Vibration-Compensated Interferometer for Surface Metrology

Abstract

An advanced interferometer was built for surface metrology in environments with severe vibration. This instrument uses active control to compensate for effects of vibration to allow surface measurement with high-resolution phase-shifting interferometry. A digital signal processor and high-speed phase control from an electro-optic modulator allows phase measurements at 4000 Hz. These measurements are fed back into a real-time servo in the digital signal processor that provides a vibration-corrected phase ramp for the surface measurements taken at video rates. Unlike fringe locking, which compensates vibration to keep the phase constant, we show a true phase servo that allows the phase to be stabilized while it is ramped, enabling surface measurements using phase-shifting interferometry that requires multiple images with controlled phase shifts.

Full text

Vibration-compensated interferometer for
surface metrology
Chunyu Zhao and James H. Burge
An advanced interferometer was built for surface metrology in environments with severe vibration.
This instrument uses active control to compensate for effects of vibration to allow surface measurement
with high-resolution phase-shifting interferometry. A digital signal processor and high-speed phase
control from an electro-optic modulator allows phase measurements at 4000 Hz. These measurements
are fed back into a real-time servo in the digital signal processor that provides a vibration-corrected phase
ramp for the surface measurements taken at video rates. Unlike fringe locking, which compensates
vibration to keep the phase constant, we show a true phase servo that allows the phase to be stabilized
while it is ramped, enabling surface measurements using phase-shifting interferometry that requires
multiple images with controlled phase shifts. © 2001 Optical Society of America
OCIS codes: 120.3180, 120.3930, 120.5050, 220.4830.
1. Introduction
The phase-shifting interferometer is frequently used
for surface measurements because of its high accu-
racy and high spatial resolution. The fact that it has
to measure sequential frames while controlling phase
during a period of time makes it vulnerable to exter-
nal vibration. Vibration can cause errors in surface
measurement and if severe enough will completely
wash out fringes. This is particularly true during
testing of large mirrors where it is impossible to put
the interferometer and the mirror on the same optical
table.
There are a number of ways to reduce the effect of
vibration on surface measurement with a phase-
shifting interferometer. The most straightforward
one is to reduce the time over which the data frames
are collected.
1
A high-speed CCD camera and
frame grabber are used in this method. Another
solution to the vibration problem is to take all the
required data frames simultaneously
2
; then a sur-
face map is generated. Yamaguchi et al.
3
used a
spatial filter and a detector to detect fringe move-
ment, and the signal is fed back into a piezoelectric-
transducer-driven reference mirror as part of a
servo loop. Although good results are produced,
the technique is limited to only certain fringe spac-
ings.
3
Deck developed another solution.
4
Here
the fringes are amplitude split into two detectors,
one with high temporal and low spatial resolution
and the other with low temporal and high spatial
resolution. The high-temporal-resolution data are
used for accurate calculation of the phase-step in-
crement in the high-spatial-resolution data; then a
special algorithm is used to generate the surface
map. The topographic maps benefit from the best
qualities of both data sets—vibration insensitivity
and high spatial resolution.
This paper presents an effective and inexpensive
solution to the vibration problem. We use a closed-
loop servo system for actively measuring the instan-
taneous phase. If vibration is detected, then a
command is given to the phase shifter to compensate
for it. The servo maintains the phase at a specified
value rather than just stabilizing the fringes. Sec-
tion 2 of this paper gives a detailed description of the
vibration-compensation method. Section 3 shows
the layout of the interferometer we built with the
vibration-compensation servo. We use an electro-
optic modulator 共EOM兲 as the phase shifter in the
interferometer, and Section 4 explains how it works
and why we use it. In Section 5 the high-speed
phase measurement—the core of the vibration-
compensation method—is justified. We analyze the
performance limitations and error sources of the in-
terferometer in Section 6. The experimental result
C. Zhao 共czhao@u.arizona.edu兲 and J. H. Burge 共jburge@as.
arizona.edu兲 are with the Optical Sciences Center, University of
Arizona, 1630 E. University Boulevard, Tucson, Arizona 85721.
Received 21 February 2001; revised manuscript received 30 July
2001.
0003-6935兾01兾346215-08$15.00兾0
© 2001 Optical Society of America
1 December 2001 兾 Vol. 40, No. 34 兾 APPLIED OPTICS 6215

that proves that the interferometer can successfully
compensate for vibration is shown in Section 7.
2. Method Description
The interferometer we describe uses two nested
phase-shifting systems that use the same phase
shifter, but at different frequencies. One is the
high-speed system, which conducts a five-bucket
measurement of phase with a single photodiode.
This is used for the phase servo to measure and cor-
rect for vibration. The other is a low-frequency
surface-measurement system, which initiates a ramp
and measures the surface figure with use of a CCD
array. The high-speed servo drives the phase to
track this low-frequency ramp. Figure 1 shows the
block diagram of the two systems. Both systems use
the same phase shifter.
The vibration-compensation servo system consists
of three main components—phase shifter, photo-
diode, and digital signal processor 共DSP兲. The es-
sential feature of the servo is the high-frequency
phase measurement. The DSP sends out a phase-
shifting control voltage for quick ramping of the
phase by 2␲, during which the intensities are mea-
sured and digitized every ␲兾2 with the high-speed
photodiode. These intensities are fed back into the
DSP, and a five-step algorithm is used to calculate
the phase. This cycle takes 100 ␮s. If deviation
共due to vibration兲 from the phase command is de-
tected, a correction is made to the phase-shifting con-
trol voltage to compensate for the deviation. Next
the phase control voltage is held constant for 150 ␮s
followed by a 2␲ shift in the opposite direction; the
phase is then calculated, and vibration is compen-
sated just like in the previous cycle 共see Fig. 2兲.In
this way the system is measuring the phase and then
compensating for vibration at a frequency of 4000 Hz.
For making surface measurements with phase-
shifting interferometry the servo tracks a command
signal from a computer with the surface-
measurement software. The surface-measurement
software does not have any interaction with the
vibration-compensation system other than supplying
the control signal and using the stabilized images.
The high-speed servo compensates for the phase
noise from vibration and drives the phase to track the
command. The vibration-compensation servo is
running all the time; when a surface-measuring cycle
starts, the phase-shifting control signals for the two
systems are superimposed 共see Fig. 3兲.
The surface-measurement system, which uses a
CCD, does not see the high-frequency phase shift,
because the pixel intensities are integrated for 16 ms
and high-frequency measurement occurs every 0.25
ms. The high-frequency part is just averaged out
and causes a reduction in fringe contrast only. Since
the surface measurement never sees the servo, the
interferometer can be used with any phase-
measuring software. Thus we can use standard
phase-shifting interferometry for surface measure-
ments by calculating phase at each point in the array.
See Fig. 3 for the phase command structure when the
surface measurement is in process.
Since high-frequency phase measurements are
needed to close the servo loop, both the phase shifter
and the photodiode require rapid response. We use
a ConOptics 380c EOM as the phase shifter. Its
3-dB cutoff frequency is 250 kHz. The photodiode is
a New Focus 2001. It has a frequency response
greater than 200 kHz.
3. System Description
We implement the vibration-compensation mecha-
nism in a Twyman–Green interferometer. Figure 4
shows the layout of the interferometer. The
vibration-compensation servo loop and the surface-
measurement loop are clearly marked. With differ-
ent choice of diverger lens, the interferometer can
test different types of optics.
4. Electro-Optic Modulator
The key component of this interferometer is the
EOM. Cole et al. used an acousto-optic modulator
共AOM兲 as the phase shifter to build his interferome-
ter with the same vibration-compensation device.
5,6
The AOM shifts phase by changing the frequency of
the light. Compared with the AOM, the EOM has
advantages of higher light efficiency, easier align-
Fig. 1. Illustration of basic vibration servo. The active element,
EOM, makes a correction in response to external inputs 共from the
surface-measurement computer and vibration兲 to the system.
Fig. 2. Traces of the phase-shifter control signal and the signal on
the photodiode for the vibration-compensation servo.
6216 APPLIED OPTICS 兾 Vol. 40, No. 34 兾 1 December 2001

ment, and phase shift’s independence on the path-
length difference between reference and test arms.
The ConOptics 380C EOM used in this interferom-
eter is made of ammonium dihydrogen phosphate
共ADP兲 crystal and works on transverse modulation
共Fig. 5兲. The crystal is characterized by the follow-
ing parameters: height W, length d, index of refrac-
tion for the ordinary light n
o
, and index of refraction
for the extraordinary light n
e
. When a voltage V is
applied across the ADP crystal along the z axis 共the
Fig. 3. Basic phase command structure for the vibration-compensation method. The photodiode intensities are used to calculate phase
errors at 4000 Hz. During a surface measurement, a ramp is initiated and an integrating bucket technique is used to calculate a phase
map 共this picture is taken from Cole’s thesis
5
兲.
Fig. 4. Layout of the interferometer.
1 December 2001 兾 Vol. 40, No. 34 兾 APPLIED OPTICS 6217

optical axis of the crystal兲, for the x-polarized light,
the index of refraction is then
7
n
x
⫽ n
o
⫺
1
2
n
o
3
p
63
V
W
, (1)
where p
63
is one element of the Pockels coefficient
array of the crystal. For the z-polarized light the
index of refraction does not change; i.e.,
n
z
⫽ n
e
. (2)
Then the phase change between the x- and the
z-polarized light that is due to the electrical field is
⌬␾ ⫽ ␾
z
⫺ ␾
x
⫽
2␲
␭
1
2
n
o
3
p
63
d
W
V. (3)
So, for a given EOM, the phase modulation is propor-
tional to the external voltage V. The voltage that
makes ⌬␾⫽␲is called the half-wave voltage, de-
noted as V
1兾2
. For the ConOptics 380c, V
1兾2
⫽ 116 V
for 633-nm He–Ne laser light. The EOM driver’s
output ranges from ⫺400 to ⫹400 V, which allows
more than three waves phase shift. The dynamic
range is large enough for our application.
5. High-Speed Phase Measurement
The vibration is detected by a high-speed phase mea-
surement using a photodiode. Even though the pho-
todiode is a single-point detector, it looks at the whole
interferogram or a large subarea of it 共see Figs. 3 and
4兲. In this section we show that we can still use
phase-shifting interferometry to conduct the mea-
surement.
Assume that the irradiance at a point 共x, y兲 in the
interferogram is A共x, y兲, and
A共 x, y兲 ⫽ A
1
共 x, y兲 ⫹ A
2
共 x, y兲cos关␾共x, y兲 ⫹ ␣兴, (4)
where ␾共x, y兲 is the phase difference between the two
interfering beams, ␣ is the phase-shift angle, A
1
共x, y兲
is the average irradiance, and A
2
共x, y兲 is the modu-
lation as the phase is shifted. Then the signal the
photodiode sees is I,
I ⫽
兰兰
A共 x, y兲dxdy
⫽
兰兰
A
1
共 x, y兲dxdy
⫹
再
cos ␣
兰兰
A
2
共 x, y兲cos关␾共x, y兲兴dxdy
⫺ sin ␣
兰兰
A
2
共 x, y兲sin关␾共x, y兲兴dxdy
冎
⫽ I
1
⫹ I
2
cos共␥ ⫹ ␣兲, (5)
where
I
1
⫽
兰兰
A
1
共 x, y兲dxdy, (6.1)
I
2
⫽
(
再
兰兰
A
2
共 x, y兲cos关␾共x, y兲兴dxdy
冎
2
⫹
再
兰兰
A
2
共 x, y兲sin关␾共x, y兲兴dxdy
冎
2
)
1兾2
,
(6.2)
tan共␥兲 ⫽
兰兰
A
2
共 x, y兲sin关␾共x, y兲兴dxdy
兰兰
A
2
共 x, y兲cos关␾共x, y兲兴dxdy
. (6.3)
So the signal on the photodiode is modulated in the
same way as if it represented the intensity at a single
point in the interferogram. It has a phase angle ␥
and sees the same phase-shift angle ␣. Then we can
use phase-shifting interferometry to determine ␥.If
the measurement is different from the command
phase, then the difference is regarded as the vibration,
and a correction is made to cancel the vibration effect.
The magnitude of I
2
depends on the number of
fringes in the aperture. When I
2
⫽ 0, there is no
contrast and the high-frequency measurement does
not work. For example, if the aperture is a unit
square and there are only tilt fringes, let the fringe
visibility be 1 and ␾共x, y兲⫽2␲␣x where ␣ is the
amount of tilt in waves; we can then calculate the
contrast I
2
兾I
1
of the signal that the photodiode sees:
I
2
I
1
⫽
冏
sin共␣␲兲
␣␲
冏
. (7)
We plot the contrast as a function of ␣ in Fig. 6. For
the vibration compensation to work, the zero contrast
must be avoided. This can always be done by means
of increasing or decreasing the tilt in the interfero-
gram. In practice, the vibration compensation was
never limited by the contrast of the photodiode signal
when we used the interferometer to test optics. A
contrast of 5% or so is adequate, and we were always
able to maintain it without any extra effort.
Fig. 5. EOM is the phase shifter of the interferometer.
6218 APPLIED OPTICS 兾 Vol. 40, No. 34 兾 1 December 2001

6. System-Performance Analysis
Since the high-speed phase measurement measures
the instantaneous phase at a frequency of 4 kHz, it is
unable to determine and then compensate for higher-
frequency vibration. In Subsection 6.A we give the
limit of this vibration-compensation mechanism.
The high-speed phase measurement is superimposed
with the surface measurement; then the fringe con-
trast in surface measurement is reduced. Subsec-
tion 6.B gives the theoretical contrast reduction in
the surface measurement. A polarizing beam split-
ter 共PBS兲 is used to separate the test beam and the
reference beam. The finite extinction ratio of the
PBS will cause error in phase measurement. Sub-
section 6.C shows that the error is negligible for the
PBS we use. Subsection 6.D analyzes the effect of
noise in the photodiode on the surface measurement.
A. Vibration-Rejection Percentage
It takes time for the servo to complete the phase
measurement and then make the compensation; so
the vibration compensation is not perfect. The per-
formance of the system is approximately calculated
below.
If the phase change caused by a sinusoidal vibra-
tion is ␾
1
,
␾
1
⫽ A sin共2␲ft兲, (8)
where A is the amplitude in waves, f is the vibration
frequency, and t is time. Assuming that the high-
frequency phase measurement accurately deter-
mines the phase change caused by vibration, then
because of the time delay in the compensation, the
compensation ␾
2
is
␾
2
⫽ A sin关2␲f 共t ⫺ ⌬t兲兴, (9)
where ⌬t is the time delay and ⌬t ⬵ 200 ␮sinour
case. Then the residual phase change due to vibra-
tion is
⌬␾ ⫽ ␾
1
⫺ ␾
2
⫽ 2 A sin共␲f⌬t兲cos
冋
2␲f
冉
t ⫺
⌬t
2
冊
册
,
(10)
whose amplitude is 2A sin共␲f⌬t兲. So the percentage
vibration rejection is
Rejection ⫽ 100关1 ⫺ 2 sin共␲f⌬t兲兴. (11)
Figure 7 shows the measured and the theoretical
vibration-rejection percentage 共measured vibration-
rejection data is taken from Cole’s thesis
5
兲. When
the vibration frequency is low, the theoretical
vibration-rejection percentage agrees with the mea-
sured one. When the vibration frequency is high,
since the measured phase change due to vibration is
not accurate, Eq. 共11兲 no longer holds, and this ex-
plains the discrepancy between the measured and the
theoretical vibration-rejection percentage at the
high-frequency end.
Ultimately, the finite sampling frequency limits
the performance of the servo. The servo measures
the vibration and then compensates it at a frequency
of 4 kHz; then the vibration of frequency close to or
higher than 4 kHz is beyond the compensation capa-
bility of this interferometer. Even for vibration fre-
quency less than 4 kHz, the interferometer has
limitations. An error in the phase shift, which is
proportional to the first derivative of the vibration,
causes errors in the high-speed measurement, which
directly affect the servo. For sinusoidal vibration
described by Eq. 共8兲 the first derivative is propor-
tional to the vibration’s amplitude-frequency prod-
uct, Af. This interferometer can reject more than
75% of the vibration with amplitude-frequency prod-
uct less than 200 wave 䡠 Hz and frequency less than
200 Hz.
B. Fringe-Contrast Reduction in Surface Measurement
Because of the averaging effect of the high-frequency
phase measurement, the fringe contrast on the
surface-measurement camera is reduced. With the
Fig. 6. Plot of the contrast of the photodiode signal for square
detector versus the amount of tilt in the interferogram. We have
shown system performance with no degradation with a contrast of
0.05.
Fig. 7. Comparison of the theoretical and the measured vibration-
rejection percentages for 4-kHz system operation 共measured vibra-
tion rejection data is taken from Cole’s thesis
5
兲.
1 December 2001 兾 Vol. 40, No. 34 兾 APPLIED OPTICS 6219

vibration servo off, the contrast of the signal on the
CCD is ␥; then the signal on a specific pixel is
I ⫽ I
1
共1 ⫹ ␥ cos ␾兲. (12)
When the vibration servo is on, during the high-speed
phase-ramping period, the signal is just averaged
out. If the duty cycle of the high-speed phase mea-
surement is ␣共the ratio between the phase-ramping
time and the sampling period兲, then the signal on the
same pixel of the CCD is now
I⬘ ⫽ ␣I
1
⫹ 共1 ⫺ ␣兲I
1
共1 ⫹ ␥ cos ␾兲
⫽ I
1
关1 ⫹ 共1 ⫺ ␣兲␥ cos ␾兲兴. (13)
So the signal contrast is now
␥⬘ ⫽ 共1 ⫺ ␣兲␥, (14)
which is reduced from the original contrast by a fac-
tor of 共1 ⫺␣兲. In our case, phase ramping takes 100
␮s and the sampling period is 250 ␮s 共4 kHz兲;so␣⫽
0.4, and the fringe contrast drops 40% when the vi-
bration servo is on compared with when it is off.
This has been shown in the lab.
5
C. Effect of Polarization Leakage of the Polarizing Beam
Splitter
Several polarization components are used in this in-
terferometer 共see Fig. 4兲. First, the EOM shifts the
phase of one polarization relative to the other; then
the PBS separates the two polarizations. The PBS
sends one polarization to the test arm and the other
to the reference arm. Because the extinction ratio of
the PBS is finite, polarization leakage occurs, which
causes errors in surface measurement. Figure 8 il-
lustrates the separation and leakage of the polariza-
tions at the PBS. The signal the camera sees is the
light irradiance I:
I ⫽ 兩E兩
2
⫽
冏
␣
1
⬙
冋
1 ⫹
1
m
2
exp共i␾兲
册
⫹ ␣
2
⬙
⫻ exp共i␣兲
冋
exp共i␾兲 ⫹
1
m
2
册
冏
2
, (15)
where ␣
1
⬙ and ␣
2
⬙ are the amplitudes of the electric
fields contributed by the two polarizations, m is the
amplitude extinction ratio of the PBS, ␾ is phase
angle to be determined, and ␣ is the phase-shift an-
gle. For typical PBS, m
2
ranges from 100 to 10
5
.
Since m
2
⬎⬎ 1, we can simplify Eq. 共15兲:
I ⬵
冏
␣
1
⬙ exp
冉
i
sin ␾
m
2
冊
⫹ ␣
2
⬙
⫻ exp
冋
i
冉
␣ ⫹ ␾ ⫺
sin ␾
m
2
冊
册
冏
2
⫽ 兩␣
1
⬙兩
2
⫹ 兩␣
2
⬙兩
2
⫹ 2␣
1
⬙␣
2
⬙ cos
冉
␣ ⫹ ␾ ⫺ 2
sin ␾
m
2
冊
.
(16)
Apparently the finite value of m
2
results in an error
⌬␾ when ␾ is measured. Equation 共16兲 shows that,
as a function of ␾,
⌬␾共␾兲 ⫽ ⫺
2 sin ␾
m
2
共rad兲. (17)
Fig. 8. Illustration of how PBS separates and then recombines the two polarizing beams. Polarization leakage is shown. ␣, phase-shift
angle; ␾, phase difference between the test and reference beams; m
2
, intensity extinction ratio of the PBS. Dots and short bars indicate
the polarizations.
6220 APPLIED OPTICS 兾 Vol. 40, No. 34 兾 1 December 2001

For the PBS we use, m
2
⬎ 1000, the peak measure-
ment error is less than 0.001 wave. So the polariza-
tion leakage has little effect on the measurement
accuracy.
D. Error in Vibration Compensation Due to Noise in the
Photodiode Signal
In Section 5 we showed that the contrast of the pho-
todiode signal is a function of the number of fringes in
the interferogram and that it might be small some-
times. The low contrast coupled with noise will re-
sult in error in the calculation of the instantaneous
phase, and the effectiveness of vibration compensa-
tion will diminish. Assume that the contrast of the
photodiode signal is C, the average signal is I, and the
noise is N; then the signal-to-noise ratio 共SNR兲 of
the high-frequency measurement is
8
SNR ⫽
CI
N
⫽
CI
NEP
冑
B
, (18)
where NEP is noise-equivalent power for the photo-
diode and B is the noise-equivalent bandwidth. In
our case, NEP ⫽ 1pW兾
公
Hz
and B ⫽ 200 kHz. For
the five-step Hariharan algorithm that is used for
high-frequency phase measurement, this SNR will
cause a phase-calculation error whose standard de-
viation is
9
␴
p
⫽
1
冑
5
1
SNR
. (19)
In an extreme case in which contrast C ⫽ 0.05 and
power I ⫽ 5 ␮W, ␴
p
is 0.0008 rad. This phase cal-
culation is done every ⌬t ⫽ 0.25 ms. If the CCD
camera’s integration time for one frame is ⌬T, which
includes many cycles of phase calculation and vibra-
tion compensation, then the random phase-
calculation error at each cycle will be averaged out.
Assuming that the noise is uncorrelated from one
measurement to the next, the residual error in the
average will fall with the square root of the number of
measurements averaged together. The residual
noise in phase for the average is then
⑀
r
⫽
冉
⌬t
⌬T
冊
1兾2
␴
p
, (20)
which is the phase-step error for surface measure-
ment. When the integration time of the CCD is T ⫽
16 ms, the error in phase step seen by the CCD is ⑀
r
⫽
1 ⫻ 10
⫺4
rad. If a five-bucket algorithm is used for
the surface measurement, the error will be 1 ⫻ 10
⫺8
rad, which is completely negligible.
9
Table 1 gives
the numbers used for the above calculations.
7. Result
The final system specifications of the interferometer
are listed in Table 2. To demonstrate the perfor-
mance of the servo system, we tested a flat mirror in
a room with severe vibration. The air handlers were
loud, and the interferometer and the flat mirror were
mounted on a nonfloating table. We used Durango
software developed by Diffraction International In-
corporated for the surface measurements. It was
impossible to get meaningful data with the vibration
servo turned off. A typical measurement without
the servo, shown in Fig. 9共a兲, has errors of the order
of 0.5 waves rms. With the vibration servo turned
on, we obtain high-quality, repeatable measurements
in this same environment. The map shown in Fig.
9共b兲 was taken in the same environment as that in
Table 1. Typical Values of the Parameters in Calculating the Residual
Phase-Step Error of Surface Measurement
Parameter Value
Laser
Wavelength 632.8 nm
Output power 2 mW
Photodiode
NEP 1 pW兾
公
Hz
B 200 kHz
Servo period 共⌬t兲 0.25 ms
CCD integration time 共⌬T兲 16 ms
Power on photodiode 共I兲⬎5 ␮W
Contrast on photodiode 共C兲⬎0.05
⑀
r
In radians ⬍1 ⫻ 10
⫺4
In waves ⬍1.6 ⫻ 10
⫺5
Table 2. System Specifications
Parameter Value Remarks
System Specifications
Source wavelength 共nm兲 633 He–Ne laser
Phase sampling frequency 共Hz兲 4000 Limited by DSP
Operational limitations 140 Hz 80% Rejection for vibrations at this frequency
300 wave 䡠 Hz For vibration frequency ⬍200 Hz
Duty cycle 共%兲 40 Time spent ramping 2␲
Contrast on CCD 共%兲 60 Maximum
EO Modulator
Half-wave voltage 共V兲 116
Maximum phase shift allowed 共wave兲 3.4
1 December 2001 兾 Vol. 40, No. 34 兾 APPLIED OPTICS 6221

Fig. 9共a兲, but with the vibration-compensation sys-
tem. This measurement compares well with a mea-
surement of the same optic, made under quiet
conditions, shown in Fig. 9共c兲. The difference of the
two results is less than ␭兾1000 for rms deviation.
8. Summary
We built a phase-shifting Twymann–Green inter-
ferometer that is capable of compensating for vibra-
tion. The basic idea is to perform high-frequency
phase measurement on top of the surface measure-
ment to detect the vibration. If vibration is de-
tected, the phase-shift angle is either increased or
decreased to cancel the vibration effect. This
vibration-compensation mechanism was demon-
strated to work as predicted. In an environment
with severe vibration, we successfully made surface
measurements with the vibration-compensation
servo on. When the servo is turned off, we were not
able to make measurements in the same situation.
References
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rates,” Ph.D. dissertation 共Optical Sciences Center, University
of Arizona, Tucson, Ariz., 1998兲.
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6. G. C. Cole, J. H. Burge, and L. Dettmann, “Vibration stabiliza-
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Fig. 9. Surface map made with phase-shifting interferometry for
three cases: 共a兲 in the presence of severe vibration with the servo
turned off, 共b兲 in the same severe environment with the servo
operational, and 共c兲 the same surface, measured with standard
phase-shifting interferometry in a quiet environment.
6222 APPLIED OPTICS 兾 Vol. 40, No. 34 兾 1 December 2001