2. real-time information about relative segment displa-
cements, segment actuators compensating for these
displacements, and a phasing camera. An inner fast
control loop using the edge sensors and the actuators
provides fast correction for segment displacements. It
runs continuously. Since the edge sensors only mea-
sure displacements relative to a zero reference, this
reference must be provided by another sensor at reg-
ular intervals (for example, at the beginning of each
night of observation). Typically this task is performed
byaphasingcamerathatuses thelightofabrightstel-
lar source to measure the shape of the wavefront. This
method is called optical phasing or on-sky calibration.
B. Active Phasing Experiment
Within the scope of a design study for the E-ELT, the
Active Phasing Experiment (APE) aims at demon-
strating active wavefront control technologies for a fu-
ture ELT [2]. The experiment will validate different
stellar light-based phasing camera concepts by
testing them on a scaled-down segmented mirror,
the so-called Active Segmented Mirror (ASM)
(D ¼ 154 mm, 61 segments). Figure 1 shows a sche-
matic drawing of the ASM. Each hexagonal segment
has a flat-to-flat distance of d ¼ 17 mm. The gap
width is approximately 70–130 μm. The total dia-
meter (corner to corner) is D ¼ 154 mm. The optical
surfaces of the segments are flat [surface quality of
approximately 50 nm peak to valley (PTV)]. Each
segment is controllable in three degrees of freedom
(piston, tip, and tilt) by means of three piezo actuators
with mechanical strokes of Æ7:5 μm and a precision in
displacement of better than 0:2 nm. More details
about the ASM can be found in [3].
Closed-loop control with a bandwidth of approxi-
mately 0:2 Hz is used to apply a given pattern of pis-
ton, tip, and tilt to the segments and to maintain the
ASM in this shape for a time period longer than
the integration times of the phasing cameras
(typically 1–30 s).
We present a high-precision optical metrology sys-
tem that serves as the sensor in the control loop,
measuring the piston, tip, and tilt of the 61 segments
at a frequency of 4:44 Hz. In addition to its role as
sensor in the closed-loop control, it will be used as
a reference to qualify the precision of the different
phasing cameras. Following the ongoing tests at
the laboratories of the European Southern Observa-
tory in Garching, Germany, APE will be tested on-sky
at a Nasmyth focus of a Unit Telescope of the Very
Large Telescope (VLT) array on Cerro Paranal, Chile.
2. System Description of the Metrology
A. Metrology Specifications
Table 1 lists the metrology specifications. All mea-
surements (piston, tip, and tilt) are performed rela-
tive to the central segment whose piston is not
controlled in closed loop. The specified measurement
range for piston (Æ7:2 μm) is covered by the range of
displacement for the piezo actuators (Æ7:5 μm).
The measurement is performed through a beam
expander optic (magnification factor 1=10) that is
not part of the metrology’s optical system. The beam
expander is placed between the exit pupil of the me-
trology and the target (ASM). The optical path length
between the exit pupil and the ASM is approximately
2200 mm. The optical layout of the complete APE
configuration is shown in [2].
B. Metrology Concept
The system combines three well-established concepts
of interferometric metrology. Those are (1) instanta-
neous phase-shifting interferometry, (2) low-
coherence interferometry, and (3) dual-wavelength
interferometry. We briefly explain the basics of these
three techniques and why they have been chosen for
the present application.
Fig. 1. ASM with 61 segments, each equipped with three piezo
actuators. The flat-to-flat distance of a single segment is
d ¼ 17 mm. The total diameter of the mirror (corner to corner)
is D ¼ 154 mm.
Table 1. Achieved Specifications of the Metrology
Item Specification
Measurement range for pistona
Æ7:2 μm
Measurement range for tip/tiltb
Æ250 μrad
Working distancec
2200 mm
Piston precision (closed loop) 0:48 nm rms
Tip/tilt precision (closed loop) 74 nrad rms
Measurement frequency 4:44 Hz
Closed-loop bandwidth 0:2 Hz
Frame-to-frame time difference 25 ms
a
The piston range is defined with respect to the piston of the
central segment.
b
The tip/tilt range is defined at the level of the target
(i.e., the ASM).
c
The working distance is the distance between the exit pupil of
the metrology and the target (ASM).
5474 APPLIED OPTICS / Vol. 47, No. 29 / 10 October 2008
3. 1. Instantaneous Phase-Shifting Interferometry
Phase-shifting interferometry (PSI) determines the
phase difference between two interfering beams by
measuring the irradiance of the interference fringes
as the phase difference between the beams’ changes
in a known manner. Besides the phase difference
there are two other unknowns to be determined by
a PSI algorithm: the amplitude of the probe beam
and the amplitude of the reference beam. Since there
are three unknowns, at least three irradiance mea-
surements have to be made. To reduce measurement
errors, four or more measurements are made. Typi-
cally the phase is changed by 90° between two conse-
cutive measurements, but there also exist PSI
algorithms using phase shifts different from 90°.
The phase shifts between consecutive measurements
are created by a phase shifter—typically a moving re-
ference mirror in a delay line or an electro-optic mod-
ulator that varies the phase of the reference beam in a
controlled way. A wide variety of PSI algorithms exist
and are described in the literature [4–6].
In the presence of vibrations and air turbulence,
changing the reference beam phase in a controlled
manner can become very difficult. The mentioned
perturbations, which will certainly be present on a
telescope platform, can change the phase difference
between two beams in unknown ways and therefore
can cause large errors in the measurement. This pro-
blem can be avoided by using the technique of instan-
taneous PSI (also referred to as snapshot or single
shot interferometry). Four interferograms at phase
shifts of 90° are simultaneously acquired, i.e., with-
out any time lag between measurements. Several of
such snapshot interferometers have been demon-
strated [7–9]. We have chosen a concept where the
phase shifts are induced by polarizing optical ele-
ments such as polarizing beam splitters and phase
retarders (wave plates). The four interferograms
are detected on four CCD cameras. An accurate pixel-
to-pixel correlation between the different cameras is
obtained by an image-matching method.
2. Low-Coherence Interferometry
Low-coherence interferometry performs interfero-
metric measurements using the light emitted from
a relatively broadband source with a coherence
length of typically a few tens of micrometers. The
term “coherence” refers to the temporal coherence
of the light. The most frequently used light source
for this type of interferometer is a superluminescent
diode (SLD). The metrology system also uses this
type of source.
A low-coherence interferometer works as a com-
parator of optical group delays, i.e., products of group
refractive index and geometrical distance. The group
delay in the probe arm of the interferometer is com-
pared to the group delay of the reference arm, whose
length can be varied using a movable reference mirror
in a delay line. Expressed as a function of the refer-
ence mirror position, the interferometric signal is a
sinusoidal fringe pattern with a fringe period equal
to half the optical wavelength λ. The pattern is modu-
lated by an envelope that is proportional to the Four-
ier transform of the source frequency spectrum. For
the approximately Gaussian spectrum of an SLD,
the envelope has a Gaussian shape too. Its width is
the round-trip coherence length lc ¼ 2 lnð2Þλ2
=ðπδλÞ,
where λ and δλ are the center wavelength and the full
spectral width at half maximum of the source, respec-
tively. The maximum of the fringe envelope occurs for
a group delay difference (GDD) of zero between the
two arms.
The two SLDs used in the metrology system operate
at wavelengths λ1 ≈ 834:6 nm and λ2 ≈ 859:6 nm, each
with a spectral bandwidth of δλ ≈ 13:5 nm. This corre-
sponds to a round-trip coherence length of lc ≈ 23 μm.
Since during an extended measurement run lasting
several hours the group delays in the two interferom-
eter arms might vary in a different way due to tem-
perature variations, the operational point of the
system might drift away from the initial optimum po-
sition of GDD zero to a position with lower fringe con-
trast. To increase this working range, we use spectral
filters to reduce the bandwidth of each SLD to a value
of δλ ≈ 3:2 nm. This yields a significantly larger round-
trip coherence length of lc ≈ 100 μm.
The advantages of a low-coherence interferometer
when compared to a classical laser interferometer
are the fact that the measurement is less affected
by speckle noise and that parasitic internal reflec-
tions do not perturb the coherent signal.
3. Two-Wavelength Interferometry
AllPSI algorithms measure theinterferometricphase
modulo 2π. Phase unwrapping is necessary to gener-
ate a continuous phase distribution that can be
converted into a surface height profile. When measur-
ing at a single wavelength λ, the range of nonambigu-
ity for distance measurements is given by λ=2 since a
displacement of the target or the reference mirror by
Δz ¼ λ=2 creates an optical path difference (OPD) of λ,
i.e., creates a phase shift of 2π. An important precon-
dition for any phase unwrapping algorithm is that the
phase difference between two measurements (either
separated in space or in time) is smaller than the
threshold of the phase unwrapping algorithm (which
is typically equal to π). This means that steps or dis-
continuities (in space or in time) larger than λ=4—the
path difference corresponding to a phase difference of
π—will cause the phase unwrapping algorithm to
break down. The fringeorder is lostover thestep (tem-
poralor spatial). In oursegmentedmirrorapplication,
spatial discontinuities larger than λ=4 exist at the
segments’ borders. Two-dimensional spatial phase
unwrapping is not suitable.
The technique of two-wavelength interferometry
allows the fringe order to be determined and can
measure step heights and discontinuities much lar-
ger than a quarter optical wavelength. The range
of nonambiguity increases to Λs=2, where Λs is the
10 October 2008 / Vol. 47, No. 29 / APPLIED OPTICS 5475
4. synthetic wavelength defined as
Λs ¼
λ1λ2
jλ1 − λ2j
; ð1Þ
with λ1 and λ2 denoting the two individual wave-
lengths.
With λ1 ≈ 834:6 nm and λ2 ≈ 859:6 nm, the range of
nonambiguity of the interferometer becomes
Λs=2 ≈ 14:4 μm, which corresponds to twice the mea-
surement range for piston (see Table 1). The measure-
ment range is compatible with the range of the piezo
actuators.
C. Functional Description of the Metrology System
Figure 2 shows the functional block diagram of the
metrology system, which can be subdivided into four
subunits: light source, phase modulator, signal deco-
der, and computer. We describe the four subunits and
their interfaces from a system point of view.
1. Light Source
The light source subunit consists of two modules: the
light source module (LSM) and the beam coupler mod-
ule (BCM) (see Fig. 2). The LSM is a crate containing
the two SLDs and their electronic drivers. The SLDs
are piloted by the light source control board (LSM
Ctrl) installed in the computer (see Fig. 2). A rectan-
gular modulation with a frequency of 20 Hz is applied
to each source. The two modulations are in phase op-
position. The LSM Ctrl also generates an electronic
trigger signal (rectangular modulation at 40 Hz) that
is fed to the image acquisition board in the computer
to synchronize the acquisition of the four CCD frames
to the modulation of the two sources. Sets of 2 × 4
frames at λ1 and λ2 are acquired consecutively and se-
parated in time by 25 ms. Not every set of eight frames
is used to determine piston, tip, and tilt. While the
computer is busy with computing instantaneous
phases, piston, tip, and tilt, acquired images are ig-
nored. The final frequency at which measurements
are delivered is 4:44 Hz.
The LSM is optically connected to the BCM by two
single-mode polarization-maintaining (PM) fibers.
The latter injects the beams from both sources at
orthogonal linear polarization states into one com-
mon single-mode PM fiber that feeds the light into
the phase modulator (see Subsection 2.C.2). Figure 3
shows the optical layout of the BCM. It is installed on
a compact-sized breadboard (dimensions 100 mm
×100 mm) that is fixed on the main breadboard. Fara-
day isolators prevent feedback into the SLDs. The
spectral filters (labeled δλ in Fig. 3) are used to in-
crease the coherence length as explained above in
Subsection 2.B.2. The polarization beam splitter
(PBS) combines the two beams. The half-wave plate
(labeled λ=2) is rotated to align the two orthogonal
polarization directions onto the slow and fast axis
of the PM fiber.
Taking into account the loss at all optical elements,
the required optical power for the two SLDs is ap-
proximately 5 mW (at fiber output). Both sources
are able to deliver more than 20 mW.
2. Phase Modulator
The phase modulator and signal decoder (see
Subsection 2.C.3) form the core of the metrology sys-
tem. Their optical layout is shown in Fig. 4. While the
phase modulator contains the illumination optics
and the two arms of the interferometer, the signal de-
coder comprises the beam-combing optics creating
the interferograms on the four cameras.
The phase modulator uses a polarization-
dependent beam-splitting mechanism. By adjusting
the angle of half-wave plate WP1, the linear polari-
zations of the beams at λ1 and λ2 emitted by the fiber
(labeled SLD1 and 2 in Fig. 4) are aligned to angles of
45° (λ1) and 135° (λ2) with respect to the breadboard
plane, respectively. Figure 4 only shows the 45°
orientation of SLD1.
Fig. 2. Functional block diagram of the metrology system.
Optical signals (in fibers or free space) are represented by dashed
lines. Electrical signals are represented by solid lines. SPA and SRA
are the probe and reference signals (electric field) whose phase dif-
ference φ carries information about the OPD between the two arms.
S1…S4 are the four CCD images (interferograms) that are acquired
by the image acquisition board in the computer. h, α, and β are the
piston, tip, and tilt measured for each mirror segment, respectively. Fig. 3. Optical layout of the BCM (compare with Figs. 2 and 4).
5476 APPLIED OPTICS / Vol. 47, No. 29 / 10 October 2008
5. The polarizing beam splitter cube PBS1 splits the
beam into the two arms of the interferometer, i.e., the
probe arm (toward the ASM) and the reference arm
(toward the reference mirror on the translation
stage). The external distance of approximately
2200 mm between the exit pupil (top right in Fig. 4)
and the ASM is reproduced internally in the delay
line with its folded mirror design.
Quarter-wave plates (WP2 and WP3), both or-
iented at 45°, transform the outgoing linear polariza-
tions [perpendicular (s) for the reference arm and
parallel (p) for the probe arm] into circular ones. Re-
flection on the final surface in each arm (ASM in the
probe arm and reference mirror in the reference arm)
inverses the handiness of the polarization. After hav-
ing passed the quarter-wave plates for a second time,
the polarizations are converted back to linear, ro-
tated by 90° relative to the outgoing beams, before
recombination on PBS1. This circulator-type setup
optimizes the total power throughput.
After recombination on PBS1 the probe and refer-
ence beams have mutually perpendicular polariza-
tions. The pair of beams with a phase shift φ carry
the information about the OPD between the two arms
that one wants to measure.
A scan of the linear translation stage carrying the
reference mirror is performed for calibrationpurposes
(see Section 3.D). During the actual measurements
the reference mirror is kept fixed at one position that
corresponds to equal optical group delays in both
arms, i.e., to the maximum of the fringe envelope.
3. Signal Decoder
The signal decoder directs the recombined beams
leaving the phase modulator at orthogonal linear po-
larizations via respectively separate paths onto four
CCDs. Thanks to the optical design each CCD has a
different “sensitivity” to the information contained in
the combined beams. The first step consists of rotat-
ing linear polarizations s and p leaving the phase
modulator by 45° using the half-wave plate WP4 or-
iented at 22:5°. Then the nonpolarizing beam splitter
BS cube directs the pair of beams to the two pairs of
CCDs (CCD1 & 2 and CCD3 & 4). The quarter-wave
plate WP5 has its crystal axis oriented parallel to one
of the two beams, i.e., it introduces an additional
phase shift of 90° between probe and reference
beams for the interferograms on CCD1 and 2. Since
at the entrance of PBS2 and PBS3 the linear polar-
izations of the two beams are oriented at 45° and
135°, each beam is equally divided into a parallel
(p) and a perpendicular (s) component, propagating
toward the two CCDs assigned to each beam splitter
cube. Finally the two interferograms observed on
each pair of CCDs are opposite in phase. The four in-
terferograms on CCD1; …; 4 are in phase quadra-
ture, which means that they are shifted in phase
Fig. 4. Optical layout of the interferometer. The dimensions of the rectangular breadboard are 1200 mm × 600 mm. The internal
magnification between the exit pupil (top right) and each of the CCDs is Mint ¼ 1=3. The segmented mirror is imaged onto the exit pupil
by a beam expander (not shown here) with magnification of Mext ¼ 1=10. The gray squares represent the local state of polarization. The
polarization of the probe beam is represented by a solid line, while the polarization of the reference beam is represented by a dashed line.
A closer look to the BCM is provided in Fig. 3.
10 October 2008 / Vol. 47, No. 29 / APPLIED OPTICS 5477
6. by 90° with respect to each other. Their image acqui-
sitions are synchronized.
4. Computer
The computer hosts the three electronic boards to
control the light sources, the translation stage in
the delay line, and the four cameras (see Fig. 2). It
also runs the algorithms for image matching,
calibration, and phase detection.
D. Optical Design
Figure 4 shows the optical layout of the interferom-
eter. Optics and optomechanics are installed on an op-
tical breadboard of dimensions 1200 mm × 600 mm.
For the APE experiment the metrology breadboard
is fixed by a three-point kinematic support on top of
the main APE optical table. To avoid the disturbing
effects of air turbulence, protective covers enclose
both the metrology and the whole APE setup (for de-
tails about APE see [2]).
The optical design uses all lenses in telecentric re-
lay configurations. This ensures a homogeneous illu-
mination on the segmented mirror (within the limits
of a clipped Gaussian beam profile) and makes the
alignment of the system easier. Most of the optical
components have either a 1 or 2 in: diameter
(1 in: ¼ 2:54 cm). All lenses are 2 in: in diameter to
ensure diffraction-limited imaging quality.
The optical interface between the metrology sys-
tem and the external optical system of the APE ex-
periment is an image of the segmented mirror (scale
factor 1=10). A beam expander, formed by the combi-
nation of an off-axis parabola and a lens (not shown
in Fig. 4), creates this image that is located in the exit
pupil of the metrology at the upper right border of its
breadboard.
The target, i.e., the segmented mirror, is illumi-
nated by a collimated beam with perpendicular inci-
dence. The circular aperture stop (diameter 15:4 mm)
shown in Fig. 4 (in front of WP1) is imaged on the exit
pupil and on the target where it has a diameter of
154 mm. It clips the illuminating Gaussian beam
at approximately half its waist radius, corresponding
to an irradiance level equal to 60% of the peak irra-
diance. The metrology-internal magnification factor
between the exit pupil and each of the CCDs is
1=3. With a magnification factor of 1=10 between
the target and its image in the exit pupil, the total
magnification becomes 1=30.
The segmented mirror is imaged onto the four
CCDs with diffraction-limited quality. Each CCD
has 768 × 580 square pixels. The number of pixels con-
tained in the image of a single mirror segment is 3581.
The Airy disk diameter is DAiry ≈ 20:8 μm, correspond-
ing to 2.5 pixels on the CCD (at the mean wavelength).
Image distortion is negligible with a maximum value
of −0:05% at the corner of the CCD image, which
corresponds to −0:0005 × 15:4=2 mm ≈ 3:9 μm—
approximately half a pixel’s dimension. The five
Seidel aberrations are all below 0:3λ with astigma-
tism as the most important contribution.
Diffraction effects on the reference beam are signif-
icantly reduced by creating an intermediatepupil(im-
age of the aperture stop) using two lenses in afocal
configuration (see Fig. 4). Hence the collimated beam
propagation distance responsible for diffraction ef-
fects on the reference beam is significantly smaller
than 2 × 2200 mm (round-trip). With an initial beam
diameter of 15:4 mm, near-field diffraction effects (in-
crease in beam diameter, ripples on the electric field
amplitude profile, and curvature of wavefront) of the
clipped Gaussian beam remain small.
The numerical aperture of the system is large en-
ough to capture a maximum tip/tilt angle of
γmax ¼ 0:8 mrad—a range significantly larger than
the maximum measurable tip/tilt of Æ250 μrad (see
Table 1).
3. Algorithms for Interferometric Image Processing
Starting with a description of the interferometric sig-
nal, we discuss the methods and algorithms used to
obtain a surface height measurement from the 2 × 4
interferograms. Our description includes topics such
as calibration, image matching, instantaneous phase
measurement, and determination of piston, tip, and
tilt by a dual-wavelength algorithm.
A. Interferometric Signal and Instantaneous Phase
We use notations ðx; yÞ and zðx; yÞ for a position in the
transverse plane of the target, i.e., the segmented
mirror, and the corresponding surface height. Eight
interferometric signals fIk
i g are created for surface
height z. The signals are acquired by four cameras
CCDi (i ¼ 1; …; 4) for two wavelengths λk (k ¼ 1
and 2). The irradiance of each interferometric signal
can be written as
Ik
i ðx; yÞ ¼ Bk
i ðx; yÞ þ Ak
i ðx; yÞ sin½φk
ðx; yÞ þ θk
i ðx; yÞŠ;
ð2Þ
where φkðx; yÞ ≡ 4πzðx; yÞ=λk mod 2π is the wrapped
interferometric phase measured at wavelength λk;
Bk
i ðx; yÞ is the background, i.e., the incoherent sum
of irradiances of both reference and probe arm;
Ak
i ðx; yÞ is the amplitude of the sinusoidal interfero-
metric signal; and θk
i ðx; yÞ is the reference phase as-
sociated to CCDi. Among these four quantities only
the phases φk
ðx; yÞ contain information about surface
height zðx; yÞ. Since the phase φk
ðx; yÞ is measured at
the same time on the four CCDs, we refer to it as the
instantaneous phase. Note that the term “instanta-
neous” refers to the four cameras but not to the
two wavelengths. As stated in Subsection 2.C.1,
the acquisitions at λ1 and λ2 are separated by a time
lag of 25 ms.
B. Operational Procedures Overview
During the measurement procedure the reference
mirror in the delay line is kept at a fixed position cor-
responding to a GDD of zero between the two arms
(see Subsection 3.E). The system acquires the 2 × 4 in-
terferograms Ik
i [Eq. (2)] and computes instantaneous
5478 APPLIED OPTICS / Vol. 47, No. 29 / 10 October 2008
7. phases φk
ðx; yÞ for both wavelengths (k ¼ 1 and 2) at
the measurement frequency of f ¼ 4:44 Hz. The two
phase maps are then used to calculate the piston
(h), tip (α), and tilt (β) of each mirror segment. In
closed-loop operation these data ðh; α; βÞ are sent to
the controller computing the commands applied to
the piezo actuators of each segment in order to bring
the mirror into the desired shape. The algorithm to
derive the piston, tip, and tilt from the two phase
maps has been specifically developed for the present
application of segmented mirror metrology and is
presented in Subsection 3.E.4.
The three other quantities, Bk
i ðx; yÞ, Ak
i ðx; yÞ, and
θk
i ðx; yÞ, affecting interferograms Ik
i vary rather slowly
during the measurement, at a typical time scale of a
few hours. The variations are mainly due to tempera-
ture drifts. They do not depend on surface height z.
Both the background and the amplitude depend on ir-
radiances Ik
i;probe and Ik
i;ref of the probe and reference
arms, respectively:
Bk
i ðx; yÞ ¼ Ik
i;probe þ Ik
i;ref ; ð3Þ
Ak
i ðx; yÞ ¼ 2ðIk
i;probeIk
i;ref Þ1=2
: ð4Þ
The fringe visibility (or contrast) Vk
i ðx; yÞ is defined as
the ratio of fringe amplitude over background: V ¼
A=B with a maximum value of V ¼ 1 for equal irra-
diances Ik
i;probe ¼ Ik
i;ref . To maximize the signal-to-
noise ratio, it is important to obtain the highest
possible fringe visibility, i.e., to equalize as much as
possible the contributions of the probe and reference
arms. This equilibrium must be respected simulta-
neously for all four CCDs and at both wavelengths.
In our case we obtain a fringe visibility varying
between 0.6 and 0.7 depending on pixel location,
CCD, and wavelength.
In instantaneous PSI it is not necessary to know
the absolute reference phase θk
i ðx; yÞ for each CCD;
nevertheless the three phase shifts between the four
interferograms must be known precisely. By conven-
tion we assume the reference phase of the first CCD
to be zero at both wavelengths, i.e., θk
1ðx; yÞ ¼ 0 for
k ¼ 1 and 2. Hence the reference phases of the re-
maining three CCDs correspond to the phase shifts
with respect to the interferogram on CCD1.
An accurate determination of the three quantities
background, amplitude, and phase shift is an essen-
tial step to be performed prior to the measurement.
This is the main purpose of the calibration procedure
described in Subsection 3.D. In contrast to the mea-
surement procedure, the calibration is performed
using a series (stack) of images (each of which is called
an image frame) acquired at different positions of the
reference mirror in the delay line. Notice that back-
ground Bk
i ðx; yÞ and amplitude Ak
i ðx; yÞ also depend
on the power of the light sources. This dependency
is essentially a first-order law for both background
and amplitude, and the remaining variations will
be further removed during the measurement. In ad-
dition the amplitude depends on the GDD between
the two arms, i.e., the position within the fringe envel-
ope. The amplitude is monitored continuously during
the measurement. Differential temperature varia-
tions occurring during a measurement run can
change the GDD. The operational points move away
from the envelope’s maximum, leading to a decrease
in fringe amplitude (and contrast). If the contrast gets
below a predefined threshold, a recentering is per-
formed: After a scan to determine the new position
of GDD zero, the reference mirror is moved to this
position.
Finally the four CCD images must be matched by
software, i.e., finding the pixels on the four different
CCDs that correspond to same physical location on
the target. This matching procedure must be per-
formed at a very high accuracy (about a tenth of a
pixel size). We discuss the requirement on matching
accuracy in more detail in Subsection 3.D.3.
C. Determining the Optimum Delay Line Position
The search for fringes is performed in two steps. Dur-
ing a first coarse scan, the reference mirror is moved
at a faster speed over a course of a few centimeters to
detect the position corresponding to a large irradiance
variation. A second fine scan at a slower speed is per-
formed in the interval around the position deter-
mined during the coarse scan. The length of this
interval is chosen to match the coherence length,
i.e., 100 μm. The scan velocity is adjusted such that
the fringes are acquired with a sampling of λ=8 ¼
π=2 between two consecutive images. A PSI demodu-
lation method (as described in Subsection 3.D.2) is
then applied to compute the fringe amplitude, i.e.,
fringe envelope. The maximum of the Gaussian envel-
ope (zero GDD) is detected by standard curve fitting
procedures. There exists a GDD zero position for each
wavelength. They are separated by approximately a
few tens of micrometers due to a differential group
dispersion in the two arms of the interferometer:
The types and thicknesses of optical glasses in the
two arms do not match each other exactly. The probe
arm contains the beam expander, which itself com-
prises a lens, while the reference arm contains two
lenses. The delay line position is then chosen as the
average of the two optimum positions at λ1 and λ2.
A planned future improvement is a full dispersion
compensation by inserting a flat piece of glass in
the reference arm. This would let the two GDD zero
positions for λ1 and λ2 coincide, i.e., further optimize
the fringe contrast.
D. Calibration Methods and Algorithms
1. Calibration Workflow
The calibration starts with the reference mirror in the
delaylineatitszeroGDDposition(seeSubsection3.C).
First the optical power of both SLDs is adjusted such
that the full dynamical range of the CCDs is exploited.
Then the four cameras acquire a stack of images while
10 October 2008 / Vol. 47, No. 29 / APPLIED OPTICS 5479
8. the translation stage performs a fine scan centered
around the zero GDD position. The matching algo-
rithm determines the matching parameters for the
four images. Camera phase differences θiðx; yÞ (i ¼ 2,
3, and 4) are calibrated using a PSI algorithm. Back-
ground Bk
i ðx; yÞ and fringe amplitude Ak
i ðx; yÞ are cali-
brated for each pixel of all four cameras. A tessellation
by software on the matched image is carried out by de-
termining the subset of pixels belonging to each
segment.
2. Temporal Demodulation by Delay Line
Scanning
As mentioned in Subsection 3.D.1, during the cali-
bration a stack of images is acquired at different po-
sitions of the reference mirror. The step size or
distance between two adjacent positions of the refer-
ence mirror is imposed by the PSI algorithm used to
estimate the phases of the four interferograms by
temporal demodulation of the fringes. The most com-
mon phase shift in PSI is π=2, corresponding to a step
size of λ=8 ¼ 104:3 nm at λ1 ¼ 834:6 nm.
Two solutions could be envisaged: (1) moving the
reference mirror to a series of standstill positions se-
parated by the required step size or (2) continuous
acquisition of image frames while the reference mir-
ror is moving at constant velocity. To our knowledge
there exists to date no commercially available trans-
lation stage that is able to move to a standstill posi-
tion with a positioning accuracy of a few nanometers
(a small fraction of the step size). Therefore our
design choice is of the second option (2).
With an image acquisition frequency of f ¼ 25 Hz,
to acquire two consecutive frames separated by a
phase of π=2 (i.e., for a displacement of Δz ¼ λ=8),
the translation stage motor must be able to move
seamlessly at the speed of v ¼ λ=8 × f. For
λ1 ¼ 834:6 nm, v ¼ 2:61 μm s−1, and λ2 ¼ 859:6 nm,
v ¼ 2:69 μm s−1
. The chosen translation stage
(MICOS PLS-85) has been successfully tested with
a velocity as low as 0:1 μm s−1
[10].
Even though the translation stage is commanded
to move at constant velocity, the real instantaneous
velocity might fluctuate significantly around the
nominal value. Because of this velocity variation,
the positioning error of the reference mirror and
its effect on the fringe demodulation has to be stu-
died with great attention.
The displacement of the MICOS PLS-85 at small
velocities in the required range has been measured
with a high resolution of a few nanometers using a
laser interferometer [10]. The residual linearity error
(i.e., the deviation of the position-versus-time curve
from a straight line fit through the data) has a PTV
amplitude of approximately Æ100 nm.
We have considered the Hariharan five π=2 step al-
gorithm [4] for the demodulation. Among multistep
phase-shifting algorithms with maximum of five
steps, Hariharan’s method is much less sensitive to
the error of phase sampling (even for a fairly large
sampling error), i.e., to position error of the transla-
tion stage. In practice more than five frames are ac-
quired during calibration. The demodulation
algorithm will be applied on a sliding five-step win-
dow moving over all frames. The phase shifts between
CCDs are estimated by averaging the demodulation
results applied on the individual sliding window.
We used the measured typical-linearity error of the
translation stage to numerically analyze the error of
the phase shifts estimated by demodulation. The sys-
tematic errors of the phase shift with respect to the
number of CCD frames is represented in Fig. 5. We
observe that the systematic errors do not improve
significantly by using more than 12 frames. In our
case, including a safety margin, we chose to acquire
20 frames.
Additionally we have studied a more sophisticated
demodulation algorithm also using five π=2 steps. Ap-
pendix A explains why we have chosen this algorithm
as the preferred solution in the final implementation
of the software for the fringe demodulation during the
calibration.
As mentioned above, the phase shifts between the
four cameras will be calibrated by continuously mov-
ing the reference mirror. The optical design should
yield phase shifts that are spatially constant over
the beam diameter, i.e., that do not depend on posi-
tion ðx; yÞ. Their theoretically expected values are
ðθ1; θ2; θ3; θ4Þ ≈
0; π;
π
2
; −
π
2
for both wavelengths since the polarizing elements
(beam splitters and wave plates) are achromatic over
this range.
Experimentally we observe, however, noticeable
spatial variations of phase shifts θk
i ðx; yÞ across the
ðx; yÞ plane. The variations have a PTV amplitude
Fig. 5. Systematic error for the calibration of phase differences
between CCDs due to the nonuniformity of the translation stage
movement for the three phase differences as a function of the num-
ber of CCD frames used.
5480 APPLIED OPTICS / Vol. 47, No. 29 / 10 October 2008
9. of a few degrees. For example, Fig. 6 shows phase
shift θ1
2ðx; yÞ (at λ1) between CCD2 and CCD1. On
the left-hand side, the raw phase shift is shown
(PTV ≈ 20°). The large variations only occur at high
spatial frequencies. Smooth low spatial frequency
components of the phase-shift distribution have
PTV values from 2° to 7° depending on which pair
of CCDs is considered. In our software implementa-
tion we project the phase-shift map into a piecewise
constant phase shift having a one single phase-shift
value per segment and per CCD. The piecewise
constant map will be used during the measure-
ment to compute the instantaneous phases (see
Subsection 3.E.2).
We also considered a self-calibration method based
on a two-dimensional Fourier-Hilbert demodulation
technique. It determines the phase shifts directly
from the interferograms without the need to step
through the phase of each one [11]. Even though this
attractive method works well most of the time, our
simulations indicate that it can fail to estimate with
sufficient accuracy when using four interferograms,
sometimes with an error of estimated phase shift
that can be as high as 6°. Note that the self-calibra-
tion method would work seamlessly with a minimum
of five interferograms, which unfortunately is not
compatible with the architecture of our system using
four cameras.
3. Matching of the Four Images
The calibration and measurement algorithms rely on
the knowledge of the matching of the four-pixel grids
on the four CCDs with physical positions ðx; yÞ on the
segmented mirror. Even after very careful alignment
of the optical elements, the images on the four CCDs
will not be completely matched, i.e., four pixels with
the same pixel coordinates ði; jÞ on the four CCD
frames will not correspond to the same physical loca-
tion ðx; yÞ on the target. The four images will differ in
rotation, translation, and their relative scaling.
While rotation and translation are mainly due to re-
sidual imperfections of the mutual alignment of the
four cameras, the different image sizes correspond to
slightly different magnification factors that are
caused by tolerances on the focal distances of the
imaging lenses (typically of the order of 1%). The op-
tical design ensures that image distortion is negligi-
ble (see Subsection 2.D). Therefore an affine
transformation (a combination of rotation, trans-
lation, and scaling) is enough to match the images
accurately.
During the calibration the images are matched
automatically by detecting the polygonal contour of
the fringe visibility V ¼ A=B. The visibility maps
are computed separately for each stack of images ac-
quired for each CCD. Then the polygonal vertices are
numerically extracted and compared with the theo-
retical locations of vertices (i.e., without affine trans-
formation). The affine transformation parameters
(there are four for each CCD: scaling, rotation angle,
and two lateral shifts) of the image are computed
from the absolute differences in position for two sets
of vertices, numerically detected and theoretical. See
Fig. 7 for an illustration of matching using the con-
tour of the fringe visibility.
The precision of the matching is directly linked
to the maximum measurable tip/tilt (compare with
Table 1). The smaller the matching error, the larger
the maximum measurable tip/tilt. For a given combi-
nation of matching error and tip/tilt angle, the sys-
tematic error obtained for the interferogram phase
on one CCD is
Fig. 6. Phase shift (θ2) between CCD2 and CCD1. On the left-hand side is the raw phase-shift, and on the right-hand side is a constant
piecewise phase shift per segment that is effectively used to compute the instantaneous phases during the measurement step.
10 October 2008 / Vol. 47, No. 29 / APPLIED OPTICS 5481
10. δθ ¼
4π
λ
Δp
Mtotal
δx · GTT; ð5Þ
where Δp is the CCD pixel linear dimension (here
Δp ¼ 8:3 μm), Mtotal is the optical magnification of
metrology and external beam expander (here
Mtotal ¼ 1=30), δx is the matching error vector in unit
of pixels, and GTT is the height gradient vector of the
target, which equals to GTT ¼ ðα; βÞ for small tip/tilt
angles. The dot denotes a scalar product of two
vectors.
A too large product δx · GTT degrades the precision
of the metrology system in two ways. First the phase
differences between the different pairs of CCDs can-
not be accurately estimated during the calibration.
The phase error δθ affects reference phase θk
i in
Eq. (2) associated to each CCD. Second the determi-
nation of the instantaneous phase φk
from four inter-
ferograms is also affected by the matching-induced
phase error δθ. Consequently, periodic systematic er-
rors in the OPD measurements at both wavelengths
will increase (compare with Section 4). The metrolo-
gic precision degrades.
By means of a specific test using a flat reference
mirror on a tip/tilt stage that replaced the segmented
mirror as target, we have determined the maximum
measurable tip/tilt angle to be 250 μrad (see Table 1).
Simulations using images with typical noise levels
have shown that our algorithm can achieve an error
of matching of jδxj ¼ 0:1 pixel. Inserting those two
quantities into Eq. (5), we can estimate the phase-
shift error in this border case to be 5:4° (single
CCD at λ1 ¼ 834:6 nm).
Once the matching parameters have been found,
the images are matched using an interpolation
scheme. An interpolated value depends on the values
of 16 ¼ 4 × 4 neighboring pixels (see Fig. 8). The un-
derlying irradiance function Iðx; yÞ is approximated
with a tensorial bicubic polynomial interpolation
scheme that provides a good accuracy of interpola-
tion of the oscillating part of the signal intensity:
Iðx; yÞ ¼
X
p;q
apqxp
yq
; with ðp; qÞ ¼ 0; …; 3:
The 16 coefficients fapqg are computed uniquely so as
to interpolate image irradiances Iði;jÞ at the 16 neigh-
boring pixels ði; jÞ.
Fig. 7. Matching of the images using the contour line of the fringe contrast (white polygonal curves). The vertices of this polygon are
extracted and matched to ideal positions of the vertices (cross markers). The original image (left) is matched to an ideal image (without
transformation) by using a bicubic polynomial scheme (right).
5482 APPLIED OPTICS / Vol. 47, No. 29 / 10 October 2008
11. E. Measurement Algorithms
1. Workflow in the Measurement Loop
The first task in the measurement loop is the acqui-
sition of eight images Ik
i ðx; yÞ at both wavelengths
(k ¼ 1 and 2) on the four CCDs (i ¼ 1; …; 4). This
is followed by a bicubic interpolation of the images
to match the pixels to the same physical location
on the segmented mirror. Then instantaneous phases
φkðx; yÞ are computed for both wavelengths. Finally
the system calculates the synthetic phase (piston),
tip, and tilt for all segments using a dual-wavelength
algorithm. In closed-loop operation the data are sent
continuously to the controller via the main network
of the APE experiment (see Fig. 9).
2. Determining the Instantaneous Phase from
Four Interferograms
From Eq. (2) we first compute the normalized irra-
diances for each camera CCDi (i ¼ 1; …; 4) and each
wavelength λk (k ¼ 1 and 2) using previously cali-
brated amplitude Ak
i;cal and background Bk
i;cal:
~Ik
i ðx; yÞ ¼
Ik
i ðx; yÞ − Bk
i;calðx; yÞ
Ak
i;calðx; yÞ
≈
Ik
i ðx; yÞ − Bk
i ðx; yÞ
Ak
i ðx; yÞ
¼ sinðφk
ðx; yÞ þ θk
i ðx; yÞÞ: ð6Þ
In practice, due to temperature variations and an al-
ways present calibration error, there is a drift of am-
plitude and background between the calibration and
the measurement. Because of this drift, the normal-
ized signal ~Ik
i cannot be exactly represented by a pure
sine function with unit amplitude but has a small
background and is not quite perfectly unitary nor-
malized:
~Ik
i ðx; yÞ ¼ ~Ak
ðx; yÞ sinðφk
ðx; yÞ þ θk
i ðx; yÞÞ þ ~Bk
ðx; yÞ;
ð7Þ
where ~Ak
≈ 1 is a normalized amplitude (close to 1),
and ~Bk
≈ 0 is a residual background (close to zero).
Quantities ~Ak
and ~Bk
are significantly similar for
all four CCDs provided a good linear response of the
CCDs and a negligible dark signal. Therefore we
dropped CCD index i for these two parameters
in Eq. (7).
Having inserted previously calibrated phase-shift
maps θk
i ðx; yÞ (constant by segment) into Eq. (7), the
wrapped instantaneous phase φk
ðx; yÞ as well as am-
plitude ~Ak
and background ~Bk
(now common to all
CCDs) are computed from the four normalized irra-
diances ~Ik;data
i ðx; yÞ by a least-squares method:
ðφk
ðx; yÞ; ~Ak
; ~Bk
Þ ¼ argmin
X
i¼1;…;4
× ðIk;model
i ðφk; ~A; ~BÞ − ~Ik;data
i Þ2; ð8Þ
where Ik;model
i is defined by Eq. (7). The notation
argminðQÞ refers to the set of parameters that mini-
mizes the quantity Q.
The permanent monitoring of amplitude ~Ak
serves
to constantly verify the GDD (position in the fringe
envelope) during a measurement run. If ~Ak
drops be-
low a given threshold, a recentering of the fringes is
triggered. A fine scan (see Subsection 3.C) is carried
out, and the translation stage is moved to a new
optimum position.
3. Removing the Optical Aberration
The measured wrapped instantaneous phase map
φk
ðx; yÞ (at both wavelengths k ¼ 1 and 2) not only
contains information about the height profile of
the segmented mirror but also includes a wavefront
error (aberration) due to the nonperfect optical sys-
tem of the metrology and the beam expander. This
aberration must be subtracted from the phase maps
before calculating the target’s height profile.
We measure the aberration at both wavelengths se-
parately on a flat reference mirror of very high surface
quality (λ=10 PTV) that is placed at the same position
as the ASM. The measured aberrations show a PTV
amplitude of approximately 280 nm. They include
the flatness error of the reference mirror itself. When,
during a measurement on the segmented mirror, the
aberrationsaresubtractedfromthephasemapsatthe
twowavelengths,thisflatnesserrorisalsosubtracted,
(x,y)
(i−1, j−1)
(i−1, j)
(i−1, j+1)
(i−1, j+2)
(i, j−1)
(i, j)
(i, j+1)
(i, j+2)
(i+1, j−1)
(i+1, j)
(i+1, j+1)
(i+1, j+2)
(i+2, j−1)
(i+2, j)
(i+2, j+1)
(i+2, j+2)
Fig. 8. Patch of 4 × 4 neighboring pixels used for tensorial bicubic
polynomial interpolation at point ðx; yÞ. Interpolation is performed
to match the fringe images acquired by four CCDs.
Metrology
(h,α,β)MEAS
(h,α,β)REF
− Controller ASM
(h,α,β)REAL
Fig. 2
Fig. 9. Block diagram of the control loop of the APE experiment.
The part of the diagram highlighted by the dashed line corre-
sponds to the block diagram in Fig. 2.
10 October 2008 / Vol. 47, No. 29 / APPLIED OPTICS 5483
12. whichleadstoasystematicerrorforthepistonandtip/
tilt of each segment. This static and low spatial fre-
quency error is not of importance for the purpose of
the APE experiment, hence is neglected here.
4. Determining Piston, Tip, and Tilt
For a small inclination the surface height of each
mirror segment number m is modeled as
zmðx; yÞ ¼ hm þ αmx þ βmy þ δzmðx; yÞ; ð9Þ
where ðx; yÞ are the transverse coordinates of a point
on segment number m, here relative to its center; hm
is the piston; and ðαm; βmÞ are the tip and tilt angles
of the segment. The quantity δzmðx; yÞ is the flatness
error of the segment defined as the deviation of the
segment surface from a nominal flat surface.
On the ASM, the flatness error δzm can reach a
PTV amplitude of approximately 50 nm (see Subsec-
tion 1.B). It is characterized by an independent cali-
bration measurement and then subtracted from the
instantaneous phases at both wavelengths (λk, k ¼ 1
and 2) at each time step:
~φm
k ðx; yÞ ¼ φm
k ðx; yÞ −
4πδzmðx; yÞ
λk
: ð10Þ
This yields the equation of a flat surface for segment
number m:
~zmðx; yÞ ¼ zmðx; yÞ − δzmðx; yÞ ¼ hm þ αmx þ βmy:
ð11Þ
To reduce the computation effort, a combined surface
error is generated for each wavelength by adding the
flatness error of target δzm to the system aberration
(see Subsection 3.E.3). This combined error is then
subtracted at each time step from the instantaneous
phases prior to the computation of piston, tip,
and tilt.
We evaluated several approaches to compute the
piston, tip, and tilt from instantaneous phases ~φm
k
and combined the results for the two wavelengths
to resolve the piston ambiguities.
Unwrapping approach: The first approach applies
a two-dimensional-unwrapping method on the
wrapped instantaneous phase. Standard approaches
are, for example, the Goldstein, gradient-guided, or
variance-guided algorithms [12]. Linear phase maps
on each segment for each wavelength λk can then be
calculated by fitting a plane through the unwrapped
phase map. Average phases hφk
mi (k ¼ 1 and 2) on seg-
ment m and the slope of the phase maps are esti-
mated. The piston of the segment is calculated by
resolving the ambiguities using two average phases
hφk
mi corresponding to the wavelengths using a stan-
dard dual-wavelength algorithm [5]. Unfortunately
two-dimensional-unwrapping is very intensive com-
putationally, and it is not possible to apply this tech-
nique at the required measurement frequency for our
metrology system.
Pixel-based dual-wavelength approach: The second
approach consists of solving the phase ambiguities
for each pixel of the instantaneous phase map using
the same standard dual-wavelength algorithm as for
the unwrapping approach. The resolution of the am-
biguities using phases at two wavelengths yields to a
synthetic phase map. The synthetic phase map on
each segment is next fitted by a plane, and the
piston, tip, and tilt of the segment can be estimated.
Unfortunately the noise on the measured phase of
each pixel (approximately 12 nm rms surface at each
wavelength) is too large for exact determination of
the integer ambiguities (see Appendix B). When
averaging the synthetic phase map, the noise of
the resulting piston is equal to the noise of the mono-
wavelength phase amplified by dual-wavelength
scaling factor s ¼ 0:5ðλ1 þ λ2Þ=jλ2 − λ1j ≈ 34, i.e., in ad-
dition to not being able to resolve the integer ambi-
guities, we would obtain a fairly large piston noise of
approximately 6:7 nm (per segment).
Flat surface dual-wavelength algorithm: Since both
above-described methods are not acceptable in our
case because of either too demanding requirements
on the computer performance (two-dimensional-
unwrapping) or a too large noise (pixel-based dual-
wavelength method), we have developed an algorithm
to estimate piston, tip, and tilt for a discontinuous and
piecewise flat surface (such as a segmented mirror).
This so-called flat surface dual wavelength (FSDWL)
algorithm can be split into two steps. The first step
consists of computing the slopes of the instantaneous
phase maps (at both wavelengths) on each segment.
Then, in the second step, the mean instantaneous
phases for both wavelengths are computed, which
serve to determine the synthetic phase (i.e., piston)
by resolving the phase ambiguities.
The calculation of the slopes (in the x and y direc-
tions) of the instantaneous phase maps ~φk
mðx; yÞ—see
Eq. (9)—is performed as follows:
∂~φk
m
∂x
¼
Mtotal
ΔpNpixel
X
i;j
ð~φk
mðxiþ1; yjÞ − ~φk
mðxi; yjÞ þ f i;j2πÞ;
ð12Þ
where Mtotal is the total magnification of the optical
system (metrology and beam expander), Δp is the
linear dimension of a CCD pixel, Npixel is the number
of pixels on segment m, and f i;j is a pixel-dependent
integer number (0; Æ1; Æ2; …), uniquely determined
such that
j~φk
mðxiþ1; yjÞ − ~φk
mðxi; yjÞ þ f i;j2πj ≤ π: ð13Þ
Similarly the slope in the y direction is calculated as
∂~φk
m
∂y
¼
Mtotal
ΔpNpixel
X
i;j
ð~φk
mðxi;yjþ1Þ−~φk
mðxi;yjÞþgi;j2πÞ;ð14Þ
5484 APPLIED OPTICS / Vol. 47, No. 29 / 10 October 2008
13. with integer parameter gi;j defined in an analogous
way to f i;j [Eq. (13)].
The x and y slopes are then subtracted from the
instantaneous phases to obtain information about
mean value hφk
mi of the phase at each wavelength
λk (k ¼ 1 and 2):
4πhm
λk
¼ hφk
mi ≈
~φk
mðxi; yjÞ −
∂~φk
m
∂x
xi −
∂~φk
m
∂y
yj
× mod 2π; ∀ pixeli;j: ð15Þ
In the case of zero measurement noise and a perfect
knowledge of the system aberration and flatness
error of the target, the right-hand side of Eq. (15)
would no longer be dependent on pixel indices
ði; jÞ. The approximately equal sign is used to denote
the residual dependence on pixel coordinates ði; jÞ.
The following formula averages mean phase hφk
mi
over all pixels while removing the 2π ambiguity:
hφk
mi ≈ tan−1
2
6
4
P
i;j sin
~φk
mðxi; yjÞ − ∂~φk
m
∂x
xi − ∂~φk
m
∂y
yj
P
i;j cos
~φk
mðxi; yjÞ − ∂~φk
m
∂x
xi − ∂~φk
m
∂y
yj
3
7
5:
ð16Þ
The mean tip/tilt angles of segment m are computed
as mean values of the slopes computed at both wave-
lengths:
αm ¼
1
8π
λ1
∂~φ1
m
∂x
þ λ2
∂~φ2
m
∂x
; ð17Þ
βm ¼
1
8π
λ1
∂~φ1
m
∂y
þ λ2
∂~φ2
m
∂y
: ð18Þ
Finally we estimate piston hm of segment number
m by determining two integer fringe orders ck
m from
two mean phases hφ1
mi and hφ2
mi [5]:
4πhm
λk
¼ hφk
mi þ ck
m2π; ðk ¼ 1; 2Þ: ð19Þ
One can show that the mean phase estimated by the
nonlinear Eq. (16) is less affected by noise compared
to a direct linear estimation from the pixel-based
phases. This is why integer fringe orders ck
m can be
accurately determined (see Appendix B). The
FSDWL algorithm allows a very stable and robust
piston estimation.
The correction applied to the phase maps in the
FSDWL algorithm is to compute the local anisotropic
(i.e., different in x and y directions) ambiguities f i;j
and gi;j in the tip/tilt calculation [see Eqs. (12) and
(14)]. This is considerably faster than any two-
dimensional-unwrapping method. A time-consuming
calculation is the evaluation of the sin and cos func-
tions to estimate mean phase hφk
mi in Eq. (16). Finally
the FSDWL algorithm is compatible with the re-
quired measurement frequency of 4:44 Hz. In a future
application this algorithm could be extended to a mul-
tiple (more than two) wavelengths configuration.
Remark on vibration: Because the two dual-
wavelength sets of images are not acquired simulta-
neously but with a time delay of 25 ms in between
(see Subsection 2.C.1), mechanical vibrations and
air turbulence during this time period can contribute
significantly to the OPD error between pistons mea-
sured at two wavelengths (see Appendix B). In order
to minimize the sensitivity to vibration, the mean
phases of the central mirror segment (m ¼ 0) hφk
0i
(k ¼ 1 and 2) are subtracted from the mean phases
of all other segments. Thus a common part of the
OPD error due to vibration is eliminated.
4. Experimental Results
We present results of closed-loop measurements on
the ASM. The loop between the control unit and
the metrology is closed (see Fig. 9). The proportional
integral controller implemented in the control unit
computes the corrections applied to the piezo actua-
tors of the segments. The bandwidth of the closed-loop
control system is approximately 0:2 Hz. The sampling
frequency of the loop is 4:44 Hz, i.e., the metrology
sends its measured data (piston h, tip α, and tilt β
of 61 segments) at this rate. The reference of the loop
is zero for piston, tip, and tilt (i.e., ðh; α; βÞREF ¼
ð0; 0; 0Þ in Fig. 9). This means that the loop tries to
keep all segments at the same height and parallel
to each other and to the reference beam of the metrol-
ogy. The piston mode of the central segment (number
m ¼ 0) is not controlled. Hence we do not try to correct
for the fluctuations of the OPD between the two inter-
ferometer arms. During a measurement of long dura-
tion (for example, during one night) this OPD varies
by typically several micrometers. This is caused by
differential thermal expansion of the two arms, each
having a length of approximately 2257 mm. The mea-
surement range (in piston) for a given measurement is
ÆΛs=4 ≈ Æ7:2 μm, centered around the piston of the
central segment as of the previous measurement.
This means that the measurement range dynamically
follows the movement of the central segment.
To reinforce the robustness of the metrology during
closed-loop operation, the measured piston (the so-
called raw piston) is corrected using a two stage cor-
rection to avoid occasionally occurring piston steps.
Those are glitches caused by external disturbing fac-
tors such as vibration and air turbulence (see Appen-
dix B for an explanation of this effect). The first stage
of correction is based on the analysis of a history pis-
ton vector of previous measurements whose length is
configurable. All tests are performed with a history
vector of length 20. This correction effectively elim-
inates occasional piston steps of λ=2 ≈ Æ423:5 nm.
The second stage of correction is activated once the
segmented mirror has converged toward the desired
target position, and this position is maintained by
10 October 2008 / Vol. 47, No. 29 / APPLIED OPTICS 5485
14. the control loop. During this time the metrology will
eliminate any piston step of λ=2 based on its knowl-
edge that this step does not correspond to the reality.
The results of the closed-loop experiment pre-
sented here are continuously recorded during a per-
iod of time of approximately 7 h. This experiment not
only demonstrates the functionality of the metrology
system but also serves to characterize the OPD error.
Because the segmented mirror is maintained at a
fixed position, the mean values of instantaneous
phases hφk
mi of all segments defined in Eq. (16) re-
main stable during that period. Nevertheless a slow
change still occurs during the measurement period
because of the differential thermal expansion of
the two interferometer arms. For each segment
m ¼ 0; …; 60, the phase hφk
mi at each wavelength λk
can thus be unwrapped along the time axis and then
converted to an OPD:
OPDk
m ¼
λk
2π
unwraphφk
mi; for m ¼ 0; …; 60 and
k ¼ 1; 2: ð20Þ
The differential OPD is defined as the difference
between the OPD at λ2 and λ1:
δOPD21
m ¼ OPD2
m − OPD1
m; for m ¼ 0; …; 60: ð21Þ
This quantity combines the OPD errors of the metrol-
ogy at both wavelengths. In an ideal system it would
be equal to zero.
Figure 10 (left) shows the quantity OPD1
1 (OPD of
segment number m ¼ 1 measured at λ1) as a function
of time. The differential thermal expansion of probe
and reference arms yields an OPD variation reaching
8 μm. The OPDs of the remaining segments and
those measured at the second wavelength λ2 are si-
milar to OPD1
1 and are not shown.
On the right-hand side of Fig. 10, the differential
OPD of segment 1 relative to the central segment
(δOPD21
1 − δOPD21
0 ) is plotted against OPD1
1. The dif-
ferential OPD of central segment δOPD21
0 is sub-
tracted such that the common piston variation of
the two segments (m ¼ 0 and 1) during the time
lag between the acquisitions at two wavelengths is
removed. This subtraction reflects the fact that in
the measurement algorithm, the mean phases of cen-
tral segment hφk
0i (k ¼ 1 and 2) are subtracted from
the mean phases of all other segments prior to resol-
ving the phase ambiguities (see Subsection 3.E.4).
There exist various causes for a nonzero differential
OPD as depicted in Fig. 10. We distinguish four types
of error sources: (1) optics-related, (2) calibration-
related, (3) environment-related, and (4) optoelectro-
nics-related. An error source of the optics-related type
is a variation of the optical aberration (at λ1 and λ2)
between the moment it is measured on the reference
0 2 4 6
0
1
2
3
4
5
6
7
8
−3 −2 −1 0 1
Segment 1 – Segment 0 (Central Segment)
847.1 nm
Drift of OPD in Closed Loop
Time [h] δOPD21
[nm]
OPD1
[µm]
Fig. 10. Left: Variation of the OPD at λ1 during approximately 7 h of measurement. Right: Differential OPD (δOPD21
1 − δOPD21
0 ) of
segment 1 relative to the central segment as a function of the OPD at λ1.
5486 APPLIED OPTICS / Vol. 47, No. 29 / 10 October 2008
15. mirror and the moment it is subtracted from the in-
stantaneous phases during the measurement (see
Subsection 3.E.3). Such a variation can be caused
by temperature-induced deformation of the optical
and mechanical components. Optics-related error
sources are also contributions to δOPD21
due to polar-
ization crosstalk in the circulator-type setup of the
phase modulator (see Subsection 2.C.2). A nonperfect
PBS (PBS1 in Fig. 4) causes this error. The polariza-
tions of probe and reference beams are not purely lin-
ear when entering the signal decoder (i.e., in front of
WP4 in Fig. 4). Those optics-related error sources
cause a periodic variation of the differential OPD with
the phase or OPD.
The calibration-related error sources causing a
nonzero δOPD21
are variations in the delay line ve-
locity and errors in the matching and the interpola-
tion of the images (see Subsection 3.D and
Appendix A). Typically calibration errors yield a per-
iodic behavior of δOPD21 with respect to absolute
phase as shown in Fig. 12.
Environment-related error sources are mechanical
vibrationandairturbulencethatactinadifferentway
on the two OPDs at λ1 and λ2 since the image acquisi-
tions are separated in time by 25 ms. The optoelectro-
nic noise-related error sources are detection noise of
the CCDs and fluctuations of the SLD powers. Both
errors due to external environmental factors and op-
toelectronic noiseshow random signatures inδOPD21
.
In the right plot we notice that differential error
δOPD21
has a clearly periodic signature (variation
at the period of mean optical wavelength
λ ≈ 847:1 nm). This indicates that the main contribu-
tions to δOPD21
are calibration and optical errors.
The random contribution remains small compared
to these systematic error sources.
Note that the differential OPD error is not quite
centered around zero. This offset of −1 nm could be
due to the phase error of the central segment at in-
itial time t ¼ 0 that is reset to zero by software con-
vention or because of the variation of the aberrations
at two wavelengths. The peak of the OPD error is less
than 3 nm, which is smaller by a comfortable margin
than the maximum tolerance of Æ12:48 nm, a limit at
which the dual-wavelength cyclic ambiguities can no
longer be correctly resolved (see Appendix B). This
means that no piston steps of λ=2 ≈ Æ423:5 nm oc-
curred during this measurement run. The processing
of the raw piston by the above-described two-stage
correction algorithm was not required.
The differential OPD errors of other segments (not
shown) all have a similar type of periodic behavior as
with segment number 1.
Figure 11 illustrates the metrology performance of
the system working in closed loop. For each segment,
N ¼ 105
, piston and tip/tilt data are recorded during
approximately 7 h. The set point command for the
loop corresponds to a flat segmented mirror, i.e.,
ðh; α; βÞREF ¼ ð0; 0; 0Þ in Fig. 9.
The standard deviation σm;h of piston hm of each
segment is computed from all piston data (index j) as
σm;h ¼
1
N
X
j¼1;…;N
h2
m;j
1
2
: ð22Þ
A similar formula is used to compute standard devia-
tion σm;γ of the combined tip and tilt γm;j:
σm;γ ¼
1
N
X
j¼1;…;N
γ2
m;j
1
2
; ð23Þ
γm;j ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
α2
m;j þ β2
m;j
q
: ð24Þ
Fig. 11. Standard deviation of piston and tip/tilt in closed loop during 7 h. While the piston is measured for segments 0; …; 60 relative to
the central segment, the tip and tilt are measured for all 61 segments. The standard deviation of pistons of all segments all together is
0:48 nm rms. The standard deviation of tip/tilt of all segments all together is 74 nrad rms.
10 October 2008 / Vol. 47, No. 29 / APPLIED OPTICS 5487
16. Standard deviations σm;h and σm;γ are respectively
plotted in the left and right graphics of Fig. 11. We no-
tice that the standard deviation of piston is smaller in
the center of the mirror and increases monotonically
with the radius. The explanation is as follows: solely
the motion of the mirror perpendicular to the wave-
front is eliminated when the instantaneous phases
of the central segment are subtracted from the phases
of other segments. The effect of a global tip/tilt motion
of the target mirror about the central segment still
prevails and thus in turn implies a larger noise at
outer rings. In the right-hand graphic, we observe
three segments that have noticeably worse standard
deviations in tip and tilt. Two of these segments (both
on the horizontal line passing through the mirror’s
center) have physical holes (used for alignment pur-
poses of the APE experiment) and hence have a smal-
ler number of pixels that can be used for image
processing. The other segment suffered from a minor
issue linked to one of its three piezo actuators that will
soon be repaired. Finally the rms of piston, tip, and tilt
standard deviations of all segments are computed.
The values are 0:48 nm for the piston and
0:074 μrad for the tip/tilt. These numbers include
the noise of the actuators. Intrinsical noises (not
shown here) of the metrology system alone (i.e., with-
out control loop, thus piezo actuator error excluded)
are characterized, and they are about half the noise
in closed-loop run presented here.
5. Summary and Conclusion
We present a noncontact optical metrology used for
the APE that aims at demonstrating optical phasing
technologies and methods for a future ELT. The
0 100 200 300
−2
−1
0
1
2
PhaseError[deg]
Hariharan’s estimation; ∆A = 10
0 100 200 300
−2
−1
0
1
2
Larkin’s estimation; ∆A = 10
0 100 200 300
260
270
280
290
300
310
EstimateA
Hariharan’s estimation (true=300)
0 100 200 300
260
270
280
290
300
310
Larkin’s estimation (true=300)
0 100 200 300
300
350
400
Phase [deg]
EstimateB
B=(I1+I2+2*I3+I4+I5)/6 (true=350)
0 100 200 300
300
350
400
Phase [deg]
B=(I1+2*I3+I5)/4 (true=350)
∆V=−0.2 ∆V=−0.1 ∆V=0 ∆V=0.1 ∆V=0.2
Fig. 12. Errors of estimation of phase (top row), amplitudes (middle row), and background (bottom row). The left-hand column shows the
error obtained when using Hariharan’s method. For phase and amplitude the right-hand column refers to Larkin’s method, whereas for the
background it shows the error related to the background estimation formula [Eq. (A2)].
5488 APPLIED OPTICS / Vol. 47, No. 29 / 10 October 2008
17. metrology system measures piston, tip, and tilt of the
61 hexagonal flat segments of the ASM, a scaled-
down version of a segmented primary mirror with
a diameter (corner to corner) of 154 mm. Each seg-
ment is equipped with three piezo actuators. The sur-
face shape (i.e., piston, tip, and tilt for all segments)
is stabilized by a control loop using the metrology as
reference sensor.
Since the system shall be operated on a real tele-
scope platform (the VLT on Cerro Paranal, Chile), the
concept of instantaneous PSI has been chosen: The
phase measurements at each wavelength are inher-
ently insensitive to vibrations and air turbulence.
The presented work demonstrates the efficiency of
such an instantaneous phase-shifting interferometer
tomeasureadiscontinuoussurface.Theexperimental
results show the possibility to achieve subnanometer
precision for piston, tip, and tilt measurements from a
working distance of more than 2 m in air. During a
measurement in closed loop with a duration of several
hours, the standard deviations (of all segments) for
piston and tip/tilt are 0:48 nm rms and 74 nrad rms,
respectively. These standard deviations include the
noise of the actuators, i.e., are larger than the noise
that can be attributed to the metrology.
The image acquisition on four CCDs and the calcu-
lations can be performed at a frequency of 4:44 Hz, al-
lowingtheuseofthemetrologyinclosed-loopcontrolof
the ASM. Surface discontinuities at segment borders
canbemeasuredwithoutambiguitybyusingthedual-
wavelength technique. An algorithm robust to phase
noise and specifically designed for piecewise flat sur-
faces has been developed and tested successfully.
Using SLDs operating at two near-infrared wave-
lengths with (additionally filtered) spectral band-
widths of approximately 3:2 nm, i.e., round-trip
coherence lengths of approximately 100 μm, the
metrology also profits from the advantage of low-
coherence interferometry: Its measurements are less
affected by speckle noise and are not perturbed by
parasitic internal reflections.
For the implementation of the system, high atten-
tion must be paid to errors in the OPD measure-
ments at the two wavelengths. The differential
OPD between λ1 and λ2 must be kept small enough
(around Æ12:5 nm) to avoid steps in the measured
piston. This is achieved by use of an enclosure that
protects the probe and reference beams from air tur-
bulence and by minimizing the time lag between the
image acquisitions at the two wavelengths (mini-
mum value equal to 25 ms determined by the image
acquisition frequency of 40 Hz).
A possible future improvement could be the use of
eight CCDs instead of four—with simultaneous ac-
quisition of four images at λ1 and four images at
λ2. This would further reduce the system’s sensitivity
with respect to vibrations and air turbulence. A sec-
ond possible improvement could be to use a concept
that employs more than two wavelengths to increase
the unambiguous measurement range.
Regarding other possible applications the metrol-
ogy could also be used as a sensor for optical phasing
of a real-size segmented mirror. However, in practice
this application is restricted to spherical surface
shapes that allow operation without a gigantic beam
expander. Another promising idea is use as a sensor in
an instrument for the calibration of capacitive or in-
ductive edge sensors for segmented mirror phasing.
This activity was supported by the European Com-
munity (Framework Programme 6, Extremely Large
Telescope Design Study, contract 011863). The
authors thank A. Bietti for providing the basic design
of the software, P. Millepied and V. Seiller for their
contributions during their student internships, B.
Sedghi and R. Frahm for closing the loop, C. Dupuy
for the final alignment on the APE bench, and K.
Larkin for fruitful discussions about phase calibra-
tion algorithms.
Appendix A: Choice of Demodulation Algorithm for the
Calibration
As explained in Subsection 3.D, the calibration of
phase shifts between CCDs, fringe amplitude, and
background is performed by temporal demodulation
of the fringe patterns detected during a scan of the
delay line. We present the results of a comparison
of different demodulation methods that led us to
the choice of the preferred algorithms to be imple-
mented in the software.
We consider a given position ðx; yÞ on the target
and the CCD pixel corresponding to this physical lo-
cation (for all CCDs). If we assume that the delay line
moves along the z axis, the fringe irradiance depends
on z as written in Eq. (2), where θk
i is the phase of the
interferogram detected on CCDi for wavelength λk
(k ¼ 1 and 2). We consider a single wavelength λ,
i.e., θk
i ¼ θi. We assume that amplitude AðzÞ is con-
stant or varies linearly with respect to z (with, how-
ever, a small slope, i.e., its variation is less than 3%
within a cycle of a fringe period of λ=2 in the z
direction).
We sample the pixel irradiance at five points, ide-
ally with a π=2 phase shift between two consecutive
samples. In practice, since the velocity of the motor
could have a (constant) error, the sampling points—
numbered by j—are stretched as follows:
zj ¼ ðj − 3Þ
λ
8
ð1 þ ΔvÞ; for j ¼ 1; …; 5: ðA1Þ
Relative velocity error Δv is typically within the in-
terval Æ0:2 (¼ Æ20%). We assume that the velocity
error is unknown, so are the delay-line positions zj.
Given Ij ¼ IðzjÞ for j ¼ 1; …; 5, the quantities
fθ; Aðz3Þ; Bg (i.e., phase, amplitude at midpoint
z3 ¼ 0, and background) can be determined by a
PSI demodulation technique. The objective is to
choose a set of demodulation formulas that is robust
with respect to the three quantities: (1) random
noise, (2) linear slope of amplitude AðzÞ, and espe-
cially (3) velocity error Δv.
10 October 2008 / Vol. 47, No. 29 / APPLIED OPTICS 5489
18. In addition it is preferable to use formulas that
have a phase error that depends as weakly as possi-
ble on absolute phase θ. We refer to this as phase-
independency property P. The reason why P is
important is that we are interested in the phase shift
between two cameras. Their absolute values have
little relevance in our application.
In Fig. 12 we have compared by simulation the de-
modulation errors for the following two five-step
methods: the well known Hariharan algorithm [4]
and Larkin’s five samples adaptive nonlinear algo-
rithm [13]. For this analysis we have simulated a
linear amplitude variation AðzÞ ¼ A0 þ ϵðzÞ, with
A0 ¼ 300 (in CCD digitalizationunits), and linear var-
iation ϵðzÞ with a PTV amplitude of Æ10 over one
fringe period (i.e., ϵðzÞ ¼ ΔA × 2z=λ, with ΔA ¼ 10).
The background is assumed as constant (B ¼ 350).
We have simulated with five velocity errors varying
from −0:2 to 0.2: Δv ¼ f−0:2; −0:1; 0; 0:1; 0:2g; Fig. 11
shows the results of the simulation. The five curves in
each graphic correspond to the five velocity errors.
Phase detection: The top row of Fig. 11 shows the
difference between true absolute phase θ and the es-
timated phase given by both demodulation methods
(left: Hariharan; right: Larkin). With Hariharan’s al-
gorithm the error (3:3° PTV) shows a clear depen-
dence on the absolute phase, hence the criterion P
is not satisfied. With Larkin’s formula the absolute
phase error is smaller (0:62° PTV), and the property
P is satisfied almost everywhere, except for the cases
where absolute phase θ gets close to π=2 or 3π=2
(where the cosine of absolute phase θ vanishes). In
this figure the difference of estimated phases for
two cameras can be read as differential ordinate of
two points on a curve separated in their abscissas
by the true (unknown) phase difference between
these cameras. Absolute phases, i.e., abscissas of
the points on the error curve, vary arbitrarily each
time a calibration is performed. Therefore, to get a
reliable estimate of the phase shift between two cam-
eras, it is important to have a flat error curve. The
absolute level of the curve is not important.
Envelope detection: In the middle row of Fig. 11, we
plot the estimated amplitude A with respect to the
phase θ given by both methods. The true amplitude
assumed for the simulation is A0 ¼ 300. Larkin’s
method satisfies well property P, whereas Harihar-
an’s algorithm clearly does not. However, the maxi-
mum error produced by Larkin’s method matches
the maximum error obtained by using Hariharan’s
method. As with the phase detection (see above), it
is preferable to select a method that well satisfies
property P. This is because the estimated amplitudes
will be used later to normalize the fringe irradiances
among the cameras before computing the instanta-
neous phase, and property P ensures that the irra-
diances on the four CCDs will be uniformly
normalized (see Subsection 3.E.2).
Background detection: For the estimation of back-
ground B, we have compared Hariharan’s method
with the following self-developed linear estimation
formula:
Bestimate ¼ ðI1 þ 2I3 þ I5Þ=4: ðA2Þ
One can show that this is a unique linear formula
providing an estimation that is not affected by linear
variation of the amplitude during the scan. More
precisely, one can show that
Bestimate ¼ Btrue þ sinðθÞACΔv2
;
where C is a constant. Thus the estimation of B using
Eq. (A2) has an error that satisfies the following
three properties: (1) the error is independent of
the slope of a linear amplitude variation AðzÞ, (2)
it depends only in the second order of velocity error
Δv, and (3) it depends in the first order on the sinus
of absolute phase θ. Because of (3) the background
estimation with Eq. (A2) does not well satisfy P un-
less the translation velocity exactly matches its ideal
value (Δv ¼ 0).
To our knowledge there exists no (nonlinear) for-
mula for estimation of background B that combines
the properties (1) and (2) with P—by the same token
as Larkin’s formulas for the phase and amplitude.
The bottom row of Fig. 11 compares the error in
background estimation when using Hariharan’s al-
gorithm (left) and Eq. (A2) (right). For the true back-
ground we assumed the value Btrue ¼ 350.
As a conclusion of this analysis, we have chosen to
use Larkin’s algorithm both for amplitude and phase
and Eq. (A2) for the background.
Appendix B: Tolerance to Differential Optical Path
Difference Errors
The metrology system uses a dual-wavelength
technique that allows the performance of absolute
distance measurements within a range of unambigu-
ity equal to Λs (in OPD), where Λs is the synthetic
wavelength defined in Eq. (1).
The dual-wavelength measurement uses two phase
measurements, φ1
and φ2
, measured at λ1 and λ2. The
fringe orders c1
and c2
¼ 0; Æ1; Æ2; … can be deter-
mined from the condition that the OPDs measured
at both wavelengths must be equal:
φ1
2π
þ c1
λ1 ¼
φ2
2π
þ c2
λ2: ðB1Þ
Errors δφ1 and δφ2 for the phases measured, respec-
tively, at λ1 and λ2 that lead to an change of Δc in both
fringe orders can be calculated from
φ1
þ δφ1
2π
þ c1
þ Δc
λ1 ¼
φ2
þ δφ2
2π
þ c2
þ Δc
λ2:
ðB2Þ
Subtracting Eq. (B1) from Eq. (B2) and replacing
δφk=2π × λk by the error δOPDk (k ¼ 1 and 2), one
obtains
5490 APPLIED OPTICS / Vol. 47, No. 29 / 10 October 2008
19. δOPD2 − δOPD1 ¼ −Δcðλ2 − λ1Þ;
or, expressed with the differential OPD errors
between two wavelengths [defined in Eq. (19)],
−
δOPD21
λ2 − λ1
¼ Δc: ðB3Þ
Since Eq. (B1) is numerically solved for the pair of
integers fckg that matches best the two phase mea-
surements, the error in fringe order Δc in Eq. (B3) is
equal to 1 (one fringe order) when
−
3
2
ðλ2 − λ1Þ ≤ δOPD21
≤ −
1
2
ðλ2 − λ1Þ:
More generally the value of Δc can be deduced from
the position of δOPD21
within the subdivision:
… −
5
2
ðλ2 − λ1Þ −
3
2
ðλ2 − λ1Þ −
1
2
ðλ2 − λ1Þ
þ
1
2
ðλ2 − λ1Þ þ
3
2
ðλ2 − λ1Þ …
In our setup ðλ2 − λ1Þ ¼ 24:97 nm. The critical step
boundaries for the differential OPD are Æ12:5 nm,
Æ37:45 nm, Æ62:42 nm; …. An error in the determi-
nation of the fringe order results in an OPD step that
equals a multiple of the mean optical wavelength λ ≈
847:1 nm (or 423:5 nm in piston). This is because the
resulting OPD is computed as the mean value of the
two OPDs at λ1 and λ2.
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based on the natural demodulation of two-dimensional fringe
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10 October 2008 / Vol. 47, No. 29 / APPLIED OPTICS 5491
![Dual-wavelength low-coherence instantaneous
phase-shifting interferometer to measure
the shape of a segmented mirror
with subnanometer precision
Rainer Wilhelm,1,3
Bruno Luong,1,
* Alain Courteville,1
Sébastien Estival,1
Frédéric Gonté,2
and Nicolas Schuhler2
1
FOGALE nanotech, 125, Rue de l’Hostellerie, 30900 Nîmes, France
2
European Southern Observatory, Karl-Schwarzschildstrasse 2, 85748 Garching b. München, Germany
3
Current affiliation: EADS Astrium GmbH - Satellites, 88039 Friedrichshafen, Germany
*Corresponding author: b.luong@fogale.fr
Received 1 July 2008; accepted 6 August 2008;
posted 20 August 2008 (Doc. ID 98097); published 8 October 2008
We present a noncontact optical metrology measuring the pistons and tip/tilt angles of the 61 hexagonal
segments of a compact-sized segmented mirror. The instrument has been developed within the scope of a
design study for the European Extremely Large Telescope (E-ELT). It is used as reference sensor for
cophasing of the mirror segments in closed-loop control. The mirror shape is also measured by different
types of stellar light-based phasing cameras whose performances will be evaluated with regard to a fu-
ture E-ELT. Following a description of the system architecture, the second part of the paper presents
experimental results demonstrating the achieved precision: 0:48 nm rms in piston and 74 nrad rms in
tip/tilt. © 2008 Optical Society of America
OCIS codes: 120.3180, 120.3930, 120.3940, 120.5050.
1. Introduction
A. Cophasing of an Extremely Large Telescope
With several telescopes and stellar interferometers of
the 8–10 m diameter class in routine operation today,
the research in telescope design is now focusing on so-
called Extremely Large Telescopes (ELT) having
segmented mirrors with diameters D ranging from
30 to 100 m. Two examples for currently running
ELT projects are the European ELT (E-ELT)
(D ¼ 42 m, 984 segments) and the Thirty Meter
Telescope (TMT, D ¼ 30 m, 492 segments).
A very difficult problem common to all ELTs is the
active control of the segmented primary mirror. If not
actively controlled, the segments will move in a ran-
dom manner, being exposed to perturbations such as
gravitational forces, wind load, and other mechanical
forces. Without controlling the segments to maintain
the shape of the mirror defined by the optical design,
the spatial resolution of the telescope would be re-
duced to the one of a telescope whose primary mirror
had the diameter d of a single segment. To achieve a
spatial resolution comparable to a monolithic tele-
scope of large diameter D, the segmented mirror sur-
face must be controlled, i.e., the segments must be
phased, with an accuracy better than λ=13 rms (wave-
front) [1].
During the phasing procedure, three degrees of
freedom of each segment are actuated: translation
along the local optical axis (piston h) and rotation
about two axes perpendicular to the local optical axis
(tip α and tilt β).
Three hardware systems are required for phasing:
capacitive or inductive edge sensors providing
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![real-time information about relative segment displa-
cements, segment actuators compensating for these
displacements, and a phasing camera. An inner fast
control loop using the edge sensors and the actuators
provides fast correction for segment displacements. It
runs continuously. Since the edge sensors only mea-
sure displacements relative to a zero reference, this
reference must be provided by another sensor at reg-
ular intervals (for example, at the beginning of each
night of observation). Typically this task is performed
byaphasingcamerathatuses thelightofabrightstel-
lar source to measure the shape of the wavefront. This
method is called optical phasing or on-sky calibration.
B. Active Phasing Experiment
Within the scope of a design study for the E-ELT, the
Active Phasing Experiment (APE) aims at demon-
strating active wavefront control technologies for a fu-
ture ELT [2]. The experiment will validate different
stellar light-based phasing camera concepts by
testing them on a scaled-down segmented mirror,
the so-called Active Segmented Mirror (ASM)
(D ¼ 154 mm, 61 segments). Figure 1 shows a sche-
matic drawing of the ASM. Each hexagonal segment
has a flat-to-flat distance of d ¼ 17 mm. The gap
width is approximately 70–130 μm. The total dia-
meter (corner to corner) is D ¼ 154 mm. The optical
surfaces of the segments are flat [surface quality of
approximately 50 nm peak to valley (PTV)]. Each
segment is controllable in three degrees of freedom
(piston, tip, and tilt) by means of three piezo actuators
with mechanical strokes of Æ7:5 μm and a precision in
displacement of better than 0:2 nm. More details
about the ASM can be found in [3].
Closed-loop control with a bandwidth of approxi-
mately 0:2 Hz is used to apply a given pattern of pis-
ton, tip, and tilt to the segments and to maintain the
ASM in this shape for a time period longer than
the integration times of the phasing cameras
(typically 1–30 s).
We present a high-precision optical metrology sys-
tem that serves as the sensor in the control loop,
measuring the piston, tip, and tilt of the 61 segments
at a frequency of 4:44 Hz. In addition to its role as
sensor in the closed-loop control, it will be used as
a reference to qualify the precision of the different
phasing cameras. Following the ongoing tests at
the laboratories of the European Southern Observa-
tory in Garching, Germany, APE will be tested on-sky
at a Nasmyth focus of a Unit Telescope of the Very
Large Telescope (VLT) array on Cerro Paranal, Chile.
2. System Description of the Metrology
A. Metrology Specifications
Table 1 lists the metrology specifications. All mea-
surements (piston, tip, and tilt) are performed rela-
tive to the central segment whose piston is not
controlled in closed loop. The specified measurement
range for piston (Æ7:2 μm) is covered by the range of
displacement for the piezo actuators (Æ7:5 μm).
The measurement is performed through a beam
expander optic (magnification factor 1=10) that is
not part of the metrology’s optical system. The beam
expander is placed between the exit pupil of the me-
trology and the target (ASM). The optical path length
between the exit pupil and the ASM is approximately
2200 mm. The optical layout of the complete APE
configuration is shown in [2].
B. Metrology Concept
The system combines three well-established concepts
of interferometric metrology. Those are (1) instanta-
neous phase-shifting interferometry, (2) low-
coherence interferometry, and (3) dual-wavelength
interferometry. We briefly explain the basics of these
three techniques and why they have been chosen for
the present application.
Fig. 1. ASM with 61 segments, each equipped with three piezo
actuators. The flat-to-flat distance of a single segment is
d ¼ 17 mm. The total diameter of the mirror (corner to corner)
is D ¼ 154 mm.
Table 1. Achieved Specifications of the Metrology
Item Specification
Measurement range for pistona
Æ7:2 μm
Measurement range for tip/tiltb
Æ250 μrad
Working distancec
2200 mm
Piston precision (closed loop) 0:48 nm rms
Tip/tilt precision (closed loop) 74 nrad rms
Measurement frequency 4:44 Hz
Closed-loop bandwidth 0:2 Hz
Frame-to-frame time difference 25 ms
a
The piston range is defined with respect to the piston of the
central segment.
b
The tip/tilt range is defined at the level of the target
(i.e., the ASM).
c
The working distance is the distance between the exit pupil of
the metrology and the target (ASM).
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