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Abstract
In this work, we focus on the polarization state management in optical devices using optical elements based on circular polarization. As an example, we point out the issue in a waveguide display using circular polarization optical elements as input/output couplers, where the polarization state of the light can change as it propagates in the waveguide due to total internal reflection (TIR). This has a negative effect on the waveguide output coupler efficiency, image uniformity, and the polarization multiplexing capability. To address this problem, we discussed two different methods to compensate the polarization state change. With the compensator applied to correct the polarization state change in the waveguide, the optical elements based on circular polarization can be used with their advantages as input/output couplers for waveguide displays.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Diffractive optical elements offer an attractive path to optics miniaturization. In the diffractive optical elements family, liquid crystal (LC) based polarization optical elements, (such as Pancharatnam-Berry Phase (PBP) devices [1–5] and polarization volume holograms (PVH)/ polarization volume gratings (PVGs) [6–12]), provide high diffraction efficiency with large angle. These devices also provide polarization state selectivity, facilitating more advanced device designs. A number of groups are now trying to utilize LC polarization optical elements in waveguide displays, retinal projection, and thin VR systems [13–20].
For example, the reported waveguide displays [13–18] using polarization volume gratings (PVGs) as input and output couplers have many benefits, such as ease of fabrication, low cost, the possibility of polarization multiplexing to widen the field of view (FOV), and pupil replication by polarization state management [14,15]. However, one problem remains: the diffraction efficiency of the PVGs is highly sensitive to the polarization state [12,21]. When light propagates in the waveguide, the TIR changes the circular polarization state [22], which leads to a negative effect on the waveguide output coupling efficiency [12], image uniformity, and the polarization multiplexing capability. We haven't yet found a corresponding compensation scheme in the literature.
In this work, we consider two compensators that allow the use of input and output couplers based on circularly polarized light in a waveguide. First, we discuss applying a compensator throughout a waveguide surface to ensure circularly polarized light after every TIR. This design works for up to several TIR reflections but is less effective for long-distance propagation since small compensation errors accumulate for every TIR. Next, we discuss using linearly polarized light propagating in the waveguide. In this method, we convert light from circular to linear polarization after the input coupler. Then, after light propagates through the waveguide, we return it to circular polarization before the output coupler. The resulting polarization state of light in the waveguide will ideally be independent of the number of TIR reflections, as TIR does not alter an S or P linear polarization state. We evaluate both designs below for single and multiple reflections.
2. Problems of the polarization state change
A well-documented change in the polarization state of circularly polarized light results from the differing phase shift of the S and P modes during TIR. The phase shifts of the S (${\phi _S}$) and P modes (${\phi _P}$) depend strongly on the incident angle theta based on the Fresnel equations shown below [23].
In addition to the differing phase shifts, light from different angles will experience a different number of TIR reflections, as shown in Fig. 1. The incident light with k vector ${\boldsymbol k}\_{\boldsymbol{air}}$ is diffracted by the input coupler. The diffracted k vector in the waveguide is$\; {\boldsymbol k}\_{\boldsymbol{def}}$. There are three rays drawn in Fig. 1 used to describe the on axis and positive/negative off axis incident light. The diffracted light propagates in the waveguide, has TIR in the waveguide-air interface, and changes the polarization state when it is reflected. However, the unique properties of PVGs as input and output couplers motivate us to confront these polarization state management complications.
3. Compensator design to use circular polarized light in the PVG waveguide
In this section, we discuss a method to keep light being circular polarized after every TIR in waveguide by a compensator. We applied the compensator on top and bottom of waveguide, so that light from all directions and with different TIR numbers is in the same circular polarization state at the end of waveguide.
To have wide working angle at least for a single wavelength in the waveguide using circular polarized input and output couplers, the compensator needs to compensate the polarization state change due to TIR over a wide-angle range. By looking thought several compensator options, we found that there is a simple way to achieve circular polarization reflection, is to use two positive A plates with optical axis along X axis ($Ax + $) seen in Fig. 1. One plate is on the top of the waveguide, the other same A plate is on the bottom of the waveguide. As shown below in Fig. 2, the retardation curve of $Ax + $ is matching with phase shift curve of TIR over a wide range of angles, with sum total polarization state change of 180 degrees. This yields the orthogonal circular polarization state, which could work with output couplers using PVGs with high efficiency and polarization selectivity.
Fig. 2. Single positive A plate ($Ax + $) compensator working mechanism. The Sum of phase shift due to TIR in compensator-air interface (blue) and retardation of compensator (green) is around 180 degrees (black dashed line). ${n_e} = 1.85$, ${n_o} = 1.55$. The calculation is done by Fresnel equation [23].
We used the Berreman method, which is a precise solution to Maxwell Equations for structures with 1D periodicity [24], to investigate the compensator performance in a waveguide over the field of view (FOV). The coordinates of our modeled system are shown in Fig. 1. Starting with wave vector k in air, ${\boldsymbol k}\_{\boldsymbol{air}}\; ({kx\_air,\; ky\_air,\; kz\_air} )$, according to k vector analysis [25] we got the relation of FOV seen by at the output of the waveguide, and the light direction in waveguide. We calculated the Stokes parameter $S3 = (2Es \times Ep \times sin\delta /\; ({Es \;\hat{}2 + Ep \;\hat{}2} )$ where $\delta $ is phase leading from $Es$ to $Ep$) of light after the TIR reflection and passing through the compensator by Berreman method.
In Fig. 3, we plot the graph of S3 in polar coordinates of light in air using polar angle $= acos({kz\_air/|{k\_air} |} )$; and azimuthal angle $= atan({ky\_air\; /kx\_air} )$. The plot shows the S3 distribution over the FOV that a user would see given consideration of the angle of light propagating in the waveguide, but not the number of reflections. The two dashed lines in graph correspond to polar angles 40 and 70 degrees in the waveguide that are the assumed limiting angles. The color bar was set for S3 increments of 0.2, over the range from 0 to 1. The color black (S3 less than 0) corresponds the output coupling efficiency of <50% of what would result for the case of S3 = 1.
Fig. 3. (a) The field of view of a user. The coordinate shown has polar angle marked by the number with cyan background, and the azimuthal angle marked with the yellow background. (b) Polarization state with one TIR reflection in a waveguide without a compensator. (c) Polarization state with 9 TIR reflections in a waveguide without a compensator. The two dashed lines in graph corresponds with polar angles 40 and 70 degrees in waveguide.
Without a compensator applied, left hand circular polarized light (LHC) from the input coupler, after one-time TIR reflection, causes S3 to shift from -1 to 0.53∼0.91 in Fig. 3(b) with different incident angles by the calculation methods shown above. With 9 TIR reflections, there is only a small angular range with S3 > 0.8 seen in Fig. 3(c). When PVGs are applied as the output couplers, the angular range with high efficiency is limited even in this case where the number of the reflections is assumed to be independent of angle.
The results above show the necessity for polarization management in a waveguide system that uses PVGs as output couplers which requires circularly polarized light. As an example of the first method to solve this problem, we show a compensator design using a positive A plates with optical axis along X axis (Ax+),${n_e} = 1.85$, ${n_o} = 1.55\; $and thickness=0.24 micron, and ${n_{waveguide}} = 1.7$. In this case the critical TIR angle in waveguide-compensator interface is $\textrm{asind}({1.55/1.7} )\, = \,65$ degrees, so that is the maximum polar angle in waveguide. The resulting S3 with one-time TIR, 9 TIR, and 49 TIR reflections are shown in Fig. 4 below. The results show that for few TIRs, there is large FOV with S3 > 0.8 which corresponding with >90% coupling out efficiency (where 100% is the efficiency for perfect circularly polarized incident light) However, for more times TIR, the small error is accumulated and the polarization state in FOV is out of the control.
Fig. 4. (a) Single layer compensator (Ax+) result with one TIR reflection. (b) 9 TIR reflections. (c) 49 TIR reflections.
4. Compensator design to use linear polarized light in the PVG waveguide
In this section, we reported one more method achieve the polarization state management, using the linear polarized light (LP) propagating in waveguide. We convert circular polarized light (CP) to LP after the input coupler, and back to CP before the output coupler as shown in Fig. 5. The advantage is that the polarization state of the light in waveguide is almost independent with the number of TIR reflections.
We have found that a critical design characteristic to have wide working angular spectrum, is that the optical axis of compensator should be out of the substrate plane. The reason is explained below: We consider the normal incident light from image source into the input coupler. After this beam is diffracted into the waveguide seen the central beam below (red arrow), it has azimuthal angle zero which is in X-Z plane, and polar angle 50-60 degrees. To optimize the optical axis of the compensator that has good and uniform performance with off axis incident, we prefer to set the compensator working perfect with the central beam, and set the optical axis in the plane (blue plane) that perpendicular with the central beam as shown in Fig. 6(a). The optical axis plane crossed with the incident plane (gray) along the dashed line. The optical axis (yellow line) is set in the blue plane, and with ∼45 degrees to the S or P modes. Noticed that it is a general analysis to optimize the optical axis. Later we slightly vary the optical axis by polar and azimuthal angles from the analytical optical axis direction, and found the one gives the best performance in the FOV.
The numerically optimized compensator has optical axis in 3D with azimuthal angle 52 degrees and polar angle 116 degrees shown in Fig. 6(a) where ${n_e} = 1.70$, ${n_o} = 1.55$, and thickness=0.53 microns. The fabrication could be done using a 3D holographic alignment material that we have previously discussed [21].
Fig. 5. The basic waveguide structure, where a compensator just after the input coupler and before output coupler is shown.
Fig. 6. (a) the optical axis of the compensator in 3D space. (b) FDTD simulation of the compensator: convert linear polarized light to circular polarized light. (c) evaluation of this compensator to convert S mode linear polarized light to right hand circular polarized light in FOV.
Using the FDTD simulation [21] in Fig. 6(b) and the Berreman calculation in Fig. 6(c), we verified the on-axis and off-axis performance of the compensator. In the bottom of the Fig. 6(b), the S mode linear polarized light incident along the red arrow, only the Ey which is out of the screen is shown. After the beam goes through the retarder layer (yellow), we see the output has both in plane and out of the plane components. The phase and amplitude information read from the graph Fig. 6(b) [21] show that the output light is right hand circular polarized. The off axis performance is verified in Fig. 6(c) of the compensator to convert linear polarized light to circular polarized. The results show that a S mode linear polarized light could be converted to circular polarized light after this compensator independent of angle. Since light is reversable, it also works to convert circular polarized light to linear polarized light in the input of waveguide.
To consider the effect of many reflections in a system using circular polarized input and output couplers and the proposed compensator, we simulated the performance of a waveguide that has 100 reflections between the input and output coupler.
As the results are expected to be wavelength dependent, we did the simulation for wavelengths of 633 nm, 532 nm and 457 nm as shown in Figs. 7(a), 7(b), and 7(c) without dispersion. With shorter wavelengths, the S3 drops at right edge of the figure which corresponds to a large polar angle in waveguide. Since the thickness of compensator was optimized for 532 nm light, shorter wavelengths of light have a larger phase retardation for all angles. The larger polar angle it has, the larger retardation it has, so on the right edge of figure, the retardation is larger than required to convert S mode LP to RHC, and shows a slightly degraded circular polarization state. With the longer wavelength of 633 nm, the low S3 appears in left edge of figure. However, we see that for all wavelengths the polarization state being delivered to the output coupler close to perfect.
Fig. 7. (a) Output polarization state S3 in 457 nm after 100 TIR reflections, (b) in 532 nm after 100 TIR reflections, (c) in 633 nm after 100 TIR reflections.
The same device also works for RHC in, converted by the first compensator to P mode LP, then to LHC by the second compensator at the output. In summary, with the design of compensator with optical axis out of substrate plane, we can convert circular polarized light to linear polarized light for propagation in a waveguide without polarization state change.
5. Conclusions
In this work, we demonstrated polarization state management upon TIR with circularly polarized incident light. As an example, we discussed the changing polarization state of light propagating in a PVG waveguide display. Then we presented two methods to compensate this polarization state change: (1) compensate the light after every TIR to maintain circular polarization (shown to work well for a limited number of reflections); and (2) convert circular polarized light from the input coupler to be linear polarized for propagation in the waveguide and revert to circular polarized light before the output coupler. This second method has the advantage that the compensation result is almost independent of the number of total internal reflections.
Funding
Facebook.
Acknowledgments
The work was done with meaningful discussions with Dr. Scott McEldowney from Facebook Reality Labs.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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