1. INTRODUCTION

Polarization gratings (PGs) [1,2] are generally described as diffractive elements with linearly varying, in-plane, optical anisotropy orientation and constant anisotropy magnitude. Unlike conventional phase or amplitude gratings, they operate by locally modifying the polarization state of light waves passing through them. Nevertheless, we can still classify PGs into different diffraction regimes [3–5] based on their diffraction behaviors. Most PGs [6,7] reported experimentally until recently manifest small diffraction angle and weak angular selectivity and thus can be classified into the Raman–Nath regime [8]. Several early works realized PGs with a small grating period using azo-polymers [4] and studied them using a nonrigorous simulation method [9]. Recently, surface-aligned nematic liquid crystal Bragg PGs with small subwavelength period have been reported [5,10], as well as a bulk-aligned alternative [11]. Those PGs are optically thick with large diffraction angle and nearly 100% first-order diffraction efficiency at the Bragg condition and thus can be classified into the Bragg regime. Different from conventional Bragg gratings made by isotropic materials, Bragg PGs [12] are verified to have large angular acceptance defined with 30% threshold [13] and spectral bandwidth defined by the full width at half-maximum (FWHM) in the diffraction efficiency curve. The high-efficiency, large deflection angle and bandwidths make Bragg PGs particularly useful for numerous applications in various fields, including emerging waveguide-based wearable displays in head-mounted-display systems [14].

A frequently used method to record the profile of a PG is polarization holography [15], where the interference pattern is created by two circular polarized beams with opposite handedness. After the photo-alignment layer (PAL) [16] is recorded, the liquid crystal polymer (LCP) network material, also known as reactive mesogens [17], is spin-coated and polymerized to form the grating. Bragg PGs with subwavelength 335 nm period can also be realized by this approach, and the grating slant is achieved by inducing a nonzero twist of the liquid crystal director profile along the vertical direction by chiral dopants [5].

Bragg PGs as a new category of PGs have not been well studied yet. A previous rigorous numerical study [3] was done mainly for the PGs in the Raman–Nath regime. Other prior numerical simulations based on the extended Jones matrix method [18], finite-difference time-domain (FDTD) [9], and finite-element method (FEM) [19] provided only basic angular and spectral responses. Recently, we applied the rigorous coupled-wave analysis (RCWA) method [20] to rigorously simulate the PGs where arbitrary polarization inputs were enabled and validated their accuracy by comparing them with the FDTD method [21]. In this work, for a better understanding of Bragg PGs, we focus on the optical physics, including their angular and spectral bandwidths, effects of material index, and nonplanar director profile. We also simulate the complex polarization response of Bragg PGs and show good agreement with prior experimental results.

2. BRAGG REGIME PGs

PGs are formed by periodic profiles of spatially varying optical anisotropy. For those fabricated by liquid crystals, the in-plane varying optic axis can be described by the orientation angle of the liquid crystal director [5], as shown in Fig. 1:

 

Fig. 1. Director profile of a single-slant (i.e., single-twist) liquid crystal PG with nonzero twist.

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Following the previous work [5], we classify PGs into the Bragg regime by calculating $Q=2\pi \lambda d/(\overline{n}{\mathrm{\Lambda }}^2)$ with volume period $\mathrm{\Lambda }={\mathrm{\Lambda }}_x\cos {\theta }_G$ and average index $\overline{n}$. A grating is classified into the Bragg regime when $Q\gg 1$. The peak efficiency of Bragg PGs is achieved at the Bragg condition that corresponds to a Bragg angle $|\sin {\theta }_B|=\lambda /(2\overline{n}\mathrm{\Lambda })$ for certain operation wavelength. The grating slant is induced to shift the peak efficiency angle in the incident medium:

A general study on slant angles was performed in our prior work [22]. Here, as a specific and useful case, we are interested in the slanted Bragg PG optimized for normal incidence where ${\theta }_P=0$, and we can analytically solve the twist rate to be

We further examine two Bragg PGs, assuming ${\mathrm{\Lambda }}_x=400\mathrm{nm}$, $\lambda =520\mathrm{nm}$, and $\overline{n}=1.65$. The parameters are chosen to diffract the green light at normal incidence into the high-index glass substrate ($n=1.7$), which is important for efficient in-coupling with wide angular acceptance [23]. The twist rates of corresponding transmissive and reflective Bragg PGs are calculated from Eq. (4) to be 217°/μm and 929°/μm, respectively. The slant angles are 26° and 64°, which are illustrated with the two diffraction configurations (transmissive and reflective) in Fig. 2. We note that both types of Bragg PG (transmissive and reflective) can be operated in both transmissive and reflective configurations, depending on the out-coupling of the exit medium. In short, total internal reflection (TIR) can cause the transmitted orders to reflect and vice versa. The condition of the TIR can be determined analytically as

 

Fig. 2. Transmissive and reflective Bragg PGs operated in (a) and (b) transmissive and (c) and (d) reflective configurations where $\mathbf{G}$ indicates the direction of grating vectors. Note that both grating types can be configured in either transmissive or reflective operation, depending on the incident and output media, due to the effect of TIR.

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Whenever $|\sin {\theta }_m|<1$ is satisfied, the output wave will propagate without TIR. On the contrary, when the right side of Eq. (5) has a large enough magnitude that $|\sin {\theta }_m|<1$ is not satisfied, then TIR will occur, and the wave will effectively reflect at the PG/exit-medium interface. Note that since this condition is directly derived from the grating equation, it is valid for both transmissive and reflective PGs.

Next, we simulate the two Bragg PGs mentioned above [Figs. 2(a) and 2(b)], both operating in the transmissive configuration. Note that the grating thicknesses are set to be 1 and 3 μm for the transmissive and reflective Bragg PGs, respectively, to achieve maximum diffraction efficiency. The simulated angular and spectral responses are shown in Fig. 3. Generally, the transmissive grating provides a moderate angular bandwidth (25° at 520 nm) and broad spectral bandwidth (200 nm at normal incidence). For the reflective counterpart, the bandwidths are around 30° and 50 nm, respectively, with sharp band edges. Note that that the reflective grating achieves nearly 100% diffraction efficiency within the band, compared to about 75% for the transmissive grating.

 

Fig. 3. Simulated angular and spectral responses of (a) a transmissive Bragg PG and (b) a reflective slanted Bragg PG with ${\mathrm{\Lambda }}_x=400\mathrm{nm}$ for circular polarized input with $\mathrm{\Delta }n=0.25$ at 520 nm.

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The bandwidths are related to the deviation of the Bragg condition in the diffraction as pointed out by coupled wave analysis [24]. Transmissive Bragg PGs are physically thin and typically operated at the half-wave condition, which will respond to a wide range of wavelengths. On the contrary, reflective Bragg PGs [19] manifest high twist rate and grating thickness, which establishes stronger spectral selectivity within the grating. Therefore, the resulting spectral bandwidth is significantly smaller. However, given the high birefringence of the liquid crystal, stronger coupling between the incident and diffractive waves is expected than conventional isotropic volume gratings [25,26], which enables large bandwidths in general.

3. ANGULAR AND SPECTRAL BANDWIDTHS

In this section, we focus on the angular and spectral bandwidths of both transmissive and reflective Bragg PGs because they are critical for optical performance in most applications. We first expect strong correlation between the material birefringence and bandwidths of the grating and investigate it for the Bragg PGs.

We begin by varying $\mathrm{\Delta }n$ from a small value (0.05) to a large value (0.35). In Fig. 4, we can observe a monotonically increasing bandwidth as the birefringence is increased. The simulated bandwidths of both Bragg PG types increase, while the actual magnitude and slope may differ. These results correspond well to the realized bandwidths of Bragg PGs [5,10]. Since transmissive PGs achieve maximum efficiency at the half-wave thickness [6], lower physical thickness is needed for materials with higher birefringence, which contributes to stronger coupling for a wide range of wavelengths and enables larger spectral bandwidth. As the bandwidths increase, the average first-order diffraction efficiency is well preserved to be $70\sim 80\%$ for transmissive gratings within the bandwidth, as shown in Fig. 4. The reflective grating has smaller spectral bandwidth but provides larger angular bandwidth and higher average efficiency, which can be more than 90%. Since the higher grating slant potentially reduces the bandwidth as demonstrated in the experimental work [5], we believe materials with higher birefringence can be applied to compensate this effect.

 

Fig. 4. Simulated (a) angular bandwidth at $\lambda =520\mathrm{nm}$ and (b) spectral bandwidth at ${\theta }_{\mathrm{in}}={\theta }_P=0°$ of the transmissive and reflective Bragg PGs with ${\mathrm{\Lambda }}_x=400\mathrm{nm}$ and $\overline{n}=1.65$ for circular polarized input.

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In another prior experimental work [12], we broadened the angular bandwidth by layering two slanted gratings in a monolithic film and pointed out that the average index was helpful in achieving larger bandwidth. In order to directly verify the effect of $\overline{n}$, we further investigate and visualize the angular and spectral responses with varying average index in Fig. 5. Note that the twist rate is adjusted accordingly to preserve ${\theta }_P=0°$, and perfect index matching is assumed for the exit medium ($n_{\mathrm{out}}=\overline{n}$). Other parameters are kept the same as the previous section.

 

Fig. 5. Simulated (a) angular response and (b) spectral response of the slanted transmissive Bragg PG with varying $\overline{n}$ and $\mathrm{\Delta }n=0.25$ at 520 nm.

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Generally, both the angular and spectral bandwidths are significantly broadened with increasing average index. With higher index, the diffracted wave propagates at smaller angle in the grating and the negative band edge corresponding to the evanescent condition is extended, which is consistent with the discussion in prior work [12]. Additionally, we also note that the spectral bandwidth in Fig. 5(b) is broadened mainly at the long-wave side as the effective wavelength ${\lambda }_G$ becomes comparable to the volume period. It is worth noting that the bandwidths are mainly broadened at one side with increasing average index compared to a generally wider response with higher material birefringence.

The results for reflective Bragg PGs are shown in Fig. 6. Similar to the transmissive case, the angular response is significantly broadened at the negative side. The angular bandwidth is about 50% larger compared to the transmissive counterpart given the same index. However, the spectral bandwidth generally is preserved at about 50 nm without strong dependence on the refraction of index. The different dependence is expected since the spectral selectivity originates from the scattering from the periodic structure along the propagation direction, where the average index has limited influence.

 

Fig. 6. Simulated (a) angular response and (b) spectral response of the reflective Bragg PG with varying $\overline{n}$.

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For both Bragg PG types, higher average index generally provides wider angular response, which can be particularly important for applications where large angular acceptance is desired. On the other hand, the wide and narrow spectral response provides additional options for different applications. It is worth pointing out that the dependence on average index is consistent with the large angular bandwidth realized in silicon-based metasurface grating [13], where 40° angular acceptance was claimed.

4. NONPLANAR DIRECTOR PROFILE

For the analysis in previous sections, the planar director profile is assumed, which corresponds to ideal PGs. Nevertheless, in the fabricated PGs, the liquid crystal director can be out of plane where the tilt angle $\mathrm{\Theta }>0$, as illustrated in Fig. 7.

 

Fig. 7. Director profile of a slanted Bragg PG with nonzero tilt.

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This deviation is potentially caused by the transition between planar and nonplanar director profiles [27] in the liquid crystal (LC) alignment process, making the resulted tilt angle hard to estimate and measure. We are interested in the effects of the nonplanar director profile on the responses of the Bragg PGs. Assuming the same grating parameters, the simulated angular and spectral response of the transmissive and reflective gratings is shown in Figs. 8 and 9. In all cases, the bandwidths and average efficiency tend to decrease for higher tilt. The degradation in performance seems negligible for a uniform tilt $\mathrm{\Theta }=25°$, but it becomes significant for $\mathrm{\Theta }=50°$, where the peak efficiency drops by 20%.

 

Fig. 8. Simulated (a) angular and (b) spectral response of a transmissive Bragg PG with varying tilt of the LC director profile.

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Fig. 9. Simulated (a) angular and (b) spectral response of a reflective Bragg PG with varying tilt of the LC director profile.

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For simplicity, we assume the tilt of the LC director to be constant everywhere in the grating, but the actual director profile can be complicated. As an example, multiple LC polymer layers are usually deposited onto the PAL to form the grating, where the tilt can vary from one layer to another. In that situation, it can be challenging to preserve the grating profile and achieve high efficiency due to the naturally occurring preference of homeotropic alignment.

5. POLARIZATION RESPONSE

In this section, we study the polarization response of both transmissive and reflective-type Bragg PGs. First, we simulate the first-order diffraction efficiency varying input angle of incidence and ellipticity angle, as shown in Fig. 10. As expected, the efficiency reaches maximum for one-handedness of the circular polarization (CP) and decreases to zero for other-handedness. Note that the simulated dependence on input polarization is highly consistent with the experimental results demonstrated in prior work [5].

 

Fig. 10. Simulated first-order efficiency versus angle of incidence and ellipticity angle for the (a) transmissive and (b) reflective Bragg PG.

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Second, we examine the zero- and first-order polarization with the circular polarized input. The result of ellipticity angle ($\chi$) is shown in Fig. 11(a), which matches the measurement of realized Bragg PGs [5]. The polarization of the first-order wave shows a transition from linear to elliptical for increasing angle of incidence, while the zero-order generally preserves the same polarization as the input. We anticipate the strong angular dependence from the varying retardation of the birefringent grating. As the first-order polarization can be very linear, we further investigate its orientation angle ($\psi$) in Fig. 11(b), and a clear angular dependence is also observed. We expect more complex response for reflective Bragg PGs, where the additional grating-air interface in the transmissive configuration may have additional effects.

 

Fig. 11. Simulated (a) zero-order and first-order ellipticity angles and (b) first-order orientation angles for both the transmissive and reflective Bragg PGs.

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Finally, we perform a best-fit to the parameters of the realized grating from the measured first-order diffraction efficiency and zero-order ellipticity angle with the linear polarized input. Note that the single slanted transmissive Bragg PG was fabricated and measured following the same process as the prior work [5,12], except for the minor changes in the material mixture to achieve the desired slant angle. The grating film was formed by coating and curing 12 sublayers of the reactive mesogen solution, comprising 4.4% nonchiral nematic RMM-A (Merck KGaA, $\mathrm{\Delta }n=0.16$ at 520 nm), 0.6% chiral nematic RMM-C (Merck KGaA), and 95% solvent propylene-glycol-methyl-ether-acetate (PGMEA from Sigma-Aldrich). The coating and curing conditions were the same as the prior work [5], except for 700 rpm spin speed and 30 s cure time. As shown in Fig. 12, we fit the realized twist angle ($\varphi$), grating thickness ($d$), and material birefringence ($\mathrm{\Delta }n$) by minimizing the merit function $f=\mathtt{mean}(|{\eta }_1{({\theta }_{\mathrm{in}})}^{\mathrm{fit}}-{\eta }_1{({\theta }_{\mathrm{in}})}^{\mathrm{data}}|+|{\chi }_0{({\theta }_{\mathrm{in}})}^{\mathrm{fit}}-{\chi }_0{({\theta }_{\mathrm{in}})}^{\mathrm{data}}|)$. We can verify good agreement between the data and fitting with the average deviation in efficiency (%) and ellipticity angle (°) to be 2.7. For instance, the fitted twist rate turned out to be 226°/μm and almost perfectly matches the predicted value given by Eq. (4). Most notably, the first-order efficiency for CP input is also plotted, given the retrieved parameters, and it matches the measured data very well, which demonstrates the excellent quality of fitting.

 

Fig. 12. Measured and fitted (a) first-order angular response and (b) zero-order ellipticity angle of the transmissive Bragg PG for linear polarization (LP) and CP inputs.

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The fitting process is particularly useful for closing the gap between the design and fabrication. As an example, the realized material birefringence, twist angle, and thickness depends on the fabrication process and the concentration of chiral dopants [17]. Therefore, we expect reduced fabrication error after retrieving the parameters from fitting and take them into account at the design stage, which in return enables a more precise design ability.

6. CONCLUSION

We comprehensively investigated the diffraction properties of newly reported Bragg regime PGs via rigorous numerical simulations. Both transmissive and reflective Bragg PGs are studied for their angular and spectral bandwidths and polarization outputs. In the transmissive configuration, both gratings can achieve high average efficiency within the bandwidths but differ in the magnitude of the bandwidths. Generally, transmissive Bragg PGs demonstrate large spectral bandwidth ($\sim 200\mathrm{nm}$) and moderate angular bandwidth ($\sim 20°$) compared to $\sim 50\mathrm{nm}$ and 25° for their reflective counterparts, which enables great flexibility for different applications. The study of nonplanar director profiles in the LC-based Bragg PGs revealed degraded grating performance, including reduced bandwidths and lower efficiency. The simulated polarization output matches the experimental results in prior work well and shows a strong angular dependence. Finally, the fitting based on the diffraction efficiency and the polarization output exhibits excellent accuracy, which we believe to be especially useful in realizing gratings with desired performance.