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Alignment tolerances and optimal design of MEMS-driven Alvarez lenses
Abstract and Figures
The Alvarez lens offers variable focal lengths through lateral shifting
of two lens elements. Imaging characteristics and alignment tolerances
of the Alvarez lens are studied, and one analytic method for optimal
coefficient selection is proposed. Results show that the performance of
the Alvarez lens decays with increasing element displacements, and
dominant optical aberrations are spherical aberrations, defocus and
coma. The lens performance is most sensitive to misalignments in the y
direction. If the tolerance is defined as a 10% rise of normalized RMS
spot radius, the misalignment should be smaller than 0.01 mm and the
tilt angle about any axis can never exceed 1°. By adopting the new
coefficient-selection method, lens performance is improved evidently and
the degrading trend with increasing displacements is mitigated markedly.
In addition, alignment tolerance is increased as well.
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... We expect to extend applications of Alvarez lenses from existing tunable glasses [13] to future miniature optical systems, including accompanied cameras of portable electronic devices, surveillance systems and medical imaging systems. In 2013, we first demonstrated a miniaturised MEMS-driven tunable Alvarez lens [14], and, subsequently, a theoretical method for lens coefficient selection was proposed to improve the lens performance [15]. ...
... Nevertheless, once there is a lateral displacement δ in x/-x direction for each optical element as shown in Figure 1(b), the corresponding thickness of each lens element in the optical path is changed, leading to an overall phase delay of -2A(n -1)(x 2 + y 2 ). Obviously, this phase delay makes the whole configuration equivalent to an optical lens with a focal length given by [14,15]: ...
... From equations (1) and (2), it can be noted that constants D and E have no direct effect on focal lengths. Alvarez pointed out that D can be used to reduce the overall thickness of the lens element for performance optimisation, based on which Barbero [16] proposed a method to determine the value of D. However, we found that it is the air gap between two optical elements rather than the overall element thickness that is the dominant factor affecting lens performance, according to which we introduced a new method to determine D, as shown below [15]: ...
Alvarez lenses can achieve focal length tuning by shifting a pair of optical elements with cubic surfaces in a direction perpendicular to the optical axis, offering us a series of advantages including compact structures with large varifocal ranges, ease of packaging, and high stability against external disturbances. In this paper, we present a miniature tunable Alvarez lens driven by a piezo actuator integrated with a compact mechanical displacement amplifier. A flexure-guided structure with a displacement amplification ratio of ~16 is adopted in the mechanical design and a novel method to determine surface profile coefficients in the optical design is employed to optimise the lens performance. Results show that the displacement vs. voltage curve of the mechanical displacement amplifier is nearly linear. A maximum displacement of ~100 µm is obtained with a 130 V input voltage applied to the piezo actuator. Dynamic tuning of focal length about 2.3 times (from 28 mm to 65 mm) is experimentally demonstrated with the assembled Alvarez lens. Images of a USAF 1951 resolution target through the lens at different focal lengths are captured by a high-resolution camera to evaluate the lens performance. No obvious distortion or blurring of images or performance degradation while tuning are noticed within the whole varifocal range.
... The optical power is modulated by the lateral displacement, which means that various focal lengths can be achieved by shifting the elements to different lateral positions. Alvarez initially pointed out that the two optical elements should have identical freeform surface profiles and be governed by one and the same cubic polynomial [20,21]. However, recent studies have indicated that polynomials with higher-order terms should be employed to describe the freeform surface for optimal lens performance [22]. ...
... The design constraints and goals are summarized in Table 1. The optimization process is performed in ZEMAX and starts with the conventional Alvarez cubic expression, where the coefficients of x 3 , xy 2 and x are initially determined according to the required optical power tuning range and maximum displacement of the actuators [21][22][23]. The axial and off-axial performances are optimized in sequence by gradually increasing the number of terms in the two polynomials set as variables. ...
... Reducing the f-number values can help to improve the resolution of the lens. Moreover, the NRSRs (normalized RMS spot radius, defined as the ratio of ray-tracing spot RMS size to that of the Airy spot) [21] are controlled to be below 1 while the SRs (Strehl ratio) are maintained at values greater than 0.9 within the whole tuning range, which indicates that the designed lens behaves as a near-diffractionlimited device. Figure 3 further reveals the performance of the tunable lens by presenting its imaging simulation results, where we find that there is no obvious distortion or blurring. ...
The design, fabrication and characterization of a miniature adjustable-focus endoscope are reported. Such an endoscope consists of a solid tunable lens for optical power tuning, two slender piezoelectric benders for laterally moving the lens elements perpendicular to the optical axis, and an image fiber bundle for image transmission. Both optical and mechanical designs are presented in this paper. Dynamic tuning of optical powers from about 135 diopters to about 205 diopters is experimentally achieved from the solid tunable lens, which contains two freeform surfaces governed by 6-degree polynomials and optimized by ray tracing studies. Results show that there is no obvious distortion or blurring in the images obtained, and the recorded resolution of the lens reaches about 30 line pairs per mm. Three test targets located at various object distances of 20 mm, 50 mm and 150 mm are focused individually by the endoscope by applying different driving DC voltages to demonstrate its adjustable-focus capability.
... However, the imaging performance of this MEMS Alvarez lens was not satisfactory. In order to improve its performance, a theoretical study on the imaging characteristics and alignment error tolerances of Alvarez lenses was conducted and reported in the same year [14]. An analytical method for freeform surface coefficient selection to achieve an enhanced imaging performance was proposed and compared with the results reported by S. Barbero in 2012 [15]. ...
... With a conventional two-element Alvarez lens, it is inevitable that, for a given limited displacement δ from a micro/miniature actuator, A must be large enough to achieve a large optical power variation. However, a large A leads to a poor imaging performance [14]. Hence, it is quite challenging for the conventional two-element lens configuration to achieve a desirable optical power variation and at the same time maintain a satisfactory imaging performance due to the limited lateral displacements offered by micro/mini actuators [17], [18] and the restrained A values from the practical consideration of lens performance. ...
... This is clear by comparing Eq. (3) and (4). However, a larger value of A brings us deteriorated lens performance [14]. Thanks to the accumulation of optical powers from multiple pairs of elements in a multi-element Alvarez lens, the value of A can be greatly reduced. ...
The design, fabrication, and characterization of miniature tunable multi-element Alvarez lenses are reported in this paper. Optical design including optimal lens configuration and surface coefficient selection, together with the effects of assembly errors on lens performances, is numerically studied using the ray-tracing method. To demonstrate the working principle, a four-element Alvarez lens is experimentally demonstrated in comparison with a conventional two-element lens. Lens elements are fabricated by a diamond turning and replication molding process and are driven by two piezo actuators integrated with displacement amplification mechanisms. Dynamic tuning of optical power is experimentally determined to be around 43.2 diopters (from 50.9 to 94.1 diopters) for the four-element lens and 21.1 diopters (from 25.3 to 46.4 diopters) for the two-element lens. Lens imaging performance is also characterized comparatively, indicated by images of the 1951 USAF resolution test chart through the lens and Modulation Transfer Function curves calculated accordingly.
... Recently, there is a revival of interest in Alvarez lenses since modern micro machining techniques allo w the realizat ion of high-performance and low-cost freeform optical surfaces and, accordingly, overcome the manufacturing restrictions of A lvarez lens elements [1][2][3][4][5]. A series of applications based on the concept of Alvarez lens were reported [2][3][4]. ...
... A series of applications based on the concept of Alvarez lens were reported [2][3][4]. Our group focuses on the min imization and integration of Alvarez lenses, and explores their applications in miniature optical systems including cameras in portable electronic devices, medical imaging systems and surveillance systems [4,5]. To achieve a focal length tuning range from a few millimeters to tens of millimeters suitable for the above mentioned miniature imaging systems, a challenging problem for min iature Alvarez lenses is the limited optical power variat ion due to the small lateral displacements offered by micro/min i actuators. ...
We report for the first time a miniature mult i-element Alvarez lens driven by a p iezo actuator integrated with mechanical displacement amplifiers. With an input DC voltage varying fro m 0 to 130 V, a dynamic focal length tuning range from 16 mm to 32 mm is experimentally demonstrated. There is no noticeable distortion in presented images formed by such an Alvarez lens. Within the whole tuning range, no performance degradation during operation is observed.
... The approach has now been modified with two mirrors instead of refractive plates [ Fig. 1(C)]. By doing so, one major disadvantages of refractive Alvarez systems is avoided: the dispersion of the materials with wavelength [25]. On the other hand, a larger finite distance between the Alvarez components causes a certain dependency between the mirror surfaces. ...
A novel, to the best of our knowledge, dual-state reflective optical relay system based on the Alvarez system is proposed, which can be used for remote sensing applications. By keeping the image and pupil positions constant, it can be combined with a telescope to achieve two different magnifications. As a compact structure with only two moving parts, freeform optical mirrors and a nearly diffraction limited performance for the infrared wavelength 8 µm make it an attractive subsystem for space applications. Different design tradeoffs and the preferred layout properties are discussed in detail.
... With the development of freeform tooling techniques like precision diamond milling and turning or lithography for diffractive optical elements, those Alvarez-Lohmann (A-L) lenses became a popular research topic. Applications range from ophtalmologic devices like tunable intraocular lenses [4,5] and tunable low-cost glasses [6,7] to compact focusing optics [8,9], tuning elements in hyperspectral imaging systems [10], microscope optics [11], zoom systems [12][13][14][15], investigating depth of field effects [16][17][18], or for the compact implementation of (fractional) Fourier transformation systems [19]. Already, Lohmann and Paris noted that this principle is not limited to the tuning of quadratic functions [20]. ...
In this work, we apply the Alvarez–Lohmann principle for varifocal lenses to diffractive off-axis elements tuned by rotation. Two methods to combine multiple elements into arrays are presented. Further, we show that inverse phase sections result from a 2 π ambiguity of the rotation. Quantization techniques are applied to eliminate these unwanted sections. As proof of concept, a retrofocus lens using a tunable diffractive lens array is presented.
A remarkable feature of Alvarez lenses is that a wide focal length tuning range can be achieved using lateral displacement rather than commonly used axial translation, thus, reducing the overall length of varifocal imaging systems. Here, we present novel lens elements based on Alvarez lenses actuated by a dielectric elastomer (DE). The proposed lens elements are composed of the varifocal component and the scanning component. Based on the proposed lens elements, an imaging system is built to realize ultra-wide varifocal imaging with a selectable region of interest. The lens elements have a variable focus function based on an Alvarez lens structure and a DE actuator and a scanning function based on the DE-based four-quadrant actuators. The large deformation generated by the DE actuators permits the lateral displacement of the Alvarez lenses up to 1.145 mm. The focal length variation of the proposed varifocal component is up to 30.5 times, where the maximum focal length is 181 mm and the minimum focal length is 5.94 mm. The rise and fall times of the varifocal component are 160 ms and 295 ms, respectively. By applying different voltages on four-quadrant actuators, the scanning component allows the varifocal component to move in different directions and endows the varifocal component with a selectable region of interest imaging capability. The scanning range of the scanning component is 17.57°. The imaging resolution of the imaging system is approximately 181 lp/mm. The system developed in the current study has the potential to be used in consumer electronics, endoscopy, and microscopy in the future.
We report an ultra-compact optical zoom endoscope containing two tunable Alvarez lenses. The two tunable lenses are controlled synchronously by piezoelectric benders to move in directions perpendicular to the optical axis to achieve optical zoom while keeping images in clear focus without moving the scope. The piezoelectric benders are arranged circumferentially surrounding the endoscope optics with a diameter about 2 mm, which results in an ultra-compact form. The demonstrated endoscope is capable of optical zoom close to 3 × from field of view (FOV) 50° to 18° continuously with the required movements for its constituent optical elements less than 110 μm. Such optical zoom endoscopes may find their potential uses in healthcare and industrial inspection systems.
We present an anamorphotic imaging system with a separately tunable magnification in horizontal and vertical direction in order to change the aspect ratio of the image. The design is based on cylindrical membrane lenses made of aluminum nitride with tunable focal power to realize a vario system without moving elements. We demonstrate the design concept of cylindrical lenses with a very compact geometry, the optical system as well as experimental results.
The need for affordable and sustainable ophthalmic systems for measurement and correction of refraction is well recognized. Power-adjustable spectacles based on the Alvarez principle (transversal lateral movement of two lenses) have emerged as an innovative technology for this purpose. Within this framework, our aim is to design a new power-adjustable sphero-cylindrical refractor. The system is comprised of two lenses and three independent lateral movements. The lenses have a planar and a third-degree polynomial surface. They are arranged with their planar surfaces in contact, so that the incoming light is only refracted by two surfaces. First, we present the theory of such a system. Second, we propose an optical design methodology. Third, we provide a design example capable of measuring sphere powers ranging from −5.00 D to +5.00 D and cross-cylinders from −2.00 D to 2.00 D. Finally, a prototype of the lenses was manufactured using free-form machining.
The meniscus between two immiscible liquids can be used as an optical lens. A change in curvature of this meniscus by electrowetting leads to a change in focal distance. It is demonstrated that two liquids in a tube form a self-centered lens with a high optical quality. The motion of the lens during a focusing action was studied by observation through the transparent tube wall. Finally, a miniature achromatic camera module was designed and constructed based on this adjustable lens, showing that it is excellently suited for use in portable applications.
The theory of third-order aberrations for a system of rotationally symmetric thin tunable-focus fluidic membrane lenses with parabolic surfaces is described. A complex analysis of the third-order design of tunable fluidic lenses is performed considering all types of primary aberrations. Moreover, formulas are derived for the calculation of the change of aberration coefficients of the parabolic tunable fluidic membrane lens with respect to the wavelength. It is shown that spherical aberration of a simple tunable-focus fluidic membrane lens with parabolic surfaces can be corrected, which is not possible with a classical spherical lens. The presented analysis is explained on examples. Derived formulas make possible to calculate parameters of optical systems with fluidic membrane lenses with small residual aberrations.
A miniature solid-state varifocal lens based on Alvarez principle with lens elements having free-form surfaces is reported. The Alvarez lens elements are implemented with diamond-turning and replication molding processes. They are integrated with electrostatically-driven MEMS comb-drive actuators fabricated using SOI micromachining. Dynamic tuning of focal length more than 1.5 times (from 3 mm to 4.65 mm) is experimentally demonstrated with only small MEMS-driven lateral movements of 40 μm. Such varifocal lens may be useful in miniature cameras for autofocus and zooming due to its advantages including ease of packaging and fast tuning speed.
A tunable biconvex microlens of 1000 μm diameter is micromachined and
pressure-actuated for variable-focusing applications. The microlens
consists of two thin membranes with reconfigurable shapes: a fluid
chamber and the interconnected microchannel. The back focus length
tuning range is demonstrated from 35 mm to 250 mm and zoom ratio of up
to seven without any mechanical moving components. Aberration
characterization is carried out systematically by the
Shack-Hartmann measurements to clarify the potential adverse
effect associated with focal length tunability, with particular
attention placed on interpretation of the Zernike modes. Experimental
results show that spherical mode (Z13) can be significantly degraded
from -0.037 μm to -0.095 μm by injecting DI water of
1.11-4.43 μL. A facile and cost-efficient approach has been
proposed to compensate spherical aberration based on differential
thickness of elastic membranes. The thickness variation for the biconvex
microlenses can be manipulated as the difference in deformation contour
as well as the resultant surface profile when subjected to uniform
applied pressure. Both ZEMAX simulation and experiment are used to
validate the design concept and search for the optimal thickness ratio.
By injecting the fixed fluid volume of 3.32 μL, the optimal thickness
ratio of 1:4 can be experimentally obtained and the measured spherical
aberration is -0.023 μm, or 56% improvement compared with 1:1
thickness ratio of -0.053 μm. The proposed microlens is robust
and can be used potentially in medical imaging systems, for example, a
dynamic environment and adaptive optics.
While Alvarez lens prototypes have recently been manufactured and tested
for visible wavelengths, there is little discussion of these types of
components for infrared applications in the published literature. We
present and characterize a germanium Alvarez lens for infrared imaging.
Mathematical analysis for determining the required cubic surfaces is
presented, and ray-based and wave-based optical simulations are
performed to confirm and refine the expected variable-focus behavior. As
part of the design study, we examine the effects of effective f-number
of the Alvarez lens and gap between the freeform surfaces on image
quality, modulation transfer function, and Strehl ratio. The germanium
Alvarez lens pair is fabricated through freeform diamond micro-milling,
and characterized using a custom-built imaging test station in the
mid-infrared. The variable-focus and imaging capabilities of this lens
are demonstrated experimentally and compared to predicted results with
good agreement.
We present a comprehensive approach for the design of adjustable-focus lenses based on the
Alvarez principle. The design methodology consists of dividing the lens into two parts: the
inner, optical part, where the optical quality is optimized, and the outer, mechanical
part, connecting the optical part to the frame. For the optical part, we present a
complete optical design methodology to minimize common optical aberrations,
considered in ophthalmic lens design, for different focus adjustments. For the
mechanical part, we show how to extend the lens surfaces to connect the optical
zone with the frame, such that the entire surface is smooth and has acceptable
thickness.
In 1967, Luis Alvarez introduced a novel concept for a focusing lens whereby two transmitting elements with cubic polynomial surfaces yield a composite lens of variable focal length with small lateral shifts. Computer simulations have demonstrated the behavior of these devices, but fabricating the refractive cubic surfaces of the types needed with adequate precision and depth modulation has proven to be challenging using standard methods, and, to the authors' knowledge, Alvarez lens elements have not been previously machined in infrared materials. Recent developments in freeform diamond machining capability have enabled the fabrication of such devices. In this paper, we discuss the fabrication of freeform refractive Alvarez elements in germanium using diamond micro-milling on a five-axis Moore Nanotech® 350FG Freeform Generator. Machining approaches are discussed, and measurements of surface figure and finish are presented. Initial experimental tests of optical performance are also discussed.
A tunable liquid microlens capable of adjusting its focal length and lateral position is demonstrated. The microlens consists of a droplet of a transparent conductive liquid placed on a dielectric substrate with a low surface energy coating. By varying the voltage applied to a set of electrodes positioned underneath of the dielectric substrate, both the position and curvature of the microlens can be reversibly changed. The dependence of the microlens behavior on the properties of the materials involved is experimentally investigated and supported by theoretical calculations. Potential limitations of the microlens performance associated with the contact angle hysteresis and stick–slip phenomena are outlined and possible ways to alleviate them are proposed. Possible extensions of the proposed approach are also discussed. © 2003 American Institute of Physics.
A large-displacement thermal actuator designed for the MEMS pitch-tunable grating device has been designed, simulated, fabricated and characterized in this paper. To avoid the inevitable limits of common thermal actuators developed in the MEMS community, this paper concludes with two design rules and proposes a novel large-displacement thermal actuator based on these two rules. The characterized performance of this thermal actuator shows output displacement at more than 300 µm within the 19 V driving voltage. The grating device with the assistance of this thermal actuator can adjust the pitch continuously from 30 µm to 38 µm with an extension ratio of more than 25%. This thermal actuator could also be integrated easily with other micro devices without the delicate micro-assembly process since its fabrication is compatible with the general semiconductor process.