U.S. patent application number 09/878981 was filed with the patent office on 2001-12-20 for method of detecting aberrations of an optical imaging system.
This patent application is currently assigned to PHILIPS CORPORATION and ASM LITHOGRAPHY B.V.. Invention is credited to Dirksen, Peter, Juffermans, Casparus A.H..
Application Number | 20010053489 09/878981 |
Document ID | / |
Family ID | 27239429 |
Filed Date | 2001-12-20 |
United States Patent
Application |
20010053489 |
Kind Code |
A1 |
Dirksen, Peter ; et
al. |
December 20, 2001 |
Method of detecting aberrations of an optical imaging system
Abstract
Aberrations of an imaging system (PL) can be detected in an
accurate and reliable way by imaging, by means of the imaging
system, a circular phase structure (22) on a photoresist (PR),
developing the resist and scanning it with a scanning detection
device (SEM) which is coupled to an image processor (IP). The
circular phase structure is imaged in a ring structure (25) and
each type of aberration, like coma, astigmatism, three-point
aberration, etc. causes a specific change in the shape of the inner
contour (CI) and the outer contour (CE) of the ring and/or a change
in the distance between these contours, so that the aberrations can
be detected independently of each other. Each type of aberration is
represented by a specific Fourier harmonic (Z-), which is composed
of Zernike coefficients (Z-), each representing a specific lower or
higher order sub-aberration. The new method enables to determine
these sub-aberrations The new method may be used for measuring a
projection system for a lithographic projection apparatus.
Inventors: |
Dirksen, Peter; (Eindhoven,
NL) ; Juffermans, Casparus A.H.; (Eindhoven,
NL) |
Correspondence
Address: |
Corporate Patent Counsel
Philips Electronics North America Corporation
580 White Plains Road
Tarrytown
NY
10591
US
|
Assignee: |
PHILIPS CORPORATION and ASM
LITHOGRAPHY B.V.
|
Family ID: |
27239429 |
Appl. No.: |
09/878981 |
Filed: |
June 12, 2001 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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09878981 |
Jun 12, 2001 |
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09447542 |
Nov 23, 1999 |
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09447542 |
Nov 23, 1999 |
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09407532 |
Sep 29, 1999 |
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6248486 |
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Current U.S.
Class: |
430/30 ; 355/18;
382/141 |
Current CPC
Class: |
G03F 7/70358 20130101;
Y10S 430/143 20130101; G03F 7/70241 20130101; G03F 7/706
20130101 |
Class at
Publication: |
430/30 ; 355/18;
382/141 |
International
Class: |
G03C 005/00; G03B
027/00; G06K 009/00 |
Foreign Application Data
Date |
Code |
Application Number |
Nov 23, 1998 |
EP |
98203945.5 |
Claims
1. A method of detecting aberrations of an optical imaging system,
comprising the steps of arranging a test object, which comprises at
least one closed single figure having a phase structure, in the
object plane of the system; providing a photoresist layer in the
image plane of the system; imaging the test object by means of the
system and an imaging beam; developing the photoresist layer;
observing the developed image by means of a scanning detection
device having a resolution which is considerably larger than that
of the imaging system; subjecting the observed image to a Fourier
analysis in order to ascertain at least one of different types of
changes of shape in the image of the single figure, each type of
shape change being indicative of a given kind of aberration, which
is represented by a specific Fourier harmonic being a combination
of a number of Zernike polynominals each preceded by a weighting
factor, the measurement of the Zernike coefficients being carried
out by the steps: setting at least one of the illumination
parameters successively at a number of different values, the number
being at least equal to the number of Zernike polynomials to be
determined; measuring a same Fourier harmonic for each of said
different values, and calculating the Zernike coefficients from of
the measured values for the said Fourier harmonic and by means of
stored weighting factors which have been obtained by a previously
carried out simulation program.
2. A method of detecting aberrations of an optical imaging system,
comprising the steps of: arranging a test object, which comprises
at least one closed single figure having a phase structure, in the
object plane of the system; providing a photoresist layer in the
image plane of the system; imaging the test object by means of the
system and an imaging beam; developing the photoresist layer;
observing the developed image by means of a scanning detection
device having a resolution which is considerably larger than that
of the imaging system; subjecting the observed image to a Fourier
analysis in order to ascertain at least one of different types of
changes of shape in the image of the single figure, each type of
shape change being indicative of a given kind of aberration, which
is represented by Fourier harmonics each composed of a combination
of Zernike coefficients, and determining the Zernike coefficients
of an observed image by comparing the observed image with an number
of reference images, which are stored together with data about
their Zernike coefficients in a look-up table, to determine which
of the reference images fits best to the observed image, the
look-up table having been obtained by a previously carried out
simulation program.
3. A method as claimed in claim 1 or 2, characterized in that a
scanning electron microscope is used as a scanning detection
device.
4. A method as claimed in claim 1, 2 or 3, characterized in that
every single figure is constituted by an area in a plate located at
a different height than the rest of said plate.
5. A method as claimed in claim 4, characterized in that the height
difference between the area of the single figure and the rest of
the plate is such that a phase difference of 180.degree. is
introduced in the imaging beam.
6. A method as claimed in claim 4 or 5, characterized in that the
diameter of the area is proportional to .lambda./(NA.M), in which
.lambda. is the wavelength of the imaging beam, NA is the numerical
aperture of the imaging system at the image side and M is the
magnification of the imaging system.
7. A method as claimed in claim 1, 2, 3, 4, 5 or 6 of detecting
aberrations of a projection system in a lithographic projection
apparatus intended to project a mask pattern, present in a
production mask, on a production substrate provided with a
photoresist layer, characterized in that a mask having at least a
single figure with a phase structure is arranged at the position of
the production mask in the projection apparatus, and in that a
photoresist layer with a support is provided at the position of a
production substrate.
8. A method as claimed in claim 7, characterized in that use is
made of an empty test mask having at least a single area with a
phase structure.
9. A system for performing the method as claimed in any one of the
preceding claims, which system is constituted by the combination
of: an apparatus of which the imaging system forms part; a test
object having at least a single figure with a phase structure; a
photo resist layer in which the test object is imaged; a scanning
detection device for scanning at least a test object image formed
and developed in the photo resist layer, and an image processor,
coupled to the scanning detection device, for storing and analyzing
the observed images, characterized in that the image processor
comprises analysis means for detecting at least one of different
types of shape changes in the formed image of the single
figure.
10. A lithographic projection apparatus for imaging a mask pattern,
present in a mask, on a substrate, which apparatus comprises an
illumination unit for supplying a projection beam, a mask holder
for accommodating the mask, a substrate holder for accommodating
the substrate, and a projection system arranged between the mask
holder and the substrate holder, said apparatus being suitable for
performing the method as claimed in any of claims 1-8,
characterized in that, in the implementation of the method, the
projection beam is used as an imaging beam and in that the
illumination unit comprises means for reducing the diameter of the
projection beam cross-section for the method to a value which is
smaller than the diameter of the projection beam cross-section
during projection of the mask pattern on the substrate.
Description
[0001] The invention relates to a method of detecting aberrations
of an optical imaging system, comprising the steps of:
[0002] arranging a test object in the object plane of the
system;
[0003] providing a photoresist layer in the image plane of the
system;
[0004] imaging the test object by means of the system and an
imaging beam;
[0005] developing the photoresist layer, and
[0006] detecting the developed image by means of a scanning
detection device having a resolution which is considerably larger
than that of the imaging system.
[0007] The fact that the resolution of the scanning detection
device is considerably larger than that of the imaging system means
that the detection device allows observation of details which are
considerably smaller than the details that can still be separately
imaged by the imaging system.
[0008] An optical imaging system in the form of a projection lens
system having a large number of lens elements is used in
photolithographic projection apparatuses which are known as wafer
steppers or as wafer step-and-scanners. Such apparatuses are used,
inter alia, for manufacturing integrated circuits, or ICs. In a
photolithographic projection apparatus, a mask pattern present in
the mask is imaged a large number of times, each time on a
different area (IC area) of the substrate by means of a projection
beam having a wavelength of, for example, 365 nm in the UV range,
or a wavelength of, for example, 248 mn in the deep UV range, and
by means of the projection lens system.
[0009] The method mentioned above is known from the opening
paragraph of EP-A 0 849 638, relating to a method of measuring the
comatic aberration of projection lens systems in lithographic
projection apparatuses.
[0010] The aim is to integrate an ever-increasing number of
electronic components in an IC. To realize this, it is desirable to
increase the surface area of an IC and to decrease the size of the
components. For the projection lens system, this means that both
the image field and the resolution must be increased, so that
increasingly smaller details, or line widths, can be imaged in a
well-defined way in an increasingly larger image field. This
requires a projection lens system, which must comply with very
stringent quality requirements. Despite the great care with which
such a projection lens system has been designed and the great
extent of accuracy with which the system is manufactured, such a
system may still exhibit aberrations such as spherical aberration,
coma and astigmatism which are not admissible for the envisaged
application. In practice, a lithographic projection lens system is
thus not an ideal, diffraction-limited system but an
aberration-limited system. Said aberrations are dependent on the
positions in the image field and are an important source of
variations of the imaged line widths occurring across the image
field. When novel techniques are used to enhance the resolving
power, or the resolution, of a lithographic projection apparatus,
such as the use of phase-shifting masks, as described in, for
example, U.S. Pat. No. 5,217,831, or when applying an off-axis
illumination as described in, for example, U.S. Pat. No. 5,367,404,
the influence of the aberrations on the imaged line widths still
increases.
[0011] Moreover, the aberrations are not constant in modern
lithographic projection lens systems. To minimize low-order
aberrations, such as distortion, curvature of the field,
astigmatism, coma and spherical aberration, these systems comprise
one or more movable lens elements. The wavelength of the projection
beam or the height of the mask table may be adjustable for the same
purpose. When these adjusting facilities are used, other and
smaller aberrations are introduced. Moreover, since the intensity
of the projection beam must be as large as possible, lithographic
projection lens systems are subject to aging so that the extent of
the aberrations may change with respect to time.
[0012] Based on the considerations described above, there is an
increasing need for a reliable and accurate method of measuring
aberrations.
[0013] It has also been proposed to use for the projection beam a
beam of extreme UV (EUV) radiation, i.e. radiation at a wavelength
in the range of several nm to several tens of nm. The resolution of
the projection lens system can thereby be enhanced considerably
without increasing the numerical aperture (NA) of the system. Since
no suitable lens material is available for EUV radiation, a mirror
projection system instead of a lens projection system must then be
used. A lithographic mirror projection system is described in,
inter alia, EP-A 0 779 258. For reasons analogous to those for the
lens projection system, there is a need for an accurate and
reliable method of measuring aberrations for this EUV mirror
projection system as well.
[0014] The opening paragraph of said EP-A 0 849 638 rejects the
method in which the image of a test mask formed in the photoresist
layer is scanned with a scanning detection device in the form of a
scanning electron microscope. Instead, it is proposed to detect
said image with optical means. To this end, a test mask having one
or more patterns of strips which are alternately
radiation-transmissive and radiation-obstructive, i.e. an amplitude
structure, is used. The comatic aberration of a projection system
can be detected with such a pattern. The detection is based on
measuring the widths of the light or dark strips in the image
formed and/or measuring the asymmetry between the strips at the
ends of the image of the patterns.
[0015] It is an object of the present invention to provide a method
of the type described in the opening paragraph, which is based on a
different principle and with which different aberrations can be
measured independently of each other. This object is met by a
method which comprises the steps of:
[0016] arranging a test object, which comprises at least one closed
single figure having a phase structure, in the object plane of the
system;
[0017] providing a photoresist layer in the image plane of the
system;
[0018] imaging the test object by means of the system and an
imaging beam;
[0019] developing the photoresist layer;
[0020] observing the developed image by means of a scanning
detection device having a resolution which is considerably larger
than that of the imaging system;
[0021] subjecting the observed image to a Fourier analysis in order
to ascertain at least one of different types of changes of shape in
the image of the single figure, each type of shape change being
indicative of a given kind of aberration, which is represented by a
specific Fourier harmonic being a combination of a number of
Zernike polynominals each preceded by a weighting factor, the
measurement of the Zernike coefficients being carried out by the
steps:
[0022] setting at least one of the illumination parameters
successively at a number of different values, the number being at
least equal to the number of Zernike polynomials to be
determined;
[0023] measuring a same Fourier harmonic for each of said different
values, and
[0024] calculating the Zernike coefficients from of the measured
values for the said Fourier harmonic and by means of stored
weighting factors which have been obtained by a previously carried
out simulation program.
[0025] A single figure is understood to mean a figure having a
single contour line which is closed in itself. The contour line is
the boundary line between the figure and its ambience.
[0026] The method uses the fact that the contour line of a figure
having a phase structure is not imaged in a single line but in a
first and a second image line, the second image line being located
within the first image line, and the distance between the first and
the second image line is determined by the point spread function,
or Airy distribution, of the imaging system. In the method useful
use is thus made of the point spread function, or Airy
distribution, of the imaging system. If this system has given
aberrations, given deviations of the ideal image occur, such as
deviations of the shape of the image lines themselves and/or
changes of the mutual position of the two image lines. The method
thus allows detection of aberrations which cannot be detected when
using a test object in the form of an amplitude, or black-white,
structure. When using a test object with an amplitude structure,
its contour line is imaged in a single line. Consequently, only the
aberrations of the imaging system which cause deviations of the
imaged single contour line can be detected when using such a test
object, and this even less accurately. When using a test object
having a phase structure, different aberrations occurring
simultaneously can be detected separately because the effects of
the different aberrations remain well distinguishable in the image
formed, in other words, the different aberrations do not exhibit
any mutual crosstalk. The method uses a Fourier analysis, which
operates with sine and cosine functions and is eminently suitable
to directly analyze the contour lines of the image. Each
aberration, for example, astigmatism is composed of a number of
sub-aberrations of lower and higher order. Each of these
sub-aberrations are usually represented by a Zernike coefficient,
i.e. an amount of a specific Zernike polynomial from the "fringe
Zernike code" which has a maximum of 37 polynomials. The novel
method is based on the insight that the Zemike coefficients of a
given aberration can be determined by determining the Fourier
harmonic related to this aberration for different illumination
conditions. It thus becomes possible to measure the
sub-aberrations. Thereby use is made of the linearity of the
measuring method.
[0027] It is to be noted that, in one embodiment described in U.S.
Pat. No. 5,754,299, relating to a method and a device for measuring
an asymmetrical aberration of a lithographic projection system, the
test object is denoted as phase pattern. However, this pattern is
not a closed single figure, but a phase grating, for example, an
alignment mark. The image formed of this grating has the same
appearance as the grating itself, i.e. each grating line is imaged
in a single line. Moreover, for measuring the aberration, an image
of the grating is formed every time at different focus settings,
and the detection is based on measuring the asymmetries between
these images, rather than on detecting changes of shape and/or
positions in an image itself.
[0028] According to the invention the above-mentioned object can
also be met with an alternative method, which comprises the steps
of:
[0029] arranging a test object, which comprises at least one closed
single figure having a phase structure, in the object plane of the
system;
[0030] providing a photoresist layer in the image plane of the
system;
[0031] imaging the test object by means of the system and an
imaging beam;
[0032] developing the photoresist layer;
[0033] observing the developed image by means of a scanning
detection device having a resolution which is considerably larger
than that of the imaging system;
[0034] subjecting the observed image to a Fourier analysis in order
to ascertain at least one of different types of changes of shape in
the image of the single figure, each type of shape change being
indicative of a given kind of aberration, which is represented by
Fourier harmonics each composed of a combination of Zemike
coefficients, and
[0035] determining the Zernike coefficients of an observed image by
comparing the observed image with an number of reference images,
which are stored together with data about their Zernike
coefficients in a look-up table, to determine which of the
reference images fits best to the observed image, the look-up table
having been obtained by a previously carried out simulation
program.
[0036] The methods are further preferably characterized in that a
scanning electron microscope is used as a scanning detection
device.
[0037] Such a microscope, which is already frequently used in
lithographic processes, has a sufficient resolution for this
application. Another and newer type of scanning detection device is
the scanning probe microscope which is available in several
implementations such as the atomic force microscope (AFM) and the
scanning optical probe microscope.
[0038] The phase structure of the test object may be realized in
various ways. For example, the single figure may be constituted by
an area in a transparent plate having a refractive index, which is
different from that of the rest of the plate.
[0039] A preferred embodiment of the novel method is characterized
in that every single figure is constituted by an area in a plate
located at a different height than the rest of said plate.
[0040] Said area may be countersunk in the plate or project from
the plate. This plate may be transparent to the radiation of the
imaging beam, or reflective.
[0041] The single figure may have various shapes, such as the shape
of square or of a triangle. A preferred embodiment of the novel
method is characterized in that said area is circularly shaped.
[0042] The shape of the single figure is then optimally adapted to
the circular symmetry of the imaging system, and the image of this
figure consists of two circular image lines. A change of the shape
and a mutual offset of these image lines can be observed easily.
Even if a square single figure is used, the novel method yields
good results because the image lines of this figure formed by the
projection system are a sufficient approximation of the circular
shape.
[0043] Each single figure is preferably further characterized in
that the height difference between the area of this figure and the
rest of the plate is such that a phase difference of 180.degree. is
introduced in the imaging beam.
[0044] For a transmissive, or reflective, test object, this means
that the height difference must be of the order of
.lambda./(2(n.sub.2-n.sub.1)), or of .lambda./4n, in which .lambda.
is the wavelength of the imaging beam, n.sub.2 is the refractive
index of the material of the test object and n.sub.1 is the
refractive index of the surrounding medium. At this height
difference, the phase difference between the part of the imaging
beam originating from the area of the single figure and the part of
the imaging beam originating from the surroundings of this area is
maximal, and the contrast in the image formed is maximal. If the
diameter of the area is of the order of the wavelength of the
imaging beam, or of a larger order, the optimal height difference
is equal to .lambda./(2(n.sub.2-n.sub.1)) or .lambda./4n. At a
smaller diameter, polarization effects must be taken into account,
and the optimal height difference deviates by several percent from
the last-mentioned values.
[0045] In accordance with a further preferred embodiment, the
diameter of the area is proportional to .lambda./(NA.M), in which
.lambda. is the wavelength of the imaging beam, NA is the numerical
aperture of the projection system at the image side and M is the
magnification of this system.
[0046] The size of the test object is then adapted to the
resolution of the projection system, allowing measurements of
aberrations of the smallest images that can be made with the
projection system.
[0047] The method may be used, inter alia, for detecting
aberrations of a projection system in a lithographic apparatus
intended to image a mask pattern, present in a production mask, on
a production substrate which is provided with a photoresist layer.
This method is further characterized in that a mask having at least
a single figure with a phase structure is used as a test object,
which mask is arranged at the position of a production mask in the
projection apparatus, and in that a photoresist layer with a
support is provided at the position of a production substrate.
[0048] This method provides the advantage that aberrations of the
projection system can be detected under circumstances, which
correspond to those for which this projection system is intended.
The number of single figures may vary from one to several tens.
Since these figures are imaged at different positions within the
image field of the projection system, insight is obtained into the
variations of the aberrations across the image field. Since the
single figures are small, they may be provided in the production
mask at positions outside the details of the mask pattern.
[0049] However, the method is preferably further characterized in
that use is made of an empty test mask having at least a single
figure.
[0050] The test object is now constituted by a recessed or a raised
part of a transparent plate of the same material and having the
same thickness as a production mask, but without a mask pattern or
parts thereof, which plate may be denoted as empty test mask.
[0051] The invention further relates to a system for performing the
method described above. The system comprises an optical apparatus
of which the imaging system forms part, a test object having at
least a single figure with a phase structure, a scanning detection
device for scanning at least a test object image formed by the
imaging system, and an image processor coupled to the scanning
projection device, for storing and analyzing the observed images,
and is characterized in that the image processor comprises analysis
means for detecting at least one of different types of changes of
the shape of said image.
[0052] The invention also relates to a lithographic projection
apparatus for imaging a mask pattern, present in a mask, on a
substrate, which apparatus comprises an illumination unit for
supplying an projection beam, a mask holder for accommodating the
mask, a substrate holder for accommodating the substrate and a
projection system arranged between the mask holder and the
substrate holder, which apparatus is suitable for performing the
method described above. This apparatus is characterized in that, in
the implementation of the method, the projection beam is used as an
imaging beam, and in that the illumination unit comprises means for
reducing the diameter of the projection beam cross-section for the
method to a value which is smaller than the diameter of the
projection beam cross-section during projection of the mask pattern
on the substrate.
[0053] These and other aspects of the invention are apparent from
and will be elucidated, by way of non-limitative example, with
reference to the embodiments described hereinafter.
[0054] In the drawings:
[0055] FIG. 1 shows diagrammatically an embodiment of a
photolithographic projection apparatus with which the method can be
performed;
[0056] FIG. 2 is a block diagram of the system for performing the
method;
[0057] FIG. 3a is a bottom view of a test object with a single
figure in the form of a recess;
[0058] FIG. 3b is a cross-section of this test object;
[0059] FIG. 4 shows the annular image formed of said recess;
[0060] FIG. 5 shows the theory of the image formation;
[0061] FIG. 6 shows an annular image without aberrations;
[0062] FIG. 7 shows an annular image with coma;
[0063] FIG. 8 shows an annular image with astigmatism;
[0064] FIG. 9 shows an annular image with three-point
aberration;
[0065] FIG. 10 shows the variation of the ring width of an annular
image with spherical aberrations for different focus settings;
[0066] FIG. 11 shows this variation in a graphic form;
[0067] FIG. 12 shows an annular image picked up under the best
focus condition;
[0068] FIG. 13 shows the different Fourier terms associated with
this image;
[0069] FIG. 14 shows the variation of a spherical aberration across
the image field of the projection system;
[0070] FIG. 15 shows annular images with coma formed at different
positions in the image field;
[0071] FIG. 16 shows such an image on a larger scale, formed at an
angle of the image field;
[0072] FIG. 17 shows the different Fourier terms associated with
this image;
[0073] FIG. 18 shows a chart of the coma measured at 21 positions
in the image field;
[0074] FIG. 19 shows annular images with astigmatism formed at
different positions in the image field;
[0075] FIG. 20 shows such an image on a larger scale, formed at an
angle of the image field;
[0076] FIG. 21 shows the different Fourier terms associated with
this image;
[0077] FIG. 22 shows a chart of the astigmatism measured at 21
positions in the image field;
[0078] FIG. 23 shows the variation of a three-point aberration
across the image field of the projection system;
[0079] FIG. 24 shows the influence of spherical aberration and
astigmatism on the measured coma across the image field;
[0080] FIG. 25 shows the influence of spherical aberration and coma
on the measured astigmatism across the image field;
[0081] FIG. 26 shows, for different values of NA of the imaging
system, the contribution of different Zernike coefficients to the
x-coma aberration;
[0082] FIG. 27 shows an example of the variation of the x- and
y-coma aberration by varying NA of the imaging system;
[0083] FIG. 28 shows a small part of an embodiment of a test mask
with a detection mark and a further mark, and
[0084] FIG. 29 shows an embodiment of a lithographic projection
apparatus with a mirror projection system.
[0085] FIG. 1 only shows diagrammatically the most important
optical elements of an embodiment of a lithographic apparatus for
repetitively imaging a mask pattern on a substrate. This apparatus
comprises a projection column accommodating a projection lens
system PL. Arranged above this system is a mask holder MH for
accommodating a mask MA in which the mask pattern C, for example,
an IC pattern to be imaged is provided. The mask holder is present
in a mask table MT. A substrate table WT is arranged under the
projection lens system PL in the projection column. This substrate
table supports the substrate holder WH for accommodating a
substrate W, for example, a semiconductor substrate, also referred
to as wafer. This substrate is provided with a radiation-sensitive
layer PR, for example a photoresist layer, on which the mask
pattern must be imaged a number of times, each time in a different
IC area Wd. The substrate table is movable in the X and Y
directions so that, after imaging the mask pattern on an IC area, a
subsequent IC area can be positioned under the mask pattern.
[0086] The apparatus further comprises an illumination system,
which is provided with a radiation source LA, for example, a
krypton-fluoride excimer laser or a mercury lamp, a lens system LS,
a reflector RE and a condenser lens CO. The projection beam PB
supplied by the illumination system illuminates the mask pattern C.
This pattern is imaged by the projection lens system PL on an IC
area of the substrate W. The illumination system may be implemented
as described in EP-A 0 658 810. The projection system has, for
example, a magnification M=1/4, a numerical aperture NA=0.6 and a
diffraction-limited image field with a diameter of 22 mm.
[0087] The apparatus is further provided with a plurality of
measuring systems, namely an alignment system for aligning the mask
MA and the substrate W with respect to each other in the XY plane,
an interferometer system for determining the X and Y positions and
the orientation of the substrate holder and hence of the substrate,
and a focus error detection system for determining a deviation
between the focal or image plane of the projection lens system PL
and the surface of the photoresist layer PR on the substrate W.
These measuring systems are parts of servosystems which comprise
electronic signal-processing and control circuits and drivers, or
actuators, with which the position and orientation of the substrate
and the focusing can be corrected with reference to the signals
supplied by the measuring systems.
[0088] The alignment system uses two alignment marks M.sub.1 and
M.sub.2 in the mask MA, denoted in the top right part of FIG. 1.
These marks preferably consist of diffraction gratings but may be
alternatively constituted by other marks such as squares or strips
which are optically different from their surroundings. The
alignment marks are preferably two-dimensional, i.e. they extend in
two mutually perpendicular directions, the X and Y directions in
FIG. 1. The substrate W has at least two alignment marks,
preferably also two-dimensional diffraction gratings, two of which,
P.sub.1 and P.sub.2, are shown in FIG. 1. The marks P.sub.1 and
P.sub.2 are located outside the area of the substrate W where the
images of the pattern C must be formed. The grating marks P.sub.1
and P.sub.2 are preferably implemented as phase gratings, and the
grating marks M.sub.1 and M.sub.2 are preferably implemented as
amplitude gratings. The alignment system may be a double alignment
system in which two alignment beams b and b' are used for imaging
the substrate alignment mark P.sub.2 and the mask alignment mark
M.sub.2, or the substrate alignment mark P.sub.1 and the mask
alignment mark M.sub.1 on each other. After they have passed the
alignment system, the alignment beams are received by a
radiation-sensitive detector 13, or 13', which converts the
relevant beam into an electric signal which is indicative of the
extent to which the substrate marks are aligned with respect to the
mask marks, and thus the substrate is aligned with respect to the
mask. A double alignment system is described in U.S. Pat. No.
4,778,275 which is referred to for further details of this
system.
[0089] For an accurate determination of the X and Y positions of
the substrate, a lithographic apparatus is provided with a
multi-axis interferometer system which is diagrammatically shown by
way of the block IF in FIG. 1. A two-axis interferometer system is
described in U.S. Pat. No. 4,251,160, and a three-axis system is
described in U.S. Pat. No. 4,737,823. A five-axis interferometer
system is described in EP-A 0 498 499, with which both the
displacements of the substrate along the X and Y axes and the
rotation about the Z axis and the tilts about the X and Y axes can
be measured very accurately.
[0090] A step-and-scan lithographic apparatus does not only
comprise a substrate interferometer system but also a mask
interferometer system.
[0091] As is diagrammatically shown in FIG. 1, the output signal Si
of the interferometer system and the signals S.sub.13 and S'.sub.13
of the alignment system are applied to a signal-processing unit
SPU, for example, a microcomputer which processes said signals to
control signals S.sub.AC for an actuator AC with which the
substrate holder is moved, via the substrate table WT, in the XY
plane.
[0092] The projection apparatus further comprises a focus error
detection device, not shown in FIG. 1, for detecting a deviation
between the focal plane and the projection lens system PL and the
plane of the photoresist layer PR. Such a deviation may be
corrected by moving, for example, the lens system and the substrate
with respect to each other in the Z direction or by moving one or
more lens elements of the projection lens system in the Z
direction. Such a detection device which may be secured, for
example, to the projection lens system, is described in U.S. Pat.
No. 4,356,392. A detection device with which both a focus error and
a local tilt of the substrate can be detected is described in U.S.
Pat. No. 5,191,200.
[0093] Very stringent requirements are imposed on the projection
lens system. Details having a line width of, for example 0.35 .mu.m
or smaller should still be sharply imaged with this system, so that
the system must have a relatively large NA, for example, 0.6.
Moreover, this system must have a relatively large, well-corrected
image field, for example, with a diameter of 23 mm. To be able to
comply with these stringent requirements, the projection lens
system comprises a large number, for example, tens of lens
elements, and the lens elements must be made very accurately and
the system must be assembled very accurately. A good control of the
projection system is then indispensable, both for determining
whether the system is sufficiently free from aberrations and is
suitable to be built into the projection apparatus, and to be able
to ascertain whether aberrations may as yet occur due to all kinds
of causes so that measures can be taken to compensate for these
aberrations.
[0094] For detecting the aberrations, the projection apparatus
itself may be used as a part of a measuring system for performing a
detection method. In accordance with this method, a test mask
having a given test pattern is arranged in the mask holder, and
this test pattern is imaged in the radiation-sensitive, or
photoresist, layer in the same way as a production mask pattern is
imaged in the radiation-sensitive layer during the production
process. Subsequently, the substrate is removed from the apparatus
and is developed and etched so that an image of the test pattern in
the form of a relief pattern in the substrate is obtained. This
relief image is subsequently scanned by a scanning detection
device, for example, a scanning electron microscope. The electron
microscope converts the observed image into image data which are
processed in an image processing device, using a special image
processing program. Its results may be visualized in diagrams or
graphs. It is alternatively possible to show visual images of the
structures observed by the electron microscope on, for example, a
monitor.
[0095] This method is shown in a block diagram in FIG. 2. In this
Figure, the projection apparatus is denoted by PA, the developing
and etching apparatus is denoted by ED, the electron microscope is
denoted by SEM, the image processing device is denoted by IP and
the monitor is denoted by MO.
[0096] The test object has a phase structure and a small part of
this test object is shown in a bottom view in FIG. 3a and in a
cross-section in FIG. 3b . This test object comprises at least one
closed figure with a phase structure in the form of a circular
recess 22 in a transparent test mask of, for example, quartz. This
recess has a diameter D and a depth d. Instead of a recess, a
figure of the test object may be alternatively constituted by a
raised part having the same diameter and the same height difference
with respect to the rest of the mask as said recess. Since the test
mask is satisfactorily transparent to the projection beam with
which the test figure is imaged on the photoresist layer, this
figure forms a phase structure for this beam. This means that,
after passage through the test mask, the part of the projection
beam PB incident on the circular area 22 has obtained a different
phase than the rest of the beam. The phase difference .phi. (in
rad.) between the beam portions is defined by 1 = ( n 2 - n 1 ) d
2
[0097] in which n.sub.2 is the refractive index of the mask
material, n.sub.1 is the refractive index of the surrounding medium
which is generally air, with n =1, and .lambda. is the wavelength
of the projection beam PB. The circle 22 is imaged by the
projection lens system in a ring 24 shown in FIG. 4. It can be
explained with reference to FIG. 5 how this ring is obtained.
[0098] In this Figure, the reference numeral 22 denotes a circular
area of the test mask on which the projection beam PB, a beam of
electromagnetic radiation, is incident. After passage through the
phase pattern 22, the size of the electric field vector E of this
beam shows the variation as a function of the position p of graph
25. The perpendicular slopes in this graph are located at the
position of the contour line of the phase pattern 22. After passage
through the projection lens system PL shown diagrammatically by
means of a single lens in FIG. 5, the size of the electric field
vector E' shows the variation as a function of the position in
graph 29. The perpendicular slopes have changed to oblique slopes.
This is a result of the fact that the projection lens system is not
an ideal system but has a point spread function, i.e. a point is
not imaged as a point but is more or less spread across an Airy
pattern during imaging. If the projection system were ideal, the
electric field vector would have the variation as shown in the
broken line graph 30. The size of the electric field vector
represents the amplitude of the projection beam, so that the graph
29 shows the amplitude of the beam as a function of the position in
the plane of the photoresist layer PR. Since the intensity I of the
beam is equal to the square of the amplitude (I=E'.sup.2), this
intensity shows the variation as a function of the position in
graph 31. Each edge in the graph 29 has changed over to two edges
with opposite slopes, which means that the single contour line of
the phase pattern is imaged in two contour lines, i.e. the circle
is imaged in a ring 24 as shown in FIG. 4. The width wi of this
ring is determined by the point spread function and its diameter di
is determined by the resolution of the projection lens system. If
the projection lens system did not have any point spread, the
intensity of the projection beam in the photoresist layer would
have the variation as shown by way of the broken line graph 32, and
the phase pattern 22 would be imaged as a circle. In the method
according to the invention, deliberate use is made of the point
spread, though being small, of the projection lens system.
[0099] Upon use of this method in a given projection apparatus, the
ring 24 had a width wi of 80 nm and a diameter di of 350 nm. The
projection lens system had a magnification M=1/4 so that the phase
pattern in the mask had a diameter D=1.4 .mu.m. The diameter di of
350 nm appeared to be an optimal value and corresponded to the
resolution of the apparatus whose projection lens system had an NA
of 0.63 and the projection beam had a wavelength of 248 nm. For
other projection apparatus, di will have a different optimal value.
Even if di has a value which is different from the optimal value,
aberrations can still be measured.
[0100] For obtaining a good contrast in the image, the phase
difference between the beam portion which has passed through the
circular area 22 and the rest of the beam must be .phi.=.pi. rad.
This means that the depth d of the recess must be equal to the
wavelength of the beam PB if the refractive index of the mask
material is 1.5 and the surrounding medium is air having a
refractive index of 1. For a practical embodiment, the optimal
depth d is, for example, 233 nm. Usable results can still be
obtained at depths different from the optimal depth.
[0101] If use is made of a test mask in which both the circular
area 22 and its surroundings are reflecting, the optimal depth, or
height, of the circular area is equal to a quarter of the
wavelength.
[0102] If the projection lens system does not exhibit any
aberrations, the inner circle ci and the outer circle ce of the
ring in FIG. 4 are concentric and, during scanning through focus,
this ring has a symmetrical behavior. Scanning through focus is
understood to mean the movement of the image along the optical axis
of the projection lens system in the +Z direction and the -Z
direction with respect to the photoresist layer. This movement of
the image with respect to the layer can be realized by changing the
focus of the projection system or by moving this system and the
photoresist layer with respect to each other in the Z
direction.
[0103] When aberrations occur, said symmetry is disturbed. Each
kind of aberration results in a characteristic deformation of the
ring, as will be explained hereinafter.
[0104] To be able to satisfactorily observe the inner circle ci and
the outer circle ce which are located close together, a scanning
microscope may be used with a resolution which is larger than that
(.lambda./NA) of the projection system. A scanning electron
microscope, which may have a magnification of the order of 100,000
and can observe details of the order of 3.5 nm, is eminently
suitable for this purpose, particularly if a large number of images
must be detected. It is alternatively possible to use other
scanning microscopes in the form of, for example, probe microscopes
such as an optical probe microscope or an AFM (Atomic Force
Microscope) or hybrid forms thereof, particularly if only a small
number of images must be detected.
[0105] The image data obtained by scanning are subjected to a
special image-processing method. This method may consist of, for
example, two operations. The first operation comprises a
determination of the contours of the ring in accordance with the
steps of:
[0106] removing noise from the incoming image data;
[0107] determining the contours of the image, for example, by
differentiation, or by determining in how far the intensity of each
observed pixel is under a given threshold;
[0108] determining the point of gravity of the intensity
distribution of the observed image;
[0109] measuring the distances between the pixels and this point of
gravity, and
[0110] plotting the measured distances in a histogram which then
shows two peaks, the inner edge of the peak, where the smaller
distances are clustered, representing the inner contour of the
ring, and the outer edge of the peak, where the larger distances
are clustered, representing the outer contour.
[0111] The second operation consists of a Fourier analysis
comprising the steps of:
[0112] decomposing radii of these contours each time extending at a
different angle .THETA. to the X axis into sine and cosine
functions of these angles, and filtering the contours, and
[0113] visualizing the intensities of the Fourier components thus
obtained in graphs.
[0114] Analysis methods which are different from this Fourier
analysis may be used instead. It is essential that the radii of the
contours are measured as a function of the angle .THETA.. The
advantage of the Fourier analysis is that it has sine functions and
cosine functions as basic functions. Determining the radii of the
contours as a function of the angle .THETA. can be most easily done
by way of the sine function and the cosine function. The
aberrations can thereby be detected in a direct manner. More
operations must be performed in other analysis methods.
[0115] If the projection lens system does not have any aberrations,
the inner contour and the outer contour of the annular image are
satisfactorily circular, and the circles are satisfactorily
concentric throughout their circumference, as is shown in FIG. 6.
Moreover, the rotational symmetry is then maintained upon scanning
through focus.
[0116] FIG. 6 shows a SEM image obtained by means of an
aberration-free lens system and with the following imaging
conditions: .lambda.=248 nm, NA=0.63, .sigma.=0.3, the thickness of
the photoresist layer =280 nm. .sigma., also referred to as the
degree of coherence, indicates the extent to which the imaging beam
fills the pupil of the lens system. A .sigma. of 0.3 means that the
imaging beam has a cross-section which is equal to 0.3 of the pupil
cross-section.
[0117] The major aberrations of the projection lens system are
coma, astigmatism, the three-point, or three-leaf, aberration, and
spherical aberration. If the projection lens system has coma, the
image formed therewith and observed by the SEM has the shape as
shown in FIG. 7. The coma in this example is obtained artificially
by deliberately changing the wavelength of the imaging beam to some
extent. The other imaging conditions are equal to those mentioned
with reference to FIG. 6. The image formed in FIG. 7 is an image
formed in the top right angle of the image plane if this image
plane is considered to coincide with the plane of the drawing,
likewise as in FIG. 15 to be described hereinafter. This image has
a coma of 45.degree.. The inner contour and the outer contour are
circles which are no longer centered with respect to each other but
are offset with respect to each other in the direction of the coma,
hence in the direction of 45.degree..
[0118] FIG. 8 shows an image formed by a projection lens system
having astigmatism. The other imaging conditions are again equal to
those mentioned with reference to FIG. 6. The contour lines of the
astigmatic image are elliptical, while the distance between these
lines, i.e. the width wi of the ring, is constant throughout the
circumference. The major axis of the ellipse is parallel to the
direction of the astigmatism. Since the image in FIG. 8 is again an
image formed in the top right angle of the image plane, the major
axis of the ellipse extends under 45.degree.. The astigmatism of
the projection lens system has been obtained artificially by
deliberately displacing a movable lens element of this system to
some extent with respect to its nominal position.
[0119] Generally, the points of the contour lines may be
represented by the series: 2 r ( ) = m = 0 ~ r m ( c ) cos ( m ) +
r m ' ( c ) sin ( m ) ( 1 )
[0120] in which r.sub.m(r) is the nominal distance of the relevant
point to the center of the image, c indicates whether the relevant
point is associated with either the inner contour or the outer
contour, cos (m.THETA.) and sin (m.THETA.) is the angle dependence
of the real distance between the relevant point and the center and
m is determined by the type of aberration.
[0121] For spherical aberration, m=0. This aberration is not
dependent on the angle .THETA., and an image formed with an imaging
system having spherical aberration is rotationally symmetrical
around the optical axis of the imaging system, i.e. around the Z
axis in FIG. 1. The change of the image due to spherical aberration
is dependent on the position along the Z axis.
[0122] For the comatic aberration, m=1. An image formed with an
imaging system having this aberration has a single axis of
symmetry, in the example of FIG. 7 the axis under 45.degree. along
which the circles are displaced with respect to each other.
[0123] For the astigmatic aberration, m=2. When this aberration
occurs, the formed, elliptic image has two axes of symmetry, namely
the major axis and the minor axis of the ellipse. In the example of
FIG. 8, these are the axis shown under 45.degree. and an axis
perpendicular thereto.
[0124] For the three-leaf, or three-point, aberration, m=3. When
this aberration occurs, the image shows three axes of a symmetry.
The image of FIG. 7 does not only have comatic aberration but also
a small three-point aberration. An image having a larger
three-point aberration is shown in FIG. 9.
[0125] The description has hitherto been based on a single test
pattern. However, a test mask may have a large number of test
patterns, for example 121, so that the aberrations can be measured
at an equally large number of positions in the image field of the
projection lens system. In practice, not all of these test
patterns, but a smaller number will be used, for example 21, in
which these test patterns are located at such positions that most
information about aberrations can be obtained therefrom. Since the
test patterns are so small, they may also be provided in a
production mask, i.e. a mask with an IC pattern, without this being
at the expense of the details of the relevant IC pattern. Then it
is not necessary to manufacture separate test masks and to exchange
masks for measuring aberrations.
[0126] For performing the detection method by means of a
lithographic projection apparatus, the projection beam preferably
has a small beam cross-section at the location of the mask so that
a maximal quantity of projection radiation is concentrated on the
test pattern and a clear image is obtained. Novel generations of
lithographic projection apparatuses have special illumination
systems which provide, inter alia, the possibility of adapting the
cross-section of the projection beam, with the total radiation
energy of the beam being maintained. Such an illumination system is
described, for example, in the article: "Photolithography using the
AERIAL illuminator in a variable NA wafer stepper" SPIE Vol. 2726,
Optical Microlithography IX, Mar. 13-15 1996, pp. 54-70. The ratio
between the cross-section of the projection beam and the pupil
cross-section is denoted by .sigma., or degree of coherence. For
projecting the mask pattern, .sigma. values of between 1 and 0.3
are currently used. In accordance with the invention, such a
lithographic apparatus can be made eminently suitable for
performing the novel method of measuring aberrations if the means
for limiting the beam cross-section are implemented in such a way
that the .sigma. values can be set at the order of 0.2 or less.
These means can be obtained by adapting the beam-limiting means
already present in the lithographic apparatus in such a way that
the cross-section of the projection beam can be made considerably
smaller than the beam cross-section which is used for projecting
the mask pattern on the substrate. This further reduction of the
beam cross-section can then be realized while maintaining the total
energy in the beam. For the aberration measurements, it is
alternatively possible to arrange an extra diaphragm in the
radiation path between the radiation source and the mask holder,
the aperture of said diaphragm being adjustable in such a way that
.sigma. values of between 1 and, for example, 0.1 can be
adjusted.
[0127] The use of the invention in a stepping lithographic
apparatus has been described hereinbefore, i.e. in an apparatus in
which the whole mask pattern is illuminated and imaged in a first
IC area, and subsequently the mask pattern and the substrate are
moved with respect to each other until a subsequent IC area is
positioned under the mask pattern and the projection system, hence
one step is made, whereafter this IC area is illuminated with the
mask pattern, another step is made again, and so forth until the
mask pattern has been imaged on all IC areas of the substrate. To
alleviate the requirements of a large NA and a large image field
imposed on the projection lens system and/or to increase the
resolution and the image field of the apparatus, a
step-and-scanning apparatus is preferably used. In this apparatus,
a mask pattern is not imaged as a whole in one step, but the mask
pattern is illuminated by a beam having a narrow, rectangular or
circularly segment-shaped beam cross-section, and the mask pattern
and the substrate are moved synchronously with respect to the
system, while taking the magnification of the projection system
into account, so that all sub-areas of the mask pattern are
consecutively imaged on corresponding sub-areas of the substrate.
Since the cross-section of the projection beam in one direction,
for example, the X direction, is already small in such an
apparatus, only the beam cross-section in the other direction, for
example, the Y direction should be decreased so as to obtain an
optimal illumination for the novel method.
[0128] FIGS. 10-25 show a number of examples of measuring results
obtained by means of the method.
[0129] FIGS. 10-14 relate to spherical aberration. As already
noted, the annular image remains rotationally symmetrical when this
aberration occurs, but the width wi of the ring is dependent on the
extent of defocusing. In the experiment performed, a spherical
aberration was introduced artificially by readjusting the height,
the Z position, of the mask table by 40 .mu.m with respect to the
nominal height. FIG. 10 shows the annular images obtained by
readjusting the focus of the projection lens system from -0.3 .mu.m
to +0.3 .mu.m with respect to the nominal focus. FIG. 11 shows the
then occurring change of the width of the ring in a graphic form.
In this Figure, the defocusing DEF is plotted in .mu.m on the
horizontal axis and the ring width wi is plotted on the vertical
axis. As is shown in FIG. 11, the ring width at nominal focus
setting has increased from the above-mentioned 80 nm to
approximately 130 nm, while a ring width of 80 nm is obtained at a
defocusing of 0.4 .mu.m.
[0130] FIG. 12 shows the shape and the location in the XY plane of
an annular image in the best focus position. The origin of the XY
system of co-ordinates is located on the optical axis of the
projection system. FIG. 13 shows the Fourier analysis data of this
image. The Fourier terms FT expressed in frequencies of the angle
.THETA. are plotted on the horizontal axis. The Fourier term at the
position 1 represents coma which is proportional to cos .THETA.,
that at position 2 represents astigmatism which is proportional to
cos 2.THETA.), that at position 3 represents three-point aberration
which is proportional to cos 3.THETA., and that at positions 4, 5
and 6 represents other aberrations which are negligibly small for
the example given. The amplitudes of the deviations of the circle
are plotted in nm on the vertical axis. For the example shown in
FIG. 12, there is some coma at an angle .THETA.=124.degree., some
astigmatism at .THETA.=178.degree. and some three-point aberration
at .THETA.=-2.degree..
[0131] The three-dimensional FIG. 14 shows an example of the
variation throughout the image field, in this example 20.times.20
mm large, of a spherical aberration. The X and Y positions in the
image field are plotted on the axes of the base plane and the
spherical aberration is plotted on the vertical axis. This
aberration is expressed in the number of nm change of the ring
width wi per .mu.m offset of the focus. The average spherical
aberration across the image field is equal to -85 nm/.mu.m in this
example.
[0132] FIGS. 15-18 relate to a comatic aberration which has been
introduced artificially by imaging the test object with radiation
whose wavelength is 40 pm larger than the nominal wavelength, i.e.
the wavelength for which the projection system has been designed.
FIG. 15 shows the annular images 40-48 which are then formed at
different positions in the image field. As already noted, the inner
contour and the outer contour are offset with respect to each other
when coma occurs, so that these contours are no longer centered
with respect to each other. The coma is relatively small in the
center of the image field, as is shown by the central image 40.
Upon a displacement from the center, the coma increases, while the
direction of the coma coincides with the direction of the
displacement. The coma directions are approximately +45.degree.,
+135.degree., -135.degree. and -45.degree. for the images 45, 46,
47 and 48, respectively.
[0133] The coma is not only dependent on the position in the image
field but also on the extent to which the imaging beam is focused
on the photoresist layer. If at a fixed position in the image field
scanning through focus takes place, then the coma has a parabolic
variation as a function of the defocusing, with the smallest coma
occurring if the focusing is optimal. FIG. 16 shows a magnification
of the image 48 for the best focus condition.
[0134] FIG. 17 shows the Fourier graph associated with the image of
FIG. 16. It is apparent from FIG. 17 that the direction of the coma
is -48.degree. and its amplitude is 30 nm. The projection system
with which this image is made also has an astigmatism of
approximately 7 nm at an angle .THETA. of 118.degree. and a
three-point aberration of approximately 5 nm at an angle .THETA. of
17.degree..
[0135] FIG. 18 shows a coma chart obtained by imaging the test
pattern at 21 different positions in the image field, the XY plane.
The direction of the coma at a measured position is indicated by
the direction of the arrow shown at that position and the size of
the coma is indicated by the underlined number near this arrow.
Each number in FIG. 18 is the average of the coma numbers
associated with the relevant field position and obtained by
scanning through focus. The average coma throughout the image field
of the example shown in FIG. 18 is 18 nm.
[0136] FIGS. 19-22 relate to an astigmatic aberration. Instead of a
single focal point, an imaging system having astigmatism has a
first and a second, astigmatic, focal line, which focal lines are
perpendicular to each other. The length of these focal lines is
dependent on the position along the optical axis of the imaging
system. In the position where the beam has its narrowest
constriction, the focal lines are equally long and the image is
circular. At positions located before the position of narrowest
constriction, the first focal line is longer than the second, and
the image is elliptical, with the major axis of the ellipse
extending in the direction of the first focal line. At positions
located behind the position of the narrowest constriction, the
second focal line is longer than the first, and the image is
elliptical, with the major axis of the ellipse extending in the
direction of the second focal line. To determine the astigmatic
aberration of a lens system, it is necessary to scan through focus.
In accordance with the novel method, the astigmatism is detected by
determining the change of the second harmonic (2.THETA.) as a
function of the defocusing. This astigmatism is expressed in nm per
.mu.m defocusing.
[0137] FIG. 19 shows images 50-58 which are formed at nine
different positions in the image field by a projection system
having astigmatism. This astigmatism has been introduced
artificially by displacing a movable lens element of the projection
system by 40 .mu.m with respect to its nominal position. In the
center of the image field, the astigmatism is relatively small as
is shown by the central image 50. Upon a displacement from the
center, the astigmatism increases, with the direction of the
astigmatism coinciding with the direction of the displacement. The
directions of astigmatism are approximately +45.degree.,
+135.degree., -135.degree. and -45.degree. for the images 55, 56,
57 and 58, respectively.
[0138] FIG. 20 shows a magnification of the image 58 and FIG. 21
shows the associated Fourier graph. It is apparent from the latter
Figure that the direction of the astigmatism is 136.degree. and its
size is approximately 18 nm/.mu.m. The projection system with which
this image is formed also has a coma of 11 nm at an angle .THETA.
of 51.degree., a three-point aberration of 4 nm at an angle .THETA.
of 11.degree. and a four-point aberration of 6 nm, proportional to
cos 4.THETA., at an angle .THETA. of 3.degree.. An image formed by
a projection lens system with four-point aberration has four axes
of symmetry.
[0139] FIG. 22 shows an astigmatism chart obtained by forming
images of the test pattern at 21 different positions of the image
field, the XY plane. The direction of astigmatism at a measured
position is indicated by the direction of the arrow at that
position, and the amount of the astigmatism is indicated by the
underlined number at that position. Each number in FIG. 22 is the
average of the astigmatic numbers associated with the relevant
position and obtained by scanning through focus. For the given
example, the average astigmatism throughout the image field, i.e.
the average of the numbers of FIG. 22 is 31.1 nm.
[0140] An example of measured three-point aberrations is shown in
the three-dimensional FIG. 23. The X and Y positions in the image
field are plotted along the axes of the base plane in this Figure
and the size of the aberration is plotted in nm on the vertical
axis. Also this aberration is maximal at the angles of the image
field. The aberration is relatively small; the average value of
this aberration in this example is 4.7 nm.
[0141] FIGS. 24 and 25 illustrate that simultaneously occurring
aberrations of different types can be measured separately by means
of the method according to the invention. FIG. 24 shows a coma
chart which is similar to that of FIG. 18. Not only first arrows
representing the pure coma are shown at the 21 different positions
in the image field, but also second arrows representing the
measured coma in the presence of spherical aberration, and third
arrows representing the measured coma in the presence of
astigmatism. It is apparent from this Figure that the measured coma
size and direction generally changes to only a small extent when
said two other aberrations occur.
[0142] FIG. 25 shows an astigmatism chart which is similar to that
of FIG. 22. Not only first arrows representing the pure astigmatism
are shown at the 21 different positions in the image field, but
also second arrows representing the astigmatism in the presence of
spherical aberration, and third arrows representing the astigmatism
in the presence of coma. It is apparent from this Figure that the
measured size and direction of the astigmatism generally changes to
only a small extent when spherical aberration and coma occur
simultaneously.
[0143] As explained above, the three aberrations coma, astigmatism
and three-point aberration can be measured independently from each
other, and each of these aberrations is represented by a specific
harmonic in the Fourier spectrum. For a perfect ring having
concentric contours, the Fourier spectrum only contains the zero
frequency that is equivalent to the mean radius. For the outer
contour of the ring, the aerial image of the test circle, the
expression (1) can be written as:
r(.PHI.,z)=r.sub.0(z)+.SIGMA.(r.sub.m,c(z). cos
(m.PHI.)+r.sub.m,s(z). sin (m.PHI.)) (2)
[0144] whereby the summation is over m, from m=1 on. In this
expression r is the nominal distance of the relevant contour point
to the center of the image, z is the position along the axis of the
imaging system, .PHI. is the angular position of the contour point,
the subscript m is the number of the Fourier harmonic and the
subscripts c and s relates to the cosine and sinus function. The
inner contour of the ring can be represented by a similar
expression. Thereby the origin has been chosen to coincide with the
center of mass of the inner contour. Under this condition,
.SIGMA.x=.SIGMA.y=0 and, by definition, the first harmonic of the
inner contour is zero.
[0145] Spherical aberration can only be related to r.sub.0 in the
above expression, because with this aberration the image remains
rotationally symmetrical. Coma is related to r.sub.1, astigmatism
to r.sub.2 and three-point aberration to r.sub.3. However each
aberration that has the same cos(m(p) term will contribute to the m
Fourier harmonic, i.e. each type of aberration is composed of a
number or sub-aberrations of lower and higher order. Each of these
sub-aberrations are usually represented by a Zernike coefficient,
i.e. an amount of a specific Zernike polynomial from the "fringe
Zemike code" which has a maximum of 37 polynomials. A list of the
Zemike terms Z with their polynomials is given in table I. This
table is known from the manual for the optical lithography
simulation program "Solid C", propriety of the firm Sigma C. Such
table can also be deduced from the well-known handbook: "Principles
of Optics" of Born & Wolf. The contribution of the Zernike
coefficients to the total aberration decreases with increasing
number of these coefficients. For spherical aberration (m=0) the
low order Zernike coefficient is Z9 and the higher order
coefficients are Z16, Z25, Z36 and Z37. For X-coma the low order
coefficient is Z7 and the higher order coefficients are Z14, Z23
and Z34, whereas for Y-coma these coefficients are Z8 and Z15, Z24,
Z35 respectively. For astigmatism, i.e. H/V or horizontal/vertical
astigmatism, the low order coefficient is Z5 and the higher order
coefficients are Z12, Z21 and Z32, whereas for the astigmatism at
45.degree. these coefficients are Z6 and Z13, Z22, Z33,
respectively. For three-point aberration in the x direction the low
order coefficient is Z10 and the higher coefficients are Z19 and
Z30, whereas for the three-point aberration in the y direction
these coefficients are Z11 and Z20,Z31 respectively.
[0146] When measuring coma, astigmatism and three-point aberration
in the way described above at the hand of, amongst others, FIG. 18,
FIG. 22 and FIG. 23, respectively, which FIGS. show these
aberrations for different points in the image field, the sum of the
lower order and the higher orders Zernike coefficients for these
aberrations are determined and not the individual Zemike
coefficients. In other words, what is obtained is for each of these
aberration is the sum of the sub-aberrations in terms of shape
deviations of the phase pattern expressed in nm or nm/.mu.m, rather
than the individual sub-aberrations (Zernike coefficient) in terms
of wave front aberrations expressed in m.lambda..
[0147] According to the present invention also the individual
Zernike coefficients can be determined. Thereby use is made of the
fact that the aberrations are dependent on the imaging parameters,
such as the numerical aperture, NA, of the imaging system and the
coherence degree and the diameter .PHI. of the test circle on the
mask. The coherence degree is the ratio of the radius of the
cross-section of the imaging beam in the plane of the entrance
pupil of the imaging system and the radius of this pupil. In
general a Fourier harmonic is a combination of all Zernike
coefficients, which combination can be expressed in a Taylor
series. Because the manufacture of lithographic projection lens
systems is carried out with such high precision, the aberrations of
this kind of lens systems are already rather small, so that the
Taylor series reduces to a linear relationship between the Fourier
harmonic and the Zernike coefficients belonging thereto. Although
the method to determine the Zernike coefficients is described
hereunder for the case that there is a linear relationship between
the Fourier harmonic and the Zemike coefficients, this method can
be applied more generally in combination with a best-fit method.
Modern lithographic apparatus have an imaging system the NA of
which can be set at different values and an illumination system
offering the possibility to change the coherence degree. In such
apparatus a relationship between the NA and the coherence degree a
may exist, for example the product of NA and .sigma. is constant.
An attractive scenario to set the illumination parameters for
determining the Zemike coefficients is, that first an NA is chosen
and that the coherence degree .sigma. is adapted to the selected
NA. In order to make the aberration ring test method as sensitive
and accurate as possible, also the diameter .PHI. of the test
circle on the mask and that of the test ring on the substrate are
preferably adapted to the NA of the illumination system. It is
however also possible that only one or two of the illumination
parameters are varied when determining the Zernike
coefficients.
[0148] The above mentioned linear relationship between the
aberration of a specific kind and the corresponding sub-aberrations
is for coma in the x-direction:
F1(x)=.alpha..Z7+.beta..Z14+.gamma..Z23 (3)
[0149] And for coma in the y-direction:
F1(y)=.alpha..Z8+.beta..Z15+.gamma..Z24 (4)
[0150] In these equations F1l(x) and F1(y) are the measured Fourier
terms, Z7, Z14, Z23 and Z8, Z15, Z24 are the unknown Zernike
coefficients which are specific for a given imaging system. The
factors .alpha., .beta. and .gamma. are weighting factors which are
dependent on the value of the NA of the imaging system and of the
values of .sigma. and .PHI.. If, for example, in addition to a
change of the NA value, also the values of .sigma. and .PHI. are
changed, the latter changes are preferably not independently but
simultaneously with and dependent from the change of NA. In
equations like (3) and (4) higher order Zernike coefficients can be
omitted if their weighting factors are small and/or the Zernike
terms are hardly present in the imaging system. That is the reason
why the Zernike terms Z34 and Z35 are not present in equation (3)
and (4) respectively. To measure the Zernike coefficients Z7, Z14
and Z23, for example NA is varied and F1(x) is measured for at
least three different values of NA: NA1, NA2, NA3. From the
equations for these values:
F1(x).sub.NA1=.alpha..sub.NA1.Z7+.beta..sub.NA1.Z14+.gamma..sub.NA1.Z23
F1(x).sub.NA2=.alpha..sub.NA2.Z7+.beta..sub.NA2.Z14+.gamma..sub.NA2.Z23
F1(x).sub.NA3=.alpha..sub.NA3.Z7+.beta..sub.NA3Z14+.gamma..sub.NA3.
Z23
[0151] the individual Zemike coefficients Z7, Z14 and Z23 can be
determined, provided that the values of the weighting factors
.alpha., .beta. and .gamma. for NA1, NA2 and NA3 are known.
[0152] The weighting factors for the different NA's can be
determined by a simulation process using one of the known optical
lithography computer simulation programs. In general such program
is supplied with parameters of the wafer stepper or -scanner, like
NA, .sigma., focus, known aberrations of the imaging system and
with parameters of the wafer and data of the of development
process. The output of the program is data describing a
three-dimensional profile. These data are supplied to another
computer program that calculates the Fourier harmonics. To
determine the value of the weighting factor .alpha. for different
values of NA, a given value of Zernike coefficient Z7 and
successively different values of NA are supplied to the simulation
program while keeping constant the other parameters supplied to the
program and for each value of NA the value of the first Fourier
harmonic F1 is determined. In the same way the value of .beta. for
different NA values can be determined by supplying the simulation
program with a given Zernike coefficient Z14 value and with
successively different NA values. To obtain the values of .gamma.
for different NA's a given Zernike coefficient Z23 value and the
different NA values should be supplied to the program.. The
.alpha.-, .beta.- and .gamma. values for the different NA's thus
obtained can be put in a table which can be stored in an electronic
processor for determining the Zemike coefficients. Such table can
be used to measure the aberrations of a lot of lithographic
projection lenses, or imaging systems.
[0153] If it is expected that also the Zemike coefficients Z34 and
Z35 give a non-negligible contribution to coma, a fourth weighting
factor .delta. is to be used and the different values of this
factor for different NA values can then be determined with the
simulation program. If during determination of the Zernike
coefficients, instead of or in addition to NA, other parameters
like .sigma. and/or .PHI. are varied, the dependency of the
weighting factors .alpha., .beta. and .gamma. from the parameter
.sigma. and/or .PHI. can be determined by supplying the simulation
program with successively different values of .sigma. and/or .PHI.
in the same as described for NA.
[0154] In principle three, or in general n, weighting factors can
be determined by using three, or in general n, different values for
NA or .sigma.. It is however also possible to use more than three,
for example six, or in general 2n, different values for NA and/or
.sigma.. This results in a more accurate and reliable determination
of the weighting factors.
[0155] As an example, in FIG. 26 is indicated what the contribution
to the first Fourier harmonic F1, plotted on the vertical axis and
expressed in nm, will be of the Zernike coefficients Z7, Z14 and
Z23 for different NA's, plotted on the horizontal axis, if each of
these coefficients is given a value of 50 m.lambda. From this FIG.
the value of the weighting factor .alpha. for different NA values
can be obtained by dividing the Z7 values for these NA values by 50
m.lambda.. In the same way the values of the other weighting
factors .beta. and .gamma. can be obtained by dividing the values
for Z14 and Z23, respectively by 50 m.lambda.. The weighting
factors for the second and third Fourier harmonic F2 and F3 show
dependencies on NA which are rather similar to the dependencies
shown in FIG. 26 for the weighting factors of the first Fourier
harmonic F1. The Zernike coefficients Z8, Z15 and Z24 of the
Fourier harmonic F2(y) can be determined from equation (4) in a
similar way as the coefficients Z8, Z15 and Z24 of the Fourier
harmonic F1(x) from equation (3).
[0156] By way of example FIG. 27 shows the measured values for the
two coma terms F1(x) and F1(y) at a given position in the image
field for six different values of NA. The Zemike coefficients
belonging to this coma and expressed in m.lambda. are: Z7=19,
Z14=-10, Z23=-20, Z8=17, Z15=13 and Z24=-23 For H/V astigmatism,
represented by the second Fourier harmonic F2(H/V), the
relationship between this harmonic and Zernike coefficients is:
F2(H/V)=a..Z5+b.Z12+c.Z21
[0157] and for 45.degree. astigmatism, represented by the second
Fourier harmonic F2(H/V) this relationship is:
F2(45)=a.Z6+b.Z13+c.Z22
[0158] The Zernike coefficients Z32 for F2 and Z33 for F2(45) have
been omitted because it is assumed that the their contribution to
the Fourier harmonics is negligible. The weighting factors a, b and
c are dependent from NA, .sigma. and .PHI. and can be determined by
a simulation process in a similar way as the factors .alpha.,
.beta. and .gamma. for coma. By choosing at least three different
values for an imaging parameter, for example the NA of the imaging
system and measuring F2 and F2(45) for these values the Zemike
coefficients Z5, Z12, Z21 and Z6, Z13 and Z 22 of astigmatism can
be determined in the, same way as the Zernike coefficients of
coma.
[0159] In the same way as described for the Zemike coefficients of
coma and astigmatism, the Zemike coefficients Z10, Z19, Z30 for the
x-three point aberration F3(x) and Z11, Z20, Z31 for the y-three
point aberration F3(y) can be determined by means of the
relationships:
F3(x)=e.Z10+f.Z19+g.Z30
F3(y)=e.Z11+f.Z20+g.Z31
[0160] Whereby the weighting factors e, f and g are dependent on NA
and/or .sigma. or .PHI. in a similar way as the factors .alpha.,
.beta. and y for coma and the factors a, b and c for astigmatism
are.
[0161] Instead of determining the individual Zemike coefficients by
means of the, for example linear, equations for the Fourier
harmonics such as equations (3) and (4) and of a simulation program
as a tool for obtaining the weighting factors, it is also possible
to supply a simulation program with a large number of values for
the Zemike coefficients, starting from a ring-shaped image, and to
calculate the profiles resulting from the different values of the
Zernike coefficients. These profiles can be stored in a look-up
table, either in the form of data describing the profile or as
graphic representations, which look-up table can be put in the
image processor, coupled to scanning device, for storing and
processing the observed ring-shaped images. Then during detection
of aberrations of an imaging system an observed ring-shaped image
can be compared with the reference images of the look-up table to
determine which of the reference images fits best to the actual
observed image. As the Zernike coefficients of the best-fit
reference image are known, the Zernike coefficients of the observed
image are known then.
[0162] The circular phase structure(s) cover(s) only a very small
part of the mask surface area. If an entirely transparent test mask
is used, the radiation passed by the mask outside the area of the
phase structure may have the effect of interference radiation and
reduce the quality of the image of the phase structure. To prevent
this, a test mask is preferably used in which only the circular
phase structure, further referred to as the figure, and a
relatively small area around it, hereinafter referred to as figure
area, are transparent, while the rest of the mask, hereinafter
referred to as outer area, has been made opaque, for example by
coating it with chromium. FIG. 26 shows a part of a test mask TM
having a circular phase structure, or area, denoted by the
reference numeral 22 again. The transparent figure area around the
circle 22 is denoted by the reference numeral 80. This area
consists of transparent mask material (20 in FIG. 3b). Outside the
figure area, the mask is coated with a chromium layer 82.
[0163] To achieve that a scanning electron microscope, or another
scanning detection device, can easily find the small image of the
FIG. 22, a recognition mark 84 is provided in the test mask and in
the outer area of each phase pattern, as is shown in FIG. 26. This
mark, which is formed by an F-shaped opening in the chromium layer
in the example shown, may be an arbitrary mark, provided that it
has details extending in the X direction as well as details
extending in the Y direction. As is shown by FIG. 26, the strips
extending in the X direction and the strips extending in the Y
direction of the recognition mark are considerably larger than the
FIG. 22 so that this mark is more easily observable and is suitable
for navigation of the detection device. As soon as this mark has
been observed, the detection device can be directed within the area
on the substrate which corresponds to the outer area 82 of the test
mask to the image of the figure area 80 and start searching the
image of the FIG. 22 located within this area. Opaque,
chromium-coated strips 86 in the X direction and strips 88 in the Y
direction may be present within the figure area 80 so as to
simplify the navigation of the detection device within the area on
the substrate corresponding to the figure area 80.
[0164] Further information may be provided, as is denoted by the
reference numeral 90, in each outer area 82 of the test mask. In
this example, the information relates to the diameter of the imaged
ring (d in FIG. 4) chosen for the relevant area 82. This
information may also be, for example, position information and
indicate the X and Y co-ordinates of the relevant figure area 80 on
the test mask. Further information, which may be useful for
performing the method, may also be provided in the recognition mark
84.
[0165] Since the marks 84 and 90 have relatively large details,
these details will always be imaged in such a way that they are
still reasonably recognizable for the scanning detection device,
even if the imaging circumstances are not ideal, for example, if
the quantity of illumination used is not optimal. If, for example,
a too small quantity of illumination were used, the quality of the
image of the phase FIG. 22 would be reduced to such an extent that
the method can no longer be used satisfactorily. By observing the
mark 84 and/or 90, the cause of the poor image quality can be
ascertained, so that the circumstances can be adapted thereto in
such a way that a usable image of the phase pattern is as yet
obtained and the method can still be used.
[0166] It has hitherto been assumed that the phase FIG. 22 is
formed by an area located higher or lower than the rest of the
plate or test mask 20. The phase figure may, however, also consist
of an area having a different refractive index than the rest of the
plate. Such an area introduces also a phase jump in a beam passing
through the plate. If a reflecting production mask is used in the
lithographic apparatus, and if the detection method is performed
with a reflecting test mask, the FIG. 22 and the figure area 80
will have to be transparent to this test mask so as to cause this
FIG. 22 to be active as a phase structure with a deviating
refractive index. To reflect the imaging beam which has passed
through the test mask at the location of the FIG. 22 and the figure
area 80, the test mask may be provided with reflecting means at the
relevant locations.
[0167] The text hereinbefore only describes measurements on a
projection lens system for a lithographic apparatus. However, the
projection system for such an apparatus may also be a mirror
projection system. Such a projection system must be used if EUV
radiation is used as projection radiation. EUV, or extreme
ultraviolet, radiation is understood to mean radiation at a
wavelength in the range of several nm to several tens of nm. This
radiation is also referred to as soft X-ray radiation. The use of
EUV radiation provides the great advantage that extremely small
details, of the order of 0.1 .mu.m or less, can be imaged
satisfactorily. In other words, an imaging system in which EUV
radiation is used has a very high resolution without the NA of the
system having to be extremely large so that also the depth of focus
of the system still has a reasonably large value. Since no suitable
material, which is sufficiently transparent and suitable for making
lenses, is available for EUV radiation, a mirror projection system
instead of a conventional projection lens system must be used for
imaging a mask pattern on the substrate. Different embodiments of
such mirror projection systems are known, which may comprise three
to six mirrors. As the number of mirrors increases, the quality of
the image is enhanced, but due to reflection losses, this is at the
expense of the quantity of radiation on the substrate. A mirror
projection system with six mirrors is described in, for example
[0168] FIG. 29 shows an embodiment of another type of mirror
projection system with six mirrors for a step-and-scanning
lithographic projection apparatus having an NA (at the image side)
of the order of 0.20, a magnification M of 0.25, a circular
segment-shaped image field having a width of 1.5 mm and a
relatively large free working distance fwd. The apparatus comprises
an illumination unit 60, shown diagrammatically, accommodating an
EUV radiation source and an optical system for forming a projection
beam PB whose cross-section has the shape of a circular segment. As
is shown in the Figure, the illumination unit may be positioned
close to the substrate table WT and the imaging section 69, 70 of
the projection system so that the projection beam PB can enter the
projection column closely along these elements. The mask MA' to be
imaged, which is a reflective mask in this example, is arranged in
a mask holder MH which forms part of a mask table MT by means of
which the mask can be moved in the scanning direction 62 and
possibly in a direction perpendicular to the scanning direction,
such that all areas of the mask pattern can be arranged under the
illumination spot formed by the projection beam PB. The mask holder
and mask table are shown only diagrammatically and may be
implemented in various ways. The substrate W is arranged on a
substrate holder WH, which is supported by a substrate table WT.
This table may move the substrate in the scanning direction, the X
direction, but also in the Y direction perpendicular thereto. In
this embodiment, the mask and the substrate move in the same
direction during scanning. A block 64 supports the substrate
table.
[0169] The projection beam reflected by the reflective mask MA is
incident on a first, concave, mirror 65. This mirror reflects the
beam as a converging beam to a second mirror 66 which is slightly
concave. The mirror 66 reflects the beam as a more strongly
converging beam to a third mirror 67. This mirror is convex and
reflects the beam as a slightly diverging beam to the fourth mirror
68. This mirror is concave and reflects the beam as a converging
beam to the fifth mirror 69 which is convex and reflects the beam
as a diverging beam to the sixth mirror 70. This mirror is concave
and focuses on the photoresist layer PR provided on the substrate
W. The mirrors 65, 66, 67 and 68 jointly form an intermediate image
of the mask, and the mirrors 69 and 70 produce the desired
telecentric image of this intermediate image on the photoresist
layer PR.
[0170] Also the mirror projection system described above and other
projection systems may have said aberrations: spherical aberration,
coma, astigmatism, three-point aberration and possible further
aberrations, and also these aberrations can be measured accurately
and reliably by means of the novel method. In the EUV lithography,
a reflective mask is preferably used, inter alia, because such a
mask can be better supported than a transmissive mask. The test
pattern required for the novel method in a reflective test mask or
production mask must then have a depth of one quarter of the
wavelength if the surrounding medium is air. This implies that a
depth of 3.25 nm is necessary for the wavelength of 13 nm preferred
in EUV lithography, which depth is very small. In that case, the
FIG. 22 with the phase structure may also consist of an area in the
plate or test mask 20 having a different refractive index than the
rest of this plate.
[0171] As is apparent from the examples described above, the
aberrations are relatively small for the measured lithographic
projection systems. In practice, it is therefore as yet unnecessary
to measure higher-order aberrations. However, as is apparent from
the Fourier graphs of FIGS. 13, 17 and 21, the novel method is also
suitable for measuring these higher-order aberrations.
[0172] The fact that the invention has been described with
reference to the measurements on a projection lens system or a
mirror projection system for a lithographic projection apparatus
does not mean that its application is limited thereto. The
invention may be used wherever the aberrations of an imaging system
must be measured independently of each other and with great
accuracy and reliability. An example of such an imaging system is a
space telescope. When using the novel method in a lithographic
projection apparatus, an optimal use is, however, made of the fact
that this apparatus itself is already intended for imaging patterns
on substrates and that the imaging and servosystems of this
apparatus may also be used for performing the novel method.
Moreover, possible means desired for performing the method, such as
said extra diaphragm, can easily be arranged in the apparatus.
1 TABLE I Z1 1 Z2 Rcos.phi. Z3 Rsin.phi. Z4 2r.sup.2 - 1 Z5 r.sup.2
- 1 Z6 r.sup.2cos2.phi. Z7 (3r.sup.3 - 2r) cos.phi. Z8 (3r.sup.3 -
2r) sin.phi. Z9 6r.sup.4 - 6r.sup.2 + 1 Z10 r.sup.3cos3.phi. Z11
r.sup.3sin3.phi. Z12 (4r.sup.4 - 3r.sup.2) cos2.phi. Z13 (4r.sup.4
- 3r.sup.2) sin2.phi. Z14 (10r.sup.5 - 12r.sup.3 + 3r) cos.phi. Z15
(10r.sup.5 - 12r.sup.3 +3r) sin.phi. Z16 20r.sup.6 - 30r.sup.4 +
12r.sup.2 - 1 Z17 r.sup.4 cos4.phi. Z18 r.sup.4 sin4.phi. Z19
(5r.sup.5 - 4r.sup.3) cos3.phi. Z20 (5r.sup.5 - 4r.sup.3) sin3.phi.
Z21 (15r.sup.6 - 20r.sup.4 + 6r.sup.2) cos2.phi. Z22 (15r.sup.6 -
20r.sup.4 + 6r.sup.2) sin2.phi. Z23 (35r.sup.7 - 60r.sup.5 +
30r.sup.3 - 4r) cos.phi. Z24 (35r.sup.7 - 60r.sup.5 + 30r.sup.3 -
4r) sin.phi. Z25 70r.sup.8 - 140r.sup.6 + 90r.sup.4 - 20r.sup.2 + 1
Z26 r.sup.5 cos5.phi. Z27 r.sup.5 sin5.phi. Z28 (6r.sup.6 -
5r.sup.4) cos4.phi. Z29 (6r.sup.6 - 5r.sup.4) sin4.phi. Z30
(21r.sup.7 - 30r.sup.5 + 10r.sup.3) cos3.phi. Z31 (21r.sup.7 -
30r.sup.5 + 10r.sup.3) sin3.phi. Z32 (56r.sup.8 - 105r.sup.6 +
60r.sup.4 - 10r.sup.2) cos2.phi. Z33 (56r.sup.8 - 105r.sup.6 +
60r.sup.4 - 10r.sup.2) sin2.phi. Z34 (126r.sup.9 - 280r.sup.7 +
210r.sup.5 - 60r.sup.3 + 5r) cos.phi. Z35 (126r.sup.9 - 280r.sup.7
+ 210r.sup.5 - 60r.sup.3 + 5r) sin.phi. Z36 25r.sup.10 - 630r.sup.8
+ 560r.sup.6 - 210r.sup.4 + 30r.sup.2 - 1 Z37 924r.sup.12 -
277r.sup.10 + 3150r.sup.8 - 1680r.sup.6 + 420r.sup.4 - 42r.sup.2 +
1
* * * * *