UC Davis
UC Davis Previously Published Works
Title
Characterization of a bimorph deformable mirror using stroboscopic phase-shifting
interferometry.
Permalink
https://escholarship.org/uc/item/9cs3660x
Journal
Sensors and actuators. A, Physical, 134(1)
ISSN
0924-4247
Authors
Horsley, David A
Park, Hyunkyu
Laut, Sophie P
et al.
Publication Date
2007-02-01
DOI
10.1016/j.sna.2006.04.052
Peer reviewed
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University of California
Sensors and Actuators A 134 (2007) 221–230
Characterization of a bimorph deformable mirror using
stroboscopic phase-shifting interferometry
David A. Horsley
a,∗
, Hyunkyu Park
a
, Sophie P. Laut
b
, John S. Werner
b
a
Department of Mechanical, Aeronautical Engineering, University of California, Davis, United States
b
Department of Ophthalmology, Vision Science, University of California, Davis, United States
Received 24 February 2006; received in revised form 26 April 2006; accepted 27 April 2006
Available online 15 June 2006
Abstract
The static and dynamic characteristics of a bimorph deformable mirror (DM) for use in an adaptive optics system are described. The DM is
a 35-actuator device composed of two disks of lead magnesium niobate (PMN), an electrostrictive ceramic that produces a mechanical strain in
response to an imposed electric field. A custom stroboscopic phase-shifting interferometer was developed to measure the deformation of the mirror
in response to applied voltage. The ability of the mirror to replicate optical aberrations described by the Zernike polynomials was tested as a
measure of the mirror’s static performance. The natural frequencies of the DM were measured up to 20 kHz using both stroboscopic interferometry
as well as a commercial laser Doppler vibrometer (LDV). Interferometric measurements of the DM surface profile were analyzed by fitting the
surface with mode-shapes predicted using classical plate theory for an elastically supported disk. The measured natural frequencies were found to
be in good agreement with the predictions of the theoretical model.
© 2006 Elsevier B.V. All rights reserved.
Keywords: Adaptive optics; Deformable mirrors; Interferometry; Micromechanical devices
1. Introduction
Originally developed to remove atmospheric distortion from
astronomical imaging systems, adaptive optics (AO) has seen
more recent application to ophthalmologic instruments and
free-space optical communication systems. In each of these
applications, the AO system uses a deformable mirror (DM)
to correct for optical aberrations by removing phase distor-
tions from the incident wavefront. Since the existing DM
technology developed for astronomy is expensive and bulky,
recent research has focused on using micro-electromechanical
(MEMS) technology to create a more compact, low-cost DM.
Several MEMS DM designs have been demonstrated, includ-
ing: membrane-based (OKO Technologies Inc.) [1]; polysilicon
surface-micromachined (Boston Micromachines Inc.) [2]; bulk
silicon (Iris AO Inc.) [3]; and piezoelectric monomorphs Jet
Propulsion Laboratory [4].
The ability of a DM to correct for a particular optical aber-
ration is determined by the aberration’s spatial frequency and
∗
Corresponding author. Tel.: +1 530 752 1778; fax: +1 530 752 4158.
E-mail address: dahorsley@ucdavis.edu (D.A. Horsley).
amplitude. Roughly speaking, the maximum spatial frequency
that a DM may correct is determined by the number of actu-
ators, while the maximum correctable amplitude is dependent
on the type of DM employed. For a segmented DM, the maxi-
mum correction amplitude depends only on actuator stroke and
is independent of spatial frequency, whereas for continuous face-
sheet and bimorph DMs, the correction amplitude depends on
the imposed spatial frequency.
Many MEMS DM designs have been driven by the moti-
vation to produce wavefront correctors with hundreds or
thousands of actuators. For applications which require the
correction of only low-order aberrations (such as defocus,
astigmatism, coma, and spherical aberration), a DM with less
than 100 actuators may be the best choice, as such a device
can be lower in cost and complexity than a DM with higher
actuator count. There is some evidence that such a low order
DM may provide sufficient correction for opthalmological AO
applications. Defocus and astigmatism, which are second-order
aberrations, represent 92% of the total wavefront aberrations
found in subjects with normal vision [5], and the aberration
magnitude diminishes with increasing radial order. The domi-
nant higher order aberrations are coma and spherical aberration,
which are third- and fourth-order aberrations and account for
0924-4247/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.sna.2006.04.052
222 D.A. Horsley et al. / Sensors and Actuators A 134 (2007) 221–230
1.8% and 1.6% of the total RMS wavefront error, respectively
[6].
We have chosen to investigate the characteristics of bimorph
DMs in an effort to understand their suitability for opthalmolog-
ical AO applications. This class of DM is capable of correcting
very large amplitude, low-order aberrations, and is simple to
construct, a fact that should ultimately result in a low-cost DM
[7]. The drawback of the bimorph design is that the maximum
correctable amplitude diminishes strongly with increasing spa-
tial frequency [8]. For this reason, the bimorph DM is unlikely to
be suitable for systems requiring correction of very high-order
aberrations. However, the bimorph may satisfy the requirements
of opthalmological AO, where correction of aberrations up to the
fifth radial-order may be sufficient.
We have tested three bimorph DMs manufactured by AOptix
(Campbell, CA), each having similar characteristics with the
exception of small manufacturing differences [9,10]. One of
these units has been successfully employed in an AO system for
in vivo retinal imaging [11]. Additionally, Dalimier and Dainty
recently showed that the AOptix DM was superior to two other
DMs when used to correct for synthetic aberrations typical of
those found in the human eye [12].
2. Methods
Tests were performed using a commercial laser-Doppler
vibrometer (LDV) and a custom-built stroboscopic phase-
shifting interferometer, described below. Our first objective was
to characterize the maximum correction amplitude that the DM
could produce as a function of spatial frequency. To this end, we
developed a method to use the DM to reproduce optical aber-
rations described by the Zernike polynomials [13]. Next, we
studied the dynamic characteristics of the DM and experimen-
tally measured the natural vibration modes.
2.1. Bimorph DM design
The layout of the AOptix DM is illustrated in plan view and
cross-section in Fig. 1. The device is composed of two 160 m
thick layers of the electrostrictive ceramic lead magnesium nio-
bate (PMN). Metal electrodes are deposited onto the PMN and
the two layers are bonded together with a 25 m thick layer
of conductive adhesive. The metallization on the back face of
the DM is patterned to produce 36 electrodes, while the uniform
metallization on the front face of the DM produces a single front
face electrode. The DM is 20 mm in diameter, with only the cen-
ter 10.2 mm of the DM used as an optical surface in order to
reduce the effect of the edge supports on the DM surface profile.
As illustrated in the figure, voltage is applied to the elec-
trodes on the front and back faces of the DM, with the inner
bonded electrodes serving as ground contacts for both layers.
The electrodes on the back face of the DM consist of a central
pad surrounded by four annular rings of electrodes. The central
pad and the electrodes in the two inner rings (channels 1–19) are
used to generate local curvature in the mirror surface, while the
electrodes in the outer ring (channels 20–35) produce a slope
at the edge of the DM. The curvature and slope electrodes are
Fig. 1. Layout of the AOptix DM: plan view of electrodes (top) and cross-
sectional view (bottom). The numbering of the actuator channels is indicated on
the plan view.
separated by the third annular electrode ring which defines the
10 mm pupil. The completed mirror assembly is mounted in a
housing with manual tip-tilt adjustment and a 10.2 mm clear
aperture.
PMN is a relaxor ferroelectric material that displays elec-
trostrictive behavior near room temperature [14]. Like piezo-
electric materials such as lead zirconium titanate (PZT), elec-
trostrictive materials deform mechanically when an electric field
is applied to the material. In contrast to piezoelectrics, in which
the direction of deformation reverses with the polarity of the
applied field, in electrostriction the deformation direction is
independent of the sign of the applied electric field. Although
a wide variety of dielectrics possess electrostrictive properties,
the effect is particularly large in the relaxor ferroelectrics like
PMN. When an electric field is applied to PMN, the material
contracts along the transverse axes. In comparison with PZT,
PMN has the advantage of greater linearity and lower hysteresis
at room temperature.
In the bimorph structure, voltage applied across the top layer
generates a tensile stress in the top layer, causing the bimorph to
undergo a concave curvature. Similarly, voltage applied across
the bottom layer results in convex curvature. Because the front
face electrode has a capacitance that is more than 36 times greater
D.A. Horsley et al. / Sensors and Actuators A 134 (2007) 221–230 223
Fig. 2. Cross-sectional view of the DM mount (top) and simplified mechanical
model for the DM edge supports (bottom).
than the capacitance of the individual back face electrodes,
driving this electrode at high frequencies requires considerably
greater power and current from the high voltage drive amplifiers.
As a result, when employed in a high bandwidth AO system, the
front face electrode is normally biased at a constant voltage, and
only the back-face electrode voltages are varied. In the absence
of any residual stress in the two layers, the DM surface is flat
whenever an equal voltage is applied across both the top and
bottom layers. Local convex (or concave) surface deformations
are then produced by setting the potential on the individual back
face electrodes above (or below) the potential on the front face
electrode. To allow symmetric convex/concave actuation, the
front-face potential is normally set to approximately the middle
of the output span of the high-voltage amplifiers. Residual stress
from the manufacturing process results in a slight parabolic cur-
vature to the DM surface when all the electrodes are at the same
potential. This initial curvature is removed by applying a small
voltage difference between the front and back face electrodes.
We have tested three DM units and each required slightly dif-
ferent voltage settings to flatten the DM.
Mounting a bimorph DM is a challenging problem, since
any constraint at the edges of the DM will reduce the curva-
ture which can be achieved. The AOptix DM is mounted using
two rubber o-rings which are preloaded by multiple set-screws
arrayed at the outer edge of the mirror [15], as illustrated in
Fig. 2. The o-ring mount produces translational compliance and
near-zero rotational compliance at the DM edge, as described
further below.
2.2. Phase-shifting interferometer
Dynamic and static measurements of the surface profile of the
DM were collected using a stroboscopic phase-shifting interfer-
ometer. A block-diagram of the instrument is illustrated in Fig. 3.
The instrument is a Twyman–Green interferometer in which a
piezoelectric stage (Polytec PI P-753.11C) translates a reference
mirror in order to introduce a controlled phase-shift between the
Fig. 3. Stroboscopic interferometer block-diagram. RM: reference mirror,
HWP: half-wave plate, QWP: quarter-wave plate, PBS: polarizing beam-splitter,
POL: polarizer, DM: deformable mirror, HVA: high voltage amplifier, DD: dig-
ital delay.
light passing through the reference and measurement arms of
the interferometer. Surface height variations in the DM create
interference fringes when the reference and measurement beams
are recombined, producing an interferogram that is captured and
digitized using a CCD camera (Cohu 6612-1000) and a frame-
grabber card (Matrox Meteor-II). The surface profile of the DM
is reconstructed using five interferograms collected at four dis-
tinct phase shifts (0, /2, ,3/2) using Hariharan’s algorithm
[16]. The use of a similar instrument for dynamic characteriza-
tion of millimeter-sized MEMS devices was first described by
Hart et al. [17]. The instrument is capable of measuring the sur-
face profile with an RMS accuracy of approximately 6 nm and
an absolute accuracy of ±60 nm across a 10 mm pupil.
The interferometer was outfitted with a pulsed diode laser to
allow the DM surface profile to be measured in response to time-
varying voltage inputs. The 10 mW laser (Hitachi HL6320G) is
driven with a custom current source capable of producing opti-
cal pulses of less than 1 s duration. Strobing the illumination
source gates the image, allowing motion at frequencies much
faster than the CCD frame rate (30 Hz) to be measured. The
strobed illumination is synchronized to the high voltage line used
to drive one of the actuators on the DM, and a programmable
digital delay unit (Directed Energy model PDG-2510) is used to
control the time delay (t) between the applied voltage and the
optical pulse. By varying t, images of the DM surface at var-
ious times throughout the actuation cycle are obtained. Ideally,
the optical pulse is sufficiently fast that the DM is essentially
motionless during the measurement interval; any motion of the
mirror surface during this interval reduces the contrast of the
interference fringes and introduces error in the surface mea-
surement. Assuming that the DM undergoes a 1 m amplitude,
10 kHz sinusoidal oscillation, the maximum motion of the mir-
ror surface over a 1 s interval is approximately 63 nm. This
224 D.A. Horsley et al. / Sensors and Actuators A 134 (2007) 221–230
is approximately λ/10 for a 635 nm diode laser, representing a
small but acceptable reduction in contrast.
2.3. Open-loop generation of Zernike modes
To characterize the dependence of the DM stroke on the
spatial frequency of deformation, a simple open-loop control
method was developed. In this approach, the deformation of
the DM surface is modeled as a weighted combination of the
deformations contributed by each actuator channel, known as
the actuator influence function. The deformation of the DM sur-
face, w(r, θ ), is described by
w(r, θ ) =
N
i=1
f (v
i
)ϕ
i
(r, θ ) (1)
where N represents the number of actuators on the DM, ϕ
i
(r, θ)
is the influence function and v
i
is the control voltage applied to
the i-th actuator, while f (v
i
) represents the normalized actuator
displacement as a function of applied voltage. In the special
case that the actuator displacement varies linearly with applied
voltage, f (v
i
) may be replaced by (v
i
/v
max
)in(1). In vector-
matrix form, (1) becomes:
w(r, θ ) = f (v)ϕ(r, θ )
T
(2)
where ϕ(r, θ ) = [ϕ
1
(r, θ ),ϕ
2
(r, θ ),...,ϕ
N
(r, θ )] and
v = [v
1
,v
2
,...,v
N
]. In electrostrictive and piezoelectric
bimorphs, the displacement is a nonlinear function of applied
voltage. To approximate f (v
i
), we drove all the actuators at
the same voltage and measured the DM displacement as a
function of applied voltage. This method does not account for
any variation between different actuators, but does account for
the general saturation characteristics of the PMN ceramic.
The influence function for each of the N = 35 back face elec-
trodes was measured by applying v
max
to the desired channel and
interferometrically measuring the resulting DM surface shape.
The guard ring was not utilized in these tests and was held at
a constant potential equal to that of the front face electrode.
Each measured influence function was then approximated with
an M-dimensional combination of Zernike polynomials through
a least-squares fit [18]:
ϕ
i
(r, θ ) =
M
j=1
a
ij
z
j
(r, θ ) (3)
where z
j
(r, θ)isthej-th Zernike polynomial, and the a
ij
’s are
the coefficients used to fit the i-th influence function. Piston, the
0-th order polynomial, was discarded after the least-squares fit.
Expressed in vector-matrix form:
ϕ(r · θ)
T
= Az(r · θ)
T
(4)
where z(r · θ) = [z
1
(r · θ),z
2
(r · θ),...,z
M
(r · θ)] is the 1 × M
vector of Zernike polynomials and A is the N × M influence
matrix. The present work explores using the DM to generate 20
Zernike mode shapes up to the fifth radial order, so the maximum
fit length was M = 20, resulting in a 35 × 20 A matrix. When A is
decomposed using the singular-value decomposition (SVD), the
magnitude of each singular value provides an indication of the
DM’s ability to reproduce each Zernike mode—singular values
near zero indicate modes which are not controllable with the
DM. The SVD is used to compute A
*
, the pseudo-inverse of A,
which in turn is used to calculate the vector of control voltages, v,
required to reproduce the desired combination of Zernike mode
shapes. If the 1 × M vector e represents the desired combination
of Zernike modes, the required control voltages can be calculated
from:
v = f
−1
(eA
∗
) (5)
Since f (v
i
) is not generally invertible, the function is approxi-
mated using a fifth-order polynomial fit of the measured voltage
to displacement curve and the inversion is performed using a
look-up table which maps actuator displacement to control volt-
ages. In the case where the actuator displacement varies linearly
with applied voltage, (5) becomes:
v = v
max
eA
∗
(6)
2.4. Modal response model
To study the dynamic response of the mirror surface to input
voltage pulses, the free vibration of the DM was analyzed using
classical plate theory. Approximating the DM as a thin, uniform,
circular plate of PMN, the free vibration of the DM, w(r, θ , t ),
is described by the following fourth-order differential equation:
D∇
4
w(r, θ , t ) + ρh
∂
2
w(r, θ , t )
∂t
2
= 0 (7)
where D = Eh
3
/(12(1 − ν
2
)) denotes the flexural stiffness, and
E, ν, ρ, h are the Young’s modulus (61 GPa), Poisson’s ratio
(0.3), density (7.8 g/cm
3
), and thickness of the DM (345 m).
When the plate deflection is decomposed into a spatially vary-
ing and a temporally varying component, so that w(r, θ , t ) =
u(r, θ ) exp(jωt), the spatial solutions to (7) are mode shapes of
the following form:
u
mn
(r, θ ) = A
mn
[J
m
(β
mn
r) + B
mn
I
m
(β
mn
r)]cos mθ (8)
where A
mn
is a normalization constant, B
mn
a mode shape param-
eter, β
mn
an eigenvalue, J
m
and I
m
denote the m-th order Bessel
function of the first kind and the m-th order hyperbolic Bessel
function of the first kind, respectively. The eigenvalues of (8)
are related to the natural frequencies of the DM by
f
mn
=
1
2π
β
mn
R
2
D
ρh
(9)
where R is the radius of the DM.
Values for the constants A
mn
, B
mn
, and β
mn
can be calculated
by applying the boundary conditions imposed by the DM mount.
These constants were computed using the solution developed
by Zagrai and Donskoy for a plate with elastic supports [19].
The translational stiffness of the o-ring support was modeled
by first computing the deflection, δ, of the o-ring due to a load
per unit length, p, using a Hertzian contact model of a cylinder
