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Reconstruction of sub-wavelength features and nano-positioning of gratings using coherent Fourier scatterometry

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Abstract and Figures

Optical scatterometry is the state of art optical inspection technique for quality control in lithographic process. As such, any boost in its performance carries very relevant potential in semiconductor industry. Recently we have shown that coherent Fourier scatterometry (CFS) can lead to a notably improved sensitivity in the reconstruction of the geometry of printed gratings. In this work, we report on implementation of a CFS instrument, which confirms the predicted performances. The system, although currently operating at a relatively low numerical aperture (NA = 0.4) and long wavelength (633 nm) allows already the reconstruction of the grating parameters with nanometer accuracy, which is comparable to that of AFM and SEM measurements on the same sample, used as reference measurements. Additionally, 1 nm accuracy in lateral positioning has been demonstrated, corresponding to 0.08% of the pitch of the grating used in the actual experiment.
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Reconstruction of sub-wavelength
features and nano-positioning of gratings
using coherent Fourier scatterometry
Nitish Kumar,1,Peter Petrik,1,2Gopika K P Ramanandan,1Omar El
Gawhary,1,3Sarathi Roy,1Silvania F Pereira,1Wim M J Coene,4and
H. Paul Urbach1
1Optics Research Group, Department of Imaging Physics, Faculty of Applied Sciences, Delft
University of Technology, Van der Waalsweg 8, 2628CH Delft, The Netherlands
2Research Centre for Natural Sciences, Institute for Technical Physics and Materials Science,
Hungarian Academy of Sciences, H-1121 Budapest, Konkoly Thege Mikl´
os ´
ut 29-33, Hungary
3VSL Dutch Metrology Institute, Thijsseweg 11, 2600 AR Delft, The Netherlands
4ASML Netherlands B.V., De Run 6501, 5504 DR Veldhoven, The Netherlands
Abstract: Optical scatterometry is the state of art optical inspection
technique for quality control in lithographic process. As such, any boost in
its performance carries very relevant potential in semiconductor industry.
Recently we have shown that coherent Fourier scatterometry (CFS) can lead
to a notably improved sensitivity in the reconstruction of the geometry of
printed gratings. In this work, we report on implementation of a CFS instru-
ment, which confirms the predicted performances. The system, although
currently operating at a relatively low numerical aperture (NA =0.4)
and long wavelength (633 nm) allows already the reconstruction of the
grating parameters with nanometer accuracy, which is comparable to that
of AFM and SEM measurements on the same sample, used as reference
measurements. Additionally, 1 nm accuracy in lateral positioning has been
demonstrated, corresponding to 0.08% of the pitch of the grating used in
the actual experiment.
© 2014 Optical Society of America
OCIS codes: (120.3940) Metrology; (290.5820) Scattering measurement; (180.5810) Scanning
microscopy; (050.1950) Diffraction gratings.
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6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024678 | OPTICS EXPRESS 24678
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1. Introduction
The demand for faster, smaller, lighter and, at the same time, high-data density electronic de-
vices sets stringent requirements for nanolithography, the science of writing small features into
a photo-sensitive resist layer on top of a silicon wafer [1]. Already for the current 45 and 32 nm
technology nodes, the uniformity of the line-width or critical dimension (CD) over the wafer as
produced by lithographic scanners must be improved for an optimal yield and performance of
the electronic components. In order to obtain the intended line shapes and sizes, a reliable in-
line process control has to take place. This is achieved by printing special targets on the wafer,
typically gratings, which are successively measured in order to adjust dose, exposure time, over-
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lay/alignment and other relevant process parameters of the photo-lithographic machine [2,3].
Currently, the metrology task of this process control is achieved by means of Incoherent Optical
Scatterometry (IOS). In this technique, which is a very well established method for the inspec-
tion of periodic structures like gratings, an incoming beam is shone on the target and the part of
the light which is scattered by it in reflection is measured in the far field. Given some a priori
knowledge of the target, one can achieve high accuracies in the reconstruction of the shape of
the grating. The advantage over other competing inspection techniques, such as imaging, Scan-
ning Electron Microscopy (SEM) or Atomic Force Microscopy (AFM), is that IOS is a very
fast technique which does not suffer from the Rayleigh diffraction limit, is easily integrable in a
lithographic machine, and can cope with the high throughputs of today’s scanners, in the order
of 200 wafers per hour.
The way near field techniques utilize coupling of probe and sample cannot be utilized in
production environment without decreasing the throughput. Rigorous far field techniques are in
this sense a good alternative. However, such advantages also have a price since scatterometry
falls into the category of inverse problems in electromagnetism, which are known for being
severely ill-posed. Inverse problems occur in many other branches of science and technology
as well. Ill-posedness in this context means that the successful reconstruction of grating param-
eters from the far field may not be possible, or may not be unique [4,5]. This implies that even a
very precise and accurate experimental far field signal does not always provide enough informa-
tion content for reconstruction. It is the presence of some
a priori
information (as for example
an approximate grating structure) that enormously reduces the impact of the ill-posedeness and
makes scatterometry feasible in practice.
Many variations of the idea behind a scatterometer have emerged in the last decades [6].
Some of the most widely used configurations are single incidence angle reflectometry, 2-Θ
scatterometry, spectroscopic ellipsometry, Fourier scatterometry, interferometric Fourier scat-
terometry,etc., [7–15] with a wide range of applications [16–20]. In an earlier paper, it has been
predicted theoretically how, and under which conditions, CFS can be more sensitive than the
classical IOS [21]. In this paper, we demonstrate the reconstruction of the parameters of a peri-
odic grating using Coherent Fourier Scatterometry (CFS), which represents a step towards the
further improvement of sensitivity of scatterometry in the sub-nanometer regime. Beside pro-
viding an accurate reconstruction of the grating’s parameters, CFS also carries the strong po-
tential of being used as subnanometer wafer alignment tool since the scattered signal is highly
sensitive to the grating position as well. Last, but not least, CFS with focused spot is not limited
to measuring periodic structures but can be applied to analysis of multilayer structures, mate-
rial sciences, photonics industry, biosensing, detection of isolated structures and other formsof
non-contact metrology.
2. Simulation and experimental considerations
2.1. Coherent Fourier scatterometry
In CFS, light from a coherent source is focused on periodic structures (typically gratings) on the
wafer. The focused spot interacts with the grating and the farfield is recorded [22]. In this way,
the angular spectrum for all scattered waves is recorded at once for all incident plane waves
within the focused spot. In the event of overlapping reflected orders in the lens pupil, there is
interference between them and some phase information is also registered [23]. This is achieved
by scanning the grating by means of a tightly focused beam, which allows resolving the phase
information in practice. The number of scanning positions needed to resolve the phase depends
upon the number of overlapping orders in the pupil of the focusing lens. Since the technique
relies on the acquisition of the diffracted far field, methods to model the interaction between
incident focused spot and the grating, which gives rise to such far field, must be implemented.
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6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024678 | OPTICS EXPRESS 24680
This task, called forward problem, is typically performed by means of a rigorous electromag-
netic solver for a set of geometrical and physical parameters of the grating (Fig. 1.) under the
predefined illumination conditions. In this way, the overall expected sensitivities of IOS and
CFS have been computed and compared [24,25].
2.2. Grating model and the illumination scheme
Let us consider an infinitely long one-dimensional grating with period Λ, which is invariant
along the y-axis as shown in Fig. 1. The geometrical shape of the grating is defined by the pa-
rameters midCD (width of the grating lines at half of the total height), height and side wall angle
(SWA). An additional metrology parameter, which we call bias, is also defined here. Bias is the
measure of the grating displacement from its nominal position with respect to the illumination
spot. Defined in this way, such parameter directly provides information on the alignment of the
wafer which the grating has been printed on. The zero bias position can be chosen arbitrarily,
and in our case, we choose it at the position where the optical axis of the microscope objective
bisects the midCD (assuming a symmetrical grating profile). These parameters and the basic
principle of data acquisition in CFS are shown in Fig. 1.
CCD Beam
Translation of the grating
Fig. 1. Scheme of the CFS illumination, data acquisition system and the grating parameters.
A collimated light beam with a well-defined polarization state in the pupil of the lens is
focussed on the grating and the scattered light is collected and collimated by the same lens, and
detected by the CCD as illustrated in Fig. 1. Any change in the grating parameters results in a
nonlinear change of the reflected far field. The total number of detected propagating diffracted
orders depends on the wavelength (
) of the incident light, the numerical aperture (NA)ofthe
lens and the period (Λ) of the grating. The amount of overlap between the diffracted orders in
the lens pupil is given by the overlap parameter F, defined as:
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6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024678 | OPTICS EXPRESS 24681
F<1 1<F<2 F>2
+1 +2
-1-2 0-1 +1
0-1 +1
Fig. 2. Overlap between the diffracted orders depending upon the value of the overlap
parameter F. The NA of the lens is marked with black circles.
As shown in Fig. 2, at the lens pupil, for 1 <F<2, there is an overlap between the 0th and
±1st orders of the grating but no overlap between the 1st and +1st order. For F<1, however,
there is an overlap between the higher diffracted orders, and for F>2, there is no overlap
between the orders.
2.3. Grating fabrication
The periodic structure used in the experiment is an etched silicon grating. To fabricate the grat-
ing, a cleaned silicon wafer was spin-coated with e-beam sensitive resist polymethyl methacry-
late (PMMA). The grating pattern was written into the resist using electron beam lithography.
After development of the resist, the grating pattern was etched into the silicon wafer using a
reactive ion etching system (F1 Leybold Fluor ethna), with SF6gas as the etchant. The remain-
ing resist layer was removed using dry oxygen plasma etch. The target parameters of grating
fabrication are pitch=1300 nm, midCD=560 nm, height=115 nm, and SWA=90 degrees.
2.4. Experimental setup
L3 L4L5
Fig. 3. Schematics of the experimental setup. (a) Ray diagram of the experimental scheme
(S1: He-Ne laser, FC: Fiber coupler, SMF: Single mode fiber, LED: Light emitting diode,
BS: Beam splitter, P: Polarizer, L: lens, DP: Detector plane, BFP: Back focal plane, OP:
Object plane (grating), MO: Microscope objective, TS: Piezo-controlled translation stage).
(b) 3D illustration of the laboratory setup.
Figure 3 represents the schematics of the coherent Fourier scatterometer. Light from a He-
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6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024678 | OPTICS EXPRESS 24682
Ne laser (
=633 nm) is coupled into a fiber which is then divided into two arms by a fiber
coupler (FC). SMF1 and SMF2 are the illumination and the alignment arm, respectively. Light
from SMF1 is collimated and polarized to provide a well-defined illumination of the sample
through the microscope objective (MO). The incident light is selected to be either in TE or TM
polarization configuration in the entrance pupil of the lens. The denomination TE(TM) here
refers to the incident electric (magnetic) field in the pupil being oriented parallel to the grooves
of the grating. The incident focused spot can be decomposed into plane waves with varying
incident angles. The maximum angle of incidence is limited by the numerical aperture NA of
the microscope objective MO. In the actual experiment, we used an objective with NA =0.4.
Each allowed incident angle contributes to the reflected diffraction orders which propagate
back through the MO to the CCD. The diameter of the collimated reflected beam is reduced by
a telescopic system, which images the back focal plane (BFP) of the MO onto the CCD with
a demagnification of 2.5X to fit into the CCD area (1600 ×1200 pixels, size of 3.75
m per pixel). In the Fourier or back focal plane (BFP) there is an interference between
the reflected orders for the chosen Fnumber (1 <F<2, see Fig. 2). The polarizer P2 in the
experimental setup can be used to detect a selected polarization at the CCD. SMF2 is used to
align the telescopic system. The red LED light source is only used to image the grating on the
CCD camera for alignment purposes during the preparation of the experiment. Components
BS2 and L6 are removed during data acquisition. The solid red line in the ray diagram of Fig.
3(a) is the data acquisition path and the black dotted paths are used only for alignment and
imaging. In the setup, the telescopic lenses L4 and L5 are used for data acquisition; and lens
L6, beam splitter BS2, and lenses L4 and L5 are used for alignment and imaging.
3. Results and discussion
3.1. Diffracted far field intensity maps
Along with the acquisition of experimental data, scatterometry also requires an accurate mod-
elling of the interaction between field and sample. We used the rigorous coupled wave analysis
(RCWA) as the rigorous solver to compute the field diffracted by the grating [26–28]. In order
to make the simulations as faithful as possible, the measured amplitude and phase distribution
of the incident field is included in the RCWA simulations as well. The amplitude for the in-
coming beam is practically uniform (measured by CCD and SHS) and the phase in the entrance
pupil of the lens is measured with a Shack-Hartmann wavefront sensor (SHS) [29]. In Figs.
4(a) and (a) the measured wavefronts for TE and TM polarizations on the lens pupil expressed
in units of wavelength of the incident light are shown. The far field intensity maps for a fixed
bias shown in Fig. 4 are the simulated and measured data obtained for a silicon etched grating
having an overlap parameter F=1.2 (see Fig. 2) for NA =0.4 at the wavelength of 633 nm.
Far field intensity maps b, c and d (b,c
and d) represent respectively the simulation, exper-
iment and the difference between them for best matched fit for TE (TM) incident light on the
lens pupil and mixed output polarization (i.e., no polarizer is used in the detection path). In the
simulations, the measured wavefront with a Shack-Hartmann sensor (SHS) is used to compute
the diffracted far field intensity maps. The camera has been tested for noise by measurements,
where each far field intensity map is averaged over 10 frames. Also, we consider the measured
noise of the CCD camera as normally distributed with standard deviation given by measured
uncertainties of
=1×103. While the energy in the diffracted order depends upon the grat-
ing parameters, the extent of overlap between the diffracted orders in the far field is given by
the overlap parameter F.
It is to be noted that the sensitivity of grating parameters (change of the far field maps) is dif-
ferent for TE and TM incident polarized light. In CFS,in the overlap region there is interference
between the grating orders and the phase change due to the shift of the grating can be related to
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(a) (b) (c) (d)
(a’) (b’) (c’) (d’)
Fig. 4. Simulated and experimental far field intensity maps for a fixed bias value and the
difference between the simulation and experiments. Wavefront for TE (a) and TM (a)
incident polarizations on the lens pupil. Far field intensity maps b, c and d (b,c
and d)
represent the simulation, experiment and the difference between them for best matched fit
for TE (TM) incident light on the lens pupil and mixed output polarization. The diameter
of the pupil in a and ais 8 mm.
the change in far field intensity maps for different bias values. The amount of phase shift in the
non-zero order due to translation distance
xis given by
where mis the order number of the reflected order.
3.2. Bias correlation
Wafer positioning is an important issue in the industrial manufacturing process. Being a phase
sensitive technique CFS introduces a change in the far fields with scanning position on the
grating. In the experiments, far field intensity maps for consecutive scan positions (bias) of
20 nm difference were recorded over the length of several periods of the grating. The ability to
distinguish between the intensity maps defines the sensitivity to bias of the present experimental
setup. The degree of correlation ’r’ (correlation coefficient) is used as a measure to distinguish
experimental images:
xy ¯
Ire f (Ixy ¯
xy ¯
Ire f 2xy(Ixy ¯
Here, Iref and I(xand yare pixels) are the far field intensities corresponding to the starting
bias position called reference and the other scanning positions. ¯
Iref and ¯
Iare the corresponding
mean values for Iref and I. The correlation coefficient rhas values between 1 to 1. When
the intensity maps are completely correlated, r=1, while r=0 implies no correlation and there
is complete anti correlation between the intensity maps for r=-1. In Fig. 5, value of correlation
coefficient derived from the experimental far field intensity maps is shown for bias values rang-
ing from 0 to 1.3
m (one period of the grating). Figure. 5(a) plots the correlation coefficients
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for scanning positions within one period of the grating. Figure 5(b) is the color label adjusted
plot to highlight the sensitivity of bias.
Bias (nm)
Bias (nm)
200 400 600 800 1000 1200
1200 −0.2
Bias (nm)
Bias (nm)
200 400 600 800 1000 1200
Fig. 5. Degree of correlation between experimental far field intensity maps. (a) Correlation
coefficients for positions separated by 20 nm for bias values within one period of the grating
(b) The same as a), but only showing the range points where 0.9<r<1.
From the above analysis of the experimental data, it is evident that CFS is highly sensitive to
grating position. However, for a symmetric grating, absolute positions may also be determined
regarding the symmetric position within a period of the grating (like the middle of the midCD,
for which the far fields are also symmetric). In CFS, nm and sub-nm positioning within a single
period of the grating can be reached in wafer alignment in semiconductor industry, also shown
in the reconstructed values (see Table 1 further in the text). This feature can also be used in
other application as in imprint technology.
3.3. Model based optimization
Solving the inverse problem of grating reconstruction with CFS amounts to the determination
of the values of the grating parameters for which the computed scattered far field maps fit best
the experimentally measured images. The grating parameters defined in Fig. 1 lie in certain
intervals obtained from a priori information and from the design specification of the grating.
Starting from the nominal values of the grating parameters, the deviation between the experi-
mental and simulated images are minimized, using a least square function (merit function) by
varying the grating parameters. The diffracted far field of the grating depends on the known
experimental conditions and the unknown grating parameters. Let adenote the grating param-
eters and I(m)
i,jand I(s)
i,jthe measured and simulated far field intensities at the ith CCD pixel and
jth scan position. The merit function to be minimized is the difference between the simulated
and experimental far field intensities summed over all the incident angles in the entrance pupil
and pixels over the detector. The merit function is thus given by
where Sis the total number of scan positions and Nis the total number of pixels at the detector
for a single far field intensity map. We use a gradient-based non-linear optimization method to
minimize the merit function defined in Eqn. 4 [30]. It can be minimized using library search or
real time optimization methods [31]. In library search, several sets of far fields for approximate
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grating parameters lying in the defined intervals are computed. Subsequently, the set of grating
parameters for the minimal value of the merit function (Eqn. 4) is selected as the desired value
of the grating parameters. It is to be noted that the optimization algorithm finds only the local
minima. Here we use a gradient based non-linear least squares real time optimization method
implemented in MATLAB. As a priori information in optimization, the target parameters of
grating fabrication (listed in the text in section 2.3) are used. Figure 6 shows a set of simulated
and experimental far fields for bias values lying within a single period of the grating obtained
with TE incident light on the lens pupil and no polarizer at the detector for the grating parame-
ters corresponding to the minimized merit function. Consecutive far fields (numbered 1 to 12)
correspond to consecutive grating positions for a bias difference of 100 nm.
Fig. 6. Simulated and experimental far field for TE incident polarization on the lens pupil
and no polarizer at the detector for the grating parameters corresponding to the minimized
merit function. The bias position is changed by 100 nm between consecutive far fields
(numbered 1 to 12).
3.4. Parameters reconstruction and discussion
The reconstructed grating parameters are listed in Table 1 (for all the grating parameters includ-
ing bias are fitted together in the optimization). The far field intensities used for the reconstruc-
tion algorithm is a set of data such as shown in Fig. 6 but the incident fields are TE and TM
polarized and no polarizer at the detector side. To verify the results, the grating was also meas-
ured by SEM and AFM. The SEM measurement was performed with a Hitachi S 400 scanning
electron microscope at 4 kV with a magnification of 35000. The uncertainties of the CFS and
SEM were determined from repeated evaluations. In the case of the SEM, the uncertainty of
the measured midCD was determined from measurements at different parts of the image (so the
uncertainties are partly caused by line edge roughness). The bright edges of the grating lines
were the main cause of the uncertainties. This edge is clearly seen in Fig. 7(a). The uncertainty
of the height measurement with the AFM can be estimated using the histogram of Fig. 7(c). The
uncertainties are in the “few nanometers” range for all techniques. A low-uncertainty measure-
ment with AFM (without the 3D option) and SEM (without cutting the sample) is only possible
for height and CD, respectively. The 3
uncertainty in the grating parameters reconstruction
by means of CFS is lower compared to that obtained through SEM and AFM measurements. We
notice higher relative uncertainties in SWA reconstruction as compared to height and midCD
uncertainties, which are also reported by others [13].
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0 1 2 3
x (μm)
Height (nm)
0 50 100
Height (nm)
No. of Pixels
561nm 1.3μm
Fig. 7. (a) Top view of the SEM image. (b) AFM cross section in the direction perpendicular
to the grating lines. (c) Histogram of heights for each pixel of the AFM measurement.
Table 1. Comparative measurements of the grating parameters using different techniques.
Parameters CFS SEM AFM
MidCD (nm) 563±2 562±4 –
Height (nm) 116±1 – 116±3
SWA() 89±3 – –
Bias (nm) 1190±1 – –
The results are compiled in table 1 for NA=0.4 and at
=633 nm. Interestingly, the lateral
position of the grating can be retrieved with an accuracy at 1 nm level, which is an impressive
accuracy, considered the numerical aperture of the system and the wavelength used. Actually,
an accurate retrieval of the alignment parameter bias is a fundamental pre-requisite for CFS to
work. In fact, no reconstruction of the grating parameters would be possible without first de-
termining the relative position between incident spot and grating. We have also performed sim-
ulations studies, in order investigate to which level of accuracy the alignment can be obtained
through CFS measurements. We have found that, by using an incident field with
=250 nm,
NA =0.95 and a grating with pitch 200 nm (all these values are very well representative of a
current state-of-the art industrial IOS) positioning accuracy at 10 pm level is attainable. Simula-
tion studies show that at such high NA and shorter wavelength the uncertainty in reconstruction
of the grating parameters can be further decreased as well. All these benefits can be attributed
to the phase sensitive signal, as for F<2 CFS can be seen as a common path interferometer.
4. Conclusions
Grating reconstruction with coherent Fourier scatterometry (CFS) has been demonstrated. The
tool is capable of illuminating and measuring the response of the sample simultaneously over
a broad range of incident and reflected angles. The measurement for all radial and azimuthal
angles can be performed within one second. Due to the coherent illumination, the measuring
spot can be focused to a size smaller than one micron. Compared to incoherent scatterometry
the advantage of coherent Fourier scatterometry is that the measured response in the pupil plane
includes interference patterns caused by overlapping orders. The interference changes when the
spot is scanned perpendicularly to the grooves of the grating, and consequently the phase in-
formation contained in the overlapping orders can be determined accurately. The capabilities
#215012 - $15.00 USD
Received 30 Jun 2014; revised 17 Aug 2014; accepted 19 Aug 2014; published 1 Oct 2014
(C) 2014 OSA
6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024678 | OPTICS EXPRESS 24687
of the tool were demonstrated by reconstructing the parameters of grating with 1300 nm pitch
using a wavelength of
= 633 nm and an objective lens of NA =0.4. The reconstruction was
performed by non-linear least squares gradient fit of the grating parameters to minimize the dif-
ference between the measured and rigorously computed pupil images. The sensitivity of coher-
ent Fourier scatterometry was found to be comparable with the applied reference metrologies
(SEM and AFM). In addition, simulation studies show that positioning accuracy of the order of
10 picometer with NA =0.95 and
in the UV can be achieved. Finally, it is worth mentioning
that accurate nano-positioning in combination with the reconstruction shape parameters of the
grating can be done in a single tool which is not possible in conventional optical scatterometry.
The authors acknowledge Mark Van Kraaij from ASML, Veldhoven. Peter Petrik is grateful
for the EMRP IND17 joint research project on scatterometry for financial support. The EMRP
is jointly funded by the EMRP participating countries within EURAMET and the European
#215012 - $15.00 USD
Received 30 Jun 2014; revised 17 Aug 2014; accepted 19 Aug 2014; published 1 Oct 2014
(C) 2014 OSA
6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.024678 | OPTICS EXPRESS 24688
... A numerical optimization scheme seeks to minimize the deviation between the experimental and model results by repetitively evaluating the model for different parameter. [1][2][3][4] Depending on the numerical complexity of the model, obtaining the result for any given parameter can take from a few seconds to several hours. This creates the need for efficient and robust reconstruction algorithms that use the available resources most effectively, while still achieving excellent reconstruction results. ...
Parameter reconstruction is a common problem in optical nano metrology. It generally involves a set of measurements, to which one attempts to fit a numerical model of the measurement process. The model evaluation typically involves to solve Maxwell's equations and is thus time consuming. This makes the reconstruction computationally demanding. Several methods exist for fitting the model to the measurements. On the one hand, Bayesian optimization methods for expensive black-box optimization enable an efficient reconstruction by training a machine learning model of the squared sum of deviations. On the other hand, curve fitting algorithms, such as the Levenberg-Marquardt method, take the deviations between all model outputs and corresponding measurement values into account which enables a fast local convergence. In this paper we present a Bayesian Target Vector Optimization scheme which combines these two approaches. We compare the performance of the presented method against a standard Levenberg-Marquardt-like algorithm, a conventional Bayesian optimization scheme, and the L-BFGS-B and Nelder-Mead simplex algorithms. As a stand-in for problems from nano metrology, we employ a non-linear least-square problem from the NIST Standard Reference Database. We find that the presented method generally uses fewer calls of the model function than any of the competing schemes to achieve similar reconstruction performance.
... [22] Scatterometry is applied for mapping surfaces from very small to extremely large [23] scales. Modeling of the diffraction patterns allows to reconstruct complex grating [24] and overlay [25] structures. Point-by-point mapping of an area is also utilized in microscopy by scanning the focused spot while measuring the scattered response-for example, to measure, find, and characterize individual plasmonic nanoparticles by spectroscopy. ...
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Thin films covering large surfaces are used in a very wide range of applications from displays through corrosion resistance, decoration, water proofing, smart windows, adhesion performance to solar panels and many more. Scaling up existing thin film measurement techniques requires a high speed and the redesign of the configurations. The aim of this review is to give an overview of recent and past activities in the area, as well as an outlook of future opportunities. This article is protected by copyright. All rights reserved.
... It has been misunderstood that the proposed method can be used only for measuring the periodic structure. In addition, a misconception emerged that this method has the same principle as that of scatterometry [17][18][19][20], which is used to measure periodic structures. ...
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A method for the measurement of the shape of a fine structure beyond the diffraction limit based on speckle interferometry has been reported. In this paper, the mechanism for measuring the shape of the fine structure in speckle interferometry using scattered light as the illumination light is discussed. Furthermore, by analyzing the phase distribution of the scattered light from the surface of the measured object, this method can be used to measure the shapes of periodic structures and single silica microspheres beyond the diffraction limit.
Parameter reconstructions are indispensable in metrology. Here, the objective is to explain K experimental measurements by fitting to them a parameterized model of the measurement process. The model parameters are regularly determined by least‐square methods, that is, by minimizing the sum of the squared residuals between the K model predictions and the K experimental observations, χ2. The model functions often involve computationally demanding numerical simulations. Bayesian optimization methods are specifically suited for minimizing expensive model functions. However, in contrast to least‐square methods such as the Levenberg–Marquardt algorithm, they only take the value of χ2 into account, and neglect the K individual model outputs. A Bayesian target‐vector optimization scheme with improved performance over previous developments, that considers all K contributions of the model function and that is specifically suited for parameter reconstruction problems which are often based on hundreds of observations is presented. Its performance is compared to established methods for an optical metrology reconstruction problem and two synthetic least‐squares problems. The proposed method outperforms established optimization methods. It also enables to determine accurate uncertainty estimates with very few observations of the actual model function by using Markov chain Monte Carlo sampling on a trained surrogate model. A Bayesian target‐vector optimization scheme is presented that combines the resource efficiency of Bayesian optimization methods for scalar model functions with the reconstruction performance of established least‐square methods. It is applied to parameter reconstruction problems and shown to regularly outperform established optimization schemes. The method further enables efficient Markov chain Monte Carlo sampling of expensive models for accurate uncertainty quantification of reconstructed parameters.
Accurate determination of the physical parameters of nanostructures from optical far-field scattering is an important and challenging topic in the semiconductor industry. Here, we propose a novel metrology method to determine simultaneously the height and side-wall angle of a step-shaped silicon nanostructure. By employing an optical singular beam into a typical coherent Fourier scatterometry system, both parameters can be retrieved through analyzing the intensity profile of the far-field scattering pattern. The use of singular beam is shown to be sensitive to slight changes of the parameters of the step. By changing the relative direction between the singularity and structure, the height and side-wall angle can both be retrieved with high precision. This new method is robust, simple, and can provide valuable means for micro-and-nano- metrologies.
We demonstrate that the sensitivity of nanoparticle detection on surfaces can be substantially improved by implementing synthetic optical holography (SOH) in coherent Fourier scatterometry (CFS), resulting in a phase-sensitive confocal differential detection technique that operates at very low power level (P = 0.016 mW). The improvement in sensitivity is due to two reasons: first, the boost in the signal at the detector due to the added reference beam; and second, the reduction of background noise caused by the electronics. With this new system, we are able to detect a 60-nm polystyrene latex (PSL) particle at a wavelength of 633 nm (∼λ/10) on a silicon wafer with an improvement in the signal-to-noise ratio (SNR) of approximately 4 dB.
Substrate Conformal Imprint Lithography (SCIL) technology enables the fabrication of complex nanostructures with sub-10 nm resolution over large areas for industrial-scale production. This technology utilizes novel composite silicone rubber stamps that provides versatility in addition to high precision. To inspect the quality and reproducibility of the nanostructures that are fabricated using SCIL, a novel optical characterization method using Fourier microscopy is proposed. In this method, nanostructures are illuminated under a microscope objective using a collimated light beam at different incident angles and the properties of the reflected and/or diffracted beams are analysed to extract critical dimensions of the nanostructures. This fast and non-destructive method has the potentials of being used as an in-line inspection technology to extract the critical dimensions of the nanostructures over large areas and improve the overall properties of nanostructured surfaces.
In this paper we describe how the digital transformation (i. e., the adoption of digital technology) of society affects National Metrology Institutes like VSL. The presented ideas represent the personal viewpoint of the author, who works at the Dutch national metrology institute VSL. They don’t necessarily correspond to the vision of VSL as institute. This digital transformation has many different aspects of social, economic and technical nature. In this paper we will mainly focus on some mathematical and statistical aspects which are important for modelling measurement instruments and analyzing measurement data. We will discuss how modern techniques like artificial intelligence, digital twins, digital calibration certificates and the introduction of the new definition of the SI system of units affect national metrology institutes. Important changes are the usage of complex algorithms and models in measurement instruments, as well as the introduction of novel calibration approaches and the digitalization of the services provided by NMIs.
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Nanophotonic structures are powerful tools for sensing, with the goal of retrieving parameters accurately at maximum speed and minimum photon budget. As information on those parameters can be distributed over multiple output scattering channels that propagate to the far field, considering well-chosen combinations of far-field optical degrees of freedom could benefit measurement precision. We explore how multiplexing readout across different polarization channels enhances parameter retrieval in nanophotonic overlay sensors. We measure the relative position between layers with nanoscale scattering structures, known as overlay in semiconductor metrology, and show that multiplexing either incident or analyzed polarization leads to improved parameter retrieval in the systems studied. At fixed photon budget, we extract additional information equivalent to more than 7 dB in signal level. These results demonstrate that significant advantages in measurement performance of nano-optical sensors can be gained by exploiting the vectorial nature of optical fields.
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We present epi-diffraction phase microscopy (epi-DPM) as a non-destructive optical method for monitoring semiconductor fabrication processes in real time and with nanometer level sensitivity. The method uses a compact Mach–Zehnder interferometer to recover quantitative amplitude and phase maps of the field reflected by the sample. The low temporal noise of 0.6 nm per pixel at 8.93 frames per second enabled us to collect a three-dimensional movie showing the dynamics of wet etching and thereby accurately quantify non-uniformities in the etch rate both across the sample and over time. By displaying a gray-scale digital image on the sample with a computer projector, we performed photochemical etching to define arrays of microlenses while simultaneously monitoring their etch profiles with epi-DPM.
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There has been a substantial increase in the research and development of optical metrology techniques as applied to linewidth and overlay metrology for semiconductor manufacturing. Much of this activity has been in advancing scatterometry applications for metrology. In recent years we have been developing a related technique known as scatterfield optical microscopy, which combines elements of scatterometry and bright field imaging. In this paper we present the application of this technique to optical system alignment, calibration, and characterization for the purpose of accurate normalization of optical data, which can be compared with optical simulations involving only absolute measurement parameters. We show a series of experimental data from lines prepared using a focus exposure matrix on silicon and make comparisons between the experimental and theoretical results. The data show agreement on the nanometer scale using parametric simulation libraries and no "tunable" parameters.
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Non-interferometric phase retrieval from the intensity measurements in Coherent Fourier Scatterometry (CFS) is presented using a scanningfocused spot. Formulae to determine the state of polarization of the scattered light and to retrieve the phase difference between overlappingscattered orders are given. The scattered far field is rigorously computed and the functionality of the method is proved with experimentalresults.
A microscopic method to inspect isolated sub 100 nm scale structures made of silicon is presented. This method is based upon an analysis of light intensity distributions at defocused images obtained along the optical axis normal to the sample plane. Experimental measurements of calibrated lines (height 50 nm, length 100 μm, and widths of 40-150 nm in 10 nm steps) on top of a monocrystalline silicon substrate are presented. Library of defocused images of calibrated lines is obtained experimentally and numerically with accordance to experimental setup parameters and measurements conditions. Processing of the measured defocused images and comparison with simulated ones from library allow one to distinguish between objects with a 10 nm change in width. It is shown that influence of optical system aberrations must be taken into account in order to achieve coincidence between simulation and measured results and increase accuracy of line width inspection accuracy. The limits of accuracy for object width measurements using this optical method are discussed.
Optical imaging beyond the diffraction limit, i.e., optical superresolution, has been studied extensively in various contexts. This paper presents an overview of some mathematical concepts relevant to superresolution in linear optical systems. Properties of bandlimited functions are surveyed and are related to both instrumental and computational aspects of superresolution. The phenomenon of superoscillation and its relation to superresolution are discussed.
Spectroscopic, specular reflected light measurements (both ellipsometry-SE, and reflectometry-SR) of grating structures have relatively and recently been shown to yield very accurate information on the critical dimensions, wall-angles and detailed wall shape of deep submicron features. The technique is often called ‘scatterometry’ or optical critical dimension (OCD) measurement. This technique has been moved rapidly from initial demonstrations to significant industrial application. In this paper, we will review the development of this technique and ex situ applications. When applied in situ, this technique opens up exciting new opportunities for studying the evolution of topography in semiconductor fabrication processes and for applying real-time control methods for nanometer level feature size accuracy. We will briefly comment on limitations and challenges for this measurement technique.
A review of scatterometry technology and its relevance to high volume silicon manufacturing is presented. First, an introduction and history of the technology are provided, with the technology being broadly described in two parts known as the “forward” and “inverse” problems. In the forward problem, the scatterometry signature is acquired by some optical measurement. Both single wavelength, angle scanning and fixed angle, wavelength scanning approaches to the forward problem are described. In the inverse problem, the scatterometry signature is analyzed to determine dimensional parameters that are of interest for the application. Two common approaches to the inverse problem are summarized. The most common approach involves a direct comparison to a pre‐computed database of theoretically generated signatures. Another approach, often called a “real time” regression, uses optimization methods to compare the measured signature against modeled signatures dynamically. Strengths and weaknesses of each method are discussed. Finally, a review of scatterometry applications with high potential in mainstream volume manufacturing will be presented. In particular, the emphasis will be on lithography applications that are readily addressable from a technology perspective and compelling in terms of their value to reducing costs and increasing yields in the manufacturing process. The increasing role of scatterometry for integrated metrology applications, and limits of the technology as the silicon industry moves well into the sub‐100 nm regime, will also be discussed.
The recent reformulation of the coupled-wave method by Lalanne and Morris [J. Opt. Soc. Am. A 13, 779 (1996)] and by Granet and Guizal [J. Opt. Soc. Am. A 13, 1019 (1996)], which dramatically improves the convergence of the method for metallic gratings in TM polarization, is given a firm mathematical foundation in this paper. The new formulation converges faster because it uniformly satisfies the boundary conditions in the grating region, whereas the old formulations do so only nonuniformly. Mathematical theorems that govern the factorization of the Fourier coefficients of products of functions having jump discontinuities are given. The results of this paper are applicable to any numerical work that requires the Fourier analysis of products of discontinuous periodic functions.
A rigorous three-dimensional vector coupled-wave method is developed to study the diffraction by an arbitrarily oriented planar grating with slanted fringes. It is demonstrated that, in the resulting diffraction, coupling exists between all space-harmonic vector fields inside the grating which correspond to the diffracted orders outside the grating. It is shown that the TE and TM components of an incident plane wave are each coupled to all the TE and TM components of all the forward-diffracted and backward-diffracted waves. The diffraction efficiency for a lossless grating is found to approach 100 percent for a general Bragg angle of incidence if either the incident electric field or the magnetic field is perpendicular to the grating vector. In addition, maximum coupling occurs between the incident and diffracted waves when the incident electric field is perpendicular to the grating vector. This method can be applied to any sinusoidal or nonsinusoidal amplitude and/or phase grating, any plane-wave angle of incidence, and any linear polarization.