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818
I
nt. J. Nanotechnol., Vol. 12, Nos. 10/11/12, 2015
Copyright © 2015 Inderscience Enterprises Ltd.
Miniature tunable Alvarez lens driven by piezo
actuator
Yongchao Zou, Wei Zhang, Feng Tian,
Fook Siong Chau and Guangya Zhou*
Micro and Nano Systems Initiative,
Department of Mechanical Engineering,
National University of Singapore,
9 Engineering Drive 1, 117576, Singapore
Fax: +65-6516-1459
Email: zouyongchao@nus.edu.sg
Email: mpezw@nus.edu.sg
Email: mpetf@nus.edu.sg
Email: mpecfs@nus.edu.sg
Email: mpezgy@nus.edu.sg
*Corresponding author
Abstract: 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.
Keywords: optical design; nanotechnology; micro-optical devices; lenses.
Reference to this paper should be made as follows: Zou, Y., Zhang, W.,
Tian, F., Chau, F.S. and Zhou, G. (2015) ‘Miniature tunable Alvarez lens
driven by piezo actuator’, Int. J. Nanotechnol., Vol. 12, Nos. 10/11/12,
pp.818–828.
Biographical notes: Yongchao Zou received the BS in Optical Engineering
from Huazhong University of Science and Technology, Wuhan, China in 2009
and he is currently working towards the PhD in the Department of Mechanical
Engineering at National University of Singapore, Singapore. His research
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iniature tunable Alvarez lens driven by piezo actuato
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interests involve the design, simulation, fabrication and testing technologies of
devices in MEMS/miniature imaging systems.
Wei Zhang studied Mechanical Engineering from 2001 to 2008 at
Northwestern Polytechnical University, China, where she finished her Master’s
degree on the Design of a Microscope for Testing Microfluidics, in May 2008.
In September 2008, she joined the Micro-Optics Lab at the University of
Freiburg, where she completed her PhD, researching tunable micro lenses.
From June 2013, she became a Postdoctoral Fellow of MNSI lab of Mechanical
Department at the National University of Singapore. Her research interest
focuses on tunable micro optics systems.
Feng Tian received the BE in Electrical Engineering from Yanshan University,
Qinhuangdao, China, in 2004, and the PhD in Optical Engineering from
Zhejiang University, Hangzhou, China, in 2010. Currently, he is a Research
Fellow of the Department of Mechanical Engineering at the National
University of Singapore, Singapore. His main research interests include nano-
photonics, NEMS device and opto-mechanics.
Fook Siong Chau received the BS (Eng.) and PhD from the University of
Nottingham, Nottingham, UK, in 1974 and 1978, respectively. He is currently
an Associate Professor in the Department of Mechanical Engineering, National
University of Singapore, Singapore, where he heads the Applied Mechanics
Academic Group. His main research interests are in the development and
applications of optical techniques for non-destructive evaluation of components
and the modelling, simulation and characterisation of microsystems,
particularly bio-MEMS and MOEMS.
Guangya Zhou received the BE and PhD in Optical Engineering from Zhejiang
University, Hangzhou, China, in 1992 and 1997, respectively. He joined the
National University of Singapore, Singapore, in 2001 as a Research Fellow,
where he is currently an Associate Professor in the Department of Mechanical
Engineering. His main research interests include microoptics, diffractive optics,
MEMS devices for optical applications and Nanophotonics.
1 Introduction
By shifting a pair of optical elements with cubic surfaces in a direction perpendicular to
the optical axis, Alvarez lenses can achieve large varifocal ranges with slight component
movements [1,2]. Compared with conventional lens groups or liquid tunable lenses,
Alvarez lenses offer us a series of advantages including compact structures, ease of
packaging and high stability against external disturbances [3–5]. However, the first
prototype of Alvarez lenses was not demonstrated until recent years when the fabrication
challenge of freeform surface was overcome by modern micro-manufacturing
technologies. The first materialised Alvarez lens was experimentally demonstrated in
2000, which was fabricated with the technique of photolithography [6]. Since then,
with the help of various micromachining techniques, a number of Alvarez lenses were
experimentally presented, including both single-element and array-arranged devices
[7–12]. Our group focuses on the miniaturisation and integration of Alvarez lens
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by merging them with microelectromechanical systems (MEMS) technologies. 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].
In this paper, we report a novel miniature tunable Alvarez lens driven by one piezo
actuator. In mechanical design, compared with the reported comb-drive actuator
fabricated from a SOI wafer, one piezo actuator integrated with a compact mechanical
displacement amplifier is adopted. In optical design, a new theoretical method for lens
coefficient selection is used. Both simulation and experimental results are covered in this
paper.
2 Optical and mechanical design
2.1 Optical design of Alvarez lens
According to Alvarez’s concept, the lens consists of a pair of elements with the same
cubic polynomial surface profiles, as shown in Figure 1. The surface profile is governed
by [1,2]:
23
1
3
tAxy x DxE
=+++
(1)
where A, D and E are constants to be determined. When the optical elements stay at the
initial position as shown in Figure 1(a), the whole structure behaves as an optical plate
and no optical power is generated. 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)(x2 + y2). Obviously, this phase delay makes the whole configuration
equivalent to an optical lens with a focal length given by [14,15]:
()
1
41
fAn
δ
=− (2)
where n is the refractive index of the material. It can be found that tunable focal lengths
can be achieved by shifting the lens elements with different displacements. 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]:
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1/3
2
max 0 0
max
max
1/3 1/ 3
2
00
optimal min max
1/3
2
00
min
min
min
3
if
32 4
93
if
16 4
3
if
32 4
Ag g
A
Ag g
DA
gg
A
A
δδ
δ
δδ
δδ
δ
−− ≤
=− < <
−− ≥
(3)
where δmax and δmin denote the maximum and minimum displacements of lens elements,
respectively.
Figure 1 Model of Alvarez lenses. Lens elements are (a) at initial position and (b) shifted
with lateral displacements. R denotes the radius of the optical stop, δ presents the
displacement in x direction, g0 expresses the initial centre-to-centre gap between
two elements and f is the equivalent focal length (see online version for colours)
(a) (b)
In our design, we expect to achieve a focal length tuning range within several millimetres
to tens of millimetres suitable for most miniature optical systems. Meanwhile, we pointed
out that the value of A should be as small as possible for improving lens performance
[15]. Hence, taking into account the tuning range of interest and the maximum
displacement we can get, A is determined as 0.075 mm–2. In addition, the initial gap
between two elements g0 is set as 0.3 mm in view of the assembly difficulties. On the
other side, the actuator is expected to provide a maximum displacement of 100 µm in our
mechanical design. Accordingly, D is determined as –0.15 by equation (3). In addition,
we set an initial offset of 100 µm to these two optical elements to shift the maximum
focal length from infinite to a meaningful value located at the range of interest.
Therefore, these two lens elements would be laterally shifted from 100 µm to 200 µm by
the actuator.
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Suppose the refractive index of the lens material is 1.56, Figure 2 presents the
simulation results for this specified Alvarez lens from aspects of ray-tracing spot
diagram, ray aberration fan, modulation-transfer-function (MTF) curves with Strehl ratio
(SR) as well as wavefront PV values and image simulation with three different lateral
displacements. It can be noted that a focal-length tuning range of 30.51 mm to 64.39 mm
is achieved. Moreover, in view of the golden rule of optical design (for an ideal lens
system, the PV aberration should be <λ/4 or the size of the spot diagram should be equal
or smaller than that of the Airy disc [17]), it can be found that the lens behaves as a
near-diffraction-limit device within the whole tuning range. In addition, no obvious
increasing blurring is observed within the three simulated images, which means there is
no noticeable lens performance degradation with the increase of lateral displacements.
Figure 2 Lens performance at different focal lengths. (a) Ray-tracing spot diagrams, (b) ray
aberration fans, (c) MTF curves with equivalent focal length (FL), Strehl ratios (SR)
and wavefront PV values (PV) and (d) image simulations of the lens with lateral
displacement set as 0.1 mm, 0.15 mm and 0.2 mm. The x axis of (b) is the normalised
entrance pupil coordinate while the y axis is the ray aberrations. ‘px’ and ‘py’ denote
the sagittal fan and tangential fan, respectively. The field height of the input image in
(d) is set as 15°. From the top to the bottom, the lateral displacement for each row is
0.1 mm, 0.15 mm and 0.2 mm, respectively (see online version for colours)
(a) (b) (c) (d)
2.2 Mechanical design of actuators
As shown in Figure 3, to obtain a maximum displacement of 100 µm for lens elements, a
commercially available piezo actuator (Module P-883.51 from PI Instrumente,
dimension: 3 mm × 3 mm × 18 mm, maximum output displacement: 18 µm) [18] is
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iniature tunable Alvarez lens driven by piezo actuato
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integrated into a flexure-guided structure [19] for displacement amplification.
A symmetric four-bar topology is employed to amplify the original displacements from
the piezo actuator and drive a folded-beam suspension with a platform for lens mounting.
Figure 3(a) and (b) demonstrate different designs for ‘pulling’ and ‘pushing’ lens
elements in x and –x directions. Except driving directions as indicated by the arrows in
Figure 3(a) and (b), the dimensions of the four-bar topologies of both the ‘pulling’ and
‘pushing’ structures are critically consistent to make sure that the single piezo actuator
can drive two lens elements to move synchronously with symmetric displacements.
Figure 3 Mechanical design of Alvarez lens. Mechanical displacement amplifiers for
(a) ‘pulling’ and (b) ‘pushing’ lens element and (c) assembly of Alvarez lens
(see online version for colours)
(a) (b) (c)
Figure 4 Mechanical performance of actuators. (a) Relation between input and output
dispalcements and (b) Von Mises stress distribution with the maximum dispalcement
(see online version for colours)
(a) (b)
Dimensions of the mechanical displacement amplifier are optimised to achieve a balance
between displacement amplification ratio and mechanical strength. The final dimensions
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are indicated in Figure 3(b) and the device thickness is 0.3 mm. On the basis of finite
element analysis, Figure 4 shows the mechanical performance of the displacement
amplifiers which is made of stainless steel 304. Specifically, Figure 4(a) presents the
relation between the output displacements of the mechanical displacement amplifiers and
that of the piezo actuator, from which we can find that an amplification ratio of ~16 is
achieved. In addition, the ‘pushing’ and ‘pulling’ displacements are consistent in the
magnitude with a same input displacement of the piezo actuator, which ensures the
symmetric moving of two lens elements. Figure 4(b) demonstrates the distribution of Von
Mises stress with the maximum displacement, from which we can find that the maximum
stress occurs in the hinges and is controlled below 250 MPa to avoid yielding.
3 Experimental results
3.1 Device fabrication and assembly
For lens-element fabrication, we follow our previous process reported in reference [15].
To begin with, a single-point diamond turning technique with a fabrication accuracy of
few nanometres is utilised to create the desired freeform surface on an aluminium
substrate. For convenience of lens-element alignment and mounting, during pre-
processing of the aluminium substrate, one square base with two semicircle grooves
and a larger one is created under the cylinder, on top of which the freeform surface is to
be turned, as shown in Figure 5(a). After that, a PDMS mould with an inverse pattern,
as illustrated in Figure 5(b), is fabricated with a standard replication process from the
aluminium mould. Finally, lens elements are achieved by another replication process
from the PDMS mould. Specifically, a UV curable optical adhesive (NOA83H, Norland,
USA) is filled in the PDMS mould and then another optical flat PDMS slab obtained by
replicating the surface of one prism is used to seal the concave mould to ensure one
optical flat surface for the lens element. The whole setup is then exposed to UV light for
at least half an hour for hardening. Owing to the low surface energy of PDMS, the
hardened lens element is quite easy to be separated from the PDMS mould without any
surface disfigurements, as demonstrated in Figure 5(c). The mechanical displacement
amplifiers are fabricated by the electric discharge machining (EDM) technique, which
guarantees a minimum feature size of 0.2 mm and a machining tolerance of 10 µm. A
commercially available stainless-steel-304 plate with a thickness of 0.3 mm is adopted as
the raw substrate. Figure 6(a) illustrates the fabricated mechanical displacement
amplifiers after EDM. With the help of a microscope, lens elements are mounted to the
platforms of displacement amplifiers by adhesives, as shown in Figure 6(b). Lastly, the
piezo actuator is inset to the stacked mechanical displacement amplifiers, forming the
final Alvarez lens as presented in Figure 6(c). To mount the piezo actuator stably, a
preload for the piezo actuator is created by properly determining the mechanical
dimensions of the V-shape structure which is used to amplify the output expansion of the
piezo actuator. More specifically, the distance between both ends of the V-shape structure
is designed as 10 µm shorter than the length of the piezo actuator. Therefore, the piezo
actuator can be mounted tightly. In addition, four washers with a thickness of 0.3 mm are
used to provide the initial gap g0 between the two layers. When the two lens elements are
stacked, these four washers are inset into the corners to separate the lens elements with
the expected gap.
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Figure 5 Fabrication process of lens elements. (a) Aluminium mould, (b) PDMS mould
and (c) final lens element (see online version for colours)
(a) (b) (c)
Figure 6 Fabrication of mechanical displacement amplifiers and assembly of Alvarez lens.
Mechanical displacement amplifiers (a) before lens mounting and (b) after lens
mounting and (c) assembled Alvarez lens (see online version for colours)
3.2 Device testing and characterisation
We firstly test the performance of the mechanical displacement amplifiers. As shown
in Figure 7, the static displacements of both the ‘pushing’ and ‘pulling’ displacement
amplifiers vs. input DC voltages are characterised. Specifically, when the amplifiers are
tested separately, with an input voltage of 130 V applied to the piezo actuator, the
‘pushing’ amplifier provides us a maximum displacement of 149.1 µm while the ‘pulling’
one offers us a displacement of 151.4 µm at most, as shown by the curves with solid
symbols in Figure 7. However, when these two amplifiers are stacked together, the
maximum output displacements of the amplifiers decreases to 99.4 µm and 100.9 µm,
respectively, as presented by the curves with hollow symbols. The reason is that the
maximum output displacement of the piezo actuator is inversely proportional to the
external load and hence reduced by the combined two mechanical amplifiers. In addition,
the displacement vs. voltage relation is nearly linear and displacements of ‘pushing’ and
‘pulling’ are symmetric. Clearly, these two amplifiers are capable to drive the lens
elements to move symmetrically in x/–x direction up to ~100 µm with an input voltage
tuned from 0 V to 130 V.
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Figure 7 Relation between output displacements and input voltages (see online version
for colours)
Figure 8 Testing of Alvarez lens. (a) Relation between equivalent focal lengths and input
voltages, (b) optical setup for lens-imaging testing, captured images with input voltages
equal to (c) 0 V, (d) 65 V and (e) 130 V (see online version for colours)
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Thereafter, the performance of the Alvarez lens is characterised. Figure 8(a)
demonstrates the ability of focal length tuning and Figure 8(c)–(e) present the imaging
performance at different input voltages (i.e., different focal lengths). More specifically, a
collimated laser system and one high-resolution CCD controlled by a linear stage are
used to measure the focal length at a particular input voltage, as shown in the inset figure
in Figure 8(a). By determining the position of the focused spot of the collimated beam
passing through Alvarez lens, the relation between the equivalent focal lengths and input
voltages can be obtained. As shown in Figure 8(a), with the help of an original offset of
100 µm as mentioned above, a focal length tuning range from ~28 mm to 65 mm
(~2.3 times) is experimentally demonstrated. Lastly, an optical imaging system is set up
to test the imaging capability of this Alvarez lens, as illustrated in Figure 8(b).
One USAF 1951 resolution target is placed in front of the lens as the object and then a
microscope integrated with a high-resolution CCD is utilised to capture the images. The
distance between the Alvarez lens and microscope is fixed and positions of the target
are adjusted for various input voltages to form clear images at CCD. As shown in
Figure 8(c)–(e), optical magnifications of images vary evidently at different focal lengths.
Furthermore, no obvious distortion and blurring occur within these presented images,
and, particularly, there is no visible performance degradation within the whole tuning
range.
4 Conclusions
In this paper, we demonstrate a miniature tunable Alvarez lens driven by a piezo actuator.
Dynamic tuning of focal length ~2.3 times (from 28 mm to 65 mm) with excellent
imaging quality is experimentally presented, which agrees well with simulation results.
Both the simulation and experimental results lead us to conclude that such an Alvarez
lens driven by a piezo actuator integrated with compact mechanical displacement
amplifiers is able to vary focal lengths continuously within a large range. Furthermore,
the proposed method for lens coefficient selection is beneficial to lens imaging quality.
Compared with reported miniature Alvarez lenses, this one is superior in either tuning
ranges or image qualities. Based on this concept, Alvarez lenses with larger tuning ranges
and more compact dimensions are potentially possible with further optimisation of
optical and mechanical design.
Acknowledgements
This work is supported by Singapore MOE research grant R-265-000-496-112.
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