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A distortion aberration (DA) correction device is fabricated using a liquid crystal lens array (LCLA), which is placed at the intermediate image plane of the optical system. Without voltage, the LCLA does not work, the image is distorted due to the aberration from the optical system; with voltage, the incident light is focused by the LCLA and then the distorted image is corrected. The correction of distorted image by LCLA is attributed to the redirection of the off-axis propagated chief ray approaches the principal point of the lens element.
©2013 Optical Society of America
1. Introduction
Liquid crystal (LC) lenses have been recently developed because of their tunable focal length, compact size and light weight. The generation of a gradient refractive index profile on the LC layer through the hole-patterned electrodes is a simple method of making LC lenses [1, 2]. Numerous applications of LC lenses, such as image processing, displays, and pico projectors, have been demonstrated [3–5]. Distortion is a primary aberration that is easily observed during the development of an optical system [6]. Distortion aberration (DA) is the different transverse magnification of each off-axis image point; each point may be different from its ideal image location. Pincushion DA appears if the real image location of an off-axis point is farther from the axis than the ideal image location. By contrast, barrel DA appears if the real image location of an off-axis point is nearer from the axis than the ideal image location. A conventional approach to reduce DA is to use a stop midway between identical lens elements. The distortion from the first lens will precisely cancel the contribution from the second. Placing an aperture stop at the lens also avoids the generation of DA, because the aperture stop causes the chief ray of the object passes through the principal point of the lens element. The insertion of a diffuser or a microlens array at the intermediate image plane has been demonstrated to reduce the aberrations of an optical system [7, 8]. The effect of the diffuser or lens array on aberration reduction is attributed to two factors: the creation of the new intermediate image plane that is flatter than the original Petzval image plane; and the redirection of the partial propagated rays into the paraxial region of the imaging lens.
This paper proposes a DA correction optical device by using the LC lens array (LCLA). The LCLA is placed at the intermediate image plane of an optical system. When the LCLA is electrically turned on, the incident light is focused by the LCLA and then the distorted image is corrected. The possible mechanisms are discussed in this paper.
2. Operation principle
Figure 1 demonstrates the operation principle of the proposed optical system with the LCLA placed at the intermediate image plane. As shown in Fig. 1(a), lens 1 creates an off-axis image point at the intermediate image plane. Without voltage, the LCLA does not work and the incident light is not focused. The off-axis image point emits a bundle of rays that propagates farther away from the optical axis with a finite solid angle. The finite solid angle creates an effective aperture stop that limits the bundle of rays to propagate on the top region of lens 2. DA is generated because the chief ray of the object is far from the principal point of lens 2. As shown in Fig. 1(b), once the voltage is applied to the LCLA, owing to the short focal length of the LCLA, the finite solid angle expands and redirects the emitted rays into the antipodal region from the optical axis of the optical system (bottom region of lens 2). Therefore, the rim of lens 2 becomes the effective aperture stop, and the chief ray of the object approaches the principal point of lens 2. Accordingly, the DA is effectively reduced.
Fig. 1 Operation principle of the proposed optical system with the LCLA inserted at the intermediate image plane: (a) without voltage applied to the LCLA; (b) with voltage applied to the LCLA. In the figures, P indicates the principal point of lens 2.
3. Experimental
To confirm the validity of the proposed model, the LCLA is fabricated using a hole-patterned electrode structure, as shown in Fig. 2 . The LC layer used in this experiment is 75 μm thick. The indium tin oxide (ITO) electrode is deposited on the inner sides of both substrates with antiparallel rubbing treatment. The diameter of the etched hole at the top substrate of the LCLA is 0.2 mm. The distance between the adjacent electrode holes is 0.1 mm. The dimension of the LCLA is 40 mm x 40 mm. The LC used is E7 (Merck). The phase retardation of the LCLA is measured by placing the LCLA between a pair of cross polarizers. The wavelength of the incident light is 632.8 nm. The rubbing direction of the LCLA has an angle of 45° with respect to the transmission axis of the polarizer. A charge-coupled device is used to capture the transmitted interference ring pattern, which can be used to calculate the focal length of the LCLA using the following equation [9, 10]:
where r is the radius of the lens aperture, d is the cell thickness, and δn is the effective refractive index difference between the lens center and the border. The reciprocal of the focal length becomes the focal power of the LCLA.Figure 3 shows the optical system for observing the DA correction effects of the LCLA. The extended light source is generated using a light-emitting diode (530 nm) array that passes through a diffuser. The input object is a rectangular grid pattern with a picture size of 16 mm x 16 mm. Lens 1 is a plano-convex spherical lens with a diameter, center thickness, edge thickness, and radius of curvature of 4 inches, 15 mm, 3 mm, and 103.8 mm, respectively. Lens 2 is a biconvex lens with a diameter, center thickness, edge thickness, and radius of curvature of 5 inches, 18 mm, 3 mm, and 207 mm, −207 mm, respectively. A polarizer is placed between the grid pattern object and lens 1. Under the paraxial condition, lens 1 generates a real image with a magnification of 2 at the intermediate image plane, whereas lens 2 generates a virtual image with a magnification of 5. Therefore, the total magnification of this optical system is 10. The LCLA is placed at the intermediate image plane of the optical system. The rubbing direction of the LCLA is parallel with the transmission axis of the polarizer. The digital camera (DC, Nikon D5100) is placed behind lens 2 to record the image of the grid pattern object. The aperture number, ISO and shutter time of D5100 is F22, 200 and 1 s, respectively. The focal length of the lens mounted on D5100 is 40 mm.
4. Results and discussion
Figures 4 and 5 show the phase retardation images and focal powers of the LCLA at various voltages. As shown in Fig. 4, the interference ring pattern appears when the applied voltage exceeds 2 V [Fig. 4(b)]. LCLA has a maximum focal power of ~1200 1/m when the applied voltage is ~3 V [Fig. 4(c)]. Notably, when the applied voltage is 2 V [Fig. 4(b)], the area of the interference ring pattern reaches its maximum, which indicates that the maximum amount of incident light can be focused by the LCLA. Figure 6 shows the pincushion DA images of the object when different voltages are applied to the LCLA. The DC is placed 1060 mm behind lens 2. Figure 6(c) shows that the pincushion DA is effectively corrected, because the maximum amount of incident light can be focused by the LCLA at 2V. Figure 7 shows the recorded barrel DA images of the object. The DC is placed 300 mm behind lens 2. In a similar manner, the barrel DA is effectively corrected when the applied voltage is 2 V. The correction of DA is markedly dependent on the polarization direction of the incident light. Figures 8(b) and 8(d) show that when the polarization direction of the incident light is perpendicular to the rubbing direction of the LCLA, the distorted images cannot be corrected. Notably, the contrast of the image is decreased after the optical system is corrected by the LCLA. In this paper, the focal length of the LCLA is measured at 632.8 nm, whereas the imaging of the optical system is based on 530 nm LED light source. Consequently, the chromatic dispersion of LC material has to be considered [11]. As shown in Fig. 5, with the applied voltage of 2V, the focal power of the LCLA measured at 632.8 nm is ~1000 1/m. As the LCLA is put in the optical system with 530 nm light source, the measured focal power of the LCLA must be higher than 1000 1/m, due to the chromatic dispersion effect of the LC material. Furthermore, the cell gap is ~75 μm, which causes the LCLA to have a slow response time; the thick cell may also distort the LC alignment, resulting in the light scattering. Therefore, for practical application, the cell gap of the LCLA has to be thin, and the LC material with high birefringence and low viscosity has to be used [12, 13]. Notably, as shown in Figs. 4 and 5, when a different voltage is applied to the LCLA, not only the focal power, but also the aperture size of LCLA, is changed. The optimized distortion correction result corresponds to the applied voltage of 2V, which gives the maximum aperture size, but not the maximum focal power of the LCLA. The aperture size of LCLA determines the area of the object that can be corrected; the focal power of the LCLA determines the effective aperture stop of lens 2. The obtained result indicates that the aperture size of LCLA dominates the distortion correction effect. However, the detailed contributions of the aperture size and the focal power of LCLA on the distortion correction effects remain to be decided.
Fig. 4 Interference ring patterns of the LCLA at various voltages: (a) 0, (b) 2, (c) 2.5, (d) 3, (e) 3.5, and (f) 4 V.
Fig. 6 Pincushion DA images of the object when the LCLA is supplied with (a) 0, (b) 1.5, (c) 2, (d) 2.5, (e) 3, and (f) 3.5 V.
Fig. 7 Barrel DA images of the object when the LCLA is supplied with (a) 0, (b) 1.5, (c) 2, (d) 2.5, (e) 3, and (f) 3.5 V.
Fig. 8 DA images of the object when the rubbing direction of the LCLA has an angle of (a) 0°, (b) 90°, (c) 0°, and (d) 90° with respect to the transmission axis of the polarizer. In (a) and (b), the DC is located 300 mm behind lens 2, whereas in (c) and (d), the DC is located 1060 mm behind lens 2. The voltage applied to the LCLA is 2 V. In each picture, R and P indicate rubbing direction of the LCLA and transmission axis of the polarizer, respectively.
Figure 9 shows the DA images of the object when the DC is placed 400 mm behind lens 2. The voltage supplied to the LCLA is 2 V. The rubbing direction of the LCLA and the transmission axis of the polarizer are parallel with the vertical lines. Figure 9(a) indicates that the DA becomes more severe when the image location is farther from the optical axis of the optical system. When the voltage supplied to the LCLA is 2 V, the barrel DA is reduced. However, the correction effect is dominant at the periphery of the image [Fig. 9(b)]. The corrected and uncorrected lines appear simultaneously at the periphery of the image. The corrected lines are attributed to the focused light that passes through the LC lens units, whereas the uncorrected lines are attributed to the unfocused light that does not pass through the LC lens units. In Fig. 9(b), the appearance of the corrected and the uncorrected lines proves that the vague images at the periphery of the corrected figures, such as Figs. 6 and 7, are not attributed to the field curvature distortion of the optical system [6]. The small aperture number of F22 also causes the DC to have a deep depth of field, inhibiting the observation of the field curve distortion in the present optical system.
Fig. 9 DA images of the object: (a) with LCLA at zero voltage, and (b) with LCLA at 2 V. The DC is placed 400 mm behind lens 2.
5. Conclusions
We have demonstrated the possibility of using the LCLA as a DA correction device in an optical system. The LCLA effectively corrects the DA of an optical system that is constructed using off-the-shelf spherical lenses. The correction effect of the LCLA on DA in the optical system depends on the applied voltage, density of the LC lens unit, and polarization direction of the incident light. The previously demonstrated diffuser or lens array is a customized DA correction device for a particular optical system [7, 8] and is therefore not suitable for other optical systems. The proposed LCLA is a universal DA correction device that may have the potential to apply in various optical systems. This LCLA works by electrically adjusting the focal length, thus providing a universal and economic method to reduce the DA of different optical systems. Further studies on the optical properties of the LCLA on the aberration reductions of optical systems are ongoing.
Acknowledgments
This work was financially supported by the National Science Council of the Republic of China, Taiwan (Contract Nos. NSC 101-2112-M-018-002-MY3 and 101-2811-M-018-005).
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