Download Interferometric back focal plane microellipsometry

Interferometric back focal plane microellipsometry
Gilbert D. Feke, Daniel P. Snow, Robert D. Grober, Peter J. de Groot, and Leslie Deck
We present a technique for ellipsometric analysis of materials with high lateral resolution. A Michelsontype phase-shifting interferometer measures the phase distribution in the back focal plane of a high
numerical aperture objective. Local measurements of the ellipsometric parameter delta are performed
over the entire spectrum of angles of incidence. We show that delta is to leading order linearly proportional to the phase change on reflection of normally incident light. We furthermore invert the Fresnel
reflection equations and derive expressions for the real and imaginary parts of the refractive index as
functions of the phase change on reflection and the reflectivity at normal incidence, both of which are
measurable with the same apparatus. Hence we accomplish local measurements of the refractive
indices of our samples. Determination of the phase change on reflection permits correction of interferometric topography measurements of heterogeneous specimens. © 1998 Optical Society of America
OCIS codes: 120.2130, 120.5050, 120.3180, 180.3170, 120.5700, 120.4530.
1. Introduction
Precision topography-profiling optical interferometers are typically capable of subnanometer surface
height resolution.1 Such interferometers cannot
distinguish phase shifts that are due to sample topography from phase shifts that are due to local variations in the refractive index.2– 4 Phase change on
reflection ~f0! can lead to height errors of as much as
30 nm.5 A scheme for measuring f0 would be desirable for correction of interferometric profilometry to
obtain true topography.
Optical ellipsometry is commonly used to determine the refractive index of a material.6 Conventional ellipsometers measure the ratio of the Fresnel
reflection coefficients for a beam reflected at a single
oblique angle. However, such ellipsometers are limited in their lateral resolution to approximately 20
mm, which is inadequate in the context of surface
profiling. Furthermore, conventional ellipsometers
operate at oblique incidence, whereas interferometric
profilometers operate at normal incidence.
Diffraction-limited resolution and geometric com-
G. D. Feke, D. P. Snow, and R. D. Grober are with the Department of Applied Physics, Yale University, New Haven, Connecticut
06520-8284. P. J. de Groot and L. Deck are with the Zygo Corporation, Laurel Brook Road, Middlefield, Connecticut 064550448.
Received 15 July 1997; revised manuscript received 20 October
1997.
0003-6935y98y101796-07$15.00y0
© 1998 Optical Society of America
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APPLIED OPTICS y Vol. 37, No. 10 y 1 April 1998
patibility with interferometric profilometers may
be achieved by using back focal plane microellipsometry,5,7–9 which measures the amplitude
and phase distributions of the reflected light in the
back focal plane of an objective lens. Specific to
profilometry correction, previous work5 has addressed the measurement of f0 by interferometric
measurement of the phase distribution only.
This paper serves to develop and refine further the
technique of interferometric back focal plane microellipsometry. The basic principles of this technique
are described. We have derived an analytical expression that allows for direct determination of f0
from the phase information over the entire numerical
aperture of the objective. We further invert the
normal-incidence Fresnel reflection equations to express the refractive index in terms of f0 and the
normal incidence reflectivity. An apparatus has
been constructed to perform local measurements of
both of these quantities. Results for several test
specimens are in agreement with theoretical expectations. Experimental difficulties suffered by others
in previous measurements of this type5 are not
present in our results. Hence the accuracy of interferometric profilometry of heterogeneous specimens
may therefore be improved to the point at which it is
limited only by the accuracy in the measurement of
f0.
2. Principle of Interferometric Back Focal Plane
Microellipsometry
Although interferometric back focal plane microellipsometry has been described previously,5 a review is
Fig. 2. Theoretical reflected phase curves for aluminum ~n9 5
1.8 1 7.7i!. f0 and the average of fp and fs are roughly equal out
to ui 5 50° ~NA 5 0.77!.
Fig. 1. Principle of back focal plane microellipsometry.
appropriate here. Figure 1 depicts a linearly polarized, collimated beam of light that is focused by the
objective lens to a spot on the sample positioned in
the front focal plane. In the object space of the lens,
an angular spectrum of plane waves is incident upon
the sample. The p and s polarizations are defined,
respectively, to be parallel and perpendicular to the
plane of incidence. The p- and s-polarized photons
are therefore simultaneously incident upon the sample.
In Fig. 1 the polarization direction of the input
beam is along the x axis. Therefore the light is
purely p-reflected along the x axis, and purely
s-reflected along the y axis. The distribution of light
in the back focal plane of the objective is the Fourier
transform of the distribution in the front focal plane.
Therefore, in the back focal plane, the plane-wave
components incident upon the sample are spatially
separated. Specifically, for a lens that obeys the
sine condition,10 the angle of incidence ui ~measured
from the normal! is related to the radial coordinate r
in the back focal plane by
sin ui 5 ~ryrmax!sin~umax!,
(1)
where rmax is the radius of the back aperture and
sin~umax! is the numerical aperture ~NA! of the objective. Measurement of the amplitude and phase distributions in the back focal plane yields
determination of the ellipsometric parameters. The
phase distribution may be measured interferometrically by making the objective–sample system one arm
of a Michelson interferometer.
3. Interrelations among Optical Quantities
A. Phase Change on Reflection and the Ellipsometric
Parameter d
Consider linearly polarized light incident at an angle
ui upon the surface of a material with complex refractive index n9 5 h 1 ik. The Fresnel equations giving
the complex reflection coefficients for p- and
s-polarized light are11
rp 5
tan~ui 2 ut!
,
tan~ui 1 ut!
rs 5 2
sin~ui 2 ut!
.
sin~ui 1 ut!
(2)
The complex angle of transmission ut is given by
Snell’s law,
sin ui 5 n9 sin ut,
(3)
for an incident medium of unity refractive index ~air!.
At normal incidence, the reflection coefficient is given
by
r0 5
n9 2 1
.
n9 1 1
(4)
The ellipsometric parameter d is defined as the
difference of the phases of the p and s complex reflection coefficients:
d ; fp 2 fs
5 arg~rp! 2 @arg~rs! 1 p#.
(5)
Here fs is defined so that d 5 0 at ui 5 0, which is
consistent with the convention used by See et al.5
The phase change on reflection at normal incidence
is given by
f0 5 arg~r0!.
(6)
As shown in Fig. 2, f0 is to a good approximation the
average of fp and fs out to a large NA. Hence f0 is
the index-dependent phase error introduced into topography measurements.
The goal of the present research is to devise a
method for making local measurements of f0 for correction of topography measurements of heterogeneous specimens. f0 may be determined exactly,
provided n9 is known, by means of Eqs. ~4! and ~6!.
However, the ellipsometric determination of n9 requires measurements at one particular angle of incidence of both of the ellipsometric angles, d and c, the
latter of which is defined as
tan c ; urpuyursu.
(7)
The determination of c from urpu and ursu is complicated in a microellipsometer because the transmittance of the objective is a function of ui .9 The error
in the measurement of f0 when the refractive index
determined from d and the incorrect values for c are
used has been shown5 to be typically ;5°.
By simply plotting f0 versus d at ui 5 45° over a
1 April 1998 y Vol. 37, No. 10 y APPLIED OPTICS
1797
range of refractive indices, See et al. have reported5
an empirical linear relation between the two,
f0 > 1.41d,
(8)
which is independent of c. The value of such a simple relation in the context of a topography-correcting
measurement is directly evident. However, Eq. ~8!
limits the measurement to a single angle of incidence,
while the rest of the angular spectrum available in
the microellipsometric measurement is disregarded.
However, by using a series of approximations to the
functional forms of both d~ui ! and f0 in appropriate
limits, we have derived the analytical result
d > f0~ui 2 1 ui 4y6!,
(9)
for arbitrary angle of incidence, where ui is in radians. Substituting ui 5 py4 into Eq. ~9! yields f0 5
1.47d, which is comparable with the empirical result
of Eq. ~8!. Equation ~9! provides a computationally
simple relation between d and f0 in terms of one
known parameter ui .
B. Refractive Index Expressed as a Function of f0 and
the Normal-Incidence Reflectivity
The reflectivity at normal incidence is given by
R0 5 ur0u2.
(10)
h~R0, f0! and k~R0, f0! are found upon inversion of
Eqs. ~6! and ~10! and are
k~R0, f0! 5
F
G
4R0 cos f0 1 2 ÎR 0~1 1 R0!
sin f0,
~1 1 R0!2 sin2 f0 1 ~1 2 R0!2 cos2 f0
(11)
h~f0, k! 5 ~2k cot f0 2 k2 1 1!1y2.
(12a)
Equation ~12a! is ill-behaved in the limit k 3 0 ~i.e.,
f0 3 0, cot f0 3 `!. In this case, an alternative
expression may be used,
h~R0, k! 5
1 1 R0 1 @4R0 2 k2~1 2 R0!2#1y2
,
1 2 R0
(12b)
which is valid for k # ~h2 2 1!1y2. For k $ ~h2 2 1!1y2,
a negative sign is required in front of the square root
of Eq. ~12b! for calculation of h. To determine which
inequality holds, h would need to be known ~at least
roughly! beforehand. This condition is readily met
for dielectric materials for which k is very small @the
case where Eq. ~12b! is required# and h is typically
not smaller than 1.5.
In summary, by measuring d, we obtain f0, and by
measuring R0 further, we obtain h and k. Equations
~9!, ~11!, and ~12! allow us to obviate the measurement of c in the determination of refractive index.
4. Apparatus
A schematic of the optical system is shown in Fig. 3.
The design is similar to that described by See et al.5
The output of a 670-nm wavelength multimode laser
diode is coupled into a single-mode optical fiber,
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APPLIED OPTICS y Vol. 37, No. 10 y 1 April 1998
Fig. 3. Schematic of the interferometric back focal plane microellipsometer: NA, numerical aperture; PZT, piezoelectric transducer; BFP, back focal plane; PC, personal computer.
which is used as a spatial filter. The polarization
state of the photons is selected in the fiber with a
BT&D MPC1000 fiber polarization controller. The
fiber output is collimated by an achromatic lens.
The collimated beam is split with a wedged plate
beam splitter.
One arm of the interferometer consists of the reference mirror, which is mounted on a tilt stage. The
tilt stage is mounted on a flexure stage. Translation
of the reference mirror is achieved by pushing the
flexure stage with a piezoelectric transducer ~PZT!.
The PZT is driven by a signal from a programmable
function generator and is precalibrated to provide
accurate, constant velocity motion ~linear scan!.
The other arm of the interferometer consists of the
objective lens and sample. The objective lens is a
Zeiss Epiplan-Neofluar infinity-corrected microscope
objective with NA 5 0.9 ~umax 5 64°!. The sample is
mounted on a translation stage that allows for both
lateral positioning and focusing of the sample. A
computer-driven micropositioner is used to make fine
adjustments to the sample focus.
Two plano– convex lenses serve as the relay optics
to image the back focal plane of the objective onto a
CCD camera. A PC-based eight-bit frame-grabber
acquires consecutive frames of video from the camera
as the PZT ramps the position of the reference mirror.
The consecutive frames record interferograms that
contain the relevant phase information.
It is straightforward to derive
fdiff~V 5 0! 5 fdiff~V 5 p! 5 d,
fdiff~V 5 py2! 5 fdiff~V 5 3py2! 5 2d.
(17a)
(17b)
Therefore this measurement yields d~ui ! for 0 # ui #
sin21 ~NA!. Another feature of the difference map is
nodes along the diagonals:
fdiff~V 5 mpy4! 5 0,
Fig. 4. Subtraction of phase maps to obtain the difference map.
By convention, the first phase map has input polarization along the
x axis, and the second has input polarization along the y axis.
5. Method of Analysis
A single interferogram is associated with each pixel
position ~x, y! in the image of the back focal plane.
Seven frames record each collection of interferograms. The velocity of the PZT ramp is matched
with the frequency of the camera such that consecutive frames are shifted in phase by an increment of
py2. We use a seven-frame phase-demodulation algorithm,12
m 5 1, 3, 5, 7.
(18)
From Fig. 4 we see that in principle it is possible to
obtain d~ui ! from just one phase map. This would be
accomplished by taking the difference between the
horizontal and vertical line cuts. However, by taking two phase maps and subtracting one from the
other, we subtract out any constant phase aberrations associated with the optical system.
Normal-incidence reflectivity measurements are
made with the same apparatus. The reference mirror is blocked so that the camera detects only the
sample reflection. The intensity of the center pixel
~ui 5 0! is obtained from an average over several
frames. A reference of known reflectivity is used to
calibrate the measurements.
6. Results and Discussion
tan f 5
7~I21 2 I1! 2 ~I23 2 I3!
,
24~I22 2 I2! 1 8I0
(13)
at each pixel position to construct the phase map
from the interferograms. In Eq. ~13!, In is the measured intensity at each pixel in the nth frame.
Two sets of interferogram data are recorded per d
measurement: The first has input polarization orthogonal to that of the second. The difference in the
two resulting phase maps is calculated to obtain a
difference map, as illustrated in Fig. 4. The functional form of the difference map is obtained from the
expressions for the field distribution in the back focal
plane13:
SD
SD
Y x2pol 5 Ex
E
Ey
Y y2pol 5 Ex
E
Ey
S
S
D
D
5
E0 R1 1 R2 cos 2V
,
R2 sin 2V
2
5
E0
R2 sin 2V
,
1
2
2 R 2 R cos 2V
x2pol
y2pol
(14)
where the incident beam is assumed to have amplitude
E0, V is the azimuthal angle measured from the x axis,
and R1 [ rp 1 rs exp~ip!, R2 [ rp 2 rs exp~ip!, which
are functions of ui . Since the reference beam is purely
x or y polarized, interference occurs between only the
Ex and Ey components of the sample-reflected beam.
Therefore the two-dimensional functional forms of the
two phase maps are
fx2pol 5 arg~Ex!x2pol,
fy2pol 5 arg~Ey!y2pol.
(15)
The difference map is given by
fdiff 5 fx2pol 2 fy2pol.
(16)
Three samples were examined: a silicon wafer, a
thick aluminum film evaporated onto a glass microscope slide, and a thick gold film evaporated onto a
silicon wafer. Resultant difference maps are shown
in Fig. 5. In the Au and Al data, d increases and
decreases along the horizontal and vertical directions, respectively, as expected from Eqs. ~17!, and
the nodal lines along the diagonals as predicted in Eq.
~18! are clearly visible.
The averages of the four d lines given in Eqs. ~17!
from the difference maps of Fig. 5 are shown in Fig.
6. Averaging is done to reduce the measurement
noise. The theoretical curves are generated from
representative values as listed in the literature14 for
the refractive indices at our operating wavelength.
These theory curves are not adjustable by any free
parameter; they are generated directly from the
Fresnel formulas of Eq. ~2!. In addition, the data
sets were fitted according to Eq. ~9!, with f0 as the
sole free parameter. Table 1 lists the theoretical
@from Eq. ~6!# and experimental values for f0 for
these samples. Good agreement ~to within a few
degrees! is found between the literature and the experimental values for f0. Given that our present
experimental accuracy is roughly 3° and that f0 may
be as much as 30°, the possible improvement for the
height resolution of interferometric topography measurements of heterogeneous samples is as much as a
factor of 10.
See et al. performed microellipsometric measurements on several materials and obtained d curves
that were not in agreement with theory but were
significantly offset from the theoretical models by a
constant function.5 They attributed their anomalous results to multiple reflections occurring inside
1 April 1998 y Vol. 37, No. 10 y APPLIED OPTICS
1799
Fig. 6. Line-cut averages from Fig. 5. The data ~■! are fit ~- - - -!
according to Eq. ~9!, and the results are compared with the theoretical curves ~—! from the literature values of the refractive indices.
ther side of the front focal plane as set by the focusing
micropositioner. The results are plotted in Fig. 7.
The exact sample positions were determined from the
interferometric data by measurement of the curvature
incurred by each phase map as a result of defocus.
Particularly, Schulz and Elssner have demonstrated15
that the phase difference between the object and reference rays in a reflected-light interference microscope
is dependent on the distance H of the sample from
focus and the angle of incidence ui as
fdefocus 5 4p~Hyl!cos ui .
(19)
Table 1. Comparison of Values for f0
Sample
Literature ~deg!
Fit ~deg!
Au
Al
Si
29.4
14.1
0.1
26.4
14.4
1.2
Fig. 5. Difference maps for gold, aluminum, and silicon.
the objective lens. However, we do not find any
evidence of such anomalies in our system.
To explore the effect of sample defocus on the measurement of f0, difference maps of aluminum were
recorded for many different sample positions on ei1800
APPLIED OPTICS y Vol. 37, No. 10 y 1 April 1998
Fig. 7. Measured f0 as a function of defocus.
Table 2. Comparison of Values for Reflectivity and Refractive Index
h 1 ik
R0
Sample Literature Experimental
Au
Al
Si
0.96
0.91
0.34
0.94
0.89
0.37
Literature
Experimental
0.16 1 3.8i
1.8 1 7.7i
3.8 1 0.01i
0.29 1 4.2i
1.8 1 7.5i
4.1 1 0.16i
With ui related to the radial coordinate r by Eq. ~1!,
each of the phase maps depicted in Fig. 4 and expressed in Eq. ~16! takes on a curvature; i.e., the
measured out-of-focus phase maps are given by
ftotal, x2pol 5 fx2pol 1 fdefocus,
ftotal, y2pol 5 fy2pol 1 fdefocus.
(20)
The sample position H is extracted by fitting the expression in Eq. ~19! to our measured phase maps.
The results of Fig. 7 demonstrate that measurement of f0 is robust to within a few degrees throughout and beyond the depth of focus, which is
approximately 20.3l to 10.3l for our objective.
However, the results for negative values of defocus
~i.e., sample positions too close to the objective! are
especially near the literature value, whereas the results for positive values are somewhat less than the
literature value.
The experimental values for f0 listed in Table 1
were combined with normal-incidence reflectivity
measurements to obtain h and k, as listed in Table 2.
Again, good agreement is found between the experimental results and the literature.
In in Eq. ~13! may be written as
S
F
DG
Imax
p
1 1 V cos f 1 n
11V
2
(
n
S D
S D
DI
Imax
2
196 1 49V 2
.
512V 2
(23)
@Equation ~23! gives the variance averaged over all
possible phases from 0 to 2p; the variance as a function of f varies only slightly from its average value.#
A typical visibility in our interferograms is 0.1.
The frame-to-frame standard deviation in pixel intensity DI was measured at 0.75 out of Imax 5 256
~eight-bit digitization!. Insertion of these values
into Eq. ~23! yields a standard deviation in phase Df
5 1°. This theoretical value for the phase noise was
found to be in agreement with the experimentally
measured Df, as obtained from taking the difference
of two phasemaps recorded in identical conditions.
8. Conclusion
Local measurements of d for various samples have
been successfully achieved with an interferometric
back focal plane microellipsometer. We have derived an analytic approximation that relates d for a
given angle of incidence directly to the phase change
on reflection at normal incidence. This measurement allows for refractive-index-dependent corrections to interferometric profilometry. Also, we have
inverted the normal-incidence Fresnel reflection
equations so that we may explicitly determine the
refractive index from measurements of the normalincidence reflectivity and the phase change on reflection.
References
,
(21)
where Imax is the greatest possible pixel intensity in
the interferogram, V is the visibility, and each frame
is shifted by a phase increment of py2 as noted above.
The coefficients of the In are chosen such that Eq. ~13!
reduces to tan f.
Phase-measurement errors due to distortions in
the phase shift and low-frequency mechanical vibration have been discussed previously9 for the sevenframe algorithm. Errors may also be introduced by
electronic camera noise,16 i.e., fluctuations in Imax.
To determine the sensitivity of the seven-frame algorithm to camera noise, we calculate the phase variance as given by
~Df!2 5 ~DI!2
^~Df!2&f 5
This research was supported in part by the Zygo
Corporation.
7. Error Analysis
In 5
where f is given by Eq. ~13! and Imax fluctuates randomly in each of the seven frames. Substitution of
Eq. ~21! into Eq. ~13! and Eq. ~13! into Eq. ~22! yields
2
]f
,
]Imax,n
(22)
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APPLIED OPTICS y Vol. 37, No. 10 y 1 April 1998
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