Abstract

We present a prototype of an adaptive intraocular lens based on a modal liquid-crystal spatial phase modulator with wireless control. The modal corrector consists of a nematic liquid-crystal layer sandwiched between two glass substrates with transparent low- and high-ohmic electrodes, respectively. Adaptive correction of ocular aberrations is achieved by changing the amplitude and the frequency of the applied control voltage. The convex-shaped glass substrates provide the required initial focusing power of the lens. A loop antenna mounded on the rim of the lens delivers an amplitude-modulated radio-frequency control signal to the integrated rectifier circuit that drives the liquid-crystal modal corrector. In vitro measurements of a 5-mm clear aperture prototype with an initial focusing power of +12.5 diopter, remotely driven by a radio-frequency control unit at ~6 MHz, were carried out using a Shack-Hartmann wave-front sensor. The lens based on a 40-μm thick liquid-crystal layer allows for an adjustable defocus of 4 waves, i. e. an accommodation of ~2.51 dioptres at a wavelength of 534 nm, and correction of spherical aberration coefficient ranging from -0.8 to 0.67 waves. Frequency-switching technique was employed to increase the response speed and eliminate transient overshoots in aberration coefficients. The full-scale settling time of the adaptive modal corrector was measured to be ~4 s.

© 2007 Optical Society of America

1. Introduction

Replacing the crystalline lens of the human eye with an artificial intraocular lens (IOL) in cataract surgery or for other medical reasons is a well-established and widely used approach in modern ophthalmic practice. The invasive treatment restores clear vision, but results in a significant loss of accommodation when a fixed monofocal IOL is implanted [1, 2]. Several configurations of accommodating IOLs driven by a natural process of contraction and relaxation of the ciliary body [3–6] and pseudoaccommodating IOLs which provide multiple dioptric powers [7–9] have been proposed in the past years. Other promising models of accommodative IOLs, for example, recently reported in [10, 11], are in development. Ideally, an IOL should combine high imaging quality with sufficient accommodation for near work, e.g. reading, and compensate for corneal aberrations of the eye. Correction of aberrations caused by the anterior and posterior surfaces of the cornea may improve significantly the overall performance of the eye since the acuity of vision is limited mainly by the ocular optics [12–14].

Previously [15], we investigated the feasibility of intraocular adaptive optics by using modal liquid-crystal (LC) phase modulators [16–18]. A miniature wirelessly-controlled LC lens was proposed to be implanted into the capsular bag of the eye of a presbyopic or cataract patient. An important advantage of the LC lens in comparison with standard IOLs is that, besides defocus, it provides dynamic correction of higher-order aberrations. Several configurations of wireless control links were considered and tested. However, some of their parameters, e. g. high operation voltage, were found to be unacceptable for intraocular optics. In the experiments with electrostatic control, an accommodation of ~3 diopters (D) was obtained by application of ~70 V between two 3 cm2 electrodes in air.

In this paper, we describe the practical implementation of an adaptive LC lens with wireless radio-frequency (RF) control that can be potentially used as an adaptive IOL. The high efficiency of the RF link allows a compact receiver design – it can be integrated with the modal corrector, and safe low-voltage operation. The lens response in terms of wave-front aberrations is studied in vitro, i. e. in a water cell that reproduces the environment of the anterior chamber of the eye. The optical performance of the theoretical model eye with the adaptive LC lens is simulated.

2. Modal LC phase correctors with wireless control link

In contrast to LC phase modulators addressed by an array of discrete electrodes, modal wave-front correctors employ a distributed, or modal, control principle to provide a smooth phase profile over the whole aperture [16–19]. When an ac voltage is applied the complex impedance due to the capacitance of the LC layer sandwiched between the low- and high-ohmic control electrodes deposited on the top and bottom glass substrate gives rise to a distributed voltage. Thus, a continuous phase profile caused by reorientation of LC molecules is generated.

Liquid-crystal modal lenses have been demonstrated to be potentially suitable for intraocular correction of the wave-front aberrations of the human eye [15]. Their small size that can be adjusted to fit into the capsular bag of the eye, low power consumption of the order of several microwatts at a control voltage ranging up to ~10 V, intrinsic ability to reduce longitudinal chromatic aberration [19], and a high accommodation range of the order ~3 D make LC lenses attractive for applications in intraocular adaptive optics.

Two identical modal correctors with a clear aperture of 5 mm were used in the experiments. Each corrector consists of a 40-μm planar-aligned layer of a single-frequency nematic liquid crystal sandwiched between a pair of BK-7 glass substrates coated with a highly-conductive (50–200 Ω/ square electrode) ITO and a low-conductive (1–10 MΩ/ square electrode) metal-oxide transparent films, correspondingly, as described in [18]. Figure 1(a) below shows the LC modal lens cross section.

The convex-shaped substrates determine the initial focal distance (when no voltage is applied) of the modal correctors to be ~80 mm (+12.5 D) in an aqueous medium having a refractive index n = 1.337. The first LC modal phase corrector, as depicted in Fig. 1(b), is integrated with a circumferential antenna mounted on its rim and a rectifying circuit for demodulating the amplitude-modulated (AM) RF signal. The second sample, mostly used in the experiments owing to the simplicity of the signal measurements, includes the LC modal corrector driven by a separate receiver, i. e. antenna and demodulator are not integrated with the lens.

 

Fig. 1. The wirelessly-controlled LC lens: (a) cross section of the LC modal corrector, (b) photograph. 1, glass (BK7) substrates; 2, ITO low-ohmic layer; 3, liquid crystal; 4, contact; 5, high-ohmic layer; R, rectifying diode; A, antenna.

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Figure 2 shows the block diagram of the wireless RF control link. The signal of a radio-frequency oscillator (RFO) at ~6 MHz is amplitude modulated by a low-frequency signal at F = 0 - 50 kHz from a function generator (FG). The modulation frequency and depth can be controlled individually by a computer. After a modulator (M), the signal is amplified by an RF amplifier (A) and then is radiated by a loop transmitting antenna (TA). The output RF power does not exceed ~0.5 W, providing the lens operation at distances up to 8 cm from the TA, and can be further reduced for an adaptive LC lens with a resonance receiving antenna. In compliance with the RF exposure safety guidelines [20], the estimated specific absorption rate is well below 0.4 W/kg [20, 21].

 

Fig. 2. Wireless RF control link. FG, function generator; RFO, radio-frequency oscillator; M, modulator; A, amplifier; TA, transmitting antenna; RA, receiving antenna, D, demodulator.

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Note that the transmitting antenna is supposed to be integrated in a spectacles frame, whereas the driving electronics can be assembled as a small module attached to one arm of the frame.

In the intraocular lens, the AM RF signal received by a loop receiving antenna (RA) is demodulated by an envelope detector and the resulting low-frequency ac voltage is applied to the LC modal corrector. The optimal design of the receiver implies a capacitor in parallel with the antenna to keep it in resonance with the transmitted RF signal, and an output capacitor to block the dc voltage. This is because liquid crystals subjected to a dc voltage may exhibit hydrodynamic flows resulting in significant light scattering [22].

To sustain the RF link operating within its dynamic range and to measure the voltage applied to the modal corrector, the demodulated low-frequency signal was monitored by a digital oscilloscope.

 

Fig. 3. Transfer characteristics of the RF link at different driving voltages U LC of the LC lens.

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Typical transfer characteristics of the RF control link are presented in Fig. 3. As seen, to maintain a fixed voltage ULC on the LC modal lens, the modulation voltage Um produced by the function generator should increase as the frequency F increases. The RF frequency of the transmitter was adjusted to correspond to the resonance frequency of the receiver ~6.4 MHz. Due to the complex voltage- and frequency-dependent impedance of the LC lens [16, 18], the required control amplitude Um in Fig. 3 is not linearly scaled with the resulting ULC at a given frequency.

3. Optical setup

Figure 4 depicts the experimental setup for characterizing the wirelessly-controlled adaptive lens. The collimated and linearly-polarized light beam of a green He-Ne laser at λ= 543.5 nm passes through the RF-controlled LC modal lens placed in a 30-mm water cell, and then is imaged by a 40-mm (L1) and a 30-mm (L2) focal length lenses onto a Shack-Hartmann wave-front sensor.

 

Fig. 4. Experimental arrangement for measuring aberrations produced by the LC lens. Inset shows the wireless link. 1, glass walls of the water cell; 2, distilled water; L1, L2, lenses; TA, transmitting antenna; RA, receiving antenna.

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The lenses L1 and L2 are used to conjugate the input plane of the sensor with the LC adaptive lens and to adjust the sensor aperture with the modal lens aperture. In our experiments, we employed the Shack-Hartmann sensor with a hexagonal 127-subaperture lenslet array and the wave-front analysis software supplied by OKO Technologies [23]. The wave-front sensor aperture is D 2 ≅ 3.8 mm that corresponds to the apparent LC lens aperture D 1 ≅ 4.9 mm in the geometry of Fig. 4.

Distances between the optical parts, as depicted in Fig. 4, were: l 1 = 43 mm, l 2 = 40 mm and l 3 = 31 mm. After initial alignment, the RMS phase error was measured to be ~0.08 waves and the peak-to-valley error was ~0.29 waves in the absence of the control AM RF signal. The theoretical values obtained by ray-tracing with the Zemax optical design software (Focus Software, Inc.) were ~0.06 and ~0.21 waves, respectively. Optical simulations have shown that in the configuration of Fig. 4, taking into account the actual shapes of L1 and L2, the wave-front aberrations produced by the LC modal lens can be determined with the precision of about 2%.

In the carried out experiments, the optical aberrations produced by the wirelessly-controlled adaptive LC lens were measured versus the frequency and amplitude of modulation of the RF signal.

4. Experimental results

Defocus (Z 4) and spherical aberration (Z 11) contribute mainly to the LC lens-generated wave-front and their control offers a straightforward way to correct the corresponding aberrations of the human eye. In accordance with [25], spherical aberration exhibits the largest change with accommodation among other aberrations (except defocus) of the eye and is mainly associated with the changes in the structure of the crystalline lens.

 

Fig. 5. Dependence of the LC lens aberrations (defocus a 4, spherical aberration a 11) on the modulation frequency F at fixed voltages U LC across the LC modal corrector: (a) 2.12 V (rms), (b) 2.83 V (rms), (c) 3.53 V (rms), (d) 4.24 V (rms).

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Figure 5 presents the dependences of defocus and spherical aberration coefficients a 4 and a 11, respectively, produced by the RF-driven LC modal lens, on the control frequency F at fixed voltages ULC across the LC lens. The definition of Noll is used for Zernike polynomials [24], in which, in particular, defocus takes the form Z 4 = √3(2r 2-1) and spherical aberration becomes Z 11 = √5(6r 4 -6r 2 + 1), where r is the transverse coordinate.

As seen from Figs. 5(a)–5(d), the defocus coefficient a 4 (red curve with squares) grows along with ULC and reaches a maximum of ~4 waves at ULC =4.24 V (rms) and F = 27 kHz. The value of the induced defocus a 4 ≅ 4 waves determined at λ = 543.5 nm for a 4.9-mm aperture of the LC lens is equivalent to the change of the focusing power (Φ) by ~2.51 D. Depending on the applied voltage ULC, the maximum of a 4 is attained in the frequency range 22–27 kHz.

Apart from defocus, the spherical aberration coefficient a 11 (blue curve with circles) reveals more complex behavior with ULC and F. At low voltages ULC ≤3 V (rms), a 11 increases almost monotonously with F till approximately 35–43 kHz and reaches a maximum of ~0.67 waves. As the voltage ULC increases, a 11 decreases in the whole frequency range and its initial maximum moves to lower frequencies. At voltages ULC ≥ 4 V (rms), a 11 changes sign at F ≅ 30 kHz and drops to a minimum of -0.8 waves at F ≅ 45 kHz. Such a behavior of the spherical aberration coefficient agrees with the theoretical and experimental results obtained for directly-controlled LC modal lenses [18, 26].

 

Fig. 6. Dependences of the optimal modulation voltage U m (and corresponding voltage U LC applied to the LC lens) and the modulation frequency F versus focusing power variation ΔΦ. Interferograms of the LC lens were simulated using the measured aberrations.

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The LC modal wave-front corrector can be operated as a varifocal optical element providing controllable focusing power Φ, while maintaining low optical distortions caused by higher-order aberrations. Figure 6 above shows the optimal combination of the modulation voltage Um (green curve with triangles) and frequency F (blue curve with diamonds) producing a gradual change in focusing power ΔΦ. The calibration dependences were obtained by minimizing higher-order aberrations of the lens by adjusting Um at fixed F. As seen, ΔΦ varies in the range 0.06–2.4 D when F increases from 4 to 26 kHz at an almost constant Um of ~1 V (rms). The corresponding voltage across the lens ULC, as depicted by the red curve with triangles in Fig. 6, changes from 3.5 to 7 V (rms). So, the RF link properly corrects for the voltage ULC applied to the LC lens reducing the control parameters to only F. The total RMS wave-front error caused by the LC lens aberrations, except defocus, reaches a maximum of 0.67 waves at ΔΦ = 2.4 D. A series of interferograms simulated using the measured aberrations of the LC modal lens is shown on the right side of Fig. 6.

An adaptive intraocular lens intended for restoration of accommodation and correction of corneal aberrations should respond dynamically to changes in viewing conditions. Simultaneous measurements of transient defocus (solid red line) and spherical aberration (dashed blue line) coefficients are shown in Fig. 7. A series of transient responses obtained by applying a control voltage ULC of 2.83 V (rms) across the LC modal lens, periodically turning on (20 s) and off (20 s), with the frequency F changing stepwise from 4 to 14 kHz by 2 kHz every 40 s, is depicted in Fig. 7(a). The full settling time attributed largely to the transient overshoots weakly depends on the modulation frequency and amount to ~16 s. One possibility to decrease the LC phase corrector switching-on time is to apply a voltage prepulse with a duration of several milliseconds and having a high amplitude, see [26] and references therein.

 

Fig. 7. Transient dynamics of the LC lens aberrations (defocus a 4, spherical aberration a 11) at: (a) periodic turning on (20 s) and off (20 s) of a control voltage U LC=2.83 V (rms), F changes stepwise from 4 kHz to 14 kHz by 2 kHz every 40 s; (b) two-level stepwise frequency and amplitude control. The upper diagrams represent U LC sequences.

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To speed up the response of the modal lens based on a single-frequency nematic LC, we implemented a dual-frequency addressing technique. The LC layer sandwiched between the low and high-ohmic electrodes constitutes a complex distributed reactive system [26]. The orientation of the LC molecules which determines the spatial phase modulation depends on the applied voltage ULC and frequency F. When the frequency is low, the reactive component of the impedance diminishes and the distribution of the ac voltage over the LC layer becomes almost constant. So, by applying a low-frequency voltage the spatial phase distribution can be homogenized over the whole lens aperture and the LC modal lens will operate in piston mode.

Figure 7(b) shows the transient dynamics of defocus and spherical aberration coefficients obtained by applying the ULC sequence depicted in the upper part. The first pulse ULC =2.83 V (rms) at 10 kHz results in pronounced overshoots, analogously to those in Fig. 7(a). Before the second pulse, an ac voltage of 2.83 V (rms) at 1 kHz was applied to the LC lens leading to pedestals in aberration coefficients of a 4 ≅ 0.06 waves and a 11 ≅ 0.005 waves, respectively. As seen, the transient caused by the second ULC pulse of 2.83 V (rms) does not reveal overshoots if the low-frequency signal at 1 kHz is constantly maintained. The settling time of the LC lens does not exceed ~4 s at the control frequencies F = 4 - 26 kHz. So, with the RF-controlled LC modal lens an accommodation amplitude of up to 2.51 D can be reached for ~4 s. The third case in Fig. 7(b) corresponds to the transient process at a fixed control frequency F of 10 kHz and a stepwise change of ULC. The measurements were carried out using the RF link.

Thus, the frequency-switching technique described above offers an efficient and simple method of improving the overall dynamics of modal phase correctors based on a single-frequency nematic liquid crystal.

Since the Hartmann method can be insensitive to light scatter, additional measurements were carried out using a Mach-Zehnder interferometer aiming at verifying the overall efficiency and optical quality of the LC lens. The wirelessly-controlled LC lens and a 50-mm focal length objective (both in air), in a telescopic arrangement, were placed into the object arm of the interferometer operating at λ = 633 nm. Figure 8 shows the interferograms and the corresponding phase cross sections obtained at (a) a fixed frequency F=10.5 kHz and different U LC and (b) a fixed voltage U LC=2.12 V (rms) and various F.

 

Fig. 8. LC lens phase cross sections and the corresponding interferograms obtained in a Mach-Zehnder interferometer at: (a) a fixed frequency F=10.5 kHz and different U LC, (b) a fixed voltage U LC=2.12 V (rms) and various F.

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As seen, the directly recorded interferograms reveal a high contrast level and very low scattering. In compliance with [18], the total transmission of polarized light by the LC lens without anti-reflection coatings amounts to ~70%.

5. Discussion and further work

The prototype of an adaptive IOL based on the LC modal corrector with the wireless RF control provides an accommodation of about +2.5 D that is larger than the accommodative range of the majority of existing single-power accommodating IOLs [3–6]. To read at typical near distance of 40 cm, 2.5 D of optical power is required. An additional accommodative power of 0.3–1.9 D can be expected from pseudo-accommodation, when the IOL moves along the anterior-posterior axis of the eye due to contraction of the ciliary body [5]. Thus, we may suppose that the accommodation range of the modal lens using a 40-μm LC layer is appropriate for most cases.

 

Fig. 9. Monochromatic MTFs of the model eye with the LC modal lens: (a) based on the measured 11 aberration coefficients of the lens prototype accommodated at +2.4 D, (b) optimized shape of the LC lens glass substrates.

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As it was discussed in [15], the size of the modal lens is limited only by the LC layer, whereas the shape of the substrates can be optimized for best image quality on the retina. Figure 9(a) shows the monochromatic, average over the sagittal and tangential sections, modulation transfer function (MTF) of the wide-angle model eye [27, 28] with the LC modal lens accommodated by +2.4 D. As seen, the MTF corresponding to the LC lens with the experimentally measured aberrations at ΔΦ = 2.4 D (red curve with solid circles) differs slightly from that calculated for the LC when all aberration coefficients except defocus were set equal to zero (green curve with open squares). Both curves fall off rapidly as the spatial frequency increases. The imaging quality can be significantly improved by adjusting the shapes of the glass substrates using, for example, the conic optical surface [29]:

z=S(x,y)=r2R{1+1(1+k)×(rR)2},

where, r=x2+y2; x,y are the transverse coordinates; R is the radius of curvature; k is the conic parameter.

Figure 9(b) presents the MTFs obtained with the LC modal lens consisting of the equal-curvature, optimized shape substrates with the following parameters: (anterior substrate) R 1 = 9.17 mm, k 1 = 7.2; (posterior substrate) R 2 = -9.17 mm, k 1 = -1.8. So, depending on the parameters of the individual eye the optimal shape of the adaptive lens can be calculated. As can be concluded from Fig. 9(b), the aberrations produced by the LC modal lens accommodated by +2.4 D only slightly reduce the MTF of the model eye.

By using the frequency-switching technique, the settling time of the LC modal lens with a 40-μm nematic LC layer can be reduced to ~4 s, whereas a settling time of ~16 s is obtained at normal operation. According to Mordi and Ciuffreda [30], however, the average latency time of accommodation of a 30-year-old individual is ~0.35 s and the time constant of accommodation amounts to ~0.2 s. Thus, a response time of ~0.5 s would be ideally suited for adaptive IOLs to correct for the eye’s aberrations dynamically. A possible way to improve the overall dynamics of the LC modal lens is to use a dual-frequency LC material, see [31] and references therein, although the dynamics should be optimized together with the LC layer thickness to provide an accommodation range of up to ~+4 D.

Aside from the dynamic range, imaging quality and response time of the wirelessly-controlled LC-based IOL prototype, there are several aspects that should be taken into account in the final IOL design:

  • Biological compatibility of the LC material and the lens optics with the ocular media. Not all LC materials comply with safety regulations [32]. However, according to Merck [33], liquid crystals can be characterized as “not acutely toxic” and the tests performed with 224 LC substances indicated that 215 LCs had no toxic effect. In order to satisfy the medical device regulations and to reduce the weight of the implant, the LC lens substrates can be fabricated from approved plastics. The LC lens should remain sealed for many years in a saline environment of the capsular bag at ~37°C. This can be achieved by a proper isolation of the LC layer. In the prototype a two-component epoxy glue was employed.
  • The receiving antenna and the rectifier should be integrated with a LC modal corrector and completely isolated from all outside, i. e. aqueous solution. A “silicon-on-anything” [34] technology is one of the most promising approaches to solving this problem. This technology potentially allows making foldable LC-based IOLs and, thus, meeting the requirements of ophthalmic surgery.
  • As soon as LC-based devices are sensitive to the light polarization, linearly-polarized light is required for proper operation of the LC lens [16]. An additional linear polarizer – resulting in 50 % loss of light, or a combination of two LC correctors acting on orthogonal polarization states can be used for randomly-polarized light.
  • By increasing the number of control channels, the precision of wave-front correction can be improved. In the current prototype, a single-channel RF AM analog control is implemented which allows correction of defocus and spherical aberration. The configurations of the LC modal phase corrector, analogous to those described in [35] and [36], can be used for the multi-channel adaptive lens. Such multi-channel systems can successfully correct asymmetric aberrations.
  • A closed-loop control is needed for an adaptive LC modal lens. We suggest that the feedback signals can be obtained by measuring and subsequent real-time processing the electroencephalographic activity of the visual cortex of the human brain [37, 38].

6. Conclusion

We have demonstrated an adaptive IOL prototype based on a LC modal phase corrector with wireless RF control. A planar-aligned layer of a nematic LC sandwiched between two glass substrates coated with transparent low- and high-ohmic electrodes provides dynamic correction of the ocular aberrations when driven by an ac control voltage. The convex-shaped glass substrates produce the initial focusing power of the lens. A low-frequency ac control voltage is delivered by an envelope detector followed by a resonance loop antenna, both integrated with the LC modal corrector. The lens is driven remotely using a computer-controlled AM RF transmitter with the loop antenna that is supposed to be integrated in a spectacle frame.

The feasibility of the RF-controlled LC modal lens was demonstrated experimentally. This lens can be potentially used as an adaptive IOL. A prototype with a clear aperture of 5 mm and a focusing power of +12.5 D was fabricated and characterized in vitro, i. e. in a water cell that simulates the in vivo environment of the anterior chamber of the eye. Adaptive correction of defocus and spherical aberration was accomplished by changing the modulation depth and frequency of the RF control signal. An accommodative range of ~2.5 D was achieved at λ = 534.5 nm for an aperture of the LC modal lens of 4.9 mm, i. e. ~4 waves. Simultaneous correction of the spherical aberration coefficient ranging from -0.8 to 0.67 waves was also demonstrated. By using the frequency-switching control of the LC modal lens, a full-scale settling time of ~4 s was attained.

Further studies on the RF wirelessly-controlled implantable adaptive LC modal lenses are currently underway.

Acknowledgments

The authors thank Svetlana Kotova (Lebedev Institute of Physics, Russia) for her assistance in fabrication of liquid-crystal lenses The work was supported by the “Nederlandse organisatie voor Wetenschappelijk Onderzoek” (NWO), grant DOE 6190.

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33. W. Becker, B. Simon-Hettich, and P. Hnicke, “Toxicological and ecotoxicological investigations of liquid crystals and disposal of lcds,” Merck brochure, Merck KGaA, Liquid Crystals Division and Institute of Toxicology 64271 Darmstadt, September 25 (2001).

34. J. Bruines, “Process outlook for analog and rf applications,” Microelectr. Engineer. 54, 35–48 (2000). [CrossRef]  

35. S. P. Kotova, M. Yu. Kvashnin, M. A. Rakhmatulin, O. A. Zayakin, I. G. Guralnik, N. A. Klimov, P. Clark, G. D. Love, A. F. Naumov, C. D. Saunter, M. Yu. Loktev, G. V. Vdovin, and L. V. Toporkova, “Modal liquid crystal wavefront corrector,” Opt. Express 10, 1258–1272 (2002). [PubMed]  

36. G. V. Vdovin, I. R. Guralnik, M. Y. Loktev, A. F. Naumov, and S. V. Sheenkov, “Dynamic method for control of wavefront shape of a light beam and device for its realization,” Russian patent 2214617, December 1999 (in Russian).

37. http://www.biosemi.com/publications.htm.

38. G. Thut, A. Nietzel, S. A. Brandt, and A. Pascual-Leone, “Alpha-band electroencephalographic activity over occipital cortex indexes visuospatial attention bias and predicts visual target detection,” J. Neuroscience 26, 9494–9502 (2006). [CrossRef]  

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2006

G.  Thut, A.  Nietzel, S. A.  Brandt, and A.  Pascual-Leone, "Alpha-band electroencephalographic activity over occipital cortex indexes visuospatial attention bias and predicts visual target detection," J. Neuroscience 26, 9494-9502 (2006).
[CrossRef]

A. N.  Simonov, G.  Vdovin, and M. C.  Rombach, "Cubic optical elements for an accommodative intraocular lens," Opt. Express 14, 7757-7775 (2006).
[CrossRef] [PubMed]

2005

H. B.  Dick, "Accommodative intraocular lenses: current status," Curr. Opin. Ophthalmol. 16, 8-26 (2005).
[CrossRef] [PubMed]

S. D.  McLeod, V.  Portney, and A.  Ting, "A dual optic accommodating foldable intraocular lens," Br. J. Ophthalmol. 87, 1083-1085 (2005).
[CrossRef]

2004

H.  Cheng, J. K.  Barnett, A. S.  Vilupuru, J. D.  Marsack, S.  Kasthurirangan, R. A.  Applegate, and A.  Roorda, "A population study on changes in wave aberrations with accommodation," J. Vision 4, 272-280 (2004).
[CrossRef]

J. A.  Mordi and K. J.  Ciuffreda, "Dynamic aspects of accommodation: age and presbyopia," Vision Res. 44, 591-601 (2004).
[CrossRef]

A. K.  Kirby and G. D.  Love, "Fast, large and controllable phase modulation using dual frequency liquid crystals," Opt. Express 12, 1470-1475 (2004).
[CrossRef] [PubMed]

2003

G.  Vdovin, M.  Loktev and A.  Naumov, "On the possibility of intraocular adaptive optics," Opt. Express 11, 810-817 (2003).
[CrossRef] [PubMed]

A.  Rana, D.  Miller, and P.  Magnante, "Understanding the accommodating intraocular lens," J. Cataract. Refract. Surg. 29, 2284-2287 (2003).
[CrossRef]

2002

2000

H.  Lesiewska-Junk and J.  Kaluzny, "Intraocular lens movement and accommodation in eyes of young patients," J. Cataract. Refract. Surg. 26, 562-565 (2000).
[CrossRef] [PubMed]

J.  Bruines, "Process outlook for analog and rf applications," Microelectr. Engineer. 54, 35-48 (2000).
[CrossRef]

M. Yu.  Loktev, V. N.  Belopukhov, F. L.  Vladimirov, G. V.  Vdovin, G. D.  Love, and A. F.  Naumov, "Wave front control systems based on modal liquid crystal lenses," Rev. Sci. Instrum. 71, 3290-3297 (2000).
[CrossRef]

T. L.  Kelly, A. F.  Naumov, M. Yu.  Loktev, M. A.  Rakhmatulin, and O. A.  Zayakin, "Focusing of astigmatic laser diode beam by combination of adaptive liquid crystal lenses," Opt. Commun. 181, 295-301 (2000).
[CrossRef]

1999

1998

1997

T.  Walkow, A.  Liekfeld, N.  Anders, D. T.  Pham, C.  Hartmann, and J. A.  Wollensak "A prospective evaluation of a diffractive versus refractive designed multifocal intraocular lenses. Visual and refractive comparison," Ophthalm. 104, 1380-1386 (1997).

1994

J.  Liang, B.  Grimm, S.  Goelz, and J. F.  Bille, "Objective measurements of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor," J. Opt. Soc. Am. 11, 1949-1957 (1994).
[CrossRef]

1993

R.  Bellucci and P.  Giardini, "Pseudoaccommodation with the 3M diffractive mulifocal intraocular lens: a refraction study of 52 subjects," J. Cataract. Refract. Surg. 19, 32-35 (1993).
[PubMed]

1992

P. J.  Gray and M. G.  Lyall, "Diffractive mulifocal intraocular lens implants for unilateral cataracts in presbyopic patents," Br. J. Ophthalmol. 76, 336-337 (1992).
[CrossRef] [PubMed]

1987

P.  Röschmann, "Radiofrequency penetration and absorption in the human body: limitations to high-field whole-body nuclear magnetic resonance imaging," Med. Phys. 14, 922-931 (1987).
[CrossRef] [PubMed]

1976

1974

M. J.  Stephen and J. P.  Straley, "Physics of liquid crystals," Rev. Mod. Phys. 46, 617-704 (1974).
[CrossRef]

1971

O.  Pomerantzeff, H.  Fish, J.  Govignon, and C. L.  Schepens, "Wide angle optical model of the human eye," Ann Ophthalmol. 3, 815-819 (1971).
[PubMed]

1951

Amano, Sh.

T.  Oshika, T.  Mimura, S.  Tanaka, Sh.  Amano, M.  Fukuyama, F.  Yoshitomi, N.  Maeda, T.  Fujikado, Y.  Hirohara, and T.  Mihashi, "Apperent accommodation and corneal wavefront aberration in pseudophakic eyes," Invest. Ophthalmol. Vis. Sci. 43, 2882-2886 (2002).
[PubMed]

Anders, N.

T.  Walkow, A.  Liekfeld, N.  Anders, D. T.  Pham, C.  Hartmann, and J. A.  Wollensak "A prospective evaluation of a diffractive versus refractive designed multifocal intraocular lenses. Visual and refractive comparison," Ophthalm. 104, 1380-1386 (1997).

Applegate, R. A.

H.  Cheng, J. K.  Barnett, A. S.  Vilupuru, J. D.  Marsack, S.  Kasthurirangan, R. A.  Applegate, and A.  Roorda, "A population study on changes in wave aberrations with accommodation," J. Vision 4, 272-280 (2004).
[CrossRef]

Barnett, J. K.

H.  Cheng, J. K.  Barnett, A. S.  Vilupuru, J. D.  Marsack, S.  Kasthurirangan, R. A.  Applegate, and A.  Roorda, "A population study on changes in wave aberrations with accommodation," J. Vision 4, 272-280 (2004).
[CrossRef]

Bellucci, R.

R.  Bellucci and P.  Giardini, "Pseudoaccommodation with the 3M diffractive mulifocal intraocular lens: a refraction study of 52 subjects," J. Cataract. Refract. Surg. 19, 32-35 (1993).
[PubMed]

Belopukhov, V. N.

M. Yu.  Loktev, V. N.  Belopukhov, F. L.  Vladimirov, G. V.  Vdovin, G. D.  Love, and A. F.  Naumov, "Wave front control systems based on modal liquid crystal lenses," Rev. Sci. Instrum. 71, 3290-3297 (2000).
[CrossRef]

Bille, J. F.

J.  Liang, B.  Grimm, S.  Goelz, and J. F.  Bille, "Objective measurements of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor," J. Opt. Soc. Am. 11, 1949-1957 (1994).
[CrossRef]

Brandt, S. A.

G.  Thut, A.  Nietzel, S. A.  Brandt, and A.  Pascual-Leone, "Alpha-band electroencephalographic activity over occipital cortex indexes visuospatial attention bias and predicts visual target detection," J. Neuroscience 26, 9494-9502 (2006).
[CrossRef]

Bruines, J.

J.  Bruines, "Process outlook for analog and rf applications," Microelectr. Engineer. 54, 35-48 (2000).
[CrossRef]

Cheng, H.

H.  Cheng, J. K.  Barnett, A. S.  Vilupuru, J. D.  Marsack, S.  Kasthurirangan, R. A.  Applegate, and A.  Roorda, "A population study on changes in wave aberrations with accommodation," J. Vision 4, 272-280 (2004).
[CrossRef]

Ciuffreda, K. J.

J. A.  Mordi and K. J.  Ciuffreda, "Dynamic aspects of accommodation: age and presbyopia," Vision Res. 44, 591-601 (2004).
[CrossRef]

Clark, P.

Dick, H. B.

H. B.  Dick, "Accommodative intraocular lenses: current status," Curr. Opin. Ophthalmol. 16, 8-26 (2005).
[CrossRef] [PubMed]

Fish, H.

O.  Pomerantzeff, H.  Fish, J.  Govignon, and C. L.  Schepens, "Wide angle optical model of the human eye," Ann Ophthalmol. 3, 815-819 (1971).
[PubMed]

Fujikado, T.

T.  Oshika, T.  Mimura, S.  Tanaka, Sh.  Amano, M.  Fukuyama, F.  Yoshitomi, N.  Maeda, T.  Fujikado, Y.  Hirohara, and T.  Mihashi, "Apperent accommodation and corneal wavefront aberration in pseudophakic eyes," Invest. Ophthalmol. Vis. Sci. 43, 2882-2886 (2002).
[PubMed]

Fukuyama, M.

T.  Oshika, T.  Mimura, S.  Tanaka, Sh.  Amano, M.  Fukuyama, F.  Yoshitomi, N.  Maeda, T.  Fujikado, Y.  Hirohara, and T.  Mihashi, "Apperent accommodation and corneal wavefront aberration in pseudophakic eyes," Invest. Ophthalmol. Vis. Sci. 43, 2882-2886 (2002).
[PubMed]

Giardini, P.

R.  Bellucci and P.  Giardini, "Pseudoaccommodation with the 3M diffractive mulifocal intraocular lens: a refraction study of 52 subjects," J. Cataract. Refract. Surg. 19, 32-35 (1993).
[PubMed]

Goelz, S.

J.  Liang, B.  Grimm, S.  Goelz, and J. F.  Bille, "Objective measurements of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor," J. Opt. Soc. Am. 11, 1949-1957 (1994).
[CrossRef]

Govignon, J.

O.  Pomerantzeff, H.  Fish, J.  Govignon, and C. L.  Schepens, "Wide angle optical model of the human eye," Ann Ophthalmol. 3, 815-819 (1971).
[PubMed]

Gray, P. J.

P. J.  Gray and M. G.  Lyall, "Diffractive mulifocal intraocular lens implants for unilateral cataracts in presbyopic patents," Br. J. Ophthalmol. 76, 336-337 (1992).
[CrossRef] [PubMed]

Grimm, B.

J.  Liang, B.  Grimm, S.  Goelz, and J. F.  Bille, "Objective measurements of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor," J. Opt. Soc. Am. 11, 1949-1957 (1994).
[CrossRef]

Guralnik, I. G.

Guralnik, I. R.

Hartmann, C.

T.  Walkow, A.  Liekfeld, N.  Anders, D. T.  Pham, C.  Hartmann, and J. A.  Wollensak "A prospective evaluation of a diffractive versus refractive designed multifocal intraocular lenses. Visual and refractive comparison," Ophthalm. 104, 1380-1386 (1997).

Hirohara, Y.

T.  Oshika, T.  Mimura, S.  Tanaka, Sh.  Amano, M.  Fukuyama, F.  Yoshitomi, N.  Maeda, T.  Fujikado, Y.  Hirohara, and T.  Mihashi, "Apperent accommodation and corneal wavefront aberration in pseudophakic eyes," Invest. Ophthalmol. Vis. Sci. 43, 2882-2886 (2002).
[PubMed]

Kaluzny, J.

H.  Lesiewska-Junk and J.  Kaluzny, "Intraocular lens movement and accommodation in eyes of young patients," J. Cataract. Refract. Surg. 26, 562-565 (2000).
[CrossRef] [PubMed]

Kasthurirangan, S.

H.  Cheng, J. K.  Barnett, A. S.  Vilupuru, J. D.  Marsack, S.  Kasthurirangan, R. A.  Applegate, and A.  Roorda, "A population study on changes in wave aberrations with accommodation," J. Vision 4, 272-280 (2004).
[CrossRef]

Kelly, T. L.

T. L.  Kelly, A. F.  Naumov, M. Yu.  Loktev, M. A.  Rakhmatulin, and O. A.  Zayakin, "Focusing of astigmatic laser diode beam by combination of adaptive liquid crystal lenses," Opt. Commun. 181, 295-301 (2000).
[CrossRef]

Kirby, A. K.

Klimov, N. A.

Kotova, S. P.

Kvashnin, M. Yu.

Lesiewska-Junk, H.

H.  Lesiewska-Junk and J.  Kaluzny, "Intraocular lens movement and accommodation in eyes of young patients," J. Cataract. Refract. Surg. 26, 562-565 (2000).
[CrossRef] [PubMed]

Liang, J.

J.  Liang, B.  Grimm, S.  Goelz, and J. F.  Bille, "Objective measurements of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor," J. Opt. Soc. Am. 11, 1949-1957 (1994).
[CrossRef]

Liekfeld, A.

T.  Walkow, A.  Liekfeld, N.  Anders, D. T.  Pham, C.  Hartmann, and J. A.  Wollensak "A prospective evaluation of a diffractive versus refractive designed multifocal intraocular lenses. Visual and refractive comparison," Ophthalm. 104, 1380-1386 (1997).

Loktev, M.

Loktev, M. Yu.

Love, G. D.

Lyall, M. G.

P. J.  Gray and M. G.  Lyall, "Diffractive mulifocal intraocular lens implants for unilateral cataracts in presbyopic patents," Br. J. Ophthalmol. 76, 336-337 (1992).
[CrossRef] [PubMed]

Maeda, N.

T.  Oshika, T.  Mimura, S.  Tanaka, Sh.  Amano, M.  Fukuyama, F.  Yoshitomi, N.  Maeda, T.  Fujikado, Y.  Hirohara, and T.  Mihashi, "Apperent accommodation and corneal wavefront aberration in pseudophakic eyes," Invest. Ophthalmol. Vis. Sci. 43, 2882-2886 (2002).
[PubMed]

Magnante, P.

A.  Rana, D.  Miller, and P.  Magnante, "Understanding the accommodating intraocular lens," J. Cataract. Refract. Surg. 29, 2284-2287 (2003).
[CrossRef]

Marsack, J. D.

H.  Cheng, J. K.  Barnett, A. S.  Vilupuru, J. D.  Marsack, S.  Kasthurirangan, R. A.  Applegate, and A.  Roorda, "A population study on changes in wave aberrations with accommodation," J. Vision 4, 272-280 (2004).
[CrossRef]

McLeod, S. D.

S. D.  McLeod, V.  Portney, and A.  Ting, "A dual optic accommodating foldable intraocular lens," Br. J. Ophthalmol. 87, 1083-1085 (2005).
[CrossRef]

Mihashi, T.

T.  Oshika, T.  Mimura, S.  Tanaka, Sh.  Amano, M.  Fukuyama, F.  Yoshitomi, N.  Maeda, T.  Fujikado, Y.  Hirohara, and T.  Mihashi, "Apperent accommodation and corneal wavefront aberration in pseudophakic eyes," Invest. Ophthalmol. Vis. Sci. 43, 2882-2886 (2002).
[PubMed]

Miller, D.

A.  Rana, D.  Miller, and P.  Magnante, "Understanding the accommodating intraocular lens," J. Cataract. Refract. Surg. 29, 2284-2287 (2003).
[CrossRef]

Mimura, T.

T.  Oshika, T.  Mimura, S.  Tanaka, Sh.  Amano, M.  Fukuyama, F.  Yoshitomi, N.  Maeda, T.  Fujikado, Y.  Hirohara, and T.  Mihashi, "Apperent accommodation and corneal wavefront aberration in pseudophakic eyes," Invest. Ophthalmol. Vis. Sci. 43, 2882-2886 (2002).
[PubMed]

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J. A.  Mordi and K. J.  Ciuffreda, "Dynamic aspects of accommodation: age and presbyopia," Vision Res. 44, 591-601 (2004).
[CrossRef]

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Naumov, A. F.

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G.  Thut, A.  Nietzel, S. A.  Brandt, and A.  Pascual-Leone, "Alpha-band electroencephalographic activity over occipital cortex indexes visuospatial attention bias and predicts visual target detection," J. Neuroscience 26, 9494-9502 (2006).
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T.  Oshika, T.  Mimura, S.  Tanaka, Sh.  Amano, M.  Fukuyama, F.  Yoshitomi, N.  Maeda, T.  Fujikado, Y.  Hirohara, and T.  Mihashi, "Apperent accommodation and corneal wavefront aberration in pseudophakic eyes," Invest. Ophthalmol. Vis. Sci. 43, 2882-2886 (2002).
[PubMed]

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G.  Thut, A.  Nietzel, S. A.  Brandt, and A.  Pascual-Leone, "Alpha-band electroencephalographic activity over occipital cortex indexes visuospatial attention bias and predicts visual target detection," J. Neuroscience 26, 9494-9502 (2006).
[CrossRef]

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T.  Walkow, A.  Liekfeld, N.  Anders, D. T.  Pham, C.  Hartmann, and J. A.  Wollensak "A prospective evaluation of a diffractive versus refractive designed multifocal intraocular lenses. Visual and refractive comparison," Ophthalm. 104, 1380-1386 (1997).

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S. D.  McLeod, V.  Portney, and A.  Ting, "A dual optic accommodating foldable intraocular lens," Br. J. Ophthalmol. 87, 1083-1085 (2005).
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S. P.  Kotova, M. Yu.  Kvashnin, M. A.  Rakhmatulin, O. A. Zayakin, I. G.  Guralnik, N. A.  Klimov, P.  Clark, G. D.  Love, A. F.  Naumov, C. D.  Saunter, M. Yu.  Loktev, G. V.  Vdovin, and L. V.  Toporkova, "Modal liquid crystal wavefront corrector," Opt. Express 10, 1258-1272 (2002).
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[CrossRef]

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A.  Rana, D.  Miller, and P.  Magnante, "Understanding the accommodating intraocular lens," J. Cataract. Refract. Surg. 29, 2284-2287 (2003).
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H.  Cheng, J. K.  Barnett, A. S.  Vilupuru, J. D.  Marsack, S.  Kasthurirangan, R. A.  Applegate, and A.  Roorda, "A population study on changes in wave aberrations with accommodation," J. Vision 4, 272-280 (2004).
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M. J.  Stephen and J. P.  Straley, "Physics of liquid crystals," Rev. Mod. Phys. 46, 617-704 (1974).
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T.  Oshika, T.  Mimura, S.  Tanaka, Sh.  Amano, M.  Fukuyama, F.  Yoshitomi, N.  Maeda, T.  Fujikado, Y.  Hirohara, and T.  Mihashi, "Apperent accommodation and corneal wavefront aberration in pseudophakic eyes," Invest. Ophthalmol. Vis. Sci. 43, 2882-2886 (2002).
[PubMed]

Thut, G.

G.  Thut, A.  Nietzel, S. A.  Brandt, and A.  Pascual-Leone, "Alpha-band electroencephalographic activity over occipital cortex indexes visuospatial attention bias and predicts visual target detection," J. Neuroscience 26, 9494-9502 (2006).
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S. D.  McLeod, V.  Portney, and A.  Ting, "A dual optic accommodating foldable intraocular lens," Br. J. Ophthalmol. 87, 1083-1085 (2005).
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H.  Cheng, J. K.  Barnett, A. S.  Vilupuru, J. D.  Marsack, S.  Kasthurirangan, R. A.  Applegate, and A.  Roorda, "A population study on changes in wave aberrations with accommodation," J. Vision 4, 272-280 (2004).
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M. Yu.  Loktev, V. N.  Belopukhov, F. L.  Vladimirov, G. V.  Vdovin, G. D.  Love, and A. F.  Naumov, "Wave front control systems based on modal liquid crystal lenses," Rev. Sci. Instrum. 71, 3290-3297 (2000).
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Williams, D. R.

Wollensak, J. A.

T.  Walkow, A.  Liekfeld, N.  Anders, D. T.  Pham, C.  Hartmann, and J. A.  Wollensak "A prospective evaluation of a diffractive versus refractive designed multifocal intraocular lenses. Visual and refractive comparison," Ophthalm. 104, 1380-1386 (1997).

Yoon, G.-Y.

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T.  Oshika, T.  Mimura, S.  Tanaka, Sh.  Amano, M.  Fukuyama, F.  Yoshitomi, N.  Maeda, T.  Fujikado, Y.  Hirohara, and T.  Mihashi, "Apperent accommodation and corneal wavefront aberration in pseudophakic eyes," Invest. Ophthalmol. Vis. Sci. 43, 2882-2886 (2002).
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S. P.  Kotova, M. Yu.  Kvashnin, M. A.  Rakhmatulin, O. A. Zayakin, I. G.  Guralnik, N. A.  Klimov, P.  Clark, G. D.  Love, A. F.  Naumov, C. D.  Saunter, M. Yu.  Loktev, G. V.  Vdovin, and L. V.  Toporkova, "Modal liquid crystal wavefront corrector," Opt. Express 10, 1258-1272 (2002).
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T. L.  Kelly, A. F.  Naumov, M. Yu.  Loktev, M. A.  Rakhmatulin, and O. A.  Zayakin, "Focusing of astigmatic laser diode beam by combination of adaptive liquid crystal lenses," Opt. Commun. 181, 295-301 (2000).
[CrossRef]

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A.  Rana, D.  Miller, and P.  Magnante, "Understanding the accommodating intraocular lens," J. Cataract. Refract. Surg. 29, 2284-2287 (2003).
[CrossRef]

J. Neuroscience

G.  Thut, A.  Nietzel, S. A.  Brandt, and A.  Pascual-Leone, "Alpha-band electroencephalographic activity over occipital cortex indexes visuospatial attention bias and predicts visual target detection," J. Neuroscience 26, 9494-9502 (2006).
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Ophthalm.

T.  Walkow, A.  Liekfeld, N.  Anders, D. T.  Pham, C.  Hartmann, and J. A.  Wollensak "A prospective evaluation of a diffractive versus refractive designed multifocal intraocular lenses. Visual and refractive comparison," Ophthalm. 104, 1380-1386 (1997).

Opt. Commun.

T. L.  Kelly, A. F.  Naumov, M. Yu.  Loktev, M. A.  Rakhmatulin, and O. A.  Zayakin, "Focusing of astigmatic laser diode beam by combination of adaptive liquid crystal lenses," Opt. Commun. 181, 295-301 (2000).
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Opt. Express

Opt. Lett.

Rev. Mod. Phys.

M. J.  Stephen and J. P.  Straley, "Physics of liquid crystals," Rev. Mod. Phys. 46, 617-704 (1974).
[CrossRef]

Rev. Sci. Instrum.

M. Yu.  Loktev, V. N.  Belopukhov, F. L.  Vladimirov, G. V.  Vdovin, G. D.  Love, and A. F.  Naumov, "Wave front control systems based on modal liquid crystal lenses," Rev. Sci. Instrum. 71, 3290-3297 (2000).
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J. A.  Mordi and K. J.  Ciuffreda, "Dynamic aspects of accommodation: age and presbyopia," Vision Res. 44, 591-601 (2004).
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Figures (9)

Fig. 1.
Fig. 1.

The wirelessly-controlled LC lens: (a) cross section of the LC modal corrector, (b) photograph. 1, glass (BK7) substrates; 2, ITO low-ohmic layer; 3, liquid crystal; 4, contact; 5, high-ohmic layer; R, rectifying diode; A, antenna.

Fig. 2.
Fig. 2.

Wireless RF control link. FG, function generator; RFO, radio-frequency oscillator; M, modulator; A, amplifier; TA, transmitting antenna; RA, receiving antenna, D, demodulator.

Fig. 3.
Fig. 3.

Transfer characteristics of the RF link at different driving voltages U LC of the LC lens.

Fig. 4.
Fig. 4.

Experimental arrangement for measuring aberrations produced by the LC lens. Inset shows the wireless link. 1, glass walls of the water cell; 2, distilled water; L1, L2, lenses; TA, transmitting antenna; RA, receiving antenna.

Fig. 5.
Fig. 5.

Dependence of the LC lens aberrations (defocus a 4, spherical aberration a 11) on the modulation frequency F at fixed voltages U LC across the LC modal corrector: (a) 2.12 V (rms), (b) 2.83 V (rms), (c) 3.53 V (rms), (d) 4.24 V (rms).

Fig. 6.
Fig. 6.

Dependences of the optimal modulation voltage U m (and corresponding voltage U LC applied to the LC lens) and the modulation frequency F versus focusing power variation ΔΦ. Interferograms of the LC lens were simulated using the measured aberrations.

Fig. 7.
Fig. 7.

Transient dynamics of the LC lens aberrations (defocus a 4, spherical aberration a 11) at: (a) periodic turning on (20 s) and off (20 s) of a control voltage U LC=2.83 V (rms), F changes stepwise from 4 kHz to 14 kHz by 2 kHz every 40 s; (b) two-level stepwise frequency and amplitude control. The upper diagrams represent U LC sequences.

Fig. 8.
Fig. 8.

LC lens phase cross sections and the corresponding interferograms obtained in a Mach-Zehnder interferometer at: (a) a fixed frequency F=10.5 kHz and different U LC, (b) a fixed voltage U LC=2.12 V (rms) and various F.

Fig. 9.
Fig. 9.

Monochromatic MTFs of the model eye with the LC modal lens: (a) based on the measured 11 aberration coefficients of the lens prototype accommodated at +2.4 D, (b) optimized shape of the LC lens glass substrates.

Equations (1)

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z = S ( x , y ) = r 2 R { 1 + 1 ( 1 + k ) × ( r R ) 2 } ,

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