Pre-compensation method for optimizing recording process of holographic optical element lenses with spherical wave reconstruction

Jiwoon Yeom; Yeseul Son; Kwang-Soon Choi

doi:10.1364/OE.405408

1. Introduction

Holographic optical elements (HOEs) have been given considerable attention in research fields of augmented reality (AR) near-to-eye displays (NEDs) and three-dimensional (3D) displays by the virtue of high transparency and thin film structure [1,2]. The HOEs transform an incident wave into the wavefront with pre-defined optical functions, when it satisfies the Bragg condition of volume gratings. On the other hand, most of incident light from the real scene passes through the HOEs without the diffraction, as it deviates from the Bragg condition. Based on these advantages, HOE-based see-through displays have been actively studied for various applications such as waveguide-based NEDs for presenting virtual images at optical infinity [3,4], direct-retinal-projection displays [5], integral imaging displays [6–8], and holographic displays [9–11] with see-through properties.

Figure 1(a) shows a schematic of HOE lens-based NED with a spherical wave reconstruction, which enables a compact system for displaying AR contents, in the ideal case. However, when the incident light on the HOE deviates from the Bragg condition, a diffraction efficiency decreases and a diffracted wave directs wrong direction. As in Fig. 1(b), the Bragg mismatched probe wave hinders the observers from watching correct AR contents, and reconstructed images suffer from the color breakup problem and low diffraction efficiency. Researchers have explored the optical characteristics of HOEs for the light with deviating from the Bragg condition due to the mismatch between the recording and displaying process [12–14]. However, those studies were limited to characterizing the behavior of diffracted wave with the Bragg mismatching condition of HOE lenses for the case of being recorded with a plane reference wave, and focusing on how to utilize the diffracted light in the off-Bragg regime.

Fig. 1. Schematics for HOE lens-based NEDs with the spherical wave reconstruction, when the wavelength of probe wave is (a) identical to, and (b) differed from the recording wavelength of HOEs.

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In case of AR NEDs, light sources in the displaying process are small sized laser diodes (LDs) or light-emitting diodes (LEDs), and their wavelengths usually differ from those of diode pumped solid state (DPSS) lasers used in the recording process, which leads to the Bragg mismatched reconstruction. Hence, the solution for compensating the wavelength mismatch between the recording and displaying process is necessary. For a periodic grating generated by the interference between two plane waves, the angle shift method of probe wave was presented for solving the wavelength mismatch problem [15]. However, in practice, complicated and aperiodic gratings rather than the periodic grating are used, and it is not preferred to shift or change the probe wave for implementing compact NEDs. In [14], a parameter sweep method was utilized to compensate the wavelength mismatch in the recording scheme of HOE lens with the plane reference wave. However, a theoretical analysis was not fully conducted, and the parameter sweep method involves a high computational cost as the optical functions to be recorded in the HOEs become complicated. Also, such a plane wave reconstruction of HOE demands large space with the long optical path due to additional collimation and demagnification optics for projection systems [1], which deteriorates the compactness of AR NEDs.

In this paper, we propose a wavelength compensation strategy for optimizing the recording process of HOE lens, when the spherical waves are used in the recording and displaying process. With considering the shrinkage effects of photopolymers, the proposed method could provide full-color HOE lenses without the color breakup problem, and allow a simple projection system without the collimation optics in the displaying process by the virtue of spherical wave reconstruction. In Section 2, principles of proposed method are presented for optimizing the recording scheme of HOE lenses based on a localized approximation for the aperiodic volume gratings. Then, the simulation results using the volume hologram model of OpticStudio are presented to verify that the proposed method solves the degradation of HOE lenses due to the wavelength mismatch between the recording and displaying process. In Section 3, a wavelength compensated full-color HOE lens is fabricated using the spherical waves as both of the reference and signal waves, and displaying experiments using a compact projection system are presented to validate the principles of the proposed method.

2. Principle

2.1 Wavelength compensated recording process

Figure 2(a) shows a desirable schematic for the displaying process of HOE lens-based NED, when the probe and reconstructed wave is the spherical waves with wavelength of λ_d. The desirable gratings in a photopolymer are formed by the interference patterns between two spherical waves, and have aperiodic structures with varying grating vectors according to the position. If we divide the whole region in the gratings of HOE into sufficiently large number of sub-regions, a local grating in each sub-region could be represented as a periodic grating generated by interference between two plane waves [16,17]. Such a localized approximation is valid as the reference and signal waves in the photopolymer vary slowly within each sub-region.

Fig. 2. Diagrams for the wavelength compensated recording process: (a) a desirable displaying geometry using spherical wave reconstruction with the wavelength of λ_d, (b) a diagram for the probe and reconstructed waves in the photopolymer, (c) required reference and signal waves in the shrinkage compensated photopolymer, analysis using the wavevector space diagrams for (d) displaying process, and (e) recording process.

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For simplicity of the analysis, a two-dimensional (2D) coordinate system is used in Fig. 2, where the origin lies at the center of photopolymer. In this section, the subscript d and r represents that the parameter corresponds to the displaying and recording process, respectively. For the sub-region of photopolymer as shown in Figs. 2(a) and 2(b), we define the wavevectors of probe and reconstructed waves in the displaying process as k_1,d=[k_1,d,x, k_1,d,z] and k_2,d=[k_2,d,x, k_2,d,z], respectively. A desirable local grating at the sub-region is represented using the grating vector K_g,d=[K_g,d,x, K_g,d,z], which is expressed as

(1)$${{\boldsymbol K}_{{\boldsymbol g,d}}} = {{\boldsymbol k}_{{\boldsymbol 2,d}}} - {{\boldsymbol k}_{{\boldsymbol 1,d}}},$$

where |k_1,d|=|k_2,d| = 2πn/λ_d, and n represents the refractive index of photopolymer.

Since the desirable grating vector in the displaying process, K_g,d, is resultant one after the recording and curing process of photopolymer, the grating vector in the recording process has to be compensated considering the shrinkage effects. Figure 2(c) shows a virtual photopolymer with the extended thickness, where the extension rate is inversely proportional to the material shrinkage rate. If we assume that the photopolymer shrinks along the thickness direction [12], a grating vector to be recorded, K_g,r, is given by

(2)$${{\boldsymbol K}_{{\boldsymbol g,r}}} = [{{K_{g,d,x}},\alpha {K_{g,d,z}}} ],$$

where α denotes the shrinkage rate of material in the thickness direction, and 0≤α≤1.

In order to record the HOEs satisfying Eq. (2) by using the laser with wavelength of λ_r, the incident angles of reference and signal waves differ from those of probe and reconstructed waves in the displaying process. Figures 2(d) and 2(e) shows the wavevector space diagrams for the displaying and recording process, respectively. In Fig. 2(e), the wavevectors of signal and reference waves in the recording process have to satisfy the Bragg condition of K_g,r as in Eq. (3).

(3)$${{\boldsymbol k}_{{\boldsymbol 1,r}}} + {{\boldsymbol K}_{{\boldsymbol g,r}}} = {{\boldsymbol k}_{{\boldsymbol 2,r}}},$$

where k_1,r and k_2,r denote the wavevectors of reference and signal waves, respectively, and |k_1,r|=|k_2,r| = 2πn/λ_r.

In order to find the optimal pair of two wavevectors satisfying Eqs. (2) and (3), we introduce the bisect vector k_b,d and k_b,r corresponding to the displaying and recording process, respectively, which is expressed as

(4)$$\begin{array}{{cc}} {{{\boldsymbol k}_{{\boldsymbol b,d}}} = \frac{{{{\boldsymbol k}_{{\boldsymbol 1,d}}} + {{\boldsymbol k}_{{\boldsymbol 2,d}}}}}{2},}&{{{\boldsymbol k}_{{\boldsymbol b,r}}} = \frac{{{{\boldsymbol k}_{{\boldsymbol 1,r}}} + {{\boldsymbol k}_{{\boldsymbol 2,r}}}}}{2}} \end{array}.$$

Since a triangle defined by k_1,r, k_2,r, and K_g,r is the isosceles triangle, the bisect vector k_b,r is normal to the grating vector K_g,r. In the same manner, k_b,d is normal to K_g,d.

We could represent k_b,r using a positive coefficient β, which represents the magnitude of k_b,r as in Eq. (5).

(5)$${{\boldsymbol k}_{{\boldsymbol b,r}}} = \beta [{\cos {\theta_{g,r}},\sin {\theta_{g,r}}} ],$$

where θ_g,r is a slanting angle of grating for the recording process as in Figs. 2(c) and 2(e), and is equal to tan⁻¹(K_g,x/αK_g,z). From the relationship among k_1,r, k_2,r, and k_b,r, the coefficient β is expressed as

(6)$$\beta = \frac{{2\pi n}}{{{\lambda _r}}}\cos \left( {\frac{{\pi - ({{\theta_{1,r}} + {\theta_{2,r}}} )}}{2}} \right),$$

where θ_1,r and θ_2,r denotes incident angles of reference and signal waves in the recording process as defined in Figs. 2(c) and 2(e).

Also, from the relationship between K_g,d and K_g,r, following relationship is derived.

(7)$$\frac{{2\pi n}}{{{\lambda _d}}}\sin \left( {\frac{{\pi - ({{\theta_{1,d}} + {\theta_{2,d}}} )}}{2}} \right)\sqrt {1 - ({1 - \alpha } ){{({\cos {\theta_{g,d}}} )}^2}} = \frac{{2\pi n}}{{{\lambda _r}}}\sin \left( {\frac{{\pi - ({{\theta_{1,r}} + {\theta_{2,r}}} )}}{2}} \right),$$

where θ_1,d, θ_2,d, and θ_g,d is the incident angles of probe wave, reconstructed wave, and the slanting angle of grating for the displaying process, and cosθ_g,d=K_g,d,z/|K_g,d| as in Figs. 2(b) and 2(d). By combining Eqs. (6) and (7), β is express as

(8)$$\beta = \frac{{2\pi n}}{{{\lambda _r}}}\sin \left( {{{\cos }^{ - 1}}\left( {\frac{{{\lambda_r}}}{{{\lambda_d}}}\cos \left( {\frac{{{\theta_{1,d}} + {\theta_{2,d}}}}{2}} \right)\sqrt {1 - ({1 - \alpha } ){{({\cos {\theta_{g,d}}} )}^2}} } \right)} \right).$$

Consequently, k_1,r and k_2,r, which are the wavevectors to be recorded, are derived as Eqs. (9) and (10), respectively.

(9)$${{\boldsymbol k}_{{\boldsymbol 1,r}}} = {{\boldsymbol k}_{{\boldsymbol b,r}}} - \frac{1}{2}{{\boldsymbol K}_{{\boldsymbol g,r}}} = \left[ {\begin{array}{{cc}} {\beta \cos {\theta_{g,r}} - \frac{1}{2}({{k_{2,d,x}} - {k_{1,d,x}}} ),}&{\beta \sin {\theta_{g,r}} - \frac{\alpha }{2}({{k_{2,d,z}} - {k_{1,d,z}}} )} \end{array}} \right].$$

(10)$${{\boldsymbol k}_{{\boldsymbol 2,r}}} = {{\boldsymbol k}_{{\boldsymbol b,r}}} + \frac{1}{2}{{\boldsymbol K}_{{\boldsymbol g,r}}} = \left[ {\begin{array}{{cc}} {\beta \cos {\theta_{g,r}} + \frac{1}{2}({{k_{2,d,x}} - {k_{1,d,x}}} ),}&{\beta \sin {\theta_{g,r}} + \frac{\alpha }{2}({{k_{2,d,z}} - {k_{1,d,z}}} )} \end{array}} \right].$$

By recording the interference pattern between k_1,r and k_2,r with the wavelength of λ_r, the formed gratings after the curing process satisfy the Bragg condition for the probe wave of k_1,d and intended reconstructed wave of k_2,d.

In addition, Eq. (7) is also applicable to derivation of the relationship between the angle and wavelength deviation for providing the maximum diffraction efficiency in the HOE, which is useful to predict the field of view (FOV) for the HOE based-NEDs [4,18]. When we assume α=1, and define Δθ as the angle deviation between θ_1,d and θ_1,r due to the wavelength deviation, Eq. (7) is revised as

(11)$$\frac{{2\pi n}}{{{\lambda _d}}}\cos \left( {\frac{{{\theta_{1,d}} + {\theta_{2,d}}}}{2}} \right) = \frac{{2\pi n}}{{{\lambda _r}}}\cos \left( {\frac{{{\theta_{1,d}} + {\theta_{2,d}}}}{2} + \Delta \theta } \right).$$

The relationship of Eq. (11) is the generalized version of the incident Bragg angle range (IBAR) for a reflection volume hologram in Ref. [18]. The IBAR represents the angular and spectral range where the theoretical diffraction efficiency reaches nearly 100%, when the reference wave is normally incident on the periodic grating. In this circumstance, Eq. (11) could be simplified into Eq. (12), which is equivalent to the formula derived in Ref. [18].

(12)$$\Delta \theta = {\cos ^{ - 1}}\left( {\frac{{{\lambda_r}}}{{{\lambda_d}}}\cos \left( {\frac{{{\theta_{2,d}}}}{2}} \right)} \right) - \frac{{{\theta _{2,d}}}}{2}.$$

For deriving required wavefronts for the reference and signal waves in the recording process, it is necessary to consider synthetically all of k_1,r and k_2,r derived from every local grating of HOEs. As in Fig. 3, suppose we divide the whole HOE area into N number of sub-regions, and represent n^th sub-region as P_n with 1≤n ≤ N. P_n has two required wavevectors in air, k_1,r,air(P_n), and k_2,r,air(P_n). The effects of all sub-regions from P₁ to P_N are combined to derive the required wavefront from the individual wavevectors. If one is able to exactly provide the required wavefront in the recording process as in Fig. 3, the recorded and shrunk HOEs conduct the designed optical function in the Bragg regime, even illuminated with the differed wavelength of λ_d.

Fig. 3. Diagram for deriving required wavefronts for the wavelength compensation recording process by combining the compensated wavevectors, k_1,r,air and k_2,r,air, of all sub-regions.

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However, it is hard to implement such a complicated wavefront exactly. In practice, since the collimated laser beam is converted into the required wavefront passing after the physical optical elements, the complicated optical elements such as a free-form lens may be required, which involve high-cost. Alternatively, we could modulate the plane wave of laser into the required wavefront by using spatial light modulators (SLMs). However, the pixelated SLM structures inherently have light loss due to the high order diffraction, which requires high-power lasers for recording the compensated HOEs, and the pixel pitch of SLMs has to be fine enough for deliberate controls of wavefront.

Here, we approximate the required wavefront for the recording process into a spherical wavefront emitted from a single focal point, which enables the practical implementation of experimental setup with low-cost. In Fig. 4(a), a process of finding an optimal focal point for approximated spherical wavefront is demonstrated for the case of reference wave. Suppose a center sub-region of the HOE is P_c, and the required wavevector for the reference wave at P_c is k_1,r,air(P_c). Along the propagation direction of k_1,r,air(P_c), we place the virtual screens with various distances from the photopolymer in order to examine the footprint of required wavefront. In other words, we calculate the intersection points between the virtual screen and the required wavefront with being sampled by N numbers of sub-region. In Fig. 4(a), F_n,d denotes the intersection point between the virtual screen with the distance of d and the extension of k_1,r,air(P_n). The optimal distance of virtual screen, where the focal point for the approximated spherical waves is placed, can be found by minimizing the standard deviation of F_n,d.

(13)$${d_{opt}} = \mathop {\arg \min }\limits_d \left( {\sum\limits_{n = 1}^N {{{\left( {{F_{n,d}} - \frac{1}{N}\sum\limits_{n = 1}^N {{F_{n,d}}} } \right)}^2}} } \right),$$

where d_opt denotes the optimal distance for the virtual screen. The position of approximated focal point, P_s, is defined as the center of footprint at the virtual plane with the optimal distance, and given by

(14)$${P_s} = \frac{1}{N}\sum\limits_{n = 1}^N {{F_{n,{d_{opt}}}}} .$$

Fig. 4. Approximation of required wavefront into the spherical wave in case of the reference wave: (a) a diagram to examine the footprint of required wave field along the direction of k_1,r,air(P_c), (b) the approximated spherical wave with the focal point P_s, and (c) the angular error due to the spherical wave approximation.

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As shown in Fig. 4(b), if one realizes the recoding setup comprised of the spherical waves with the approximated focal point P_s and the wavelength of λ_r, the reconstructed wavefront in the displaying process with the wavelength of λ_d forms the intended focal point correctly.

In the perspective of n^th sub-region, P_n, a newly derived wavevector for the recording process after the spherical wave approximation becomes different from k_1,r,air(P_n), which is the wavevector designed to satisfy the Bragg condition of K_g,r. In Fig. 4(c), the amount of angular error due to the spherical wave approximation is defined as θ_err. Since θ_err leads to the Bragg mismatching, it should be within the bandwidth of angular tolerance of Bragg condition. Typically, the angular tolerance of Bragg condition is determined as the full width at half maximum (FWHM) of diffraction efficiency profile according to the incidence angle shift [14]. If θ_err is within the angular tolerance of Bragg condition, the spherical wave approximation is valid, and one can implement the recording setup for the required wavefront without the complicated optical elements.

2.2 Numerical simulation using volume hologram model

In this section, we simulate the diffraction properties of HOE lenses with on- and off-Bragg condition, and verify the concept of proposed method based on the volume hologram model of Zemax OpticStudio as proposed in [19,20]. The model is implemented based on Welford’s equation [21] and coupled wave theory [22], and useful to predict the propagation direction and efficiency of diffracted light, even though it deviates from the Bragg condition. In detail, when the incident light deviates from the Bragg condition, which is determined in the recording process of Fig. 2(e), the diffraction efficiency can be calculated as Eq. (15) by using the coupled wave theory for the lossless reflection-type thick gratings.

(15)$$\eta = {\left( {1 + \frac{{1 - {\xi^2}/{\nu^2}}}{{{{\sinh }^2}\left( {\sqrt {{\nu^2} - {\xi^2}} } \right)}}} \right)^{ - 1}},$$

where the parameters ν and ξ are real-valued parameters, and given by following formula [4,19,20,22]:

(16)$$\nu = \frac{{j\pi {n_1}t}}{{({{\lambda_r} + \Delta {\lambda_{mis}}} )\sqrt {\cos ({{\theta_{1,r}} + \Delta {\theta_{mis}}} )({\cos ({{\theta_{1,r}} + \Delta {\theta_{mis}}} )- ({{\lambda_r} + \Delta {\lambda_{mis}}} )|{{{\boldsymbol K}_{{\boldsymbol g,r}}}} |\cos {\theta_{g,r}}/2\pi n} )} }},$$

(17)$$\xi ={-} \frac{{\Delta {\theta _{mis}}|{{{\boldsymbol K}_{{\boldsymbol g,r}}}} |\sin ({{\theta_{g,r}} - {\theta_{1,r}}} )- \Delta {\lambda _{mis}}{{|{{{\boldsymbol K}_{{\boldsymbol g,r}}}} |}^2}/4\pi n}}{{2({\cos ({{\theta_{1,r}} + \Delta {\theta_{mis}}} )- ({\lambda_{1,r}} + \Delta {\lambda_{mis}})|{{{\boldsymbol K}_{{\boldsymbol g,r}}}} |\cos {\theta_{g,r}}/2\pi n} )}}t.$$

Here, Δθ_mis and Δλ_mis represent the angle and wavelength deviations compared to θ_1,r and λ_r in Fig. 2(e) for the incident light with the off-Bragg condition, respectively. The parameter n₁ represents the refractive index modulation of photopolymer.

As the groundwork, which presents the effects of Bragg mismatching on the HOE lenses, we conducted the simulations for two cases without considering the shrinkage effects: λ_d=λ_r and λ_d≠λ_r corresponding to the ideal case and the wavelength mismatched case, respectively. The system configuration of simulated HOE lens-based NED is provided in Fig. 5(a), which is implemented with the waveguide technologies for a compact system. The detailed specifications are listed in Table 1, and the wavelengths for the simulation were selected considering the specifications of experimental setup as will be addressed in the next section. In Fig. 5(a), the difference between the comprehensive ray-tracing results for λ_d=λ_r and λ_d≠λ_r cases are presented. When λ_d was equal to λ_r, the diffracted wavefront was focused at the center of detector as intended. However, in λ_d≠λ_r case, the focal point was shifted, as the diffraction direction changed due to the off-Bragg condition of incident light.

Fig. 5. Effects for wavelength mismatch on the HOE lens: (a) the simulation configuration and comprehensive ray-tracing results of HOE lens-based NED system with designing the eye-relief of 25 mm, (b) the simulated detector images placed at focal point corresponding to the eye-relief and normalized intensity profile, and (c) the spectral responses of implemented NED system based on the volume hologram model for green colors.

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Table 1. Specifications of simulated systems

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In Fig. 5(b), for λ_d=λ_r case, the focal points reconstructed from the HOE lens for red, green, and blue (RGB) point light sources were formed at center of detector plane as intended. However, for λ_d≠λ_r case, all of the focal points were not placed at the center of the detector, even though we assume the photopolymer did not suffer from the shrinkage. Furthermore, the shift of focal points showed more severe for the red color, since the wavelength difference between λ_d and λ_r was larger compared to the green and blue colors. The diffraction efficiency for λ_d≠λ_r case significantly dropped as shown in the normalized intensity profile of focal points.

Figure 5(c) shows the diffractive characteristics in the efficiency of investigated HOE lens with considering the wide range of wavelength deviations, especially for the case of green color (i.e., λ_r=532nm). Since the volume hologram model in OpticStudio is based on the aforementioned coupled wave theory, the decrease in the diffraction efficiency according to the wavelength mismatch could be calculated. In the simulation, we altered the wavelength of λ_d with maintaining the beam path of probe wave, and examined the change of diffraction efficiency at the detector plane. In the results of Fig. 5(c), when the wavelength deviated by the amount of 6.5 nm from λ_r, the diffraction efficiency decreased into half of maximum value.

Figure 6 shows simulation results with and without applying the proposed method for λ_d≠λ_r case. For the simulation of Fig. 6(a), the shrinkage effect was set to be 1.0%. The upper rows of Figs. 6(a) and (b) show the focal points at the detector plane with and without the proposed compensation method, respectively. When we applied the proposed method, the RGB focal points were placed correctly at the center of detector plane. The lower parts of Fig. 6 present simulated images for being observed through the eye model located at the focal points of HOE lens, where an image of resolution target was projected into the waveguide and HOE lens. The differently positioned focal points for RGB colors induced the color breakup problem and the chromatic aberration for the observed images as in Fig. 6(b). Meanwhile, the observed images after applying the wavelength compensated recording process showed correct full-color images.

Fig. 6. Simulation results to verify the effects of proposed wavelength compensated recording process: the simulated focal points and observed images through the eye model (a) with, and (b) without the proposed method.

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For investigating errors in the reconstructed waves, we introduce the angle deviation of diffracted wave, defined as δ, compared to the ideal case. As in Fig. 7(a) and Eq. (18), δ is represented using the inner product of normalized wavevectors.

(18)$$\delta = {\cos ^{ - 1}}\left( {\frac{{{{\boldsymbol k}_{\boldsymbol B}} \cdot {{\boldsymbol k}_{{\boldsymbol mis}}}}}{{|{{{\boldsymbol k}_{\boldsymbol B}}} ||{{{\boldsymbol k}_{{\boldsymbol mis}}}} |}}} \right),$$

where k_B and k_mis represent the wavevector of diffracted wave for λ_d=λ_r, and λ_d≠λ_r, respectively. For calculating k_B, the shrinkage effects of HOE lens were not considered, so that the wavevector k_B could be used to indicate the reference direction for the diffraction at each position corresponding to the ideal case.

Fig. 7. Investigation of angle deviation for the reconstructed waves compared to the ideal case: (a) a schematic for measuring the angle deviation δ, (b) 2D δ-plot according to the positions at the HOE lens without, and (c) with the proposed method. The presented δ-plots correspond to the HOE lens of Fig. 6, especially for green color.

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Figure 7(b) presents the angle deviation of diffracted wave for the HOE lens simulated in Fig. 6(b) in the case of green color. When the proposed compensation method was not applied, the average value of δ was 0.75° in the photopolymer medium, which made the location of focal point be shifted with the degraded diffraction efficiency in the off-Bragg regime. However, in Fig. 7(c) with the proposed method, the angle deviation compared with the ideal case significantly decreased, where the average value of δ was 0.06° in the photopolymer medium. The errors of 0.06° in the reconstructed waves resulted from the spherical wave approximation, and also led to a slight shift of focal points at the reconstructed plane. However, as verified in Fig. 6(a), the shift of compensated focal points was sufficiently small, so that the degradation in the simulated images through the eye model was not observed.

3. Experiments

3.1 Experimental verification for periodic gratings

As the groundwork, firstly, experimental results are presented to verify the analysis derived in Section 2 by investigating the periodic gratings using the interference between two plane waves. Since the localized approximation assumes that each sub-region is composed of periodic grating, the experiments in this section validates the concept of proposed method in the local grating corresponding to single sub-region of HOE lens.

We fabricated a periodic grating sample on the photopolymer with the grating slanting angle, θ_g,d, of 27.5°, and designed normally incident lights with the wavelength of 473nm, 547nm, and 628nm to be diffracted into 55° in the medium: i.e., θ_1,d=0°, θ_2,d=55°. Three layers of photopolymer films are stacked on a glass substrate in order to fabricate a full-color HOE with high diffraction efficiency as in [23]. For obtaining the diffraction characteristics according to the wavelength of probe wave, we assumed that the diffraction efficiency of HOE was proportional to the value which subtracts transmittance (%) from 100%. This assumption is useful to intuitively understand the spectral behavior of diffracted wave from the HOE with neglecting the absorption and loss of photopolymer [6].

Figure 8(a) shows the experimental setup for measuring the spectral transmittance of fabricated HOE with varying the detecting angle. At each measuring angle, the diffraction efficiency was estimated from the spectral transmittance based on aforementioned way. In Fig. 8(b), the relationship between the angle and wavelength deviation for providing the maximum diffraction efficiency is presented. In Fig. 8(b), we presented the reference lines in accordance with Eqs. (11) and (12) for the comparison, and it showed good agreements for the peak wavelengths which presented the maximum efficiency according to the angle deviation. Also, for the fixed value of angle deviation, the spectral response of Fig. 8(b) showed that the differed wavelength of probe wave degraded the diffraction efficiency as expected. For example, when Δθ is 0°, the FWHM of diffraction efficiency for RGB colors was 11 nm in average value.

Fig. 8. Experimental verification for angle and wavelength deviation of HOE with periodic gratings: (a) a measurement setup for the spectral transmittance, and (b) the measured diffraction properties according to the measuring angle deviation.

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3.2 Wavelength compensated HOE lens for spherical wave reconstruction

We built a proof-of-concept system adopting a LD-based projector, and a full-color HOE lens implemented on a thin waveguide as presented in Section 2.2. Figures 9(a) and 9(b) show the experimental setups for recording the wavelength compensated HOE lens, where both of the signal and reference waves were spherical waves. For given wavelengths, the compensated coordinates of approximated focal points to record HOE lens are listed in Table 2, where the origin of coordinate lies at the center of photopolymer as in Fig. 2. The 1-inch sized achromatic lens with the focal length of 30 mm and 40 mm was used for the reference wave and signal wave, respectively. Each achromatic lens was carefully moved using a linear translation stage to control the relative position of focal point for RGB lasers. For fabricating the full-color HOE lens in this section, we adopted the sequential wavelength multiplexing method using the single photopolymer layer in the recording process [4], so that the fabricated HOE lens had the advantages of high transparency.

Fig. 9. Experimental setups for the wavelength compensated HOE lens: (a) a schematic diagram, (b) a picture of experimental setup for the recording process, (c) the resultant HOE lens sample implemented with the waveguide, and (d) a photograph for the FOV measurements of fabricated sample.

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Table 2. Wavelength compensated focal points for recording process of fabricated HOE lens

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We placed a prism adjacent to the waveguide for injecting the reference wave with satisfying the total internal reflection (TIR) condition. Meanwhile, the one side of photopolymer was directly adjacent to the air, since the focal length of lens in the signal beam path was too short to place the additional glass substrate. The reference wave satisfied the TIR condition at the interface between the photopolymer and the air, and the undesirable gratings due to the interference between the signal wave and reflected reference wave, and between the reference wave and reflected one were also recorded in the photopolymer. Even though the undesirable gratings due to the reflected reference wave degraded the diffraction efficiency of HOE lens, they did not distort the reconstructed wavefront for the designed function of HOE lens. Hence the fabricated HOE lens could perform the optical function of see-through displays as intended. If the optical elements with the high numerical aperture and long focal length are available, the additional glass substrate can adjoin the photopolymer as in Ref. [4] for preventing the reference wave from being reflected at the interface between the photopolymer and the air, and the diffraction efficiency could be enhanced.

Figure 9(c) shows a fabricated sample for the wavelength compensated HOE lens with the waveguide. The fabricated HOE lens appeared transparent to the light from the environment, as a desktop monitor was clearly observed through the sample. The averaged transparency of fabricated HOE lens was 72% compared to the waveguide substrate with the measuring wavelength from 400 nm to 700 nm. Figure 9(d) shows the experimental setup to measure the FOV of fabricated HOE lens, where the measurement setup was inspired from Ref. [24]. The trajectory of diffracted light from the fabricated HOE lens created the footprint image on the screen with the protractor pattern, and the FOV was measured by the angular range within the trajectory. As shown in Fig. 9(d), the fabricated sample had the FOV of 30° in the horizontal direction.

Figure 10(a) shows a picture of setup for displaying experiments, where a laser beam scanning (LBS) projector without the additional collimation optics was used for the spherical wave reconstruction. The center wavelengths of LBS projector were 453 nm, 524 nm, and 643 nm. The projected images were injected into the waveguide after being reflected by the incoupler, and the HOE lens formed focal points at the eye-relief distance. In Fig. 10(b), the LBS projector presented a full-white image for the fabricated HOE lens, and the outcoupled wavefront formed a combined focal point for RGB colors. The green lines in Fig. 10(b) represent the imaging optical path in the displaying process from the RGB LDs in the LBS projector to the reconstructed focal points by the HOE lens and the waveguide. We placed a screen with the ring patterns at the eye-relief distance of HOE lens, where the smallest sized ring had a diameter of 5 mm which was the average size of optical aperture for the human eye [25]. It was found that the reconstructed focal points in Fig. 10(b) did not show the color separation according to the wavelengths, and formed the intended spot at the eye-relief distance with the size smaller than the eye pupil.

Fig. 10. Displaying experiments: (a) an experimental setup adopting the LBS projection module for the spherical wave reconstruction, photographs of (b) reconstructed focal points for RGB colors, (c) full-color images observed through the HOE lens and waveguide (Visualization 1), and (d) projected original pictures.

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Figure 10(c) shows the captured images through the HOE lens, and their original projected pictures are also presented in Fig. 10(d). We placed a camera at the focal point of HOE lens to capture the observed see-through images, and the resultant photographs did not show the color breakup problems. We placed the check board pattern and the tweezer behind the HOE lens as the background objects, and the virtual images provided from the LBS projector were successfully represented over the observed background objects in the real world. Visualization 1 presented the experimental results for the demonstration of AR concept using the fabricated HOE lens with the wavelength compensated spherical wave reconstruction.

4. Conclusion

In this paper, we proposed the wavelength compensation method for the optimal recording process of HOE lens, where both of reference and signal waves have the spherical wavefronts. In the proposed method, the aperiodic gratings in the HOE were divided into sufficiently large number of sub-regions, and the wavelength mismatch and shrinkage effects were compensated for each sub-region. For the practical realization of the experimental setup, the required wavefront was approximated into the simple spherical wave, and the error metric was investigated for the reconstructed wavefront based on the deviations in the angle of departure from the HOE. The simulation results with the volume hologram model in OpticStudio verified the principles of proposed method. As the preliminary experimental verification, we fabricated the periodic grating using the HOE with the stacked photopolymers, and the measured spectral response accorded with the analysis derived in Section 2. Also, the displaying experiments with the full-color HOE lens were presented, where the wavelength multiplexing method was applied into the single photopolymer layer for the high transparency. The reconstructed focal point and observed images verified that the proposed method compensated the wavelength mismatch and shrinkage effect, even the maximum wavelength deviation between the recording and displaying process was 17 nm.

The proposed compensation method can be extended for fabricating HOEs with various types of NED applications, and more complicated optical functions based on the interference between arbitrary wavefronts, if the waves slowly vary in the medium so that the localized approximation is valid. For example, besides the retinal-projection-type NEDs demonstrated in this paper, the proposed method could also be adopted for the waveguide-based NEDs with a larger eye-box using the exit pupil expansion. Since such the NED systems are typically implemented by the periodic gratings with varying the diffraction efficiency according to the relative positions in the HOE outcoupler [26,27], the wavelength compensation strategy in this paper could also be applicable. We believe that the proposed method is promising to practical implementation of HOE-based NEDs by alleviating the requirements for the light sources in the displaying process, which allows to adopting the compact and cost-effective projection module with diverse wavelengths of light sources.

Funding

Institute for Information and Communications Technology Planning and Evaluation grant funded by the Korea government Ministry of Science and ICT, South Korea; Development of Fundamental Technology of Core Components for Augmented and Virtual Reality Devices (No.2017-0-01803).

Disclosures

The authors declare no conflicts of interest.

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	Items	Specifications
Waveguide	Dimension	46mm×50mm×3.5mm
	Wedge angle of incoupler	30°
	Distance between center of incoupler and center of HOE lens	27.3mm
	Material	BK7
HOE	Dimension	18mm×12mm
	Focal points for recording process (Origin: center of HOE as in Fig. 2)	(−43.3, −25.0) for reference wave, (0, −45.9) for signal wave
	Recording wavelength (λ_r)	460 nm, 532 nm, 660 nm
Light sources	Displaying wavelength (λ_d)	460 nm, 532 nm, 660 nm (λ_d=λ_r case)
	Displaying wavelength (λ_d)	453 nm, 524 nm, 643 nm (λ_d≠λ_r case)
	Projection angle	18°×12°

Displaying wavelength	453 nm	524 nm	643 nm
Recording wavelength	460 nm	532 nm	660 nm
Focal point of signal beam, S(x,z)	(−0.84, −46.67)	(−0.83, −46.67)	(0.74, −47.35)
Focal point of reference beam, R(x,z)	(−43.67, −25.77)	(−43.68, −25.76)	(−44.51, −24.29)

	Items	Specifications
Waveguide	Dimension	46mm×50mm×3.5mm
	Wedge angle of incoupler	30°
	Distance between center of incoupler and center of HOE lens	27.3mm
	Material	BK7
HOE	Dimension	18mm×12mm
	Focal points for recording process (Origin: center of HOE as in Fig. 2)	(−43.3, −25.0) for reference wave, (0, −45.9) for signal wave
	Recording wavelength (λ_r)	460 nm, 532 nm, 660 nm
Light sources	Displaying wavelength (λ_d)	460 nm, 532 nm, 660 nm (λ_d=λ_r case)
	Displaying wavelength (λ_d)	453 nm, 524 nm, 643 nm (λ_d≠λ_r case)
	Projection angle	18°×12°

Displaying wavelength	453 nm	524 nm	643 nm
Recording wavelength	460 nm	532 nm	660 nm
Focal point of signal beam, S(x,z)	(−0.84, −46.67)	(−0.83, −46.67)	(0.74, −47.35)
Focal point of reference beam, R(x,z)	(−43.67, −25.77)	(−43.68, −25.76)	(−44.51, −24.29)

Pre-compensation method for optimizing recording process of holographic optical element lenses with spherical wave reconstruction

Abstract

1. Introduction

2. Principle

2.1 Wavelength compensated recording process

2.2 Numerical simulation using volume hologram model

3. Experiments

3.1 Experimental verification for periodic gratings

3.2 Wavelength compensated HOE lens for spherical wave reconstruction

4. Conclusion

Funding

Disclosures

References

Supplementary Material (1)

Cited By

Figures (10)

Tables (2)

Equations (18)

Optics Express