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We use a phase modulation method to form tunable lens arrays on liquid crystal on silicon (LCOS). With independent voltage adjustment on each pixel, LCOS generates a high-resolution gradual phase modulation profile, which makes it possible for the lens array to be freely tunable in focal length and other parameters. A tunable lens array is made on LCOS by use of this method, and we provide details of the theoretical analyses and experimental results.
©2005 Optical Society of America
1. Introduction
Liquid crystal (LC) lenses have been in existence for years to facilitate optical applications. Many types of LC lens have been proposed, for example, surface-relief profile [1], shaped electrode [2,3], and polymer network [4]. These types require modification of the LC cell such as an attached polymer lens or a cylindrical electrode [3]. Only some are suitable for use in a lens array but are not convenient for focus tuning applications. Here we use liquid crystal on silicon (LCOS) to simulate an ideal thin lens and to form a LC lens array on it. LCOS is composed of a LC pixel matrix that achieves precise phase retardation by independent voltage control on each single pixel. We use a phase transformation method to obtain the phase transformation factor of LCOS for a given focal length and radius of the lens. We formed a 6 x 6 lens array and tuned the focal length from infinity to 1m with a real-time regenerated color map. We demonstrate the advantages of a LCOS lens array by individually varying some columns of the lens array. Details are given of the theoretical analyses and results of the focusing quality.
The construction of LCOS is shown in Fig. 1. LCOS is a new type of active LC device. It consists of a twisted nematic liquid crystal (TNLC) film that is divided into LC pixels, an indium tin oxide (ITO) glass electrode, an aluminum (Al) mirror electrode, and a control circuit integrated on a silicon base. A large-scale integrated circuit is used to drive the LC pixels and every single pixel is electronically controllable to obtain the necessary phase retardation, which is why LCOS is widely used in projector displays and spatial light modulators (SLMs). We also found that it has major advantages such as a small size, high resolution, and real-time electronic control, all of which are preferable for a tunable LC lens.
2. Phase profile of a liquid crystal on silicon lens array
We took one pixel from LCOS as an example to demonstrate how polarized light is modulated in a LCOS SLM. Light incident upon LCOS at an approximate incidence (5° or smaller incident angle) passes through the twisted LC molecules and is reflected; see Fig. 2. By setting up the polarizer and analyzer for incident and reflected light we were able to obtain various modulation modes. These can be determined mainly by the following parameters: twist angle φ of LC molecules, the angle between the LC director on the LCOS surface and input polarizer α1 and analyzer α2, and the thickness birefringence product dΔn of the LC film. When we applied increased voltage, tilt angle θ of the LC molecules increased from its pretilt value to a maximum of 90°. Thus, parameter dΔn varies as shown by the following equations:
A Jones matrix was used to calculate the final result as reported in Ref. 5. The Jones matrix of the LCOS SLM is given by
where LC represents the Jones matrix of the LC film, Pol1 and Pol2 represent the polarizer and analyzer, Mir represents the reflecting mirror, Rot (ϕ) and Rot (-ϕ) represent the compensation for coordinate axis rotation when light is transmitted through the TNLC cell, and J and JR are Jones vectors of incident and reflected light. With Eq. (3) we can calculate the reflective intensity and phase retardation of a LCOS SLM. The results reported in Ref. 5 indicate that, in the configuration of α1 and α2 at 0° and -0°, in which the polarization directions of both polarizer and analyzer are parallel to the input director of the LC film, the reflective intensity is approximately 100% when dΔn increases. Hence it is in a phase-only modulation mode and satisfies the requirement of pure phase transformation for lens simulation.
Figure 3 shows the phase transformation of an ideal thin lens on LCOS. A plane light wave is incident on LCOS and converges at focus F, so reflected light is a convergent spherical wave with a radius of f. We used a complex amplitude expression to present the distribution of incident wave U(r) and reflected wave U’(r) on the front surface of the LCOS. These are written as
where A is the amplitude of an incident light wave (we ignore the energy loss in a LC film so A does not vary in transform), k is 2π/λ, z represents the distance from the light source to the LCOS surface, and r is the radius of the reflected wave on LCOS. The phase transformation factor is given by
Obviously R(r) is axisymmetric and represents the phase transformation characteristic of the LCOS lens and is determined by the distribution of phase retardation P(r), which is given by
In Eq. (7) we ignore the constant phase item in Eq. (6). If we now take into consideration that the incident light wave is a spherical wave with radius p, U(r) is given by
When we multiply U(r) by phase transformation factor R(r), the distribution of reflected light wave U’(r) is given by
where
Equation (10) is the exact imaging formula of an ideal thin lens, where p and q represent object distance and image distance and f is the focal length. According to the applied phase modulation map in Eq. (7), LCOS reflects incident light to the focus calculated by Eq. (10) such as an ideal thin lens with focal length f. The largest available radius r of a lens is limited by maximum phase retardation P max of LCOS. To solve this problem we added 2nπ compensation in Eq. (7) to extend the work area to cover the entire LCOS surface or light incident area.
LCOS uses anisotropic refraction rather than nonuniform thickness such as a glass or polymer lens to achieve phase transformation, which means that a LCOS lens does not have spherical parts. Thus a LCOS lens generates lower spherical aberration during fabrication compared with a spherical lens. It should also be noted that reflected wave U’(r) is a single spherical wave given by Eqs. (5) and (9), so there is no spherical aberration in theoretical wave transmission or at least low considering the paraxial approximation in the complex amplitude expression. High resolution and independent control of each pixel also ensure a smooth phase profile on LCOS, and this allows us to propose a more accurate method to achieve a tunable LCOS lens and, with additional research, a lens array such as a Fourier transform [6].
3. Experimental results
Pixels on LCOS are voltage driven to control phase retardation. We provided the voltage of a certain pixel by use of an extended graphics array (XGA) port of the control computer of LCOS by displaying the appropriate color value on the corresponding pixel on the computer monitor. We used a Michelson interferometer to measure [7] the relationship of phase retardation of LCOS and the input color value; the results are shown in Fig. 5. The LCOS is aurora ASI2000, designed for red light. The highest red value of 255 gives a 1.96π phase retardation. It is known that, because of inhomogeneous distribution of anchoring energy and elastic torque in LC film, the relationship of tilt angle of LC molecules and voltage cannot keep linearity at a large tilt condition. We made a 6-pixel reflective LC film and measured the phase-shift voltage curve [7]; the result is shown in Fig. 4. It can be seen that the curve is nonlinear. In LCOS the LC film and drive circuit are packaged together. By preadjustment of the relationship of the color value and drive voltage the phase- shift color value curve remains approximately linear in the whole variable range, as shown in Fig. 5.
For the experiment we used a light source with a 5-mW solid laser and 532-nm wavelength. The beam was filtered and collimated to generate parallel incident light, and we used parallel plane shear interferometry to verify the parallelism of the light. The polarization directions of the polarizer and analyzer are parallel to the input director on the LCOS front surface. Light is incident upon LCOS at a 5° incident angle and is reflected to the CCD camera positioned at the focal position. The size of LCOS aurora ASI2000 is 19.456 mm×14.592 mm with 1024×768 resolution, a 19-µm pixel pitch, a 1-µm gap, 5.5-µm thickness, and a 0.109 Δn. The radius of the laser beam was adjusted so that it would illuminate the entire LCOS surface.
First we displayed a uniform red value on the full screen and LCOS behaved like a flat mirror. As mentioned before, the LCOS with polarizer works in a pure phase modulation mode so it reflects almost 100% light intensity despite whatever red value we input. The reflected light converged at a closer distance from infinity when we decreased the focal length parameter of the red map. Finally we set the focal length at 1 m and the focusing condition is shown in Fig. 6. Figure 6(a) is the red map for a 6×6 lens array with a 1-m focal length. We displayed it on the monitor of the control computer with 1024×768 pixels. The radius of each lens is 50 pixels, which is approximately 0.98 mm. The gaps between lenses are marked in Fig. 6(a). The focusing image is shown in Fig. 6(b). Light converged at a point with 0.12-mm radius. The red value radius curve is shown in Fig. 7.
In Fig. 7 we demonstrate how the red value profile of a single lens in a lens array changed with focal length. We achieved real-time focal-length tuning by generating a corresponding red value profile in the control computer. When the focal length was shortened, the red value of the pixels increased.. In the case of f=1m the highest red value approached 255. If it were to vary over 255, for example, the focal length would be shorter or the radius of the lens would be larger, the 2π compensation would be needed.
Figure 8 shows the fine-tuning condition of a single lens near the 1-m focus. From left to right the focal length approached and was eventually accurately set to 1m. The minimum available adjustment in focal-length tuning is 1.3 mm, which was limited by a minimum change of the red value in the computer. With the method to tune focal length, we were also able to real-time tune the radius of the lens and the gaps by varying the corresponding parameters of the red map in Fig. 6(a). When we take into consideration that each pixel is completely independently controllable, the lens array is suitable for partial parameter adjustment. To demonstrate this we set up a 2×6 concave lens array at a 1-m negative focal length, aligned with a 4×6 convex lens at a 1-m focal length array; see Fig. 6(c). The result is shown in Fig. 6(d). The left two columns clearly show bright light circles that diverge because of the concave lens, near the rightmost four columns of bright focusing points.
To evaluate the focusing quality we measured the light intensity distribution in the focal zone of a single lens. Figure 9 shows that the focal intensity is 25 times higher than that of the surrounding area. A laser powermeter indicated approximately 0.045 V when LCOS acted as a flat mirror. When the focal length was set to 1m we obtained a peak value of approximately 0.43 V at the center of focus and the radius of focus was less than 6 pixels, which was 0.117mm. The intensity of the outside area remained at less than 0.018 V.
4. Conclusion
We have demonstrated a lens array on LCOS by using a phase transformation method. The focal length is tunable from 1m to infinity. Phase modulation on LCOS satisfies the following aspects of the phase profile requirements for a lens array: independent control phase retardation of each pixel, smooth adjustment of the phase map is well suited for a complex quadratic curve, use of the programmable color map is convenient for adjustment of not only focal length but also radius and other parameters or parts of the entire lens array.
Acknowledgments
This work is supported by the State “863” project No. 2002AA84TS11, the National Natural Science Foundation and China Academy of Engineering Physics No. 10176007, the National Natural Science Foundation No. 69877004.
References and links
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