Suppressing phase errors from vibration
in phase-shifting interferometry
Leslie L. Deck
Zygo Corporation, Laurel Brook Road, Middlefield, Connecticut 06455, USA ([email protected])
Received 3 March 2009; revised 16 June 2009; accepted 18 June 2009;
posted 18 June 2009 (Doc. ID 108271); published 6 July 2009
A general method for reducing the influence of vibrations in phase-shifting interferometry corrects the
surface phase map through a spectral analysis of a “phase-error pattern,” a plot of the interference
intensity versus the measured phase, for each phase-shifted image. The method is computationally fast,
applicable to any phase-shifting algorithm and interferometer geometry, has few restrictions on surface
shape, and unlike spatial Fourier methods, high density spatial carrier fringes are not required, although
at least a fringe of phase departure is recommended. Over a 100× reduction in vibrationally induced surface distortion is achieved for small amplitude vibrations on real data. © 2009 Optical Society of America
OCIS codes:
120.3180, 120.3940, 120.5060, 120.6650, 120.7280.
1. Introduction
Interferometric surface profilers incorporating
phase-shifting interferometry (PSI) techniques [1]
routinely profile surfaces with nanometer level vertical resolution but still suffer from high sensitivity
to environmental vibrations. Because of this, passive
isolation components such as air tables are common
auxiliaries in interferometric metrology systems. For
the most part, these isolation components work well
enough for interferometers to have become ubiquitous tools for precision profiling, but the residual
sensitivity of these metrology systems to vibration
can be a barrier to further improvement.
Phase-shifting techniques analyze intensity variation from a controlled phase shift to determine the
starting phase of the interferogram, and in profiling
applications the surface profile is obtained from the
spatial distribution of starting phases. Implicit in the
technique is the assumption that the phase shift
between intensity measurements is perfectly known.
Vibration produces unknown, time-dependent
changes to the phase shift, which the analysis then
falsely interprets as a surface variation. The manifestation of this error is a periodic deformation of
the measured surface, often called ripple, with a
0003-6935/09/203948-13$15.00/0
© 2009 Optical Society of America
3948
APPLIED OPTICS / Vol. 48, No. 20 / 10 July 2009
spatial frequency equal to twice the fringe frequency.
To first order in the vibration amplitude, this 2-cycle
deformation, which can be readily understood in
terms of a Fourier description of the technique [2],
occurs for all vibration frequencies and phases.
Vibration sensitivity in interferometry is such a
pervasive problem that a large number of different
techniques have been marshaled against it. Representative examples include vibration suppression
through direct measurements of cavity motion
[3,4], instantaneous interferometry [5–7] including
spatial carrier techniques [8–10], iterative, nonlinear
least-squares fits to the phase shifted sequence
[11–13], high speed and quadrature measurements
[14–16], generalized PSI methods [17–20], and various hybrids of these approaches [21]. A unique
method that does not easily fit in the categories
above was proposed by Huntley [22], who used information from the 3rd harmonic of the phase-shift
frequency to reduce 1st order vibrational errors.
Many of the techniques mentioned above carry
some disadvantages that can compromise the metrology. Instantaneous methods are optically more complicated and can suffer artifacts from misidentifying
reference and test beams (typically due to polarization leakage since polarization is often used to separate the two interferometer legs). Spatial carrier
techniques incur loss of spatial resolution and added
retrace error (if left uncorrected) from the tilts
needed to produce high-density fringes. Iterative
least-squares algorithms are time consuming and
prone to nonoptimal solutions. Generalized PSI algorithms can accommodate arbitrary cavity motions by
accounting for nonuniform phase-shift increments
between interferograms, but whether these increments are measured through additional hardware
or derived from the interferograms themselves, small
errors in these increments tend to produce significant phase distortion because generalized PSI
algorithms have very broad temporal frequency
bandwidths.
The technique described here, called vibrationcompensated PSI (VC) [23], differs from previous approaches in that it does not determine the phase map
but rather corrects the phase map already obtained
from a PSI measurement. The correction is based on
a spectral analysis of a “phase-error pattern,” a sinusoidal signal generated by plotting the intensity as a
function of the PSI measured modulo-2π surface
phase. VC can be applied to any PSI algorithm that
can be written as a windowed Fourier filter [24] and
the phase modulation can be linear or even sinusoidal [25]. In this paper, VC is specifically applied to
two linear algorithms, the well-known Schwider–
Haraharan 5-frame algorithm [26,27] and the de
Groot 13-frame algorithm [28]. There are no restrictions on surface shape or interferometer type and the
method does not require high density carrier fringes.
Though in its most simple form the method assumes
the vibration is spatially independent; extending the
analysis to account for spatial dependencies due to
test surface angular motion is straightforward and
is demonstrated later in the paper. The goal of VC
is not to achieve vibration immunity, but rather to
improve PSI instruments and techniques to reach
a higher level of performance in the presence of vibrations found in standard practice. The strategy
of calculating a correction to the PSI profile has
the advantage of retaining the temporal filtering
qualities of PSI algorithms in wide use today.
The paper first explains why phase-error patterns
are useful signatures of vibrationally induced distortion. A theoretical framework for quantifying the
phase-error pattern distortion signature is then developed and applied as a general correction to PSI
algorithms. Simulated data examples highlight the
major points, and verify the predicted theoretical
performance. Real-data examples follow to illustrate
the significant improvements obtained under realistic conditions.
2. Phase-Error Patterns in Phase-Shifting
Interferometry
A typical PSI measurement system using Fizeau
optical geometry is depicted in Fig. 1. A test surface
of arbitrary shape making up part of the interferometer cavity is imaged onto a camera while the cavity
optical path difference (OPD) is varied to shift the
interferometric phase along the Z axis in a controlled
manner. The optical system is aligned to the Z axis
Fig. 1. Typical PSI measurement system.
and the surface is imaged onto a camera so each pixel
corresponds to a unique position x in the XY plane.
The chosen PSI algorithm determines the number
of camera images (interferograms) acquired during
the phase shift. The system shown in Fig. 1 uses a
piezo-electric transducer (PZT) to produce the desired phase shift via small programmed movements
of the reference surface; however, other phaseshifting methods, such as wavelength tuning [29],
are also employed. The interference between the
light reflected from the test and reference surfaces
is sampled by the camera as a function of phase shift,
and a series of interferograms is acquired. These interferograms are subsequently analyzed with the
PSI algorithm to extract the test surface wrapped
(mod-2π) phase profile. Standard phase unwrapping
algorithms [1] extend the phase profile range assuming surface continuity, and finally the surface phase
profile is converted into physical units using the
known illumination wavelength.
In the absence of vibrations, the measured surface
phase map is undistorted. If the intensity for each
pixel in an interferogram is plotted against the corresponding measured phase modulo-2π, the points
will produce a perfect single-cycle sinusoid if the
fringe contrast is constant across the field and the
departure is at least 1 fringe. This plot is called
the “phase-error pattern” and is shown in Fig. 2
for an arbitrary simulated surface by the gray line.
Each interferogram produces a unique phase-error
pattern. If new vibrations along the Z axis are introduced during the acquisition, distortions in the measured phase laterally shift the phase positions of the
intensity points. As long as the test object acts as a
rigid body, all points with the same wrapped phase
height experience identical phase shifts and, hence,
suffer the same distortion. The phase-error pattern
then appears as a distorted sinusoid, as shown by
the dark line in Fig. 2 for one particular pure-tone
vibration amplitude and frequency. Note that if the
surface departure is less than 1 fringe, the phaseerror pattern cycle will not be fully sampled. The
10 July 2009 / Vol. 48, No. 20 / APPLIED OPTICS
3949
degree to which the phase-error pattern cycle is
sampled is referred to as “phase diversity” and
generally a phase diversity of 2π is required to fully
characterize the distortion.
The phase-error pattern distortion is related to the
error in the measured surface. This relationship is
derived in Section 3 and used to correct the measured
surface for vibration-induced errors. It is useful to
consider the phase-error pattern as a 1-dimensional
representation of the 2-dimensional interferogram of
the measured surface. The phase-error pattern is
thus a surface shape independent intensity representation with characteristics that provide a natural
way to determine the phase error as a function of
surface height.
3. Using Phase-Error Patterns to Remove
Vibration-Induced Errors
A previous publication [30] showed that the true surface phase ΦðxÞ can be written as a series expansion
in odd powers of the PSI measured surface phase
^
ΦðxÞ:
exp½iΦðxÞ ¼
∞
X
Thus, the Fourier transform of the phase-error pattern with respect to the measured phase produces a
^ and the
spectrum with peaks at odd multiples of Φ,
complex value at each peak contains a linear sum of
two of the harmonic coefficients:
Cð2k þ 1; tÞ ¼ gk exp½iβðtÞ −
gkþ1
exp½−iβðtÞ: ð5Þ
2k þ 1
Before Eq. (5) can be solved to obtain the harmonic
coefficients, an estimate of the time-dependent phase
shifts βðtÞ must be made. This is done by noting that,
since the magnitude of gk diminishes as k increases
[Eq. (A19)], one can assume the second term in
Eq. (5) becomes negligible at some appropriately
large value κ for k. Successive back-substitutions
through the set of Eqs. (5) for k ¼ κ…0, along with
the fact that Imðg0 Þ ¼ 0, then supplies the following
equation for βðtÞ:
#
− 12 !
Im
Cð2k þ 1; tÞ exp½−ið2k þ 1ÞβðtÞ k − 12 ! 2k
k¼0
"
^
gk exp½ið2k þ 1ÞΦðxÞ
κ
X
¼ 0:
k¼0
^
k exp½−ið2k − 1ÞΦðxÞ
g
;
−
2k − 1
k¼1
∞
X
ð6Þ
ð1Þ
where the complex harmonic coefficients gk depend
on the PSI algorithm sampling function and the vibration frequency and phase, but not on time. The
derivation of this result is included in Appendix A
for reference. Reference [30] did not offer detail on
how to obtain the harmonic coefficients, but it turns
out that they are easily obtained from phase-error
patterns. To see this, first derive a mathematical de^ tÞ by inserting
scription of phase-error patterns cðΦ;
Eq. (1) into Eq. (A2):
Because the harmonic amplitude generally
decreases as the frequency increases, it is rarely
necessary for κ to exceed 3.
Phase-error patterns thus provide all the signals
necessary to solve for the vibration-induced error
in the PSI measured surface phase profile. The procedure to remove these errors from the phase map is
a straightforward three-step process: First, the initi^
al surface phase map ΦðxÞ
is determined with the
chosen PSI algorithm in the standard way; second,
phase-error patterns are developed and Fourier
P∞
P∞ gk exp½−ið2k−1ÞΦ
^
I0 V
^
^
cðΦ; tÞ ¼ 2 exp½iβðtÞ
k¼0 gk exp ið2k þ 1ÞΦ −
k¼1
2k−1
P
P∞
^
∞ gk exp½ið2k−1ÞΦ
^
þ exp½−iβðtÞ
k¼0 gk exp½−ið2k þ 1ÞΦ −
k¼1
2k−1
:
ð2Þ
^ ,
Now performing a Fourier transform with respect to Φ
Z∞
CðK; tÞ ¼
^ tÞ expð−iK ΦÞ
^ dΦ;
^
cðΦ;
ð3Þ
−∞
produces
CðK; tÞ ¼
3950
I0 V
2
P∞
P∞
k δðKþ2k−1Þ
g
k¼1
2k−1
exp½iβðtÞ
k¼0 gk δðK − 2k − 1Þ þ
P∞ gk δðK−2kþ1Þ
P∞
k δðK þ 2k þ 1Þ þ k¼1
:
þ exp −iβðtÞ½ k¼0 g
2k−1
APPLIED OPTICS / Vol. 48, No. 20 / 10 July 2009
ð4Þ
Fig. 2. Phase-error patterns from a simulated surface with at
least one fringe of departure for both vibration free and 0:5 rad
amplitude vibrations.
transformed to first determine βðtÞ through Eq. (6)
and the harmonic coefficients gk with Eq. (5); finally,
these coefficients along with the initial phase map
are used in Eq. (1) to evaluate a corrected surface
phase map. It is important to note that each interferogram from the PSI acquisition produces an independent measure of the harmonic coefficients and, since
they must be time independent, can be averaged
together to further reduce errors in the correction.
VC can be applied regardless of detector integration since the harmonic coefficients derived from
the phase-error patterns automatically account for
the effect this has on the spectrum. Of course, a practical limitation occurs if the vibration is severe
enough to significantly degrade fringe contrast
during the integration time, but in that case the
PSI algorithm usually fails. The simulations in
Section 4 will serve to illustrate the mathematical
procedure outlined here.
4. Simulation Example
To illustrate the theory outlined in Section 3, interferograms are generated from a simulated line profile with nominally π=2 phase increments under a
severe vibration condition and analyzed with VC
after being processed with the Schwider–Hariharan
5-frame PSI algorithm. The line profile shape is
given by φðxÞ ¼ 10π sinðxÞ, representing a curved surface with 5 fringes of surface departure, and is spatially sampled with 1024 equally-spaced pixels over
the spatial range of 0 < x < π. Figure 3 shows the
true phase height profile of the simulated surface.
Fig. 3. Phase height profile of the simulated surface.
Fig. 4. Interferogram intensity of the simulated surface
(light gray) and measured PSI phase error due to vibration
(dark line). The phase error shows the expected 2-cycle distortion
characteristic.
In principle, any surface shape with 1 fringe or more
of departure would have been sufficient to provide
fully sampled phase-error patterns, but vibration
errors on a surface with 5 fringes of departure are
visually clearer than on a surface with one, and a
curved surface highlights the fact that VC does not
require straight line fringes.
The difference between the profile measured with
the 5-frame PSI algorithm and the true profile (the
residual) for a vibration frequency of 20% of the
sample rate and vibration amplitude of 1 rad is illustrated in Fig. 4. This represents a larger vibration
amplitude than typically encountered in standard
practice but serves to better illustrate the distortions. The residual shows the expected 2-cycle error
characteristic of vibrational distortion, with an amplitude of about 0:2 rad. Five phase-error patterns
are constructed by plotting the intensity as a function of measured wrapped phase for each of the five
frames in the acquisition, and Fig. 5 shows the
phase-error pattern obtained from the first frame.
The simulation only contained 1024 sampled points,
but in real interferometric profilers the number of
samples contained in this one cycle can be quite
large; typical imagers used today contain 100s of
thousands. The vibration-induced distortion is
clearly evident in the phase-error pattern from the
deviation from a pure sinusoid. The Fourier trans-
Fig. 5. Phase-error pattern generated using the intensities of the
first frame and the measured phase. Each data point corresponds
to an individual pixel.
10 July 2009 / Vol. 48, No. 20 / APPLIED OPTICS
3951
form of the phase-error pattern with respect to the
measured phase (Fig. 6) reveals peaks at only odd
multiples of the fundamental fringe frequency as
predicted by Eq. (4).
The estimated phase-shift variation βðtÞ and harmonic coefficients gk are determined with Eqs. (6)
and (5), respectively, using the complex spectra at
the odd harmonics. In practice, it is most convenient
to use a least-squares fit of the phase-error patterns
to obtain the harmonics rather than a Fourier transform since a least-squares approach accounts naturally for nonuniform sampling, allows convenient
data weighting, and is more robust against missing
data. The harmonic coefficients and the measured
phase map are then used in Eq. (1) to calculate
the corrected phase for each pixel. The final step is
to spatially unwrap the corrected phase profile using
standard techniques. The result after VC processing
using the first 3 harmonic coefficients is shown in
Fig. 7, which plots the residual of the corrected phase
profile (dark line), compared to the residuals from
the uncorrected profile (gray line). The peak-tovalley phase error residual has decreased by a factor
of 140 and the standard deviation by a factor of 160.
The remaining error stems from the finite number of
coefficients used and in the approximation defined
by Eq. (A21).
A. Spatially Dependent Vibrations
The description has so far been limited to vibrations
producing motion only along the optical axis, so the
phase shifts are constant across the test surface.
Though often well satisfied in practice, this can be
a limitation in some applications. Vibrations can also
induce rigid body angular motions about axes perpendicular to the measurement axis. These angular
motions produce spatially dependent optical path
changes in the interferometer that, though typically
smaller than the pure piston term, can nevertheless
degrade a PSI measurement. VC can account for
these tilt-induced phase shifts by allowing the harmonic coefficients gk to acquire a spatial dependence
and calculating the correction using a unique gk for
each pixel.
The spatial dependence of the harmonic coefficients is obtained through the simple expedient of
subdividing the image, with the obvious additional
Fig. 7. Residual phase error after applying the VC correction to
3rd order (dark line near zero) compared with the original phase
error (gray line).
requirement that each subimage has at least one
fringe of departure to develop a full cycle phase-error
pattern. At least three noncollinear regions are
sufficient to remove the distortions due to rigid body
motions found in standard practice, and requiring
one fringe per region is not especially demanding.
In practice, the image is divided into quadrants.
For each desired order, the four harmonic coefficients
obtained from the quadrants are then fit to a plane
and the appropriate harmonic coefficients are then
calculated for each pixel separately from this best
fit plane before applying Eq. (1) to calculate the
phase. This is adequate to account for test object tilts,
and higher-order dependencies can be handled with
finer image subdivisions.
To demonstrate the procedure, a linear variation
in the phase-shift amplitude was incorporated in
the simulation above, as one might expect to occur
if the test surface tilted during phase shifting. The
peak vibration amplitude is again 1 rad at a normalized frequency of 0.2, and the 5-frame PSI algorithm
is used to calculate the initial phase profile. The correction subdivides the phase profile into two halves,
calculates the harmonic coefficients separately for
each half, and then applies a linear regression to
each pair of harmonic coefficients for each order separately. The corrected phase is then recalculated
using Eq. (1) for each pixel using harmonic coefficients determined from the pixel position and the
regression analysis.
Figure 8 compares the residual phase-error profile
of the corrected surface profile for two different
values of the harmonic order, to the uncorrected
residual profile. The spatial dependence of the vibration amplitude is easily observed in the uncorrected
phase-error profile, which had an rms residual of
80 mrad. In contrast, the rms residual of the 2nd
order corrected profile was only 1:5 mrad.
5.
Fig. 6. Power spectrum of the phase-error pattern shown in Fig. 5
showing the predicted dependence on odd-order harmonics.
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APPLIED OPTICS / Vol. 48, No. 20 / 10 July 2009
VC Performance Predictions
VC performance as a function of vibration amplitude
and frequency is conveniently expressed in terms of a
phase-error sensitivity spectrum [2]; defined as the
rms surface error normalized to the vibration amplitude as a function of vibration frequency after integrat-
expðiΔVC Þ ¼ −g0 −
∞ X
k¼1
^ k expð−i2kΦÞ
g
^
gk expði2kΦÞ−
;
2k−1
ð8Þ
Fig. 8. Residual phase-error profiles for standard 5-frame PSI
phase map and the VC phase map for two values of the harmonic
order. The vibration amplitude varied linearly across the profile.
ing over all vibrational phases. A sensitivity of 1 means
that the rms surface error equals the amplitude of the
vibrational disturbance that caused it. Each PSI
algorithm will have a unique phase-error sensitivity
spectrum and the formalism developed in Appendix A
and Section 3 provides ready-made formulas for their
calculation. Equation (A12) for example can be used to
^ −Φ
derive an expression for the phase error ΔPSI ¼ Φ
for any PSI algorithm via
^ − ΦÞÞ
expðiΔPSI Þ ¼ expðiðΦ
¼
ηþμ
expð−2iΦÞ
;
jη × expðiΦÞ þ μ
expð−iΦÞj
ð7Þ
with η and μ given by Eqs. (A15) and (A16), respectively,
after FðωÞ is evaluated using the coefficients of the chosen PSI algorithm. Table 1 shows the sampling coefficients of the two PSI algorithms used in this paper.
Similarly, Eq. (1) is used for the residual phase
^ − Φ after VC correction via
error ΔVC ¼ Φ
Table 1.
Sampling Coefficients for the Two PSI Algorithms Used in This Paper
5-Frame Algorithm (π=2 Phase Increments)
1
−2i
−2
2i
P
^
with gk ¼ ∞
n¼0 gn hnþk and Φ given by Eq. (A12). The
rms phase-error sensitivity spectrum is then calculated by evaluating the standard deviation of the
computed phase errors ΔPSI ðΔVC Þ over a range of
starting interferometric phases (Φ) and vibrational
phases (α), as a function of vibrational frequency (ωv ).
Phase-error Eqs. (7) and (8) are general theoretical
predictions of measurement errors valid for any PSI
algorithm at vibrational amplitudes for which the algorithm itself does not fail, lifting the restriction to
small amplitudes characteristic of previously published analytical solutions (e.g., [2]). As an example,
Fig. 9 compares the rms phase-error sensitivity spectrum to small amplitude vibrations (≤ 0:1 rad or
equivalent to about 5 nm amplitude at 633 nm wavelengths) for the two PSI algorithms used in this
paper. A detector integration of 1 frame period is
assumed, which is standard practice for most
commercial profilers and the phase-error spectrum
is calculated for the uncorrected PSI algorithm
and the VC corrected algorithm for the first 3 harmonic orders. The uncorrected phase-error sensitivity
spectrum obtained from Eq. (7) is in excellent agreement with previous small amplitude limit predictions [2]. VC correction shows a reduction in the
rms phase error of over 2 orders of magnitude across
the spectrum relative to the uncorrected PSI algorithm if corrections greater than 1st order are used.
As the vibration amplitude increases, the effectiveness of VC is reduced, but significant improvement is
still obtained across the spectrum. Figure 10 shows
that VC provides a factor of 5 improvement for the
1
13-Frame Algorithm (π=4 Phase Increments)
−3i −4 − 4i −12 −12 þ 12i 21i 16 þ 16i 24 16 − 16i 21i −12 − 12i −12 −4 þ 4i 3i
Fig. 9. RMS vibration sensitivity for small amplitude vibrations compared to PSI algorithm sensitivity for the 5 frame (left) and 13 frame
(right) PSI algorithms. The vibration frequency is normalized to the sample rate. The vibration amplitude was 0:1 rad.
10 July 2009 / Vol. 48, No. 20 / APPLIED OPTICS
3953
5-frame algorithm, and a factor of 9 for the 13-frame
algorithm at 1 rad of vibration amplitude at the most
sensitive vibration frequencies if corrections to the
3rd order are made. The improvements are even
greater far from these peak sensitivity frequencies.
It should be noted that though the graphs extend
only to normalized vibration frequencies of 1, errors
at higher frequencies are even further attenuated
due to the low pass effect of detector integration.
6. Factors That May Affect VC Performance
A. Spatial Intensity Variation
The mathematical treatment presented here assumes that the intensity and contrast are spatially
independent, implying uniform illumination intensity and interference cavity reflectivity. Uniform cavity reflectivity occurs very often in precision profiling
applications, however, practical interferometers often have some amount of illumination nonuniformity, typically stemming from the mode profile of
the light source. Additionally, spatial nonuniformity
caused by pixel gain variations in the imaging camera can produce effects indistinguishable from intensity nonuniformity, though these are typically very
small. Both of these cases need to be accounted for
in practical applications of VC.
A simple and effective solution to account for these
illumination nonuniformities in real systems is to
measure the nonuniformity and precorrect each
interferogram before the VC analysis. The nonuniformity can be accurately measured by averaging
together a number of images of a single (not a cavity)
uniform surface to form an illumination calibration
map (ICM). The reference surface itself serves well
for this purpose since it is typically featureless and
uniformly reflecting. The ICM is normalized to a
peak value of one and each interferogram is divided
pixel-by-pixel by the ICM before phase-error pattern
processing is performed. The ICM is valid regardless
of instrument global light level and can be reused until system changes occur that affect the illumination
profile—such as changing zoom settings or replacing
transmission elements. In practice, the illumination
profiles for particular optical settings have been
found to be reliably repeatable, so ICMs for a particular configuration can be saved and reloaded when
that configuration is reestablished. This method
works particularly well and was used for all of the
real examples shown in this paper.
B.
Temporal Intensity Fluctuations
Time-dependent changes in the intensity occur in
two types: common-mode fluctuations where the intensity changes are common across the interferogram and non-common-mode fluctuations where
each pixel fluctuates randomly, typically due to shot
noise and/or electronic noise sources. Noncommonmode intensity noise has little effect on the phaseerror pattern analysis since the large number of
pixels in the phase-error pattern makes it statistically unlikely that the fluctuations resemble the
large-scale spatial frequencies used in the correction.
Common-mode fluctuations, however, can produce
distortions in the phase-error patterns, which manifest themselves as harmonics of the measured interferometric phase. Assume the interference intensity
under the influence of common-mode sinusoidal
variations is modeled as
Iðx; tÞ ¼ I 0 ½1 þ p cosðωp t þ γÞ½1 þ V
× cosðΦðxÞ þ ω0 tÞ;
ð9Þ
where p is the modulation amplitude fraction, ωp is
the angular frequency, and γ is the phase offset.
Following the analysis flow of Appendix A, it is
straightforward to calculate the influence the sinusoidal intensity modulation has on the phase-error
patterns. The result is that, to first order in p, the
^ dependence of the phase-error patterns cðΦ;
^ tÞ is
Φ
^ tÞ ≅ GðtÞ þ
cðΦ;
3
X
^
gk ðtÞ expðikΦÞ
k¼0
þ
3
X
^
k ðtÞ expð−ikΦÞ;
g
ð10Þ
k¼1
Fig. 10. RMS vibration sensitivity for large amplitude vibrations compared to PSI algorithm sensitivity for the 5 frame (left) and 13 frame
(right) PSI algorithms. The vibration frequency is normalized to the sample rate. The vibration amplitude was 1 rad.
3954
APPLIED OPTICS / Vol. 48, No. 20 / 10 July 2009
with the time dependence contained completely in
coefficients G and gk . The phase-error pattern spectrum is, thus,
CðK; tÞ ¼
3
X
gk ðtÞδðK − kÞ þ
k¼0
3
X
k ðtÞδðK þ kÞ: ð11Þ
g
k¼1
Equation (11) shows that the intensity fluctuation
introduces peaks in the phase-error pattern spec^ and,
trum CðK; tÞ at the first three harmonics of Φ
thus, will corrupt a VC analysis by introducing error
in the 1st and 3rd harmonics. The severity of the error depends sensitively on the PSI algorithm used,
the intensity fluctuation amplitude fraction p, its
phase γ, and its frequency relative to the interference
frequency, with the largest errors occurring when
ωp ¼ ω0 or ωp ¼ 2ω0.
C. Discontinuous Surfaces
As explained in Section 2, surface features do not affect the shape of the phase-error pattern. This is a
fundamental difference between VC and spatial
Fourier methods [8]. The VC correction can be applied to surfaces with substantial structure, steps,
or arbitrarily shaped regions without modification.
This is demonstrated in Section 7.
D. Phase-Shifter Miscalibration
Phase-shifter miscalibration is a well-known error
source in phase-shifting interferometry [26] and a
large amount of effort has been devoted to limiting
its influence on the measurement, especially in the
design of new PSI algorithms. From the point of view
of the phase-error pattern technique, a phase-shifter
miscalibration is identical to a very low frequency vibration, so as long as the miscalibration is not severe
enough to produce phase discontinuities in the PSI algorithm, VC will compensate for these types of errors.
E. Turbulence
Air turbulence in interferometry is characterized by
rapid variations of air pressure and temperature
over many different time and length scales. These
pressure and temperature variations introduce local
optical index changes that directly affect the interferometric measurand: the cavity OPD. To appropriately account for the surface distortions produced by
turbulence, VC must measure the OPD changes with
a spatial and temporal resolution similar to (and preferably finer than) the scales defined by the turbulence eddies, which themselves depend on the
atmospheric viscosity. The temporal resolution of
VC is defined by the interferogram sample rate
(the camera frame rate) and is typically out of the
control of the VC method. The spatial resolution,
however, can be increased by further image subdivision beyond the simple quadrants designed to handle
rigid body test surface motion mentioned above. The
price paid for this improved spatial resolution is a
greater computational burden, increased error in
the calculation of the harmonic coefficients due to
fewer data points forming the phase-error patterns,
and an increase in retrace error from the surface
departure required to provide 2π phase diversity
in each subimage. As the required spatial resolution
approaches the sampling limit, too few points are acquired to adequately sample the phase-error pattern
and the VC method breaks down. In such cases, the
Fourier method of Goldberg and Bokor [21] may be
more appropriate.
F.
Large Amplitude Vibrations
The VC method cannot repair the phase map if the
vibration is large enough to produce phase discontinuities in the PSI-generated phase map. This is of
course a failure of the PSI algorithm, not VC. The environmental conditions where this occurs depends on
the PSI algorithm and the system sample rate (typically the camera frame rate), but a good rule of
thumb is that the product of the vibration phase
amplitude with its angular frequency divided by
the sample rate must be less than the nominal phase
increment.
G.
Nonlinear Intensity Transfer Function
Phase-shifting algorithms assume a perfect 1∶1 mapping of light intensity to electronic interference signal,
and the mapping is typically done by the imager. Some
imagers, due to the physics of the light conversion process or through nonlinearities in the amplification of
the electrical signal, exhibit a nonlinear intensity
transfer function, simply referred to as camera nonlinearity. Camera nonlinearity produces distortion in
the PSI-calculated phase profile as well as the
phase-error pattern spectrum, and this affects the
harmonic coefficient accuracy. Thus, VC cannot correct for this error. If known, camera nonlinearity
can always be removed by postcorrecting the acquired
interferograms with a lookup table before phase
processing. Fortunately, the linearity of most modern
visible-light cameras is high enough that this type of
error is not a concern; however some cameras, notably
infrared imagers utilizing bolometric principles, can
exhibit unacceptable nonlinearity.
H.
Low Phase Diversity
Low phase diversity means the phase-error patterns
are not fully sampled, which occurs if the surface
phase departure is less than 2π (one fringe). Harmonic coefficients obtained from only partially sampled
phase-error patterns will have increased uncertainty
and this will adversely affect the correction, but the
uncertainty is a relatively slow function of the phaseerror pattern sampling fraction. As mentioned in
Section 4, calculating the harmonic coefficients with
a least-squares method is highly robust, and good
compensation can often be obtained with a little
more than half a cycle of phase diversity.
I.
High Finesse Cavities
The VC correction method was derived assuming low
finesse interference so the interference could be modeled as a pure cosine. Vibrations then introduce only
10 July 2009 / Vol. 48, No. 20 / APPLIED OPTICS
3955
odd harmonic distortions into the phase-error patterns. As the finesse increases, multiple reflections
produce both odd and even harmonic distortions in
the phase-error patterns, which can corrupt the
VC calculated harmonic coefficients.
In standard interferometric profilers, the level of
finesse-induced distortion depends sensitively on
the PSI algorithm but is typically much smaller than
the measurement phase noise. For example, numerical simulations show that the finesse-induced phase
distortion using the Schwider–Hariharan 5-frame algorithm in a 4%–4% cavity (a cavity composed of two
surfaces, each with 4% reflectivity) is only 1 mrad
rms, about 5 times smaller than the phase noise incurred from 8 bit digitization precision [32]. That distortion increases to 24 mrad rms with a 4%–90%
cavity. VC applied to phase profiles derived with
the 5-frame algorithm actually reduces the rms
phase error by about a factor of two regardless of test
surface reflectivity by eliminating the distortion contribution from the odd harmonics. In contrast the de
Groot 13-frame algorithm is especially designed to be
insensitive to finesse-induced distortion and produces only 0:033 mrad rms and 0:7 mrad rms phase
distortions for 4%–4% and 4%–90% cavities, respectively. Applying VC to phase profiles from this algorithm increases the finesse-induced distortion by
about a factor of 5, but this is still well below typical
phase noise.
For those applications that require the highest precision, it is relatively easy to achieve low finesse in
practice, even for highly reflective test surfaces, since
an intervening absorptive or reflective object can
usually be inserted in the cavity to reduce the effective test surface reflectivity if necessary. Commercial
references with attenuation coatings [32] are available for this purpose.
It is useful to note that, in microscope profiling
applications, multiple interference effects are rarely
a problem since incoherent illumination (both
spatially and temporally) is usually employed [33],
producing a coherence length shorter than the cavity
OPD. However the coherence should still be long
enough to produce negligible contrast variation
across the image.
J.
Fig. 11. (Color online) 13-frame PSI measurement of a vibrated
flat cavity with about 5 fringes of tilt containing ripple with 25 nm
amplitude before (left) and after (right) the VC method to 3rd order
is applied.
ual retrace distortion is unacceptable, two measurements with equal and opposite tilts can be averaged.
The tilt-induced retrace distortion (mainly coma)
changes sign between the two measurements and
largely cancels, whereas the (tilt removed) surface
profile is unaffected.
7.
Measurement Examples
The VC method has been incorporated into the Zygo
MetroPro phase retrieval software and this section
demonstrates typical measurement results of precision optical surfaces under a variety of environmental conditions. The 13-frame PSI algorithm, whose
coefficients are shown in Table 1, was used, and in
all cases the interferograms were corrected for spatial illumination intensity variations using an ICM
as described Subsection 6.A
VC performance was tested on both flat and spherical cavities for a variety of vibration conditions and
in all cases VC performance agreed well with the predictions of the phase-error transfer functions derived
in Section 5. Figures 11 and 12 compare PSI and VC
surface profiles for two representative measurements made with a large aperture interferometer
(Zygo VeriFire XP) using the 13-frame PSI algorithm
for flat and spherical objects, respectively. In both
cases the vibration amplitude was 25 nm (0:5 rad)
with a normalized frequency set to the most sensitive
part of the PSI sensitivity spectrum (1=4 of the
camera frame rate). The large amplitude 2-cycle
Retrace Error
Though retrace error does not directly influence the
performance of the VC algorithm, the requirement
for enough phase diversity to develop the phase-error
pattern means that the surface wavefront must depart from the reference wavefront, thereby creating
the necessary conditions for retrace error distortion.
The degree of retrace distortion depends sensitively
on the optical characteristics of the interferometer
and the amount of wavefront departure; however, unlike many other vibration-compensation schemes
(such as Fourier methods), VC minimizes the required departure (and thus retrace error) while still
providing excellent vibration compensation. For
those demanding applications where even this resid3956
APPLIED OPTICS / Vol. 48, No. 20 / 10 July 2009
Fig. 12. (Color online) 13-frame PSI measurement of a vibrated
spherical cavity with 4 fringes of departure containing ripple with
25 nm amplitude before (left) and after (right) the VC method to
3rd order is applied.
Fig. 13. (Color online) Surface profile of a poorly mounted flat that incurred significant tilt during the acquisition. The left figure is the
surface using 13-frame PSI, the center is with 3rd order VC without spatial dependence, and the right figure is 3rd order VC incorporating
spatial dependence.
distortion evident in the PSI measured profiles is
completely absent after the VC correction is applied,
consistent with the factor of 60 distortion reduction
predicted for this vibration amplitude and frequency.
Though the measurements shown above used VC
with spatially dependent corrections, there was actually little performance difference between spatially
dependent or independent vibration processing since
the vibrations mainly consisted of motions along the
optical axis. To demonstrate a condition where spatially dependent processing is important, a test surface was mounted nonkinematically so vibrations
made the part pivot about an arbitrarily located fulcrum during the measurement. Figure 13 shows the
surface maps after analyzing for spatially independent and dependent vibrations separately. The vibrations in this example were not induced but occurred
naturally from the environment. Note that the 2cycle ripple is only fully removed when spatially
dependent VC is enabled.
The method was also applied to data from a phasemeasuring microscope (Zygo NewView 600P). Unlike
large aperture systems, microscope applications often involve surfaces with unusually shaped features
and surface structures. Since phase-error patterns
(e.g., Fig. 2) can be constructed from arbitrary locations in the image, VC still works well with irregularly shaped regions—as long as the phase
diversity requirement is met. This is demonstrated
in Fig. 14.
A particular motion sometimes encountered in microscope applications is linear motion perpendicular
to the optical axis or angular motion about the optical
axis, which I collectively call off-axis motion. Off-axis
motion is often a result of cantilevering the profiler to
access difficult regions of the test object. Such motion
blurs the overlap between interferograms and reduces the effective lateral resolution of the measured
profile. Though VC will still provide compensation
against simultaneously occurring onaxis motion, it
cannot correct for the resolution loss.
8.
Summary
This paper describes a new technique for postprocessing phase-shifting interferometry data to reduce the
influence of vibrations and imperfect phase shifting.
The technique relies on the construction and spectral
analysis of a unique space-domain representation,
called a phase-error pattern, of each acquired interferogram. The phase-error pattern, a plot of the measured intensity versus measured phase, is an
effective signal of the remaining distortion in the surface profile after applying the chosen phase-shifting
algorithm. The technique is computationally fast, requires no additional hardware, and is applicable to
any phase-shifting algorithm. Unlike Fourier methods, there are few restrictions on the surface shape,
but at least one fringe of surface departure is recommended. Though the level of compensation depends
on the PSI algorithm and vibration amplitude and
Fig. 14. (Color online) VC applied to data from a phase-measuring microscope. The fringe pattern is shown at left. The center image
shows the measured surface profile without VC and the right image is with VC.
10 July 2009 / Vol. 48, No. 20 / APPLIED OPTICS
3957
frequency, some compensation is obtained for all
vibration conditions for which the PSI algorithm
itself does not fail.
Appendix A
Let the interference intensity in the presence of a
pure vibrational tone be modeled as
Iðx; tÞ ¼ I 0 f1 þ V cos½ΦðxÞ þ ω0 t
þ r cosðωv t þ αÞg;
ðA1Þ
where I 0 is the illumination intensity, V is the
contrast, ω0 is the fundamental interference angular
frequency of the PSI acquisition, ωv is the vibration
angular frequency, α is its starting phase, r is the vibration amplitude, x represents the ðx; yÞ surface position, and ΦðxÞ is the interferometric starting phase
(the phase to be recovered), which depends on x. Note
that this description specifically assumes uniform
illumination intensity and low finesse interference.
For convenience, Eq. (A1) is separated into a DC term
and the signal of interest sðx; tÞ:
sðx; tÞ ¼
I0 V
fexp½iΦðxÞ exp½iβðtÞ
2
þ exp½−iΦðxÞ exp½−iβðtÞg;
ðA2Þ
with βðtÞ ¼ ω0 t þ r cosðωv t þ αÞ representing the
phase evolution experienced during the PSI acquisition. With the help of the Jacobi–Anger expansion
[34] and after integrating over the detector integration period τ using
1
τ
τ
tþ
Z2
0
eigt dt0 ¼ eigt sincðgτ=2Þ;
ðA3Þ
t−2τ
the interference signal can be written as
∞
I0 V
ω0 τ
I0 V X
τðω0 þ kωv Þ
k
J ðrÞsinc
fexp½iΦðxÞ þ iω0 t þ exp½−iΦðxÞ − iω0 tg þ
i J k ðrÞsinc
sðx; tÞ ¼
2 0
2
2 k¼1
2
∞
I VX
τðω0 − kωv Þ
exp½iΦðxÞ − ikα exp½iðω0 − kωv Þt
ik J k ðrÞsinc
× exp½iΦðxÞ þ ikα exp½iðω0 þ kωv Þt þ 0
2 k¼1
2
∞
I0 V X
τð−ω0 þ kωv Þ
k
exp½−iΦðxÞ þ ikα exp½ið−ω0 þ kωv Þt
ð−iÞ J k ðrÞsinc
×
2 k¼1
2
∞
I VX
ð−iÞk J k ðrÞsinc τð−ω02−kωv Þ exp½−iΦðxÞ − ikα exp½ið−ω0 − kωv Þt :
ðA4Þ
þ 0
2 k¼1
Following standard PSI practice [35], the spectrum is first calculated with the Fourier transform:
Z∞
sðtÞ expð−iωtÞdt;
SðωÞ ¼
ðA5Þ
−∞
producing
Sðx; ωÞ ¼
3958
I0 V
ω τ
J 0 ðrÞsinc 0 fexp½iΦðxÞδðω − ω0 Þ þ exp½−iΦðxÞδðω þ ω0 Þg
2
2
∞
I0 V X k
τðω0 þ kωv Þ
i J k ðrÞsinc
þ
exp½iðΦðxÞ þ kαÞδðω − ω0 − kωv Þ
2 k¼1
2
∞
I VX
τðω0 − kωv Þ
exp½iðΦðxÞ − kαÞδðω − ω0 þ kωv Þ
þ 0
ik J k ðrÞsinc
2 k¼1
2
∞
I0 V X
τð−ω0 þ kωv Þ
k
exp½ið−ΦðxÞ þ kαÞδðω þ ω0 − kωv Þ
ð−iÞ J k ðrÞsinc
þ
2 k¼1
2
∞
I VX
τð−ω0 þ kωv Þ
exp½ið−ΦðxÞ þ kαÞδðω þ ω0 − kωv Þ;
ð−iÞk J k ðrÞsinc
þ 0
2 k¼1
2
APPLIED OPTICS / Vol. 48, No. 20 / 10 July 2009
ðA6Þ
where δðωÞ is the Dirac delta function. The PSI phase
^
profile ΦðxÞ
is then determined with
Im½Sðx; ω0 Þ
^
ΦðxÞ ¼ arg½Sðx; ω0 Þ ¼ atan
:
Re½Sðx; ω0 Þ
ðA7Þ
^
The terms in Eqs. (A6) contribute to ΦðxÞ
only if
the argument of the delta-function equals zero,
which, for the vibrationally induced spectral terms,
occurs only when the vibration frequency satisfies
ωv ¼ 2ω0 =k. Combining the nonzero terms, the spectral component at ω0 becomes
exp½−iΦðxÞ;
Sðx; ω0 Þ ¼ η exp½iΦðxÞ þ μ
where z ¼ μ=η. The positive solution is taken since
the measured phase must tend toward the true
phase as the vibration amplitude tends to zero. Note
that for an infinitely sampled intensity signal, η=jηj ¼
1 since Eq. (A9) is purely real.
Practical PSI algorithms only sparsely timesample the interference and do not necessarily
preserve the absolute phase since an overall phase
offset has no effect on the surface profile. To account
for these effects the delta functions in Eqs. (A9) and
(A10) must be replaced with the Fourier transform of
the PSI sampling function,
ðA8Þ
FðωÞ ¼
N −1
X
cj expðiωjÞ;
ðA13Þ
j¼0
where the bar over variables signifies the complex
conjugate, with
η¼
μ
¼
I0 V
ω τ
J 0 ðrÞsinc 0 δð0Þ;
2
2
ðA9Þ
I0 V
ω0 τ
η¼
J ðrÞsinc
Fð0Þ;
2 0
2
∞
I0 V X
τðkωv − ω0 Þ
ð−iÞk J k ðrÞsinc
2 k¼1
2
× expðikαÞδðωv − 2ω0 =kÞ:
ðA10Þ
Equation (A8) represents the interference
spectrum computed by an infinitely sampled PSI
algorithm in the presence of vibrations. The
vibrational contribution is carried completely in
the second term μ
exp½−iΦðxÞ and, for a single pure
vibrational tone, μ has only one nonzero term if the
vibration frequency satisfies ωv ¼ 2ω0 =k. This result
explains the 1st order sensitivity of PSI algorithms to
vibrations at twice the phase-shifting frequency and
also provides insight into the frequency dependence
of higher-order sensitivity. Higher-order contributions occur at progressively smaller vibration
frequencies, but the fact that the vibrational contribution is proportional to the amplitude tends to
increase the contribution of low frequency vibrations,
so the higher-order vibrational contributions can still
be large.
Since the surface profile is ultimately determined
from the spectral phase, Eq. (A8) provides the follow^
ing expression for the measured phase profile ΦðxÞ:
^
expðiΦðxÞÞ
¼
where the cj are the complex coefficients of the N
sample PSI algorithm, and ν is frequency normalized
to the sample rate. Thus, for finite PSI algorithms,
Eqs. (A9) and (A10) become
η exp½iΦðxÞ þ μ
exp½−iΦðxÞ
: ðA11Þ
jη exp½iΦðxÞ þ μ
exp½−iΦðxÞj
Solving for expðiΦÞ gives
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
^
η u
1 − z exp½−2iΦðxÞ
^
exp½iΦðxÞ ¼ exp½iΦðxÞ t
;
^
jηj
1 − z exp½2iΦðxÞ
ðA12Þ
ðA14Þ
∞
I0 V X
τðkωv − ω0 Þ
k
μ
¼
ð−iÞ J k ðrÞsinc
2 k¼1
2
× expðikαÞFðωv − 2ω0 =kÞ:
ðA15Þ
This substitution reduces spectral resolution and
increases leakage, making the power from a single
frequency leak across the measured spectrum. In
practice, therefore, many terms contribute to μ
[Eq. (A10)] even if only a single vibrational tone is
present, with the influence of the higher-order terms
increasing with vibration strength. Furthermore,
η=jηj may, in general, be complex depending on the
PSI sampling coefficients cj , with a value other than
unity representing a constant phase offset in the
sampling sequence. Without loss of generality, however, the value of η=jηj can be changed to unity by
multiplying the algorithm sampling coefficients by
an appropriately complex constant of unit magnitude
without changing the algorithm dynamic response
[24]; therefore, in what follows η=jηj ¼ 1.
Equation (A12) describes how to use the measured
phase, which is corrupted by the vibrational disturbance, to recover the true phase. For vibrations with
small enough amplitude, jzj ¼ jμ=ηj < 1, and the
radical can be expanded using the binomial theorem
to obtain
^
exp½iΦðxÞ ¼ exp½iΦðxÞ
∞
X
^
pn exp½−i2nΦðxÞ
n¼0
×
∞
X
^
qm exp½i2mΦðxÞ;
ðA16Þ
m¼0
10 July 2009 / Vol. 48, No. 20 / APPLIED OPTICS
3959
with coefficients
pn ¼
1
2!
ð12 − nÞ!n!
ð−zÞn and qm ¼
− 12 !
ð−zÞm :
ð− 12 − mÞ!m!
ðA17Þ
The product of the two sums produces a series of
positive and negative odd harmonics of the measured
phase
∞
X
exp½iΦðxÞ ¼
^
gk exp½ið2k þ 1ÞΦðxÞ;
ðA18Þ
k¼−∞
with the harmonic coefficients gk given by
gk ¼
P∞
pn qnþk
Pn¼0
∞
n¼0 pn−k qn
k≥0
:
k<0
ðA19Þ
Close inspection of Eq. (A19) provides the following
relation between the complex conjugates of the
harmonic coefficients:
−k ;
gk ≅ −ð2k − 1Þg
ðA20Þ
from which Eq. (A18) can be rewritten as
exp½iΦðxÞ ¼
∞
X
^
gk exp½ið2k þ 1ÞΦðxÞ
k¼0
−
∞
^
X
k exp½−ið2k − 1ÞΦðxÞ
g
k¼1
2k − 1
:
ðA21Þ
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