Super achromatic wide-angle quarter-wave plates using multi-twist retarders

The achromaticity and wide-angle property of quarter-wave plates (QWPs) are crucial for the color uniformity and image resolution of the future displays such as virtual reality (VR) pancake lens and augmented reality (AR) waveguide/focusing systems. However, most reported achromatic wide-angle QWPs designs composed by stacks of different birefringent plates are too complicated with limited achromaticity and wide-angle performance. The multi-twist retarders (MTR) QWPs presented in previous work already showed its potential to achieve high achromaticity in RGB using one monolithic film in normal incidence, but the incompetent polarization control in blue-violet limits its application in LED-based polarization-sensitive AR/VR headsets. In this work, we theoretically investigate a new type of MTR QWPs achieving super achromaticity from violet to red with average ellipticity 43° and simultaneously maintaining wide-viewing angle up to ±45°, which enables a precise polarization control within the field-of-view (FOV) of current AV/VR headset. The new proposed MTR QWP is also reported to obtain average reflection luminance leakage 0.15~% and maximum leakage 0.23~%, making it a promising element to reduce polarization leakage and enhance image resolution in the next-generation displays.


VALIDATION OF EXTENDED JONES MATRIX A. Case study of cross/parallel polarizers
We first consider the transmission of crossed polarizers. If birefringent plate complex refractive index is set as n = n e + κ e for e-wave and n = n o + κ o for o-wave, with approximation of small birefringence (n o ≈ n e ), the sheet polarizers can characterized by κ o = 0 , 0 < κ e 1 and thickness d such that 1 2πκ e d/λ. To simplify the problem, the light source is monochromatic with wavelengthλ = 550 nm, and the two O-type polarizer the small birefringence approximation is satisfied. For the cross polarizers setup, the front polarizer transmission axis is set as p 1 = 45 • , and the back polarizer transmission axis at p 2 = 135 • . For the parallel polarizers setup, both front/back polarizer is set as p 1 = p 2 = 0 • . The calculated transmission between a pair of polarizers is depicted in Fig. S1. Light source is set as monochromatic light with wavelength λ = 550 nm. polarizers refractive index n p = 1.5. Polar angle range is (0 • , 90 • ).

B. Case study of uniaxial plates
In this subsection, three most common liquid crystal alignment formats, the uniaxial a-plate c-plate and o-plate will be simulated.
First we start with uniaxial a-plate, which represents a kind of birefringent material with optic axis lies in the plane perpendicular to the incident plane ( Fig. S2(a)). We can assume the material discussed is of positive birefringence. When it's oblique incident, the birefringence is angledependent as ∆ n (θ, φ) = n(θ, φ) − n o , in which n(θ, φ) is angle dependent. The transmission between cross polarizers (front polarizer transmission axis p 1 = 45 • , back polarizer transmission axis p 2 = 135 • ) of single a-plate is depicted in Fig. S2(d). Here the transmission calculation includes the Fresnel reflection and transmission on the surface of the polarizer. It can be seen that the correct transmission of a-plate between cross polarizers has alternate constructive interference and destructive interference, representing the full wave retardation and half-wave retardation respectively.
In the case of uniaxial c-plate, it has optic axis perpendicular to the birefringent plane ( Fig. S2(b)). From the following polar pattern of c-plate, two things are worth to notice. Firstly, the center of the plate is dark fringes. Considering the thickness d = 2.5 µm and the monochromatic light source is λ = 500 µm, indicating this is a half-wave plate. Secondly, we can visualize the interference is center-symmetric, indicating that the birefringence and the retardation is also center-symmetric.

C. Case Study of Single uniaxial HWP/QWP
To validate the HWPs and QWPs design, we first use the extended Jones matrix to simulate single uniaxial a-plate HWP and QWP. With the central wavelength set as λ c = 550 nm and birefringence ∆n = 0.1, the thickness of HWP (d H ) can be written as d H = 0.5λ/∆ n , similarly thickness of QWP d Q can be written as d Q = 0.25λ/∆ n .
In this subsection, both transmission and ellipticity will be validated. For the transmission validation, the uniaxial HWP or QWP will be placed between cross polarizers. Generally, despite some Fresnel reflection loss, the ideal transmission of HWP in this polarizer setting is around 50%, while the QWP case, the ideal transmission is around 25%.
For the ellipticity validation, the HWP or QWP will be placed behind the front polarizer, and the output intensity is used to calculate the ellipticity. To find the ellipticity χ and orientation angle Ψ of the output light and the retardation of the material Φ, we can represent the Stokes vector by Jones vector at every local point of the polar surface as: The ellipticity and orientation angle can be calculated by: In Fig. S3, it can be seen that the center transmission of HWP between cross-polarizers is almost half of the incidence energy, while the QWP gives around around 25%, which meets our expectation. Regarding to the ellipticity, the central ellipticity of HWP is around 0 • , and QWP is around 45 • . In both HWP and QWP cases, the half-wave or quarter-wave performance degrades as polar angle increases.

D. Case study of TN-cell
The case of TN-cell is similar to the chiral layer in MTR structure. Therefore, if we want to use the extended Jones matrix to simulate the whole MTR structure, it's crucial to prove that the TN-cell simulation is correct. As an example, we analyze the transmission properties of the TN-LC with 20 layers, with 2 • pretilt, θ p = 88 • , φ p = 45 • twist angle, with other LC thickness d = 5.9 µm, n o = 1.487, n e = 1.568 and λ = 0.55 µm for the incident light. When there is no drive voltage Fig. S4(a), resulting the polar pattern transmission in Fig. S4(b). When there is a drive voltage U = 8V across TN-LC cell, the TN cell twist and tilt angle will be changed by the voltage (Fig. S4(c)), making TN-LC cell at its OFF state and the incident light is blocked on normal incidence, while some leakage at oblique incidence ( Fig. S4(d)). We find that our simulation of polar intensity pattern approximately matches the experimental pattern [1], therefore we can say that we successfully proved the validity of the Matlab code of the extended Jones matrix. It should be noticed that the slight difference of the OFF case between the experimental pattern and the simulation might originate to the finite sampling of the twist and tilt.

E. Case Study of Prior MTR Waveplate Design
Although the previous work [2] is not simulated by Extended Jones matrix, we assume that we can still reproduce the similar result in normal incidence. In this section we try to use the extended Jones matrix to validate the transmission, equivalent retardation, equivalent optical axis, ellipticity of MTR HWPs and QWPs designs in prior work in the case of normal incidence. The design variables of 5 achromatic HWPs and QWPs design are summarized in the Table. S1.   Firstly, we investigate the HWPs design 2TR HW-A. Generally, for the requirement of half-wave retardation with linear input, the initial Stokes vector is S i = (1, 1, 0, 0) T , and the ideal output should be S i = (1, −1, 0, 0) T . Such half-wave retardation may varies by wavelength because of the different optical path and wavelength dispersion of different wavelength. In the case of HWPs (2TR HW-A), the variation of S 1 is depicted in Fig. S5(a). It can be seen that the S 1 ∈ [−1, −0.98] from wavelength λ ∈ [440 nm, 680 nm], suggesting the half-wave performance is satisfactory within this wavelength range.
Another HWPs design 2TR HW-B with circular input is also investigated. Contrary to the 2TR HW-A, The 2TR HW-B is optimized with the input Stokes vector is S i = (1, 0, 0, 1) T , and the ideal output should be S i = (1, 0, 0, −1) T . That is the half-wave retardation brings the right-handcircular (RHC) input to left-hand-circular (LHC) output. It can be seen that the S 3 ∈ [−1, −0.98] from wavelength λ ∈ [457 nm, 650 nm], suggesting the half-wave performance is satisfactory within this wavelength range. This wavelength range is slightly narrower than the its linear input counterpart 2TR HW-A (Fig. S5(b)).
Finally, we investigate the QWPs design 2TR QW-A. The ellipticity is plotted to compare with the result in prior work. In this case, horizontal linear input of light is incident onto the 2TR QW-A. It can be seen that the ellipticity e ∈ [0.98, 1] from wavelength λ ∈ [450 nm, 620 nm], suggesting satisfactory quarter-wave retardation is imposed within this wavelength range (Fig. S5(c)).
To summarize, the result in this sections are identical to the curved in the prior work [2],

RETARDATION PATTERN OF QWPS
This section provides retardation contour and the average retardation in off-axis of four types of QWPs. The average retardation Table. S2 for polar angle θ ∈ (0 • , 30 • ) and Table. S3 for θ ∈ (0 • , 45 • ) are presented to confirm that from purple to red wavelength range, the 3TR QW-D has the least variation of average retardation (Γ) off quarter wave thus has the highest average ellipticity angle among these four. Table S2. Quarter-Wave Plates Off-Axis Retardation (polar angle θ ∈ (0 • , 30 • ))  Besides, in Fig. S6 we find that the retardation polar pattern of these four types of QWPs has the similar distribution as the ellipticity pattern. This is because the QWPs retardation is the cause of the ellipticity change from linear polarization to circular polarization.

REFLECTION LEAKAGE OF QWPS
We first simulated the HQ design with perfect polarizer, ignoring the Fresnel reflection, and assuming the input of polarizer is 1. We compared the result of the HQ design simulated using commercial software TechWiz LCD in [3], and found that the HQ design matches the polar pattern of the result simulated by TechWiz LCD within θ ∈ (0 • , 90 • ) (Fig. S7). This validation further proves that both the extended Jones matrix for the elements and the whole reflection system modelling is correct.