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Abstract
The geometric-phase lens (GPLs) with small form factor compared to traditional refractive lenses has been identified as a compelling solution in augmented-/virtual-/mixed-reality (AR/VR/MR) headsets. Formed either with liquid crystals (LCs) or metasurfaces, the GPL is a type of emerging leading technology that implements the arbitrary aspheric phase to realize low loss and minimal ghosting. However, the inherent chromatic abberation (CA) of GPLs can significantly degrade the image quality. A possible solution is the independent spectral phase implementation for RGB. In this work, we propose the design of three types of multi-twist LC based color-selective GPLs (CS-GPLs), exhibiting highly chromatic efficiency spectra with diameter 30 mm, focal length around 41.2~mm, and F −number 1.37. Through theoretical and experimental validation, each type of CS-GPL manifests high diffraction efficiency (>91%) on respective primary color of orthogonal polarization and high transmission on the complementary color of input polarization. The triplet composed by RGB CS-GPLs demonstrates relative contrast ratio and minimal ghosting. The strong color and polarization dependency of CS-GPLs not only provide a novel technique to mitigate CA but also offer more design freedom in the AR/VR/MR polarization and imaging system.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Augmented-/virtual-/mixed-reality (AR/VR/MR) head-mounted displays (HMDs) need optics that are lightweight, thin, and efficient. Geometric-phase lenses (GPLs), sometimes called Pancharatnam-Berry lenses, are compelling options because they employ wavefront phase control through patterned surfaces of either nano-structures or liquid crystals (LCs), rather than traditional optical path differences from curved interface refraction. Recently in AR displays, the GPLs facilitates large viewing angle by implementing Maxwellian view on the human eye pupil [1,2]. The nano-structure-based GPLs, which is widely known as metalens [3–5], manifests a large degree of freedom of the editable phase through the nano-structure arrangement [4], but suffers from relatively low efficiency (around $80~\%$ [3,4]), large focal length shift (around $1~$cm [4]), and tiny diameters (a few hundred $\mu$m [4]) due to the high fabrication cost of the lithography technique for large area. Another type of achromatic LC-based GPLs has been gaining more interest [6,7] for its polarization selectivity on diffraction orders, high diffraction efficiency [8,9] and relatively low cost compared to metalens. However, the chromatic abberation (CA) caused by the grating phenomenon of the focal length inverse-proportional to the wavelength, is the major hindrance for the perfect focusing of RGB in these types of near-eye displays. To mitigate the CA within the GPLs, various strategies have been reported. One study reported a lens group composed by color filters and achromatic GPLs enabling separate color focusing and the CA mitigation in some extent [10], but its complicated structure inevitably introduces some ghost images. Another strategy of combining the Fresnel lens and GPLs with complimentary CA to achieve the achromatic imaging [11] has shown some correction of CA, but the Fresnel lens inevitably introduces Fresnel-reset-based image degradation, which can get worsen in oblique incidence. Recently the holography optics element (HOE) diffusers [12,13] with compensated wavefront of the refractive lens are used to combine the GPLs to get similar spot size of RGB in normal incidence and thus correct the CA. However, the options of refractive lens unit for the exact phase compensation for holography on both normal/oblique incidence might be very limited. Besides, the addition of diffusers might further decrease the applications transparency and increase the stray light.
Instead of passively compensating the phase for RGB, another concept is to actively implement the phase for RGB separately. i.e. creating a type of GPLs with color-selectivity diffracts only a single primary color while transmitting the others. However, such CS-GPLs has never been developed yet.
In this work, we first propose a design of three types of color selective GPLs (CS-GPLs), i.e. a diffractive GPLs with color filtering function. Inspired by the early polarization interference filters (PIFs) [14–16] and more recently multi-twist-retarder (MTR) color filter [17], the half-wave (HW) retardation condition can be employed in both PGs and color filtering, and the wavelength polarization selectivity of the CS-GPLs are theoretically proved by rigorous couple wave analysis (RCWA) algorithm. Then the experimental verification of color selected diffraction efficiency (DE) is provided. Finally, the recreated RGB image with no CA is experimentally demonstrated by integrating the CS-GPLs triplet. It can be seen that the polarization dependency focusing ability of CS-GPLs makes it a novel and promising element to enhance the image resolution in the next generation displays.
2. Background
2.1 Geometric phase and polarization gratings
In optics representation, the geometric phase (or the Pancharatnam-Berry phase) manifests [18] the polarization state of light adiabatically moves in a closed path on the Poincare sphere [18]. In microscopic level, such phase ($\delta _{g}$) is achieved by local permittivity variation, which can be generated by nanostructure orientation [19], or local optics axis direction variation ($\Phi$) of birefringent material. Using liquid crystal polymer (LCP) and photoalignment layer (PAL), such variation can be easily implemented by polarization holography techniques [20–26] or direct-writing approach [27–31]. The resulted geometric phase is determined as $\delta _{g}=2\Phi$, which is wavelength independent. When an arbitrary polarized light wave $|\chi _{in}\rangle$ incident onto the spatially varying geometric phase holograms (GPHs) such as polarization gratings (PGs) [8,32–36], the optics axis periodicity induces permittivity periodic variation. In the case of PGs, three distinct output waves (waves and efficiency described $(|\chi _i, \eta _i)$: a right-hand circular (RHC) primary wave $|\chi _{+1}\rangle$, a conjugate left-hand circular (LHC) wave $|\chi _{-1}\rangle$, and a leakage wave $|\chi _0\rangle$ [37]. Considering the DE of each order ($\eta _i, ~i=-1,~0,~+1$) is generally wavelength dependent [35], therefore the diffraction wave generated by GPH can be written as:
The diffraction phenomenon of PGs is influenced by both the retardation and the input polarization. For a PGs with half-wave (HW) retardation of a specific wavelength [35], the diffraction is maximized and the transmission is minimized, i.e. $\eta _{+1}(\lambda )+\eta _{-1}(\lambda )=1$ and $\eta _0=0$. With LHC or RHC as input, the $+1$ order or $-1$ order reaches maximum with orthogonal circular polarization.
2.2 Rigorous couple wave analysis
To theoretically investigate the CS-GPLs diffraction, it is crucial to simulate the transmitted and reflected electric field of all diffractive order within such three dimension anisotropic media. The Rigorous Couple Wave Analysis (RCWA) algorithm that describes the permittivity periodicity by Fourier Series expansion is an ideal method to simulate the optics axis periodic variation of the CS-GPLs. Through solving the Maxwell equation of each layer, the transmission and reflection of each order can be found by matching the boundary condition of $\mathbf {s}$-polarization and $\mathbf {p}$-polarization. It has the advantage of including all diffraction orders interference compared to two-wave couple-wave theory (CWT) [38]. Therefore it is the most rigorous method to simulate the periodic isotropic or anisotropic structure. In this manuscript, we create the periodic liquid crystal model in three dimension, specify the periodic permittivity in Fourier Series of each sublayer, and implement RCWA algorithm to calculate the DE of CS-GPLs using self-written Matlab code. For the complex LCP layers composed of twist and non-twist structure, the enhanced transmittance matrix (EMT) [39] is used to connect the EM field of each diffractive order by imposing the boundary condition between adjacent sublayers. To evaluate the wide angle performance, the conical diffraction [40] is also investigated by incorporating conical RCWA and conical EMT algorithm. Considering the GPLs has different periodicity along the radius, the simulation of the whole structure should be of semi-periodic pattern. However, since the aperture is much larger compared to single period, we adopt the assumption that the period variation can be ignored at a local spot. Therefore, the periodic condition of RCWA can still be valid, i.e. the RCWA algorithm is still applicable for the DE calculation of a specific location on the CS-GPLs with fixed periodicity.
3. Color-selective lens design
3.1 Requirement of the lens system in AR/VR
The advantage of polarization-selective focusing makes the GPLs a promising element in near-eye displays. For VR and AR system, the system might be different. The configuration of placing displays in the focal length of the GPLs to general image at infinity is current widely used in VR system. In AR system, with the object from the waveguide output coupler as infinity, the GPLs also has the potential to form a Maxwellian view by placing the human eye at the focal spot of the lens [1,2].
To satisfy the requirement of compactness and immersive virtual experience of near-eye displays, the optical system are usually limited to tens of millimeters. In current state of art, the angle field of view (AFOV) can be up to $\pm 60^{\circ }$ for VR and $\pm 15^{\circ }$ for AR. In this manuscript, as proof of concept we limit the FOV within $\pm 20^{\circ }$. With the FOV at hand, the desired focal length can be calculated as $f=D/(2\tan \theta )$. For the lens diameter $D=30$ mm, the designed focal length is around $f=41.2~$mm, and the $F-$number is $F_{\#}=1.37$.
3.2 Geometric-phase lens
As mentioned before, to alleviate the CA of the imaging system, it is necessary to design three color of CS-GPLs with same focal length. In this way, when circular polarization white light incident on to the triplet of red, green blue CS-GPLs, high throughput of orthogonal circular polarized RGB light can be in focus simultaneously, as depicted in Fig. 1. The geometric phase at $z=0$ thus forms the parabolic phase of a conventional lens with focal length $f$ can be written as $\delta _g (x,y) =(2\pi /\lambda )(\sqrt {x^2+y^2+f^2}-f)$, and the orientation profile can be written as $\Phi (x,y,0)=(\pi /\lambda )(\sqrt {x^2+y^2+f^2}-f)$. As reference, the folded geometric phase $\delta _g$ and the orientation $\Phi$ along the radius $z=0$ are provided in Supplement 1, Section 1.
3.3 Wavelength-dependent effective retardation
In early PIFs such as Solc filter, Lyot filter realized color filtering by imposed HW retardation to the stop-band wavelength range and full-wave (FW) retardation to the pass-band wavelength range. The principle is that the retardation of these birefringent network are wavelength-dependent. Later the LCP-based multi-twist-retarder (MTR) color filter [17] proved that this principle also applies in birefringent network with both twist and non-twist layers. Inspired by the effective retardation of MTR color filter, it is possible to combine the GPLs and MTR color filter by implementing the HW retardation at the stop-band of color filter transmission, which becomes the $+1$ order of diffraction, i.e. $\Gamma (\lambda _H)=(N+1)\pi$. On the other hand, the FW retardation at the pass-band is used to maintain the $0$ order transmission, i.e. $\Gamma (\lambda _F)=2N\pi$, with $N$ as integer.
3.4 Polarization selectivity of order and efficiency
As a general property of PGs, the input polarization is still responsible for determining the ratio between $\eta _{+1}$ and $\eta _{-1}$. For circular PGs with local linear anisotropy, the eigenvectors of the Maxwell equation are waves with orthogonal circular polarization [41]. At normal incidence, the diffraction efficiency can be derived from Jones matrix, which has a simple relations with both the input polarization and the wavelength-dependent retardation. As the input white light incident onto the PGs, light with HW retardation $\lambda =\lambda _H$ will be diffracted to the $+1$ order, while the light with FW retardation $\lambda =\lambda _F$ will be transmitted as $0$ order. The diffraction efficiency of three orders in normal incidence can be written as [34,35,41]:
When the lens phase profile is implemented by the gradual variation of optics axis along radius, the above wavelength-dependent effective retardation and polarization grating properties still apply. In this aspect, the CS-GPLs can be designed by combining the GPLs profile, wavelength-dependent effective retardation principle, as well as the polarization-selectivity of PGs. Therefore, When the LHC polarized white light incident onto a CS-GPLs, the light with wavelength that undergoes HW retardation can be focused as RHC $+1$ order, while the rest of wavelength of light transmits as LHC $0$ order and minimal RHC $-1$ order occurs. Therefore, the design of CS-GPLs should impose the HW retardation on primary colors, which then becomes the $+1$ order focusing wavelength of the CS-GPLs, while the complimentary should be FW retardation and transmitted as $0$ order. This principle can be summarized as the following Table 1. For simplicity, we use the terminology $BY$ to refer the lens diffracting blue and transmit yellow. This principle also applies to GM and RC lens.
Table 1. The CS-GPLs wavelength polarization design (Input: LHC white light)
3.5 Three-dimension orientation profile
The polarization grating phenomenon is originated from periodic birefringence, which can be realized periodic optics axis orientation of LCP [8,41]. For PGs of one-dimension periodicity and $n$ layers, the orientation profile $\Phi '(x, z)$ can be expressed as:
Fig. 2. CS-GPLs with 5-layer of alternating nonchiral/chiral LCP sublayers. Parabolic lens profile geometric phase is implemented in photoalignment layer (PAL) as $\Psi (x,y)$ and LCP initial phase as $\Phi _0(x,y,z)$. When input light incident on the CS-GPLs, Primary color focuses while complementary colors transmits.
From the design requirement of CS-GPLs listed in Table 1, a high saturation chromatic spectra of the diffraction is desired. Inspired by the MTR color filter high saturation performance, we adopted the similar structure of color filter of alternating nonchiral/chiral layers as a baseline of the CS-GPLs design, i.e. nonchiral odd layer and chiral even layer. With this basic model, there are several variables for LCP structure, the thickness of non-chiral layer, twist angle of chiral layer, and layer number. To balance the fabrication complexity, wide-view properties and color selectivity, we adopt five layer structure for BY, GM and RC lens, and set the variable as follow: ($d_1, \phi _2, d_3, \phi _4, d_5$) to implement the optimization. In order to avoid parasite retardation of the chiral layer, the thickness of the even layer $d_2$ and $d_4$ has been set as $0.25~\mu$m. This architecture can be simulated using RCWA self-written Matlab code.
Based on this five-layer architecture, the optimization algorithm in Matlab is implemented to construct the merit function to find largest DE of certain primary, i.e. $f=1-\eta _{p}~~(p=r,g,b)$ and find its global minimum. The design parameter of three lens (thickness and twist angle) of consecutive layer ($L_i$, $i=1,2,3,4,5$) are listed in Table 2. To further investigate the retardation accumulated from layer to layer, the retardation represented in target wave number ($\lambda _p$) are also listed in Table 3. Since the phase shift between ordinary and extraordinary rays, or optical path difference (OPD) is equivalent to retardation $\Gamma$ in normal incidence. Therefore, for the nonchiral odd layers ($i=1,3,5$), the retardation $\Gamma _i$ of each layer can be directly calculated as $\Gamma _i=\Delta n (\lambda )d_i$, with $\Delta n (\lambda )$ as birefringence that included dispersion effect and $d_i$ the thickness of the respective layer. For the chiral layers ($i=2,4$), the retardation can be calculated using extended Jones matrix method [42]. As can be seen from Table 3, the total retardation of the respective lens is always half-wave of the target wavelength $\lambda _p$. For example, the BY lens has the retardation of $5.5\lambda _b$. The majority of retardation comes from the non-chiral thick layers, while the chiral layers contribute small portion of retardation due to the relatively small thickness.
Table 2. The CS-GPLs design parameters
Table 3. Design retardation $\Gamma$ Unit: Wave number ($\lambda _{p}$)
4. Theoretical validation
In an actual optical system, the luminance (or brightness) within AFOV is an important spec that directly determines the image contrast ratio (CR) and MTF. When dealing with a polarization imaging system incorporating GPLs, both the DE angular response and the output polarization are modulated by GPLs. The GPLs output polarization is important because the polarization state can be transferred to next polarization optics, such as circular polarizer [43,44], reflective polarizer [45], or polarization volume grating (PVG) [46–48], resulting certain amount of polarization leakage, and finally leading to CR reduction and MTF degrade.
Therefore, for an imaging system incorporating GPLs, both the diffraction and polarization evolution need to be investigated. In this section, RCWA and EMT algorithm are implemented to simulate the planar and conical diffraction of the CS-GPLs. For validation of the self-written RCWA code, refer to Supplement 1, Section 2. As mentioned above, although the GPLs is semi-periodic structure, the period, which is around few microns, is minimal compared to the lens aperture (radius $r=15~$mm). Therefore, at the specific spot of the lens, we can assume that the period is constant. Without loss of generality, we choose a spot at $r=6.5~$mm to investigate the lens performance. The period of BY, GM and RC lens is: $5.93~\mu$m, $6.67~\mu$m and $7.88~\mu$m, which can be calculated from Eq. (4b). These periods for three types of lens will be used in both planar diffraction and conical diffraction.
4.1 Planar diffraction
For imaging system incorporating GPLs, the angular/wavelength response is directly related to the AR/VR system image brightness and therefore becomes the most important specs of CS-GPLs. In the incident plane where the diffraction has the largest variation, the wavelength/angle response contour of three lens are depicted in Fig. 3. It can be seen that all the lens shows high contrast of respective primary/complementary color in $+1$ order, while the leakage of $-1$ order are minimal. We also notice that such color-selective behavior also extends to oblique incidence. The AFOV can be up to $\pm 20^{\circ }$ for all types of lens at their primary colors.
Fig. 3. Simulated Diffraction efficiency of BY, GM and RC lens respectively varied by incident angle and wavelength (within incident plane): (a)-(c) −1 th order; (b)-(d) $0$ order; (e)-(g) $+1$ order. The period corresponds to the radius $r=6.5$ mm.
Since the output polarization of the $+1$ order can significantly influence the image resolution of the system, the output polarization angle/wavelength response should also be investigated. The output polarization can be calculated by first extracting the phase of the $+1$ order complex transmission coefficient $T_s$ and $T_p$, which can be calculated by RCWA algorithm. Secondly, the phase shift is therefore the subtraction of the phase of $T_s$ and $T_p$. To address the CS-GPLs performance on the whole aperture, the similar angle/wavelength response of different position ($r=3.5/6.5/15$ mm) on CS-GPLs are also presented in Supplement 1, Section 3. The simulation result shows that the color-selected diffraction retains design FOV of $\pm 20^{\circ }$ within radius $r<6.5$ mm, which corresponds to relatively large period. For position $r=15$ mm, the FOV of the +1 order starts to shrink.
Finally, with the amplitude of transmittance and phase shift, the Stokes vector can be calculated. Assuming perfect LHC input, the $+1$ order Stokes vector $S_3$ angle/wavelength response of three types of CS-GPLs is respectively depicted in Fig. 4. It can be seen that for the respective primary color, the CS-GPLs has large field of ideal RHC polarization, which can up to $\pm 50^{\circ }$. For the other two primaries, the AFOV shrinks to around $\pm 20^{\circ }$.
Fig. 4. (a)-(c) Simulated Stokes Vector $S_3$ of $+1$ order of BY, GM and RC lens respectively varied by incident angle and wavelength. The period corresponds to the radius $r=6.5$ mm.
To demonstrate the CS-GPLs capability of wavelength and polarization selectivity, we use the BY lens as example. When the LHC polarized white light incident onto the BY lens, the diffracted light maintains good RHC polarization with $S_3=1$ across the visible wavelength range, which manifests the CS-GPLs can successfully maintain the polarization grating property of polarization selectivity within AFOV of $\pm 20^{\circ }$. However, due to the wavelength selectivity of efficiency, the light diffracted in $+1$ order has minimum portion of yellow, which is mainly retained in $0$ order with LHC polarization. Adding the geometric parabolic lens phase, the BY lens can successfully focus the blue with RHC polarization, while transmit the yellow with LHC polarization. For the other two type of lens, the principle is similar.
4.2 Conical diffraction
To further investigate the luminance uniformity within the AFOV, the conical diffraction and related output polarization are investigated. Because the focusing wavelength is set to be primary wavelength, we limit the conical diffraction discussion with the incident light as the primary color ($\lambda _c = 456/530/630$ nm) and LHC polarization. Three primary color of light, with varied polar angle ($\theta$) and azimuth angle ($\psi$), is respectively incident onto the corresponding CS-GPLs BY/GM/RC.
The conical DE is shown in Fig. 5. It can be seen that the all three types of lens can maintain relatively high DE, $\eta _{+1}(\lambda _c)\geqslant 80\%$ within AFOV $\theta \in (-20^{\circ }, 20^{\circ })$ and azimuth angle $\psi \in (0, 360^{\circ })$. However, when $\theta \geqslant 20^{\circ }$, the retardation deviation $\Delta \Gamma$ becomes noticeable, making the DE drop for all the lens, among which the RC lens drops most significantly (Fig. 5(g)-(i)), suggesting the RC lens angular performance might post a FOV limitation of the combination of the three lens. The origin of non-symmetric conical diffraction will be discussed later.
Fig. 5. Simulated diffraction efficiency polar contour varied by wavelength of BY, GM, RC lens with LHC input (a)-(c) $-1$ order; (d)-(f) $0$ order; (g)-(i) $+1$ order. The period corresponds to the radius $r=6.5$ mm.
As to the output $+1$ order polarization in. Figure 6, it can be seen that within AFOV $\pm 20^{\circ }$, each lens has almost perfect RHC polarization, and the abrupt decrease of RC lens has a more abrupt decrease of Stokes Vector $S'_3$ at polar angle $\theta \leqslant -20^{\circ }$, corresponding to the DE decrease range depicted in Fig. 5.
Fig. 6. Simulated Stokes vector $S_3$ polar contour of $+1$ order of three types of lens with LHC input: (a)BY, input wavelength $\lambda _B=456$ nm; (b) GM, input wavelength $\lambda _G=530$ nm; (c)RC, input wavelength $\lambda _R=630$ nm. The period corresponds to the radius $r=6.5$ mm.
5. Experimental
5.1 Fabrication
To fabricate the CS-GPLs, a photo alignment layer (PAL) and a series of liquid crystal reactive mesogen layers is firstly spin-coated (@1500 RPM, 60 s) on a thin 2 inch round glass substrate and baked (@$130^{\circ }C$, 120s). Second, a laser direct writing scanner with HeCd laser ($\lambda =365$ nm) and a polarization rotator [9,31] is used to create LCP optics axis profile of $\Psi (x,y)$ on the substrate which moved by a 2D translation system. This linear polarization profile $\psi (x,y)$, through the photo-alignment effect, gives rise to a continuous optic axis profile $\Phi (x,y)=(\pi /\lambda )((x^2+y^2+f^2)^\frac {1}{2}-f)$, with diffractive wavelength as $\lambda$ and the focal length as $f$. To create pattern smaller than beam size, a lens with periodicity $2\Lambda$ can be made using direct writing system and used as phase mask to produce lens with periodicity $\Lambda$ under linear polarized ultraviolet light (centeral wavelength $\bar {\lambda }=365$ nm). Third, six mixtures are prepared using reactive mesogen RMM-A ( refractive index $n_e (\lambda )= 1.62+1.83\times 10^{-14} /\lambda ^2$ , $n_o (\lambda )= 1.50+1.00\times 10^{-14} /\lambda ^2$), RMM-B ( refractive index $n_e (\lambda )= 1.67+2.87\times 10^{-14} /\lambda ^2$ , $n_o (\lambda )= 1.49+0.80\times 10^{-14} /\lambda ^2$) doped with calculated amounts of positive chiral RMM-C and negative chiral RMM-D. Fourth, A thin layer of RMM-A was spin coated on the exposed PAL layer with a $5\%$ solution in solvent Propylene glycol methyl ether acetate (PGMEA). Fifth, to start coating the MTR, use the mixture and spin-speed listed in Supplement 1, Section 4.
5.2 Characterization of the lens
To identify the actual focal length $f'$ of the CS-GPLs, the periodicity of locations on different radius $r=3.5~$mm, 6.5 mm, 15 mm (Fig. 7(a)) of the CS-GPLs are measured using polarization microscopy (Fig. 7(b)-(j)). From the measured grating period $(\Lambda _A, \Lambda _B, \Lambda _C)$, the corresponding focal length $(f'_A, f'_B, f'_C)$ can be calculated. The calculation method and result are provided in Supplement 1, Section 5. The result shows that the averaged focal length $\bar {f'}$ of three radius locations (Point $A$, $B$ and $C$) ranges from $41.1$ mm to $41.7$ mm for three types of lens, matching well with the design focal length $f=41.2$ mm. To demonstrate the visual effect of the chromaticity and focusing, the virtual image at infinity is shown by placing the object with white number at the focal length under ambient light (Fig. 7(k)-(n)). It can be seen that despite of the low CR caused by the broadband spectrum of the sunlight, the primary RGB $+1$ order diffraction are shown clearly with magnification $m>1$ on the respective CS-GPLs, while the original image transmitted with complimentary color with no magnification.
Fig. 7. (a) Three spots on different radius (Point $A$, $B$, and $C$) on CS-GPLs and the polarization microscopy of the respective positions: (b)-(d) BY lens ; (e)-(g) GM lens; (h)-(j) RC lens. The grating period from (b) to (i) : $11.00~\mu$m, $5.73~\mu$m, $2.61~\mu$m, $12.50~\mu$m, $6.67~\mu$m, $2.95~\mu$m, $15.05~\mu$m, $7.88~\mu$m, $3.5~\mu$m. (k) Object for focus and its virtual image at infinity of the CS-GPLs: (l) BY lens; (m) GM lens; (n) RC lens.
5.3 Transmission and diffraction efficiency
Because the color-selected diffraction of CS-GPLs is originated from the wavelength retardation, which can be investigated by measuring the $0$ order transmission spectrum with broadband light source. In the scenarios of transmittance spectrum, the perfect color selectivity requirement of the HW retardation of primary (Eq. (2a)) and FW retardation of complimentary (Eq. (2b)) embody minimal stop-band and maximum pass-band, respectively. For BY lens shown in Fig. 8(a), it can be seen that compared to design (yellow solid), the measured transmission (yellow dash) representing 0 order is around $80~\%$ in the pass-band, with leakage less than $20~\%$ at primary blue. The deviation from the design spectrum might be caused by the parasitic retardation of the chiral layers, resulting to lowering CR of $+1$ diffraction and transmission. Similar case also occurs of RC lens in Fig. 8(c). For the GM lens, the CR has best performance, with slight shift on the valley position. From Fig. 8(b), the leakage of the measured $0$ order transmission (purple dash) at green central wavelength is $T_0(\lambda _G)=3.87~\%$, suggesting high DE of $+1$ order with LHC input.
Fig. 8. Design and experimental $0$ order transmission and complimentary $+1$ order spectrum of (a) BY Lens; (b) GM Lens; (c) RC Lens. The dash-dot lines are RGB laser central wavelength for designing the lens.
To align with the current requirement of AR/VR system, R/G/B lasers (central wavelength $\lambda _c=456/532/630$ nm) are used to investigate the transmission and diffraction efficiency. To guarantee the perfect LHC input, three types of quarter-wave plates (QWPs) that working on R/G/B separately (RQWP, GQWP, and BQWP) is alternately aligned with the respective laser. The ellipticity angle of the resulted input light is $43.2^{\circ }$, $43.6^{\circ }$, and $41.7^{\circ }$ respectively. Three integration spheres (ISs) are placed at a distance of approximately focal length ($d\approx f$) away from the lens to measure the intensity of each order (Fig. 9(a)).
Fig. 9. (a) Configuration of transmission/diffraction efficiency measurement set-up. Three separate measurements is implemented by letting R/G/B laser transmits R/G/B quarter-wave plate (RQWP, GQWP, BQWP), then to the respective CS-GPL, and finally to the integration spheres (ISs) . (b)-(d) Transmission (bar) and diffraction efficiency (dot) of three CS-GPLs. Diffraction of the CS-GPLs: BY, GM, RC. (e)-(m) Diffraction spot image of respective CS-GPLs incident by three left-hand-circular (LHC) polarized lasers with central wavelength $\lambda _c=(456, 532, 630)$ nm.
The image of laser spots of R/G/B laser focused by the BY/GM/RC lens is shown in Fig. 9(e)-(m), and the measured transmittance and DE of three CS-GPLs are presented in Fig. 9(b)-(d). For the completed data of transmission, DE and its normalization algorithm based on the measured intensity, refer to the Supplement 1, Section 6. It can be seen that the GM lens demonstrates highest $+1$ order DE and lowest leakage of non-focusing two primaries, while RC lens has slightly lower DE and relatively same proportion of leakage. For the BY lens, although the decrease of transmission and DE are not significant, the leakage of the other two non-focusing primaries are noticeable, suggesting that the BY lens has the lowest CR in the focal plane. Such findings corresponds well with the photos taken in ambient light condition in Fig. 7(l)-(n).
5.4 Imaging of the lens triplet
The purpose of developing the CS-GPLs is to address the CA inherent to GPLs. In this subsection, three CS-GPLs designed with same focal length are combined as triplet and the related imaging system is investigated.
As depicted in Fig. 10(e), apart from the CS-GPLs, two pairs of left-hand quasi circular polarizer (LQCP) composed of linear polarizer (LP) and wide-angle achromatic quarter-wave plate (AQWP) are used. First, to enable the LHC polarization input of RGB light within FOV, LQCP comprised LP and wide-angle AQWP [42] with slow axis at $\phi _c=-45^{\circ }$ is placed ahead of the triplet (Fig. 10(f)) to create LHC polarization from the unpolarized displays. Second, after the triplet, the RHC main image is chosen and the LHC stray light is removed by a pair of right-hand quasi circular polarizer (RQCA) that comprises AQWP ($\phi _c=-45^{\circ }$) and LP. In this way, the RGB image with RHC polarization generated by each CS-GPLs can be combined at the distance of eye relief (ER) (around 25 cm).
Fig. 10. (a)-(c): Transmission image and $+1$ order diffraction in the inset of the CS-GPL: (a) BY; (b) GM; (c) RC; (d) Inset is the object.(e) Configuration of unpolarized Light is projected from LED displays to triplet lens system. LED displays is placed at distance of focal length from the triplet. The observer side is the combination of QWPs ($-45^{\circ }$) and LP ($0^{\circ }$). Human eye or camera is placed at eye relief distance (ER) from the triplet $25$ cm. The object height is $H=1.5$ cm, the field of field of the image is around $FOV=20^{\circ }$.
The image captured by the separate CS-GPLs is shown in Fig. 10(a)-(c). The primary color diffracted image has a magnification $m>1$ in these images, while the 0 order transmission is the object. It can be seen that the intensity of green and red diffraction of GM lens and RC lens are much higher than the blue intensity of the BY lens. We also noticed that the $0$ order transmission intensity of BY lens is higher than the diffracted blue. It can be explained by the fact that in colorimetry yellow has larger lightness than blue, and human eye has less sensitivity in blue. From the GM lens image, we can see the transmission image of the $0$th order is magenta, while in RC lens the transmission image of the $0$th order is blue.
It can be seen that the composed RGB image can achieve white color with less chromatic aberration at the fringes, suggesting that the focal length of these three lens are similar. It can be seen that the white image color is a bit yellowish, suggesting that intensity of the blue light focused is relatively low. We notice there is small color separation at the edge of the pattern, this might be due to the off-axis chromatic aberration. However, in the normal incidence, the chromatic aberration is corrected. The object height is $H=1.5$ cm, the field of view (FOV) of the image is around $FOV=20^{\circ }$.
In order to investigate the main beam efficiency of RGB image, the Jones calculus of the system is implemented and is provided in Supplement 1, Section 8. A. With the measured DE of the CS-GPLs, the efficiency of the primary blue, green and red is $70.9~\%$, $75.9~\%$ and $67.9~\%$.
6. Discussion
One interesting fact is that although the Stokes Vector $S_3$ pattern has some similarity with the DE pattern of $+1$ order DE in Fig. 5, it does not strictly follow its variation. The similarity occurs because the effective retardation deviation $\Gamma$ is the common origin of the non-uniformity of both the DE and polarization. Specifically, the effective retardation changes the phase shift between the $\mathbf {s}$-polarization and $\mathbf {p}$-polarization, and therefore changes the output polarization. In the case of PGs, the conical diffraction not only inherit its identity as common grating effect, but it also imposes polarization change onto the output polarization. Therefore, in the case oblique incidence of LHC, the output polarization is no longer perfect RHC. To summarize, the output polarization variation is subjected to both effective retardation deviation and the conical grating effect.
Asides from the main image composed by three main lights from the CS-GPL, ghost image shows up when 0 order of primary color of light is focused on other CS-GPL. To illustrate the ghost image origin, we use the primary blue light in the triplet as an example. The small portion of 0 order blue light which is not focused by BY lens and has the LHC polarization, can be focused by RC lens. Such polarization leakage is caused by the retardation deviation of both BY lens and RC lens. On one hand, the retardation deviation of blue in BY lens generate the non-zero leakage. On the other, the retardation deviation of blue in RC lens facilitate the focusing of the blue. Because the focal length is wavelength dependent for each CS-GPL, these ghost images are therefore not an image at infinity, but a magnified or contracted image with shorter or longer focal length. These color ghost images increase the spot size and degrade the MTF of the imaging system. The Jones calculus of ghost image is provided in Supplement 1, Section 8B.
If we define the CR of each color as the ratio of main color image intensity divided by the ghost intensity of respective color, the CR of the system can be calculated as Fig. 11. The result shows that the CR of green is relatively larger than red and blue. The reason is that the ghost of red and blue intensity is relatively higher than green, which is due to the leakage of red and blue leakage of BY lens and RC lens.
A quick possible mitigation to improve the CR imbalance of the triplet system is adding polarization color filter of blue and red to alleviate the retardation deviation. Once the leakage is minimize for BY and RC lens, the CR can be improved. In the long run, the CS-GPLs retardation deviation can be further optimized by design and fabrication iteration.
7. Conclusion
In this work, we proposed the design, theoretical and experimental validation of three types of color-selected geometric phase lens (CS-GPLs), which exhibits high diffraction efficiency based on input wavelength and polarization. The triplet system composed by these CS-GPLs simultaneously focuses the RGB image and thus significantly alleviate the chromatic abberation of the imaging system within field of view $\pm 20^{\circ }$, with minimal ghosting and high contrast ratio. Given its strong wavelength polarization dependency on the diffraction phenomenon, the CS-GPLs can be a type of novel element to mitigate the chromatic abberation in the ARVR system.
Funding
ImagineOptix Corp (2014-2450).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results are all presented either in this manuscript or Supplement 1.
Supplemental document
See Supplement 1 for supporting content.
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