A new polarization-independent and fast-response adaptive microlens array using a polymer-stabilized blue phase liquid crystal is proposed. With a curved top electrode and planar bottom electrode, gradient electric fields are generated and lens-like phase profile obtained. Optimization process leads to an ideal parabolic phase profile for suppressing spherical aberration.
©2011 Optical Society of America
Adaptive liquid crystal (LC) lens offers a tunable focal length and is useful for auto-focusing [1,2], 2D/3D switchable displays , and tunable photonic devices [4,5]. The basic operation principle of an LC lens is to generate gradient refractive index profile across the LC layer [6–11]. Most LC lenses developed so far employ nematic LCs, which offer a large birefringence for achieving a short focal length. However, two urgent issues remain to be overcome: polarization dependency and slow response time, especially when a thick LC layer is involved.
Recently, polymer-stabilized blue-phase liquid crystal (BPLC) based on Kerr effect [12–16] is emerging as a promising candidate for new photonics applications. The major attractions of BPLC are twofold: 1) its response time is in submillisecond range which is about 10X faster than that of nematic LCs, and 2) it does not require any molecular alignment layer, which greatly simplifies the fabrication process. A hole-patterned microlens using a polymer-stabilized BPLC has been recently demonstrated experimentally . It offers several attractive features, such as fast response time and no need for a polarizer. In the voltage-off state (V=0), the BPLC is optically isotropic so that the refractive index is uniform, resulting in zero optical power. When a voltage is applied across the top aluminum electrode (with a hole in the center as the aperture) and the bottom planar ITO (indium tin oxide) electrode, inhomogeneous electric fields are generated along the lens radius. Because the LC refractive index is spatially modulated by the electric fields, a lens profile is obtained. However, the performance of this lens could still be improved. For instance, this lens has a compromised image quality due to non-ideal shape of the refractive index profile, and the two orthogonal polarizations (TE and TM) will have a slightly different focal length because of the strong horizontal electric fields near the edge of the hole.
In this paper, we propose a new adaptive microlens structure using a polymer-stabilized BPLC. By varying the voltage on the concave (or convex) electrodes, we can tune the focal length continuously. Our simulation results show that indeed the device is polarization independent. Moreover, our design has a lower operating voltage and better phase profile than the hole-patterned microlens array . After optimization, an ideal parabolic phase profile is obtained which helps to suppressing spherical aberration.
2. Device structure and principle
Figure 1 shows the side-view (x-z plane) of the proposed microlens array. It consists of two glass substrates. The top glass substrate has microlens array structure, which could be concave or convex. A transparent ITO electrode is coated on the inner surface of the top substrate and then flattened by a polymer layer. The bottom substrate has a planar ITO electrode on the inner side. Sandwiched between these two substrates is a polymer-stabilized BPLC with a thickness of dLC. The thickness of the polymer layer at the center of the lens is d1, and that at the edge is d2; and the aperture radius of each individual microlens in the x-y plane is R.
The pink lines in Fig. 1 denote the generic beam path, in which the refraction between the top glass and polymer interface is neglected. At V=0, the BPLC is optically isotropic and does not contribute to the optical power. As the applied voltage increases, vertical electric fields are generated in the BPLC layer. According to Extended Kerr Effect , the electric field induced birefringence is:Fig. 1 shows, the electric field in the lens center is weaker than that near the edges because of the longer distance between top and bottom electrodes. As a result, the induced birefringence ∆nind is smaller (i.e., no is larger) in the center than that in the edges. So the phase profile over the aperture is like a positive lens. On the other hand, if the top glass substrate is convex, a tunable negative focal length could be achieved. By varying the voltage, the focal length of the LC layer could be tuned continuously. The focal length can be expressed as [10, 11]10]. Therefore, the total focal length (ft) which can be found from would change sign as fLC is tuned. This BPLC lens, similar to other nematic LC lenses, still has chromatic aberration because of the LC refractive index dispersion . Therefore, the focal length is longer for a longer wavelength. However, with the initial focal length, the achromatic aberration could be reduced by choosing the employed LC, polymer, and glass to have similar dispersive properties.
3. Simulation results
We carried out simulations to validate the device concepts. We first use commercial software Dimos (AUTRONIC-MELCHERS, Germany) to obtain the electric potential distribution and then calculate the optical properties based on extended 2x2 Jones Matrix . We assume the BPLC has a saturation birefringence ∆ns~0.2 (at λ=633 nm), saturation electric field ES~5.6x106 V/m, and Kerr constant K≈∆ns/λEs2=10 nm/V2 . In our design, R=225 µm, d1=76 µm, d2=2 µm, and dLC=17 µm. For simplicity we assume the index between the glass and polymer is matched, and therefore ft=fLC.
For the structure where the top substrate is spherical and concave as in Fig. 1, we calculate the phase profile across the lens for TM and TE waves. For comparison purpose, a similar structure using hole-patterned electrode  is also calculated with dLC=17 µm and aperture radius R=225 µm. Figure 2 depicts the relative phase profiles of both structures at V=100Vrms. The phase at the center of each microlens is offset to be zero for easy reading. The blue solid curve and red dashed curve represent TM and TE polarization in the proposed microlens array respectively, while the black and green curves represent TM and TE for the hole-patterned microlens array respectively.
In comparison to the hole-patterned microlens structure, our design exhibits following advantages:
- 1. For a given voltage (say, 100Vrms), our microlens array has a larger relative phase, resulting in a shorter focal length. This is because in the hole-patterned design, the aperture area has no top electrode and the electric fields decrease very quickly as the radius decreases. For instance, the relative phase at the radius of 150 µm is only about 1/10 of that at the edge of the lens. However, in our design the electric fields are generated between top and bottom ITO electrodes across the whole aperture, and the intensity of fields is utilized effectively. This indicates that our design has a lower operating voltage to achieve the same phase change than the hole-patterned structure.
- 2. In the hole-patterned structure, TM and TE polarizations make a noticeable difference near the edge. This undesirable polarization dependence is due to the strong horizontal field components near the edge of top aluminum electrode. The horizontal electric field component would generate birefringence in the horizontal direction. Thus, one polarization would experience an increased refractive index ne(E)≈ni+2∆nind(E)/3, while the other a decreased index no(E)≈ni-∆nind(E)/3. As a result, the TE wave will have a larger phase change, i.e., shorter focal length than the TM wave. In our design, although curved electric fields still exist near the top ITO electrode due to its curved shape, the polymer layer shields most of the horizontal components. With the electric fields almost vertical in the BPLC layer, both TE and TM experience the same ordinary index no(E)≈ni-∆nind(E)/3, and the polarization dependency is suppressed substantially.
- 3. Last but not least, the shape of the phase profile could be precisely controlled in our design by the shape of the top ITO. Figures 3(a) , 3(b) and 3(c) depict three structures with different ITO shapes; and Fig. 3(d) compares their phase profiles at 100Vrms with a perfect parabolic shape (red curve). Figure 3(a) shows a structure with spherical top ITO as we discussed previously; Fig. 3(b) depicts a structure with a cone-shape (or triangle from side view) top ITO electrode; and Fig. 3(c) has an Eiffel-Tower-like top ITO electrode. The center polymer thickness d1, edge polymer thickness d2, LC cell gap dLC and aperture radius R are all kept the same as in the previous calculations (R=225 µm, d1=76 µm, d2=2 µm, and dLC=17 µm.). Since the device is polarization independent, we only plot one polarization in Fig. 3(d) for easy reading. As shown in Fig. 3(d), the structure in Fig. 3(c) has more desirable phase profile than the other two, because a parabolic phase profile is helpful for suppressing spherical aberration which in turn improves the image quality.
Here, we briefly describe the procedures how we obtain the device structure shown in Fig. 3(c). In the low field region, extended Kerr Effect is reduced to Kerr Effect as:
In order to have a parabolic phase profile along the radius, the refractive index difference δn(r) should be proportional to r2, where r is the distance to the center of the lens. Under such a circumstance, we can writeEq. (5), we find
And accordingly, the shape of the top ITO is obtained. When high fields are involved, the calculation is more complicated. But for each operating voltage, the ITO shape could always be calculated. By optimizing our design at 100Vrms, we have kept the phase profile almost parabolic from 12Vrms to 100Vrms. However, for such a structure it is preferred that polymer and glass have the same refractive index. Otherwise, aberrations and total internal reflection might be introduced by the curved polymer-glass interface.
Figure 4 is a plot of the voltage-dependent focal length for the structure in Fig. 3(c). The blue curve is for TM polarization, and the red one is for TE polarization. As the voltage varies, the focal length of TM and TE polarization keeps the same, which further proves that this design is indeed polarization insensitive. As expected, the focal length gets shorter as the voltage increases. At 100Vrms, a 4-cm focal length is obtained. In the low voltage region, the change of focal length is more dramatic, while in the high voltage region the slope becomes flatter. The primary reason is that the focal length f is inversely proportional to the phase change as described in Eq. (3), and the secondary reason is that the induced birefringence gradually saturates at high fields.
The polymer layer thickness plays a critical role in our design. If it is too thin, the strong curved electric fields will penetrate to the BPLC layer and result in more noticeable polarization dependence. If it is too thick, then the electric field would be weakened, and a higher voltage is required in order to achieve the same focal length.
We have proposed a fast-response and polarization-independent microlens array using a polymer-stabilized BPLC. By varying the applied voltage from 0 to 100Vrms, the focal length of the LC layer could be continuously tuned from ∞ to 4 cm. If there is index mismatch between the employed polymer and glass, an initial optical power would exist, and the focal length of the whole system could be tuned from negative to positive (or from positive to negative). By optimizing the shape of top ITO electrode, a parabolic phase profile is obtained, which is desirable for achieving high image quality. Simulation results show that our device is indeed polarization independent, and has a lower operating voltage and better profile shape than the previously reported hole-patterned structure. Such a device would be very attractive for display and photonics applications.
The authors are indebted to Jin Yan, Su Xu and Dr. James E. Harvey for useful discussion and the financial support from Industrial Technology Research Institute (Taiwan).
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