Modal wavefront correctors based on nematic liquid crystals

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(1)MODAL WAVEFRONT CORRECTORS BASED ON NEMATIC LIQUID CRYSTALS PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof. dr. ir. J. T. Fokkema, voorzitter van het College voor Promoties,. in het openbaar te verdedigen. op maandag 24 october 2005 om 10.30 uur. door. Mikhail Yurievich Loktev. Fysicus. Leraar Samara State University, Rusland geboren te Kuibyshev, USSR.

(2) Dit proefschrift is goedgekeurd door de promotor: Prof. dr. P. J. French. Toegevoegd promotor: Dr. ir. G. Vdovin. Samenstelling promotiecomissie: Rector Magnificus Prof. dr. P. J. French Dr. ir. G. Vdovin Prof. dr. J. J. M. Braat Prof. dr. L. Nanver Prof. dr. P. Artal Prof. dr. M. K. Giles Dr. A. F. Naumov. voorzitter Technische Universiteit Delft, promotor Technische Universiteit Delft, toegevoegd promotor Technische Universiteit Delft Technische Universiteit Delft University of Murcia, Spanje New Mexico State University, VS Physical Optics Corporation, VS. Printed by [OPTIMA] Grafische Communicatie, Rotterdam ISBN: 90-8559-100-7 c M. Y. Loktev Copyright 2005 by All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without written permission of the publisher or the author. PRINTED IN THE NETHERLANDS.

(3) Contents 1 Introduction 1.1 Active and adaptive optics . . . 1.2 Wavefront correction . . . . . . 1.3 Liquid crystal technology . . . 1.3.1 Displays . . . . . . . . . 1.3.2 Spatial light modulators 1.4 Motivation and objectives . . . 1.5 Organization of the thesis . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 2 Performances of wavefront correctors in 2.1 Introduction . . . . . . . . . . . . . . . . 2.2 Measures of correction quality . . . . . . 2.3 Models of wavefront correctors . . . . . 2.3.1 Piston correctors . . . . . . . . . 2.3.2 Membrane mirrors . . . . . . . . 2.3.3 Continuous faceplate mirrors . . 2.4 Results . . . . . . . . . . . . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . .. . . . . . . .. 5 5 8 9 10 10 13 14. atmospheric optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17 17 18 20 21 21 22 24 28. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 3 Electro-optical characteristics of nematic liquid crystals 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Principles of phase modulation in LC devices . . . . . . . 3.2.1 Nematic LC devices . . . . . . . . . . . . . . . . . 3.2.2 Ferroelectric LC devices . . . . . . . . . . . . . . . 3.2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . 3.3 The Ericksen-Leslie theory . . . . . . . . . . . . . . . . . . 3.4 Investigation of the S-effect in statics . . . . . . . . . . . . 3.4.1 Method of calculation . . . . . . . . . . . . . . . . 3.4.2 Measurements . . . . . . . . . . . . . . . . . . . . 3.4.3 Finding of unknown parameters . . . . . . . . . . .. . . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. 29 29 30 30 33 35 35 36 36 39 39.

(4) 2. CONTENTS. 3.5 3.6. 3.4.4 Uniform electric field approximation . . . . . . . . . . . Dynamics of S-effect . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4 Simulation and control of modal liquid crystal lenses 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Overview of adaptive LC lenses . . . . . . . . . . . . . 4.1.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Construction and principles of operation . . . . . . . . . . . . 4.3 The lens equation and the constant impedance approximation 4.3.1 Cylindrical lens . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Spherical lens . . . . . . . . . . . . . . . . . . . . . . . 4.4 Numerical analysis of the voltage distribution . . . . . . . . . 4.4.1 Analysis of the static case . . . . . . . . . . . . . . . . 4.4.2 Simulation of lens dynamics . . . . . . . . . . . . . . . 4.5 Manufacturing of the MLCL . . . . . . . . . . . . . . . . . . . 4.6 Control methods . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Cylindrical lens . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Spherical lens . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Optimization of the lens performance . . . . . . . . . . . . . . 4.7.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Adaptive focusing system . . . . . . . . . . . . . . . . 4.7.3 Phase profile optimization, numerical model . . . . . . 4.7.4 Practical calibration of the MLCL . . . . . . . . . . . 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .. 41 41 44 47 47 47 49 50 52 53 54 55 56 61 64 66 66 69 73 73 74 75 77 82 91. 5 Liquid crystal modal wavefront corrector based on a voltage divider 93 5.1 Basic principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.2 Numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.3 Influence functions . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.4 Operation modes . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.4.1 Single-frequency control . . . . . . . . . . . . . . . . . . 106 5.4.2 Single-frequency control with adjustment of phase shifts 110 5.4.3 Frequency-multiplexing control . . . . . . . . . . . . . . 111 5.4.4 Non-harmonic AC voltages . . . . . . . . . . . . . . . . 118 5.4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.5 Glass-based devices . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.5.1 Manufacturing technology . . . . . . . . . . . . . . . . . 121 5.5.2 Experimental results . . . . . . . . . . . . . . . . . . . . 122 5.6 Implementation using silicon technology . . . . . . . . . . . . . 127.

(5) CONTENTS 5.7. 3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6 Silicon-based liquid crystal hybrid wavefront 6.1 Basic principles . . . . . . . . . . . . . . . . . 6.2 Numerical model . . . . . . . . . . . . . . . . 6.3 Choice of the optimum design parameters . . 6.3.1 Number of pixels . . . . . . . . . . . . 6.3.2 Matching of resistors . . . . . . . . . . 6.3.3 IC resistor type . . . . . . . . . . . . . 6.3.4 Process type . . . . . . . . . . . . . . 6.4 Manufacturing technology . . . . . . . . . . . 6.5 Experiment . . . . . . . . . . . . . . . . . . . 6.5.1 Electrical testing of the chip . . . . . . 6.5.2 Optical testing . . . . . . . . . . . . . 6.6 Summary . . . . . . . . . . . . . . . . . . . .. corrector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. 131. 133 . 133 . 134 . 137 . 137 . 139 . 139 . 140 . 142 . 144 . 144 . 145 . 147. 7 Liquid crystal modal wavefront corrector with a thick dielectric substrate 149 7.1 Principles of operation . . . . . . . . . . . . . . . . . . . . . . . 149 7.2 Experimental devices . . . . . . . . . . . . . . . . . . . . . . . . 151 7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8 Conclusions. 159. Bibliography. 163. List of abbreviations. 175. Summary. 177. Samenvatting. 180. Acknowledgments. 183. Author’s publications. 185. About the author. 190.

(6) 4. CONTENTS.

(7) Chapter 1. Introduction 1.1. Active and adaptive optics. Fixed optical components, such as lenses, mirrors and prisms, form the basis of the majority of optical imaging systems embedded in cameras, telescopes, microscopes etc. A combination of optical elements ensures that the imaging system has the required characteristics. To provide accommodation and focusing in a wide range, optical systems need to have an adjustable design. In the majority of systems adjustment is done mechanically, by variation of the configuration of the system. Another adjustment method is to use adjustable, or “active”, optical components. An example of such an active optical element in nature is the crystalline lens of the human eye, whose focusing power can be changed by the eye muscles. In optical instruments, however, active components are not yet applied very often, but there is a growing interest to this kind of components. Moreover, they have the potential to become a key element in future optical electronics. Active elements can fulfil another function - they can compensate for aberrations in the optical system, keeping them in the required range of tolerances for the whole range of operation. Aberrations are wavefront errors that lead to reduced resolution or power efficiency of the system. They are caused by imperfections of the optical elements and external conditions, such as turbulence of the ambient media and thermal effects. Aberrations can largely be compensated in the optical design, but their contribution is definitive in large-aperture high-resolution systems, systems for imaging in highly aberrated media, and laser systems which are highly sensitive to energy losses. In these cases active compensation of aberrations is crucially important, and here we often use the.

(8) 6. Introduction term “adaptive optics”. Adaptive optics (AO) is a rapidly developing branch of applied optics [1, 2] whose purpose is to increase the resolution of optical systems by active compensation of phase aberrations. Although there is no consensus about the use of the terms “active optics” and “adaptive optics”, the use of the last term normally assumes use of a higher bandwidth (> 10 Hz) and the presence of a wavefront sensor coupled with an active optical element in a feedback loop [3]. The majority of adaptive optical systems are based on the traditional phaseconjugate scheme [4]. Its operation can be explained using a simplified example from astronomy. A light wave coming from a distant star forms an image of this star in a telescope. Passing through the atmosphere, the light wave is affected by fluctuations in density of the atmosphere due to turbulence, distorting the front of the wave (Figure 1.1(a)). Due to this kind of distortion, a blurred image is produced. To compensate for this effect, two elements can be introduced. The first is called a “wavefront sensor” and measures the wavefront distortion introduced by the atmosphere; the second element, the so-called “wavefront corrector”, produces the phase function conjugated to the aberrated wavefront, thus correcting the distortion of an initial wavefront. The processing unit (computer) coordinates operation of these two elements (Figure 1.1(b)). As the atmospheric distortions evolve over time, the system should perform several hundred measurements and corrections per second. In some systems, an additional tip-tilt mirror is used to reduce the large stroke requirements for the wavefront corrector. For compensation of aberrations produced by “thick” layers of highly aberrated media, multi-conjugate adaptive systems must be used. Volume aberrations are described satisfactorily by layered models [5], and several “wavefront sensor - wavefront corrector” pairs need to be used to compensate them. The concept of adaptive optics was first proposed in 1953 by Horace Babcock [6], however, the first adaptive optical system was built only in the late 60s in Haleakala Observatory in Maui, Hawaii. An essential development in this area occurred in the 70s and 80s: the papers published in these years laid the theoretical basis of AO and contain a lot of valuable experimental material from observatories. Due to the extremely high cost of AO systems [1], the area of applications was restricted to high-budget observatories and military projects on the delivery of high-energy laser beams from ground to space for missile defense and secure communications. This is why the research in this field was mainly related to atmospheric optics [5, 7, 8]. In the last decade, however, advances in sensors and electronic processing brought down the cost of adaptive optics. The availability of silicon micromachined adaptive mirrors developed in TU Delft [9], liquid crystal wavefront correctors [10] and CCD-based Shack-Hartmann wavefront sensors has led to the introduction of many new applications in scientific, industrial and medical.

(9) 1.1 Active and adaptive optics. Figure 1.1: Image blurring induced by atmospheric wavefront distortions in astronomy (a); operation of the adaptive optical system (b).. 7.

(10) 8. Introduction fields [11, 12], also stimulating research in optics of the human eye, confocal microscopy, ultra-short laser pulse generation and machine vision. The Optical Microsystems Group of Electronic Instrumentation Laboratory, TU Delft works on development and application of low-cost adaptive optics. Here we could mention the integrated CMOS-based Shack-Hartmann wavefront sensor [13], deformable mirrors [9, 14], liquid crystal wavefront correctors presented in this thesis and the system for adaptive correction of human-eye aberrations [15]. Further development and mass production of low-cost active and adaptive optics are expected to lead to new generations of optical devices providing a higher level of resolution and image quality. Nevertheless, the cost of the whole adaptive system including electronics and control systems still remains high; besides, these systems are bulky. To resolve these problems, an integrated approach should be applied to the design of adaptive optical components and the whole system.. 1.2. Wavefront correction. “Wavefront” is a key term in adaptive optics. We shall use the following definition. The electric field vector of a monochromatic light wave can be represented in the form (1.1) E(r, t) = E(r)ei(kr+ε(r)) e−iωt , where k is the wave vector, which is perpendicular to the electric field and has a modulus related to the wavelength λ as k = 2π/λ; ω is the angular frequency related to the temporal frequency ν as ω = 2πν; E(r) describes the amplitude, and ϕ(r) = kr + ε(r) the phase of the oscillating field E. The surfaces joining all points of equal phase are known as wavefronts [16]. The wavefront surface can be described by the equation ϕ(r) = ϕ0 , where ϕ0 is a constant. The distribution of the optical phase in a certain plane, such as the pupil plane of an optical system, provides a good approximation of the wavefront in many cases. This is the reason why the measurement of the optical phase distribution is often referred to as “wavefront measurement”. However, this does not apply for strong aberrations, especially if these are accompanied by significant amplitude modulation and wavefront dislocations; in this situation both the amplitude and phase distributions should be analyzed. The wavefront shape can be corrected by modifying the optical phase profile. The core principle used here is phase conjugation, which means that the correcting phase profile should be optically conjugated to the measured phase aberration. Correction with the conjugated phase profile results in an ideal flat wavefront, as shown in Figure 1.1(b). However, if the wavefront cannot be.

(11) 1.3 Liquid crystal technology precisely determined or precisely replicated by the relay optics, or if diffraction effects dominate, the phase conjugation cannot be precisely employed [2]. Operation of phase modulation devices is based on control of the optical path difference (OPD), which can be written as OPD = n∆z, where n is the refractive index, and ∆z is the physical distance traveled by the wave. OPD is related to the phase ϕ as ϕ = 2π · OPD/λ. Deformable mirrors, which is the primary technology for wavefront correctors, modulate ∆z, operating in the reflective mode. Liquid crystal (LC) phase modulators represent a low-cost alternative to mechanically driven mirrors; their operation is based on modulation of n of the LC layer under the applied electric field in transparent or reflective mode. Wavefront correctors are traditionally subdivided into two classes according to the implemented compensation technique - zonal and modal. Zonal correctors such as segmented piston and tip-tilt mirrors allow individual control of a phase over a set of subapertures providing step-wise phase compensation, whereas modal ones such as deformable mirrors use a set of smooth functions (modes, or influence functions) to approximate the required phase function. Deformable mirrors are presented in a series of modifications: membrane [9], bimorph [17] and continuous faceplate mirrors [18] with different geometries of actuators and boundary conditions. A number of LC correctors with pixelated structure of actuators are commercially available, whose operation is similar to that of piston-type segmented mirrors. In this thesis we discuss several configurations of LC devices with modal-type operation similar to that of deformable mirrors with a continuous face plate.. 1.3. Liquid crystal technology. Liquid crystals are fluids with a certain order in the arrangement of the molecules [19]. As a result, there is anisotropy of their mechanical, electrical, magnetic and optical properties. The most important property of LCs is that the orientation of its director (the axis of preferred molecular orientation) can be changed under an external electric or magnetic field, which allows electrical control of the optical properties of the LC cell. There is a variety of LC types classified according to the symmetry of their molecular structure. For use in electro-optical devices, two LC types are of interest: nematic liquid crystals (NLCs) and ferroelectric liquid crystals (FLCs).. 9.

(12) 10. Introduction. 1.3.1. Displays. The majority of LC applications are LC displays (LCDs). The dynamic scattering effect was used in the earliest LCD prototypes at the end of the 1960s. Twisted-nematic LCDs (TN-LCDs) with matrix addressing followed in the early 70s, and later, in the 80s, super-twist nematic LCDs (STN-LCDs) were introduced. At the moment the most widely used technology is active-matrixaddressed LCDs (AM-LCDs), where every pixel with twisted-nematic LC is connected to a nonlinear element, so that the required level of voltage is maintained during the whole frame period, which provides the required level of contrast and the possibility to generate color by means of embedded RGB filters. The development of thin-film-transistor (TFT) technology resulted in the wide use of AM-LCDs for desktop and laptop PCs, liquid crystal televisions (LCTVs) and video projectors.. 1.3.2. Spatial light modulators. The most important among the non-display applications of the LCs is the spatial light modulator (SLM). An SLM is an electro-optic device capable of modulating the intensity, phase or polarization of light both in space and time [20]. Although SLMs are used mainly in optical communications and optical data processing, they are also usable for wavefront correction as an alternative to deformable mirrors. The advantages of the LC SLM technology are low control voltages (units of volts), low power consumption (∼0.1 mW/cm2 ), large dynamic range (tens of wavelengths), the small volume of a flat design, the absence of moving parts, wide interval of working temperatures (−20 . . . 1000 C) and the low cost of the materials. An important advantage is the possibility to make transparent devices, which allows the creation of much more compact systems than those based on deformable mirrors. In addition, LC SLM technology profits from the basis of a powerful LC display industry, the products of which are widely used in everyday life. Optically addressed SLMs Traditionally, there are two major types of LC SLMs: (a) optically addressed SLMs (OA-SLMs), in which a two-dimensional image controls an output intensity, phase or polarization profile; and (b) electrically addressed SLMs (EASLMs), where the modulation is controlled by electrical signals. A typical OA-SLM consists of a photoconductor and a thin LC layer situated between a pair of transparent electrodes. In the absence of recording light the photoconductor exhibits very high resistivity limiting the voltage across the LC layer. Sufficient intensity of light lowers the resistivity of the photoconductor resulting in adequate voltage being dropped across the LC. Thus,.

(13) 1.3 Liquid crystal technology local properties of the LC layer at a given point of the aperture depend on the intensity of the recording light. Due to the very high resolution of OA-SLMs, the intensity of the recording beam is directly converted to the output intensity, phase or polarization profile. It allows recording a hologram in the LC by forming an interference pattern of aberrated and reference beams on the surface of the photoconductor. Temporal variation of the aberrated beam results in change of the hologram recorded, which allows implementing real-time holography. Based on this principle, various optical schemes can be designed to compensate for aberrations of the input beam [21]. There are two basic configurations of adaptive optical systems based on real-time holography. In the first configuration, a time-reversed replica of the aberrated beam is generated, i.e., the beam with phase conjugated to the aberration. This technique is known as wavefront reversal; initially it was realized in nonlinear optical media and was extensively studied in the 70s and 80s [22]. In the second configuration, the traditional phase-conjugate scheme is implemented with purely optical feedback [23]. Research on OA-SLM-based real-time holography has been conducted since the 80s [24] and is still of importance. State-of-the-art OA-SLMs are characterized by high diffraction efficiency (up to 31%) and good temporal performance [25]. Using this method, one can efficiently compensate hundreds of waves of aberration caused by poor-quality primary mirrors in telescopes. Among other applications of OA-SLMs in adaptive optics we could mention their use for sensing and correction of small phase distortions. In [26, 27], the wavefront visualization was implemented by applying a phase OA-SLM as a nonlinear phase filter using the phase contrast technique. Wavefront correction using the phase conjugation scheme with an OA-SLM placed in the Fourier plane was demonstrated in [28]. Electrically addressed SLMs First electrically-addressed LC SLMs with pure phase modulation were used as low-order wavefront correction devices such as adaptive LC lenses [29, 30] and correctors of tilt [31]. Correction of one-dimensional wavefront distortions in an adaptive interferometer was demonstrated in 1985 [32] using a one-dimensional device with an array of transparent linear actuators. A two-dimensional phase modulator was produced in 1988 by Naumov [33] by attaching a cathode ray tube (CRT) to a phase LC SLM with optical addressing; it was applied to focusing of a laser beam into a ring, line and square and as a controllable phase diffraction grating. The same device was used in the adaptive interferometer with optical feedback [34]. Further research in this field was mostly related to serially produced liquid crystal television (LCTV) panels [35] and commercially available phase. 11.

(14) 12. Introduction SLMs with electrical addressing [10]. In brief, the state of the art in phase LC SLMs with electrical addressing is as follows. Meadowlark Optics (USA) [36] produces transparent LC modulators with direct addressing of pixels, whose optical performance is similar to that of the piston-type-segmented mirror. At the moment linear arrays with up to 256 pixels and 2D hexagonal arrays with up to 127 pixels are in small serial production. High-density reflective-type devices are produced using Liquid Crystal on Silicon (LCoS) technology [37, 38]. Holoeye (Germany) [38] makes phase modulators with 1920x1200 pixels on the basis of a Philips LCoS chip. Hamamatsu Photonics produces programmable phase modulators consisting of an EA-SLM with amplitude modulation coupled to an OA-SLM with phase modulation; their recent model X8267 allows addressing of 768x768 pixels with a 100% fill factor [39]. One of the known drawbacks of LC phase modulators is their slow speed. For nematic LC modulators, the switching speed is about 100 ms for 1 wave modulation. In its present state, this technology is not suitable for use in astronomy for correction of atmospheric turbulence, but there are a number of applications where fixed or slowly varying aberrations should be compensated, for example correction of static aberrations and heat effects in laser systems [40] and telescopes [41], adaptive focusing for machine and human vision, correction of the human eye aberrations [39], wide field-of-view imaging [42], and compression and shaping of femtosecond laser pulses [43]. Very fast (< 10µs) phase modulation can be achieved using a ferroelectric LC SLM with binary phase modulation [44]. In this case, however, a holographic technique should be applied, which results in a limited diffraction efficiency and requires the use of a spatial filter. For the future, faster LC materials seem to be more promising, such as dual-frequency nematics [45] and ferroelectric LC materials which allow analog phase modulation [46, 47]. High-resolution state-of-the-art phase LC SLMs are very powerful but expensive tools suitable for a wide range of applications. For applications like beam steering and focusing, requiring a low-cost solution, a number of LCbased active optical components have been developed. These are LC diffraction gratings, variable prisms, varifocal lenses, correctors of coma for optical pickups [48] and astigmatism for laser diodes [49]. LC SLMs also allow implementation of the adaptive wavefront sensor based on the Shack-Hartmann principle [50]. A static microlens array of a conventional Shack-Hartmann sensor is replaced by an array of Fresnel microlenses displayed by the SLM. Precision of the sensor can be improved by scanning of the wavefront, which can be achieved by shifting of the array displayed..

(15) 1.4 Motivation and objectives. 1.4. Motivation and objectives. Optical wavefronts are traditionally represented by continuous smooth functions such as astigmatism, defocus, coma, etc. Pixelated devices are badly suited for compensation of optical aberrations. For instance, to produce a high-quality lens with a variable focal distance, thousands of individually controlled pixels would be required. A modal approach to the design of LC phase modulators, which was suggested by Naumov [51, 52], makes it possible to form a smooth and continuous wavefront using a rather limited number of control channels. For instance, defocus in adaptive cylindrical and spherical lenses can be controlled using only one AC signal [51]. Modal correctors are known to provide much better wavefront fitting in comparison to the piston-type ones. On average, 4 times fewer actuators are required for a modal corrector than for a piston one to provide similar performance. Thus, the performance of general-purpose phase LC SLMs can be improved by making a modulator with modal influence functions [52]. Also, it was found that the modal LC corrector can be operated using several driving AC voltage parameters, namely, amplitude, frequency and phase delay. Therefore, the modal LC corrector can be operated with several degrees of freedom per actuator. This feature is unique among all types of modal correctors. However, new algorithms must be developed for this kind of control. This thesis targets the investigation and further development of this promising approach. Despite the fact that the first papers on this approach were published in the late 1980s, most of the theoretical and experimental results have been obtained since 1997 in a cooperation between the groups of the P. N. Lebedev Physical Institute (Samara, Russia), TU Delft (Delft, the Netherlands) and, later, the University of Durham (Durham, UK). Part of these results are presented in this thesis. Three of the previously suggested devices were studied, namely, • an adaptive cylindrical LC lens; • an adaptive spherical LC lens; • a liquid crystal modal wavefront corrector (LC-MWC) based on a distributed voltage divider; where our research focused on • the electro-optical behavior of the modal LC devices; • potentially useful control modes;. 13.

(16) 14. Introduction • algorithms for optimization and control; • alternative technologies for modal addressing of LCs. In addition, two new devices were reported, namely, • liquid crystal hybrid wavefront corrector (LC-HWC) based on silicon technology; • liquid crystal modal wavefront corrector (LC-MWC) with a thick dielectric substrate. The first of these devices, LC-HWC, was suggested as a modification of the voltage-divider-based LC-MWC, which can be implemented using silicon technology. Silicon technology allows the formation of the required configuration of actuators on the surface of a silicon chip using standard integrated circuit (IC) manufacturing processes. Another attractive feature of this technology is that on-chip implementation of the multiplexed control enables easy scaling of this technology to a very large number of actuators. The second new device is a modal LC corrector working on the principle of spreading of the electric field in a thick dielectric substrate. Compared to the voltage-divider-based LC-MWC, it can be realized on a much lower budget and has a higher efficiency. Another advantage of this method is the potential possibility to implement it in a transmissive configuration.. 1.5. Organization of the thesis. The thesis investigates and optimizes the performance of existing modal LC wavefront correctors, and describes new LC devices with modal control. It consists of eight chapters, including an introductory part and a summary. The introductory part gives a short overview of the field, introduces its basic concepts and explains the motivation behind the choice of the research topic. In the following six chapters the results of research are presented in detail. Chapter 2 describes a new method to evaluate the average optical performance of a wavefront corrector applied to compensate random phase aberrations. Chapter 3 introduces basic mechanisms of phase modulation in LC devices. A numerical method and software for simulation of the electro-optic S-effect in nematics are described. This model is used in the following chapters to analyze the electro-optics of modal LC devices. Chapter 4 presents the results of our investigation of adaptive spherical and cylindrical LC lenses. Based on both simulation and experimental results, the influence of harmonic and multi-harmonic control voltages on the performance.

(17) 1.5 Organization of the thesis of LC lenses is studied. Optimization algorithms are suggested for an experimental adaptive focusing system and for a system for practical calibration of LC lenses. Implementation of both systems is reported. Calibration results are verified by interferometric measurements. Chapter 5 is dedicated to the investigation of the multi-element liquid crystal modal wavefront corrector (LC-MWC) based on a distributed voltage divider, whose operation is similar to that of deformable mirrors. Possibilities for using various parameters for driving AC voltages as additional degrees of freedom are analyzed both theoretically and experimentally. The efficiency of a 37-channel LC-MWC is evaluated according to the method described in Chapter 2. The correction of low-order aberrations is demonstrated in an experiment. The chapter also reports on attempts to implement this type of LC-MWC using silicon technology. Chapter 6 introduces a new liquid crystal hybrid wavefront corrector (LCHWC), which combines features of both the zonal and modal approaches and can be implemented using bipolar silicon technology. The design of a first prototype and the technology used to manufacture it are described. The creation of modal-type influence functions and their superposition are demonstrated by interferometric measurements. Chapter 7 considers another implementation of the modal LC corrector, which is based on spreading of the electric field in a thick dielectric substrate with a high dielectric constant. Results of preliminary numerical simulation and the experimental investigation of a prototype are described. Interferometric testing of the prototype demonstrated that it formed smooth wavefronts and could control them with two degrees of freedom per actuator. Chapter 8 presents conclusions from this research and discusses possibilities for further development.. 15.

(18) 16. Introduction.

(19) Chapter 2. Performances of wavefront correctors in atmospheric optics 2.1. Introduction. The theory of zonal and modal wavefront correctors has essentially been developed since the 1970s. As at that time the main practical application of adaptive optics was high-resolution imaging through the turbulent atmosphere, the study focused on compensation of random wavefronts produced by the atmosphere. Kolmogorov’s model [53] is most frequently used for turbulence description because of its relative mathematical simplicity and good agreement with experiments. For zonal correctors, important evaluations were made for the case of a circular aperture covered by circular subapertures of equal diameter; the generalized case of compensation of local aberrations with an arbitrary order is considered by Kornienko [54] and Greenwood [55]. They show that the basic parameter characterizing an optical performance of zonal corrector is d/r0 , where d is the subaperture diameter, and r0 is a turbulence characteristic scale. In practice, the fill factor of a zonal corrector should be as close to unity as possible, and the most frequently used structures of actuators are hexagonal, rectangular, and ring segments of equal areas covering a circular aperture. The difference in performance for various actuator structures has not been studied well and therefore presents one of the subjects of this chapter. Modal wavefront correction is usually described in terms of complete or-.

(20) 18. Performances of wavefront correctors in atmospheric optics thogonal sets of functions, such as Zernike polynomials or Karhunen-Loève functions. The last one is a special set of functions, best suitable for representation of random wavefronts with given statistics, such as those described by Kolmogorov’s theory. For these sets, the rms wavefront aberrations were evaluated by Wang and Markey [56] for the general case of modal correctors with a Karhunen-Loève basis and by Paterson et al. [57] for the membrane-type mirror. However, we are not aware of any study comparing the performance of different types of modal and zonal correctors. A difference in performance can be expected for different mechanical properties of the mirror’s reflective plate and for different layouts of the actuator arrays. Also important is an investigation of how the practical corrector deviates from the ideal one in terms of correction performance. Finally, it is of interest to determine which of the existing mirror types gives the best correction. This chapter compares 15 different deformable mirrors represented by 5 different types of influence functions with 3 different 37-actuator structures. In the framework of Kolmogorov’s statistics of wavefronts, a statistically averaged residual error of correction and the Strehl factor were estimated numerically for each of these correctors as a function of Dp /r0 , where Dp is the diameter of the mirror’s pupil. The results obtained were compared with a theoretical limit for a given number of control channels, represented by a 37-channel KarhunenLòeve corrector, and for each corrector equivalent numbers of Karhunen-L` oeve modes resulting in the same fitting error were found. These results provide a good reference for the performance of the modal LC correctors considered in Chapters 5 and 6.. 2.2. Measures of correction quality. Let the function φ(x, y, t) be a time-dependent turbulence-induced phase distribution across the aperture of a wavefront corrector. We can decompose this function over an orthogonal set of Zernike polynomials Zi (x, y): φ(x, y, t) =. ∞ . αi (t)Zi (x, y).. (2.1). i=0. Here, the normalized Zernike polynomials with a unitary amplitude on a circle of unitary radius [58] are used:  m  Rn (r) cos (mθ) , m > 0 Zi (x, y) = Rnm (r) sin (mθ) , m < 0  m Rn (r), m = 0. (2.2).

(21) 2.2 Measures of correction quality. 19. where . (n−m)/2. Rnm (r). =. s=0. (−1)s (n − s)! rn−2s , s![(n + m)/2 − s]![(n − m)/2 − s]!. (2.3). and r and θ are polar coordinates; n is a non-negative integer and m varies from −n to n with a step of 2. The coefficients αi (t) in Equation (2.1) are random functions of time with zero average. Considering random wavefronts with Kolmogorov’s statistics, Noll [59] derived an analytical formula for cross-correlations of these coefficients (modified for the normalization chosen):   A, √ m = 0 and m = 0, (2.4) ai ai = 2A, m = 0 and m = 0, or m = 0 and m = 0,  2A, m = 0 and m = 0, 5/3 Dp A = 0.00386 (n + 1)(n + 1)δmm (−1)(n+n −m−m )/2 r0 Γ(14/3)Γ[(n + n − 5/3)/2] . × Γ[(n − n + 17/3)/2]Γ[(n − n + 17/3)/2]Γ[(n + n + 23/3)/2] Let us denote the influence functions of our corrector as Fk (x, y), k = 1 . . . N , where N is the number of control channels and make a least-squares fit of Zernike polynomials by the influence functions Fk (x, y): i (x, y) = Z. N . Cki Fk (x, y).. (2.5). k=1. If the function can be decomposed over a linear set, then the least-squares fit of this function is equal to the linear combination of least-squares fits of all its decomposition terms with the same coefficients; it can be shown from the linearity of the system of equations resulting from the least-squares method [60]. Thus, ∞ ∞ N y, t) = αi (t)Zi (x, y) = αi (t)Cki Fk (x, y) (2.6) φ(x, i=0. i=0 k=1. is the least-squares approximation of φ(x, y, t) provided by the wavefront corrector. Here we do not consider dynamic properties of the adaptive system and suppose that the correction is instant. The difference between the real and the compensated wavefronts is . ∞ N αi (t) Zi (x, y) − Cki Fk (x, y) φ(x, y, t) − φ(x, y, t) = i=0. k=1.

(22) 20. Performances of wavefront correctors in atmospheric optics ∞ . =. αi (t)∆i (x, y),. (2.7). i=0. where ∆i (x, y) is a error from fitting of the i-th Zernike polynomial. The rms error of correction can be found by taking a square of expression (2.7) and averaging it over spatial and temporal variables σ2 =. ∞ ∞ i=0 i =0. ∆i ∆i r · αi αi t ,. (2.8). ∆i (x, y)∆i (x, y)dxdy. , dxdy. (2.9). where ∆i ∆i r =. Ω. Ω. αi αi t is calculated by formula (2.4), and Ω is the area of the corrector’s pupil. It follows from Equations (2.4) and (2.8) that 2. σ =B. Dp r0. 5/3 ,. (2.10). where the constant B is device dependent and may be used to characterize the corrector’s performance. The residual rms correction error was evaluated by formula (2.8) through subsequent fitting of the Zernike polynomials by the influence functions of the wavefront corrector studied. The model was restricted to include only 230 Zernike polynomials (the maximum value of n is 20). The constant B for an uncorrected wavefront evaluated in this way was equal to 1.0271 rad, which is in good agreement with the results derived elsewhere [56]. Other common measures of the corrector’s performance are the approximate value of

(23). the Strehl factor defined as S = exp −σ 2 , where σ is given in radians, and the efficiency NKL - an equivalent number of Karhunen-Loève modes resulting in the same fitting error. Note, however that in adaptive optics systems the overall fitting error also depends on the type of the wavefront sensor employed and the geometry of its subapertures.. 2.3. Models of wavefront correctors. Modal correctors are presented in this investigation by 12 types of adaptive mirrors; these are combinations of 4 different types of active surface - (1) membrane, (2) flexible faceplate with free edge, (3) flexible faceplate with supported.

(24) 2.3 Models of wavefront correctors. 21. Figure 2.1: Structures of actuators in piston correctors, from left to right: hexagonal, squares and ring segments. Example of astigmatism compensation is shown. edge and (4) flexible faceplate with clamped edge. All the modal correctors were analyzed with three different structures of actuators. For each modal corrector, the appropriate numerical or analytical model of the mirror influence function was used. These models are described below.. 2.3.1. Piston correctors. Piston correctors of circular aperture with the three most frequently used configurations of actuators were considered: hexagonal, rectangular, and a structure of ring segments with equal areas and approximately equal linear sizes. In total 37 actuators were studied. Figure 2.1 demonstrates an example of astigmatism correction by each of these piston corrector configurations.. 2.3.2. Membrane mirrors. The influence functions of a membrane mirror were found by solving the Poisson equation numerically. This equation describes the surface deflection U (x, y) of a stretched membrane under an external load P (x, y) [61]: ∆U (x, y) = −P (x, y)/T,. (2.11). where T is the membrane tension. The successive over-relaxation algorithm described in [60] was used to solve this equation. Thin-plate bimorph mirrors are described by a similar equation [62], and in the case of a fixed edge they produce the same responses as membrane mirrors. However, from the design side, bimorph mirrors offer more freedom in choosing the boundary conditions..

(25) 22. Performances of wavefront correctors in atmospheric optics. 2.3.3. Continuous faceplate mirrors. The behavior of a mirror with a thin continuous faceplate is described by the biharmonic equation [61]: ∆2 U (x, y) = P (x, y)/R,. (2.12). where R is the cylindrical rigidity of a faceplate. Analytical solutions were obtained by A. I. Lourye for a circular plate with three types of boundary conditions: a free, clamped, and supported edge [63]. Clamped-edge faceplate Deformation of a plate with a clamped edge under action of the force P applied at a point with polar coordinates (ρ, ψ) is described by the formula. P U (r, ϕ) = (1 − r2 )(1 − ρ2 ) + [r2 + ρ2 − 2rρ cos(ϕ − ψ)] 16πR r2 + ρ2 − 2rρ cos(ϕ − ψ) × ln , (2.13) 1 + r2 ρ2 − 2rρ cos(ϕ − ψ) where r and ϕ are polar coordinates in the plane of mirror. Supported-edge faceplate For a supported edge,. P U (r, ϕ) = [r2 + ρ2 − 2rρ cos(ϕ − ψ)] 16πR. (2.14). r2 + ρ2 − 2rρ cos(ϕ − ψ) + (1 − r2 )(1 − ρ2 ) 1 + r2 ρ2 − 2rρ cos(ϕ − ψ)  1 . − 1−µ 2 [1 − srρ cos(ϕ − ψ)]s 1 − µ  2 , ds − 1+µ  1 + s2 r2 ρ2 − 2srρ cos(ϕ − ψ). × ln ×. 0. where µ is the Poisson coefficient. Expressions (2.13) and (2.14) were used to calculate the influence functions of continuous faceplate mirrors with clamped and supported edge. Free-edge faceplate Let us consider a circular plate with a free edge. As shown in [63], its deformation is caused by a set of point-like forces Pi , which act in points ζi and can be presented as a superposition of functions, accurate to within the term W0 + W1 r cos ϕ + W2 r sin ϕ. This term governs the translation and rotation of the plate as a whole: 1 Pi W (z, z¯, ζi , ζ¯i ) + W0 + W1 re z + W2 im z, 16πR i=1 N. U (z, z¯) =. (2.15).

(26) 2.3 Models of wavefront correctors where. 23. ¯ = (z − ζ)(¯ ¯ ln(z − ζ) + ln(¯ ¯ + 1−µ W (z, z¯, ζ, ζ) z − ζ) z − ζ) 3+µ 2 ¯ + ln(1 − z¯ζ)] + (1 − µ) × [ln(1 − z ζ) z z¯ζ ζ¯ (1 + µ)(3 + µ) 8(1 + µ) ¯ ln(1 − z ζ) ¯ + k(z ζ) ¯ [(1 − z ζ) + (1 − µ)(3 + µ) + (1 − z¯ζ) ln(1 − z¯ζ) + k(¯ z ζ)], (2.16). z = r cos ϕ + ir sin ϕ, ζ = ρ cos ψ + iρ sin ψ are coordinates expressed in the complex-valued form, and k is the logarithmic integral. x k(x) =. ln(1 − α) dα. α. (2.17). 0. To calculate the best approximation of an arbitrary wavefront φ(x, y) by a continuous faceplate mirror with a free edge, it is necessary to find N unknown forces Pi and 3 coefficients W0 , W1 and W2 . They can all be found from a system of N + 3 linear equations. The first N equations are obtained by minimization of the rms error using the least-squares method; the 3 supplementary equations from statics represent conditions of mechanical equilibrium of the plate: . N     P W (z, z¯, zi , z¯i )W (z, z¯, zj , z¯j )dxdy j    Ω  j=1 .     W (z, z ¯ , z , z ¯ )dxdy + W W (z, z¯, zi , z¯i )xdxdy +W  0 i i 1   Ω . Ω     W (z, z¯, zi , z¯i )ydxdy +W2   . Ω     = W (z, z¯, zi , z¯i )φ(x, y)dxdy, i = 1 . . . N, Ω (2.18) N     Pj = 0,     j=1    N      Pj xj = 0,     j=1    N     Pj yj = 0.   j=1. This system is easily solved using standard Gauss elimination..

(27) 24. Performances of wavefront correctors in atmospheric optics. 2.4. Results. The residual correction errors for all types of correctors in a percentage to the total rms wavefront variation are summarized in Table 2.1. For piston-type correctors, this error is in the range ∼ 22 . . . 24%. The best approximation is obtained by actuators shaped like ring segments. The hexagonal structure produces a very close result, whereas for square pixels, the error is about 1.5% higher. Modal-type correctors perform much better: their rms error is in the range 9 . . . 10%. The difference in performance between all types of modal correctors with all types of actuator structures considered is less than 0.7% of the total wavefront variation. The difference caused by the structure of actuators is about 0.1 . . . 0.2%. Thus, for the modal correctors considered, the geometry of the actuator structure does not play a significant role as long as the actuators homogeneously cover the effective area of the mirror. The diameter Dm of a membrane or a mirror faceplate with a supported or fixed edge must be larger than the diameter of the mirror pupil Dp , to facilitate correction by allowing non-zero phase values at the edge of the pupil. The performance for this type of corrector depends on the ratio Dm /Dp . Another design parameter that influences the correction quality is the diameter of the structure of actuators Da . We defined Da as double the distance from the center to the boundary of an actuator structure. The rms correction error as a function of Dm /Dp and Da /Dp is shown in Figure 2.2. For all cases with a fixed edge, the optimum mirror diameter is 1.6 . . . 1.8Dp (see Table 2.1), and the average diameter of the actuator structure must be equal to the mirror pupil Dp or up to 20% larger. For a continuous faceplate mirror with a free edge, the optimum correction is reached at Da = Dp , and the quality of correction depends weakly on the pupil diameter Dp . To set a good reference point for the figures obtained, it is important to compare the results for practically used modal correctors with those of an optimum modal system with the same number of corrector modes, to determine how close our results are to the theoretical limit imposed by using 37 control channels. It is known that Karhunen-Loève (K-L) functions form the best statistical approximation of a random function at a given number of approximating modes [56]. K-L functions Gi (r) are obtained as solutions of the integral equation. Gi (r ) φ(r )φ(r) d2 r = Λ2i Gi (r),. (2.19). Ω. where the kernel of integral operator φ(r )φ(r) is a covariance function of atmospheric phase distortions. As described in [56], the solution of Equation.

(28) 2.4 Results. 25. (a) Hexagonal structure Type of corector Piston Membrane Continuous faceplate, clamped edge Continuous faceplate, supported edge Continuous faceplate, free edge. Optimum Dm /Dp 1.0 1.8. Optimum Da /Dp 1.0 1.2. Error, % 22.70 9.682. B, rad2 0.05282 0.00961. Maximum Dp /r0 4.686 13.027. NKL. 1.8. 1.2. 9.333. 0.00892. 13.624. 31. 1.8. 1.2. 9.306. 0.00888. 13.657. 32. 1.8. 1.0. 9.022. 0.00834. 14.188. 33. Optimum Dm /Dp 1.0 1.8. Optimum Da /Dp 1.0 1.2. Error, % 22.55 9.468. B, rad2 0.05214 0.00918. Maximum Dp /r0 4.723 13.389. NKL. 1.8. 1.2. 9.267. 0.00880. 13.738. 32. 1.8. 1.2. 9.238. 0.00874. 13.786. 32. 1.4. 1.0. 9.107. 0.00850. 14.018. 33. Optimum Dm /Dp 1.0 2.0. Optimum Da /Dp 1.0 1.0. Error, % 24.26 9.696. B, rad2 0.06031 0.00963. Maximum Dp /r0 4.328 13.013. NKL. 2.0. 1.2. 9.351. 0.00895. 13.593. 31. 2.0. 1.2. 9.321. 0.00890. 13.641. 31. 1.8. 1.0. 9.119. 0.00852. 14.001. 33. 5 29. (b) Ring segments Type of corector Piston Membrane Continuous faceplate, clamped edge Continuous faceplate, supported edge Continuous faceplate, free edge. 6 31. (c) Squares Type of corector Piston Membrane Continuous faceplate, clamped edge Continuous faceplate, supported edge Continuous faceplate, free edge. 4 29. Table 2.1: Optimum values of parameters Dm /Dp and Da /Dp , residual correction errors (in percentage to total wavefront variation), values of parameter B, maximum values of the turbulence parameter Dp /r0 , and equivalent numbers of Karhunen-Loève modes for different types of wavefront corrector and different structures of 37 actuators..

(29) 26. Performances of wavefront correctors in atmospheric optics. Figure 2.2: Dependencies of rms correction error of (a) membrane mirror, (b) flexible faceplate mirror with free edge, (c) flexible faceplate mirror with supported edge and (d) flexible faceplate mirror with clamped edge from parameters Dm /Dp and Da /Dp . Here, Dm is the diameter of the whole mirror, Dp that of the mirror pupil, and Da that of the structure of actuators. Rms correction error values are given in waves..

(30) 2.4 Results. Figure 2.3: Dependency of Strehl factor on turbulence parameter Dp /r0 without correction and for its different types. (2.19) can be reduced to the problem of finding eigenfunctions of a symmetrical matrix. This problem was solved using the Jacobi transformation method [60]. It was found that the basis of 37 K-L functions results in a residual error equal to 8.5% of the total wavefront variation. Thus, the quality of correction that can be achieved with a free-edge plate with hexagonal structure of actuators only deviates from the optimal by 0.5%. For all configurations of modal correctors, the difference was less than 2.2% of the total wavefront variation. The efficiency of wavefront correction can be evaluated as an equivalent number of K-L modes NKL resulting in the same fitting error. The estimation of the efficiency of 37-channel piston corrector gives 4 . . . 6 K-L modes, whereas for modal correctors it is in the range 29 . . . 33 modes (Table 2.1). The dependency of the Strehl factor on the parameter Dp /r0 for all types of correctors considered, including absence of correction and K-L correction, are shown in Figure 2.3. From these dependencies we found the maximum values of the turbulence parameter that can be corrected by each corrector - see Table 2.1. Here, the value of a Strehl factor S ≥ 0.5 is considered as acceptable. Without correction, wavefront degradation appears at approximately Dp /r0 = 0.79; correction in the basis of 37 K-L functions allows an increase of this parameter up to 15.21; the piston corrector gives Dp /r0 = 4.3 . . . 4.7, and modal correctors give values in the range 13.0 . . . 14.2.. 27.

(31) 28. Performances of wavefront correctors in atmospheric optics. 2.5. Conclusion. The method suggested in this chapter compares different types of wavefront correctors in terms of their static correction performance. It is especially useful for evaluating the new LC-based modal correctors studied in this thesis. In addition it can be used to optimize the wavefront corrector design as is done for the liquid crystal hybrid wavefront corrector in Chapter 6 of this thesis. No significant differences were found in correction performance between different wavefront correctors of each class - zonal and modal correctors - resulting from the difference in electrode structure or the type of active plate. The nonoptimal set of response functions in modal correctors did not result in any essential loss of correction quality either, even when compared to the ideal K-L case. The continuous faceplate mirror with a free edge and hexagonal actuator structure gave the best approximation to the theoretical limit; but all other types of modal correctors, used in the range of their optimal parameters, also produced results that are close to the ideal case..

(32) Chapter 3. Electro-optical characteristics of nematic liquid crystals 3.1. Introduction. As phase modulation in liquid crystals is based on the electro-optic effect, we will start our investigation of LC devices by considering this effect and the relevant characteristics of LC materials. This chapter is dedicated to the analysis of the electro-optical characteristics of nematic liquid crystals intended for application in phase modulation devices. The operation of a phase modulator can be described by its phase transparency function, that is, by the spatial distribution of phase delay introduced by the modulator. The phase delay is determined by the electric field applied to the LC layer. The dependence between the electric field and the resulting phase delay is the most important characteristic of the LC and is determined by mechanical, electrical and optical properties of the LC material and by the initial alignment of the LC molecules. Most of the commercially available LC SLMs are pixelated devices, where the applied voltage is controlled individually for each pixel. For the modal LC devices presented in this thesis, the voltage distribution is formed under the influence of their capacitative and resistive characteristics. That is why these characteristics are also part of our investigation..

(33) 30. Electro-optical characteristics of nematic liquid crystals. Figure 3.1: Demonstration of electro-optic effects in a sandwich-type LC cell: (a) S-effect; (b) B-effect; (c) twist effect.. 3.2. Principles of phase modulation in LC devices. Liquid crystals are used in optics because of the anisotropy of their optical constant and the effect of reorientation of LC molecules under an external electric field, which is known as the electro-optic effect. The anisotropy of the electric constant of the material is the origin of the reorientation, whereas the dynamics of the process also depend on the viscoelastic properties of the medium and the initial orientation of the director. Currently, nematic and ferroelectric LCs are applied in phase modulators. These LC phases and the principles of phase modulation are briefly described below.. 3.2.1. Nematic LC devices. In a nematic LC phase, rod-like molecules are statistically oriented along a certain preferred axis, called the director. The director orientation may change in space, but the characteristic distance of its variation is much longer than the dimensions of a molecule. Such an LC can be treated locally as an uniaxial crystal with non-polar symmetry, whose optical axis is parallel to the director of the molecules (domains). Using special alignment techniques, it is possible to obtain a quite uniform orientation of the molecular axes in a thin layer of the material, and thus a liquid monocrystal. The operation of nematic-based LC phase modulators can be explained using the example of a sandwich-type LC cell (see Figure 3.1). A flat capillary with a thickness of 1 . . . 200µm is formed by two glass substrates with trans-.

(34) 3.2 Principles of phase modulation in LC devices parent electrodes deposited on their surfaces bounding the LC. Substrates are separated by dielectric spacers, which set the required thickness between the substrates. Special alignment techniques provide a certain orientation of LC molecules at the boundary walls of the cell. Three types of alignment are the most widely used - planar (where the molecules are parallel to the substrates), homeotropic (perpendicular to substrates) and tilted. Electrically controlled birefringence An electric field applied between the electrodes changes the director orientation. For LCs having positive dielectric anisotropy (ε > ε⊥ ) and initially a planar alignment, the electric field forces the molecules to align parallel to the field direction, causing splay deformation (the S-effect, see Figure 3.1(a)). LCs with negative dielectric anisotropy and initially a homeotropic alignment undergo bend deformation (the B-effect, Figure 3.1(b)), where LC molecules tend to rotate perpendicularly to the field. The operation of the LC cell in these two modes can be characterized as electrically controlled birefringence (ECB). The effective birefringence value of ∆n = n − n⊥ , where n and n⊥ are refractive indices along and normal to the LC director, changes due to re-orientation of the LC molecules. If linearly polarized light, whose polarization direction is parallel to the plane of the LC director, passes through the cell, its phase changes with the birefringence. The optical anisotropy of commercial nematic LC mixtures can be as high as 0.45 [64], which makes it possible to obtain continuous phase modulation in the range of tens of wavelengths for visible light with mostly negligible amplitude modulation. As due to its wide application in LCDs the technology of the planar alignment is more advanced than that of the homeotropic alignment, the majority of ECB-based phase modulators use the S-effect. Besides, the B-effect is less suitable for phase modulation because of high scattering. Twist effect Another effect which can be applied for phase modulation is the twist effect. In a twisted LC cell (Figure 3.1(c)), the molecules on the opposite electrodes are aligned perpendicularly to each other, and the material has a positive dielectric anisotropy. When no electric field is applied, the molecules are parallel to the substrates, and the director orientation smoothly changes with depth, forming a helical structure with a 90◦ twist. The twisted cell acts on linearly polarized light as a controllable rotator of the polarization plane. When the polarization of light is aligned with the director angle and there is no electric field, the angle of polarization changes along the twisted molecules by a waveguide effect with a total rotation of 90◦ .. 31.

(35) 32. Electro-optical characteristics of nematic liquid crystals When the voltage is applied, the molecules tend to align perpendicularly to the substrates. For a certain voltage which is called the optical threshold of the twist, the waveguide regime no longer applies, and no rotation of the polarization plane is induced. The primary use of the twist effect is the amplitude modulation which occurs when the cell is placed between two parallel polarizers. This effect is widely used in TN-LCDs. However, phase modulation also occurs in the twisted cell due to re-orientation of the molecules and change of the birefringence. Whereas it is more straightforward to use the S- or B-effect for pure phase modulation, twisted-nematic (TN) panels are much easier to obtain, because they are serially produced for liquid crystal television (LCTV) devices such as video projectors. Overview of techniques A number of papers was published in the 1990s investigating the phase modulation characteristics of commercially available TN-LCTV panels [65, 66]. Yamauchi and Eiju [66] showed that their applicability may be limited by the fact that thin panels with small phase retardation which are found in the most highresolution commercial video projectors not only rotate the polarization angle but also change the polarization state. To compensate for this effect, Dou and Giles [35] used a double-pass setup, which also allows for doubling of the phase modulation capability. Another compensation method suggested by Kelly and Munch [67] is to use two TN panels, the first one for phase modulation and the second one to remove coupled amplitude modulation. Several adaptive optical systems using different optical setups and different control algorithms were reported, and successful correction was achieved [68, 35, 69, 70]. At the same time, planar-nematic LC phase modulators based on the Seffect appeared on the market, stimulating research in this field [10, 71, 72]. Due to the phase-only modulation under this mode, control of these devices is straightforward, and they can be applied in a wider range of situations than twisted nematic devices. Their only serious drawback is a quite limited temporal bandwidth. However, methods to increase their speed, such as the transient nematic effect [73] and dual-frequency control [74], do exist and are extensively studied. Dual-frequency control is applied to nematic LCs with low-frequency dispersion of ε . At a certain frequency which is called the crossover frequency, a change in sign occurs of the dielectric anisotropy of the LC, ∆ε [75]. For certain nematic mixtures, the crossover frequency can be made as low as 7 kHz (LC1001 mixture produced by the NIOPIK Institute, Russia). The sign reversal of ∆ε can be used to orient the liquid crystal either with its optical axis parallel (∆ε > 0) or normal (∆ε < 0) to the direction of electric field by changing the.

(36) 3.2 Principles of phase modulation in LC devices frequency of the applied voltage. By increasing the control voltage, the driving speed of the dual-frequency LC can be increased in both directions. For example, a π-phase modulation cycle was realized in 25 µs using this method [76]. Vibrations in an interferometer with frequencies up to 1.2 kHz were suppressed using a dual-frequency LC cell [77]. A 40 Hz closed-loop correction bandwidth was achieved in an adaptive optical system by Dayton et al. using a dual-frequency modulator [78], whereas a 5 Hz bandwidth was observed in a similar system using an ordinary nematic modulator [71]. In general, the high-speed operation of dual-frequency nematics requires higher voltages and much more complicated addressing schemes compared to control of ordinary nematics [79]; these factors prevent the dualfrequency method from being widely applied. A known drawback of nematic LC phase modulators is their dependency on polarization, as operation with unpolarized light is required for most AO applications. Two solutions have been suggested for this problem. The first approach is to combine two devices operating on two different polarization components of light, which are orthogonal to each other. The second one, suggested by Love [80], is to integrate a quarter-wave plate in a reflective-type device, which provides identical phase delays by dual pass for both polarization components. This technique, however, can be applied only to monochromatic light. Customized (with a planar LC alignment instead of TN) LCTV with TFT active matrix addressing allows a device with a large number of pixels; a 640x480 phase-only modulator was reported by Hu et al. [81]. However, the optical quality of substrates used for commercial LCTVs is insufficient, so that only a part of the whole LCTV area can be used efficiently (32x32 pixels in [81]). It is important to notice that nematic LC cells are addressed by alternating current (AC), not by direct current (DC). The reason for this is that the DC addressing causes a build up of charge in the LC material and permanently damages the LC material due to electrolytical effects [82]. Note that the LC director can follow only relatively slow voltage oscillations whose period is approximately equal to or greater than the LC switching time. For voltage oscillations whose period is much smaller (frequency >1 kHz), the orientation of the LC director depends only on the rms voltage; thus, electrical and optical parameters of the LC also determined by rms voltage. In fact, DC-free addressing schemes are implemented in all TN-LCDs.. 3.2.2. Ferroelectric LC devices. Ferroelectric liquid crystal (FLC) materials are obtained by addition of a chiral dopant to smectic C*. Smectic phase is characterized by both orientational and positional order: besides their orientation along the director axis, smectic. 33.

(37) 34. Electro-optical characteristics of nematic liquid crystals molecules are arranged into layers with an average thickness which is comparable to the molecular length. The chiral dopant causes helical twisting of the molecular structure of the mixture and also induces spontaneous electrical polarization in it. Helix twist can be compensated by mixing two ferroelectric LCs with opposite signs of chirality, while keeping the nonzero value of the spontaneous polarization. This produces so-called surface-stabilized ferroelectric liquid crystal (SSFLC) mixtures. The interaction of the molecules’ spontaneous polarization with the external electric field allows very fast switching between two bistable states in a SSFLC by changing the sign of the electric field [83]. By arranging and correctly orienting crossed polarizers around the device, an FLC cell may be configured as a binary phase cell. The possibility of polarization-insensitive FLC operation was shown by Warr and Mears [84]. Continuous phase-only modulation in a SSFLC cell, although only with a maximum phase shift of π, was demonstrated by Abdulhalim [46]. The unusual regime of control is due to a very small feeding voltage and, therefore, these modulators have lower rapidity (the response time is 283 µs under an electric field of 0.5 V/µm). A promising technique for continuous phase modulation is based on the deformed helical ferroelectric (DHF) effect [85]. An electric field applied parallel to the layers of smectic C* (i.e., perpendicular to the axis of the helical structure) causes continuous deformation of the helix. Under the critical electric field strength, the helical structure unwinds and FLC molecules become oriented with their spontaneous polarization vectors parallel to the electric field. To avoid light scattering from defects, the pitch of the helix needs to be several times smaller than the wavelength of visible light. The switching time of DHF-SLMs is about 100 times shorter than that of nematic SLMs [47]. Currently, FLC spatial phase modulators are widely used to generate phase holograms in beam steering applications and free-space optical switches [86]. The holographic technique is also suitable for generating the arbitrary continuous phase distribution in non-zero diffraction orders. For instance, Kelly et al. [87] simulated dual-layer atmospheric turbulence in laboratory conditions using two binary FLC SLMs. Real-time holography can be realized in FLC SLMs with optical addressing [88]; this method is applied to correct strong aberrations of the telescope’s primary mirror. A 31% diffraction efficiency has been attained for the DHF-FLC with a 600-µs holographic write time at a spatial frequency of 18 line pairs/mm (lp/mm); this device was also capable of producing an 8% diffraction efficiency at a spatial frequency of 370 lp/mm [25]. Implementation of a fast LC lens based on the DHF effect in pure phase modulation mode was reported by Eschler et al. [89]. Although ferroelectric LCs seem to have a promising future because of extremely fast switching speed, they presently suffer from several drawbacks and difficulties in the fabrication process. It is extremely difficult to obtain perfect.

(38) 3.3 The Ericksen-Leslie theory FLC alignment over the large surface area of the cell [83]. The layered structure of smectic LCs is rather sensitive to mechanical and thermal stress and often needs a special protective housing.. 3.2.3. Conclusion. At the moment, the most mature technology for LC phase modulation is based on the electro-optic S-effect in nematics. This is the main reason why we chose to use this technology in the LC devices studied in this thesis; however, the results are also applicable to modulators based on the B-effect. Dual-frequency nematics provide a higher operating speed; however, they require much more complicated control, which is difficult for theoretical analysis. In the future, efforts can be made to combine the approaches studied in Chapters 4-7 with dual-frequency control. The electro-optic properties of the nematic LC under the S-effect will be considered further in this chapter.. 3.3. The Ericksen-Leslie theory. Several effects are produced in the LC by the electric field. These are the Freederiksz transition (reorientation of the LC director), backflow (the macroscopic flow accompanying the director reorientation), ionic impurities conduction and Debye relaxation [83]. As a result, the electro-optical characteristics of the LCs are very difficult to determine in theoretical analysis. However, to understand the behavior of the LC under an electric field at least qualitatively, it is useful to analyze the dominant effect, the Freederiksz transition. As a rule, the distribution of the LC director is analyzed based on minimization of the functional of the nematic free energy. The equation describing the dynamics of the director, which was proposed by Ericksen and Leslie [90, 91], 2 ∂Θ ∂Θ ∂ (3.1) (K11 cos2 Θ + K33 sin2 Θ) − (K33 − K11 ) sin Θ cos Θ ∂z ∂z ∂z +(α2 sin2 Θ − α3 cos2 Θ). ∂Θ ∂2Θ ∂υ + ∆εε0 E 2 sin Θ cos Θ = γ1 +I 2 , ∂z ∂t ∂t. is rather complicated and cannot be solved analytically [92]. Here, Θ is the deformation angle of the director, K11 and K33 are splay and bend Frank elastic constants, respectively, ∆ε = ε|| − ε⊥ is the dielectric anisotropy, υ is the flow velocity, α2 and α3 are viscosity coefficients in Leslie’s notation, γ1 = α3 − α2 is the rotational viscosity, I represents the inertia, and E is the electric field in the LC medium. Due to the LC director deformation, E is non-uniformly. 35.

(39) 36. Electro-optical characteristics of nematic liquid crystals distributed across the LC layer. The electric field distribution can be expressed in terms of the deformation angle Θ and the voltage V applied to the LC: V. E=. d (ε⊥ cos2 Θ + ε|| sin2 Θ) 0. ,. (3.2). dz ε⊥ cos2 Θ(z) + ε|| sin2 Θ(z). where d is the thickness of the LC layer. The most common approximation of neglecting the angular momentum is justified by the fact that the change of orientation is strongly damped by the rotational viscosity. Another approximation that greatly simplifies the problem is neglecting the backflow. As is shown in [93], this effect can partly be taken into account by using effective viscosity instead of γ1 : γS∗ = γ1 − 2α32 /(α3 + α4 + α6 ) ∗ γB. = γ1 −. 2α22 /(α4. + α5 − α2 ). for splay deformation (S-effect),. (3.3). for bend deformation (B-effect).. In the series of works [92, 94, 95], useful results for the S- and B-effects were obtained in the small-signal approximation corresponding to the case of small (< 50◦ ) deformation angles. For large voltages corresponding to maximum deformations, the most straightforward way is to solve Equation (3.1) numerically (except for the terms related to the inertia and the backflow), in combination with experimental investigation of their electro-optical static and dynamic characteristics. The next section will describe the numerical model and the software to calculate the phase delay in the LC layer, its specific capacitance and conductance. We numerically simulated the E49 nematic LC (produced by Merck, Germany) with an initial planar alignment and small pretilt of molecules on the surface.. 3.4 3.4.1. Investigation of the S-effect in statics Method of calculation. Even without the backflow effect, the Ericksen-Leslie equation is essentially nonlinear, especially if we take the electric field as non-homogeneous. However, the static distribution of the LC director can be calculated using an iterative technique based on the dynamic Equation (3.1). In the method presented, the temporal variable t is used only as an auxiliary variable. The Ericksen-Leslie equation is replaced by an equivalent presentation introducing finite differences instead of derivatives ∆t (3.4) (K11 cos2 Θij + K33 sin2 Θij ) Θi,j+1 = Θij + γ1.

(40) 3.4 Investigation of the S-effect in statics × ×. 37. Θi−1,j − 2Θij + Θi+1,j + (K33 − K11 ) sin Θij cos Θij (∆z)2. 2 Θi+1,j − Θi−1,j 2 + ε0 ∆εEij sin Θij cos Θij . 2∆z. Here, i and j are indices of coordinate and time, respectively, ∆z is the coordinate step, and ∆t is the time step. γ1 has an arbitrary value; it can be set to 1, and the value of ∆t is limited by the condition of stability of the finite-differential scheme γ1 (∆x)2 . (3.5) ∆t = 8K11 This condition is found by the method described in [96]. Boundary conditions are set according to initial alignment with pretilt angle Θ0. Θ0,j = Θ0 ΘN,j = Θ0. (3.6). where N +1 is the number of points by coordinate. Iterations start from the uniform distribution of the deformation angle Θ and uniform distribution of the electric field E = V /d and continue by calculation of the next j+1-th level until the required accuracy is reached. The electric field on each step by j is corrected by the formula Ei,j+1 =. V. d (ε⊥ cos2 Θij + ε|| sin2 Θij ) 0. .. (3.7). dz ε⊥ cos2 Θij + ε|| sin2 Θij. From the distributions of the LC deformation angle found we can calculate the electro-optical characteristics of the LC layer: the dependency of the phase delay ∆Φ, the specific capacitance c and specific conductance g on voltage. This is calculated using the formulas derived in [97]:. ∆Φ. =. 2π λ. d 0. .  n|| n⊥ n2⊥ cos2 θ + n2|| sin2 θ. α , α + β2 β , = ε0 ω 2 α + β2. c = ε0 g. . 2. − n⊥  dz,. (3.8) (3.9) (3.10).