction element based on RCWA is reliable and suitable for imaging analysis in ZEMAX.

 

Fig. 7. The energy distribution of (a)-(c) Right- and (d)-(f) Left- handed circularly polarized light (RCP or LCP) at 1-th order diffracted light, 0-th order reflected light and 0-th order transmitted light respectively.

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4.2 Realizing of imaging simulation in waveguide

Based on the realization of the RPVG diffractive element in the non-sequential mode, the one-dimensional pupil expansion holographic waveguide display system is simulated appropriately in this section. The one-dimensional pupil expansion holographic waveguide sheet established in the non-sequential mode. As shown in the Fig. 8(a), the in-coupling and out-coupling RPVGs are both rectangular for simulation illustration, and the image used for simulation input is shown in Fig. 8(b). These parameters of the in- and out-coupling grating are the same, and they are symmetrically attached to the lower surface of the waveguide.

 

Fig. 8. (a) Top view of One-dimensional pupil coupling gratings structure diagram, (b) The image used for simulation input.

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Since the symmetry of in-coupling and out-coupling grating and the image of the simulation result is reversed from the original image, so the image is rotated by 180° for pretreatment. Here, without affecting the simulation results, for simplicity, we use ideal lenses instead of collimating systems, ideal lenses and color detectors instead of human eye, and rectangular light sources and slides instead of micro-display image sources. Firstly, the single-layer waveguide structure is simulated under the green source. The real simulation process diagram can be seen from the Fig. 9(a) and there are dark stripes on the image in the Fig. 9(b). This is probably ascribed to the RPVG diffraction efficiency of similar sidebands shown in the Fig. 4(a). This overall image result color is green mainly due to the wavelength selectivity of the RPVG in Fig. 4(b). This kind of monochromatic waveguide is very useful in special scenes, such as observing the dark circumstance.

 

Fig. 9. One-dimensional expansion waveguide shaded configuration model in Zemax. (a) Single-layer waveguide shaded configuration with single green source, (b) Simulation results of the retina image with incident light at 532nm, (c) Three-layers waveguide shaded configuration with full color, (d) Full-color image output with a natural wide spectrum light source.

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As we known, the response bandwidth can be increased by increasing refractive index modulation, but it is difficult to realize refractive index modulation due to the limitation of the liquid crystal material. An effective method to increase the angular response bandwidth is to use multilayer waveguide, and in which are stacked together to form a laminated waveguide with different diffraction angles. As previously stated, it is feasible to utilize a multiple layer grating structure to increase the wavelength bandwidth. This means that the red (630nm), green (532nm) and blue (457nm) RPVG each uses a waveguide, so that achieve separate propagation of each single waveguide under the same diffracted angle, reducing the cross talk of each grating in the Fig. 9(c). With presetting the same diffraction angle, the parameters of RPVG in each waveguide layer can be determined feasibly. Eventually, the real full color image in the Fig. 9(d) when the wavelength of the incident light includes 630nm, 532nm, and 457nm is obtained. Compared with the single-layer waveguide, the imaging effect of the three-layers waveguide has been greatly improved in color display. Considerable efforts will be done to increase the field of view of RPVG-based waveguide display system by elaborately expanding the dimensional of the exit pupil.

5. Conclusion

In this work, the combination between the RCWA model for the RPVG and imaging simulation of RPVG as the coupling component is realized. Based on this RCWA model the polarization response of the RPVG to incident light is studied by introducing the tilt angle, and the RPVG polarization behavior are transferred to the Zemax software through the self-compiled dynamic link library. The energy distribution in different polarization incidence, one-dimension waveguide and three-layer waveguide system are mainly illustrated. These single green and full color images are obtained via the practical designed waveguide structure. This work briefly and closely combines theoretical modeling and imaging simulation as an entire via the RCWA, which can provide effective guidance for diffraction element design and imaging simulation for near-eye holographic waveguide displays.