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A near-diffraction-limited, low-haze and tunable liquid crystal (LC) lens is presented. Building on an understanding of the key factors that have limited the performance of lenses based on liquid crystals, we show a simple design whose optical quality is similar to a high quality glass lens. It uses ‘floating’ electrodes to provide a smooth, controllable applied potential profile across the aperture to manage the phase profile.
©2013 Optical Society of America
1. Introduction
Electro-optical LC lenses have been considered as an alternative to conventional glass lenses because of their ability to change their optical power with the application of a control voltage. They can be used in many applications such as: imaging systems of compact cameras, eye correction, and 3D display systems. A number of LC lens designs have been introduced, but high-quality performance is only discussed in very few papers [1–23]. A high-quality LC lens needs to have a parabolic phase profile in the entire tunable range, a spot profile same as theoretically predicted for diffraction-limited lens in its focal plane, a MTF (Modulation Transfer Function) closer to diffraction-limited curve with the same cutoff frequency, and a comparable detail resolving ability as industrial high-quality glass lens with no additional scattering and chromatic aberrations demonstrated in the formed images.
Of the considered designs, one with discrete ring electrodes gives the best control of the index profile, as the lens’ OPD (Optical Path Difference) can be tuned to be a perfect parabola in principle. However, based on our paper analyzing the effects of fundamental factors on LC lens performance [24, 25], LC director distortions in gap regions between the neighboring ring electrodes generate phase variations that degrades the image by inducing haze.
Therefore, in the design of a high-quality LC lens, the above issue is an important one to solve. Work has been published to eliminate the gaps by interleaving the odd-numbered and even-numbered electrodes into two layers that are separated by an insulating layer, thus there are no gaps between two neighboring electrodes [3]. However, the implemented design requires a very precise placement of 2nd layer ITO pattern relative to the first layer in order to avoid misalignment; also it needs all electrodes on both layers to be electrically addressed, which requires a complex design of bus lines and driver connections.
What is needed is a simple design that can provide a controllable, smooth electric field distribution inside of the liquid crystal cell with the potential for providing a lens with diffraction limited performance. Also, a full characterization is needed to compare its performance with a high-quality glass lens.
Therefore, in this paper, we present the design of a lens using a layer of ring electrodes floating above the addressed concentric ring electrodes to be able to demonstrate a near-diffraction-limited performance. With this approach, the voltage profile is applied on the bottom layer of addressed ring electrodes, and each of the floating electrodes is placed over each gap area overlapping with small part of the neighboring pair of addressed electrodes on both sides of the gap. The potential on the floating electrode becomes the intermediate value of that on both electrodes on the bottom layer by the dielectric coupling and capacitive voltage division. Moreover, another advantage is that the addition of the floating electrodes effectively increases the phase sampling across the lens plane, which is able to further improve the efficiency [25, 26].
In the performance evaluation, we adopt the experimental approaches such as interferometric analysis of the phase profile, the light distribution in focal plane, imaging testing, and modulation contrast measurement. Throughout the investigation, we compare the performance of the LC lens with floating electrodes to a high-quality glass lens of the same power and the LC lens with concentric rings and gaps but no floating electrodes.
2. Floating electrode design
As a quantifiable example of our lens design, we provide an example device. Previously, we have reported a LC lens design with concentric rings and inter-ring resistor network (radius r = 1.2 mm) whose focal length could be tuned from f = 400 mm to infinity [27]. The total OPD of the ideal lens in unit of the number of waves can be obtained by [28]:
If the sampling rate (number of phase steps per wave) is certain to represent the continuous phase profile, assuming the area of each ring electrode has equal amount of the fraction of the unit wave, the total number of the rings is obtained as. Therefore, if the focal length is f = 400mm, design wavelength λ = 543.5 nm, and 10 phase steps per wave is considered, there are totally 33 electrodes where the outer radius of each ring can be expressed as:Here, n is the index number of each ring electrode. In our lens, there is a 3 µm gap between any two neighboring electrodes.
Using the position and size of the electrode pattern defined as the bottom layer, the floating electrode pattern (width and position) above can be designed. Each floating ring is located over each gap on the bottom layer electrodes and has a width of the gap plus what needs to cover ¼ of area of each neighboring electrodes it overlaps on the bottom layer. In this case, the capacitive coupling of the ‘floating’ electrodes to the addressed electrodes will cause the ‘floating’ electrodes to be at the average potential of the underlying addressed electrodes.
In addition to minimizing the effects of the gaps between the addressed electrodes, the floating electrodes effectively increase the phase sampling across the lens plane, which will further improve the efficiency. In this example, the sampling of the phase profile is doubled to 20 phase steps per wave.
Numerical calculations allow the optical performance in the focal plane to be estimated. The Strehl ratio (peak intensity calculated in focal plane for the LC lens as compared to that for the ideal lens [28]) of the LC lens becomes as high as 98.64% (from 89% for the LC lens without floating electrodes calculated before [25]), compared to the peak intensity of the ideal lens. In addition, MTF of the LC lens becomes almost the same as the diffraction-limited ideal curve, and there is almost no contrast drop at low frequencies (Fig. 1). From the calculation, therefore, a great improvement and a high quality due to the floating electrodes can be expected.
Fig. 1 Numerically calculated MTF for the performance of LC lens with floating electrode, compared to an ideal lens of same power.
3. Floating electrode fabrication
Based on the LC lens design with concentric ring electrodes and inter-ring resistor network, the fabrication of the bottom layer of ITO electrode rings has been shown, together with the Ni bus lines patterning for addressing about every 4th ITO electrode of the LC lens [7, 9]. In order to add the floating electrodes above, another layer of SiO2 is needed to protect the Ni during the following ITO etching. The thickness is preferred to be small to ensure the capacitive coupling between the floating electrodes and the bottom electrode layer is much larger than that to the ground plane electrode on the opposite cell substrate. In this case, the thin layer of SiO2 with about 300 Å in thickness is deposited on the substrate. Next, a 2nd layer of ITO with an approximated thickness of 400 Å is deposited, and a standard photolithography process is used for the floating electrode pattern, followed by the standard alignment layer deposition and rubbing. The cell is made with another substrate of common ground ITO glass coated with the same alignment layer with 10 µm sphere silica spacers (Fig. 2), and filled with liquid crystal mixture 18349 (Δn = 0.27).
4. LC lens performance evaluation
4.1. Interferometry
The Mach-Zehnder interferometer is used for measuring the OPD of LC lens and optimizing the voltage profile until a parabolic profile is obtained (Voltages applied on each addressable ring electrode are shown in Table 1). Due to a controlled fabrication process, the thickness variation across the lens aperture is a small fraction of the wavelength of light. The interferogram for the entire lens is without significant distortions from the desired parabolic profile (Fig. 3).
Table 1. Voltages applied on each addressable ring electrode.
Fig. 3 (a) Interferogram of the actual LC lens; (b) Measured OPD of the actual LC lens; (c) Measured phase profile 1D along radial axis, compared with ideal lens profile.
With the help of a high power objective lens in the interferometer setup, an area of the lens aperture of several electrodes can be imaged and the phase profile can be measured. For the outermost area of the LC lens consisting 3 or 4 gaps, the phase bumps in gaps are greatly minimized for the lens with floating electrodes, compared to the LC lens without the floating electrodes (Fig. 4).
Fig. 4 (c) Measured OPD 2D in 3D view with clear and smooth phase steps in outermost area of the LC lens with floating electrodes; (d) Measured OPD 2D in 3D view for the same area of LC lens without floating electrodes, very clear phase bumps are shown; (e) Measured phase profile 1D and side view of OPD 2D for LC lens with floating electrodes (x axis is the pixel numbers, representing about 60µm in length in lens actual area, y axis is the phase in unit of number of waves); (f) Measured phase profile 1D and side view of OPD 2D of the same area of the LC lens without the floating electrodes.
4.2. Light intensity measurement in the focal plane
Light intensity distribution in the focal plane is measured with a large CCD imager (22 mm × 14 mm) with a pixel size 5 µm × 5 µm, which is small enough to resolve the spot pattern in focal plane. A linear polarized and collimated laser light (5 mw, λ = 543.5 nm) is expanded to a beam of 10 mm in width. A circular aperture of 2.4 mm diameter is placed immediately in front of the LC lens concentrically with respect to the active area of the LC lens. The alignment direction of the LC lens is parallel to the polarization of the light. The light passing through the aperture is considered as a plane wave, and focused to a spot in the focal plane. The CCD is positioned for the best focus of the light at approximately z = 400 mm away from the lens.
To exaggerate the differences between the tested lenses, the example images of light in focal plane are shown with ‘over’ exposure. Due to a large dynamic range in the focal plane, the center spot in this case is saturated, but the large-scale scattering halo of rings across the entire imager due to the phase variations in gaps is shown to be greatly reduced for LC lens with floating electrodes (Fig. 5).
Fig. 5 Maximum exposure of the light distribution at focal plane for (a) glass lens f = 400 mm; (b) LC lens f = 400 mm without floating electrodes; (c) LC lens f = 400 mm with floating electrodes.
Quantitatively, the Strehl ratio here is defined as the peak intensity measured in focal plane for the LC lens as compared to that for the glass lens. Normalized to the glass lens, the Strehl ratio is measured as 93% for the LC lens with floating electrodes, much improved from the LC lens without floating electrodes (80%). Most of the drop for the case of the floating electrode lens is caused by the additional reflection loss by the LC lens comparing to the glass lens (measured as about 16% of the total incident light is reflected back for both LC lenses, compared to the glass lens which reflects about 8%).
Typically, the MTF is obtained by taking Fourier Transform of the PSF (Point Spread Function) in the focal plane and normalizing it to the transform’s zero frequency value (equal to the area under the PSF curve) [29]. However, experimentally, if the lens being considered has large angle scattering that results in a haze in the image, this method may not provide a useful quantification of the lens performance because the scattered light may not be completely collected by the limited detector acceptance angle used to measure the PSF.
To account for this possibility, we start with the idea that the sum of the area under the PSF curve, and the intensity of reflected light for each lens should be equal to the incident light intensity, and therefore the same for each lens. With this thought, we normalize the curves for the LC lenses to a number that is equal to the area under the glass lens’s PSF (assuming no scattering measured for glass lens), corrected for the measured differences in the reflectivity of the glass and LC lenses. Specifically, the normalization factor for the LC lens with our modified normalization is: Int (PSF, glass lens) *(1-Rlc)/(1-Rg). Here, Int (PSF, glass lens) is the area under the PSF for the glass lens; Rlc is the percentage of the total light reflected by the LC lens; and Rg is the percentage of the total light reflected by the glass lens. Therefore, this method assumes that the incident light intensity is the same and the measurements of intensity of the reflected light are quite precise. Because these assumptions can be questioned, we have plotted the MTF curves normalized by the typical method, and our modified method (Fig. 6). In fact, the area of PSF of LC lens without floating electrodes is measured as about 80% of that measured for glass lens, same as its measured Strehl ratio, indicating that the light might be scattered out of the detector collection angle, the modified normalization method should give more accurate result; in the case of LC lens with floating electrodes, the area of PSF curve is measured about 95% of that measured for glass lens, 2% more than measured Strehl ratio compared to the glass lens, indicating that most of the light in the focal plane is captured by the detector, and the typical normalization approach might already be able to give accurate result.
Fig. 6 (a) Calculated MTF from measured light distribution in the focal plane with typical normalization (area under measured PSF curve) for the glass lens, LC lens without floating electrodes, and LC lens with floating electrodes; (b) Calculated MTF from measured light distribution in the focal plane with modified normalization for the glass lens (mid curve), LC lens without floating electrodes (bottom curve), and LC lens with floating electrodes (top curve).
In both cases, the cutoff frequency is measured as about 10.5 cyc/mm, close to the theoretical prediction cyc/mm (Here, f is the focal length and D is the lens diameter [30]). With the typical normalization, MTF curves of LC lenses with and without floating electrodes are very similar to glass lens. With the modified normalization approach, MTF for LC lens without floating electrodes drops to about 0.85 at zero frequency, resulting from the scattering of the light; for LC lens with floating electrodes, it is about 1.05 at zero frequency. Also, the measured MTF data sets for the same lens are very close, and the sample-to-sample variation in average is less than 0.01, considering all frequencies. Obviously, with the modified normalization method, more accurate MTF curves are obtained, and the performance of the LC lens with floating electrodes has been greatly improved from the LC lens without them, and becomes very close to the glass lens.
4.3. Imaging test grouped with a high-quality glass lens f = 125 mm and Modulation contrast measurement
To have the imaging test, a high-quality glass lens with focal length f = 125 mm is used together with the test lens (LC lens or the glass lens of same power) as a group to increase the total power. A chrome-patterned USAF 1951 resolution chart is illuminated from behind by a uniform white light table and placed 400 mm away from the lens group. The same circular aperture with 2.4 mm in diameter is aligned and attached on top of and within the active area of the LC lens, the same imager CCD used in the previous section is placed about 125 mm behind the lens group, which makes it conjugate to the chart. A linear polarizer is placed in front of the lens group, with its polarization transmission axis parallel to the LC lens rubbing direction, if the test lens is the LC lens. The f = 400 mm test lens is placed immediately next to the f = 125 mm glass lens. The test is done in dark room, with an additional hood to eliminate any remaining ambient light.
With normal exposure (8 sec), the image formed through the LC lens without floating electrodes shows haze, caused by the diffraction of the phase variation in gaps (Fig. 7(b)), which is greatly reduced by the lens with floating electrodes (Fig. 7(c)). The problem can be exaggerated with a longer ‘over exposure’ (30 sec). While the image through the LC lens without floating electrodes shows larger amount of haze, the one formed by the floating electrode lens still keeps a good level of contrast, almost the same level as the image for glass lens. For both LC lenses, the limiting resolution is similar to the glass lens (Fig. 8).
Fig. 7 Normal exposure (8 sec) of images through (a) glass lens f = 400 mm; (b) LC lens without floating electrodes; (c) LC lens with floating electrodes; (d) when the LC lens is power off.
Fig. 8 “Over” exposure (30 sec) of images through (a) glass lens f = 400 mm; (b) LC lens without floating electrodes; (c) LC lens with floating electrodes.
As the spatial periodicity for each group of elements in the resolution chart are known, the modulation contrast ((Imax-Imin)/(Imax+Imin)) for each individual group can be specifically measured from the intensity values, which is used to further assess and quantify the imaging performance. Because this measurement includes the effect of optics, the CCD, stray light, etc, only a relative comparison of the performance of LC lens and the glass lens is possible.
The modulation contrast curves for both LC lenses are shown compared to the one for the glass lens. Clearly, the imaging performance has been greatly improved with the floating electrode lens (Fig. 9). The contrast for the lens without floating electrodes is about 90% at near-zero frequency, which has been significantly improved to about 95% for the floating electrode lens, which is very close to the glass lens value of about 97%.
Fig. 9 (a) Measured image modulation contrast for glass lens f = 400 mm, LC lens without floating electrodes, and LC lens with floating electrodes (grayscale and green channel); (b) close-up at low frequencies.
As the design wavelength is green, it is interesting to measure the modulation contrast from the green channel of the images formed through LC lens with floating electrode. The performance of the green light is even more improved (Fig. 9).
With the same imaging setup and same external light source, the images of high-definition print are taken for comparison with normal exposure. The shade, contrast, and chromatic performance of LC lens are very good even for the one without floating electrodes, while noticeable haze can be noticed in the shaded area of the images. With the help of the floating electrodes, haze is greatly minimized, and the performance is much improved to be similar to that of the high quality glass lens (Fig. 10).
Fig. 10 Normal exposure (2 sec) of print images through (a) glass lens f = 400 mm; (b) LC lens without floating electrodes; (c) LC lens with floating electrodes; (d) when the LC lens is power off.
5. Conclusion
Based on the previous work, we have found that variations in the desired voltage profile applied to the liquid crystal material, due to electrode gaps, is the main underlying reason for the efficiency drop of LC lenses with concentric electrode rings. In this work, the floating electrode design has greatly improved the lens performance, particularly by eliminating the scattering due to the gaps. The new design is seen to boost the Strehl ratio in the focal plane, and increase the image contrast, especially at near-zero frequency. Since we have demonstrated that the desired parabolic phase profile and high-quality imaging can be obtained for the LC lens at its highest optical power (most challenging case), the performance of this tunable lens for any lower power would be at least as good as shown here. As a result, a near-diffraction-limited tunable LC lens with low haze is shown whose design ideas can be applied to a large number of applications.
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