1. Introduction

Liquid crystal (LC) phase modulators are used for wavefront correction in various active and adaptive optics applications [1]. Their operation is based on change of the birefringence of LC under an applied electric field. A typical LC phase modulator has a thin planar aligned nematic LC layer sandwiched between two substrates. The top substrate is transparent and has a continuous conductive coating, normally made of indium oxide doped with tin oxide (ITO). The bottom substrate, which can be either transparent or reflective, has a patterned structure of electrodes. The phase response of a LC phase modulator, ΔΦ, depends on the rms value of the voltage applied to electrodes, V_rms, following the electro-optical response characteristic of the LC ΔΦ = ΔΦ(V_rms).

Commercially available LC phase modulators use electrodes that are in direct contact with the LC layer [2, 3, 4, 5]. When the electrode size is larger than the thickness of the LC layer, the electric field is nearly constant over the actuated area, resulting in a step-like response, as shown in Fig. 1(a). The phase response of such a modulator is similar to that of a piston-type segmented deformable mirror. Due to their ability of direct control of the optical phase over a large array of pixels, high-resolution state-of-the-art LC phase modulators have become very powerful tools suitable for a wide range of applications.

For applications requiring the correction of low-order aberrations, such as beam steering and focusing, it is more advantageous to use specialized active optical components with naturally smooth responses rather than pixelated ones. For instance, the focal distance of a varifocal lens can be controlled using a single input, whereas the formation of a high-quality lens by a generic LC phase modulator requires addressing thousands of pixels. Various low-order LC active optical components have been developed [1, 6]. These are LC diffraction gratings, variable prisms, varifocal lenses [7], correctors of coma for optical pickups [6] and astigmatism for laser diodes [8].

An important advantage of LC technology is that it offers the possibility to make transparent devices, which allows the creation of more highly compact systems than those based on deformable mirrors. In particular, a transparent LC active optical component can be integrated in a contact lens or an adaptive lens eye implant to correct for the accommodation loss and higher-order aberrations of the human eye [9], which is difficult to implement using deformable mirrors. Besides, transparent components allow realization of stacked multilayered wavefront correctors for multi-conjugate adaptive optics.

Optical wavefronts are traditionally represented by continuous smooth functions, such as astigmatism, defocus, coma, etc. Modal wavefront correctors, such as deformable mirrors with continuous faceplace, provide better approximation to continuous wavefronts than pixelated devices. On average, to provide similar performance four times fewer actuators are required for a modal corrector than for a piston corrector.

 

Fig. 1. Forming of an influence function in a piston-type (a) and modal-type (b) LC phase modulators.

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Earlier, it was proposed to implement a modal-type liquid crystal wavefront corrector using a continuous high-resistance electrode as a distributed voltage divider with point-like actuators connected to it [10, 11]. The unique feature of this modal corrector is the potential to operate with several degrees of freedom per actuator. However, it was found that the performance of this device suffers from residual aberrations with high spatial frequencies and is somewhat lower than one of the membrane and continuous faceplate deformable mirrors [12]. Besides, this corrector can be implemented only in reflective configurations and uses very demanding manufacturing technology. All these factors make this approach noncompetitive with existing deformable mirror technologies.

In this paper we suggest another approach to the design of a modal liquid crystal wavefront corrector, which can be realized on a lower budget with higher efficiency. The features targeted in this approach are as follow:

2. Basic principles and simulation

The operation of the device is based on spreading of the electric field in a thick dielectric substrate with a high dielectric constant – see Fig. 1(b). We can expect an extension of the distributed electric field outside of the actuator area comparable to the thickness of the dielectric substrate. Since the ITO electrode is equipotential, the direction of the field inside the LC is perpendicular to the ITO electrode. Ceramic material with a high relative dielectric constant -up to 20000 - must be used to minimize the range of control voltages.

 

Fig. 2. 3D distribution of the electric potential in the ceramic substrate with the circular actuator on the top and the LC layer on the bottom (a); maximum electric potential at the LC-ceramic interface vs actuator diameter (b); normalized distributions of the electric potential for different actuator diameters (c).

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To model the electric field across the LC layer and to evaluate the range of control voltages, we performed finite element simulation using the ANSYS simulation package. The relative dielectric constant of our ceramic samples was in the range of 12000…15000. The LC mixture BL006 produced by Merck KGaA, Darmstadt, Germany, has a relative dielectric constant in the range 5…20, depending on the polarization state. The average values of these parameters were taken for the simulation, which were 13500 for the ceramic and 12.5 for the liquid crystal material.

An axi-symmetric configuration was considered, which consisted of a 25 μm-thick LC layer, a circular 3 mm-thick ceramic substrate with a diameter of 20 mm, and a circular actuator, centered with respect to the substrate. Zero electric potential was applied to the ITO electrode, 1 V potential to the circular actuator and 0 V potential to the ceramic substrate outside of the actuator. This situation corresponds to the case where all actuators are grounded except the central one.

An example 3D distribution of the electric potential in the ceramic substrate is shown in Fig. 2(a). The radial distributions of the potential over the LC layer were calculated for different diameters of the circular actuator; the results are presented in Figs. 2(a) and 2(b). Fig. 2(b) shows that the maximum potential depends on the electrode diameter. For a 3 mm electrode, the maximum potential was 0.18 V. Assuming that a maximum potential of 10 V is required over the LC layer, approximately 55 V will be needed to drive the LC phase modulator.

 

Fig. 3. Schematic diagram of a ceramic-based modal LC wavefront corrector and a control unit (a); layout and numbering of actuators for the 19-channel device (b).

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Comparison of the normalized distributions in Fig. 2(c) clearly shows that the potential distribution widens with the actuator diameter. This effect was not significant with the change of the diameter from 1 to 2 mm, which means that for small actuators the size of the potential distribution is mainly determined by the thickness of the dielectric.

3. Experimental devices

A 19-channel LC modal wavefront corrector (LC-MWC) and a PC-based control unit were designed and assembled based on the results of numerical simulation. The schematic of the device is shown in Fig. 3(a). A square-shaped 3 mm-thick ceramic substrate dimensioned 20×20 mm was polished to a flatness of λ/4 (peak-to-valley) for λ = 633 nm and coated with a multi-layer dielectric mirror optimized for maximum reflectivity at 633 nm. A homogeneous 25 μm spacing between the ceramic substrate and the cover glass was secured by teflon spacers. Planar alignment of the LC was provided by rubbed polyimide coatings. Birefringence of BL006 is 0.286 at λ = 589 nm, which results in a maximum range of phase variation of about 22 λ.

A 19-element hexagonal structure of actuators (Fig. 3(b)) with 3 mm pitch and 0.3 mm inter-pixel spacing was used for the back side of the ceramic. The electrode structure was manufactured by OKO Technologies on a printed circuit board (PCB), originally for micromachined membrane mirrors.

To drive the modulator, we used 24-channel digital-to-analog converter boards with 20-channel high voltage amplifier boards, both from OKO Technologies. Reliable multichannelgeneration of rectangular pulses was provided by custom control software developed using Ardence RTX, a real-time subsystem for Windows NT/2000/XP. The system generated 19 meander AC voltages with amplitudes up to 150 V and frequencies up to 5 kHz. The driving signal could be inverted in arbitrary channels, corresponding to the change of the initial phase from 0 to π. Henceforth, these phase shifts will be denoted as a “negative” value of the AC control signal.

 

Fig. 4. Influence function of the central actuator (a); simultaneous response of 2 (b), 4 (c), 7 (d) and 19 (e) actuators; amplitude is 110 V and frequency 1 kHz for all addressed actuators.

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Fig. 5. Simultaneous response of two adjacent actuators: V ₁=V ₂=110 V (a); V ₁=110 V, V ₂=-110 V (b); other actuators are biased at V_i=22 V, i = 3…19, for both patterns.

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The response of the device was investigated using a Twyman-Green interferometer relative to a flat reference. The interferometric patterns are shown for ∼15×15 mm aperture. The direction of the polarization of light from a He-Ne laser operating at λ = 633 nm was adjusted by means of a polarizer to coincide with the direction of the initial alignment of LC molecules. When observing interferometric patterns for the polarization component perpendicular to the initial LC alignment, we did not notice any changes with the signals applied. This means that the substrate did not deform in response to high voltages.

We found that in the range of 100…5000 Hz, the response of LC-MWC was frequency-independent. All further results were obtained for a 1 kHz control frequency. The actuators are numbered according to Fig. 3(b).

As Fig. 4(a) shows, addressing of a single actuator results in a nearly rotationally symmetric response. Localization of the response is due to the fact the LC does not respond to electric fields below a certain threshold level, which is about 1.2 V for BL006. As the influence function is rather broad, superposition of the influence functions of several actuators results in smooth wavefronts without spikes, as shown in Fig. 4(b)–(e).

Fig. 5 demonstrates the possibility to use the phase of the driving AC voltage as an additional control parameter. In this example, the AC voltage applied to the actuator 2 was inverted with respect to that applied to actuator 1 and the other ones, which were biased at 22 V. When the driving AC voltages for the actuators 1 and 2 are in phase, the response will correspond to the sum of the voltage distributions, resulting in an elliptically shaped response (Fig. 5(a)). When the two voltages are in opposite phase, we get a difference of the voltage distributions generated by the actuators. With the bias applied, it results in a coma-like response, which is shown in Fig. 5(b). Similar behavior can be demonstrated for three, four (Fig. 6), and more actuators.

 

Fig. 6. Simultaneous response of four actuators (central actuator encircled by three adjacent ones): V ₁=V ₂=V ₄=V ₆=110V (a); V ₁=-110V, V ₂=V ₄=V ₆=110V (b),(c); other actuators are off for both patterns. Interferometric patterns are shown in (a) and (b), and the result of the wavefront measurement with a Shack-Hartmann sensor is shown in (c).

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As both the amplitude and phase of the AC signal are used, the device can be operated with two degrees of freedom per actuator. In [12], this mode was investigated in application to the voltage-divider-based LC-MWC, but the results could also be applied to the ceramic-based corrector.

As the wavefronts generated by the ceramic-based LC-MWC are relatively smooth, these are suitable for analysis with a Hartmann-Shack sensor. An example of wavefront reconstruction using a Shack-Hartmann sensor with 127 microlenses is shown in Fig. 6(c). FrontSurfer software [13] was used to reconstruct the wavefront. In this example, the wavefront was approximated by 44 Zernike polynomials. The reconstruction result (Fig. 6(c)) was in agreement with interferometric data shown in Fig. 6(b).

Although the frequency of the AC voltage did not affect the influence functions, it is important to consider the case when the frequencies are different for two or more control voltages. In this case we deal with “incoherent” composition of the influence functions. Unlike the “coherent” case (equal frequencies), where the phase response of the modulator will correspond to the complex-valued sum of the voltage distributions generated by the actuators, in “incoherent” case it will correspond to rms composition of the influence functions. This problem was studied in [12] in application to the voltage divider-based LC-MWC. As seen from Fig. 7, this interpretation is also applicable to the ceramic-based device. The “coherent” composition of three influence functions shown in Fig. 7(a) was not dependent on frequency; we attained similar results when the common frequency was equal to 625, 1250 and 2500 Hz. However, we observed considerable change of the interferometric pattern after setting different frequencies to different actuators (Fig. 7(b)). As it was found in [12], linear control algorithms may not work in the “incoherent” case.

 

Fig. 7. “Coherent” and “incoherent” composition of the influence functions of three actuators: amplitude 110 V and frequency 625 Hz for all three actuators (a); same amplitude, frequencies 625, 1250 and 2500 Hz (b).

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While we expected 55 V to be sufficient to obtain a full response from the LC-MWC, in practice we needed to apply more than 100 V. This lower than expected sensitivity was caused by air gaps between the electrodes and the ceramic due to roughness of both the ceramic substrate and the printed circuit board and to the difference in their profiles. Both issues can be resolved by direct deposition of electrodes onto the ceramic. To further reduce the range of voltages, the ceramic layer should be made thinner.

The advantages of the reported device over mechanical deformable mirrors and the previously reported voltage divider-based LC-MWC include smooth and continuous phase profile, operation with several degrees of freedom per actuator, simplicity of design and technology, possibilities of implementation in a transmissive configuration using optically transparent ceramics and easy scaling to a large numbers of actuators. Thus, the approach proposed satisfies the requirements listed in Section 1.

The measured reflectivity of the device was 74% at λ = 633 nm. As neither antireflection coatings nor matching of the refraction coefficients between the glass and ITO were applied, this figure can be improved. Deposition of a quarter-wave plate on top of the dielectric mirror in a reflective device will make it polarization-independent for monochromatic light [14].

The response time of the ceramic-based LC-MWC was below 3 seconds, based on visual observation of the relaxation of the interferometric pattern. As the time of LC relaxation is proportional to the square of the LC layer thickness [15], the modulation range can be compromised to improve the speed. Additionally, efforts can be made in the future to implement dual-frequency control [16, 3] for this device to improve its speed.

4. Conclusions

The approach considered in this paper makes use of the distributed electric field in thick dielectric layers to form the modal response of an actuator. We have assembled and tested a 19-channel reflective liquid crystal modal wavefront corrector (LC-MWC) with a back substrate made of a ceramic material with a very high dielectric constant.

Although the influence function of this device shows no dependence on frequency, in a general way the behavior of the device is similar to that of the voltage-divider-based LC-MWC described in [10, 11, 12]. In particular the thick-dielectric-based LC-MWC can be controlled with two degrees of freedom per actuator. In addition it has the advantage of being free from high spatial frequency aberrations around the actuators that were typical for the previous implementation of LC-MWC. Based on these preliminary results, we expect this device to have superior performance in the correction of static or slowly varying phase aberrations. Compared to the voltage-divider-based LC-MWC, this seems to be a more efficient solution, deserving further study and development.