U.S. patent application number 10/714840 was filed with the patent office on 2004-05-27 for method of determining movement sequence, alignment apparatus, method and apparatus of designing optical system, and medium in which program realizing the designing method.
This patent application is currently assigned to Nikon Corporation. Invention is credited to Kobayashi, Shigenobu, Ono, Isao, Tatsuzawa, Yoshihiro, Yoshida, Koji.
Application Number | 20040102863 10/714840 |
Document ID | / |
Family ID | 27549537 |
Filed Date | 2004-05-27 |
United States Patent
Application |
20040102863 |
Kind Code |
A1 |
Yoshida, Koji ; et
al. |
May 27, 2004 |
Method of determining movement sequence, alignment apparatus,
method and apparatus of designing optical system, and medium in
which program realizing the designing method
Abstract
A determining method of movement sequence and a positioning
apparatus of the invention are arranged in such a manner that, in
order to measure positions of plural marks as being measurement
targets provided on a wafer within a shorter time, a group
including executable movement sequences is generated out of a group
of movement sequence candidates, each indicating a measurement
order of these marks, and a movement sequence that accomplishes a
movement operation between the marks within the shortest time is
obtained from the group thus generated. For efficiently searching
an optical system as a globally optimal solution within a shorter
computation time, independently of an initial solution given, a
designing method of optical system of the invention obtains the
optimal solution of the optical system to be designed, using an
evolutionary computation method (genetic algorithm) having a
genetic operator for handling continuos values explicitly.
Particularly, from a partial space defined by a predetermined
continuous occurrence probability distribution of occurrence
probabilities set based on parent individuals, child individuals to
be candidates in the next generation population are generated
according to the occurrence probabilities.
Inventors: |
Yoshida, Koji; (Kanagawa,
JP) ; Ono, Isao; (Kanagawa, JP) ; Tatsuzawa,
Yoshihiro; (Kanagawa, JP) ; Kobayashi, Shigenobu;
(Tokyo, JP) |
Correspondence
Address: |
STAAS & HALSEY LLP
SUITE 700
1201 NEW YORK AVENUE, N.W.
WASHINGTON
DC
20005
US
|
Assignee: |
Nikon Corporation
Tokyo
JP
|
Family ID: |
27549537 |
Appl. No.: |
10/714840 |
Filed: |
November 18, 2003 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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10714840 |
Nov 18, 2003 |
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09900016 |
Jul 9, 2001 |
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09900016 |
Jul 9, 2001 |
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09023204 |
Feb 13, 1998 |
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Current U.S.
Class: |
700/97 |
Current CPC
Class: |
G03F 9/70 20130101; G03F
9/7003 20130101; G03F 7/70425 20130101; G03F 9/7092 20130101; G03F
9/7046 20130101; G06N 3/126 20130101; G03F 7/70616 20130101; G03F
7/70691 20130101 |
Class at
Publication: |
700/097 |
International
Class: |
G06F 019/00 |
Foreign Application Data
Date |
Code |
Application Number |
Feb 14, 1997 |
JP |
030518/1997 |
Dec 17, 1997 |
JP |
348342/1997 |
Dec 24, 1997 |
JP |
355176/1997 |
Mar 11, 1997 |
JP |
056373/1997 |
Jun 25, 1997 |
JP |
168420/1997 |
Claims
What is claimed is:
1. A determining method of movement sequence for determining an
order of measurement of a plurality of measurement target areas,
which is executed prior to an alignment step in which while the
plurality of measurement target areas provided on a substrate are
successively moved into a preset measuring area of a measuring
system, positions of the respective measurement target areas moved
into the measuring area are measured, thereby achieving alignment
between a transfer position of a pattern of an original plate and
each chip area on the substrate, said determining method of
movement sequence comprising an arithmetic step of obtaining a
solution of a most preferable movement sequence with respect to an
overall movement time between said plurality of measurement target
areas, by using a predetermined search technique, said arithmetic
step comprising: a first step of generating a group including a
plurality of executable movement sequences out of a group of
movement sequence candidates, each indicating a measurement order
of said plurality of measurement target areas; and a second step of
selecting a movement sequence that can accomplish a movement
operation between said plurality of target areas in the shortest
time, out of said group generated.
2. The method according to claim 1, further comprising a pre-step
carried out prior to said arithmetic step, said pre-step being a
step of producing a movement time management table in which for
each of said plurality of measurement target areas, a movement time
is recorded as a time necessary for movement of the target area of
interest from a position thereof at the time of completion of
position measurement of either one of said plurality of measurement
target areas into said measuring area of the measuring system.
3. The method according to claim 2, wherein said movement time
management table includes such information that for a pair of
measurement target areas selected out of said plurality of
measurement target areas, after completion of the position
measurement of one measurement target area selected, the other
measurement target area selected is prohibited from moving from a
position thereof at the time of completion of the position
measurement of the one measurement target area selected into said
measuring area of the measuring system.
4. The method according to claim 1, wherein said search technique
includes at least one of a method based on operations-research
technique, an evolutionary computation method, and a combination
thereof.
5. The method according to claim 4, wherein said method based on
operations-research technique includes at least one of a linear
programming method, a Lin and Kernighan's approach, and a k-OPT
method.
6. The method according to claim 5, wherein said linear programming
method is a method arranged in such a manner that when there exist
plural near solutions to the best solution of a movement sequence
to be obtained, a plurality of good solutions are generated by
recomputation with change in a method for selecting one specific
solution or with change in a search start point and a most
preferable, good solution with respect to the overall movement time
between said plurality of measurement target areas is selected out
of the plurality of good solutions thus generated.
7. The method according to claim 5, wherein said combination method
including said linear programming method is a method arranged in
such a manner that, using a plurality of first good solutions
obtained by said linear programming method for a movement sequence
to be obtained, as initial solutions, a plurality of second good
solutions are generated by the Lin and Kernighan's approach or the
k-OPT method and a most preferable, second good solution with
respect to the overall movement time between said plurality of
target areas is selected out of said plurality of second good
solutions thus generated.
8. The method according to claim 1, wherein said search technique
obtains a solution of a most preferable movement sequence with
respect to the overall movement time between said plurality of
target areas by use of a genetic algorithm, using constraint
satisfying solutions generated at random, as initial solutions.
9. The method according to claim 1, wherein said search technique
obtains a solution of a most preferable movement sequence with
respect to the overall movement time between said plurality of
target areas by use of a genetic algorithm, using solutions
obtained by at least one of a linear programming method, a Lin and
Kernighan's approach, a k-OPT method, and a combination thereof, as
starting solutions.
10. The method according to claim 9, wherein an execution time of
said arithmetic step using said genetic algorithm is shortened by
improvement in solutions of movement sequences updated on occasion
during execution of said genetic algorithm by one of the Lin and
Kernighan's approach and the k-OPT method.
11. The method according to claim 9, wherein said genetic algorithm
has a mutation operator, said mutation operator having an operator
for changing an order of measurement of measurement target areas
selected from said plurality of measurement target areas.
12. A determining method of movement sequence for determining an
order of measurement of a plurality of alignment marks as becoming
measurement targets provided on a substrate, which is executed
prior to an alignment step in which while the plurality of
alignment marks are successively moved into a preset measuring area
of a measuring system, positions of the respective alignment marks
moved into the measuring area are measured, thereby achieving
alignment between a transfer position of a pattern of an original
plate and each chip area on the substrate, said determining method
of movement sequence comprising an arithmetic step of obtaining a
solution of a most preferable movement sequence with respect to an
overall movement time between said plurality of alignment marks, by
use of a predetermined search technique, said arithmetic step
comprising: at least a first step of generating a group including a
plurality of executable movement sequences out of a group of
movement sequence candidates, each indicating a measurement order
of said plurality of alignment marks; and a second step of
selecting a movement sequence that can accomplish a movement
operation between said plurality of alignment marks in the shortest
time, out of said group generated.
13. The method according to claim 12, further comprising a pre-step
carried out prior to said arithmetic step, said pre-step being a
step of producing a movement time management table in which for
each of said plurality of alignment marks, a movement time is
recorded as a time necessary for movement of the alignment mark of
interest from a position thereof at the time of completion of
position measurement of either one of said plurality of alignment
marks into said measuring area of the measuring system.
14. The method according to claim 13, wherein said movement time
management table includes such information that for a pair of
alignment marks selected out of said plurality of alignment marks,
after completion of the position measurement of one alignment mark
selected, the other alignment mark selected is prohibited from
moving from a position thereof at the time of completion of the
position measurement of the one alignment mark selected into said
measuring area of the measuring system.
15. An alignment apparatus for successively measuring positions of
a plurality of alignment marks as becoming measurement targets
provided on a substrate and performing alignment between a transfer
position of a pattern of an original plate and each chip area on
the substrate by use of a statistical arithmetic method based on
information of the positions of the respective alignment marks
obtained, said positioning apparatus comprising: a measuring device
for measuring each of the positions of said plurality of alignment
marks; a moving device for effecting relative movement between said
plurality of alignment marks and a measuring area of said measuring
device; an arithmetic section for generating a group of a plurality
of executable movement sequences out of a group of movement
sequence candidates, each indicating a measurement order of said
plurality of alignment marks, and selecting a movement sequence
that accomplishes a movement operation between said plurality of
alignment marks within the shortest time, out of said group
generated; and a control section for controlling said moving device
so as to successively move said plurality of alignment marks into
the measuring area of said measuring device, according to a
solution of the movement sequence obtained by said arithmetic
section.
16. The apparatus according to claim 15, further comprising a
memory for storing a movement time management table in which for
each of said plurality of alignment marks, a movement time is
recorded as a time necessary for movement of the alignment mark of
interest from a position thereof at the time of completion of
position measurement of either one of said plurality of alignment
marks into said measuring area of the measuring device.
17. The apparatus according to claim 16, wherein said movement time
management table stored in said memory includes such information
that for a pair of alignment marks selected out of said plurality
of alignment marks, after completion of the position measurement of
one alignment mark selected, the other alignment mark selected is
prohibited from moving from a position thereof at the time of
completion of the position measurement of the one-alignment mark
selected into said measuring area of the measuring device.
18. The apparatus according to claim 15, wherein said arithmetic
section executes a search technique of at least one of a method
based on operations-research technique, an evolutionary computation
method, and a combination thereof.
19. A designing method of optical system comprising: a selection
step of selecting at least two parent individuals from a population
consisting of a plurality of individuals, said population being an
n (.gtoreq.1) generation population and each individual being a
real vector representing a candidate of an optical system to be
designed; a child generation step of newly generating a population
of plural child individuals by applying at least one of a crossover
operator and a mutation operator as a genetic operator to said
parent individuals selected; and a survival selection step of
selecting individuals to be left as individuals in a next
generation population from said n generation population and said
population of child individuals.
20. The method according to claim 19, wherein said survival
selection step is a step of selecting as individuals of the next
generation population individuals satisfying at least either of one
or two or more evaluation criteria from said n generation
population and said population of the child individuals
generated.
21. The method according to claim 1, wherein in said child
generation step said crossover operator generates, from the inside
of a partial space defined by a predetermined continuous occurrence
probability distribution of occurrence probabilities set based on
components of real vectors of the respective parent individuals
selected, a real vector having a component of a value occurring
according to the occurrence probabilities, as a child
individual.
22. The method according to claim 1, wherein in said child
generation step said mutation operator generates, from the inside
of a partial space defined by a predetermined continuous occurrence
probability distribution of occurrence probabilities increasing
with approaching at least one parent individual out of said parent
individuals selected, a real vector having a component of a value
occurring according to the occurrence probabilities, as a child
individual.
23. The method according to claim 1, wherein said selection step,
said child generation step, and said survival selection step are
carried out in order plural times.
24. A designing method of optical system for repetitively
performing generation of a population consisting of a plurality of
individuals, each individual having a plurality of parameters
representing a candidate of an optical system to be designed, said
optical system including at least one optical element, and
selection of individuals to be left as individuals in a next
generation population, thereby optimizing the optical system to be
designed, wherein optimization of at least one selected parameter
out of said plural parameters of the individuals is effected by
selecting a plurality of parent individuals out of said individuals
generated, setting a predetermined continuous occurrence
probability distribution of occurrence probabilities, based on the
selected parameter of each of said plurality of parent individuals,
and newly generating a child individual having as a value of said
selected parameter a value occurring according to the occurrence
probabilities, from the inside of a partial space defined by said
occurrence probability distribution.
25. The method according to claim 24, wherein from a population
including at least said parent individuals and said child
individual generated, an individual having as a value of said
selected parameter a value fitting either of one or two or more
evaluation criteria is selected as an individual in the next
generation population.
26. The method according to claim 24, wherein said selected
parameter of the individual is at least one of a curvature of a
boundary surface in said optical element, a distance between
boundary surfaces, and a refractive index of a medium placed
between the boundary surfaces.
27. A designing method of optical system comprising: a parent
selection step of selecting at least two real vectors to be
parents, from a population of plural individuals each representing
a candidate of an optical system to be designed, said population
being an n (.gtoreq.1) generation population and each individual
being a real vector having a component of one or two or more
predetermined parameters featuring the optical system; a child
generation step of executing at least one of a crossover step and a
mutation step, said crossover step being a step of generating, from
the inside of a partial space defined and expressed by a
predetermined continuous occurrence probability distribution of
occurrence probabilities set based on components of the respective
real vectors of said parent individuals selected, a real vector
having a component of a value occurring according to the occurrence
probabilities, as a child individual, and said mutation step being
a step of generating, from the inside of a partial space defined by
a predetermined continuous occurrence probability distribution of
occurrence probabilities increasing with approaching at least one
parent individual out of said parent individuals selected, a real
vector having a component of a value occurring according to the
occurrence probabilities, as a child individual; and a survival
selection step of selecting individuals to be left as individuals
in a next generation population from said n generation population
and said child individual generated.
28. The method according to claim 27, wherein in said survival
selection step said individuals selected replace individuals not
selected in said n generation population, thereby generating the
next generation population.
29. The method according to claim 27, wherein in said survival
selection step the individuals to be left as individuals in the
next generation population are selected in order from an individual
fittest to a predetermined evaluation criterion and in proportion
to a fitness value of each individual from the population of said
parent individuals and said child individual generated.
30. The method according to claim 27, wherein in said survival
selection step an individual satisfying at least either of one or
two or more evaluation criteria is selected as an individual in the
next generation population from the population of said parent
individuals and said child individual generated.
31. The method according to claim 27, wherein said component of
real vector of individual is at least one of a radius of curvature
of a boundary surface of said optical element, a distance between
boundary surfaces, and a refractive index of a medium placed
between the boundary surfaces.
32. A designing apparatus of optical system comprising an
arithmetic section for repetitively executing generation of plural
parameters each representing a candidate of an optical system to be
designed, said optical system including at least one optical
element, and selection of parameters to be left out of the plural
parameters generated, thereby optimizing the optical system to be
designed, and a memory for temporarily storing the parameters
generated, wherein said arithmetic section executes at least a
parent selection step of selecting at least two real vectors to be
parents, from an n (.gtoreq.1) generation population consisting of
a plurality of real vectors given as said plural parameters; a
child generation step of executing at least one of a crossover step
and a mutation step, said crossover step being a step of
generating, from the inside of a partial space defined by a
predetermined continuous occurrence probability distribution of
occurrence probabilities set based on components of the respective
real vectors of said parent individuals selected, a real vector
having a component of a value occurring according to the occurrence
probabilities, as a child individual, and said mutation step being
a step of generating, from the inside of a partial space defined by
a predetermined continuous occurrence probability distribution of
occurrence probabilities increasing with approaching at least one
parent individual out of said parent individuals selected, a real
vector having a component of a value occurring according to the
occurrence probabilities, as a child individual; and a survival
selection step of selecting individuals to be left as individuals
in a next generation population from said n generation population
and said child individual generated.
33. The apparatus according to claim 32, wherein in said survival
selection step the arithmetic section replaces individuals not
selected in said n generation population by said selected
individuals, thereby generating the next generation population.
34. The apparatus according to claim 32, wherein in said survival
selection step said arithmetic section selects the individuals to
be left as individuals in the next generation population in order
from an individual fittest to a predetermined evaluation criterion
and in proportion to a fitness value of each individual from the
population of said parent individuals and said child individual
generated.
35. The apparatus according to claim 32, wherein in said survival
selection step said arithmetic section selects an individual
satisfying at least either of one or two or more evaluation
criteria as an individual in the next generation population from
the population of said parent individuals and said child individual
generated.
36. The apparatus according to claim 32, wherein said component of
real vector of individual handled in said arithmetic section is at
least either one of a radius of curvature of a boundary surface of
said optical element, a distance between boundary surfaces, and a
refractive index of a medium placed between the boundary
surfaces.
37. A medium in which a program is recorded, said program
comprising: a parent selection step of selecting at least two real
vectors to be parents, from a population of plural individuals each
representing a candidate of an optical system to be designed, said
population being an n (.gtoreq.1) generation population and each
individual being a real vector having a component of one or two or
more predetermined parameters featuring the optical system; a child
generation step of executing at least one of a crossover step and a
mutation step, said crossover step being a step of generating, from
the inside of a partial space defined by a predetermined continuous
occurrence probability distribution of occurrence probabilities set
based on components of the respective real vectors of said parent
individuals selected, a real vector having a component of a value
occurring according to the occurrence probabilities, as a child
individual, and said mutation step being a step of generating, from
the inside of a partial space defined by a predetermined continuous
occurrence probability distribution of occurrence probabilities
increasing with approaching at least one parent individual out of
said parent individuals selected, a real vector having a component
of a value occurring according to the occurrence probabilities, as
a child individual; and a survival selection step of selecting
individuals to be left as individuals in a next generation
population from said n generation population and said child
individual generated.
38. The medium according to claim 37, wherein said program recorded
therein is arranged so that in said survival selection step said
individuals selected replace individuals not selected in said n
generation population, thereby generating the next generation
population.
39. The medium according to claim 37, wherein said program recorded
is arranged so that in said survival selection step the individuals
to be left-as individuals in the next generation population are
selected in order from an individual fittest to a predetermined
evaluation criterion and in proportion to a fitness value of each
individual from the population of said parent individuals and said
child individual generated.
40. The medium according to claim 37, wherein said program recorded
is arranged so that in said survival selection step an individual
satisfying at least either of one or two or more evaluation
criteria is selected as an individual in the next generation
population from the population of said parent individuals and said
child individuals generated.
41. The medium according to claim 37, wherein said program recorded
is arranged so that said component of real vector of individual is
at least one of a radius of curvature of a boundary surface of said
optical element, a distance between boundary surfaces, and a
refractive index of a medium placed between the boundary surfaces.
Description
BAGROUND OF THE INVENTION
[0001] 1. Field of the Invention
[0002] The present invention relates to a determining method of
movement sequence and an alignment apparatus, for example, for
reducing the time of alignment between a pattern of an original
plate and marks on a substrate in exposure apparatus, to a
designing method and apparatus of an optical system such as a
projection optical system of the exposure apparatus or a lens
system for camera, and to a medium in which a program for realizing
the designing method is recorded.
[0003] 2. Related Background Art
[0004] In general, the exposure apparatus is arranged in such a way
that, before carrying out exposure of the second layer or a layer
thereafter into chip areas (or shot areas) on a wafer
(photosensitive substrate) in which predetermined circuit patterns
are to be formed, alignment is accomplished between a pattern of an
original plate for the second or subsequent layer and the chip
areas by use of EGA (a statistical arithmetic method). The EGA
(Enhanced Global Alignment) is a technique for measuring positions
of alignment marks (measured areas) provided mainly in the
peripheral area of a plurality of selected chip areas to obtain a
residual rotation error of the wafer, linear expansion or
contraction of the wafer, an offset of the wafer, etc. and, based
thereon, aligning the all chip areas of the wafer, for example, as
disclosed in Japanese Laid-open Patent Application No. Sho
61-44429. As another technique of a further development of the EGA,
U.S. Pat. No. 4,780,617 discloses an alignment technique for
obtaining a residual rotation error of each chip area itself, an
orthogonality error of the chip areas, and linear expansion or
contraction of each chip area itself and performing alignment so as
to minimize even these errors.
[0005] Particularly, when the technique of Japanese Laid-open
Patent Application No. Hei 6-275496 is applied, because of the many
alignment marks to be measured, the measurement time will be very
long unless the alignment marks are measured as efficient as
possible. For example, let us consider an example in which there
are 76 exposed chip areas (areas indicated by number 01 to number
76 in the figure) in the first layer of the wafer W and four
alignment marks are provided for each of the chip areas, as shown
in FIG. 1. In this case, in the alignment of wafer by the EGA, the
operator first selects a plurality of chip areas that are inside
the outermost region and at vertices of polygon (for example,
twenty chip areas hatched in FIG. 2), on an empirical basis.
Coordinates of the designed center of each chip area (representing
the position of each chip area) are stored in a memory of a main
control system. Positions of four alignment marks of each chip area
(defined by coordinates of the center thereof) are also stored in
the memory of the main control system. Accordingly, the exposure
apparatus was arranged to measure the position of each alignment
mark according to the following movement sequence empirically
seeming best, by executing the EGA.
[0006] Specifically, for example, when a measuring point of an
alignment optical system is at a start point ST (x: 186.5, y:
155.5), an XY stage with a wafer mounted thereon moves so that a
right upper alignment mark of a chip area closest to the start
point ST (the chip area 64 in FIG. 2) comes to the measuring point
of the alignment optical system (so as to be in the measuring
area). After completion of the position measurement of the
alignment mark, the XY stage moves so as to measure the positions
of the four alignment marks counterclockwise. Next, the XY stage
moves so as to measure coordinates of the right upper alignment
mark of a chip area closest clockwise (the chip area 63 in FIG. 1).
After that, the XY stage moves so as to measure coordinates of the
four alignment marks counterclockwise. Repeating this operation,
the XY stage moves to measure the positions of the alignment marks
of the all chip areas selected and return the measuring point of
the alignment optical system to the end point EN (x: 215, y: 133).
Of course, such controls of movement were also employed that the XY
stage moved so as to measure the positions of the alignment marks
of each chip area clockwise and that after completion of the
position measurement of the all alignment marks of a chip area, the
XY stage moved so as to measure the alignment marks of a chip area
closest counterclockwise.
[0007] However, the movement sequence of the XY stage in the
position measurement of each alignment mark was determined
empirically as described above, and no consideration was given to
efficient movement control of the XY stage for the position
measurement of each alignment mark.
[0008] The reason is that there arises the following problem in
obtaining the movement sequence of the XY stage for the position
measurement of alignment marks using the statistical measurement
process such as the EGA. For example, where there are n alignment
marks to be measured on the wafer, the number of conceivable stage
movements for movement between alignment marks is at most
.sub.nP.sub.2=n(n-1) (even though the turnaround time differs
depending upon the positive or negative movement direction of the
stage) and computation thereof can be done quickly. Therefore, the
overall turnaround time is determined uniquely as soon as the
measurement process order is determined. However, there are n! way
as to the order for the measurement process of n alignment marks,
and the computation time becomes too long when the all possible
solutions are computed using the producing and checking method of
the all conceivable orders. Particularly, if n>13, the
computation is practically impossible ("Practical Course:
Invitation to Traveling-Salesman Problems I, II, III," Operations
Research 39 (1994), No. 1: pp 25-31, No. 2: pp 91-96, No. 3: pp
156-162). Accordingly, the conventional alignment methods did not
involve a step of finding the optimum movement sequence under
practical operation conditions.
[0009] Now, let us focus attention on the optical system such as
the projection optical system of the aforementioned exposure
apparatus. The designing of the optical system including lens
elements has been known heretofore and is known as a very difficult
issue. This is because various factors, such as multiple
dimensions, a super-multimodal property, strong dependent relation
between variables, or complex constraints, make the issue tough. In
addition, as criteria for evaluation of the optical system to be
designed there exist numerous evaluation criteria such as the
Seidel's five aberrations, the size, or the cost.
[0010] In the conventional designing method of optical system the
basic search is a local search in the neighborhood of an initial or
starting solution. If the initial solution is not appropriate, the
result will fall into a local solution, so that the search will end
unsuccessfully. It was thus the conventional practice to employ a
method for changing the initial solution in a trial and error
manner in order to find an optical system having the aimed
performance. Since the conventional search basically allowed
optimization of only one evaluation criterion, the designing
process was changed to a single-objective process by setting a
tradeoff ratio, in spite of the many evaluation criteria (Yoshiya
Matsui: Lens Designing Method, Kyoritsu shuppan (1986); Jihei
Nakagawa: Lens Design Engineering, Tokai daigaku shuppankai (1986);
Toru Kusakawa: Lens Optics, Tokai daigaku shuppankai (1988)).
[0011] It is not possible to preliminarily know the tradeoff ratio
for obtaining the optical system having the aimed performance. It
is thus the present status that loads on experts are very heavy in
the search for the initial solution for local search and in the
search for the tradeoff ratio between the evaluation criteria.
[0012] Further, the conventional method for modifying the optical
system is a method for, with data of one optical system
preliminarily given as initial data by the designer, altering
plural parameters, including radii of curvatures of boundary
surfaces in respective optical elements (lens elements, reflectors,
etc.) belonging to this optical system, distances between the
boundary surfaces, and refractive indices of spaces (the lens
elements and aerial lenses between the lens elements) located
between the boundary surfaces, using an index of increase or
decrease of a performance function indicating the performance of
the lens optical system at that time.
[0013] Then the same improving procedure is repeated using the data
of the optical system represented by the plural parameters after
the alteration, as a new solution (i.e., the optical system to be
improved). For example, if good or bad optical performance is
reflected to increase or decrease of the performance function, the
plural parameters will be altered so as to increase the performance
function and updated each to the parameters after the alteration,
as values of new parameters.
[0014] On the other hand, the genetic algorithm (GA) is known as
one of optimization techniques, which imitates the evolutionary
process of organism on an engineering basis. This genetic algorithm
(hereinafter referred to as GA) is a generate and test method,
which is characterized in that the essential point is only that
dominance can be evaluated between two solution candidates.
Therefore, it does not require the condition of differentiability
of the performance function or the like and is thus effective to
problems with complex constraints. The GA also has the feature of
performing a search using a population of plural solution
candidates and is drawing attention as a global search technique.
Further, the GA is also drawing attention as a multi-objective
optimization technique for handling the plural evaluation criteria
explicitly and finding a Pareto optimal solution set by a single
search.
[0015] For example, M. WALK AND J. NIKLAUS, "Some Remarks on
Computer-Aided Design of Optical Lens System" (JOURNAL OF
OPTIMIZATION THEORY AND APPLICATION: Vol. 59, No. 2, pp. 173-181,
NOVEMBER 1988) and X. CHEN AND K. YAMAMOTO, "Genetic algorism and
its application in lens design" (SPIE, Vol. 2863, PP. 216-221)
describe the technology of application of the above GA to the
design of optical system.
SUMMARY OF THE INVENTION
[0016] The present invention concerns a movement control of a stage
with a wafer mounted thereon for measuring positions of plural
measurement target areas (including alignment areas) provided on
the wafer in a shorter time, in alignment between a photomask or a
reticle (hereinafter referred to generally as "reticle") and each
chip area on the wafer, and more particularly, the invention
relates to a determining method of movement sequence and an
alignment apparatus to minimize the overall turnaround time of the
stage movement associated with the sequential measurement
process.
[0017] The inventors examined the conventional alignment technology
described above and found the following issues.
[0018] First, the determining method of movement sequence in the
measurement process carried out for alignment must obtain an
optimum solution or a near-optimum solution to the permutation
optimization problem within a shorter computation time. As
described above, in the case of the movement sequence of the stage,
an ideal process is to produce all possible measurement orders
(movement sequences) of alignment mark positions and to find the
shortest turnaround time (the overall movement time excluding the
measurement times) as an optimum solution out of these candidates
generated. However, when examination is made as to the order of
measurement of the wafer having n alignment marks, n! paths must be
checked only for the order of measurement of positions of the n
alignment marks on the wafer. Especially, if n>13, the
computation time will become so enormous that it is practically
impossible to obtain a solution. Therefore, in order to increase
the throughput of the EGA, it is necessary to obtain the optimum
solution or the near-optimum solution of the movement sequence of
measurement process more efficiently.
[0019] Second, the determination of movement sequence of
measurement process needs to take account of the tradeoff between
the permissible computation time for obtaining the optimum solution
or the near-optimum solution and the quality of the obtained
solution. Specifically, the computation for obtaining the optimum
solution of the movement sequence for the position measurement of
alignment marks is carried out on the occasion of exchange of
reticles of different exposure patterns or on the occasion of
exchange of wafers from the reason that positions for the
measurement process are designated arbitrarily wafer by wafer, even
though they have a common exposure pattern. Therefore, permissible
computation times vary depending upon the circumstances. For
example, the time for exchange of reticles and alignment of the
reticle (the reticle loading time) is normally 20 sec or so. The
time for loading a first wafer in a certain lot (the wafer loading
time) is normally 5 sec or so.
[0020] An object of the present invention is, therefore, to provide
a determining method of movement sequence for obtaining a solution
of a preferable movement sequence within a short time and an
alignment apparatus provided with an arithmetic unit for carrying
out the determining method. The determining method of movement
sequence according to the present invention is a method to increase
the throughput of EGA, in which a near-optimum solution of the
movement sequence is first obtained within a very short computation
time, then solutions of movement sequences to make shorter the
movement time for the measurement process of alignment mark
positions are successively produced as long as the computation time
allows, and the optimum solution of the movement sequence is
generated finally (if the sufficient, permissible computation time
is given). This can provide a solution in a quality consistent with
the permissible computation time given (the better the longer the
permissible computation time), depending upon the circumstances of
computational resources that can be used.
[0021] The determining method of movement sequence according to the
present invention is carried out prior to an alignment step of
performing alignment between a transfer position of a pattern of an
original plate (a mask or a reticle) and each chip area on a
substrate (wafer), the alignment step being a step of measuring
positions of measurement target areas while successively moving the
measurement target areas (alignment marks) on the substrate into a
measuring area of a measuring system (an alignment optical system).
The method according to the present invention determines the
movement sequence indicating the measurement order of the alignment
marks within a shorter time, thereby drastically increasing the
throughput of the EGA.
[0022] Specifically, the determining method of movement sequence
according to the present invention comprises an arithmetic step of
obtaining a solution of a most preferable movement sequence with
respect to an overall movement time between the measurement target
areas by use of a predetermined search technique. This arithmetic
step includes at least a first step of generating a group including
a plurality of executable movement sequences out of a group of
movement sequence candidates each indicating a measurement order of
plural measurement target areas, and a second step of selecting a
movement sequence that completes the movement operation between the
plural target areas within the shortest time out of the group thus
generated.
[0023] Further, the determining method of movement sequence
according to the present invention comprises a pre-step executed
prior to the above arithmetic step, the pre-step being a step of
producing a movement time management table in which for each of the
plural measurement target areas, a movement time is recorded as a
time necessary for movement of the target area of interest from a
position thereof at the time of completion of position measurement
of either one of the plural measurement target areas into the
measuring area of the measuring system. This movement time
management table also includes such information that for a pair of
measurement target areas selected out of the plurality of
measurement target areas, after completion of the position
measurement of one measurement target area selected, the other
measurement target area selected is prohibited from moving from a
position thereof at the time of completion of the position
measurement of the one measurement target area into the measuring
area of the measuring system.
[0024] Particularly, the search technique executed in the above
arithmetic step includes at least either one of a method based on
operations-research technique, a genetic algorithm (hereinafter
referred to as GA) as an evolutionary computation method, and a
combination thereof. The method based on operations-research
technique includes at least either one of a linear programming
method, a Lin and Kernighan's approach (hereinafter referred to as
LK method), and a k-OPT method.
[0025] The above GA can include as a genetic operator a search
technique like an improving method such as the above method based
on operations-research technique (the linear programming method (NN
Method), the LK method (S. Lin and B. W. Kernighan, An Effective
Heuristic Algorithm for the Traveling Salesman Problem, Operations
Research 21 (1973) pp 498-516), or the k-OPT method (including the
2-OPT method and 3-OPT method) ("Practical Course: Invitation to
the Traveling-Salesman Problems I, II, III," Operations Research 39
(1994) No. 1: pp 25-31, No. 2: pp 91-96, No. 3: pp 156-162). By
this, a solution of a most preferable movement sequence out of
solutions found at that time can be always obtained even on the way
of computation for seeking the movement sequence. If a further
computation time is given, a solution of a more preferable movement
sequence can be obtained. Therefore, according to the present
invention, a good solution of a movement sequence consistent with
the computation time permitted can be obtained even if the
computation is interrupted or even if a limitation is preliminarily
imposed on the computation time.
[0026] On the other hand, the alignment apparatus according to the
present invention comprises at least a measuring device (measuring
system) for measuring each of positions of plural alignment marks,
a moving device for effecting relative movement between the plural
alignment marks and a measuring area of the measuring device, an
arithmetic section for carrying out the above-stated determining
method of movement sequence, and a control section for controlling
the moving device so as to move the plural alignment marks
successively into the measuring area of the measuring device,
according to a solution of a movement sequence obtained by the
arithmetic section.
[0027] Further, the alignment apparatus according to the present
invention comprises a memory for storing a movement time management
table in which for each of the plural alignment marks a movement
time is recorded as a time necessary for movement of the alignment
mark of interest from a position thereof at the time of completion
of position measurement of either one of the plural alignment marks
into the measuring area of the measuring device. The movement time
management table stored in this memory also includes such
information that for a pair of alignment marks selected out of the
plural alignment marks, after completion of the position
measurement of one alignment mark selected, the other alignment
mark selected is prohibited from moving from a position thereof at
the time of completion of the position measurement of the one
alignment mark into the measuring area of the measuring device.
[0028] The GA applied as the above search technique is used, for
example, as an approach for the case to find the shortest-length
path in visiting each city only once and all cities, but there has
been and is no example of application to optimization of the
movement sequence for the position measurement of alignment marks
in the steppers or the like.
[0029] As described above, in the present invention, the
computation of optimization (how efficiently the XY stage is moved)
of the movement sequence for the position measurement by the search
technique (the linear programming method, the LK method, the k-OPT
method, or the GA) is carried out prior to the measurement of the
positions of plural alignment marks. Based on the computation
result, the alignment apparatus according to the present invention
operates to move the XY stage so as to bring each of the alignment
marks into the measuring area of the alignment optical system. Then
the position of each alignment mark is measured by the alignment
optical system.
[0030] Since the movement sequence for alignment was determined
empirically before, it was not always the movement sequence for
good or optimal alignment. In contrast, the present invention
allows us to find a good or optimal movement sequence and can thus
increase the throughput of EGA.
[0031] Meanwhile, in the case of the conventional designing method
and modifying method of optical system described previously, once
the performance function reaches a locally maximal value, no
further improvement is made even by execution of further improving
procedures. An optical system associated with the parameters
obtained under such circumstances is a locally optimal solution or
a locally optimal optical system and a possibility that it is a
true optimal solution or a globally optimal solution, is very low.
Accordingly, the performance of the optical system as a solution
achieved by the conventional designing method and modifying method
of optical system is dependent on the optical system first given as
an initial solution by the designer. This means that a possibility
that the optimal solution will not come out is extremely high if
the modality of the optical system of the optimal solution is
greatly different from that of the optical system first given as an
initial solution by the designer.
[0032] In the conventional technology for designing the optical
system by use of the genetic algorithm, since the genetic operator
used in the GA was able to handle only discrete numbers, the plural
parameters featuring the optical system, which were originally
continuous variables, needed to be digitized as discrete values and
it was thus hard to obtain a solution with sufficient accuracy
within a practical time.
[0033] Specifically, the conventional technology for executing the
GA in a discrete fashion often utilizes the coding/crossover
procedures as shown in FIG. 3 to FIG. 5, in which each of
parameters constituting a parent individual or a child individual
is expressed by a binary code and in which the binary codes are
used in combination with one point crossover, two point crossover,
or uniform crossover. FIG. 3 is a drawing to show a step of
generating one set of parameters of child individuals from one set
of parameters of parent individuals expressed by binary codes, by
the one point crossover, FIG. 4 is a drawing to show a step of
generating one set of parameters of child individuals from one set
of parameters of parent individuals expressed by binary codes, by
the two point crossover, and FIG. 5 is a drawing to show a step of
generating one set of parameters of child individuals from one set
of parameters of parent individuals expressed by binary codes, by
the uniform crossover.
[0034] In the discrete GA as described above, the phase structure
(binary codes) of a genotype space is greatly different from that
(actual values) of a phenotype space. This means that a character
(property) of a parent individual becomes unlikely to be inherited
by a child individual newly generated. Therefore, even if parent
individuals of a certain generation approach the optimal solution,
child individuals of the next generation could deviate from the
optimal solution with an extremely high probability to be farther
therefrom than their parent individuals. This raises the problem
that wasteful searches increase considerably.
[0035] Namely, for achieving sufficient accuracy in the discrete
GA, it is necessary to make the degree of digitization of each
parameter finer. This inevitably increases the gene information and
causes considerable increase of computation time. Conversely, if
each parameter is roughly digitized in order to obtain a solution
within a practical computation time, there will arise a risk of
failing to find the optimal solution where the optimal solution
exists in a narrowest gap between discrete numbers.
[0036] When consideration is given to the evaluation criteria of
the optical system designed, there are many evaluation criteria
contradicting with each other in the actual performance aspect (for
example, resolution and distortion). In either one of the
conventional technology described above, a plurality of evaluation
criteria are integrally expressed in the form of a performance
function by preliminarily weighting the reciprocity among the
evaluation criteria uniquely.
[0037] However, whether the preset weighting is valid or not can be
first checked after various optimal solutions have been obtained
with changing weights of the respective evaluation criteria. A
solution adapted for each evaluation criterion is generally called
a Pareto optimal solution, and there are plural Pareto optimal
solutions corresponding to various weightings.
[0038] Therefore, where there exist plural evaluation criteria of
the optical system designed, multi-objective optimization needs to
be effected so as to fit each evaluation criterion. In the case of
the conventional designing method and modifying method of optical
system, the resultant solution cannot be always determined to be
the optimal solution.
[0039] The present invention has been accomplished to solve the
above problem. A principal object of the present invention is to
provide a designing method and apparatus of optical system for
obtaining an optical system as a globally optimal solution within a
practical time, independent of initial data preliminarily given,
and a medium in which a program for realizing the designing method
is recorded. Specifically, the present invention relates to a
designing method, apparatus, and so on for designing an optical
system including at least one optical element (lens element,
reflector, etc.), for example, such as a lens for photography, a
lens for microscope, or a projection optical system in the
projection exposure apparatus of the one-shot exposure method or
the scanning exposure method (e.g., a step-and-repeat type stepper
or a step-and-scan type stepper), and also enables optimal
relocation (i.e., correction for locations) where the lens elements
belonging to the optical system to be designed are not processed
exactly in designed values, and optimization of combination of lens
elements where a plurality of optical systems including such lens
elements not processed in the designed values are produced.
[0040] A designing method of optical system according to the
present invention is a method for designing an optical system
including at least one optical element (a lens element, a
reflector, or the like) by use of an evolutionary computation
method (genetic algorithm), which is characterized in that the
optical system is optimized by use of a genetic operator for
directly handling continuous values. Specifically, the designing
method of optical system comprises at least a selection step of
selecting at least two parent individuals from a generation
population of n (.gtoreq.1, wherein n=1 indicates an initial
population) having a plurality of individuals of data of the
optical system to be designed, a child generation step of newly
generating a population consisting of a plurality of child
individuals by applying a genetic operator, being at least either
one of a crossover operator and a mutation operator, to the parent
individuals selected, and a survival selection step of generating a
next generation population.
[0041] Particularly, an individual that is a candidate for the
optical system to be designed is given as a real vector having
components, for example, including curvatures of boundary surfaces
specifying the lens element, the reflector, or the like included in
the optical system to be designed, a distance between the boundary
surfaces, and a refractive index of a space located between the
boundary surfaces, whereby mapping or coding is effected from the
data of the optical system of the individual (phenotype) to the
genotype. The optimization of the optical system does not have to
be made for the all components of the real vector, but can be done
individually for a specific vector component (at least one of the
plural parameters featuring the optical system to be designed).
[0042] In the designing method of optical system according to the
present invention, the above child generation step is the step of
generating the child individuals by at least either one of the
crossover operator (a crossover step) and the mutation operator (a
mutation step). Namely, from the inside of a partial space defined
by a predetermined continuous occurrence probability distribution
of occurrence probabilities set based on components of real vectors
of the parent individuals selected, the crossover operator
generates as a child individual a real vector having a value
occurring according to the occurrence probabilities. From the
inside of a partial space defined by a predetermined continuous
occurrence probability distribution of occurrence probabilities
increasing with approaching at least one parent individual out of
the parent individuals selected, the mutation operator generates as
a child individual a real vector having a component of a value
occurring according to the occurrence probabilities.
[0043] In a search process in which the above selection step, child
generation step, and survival selection step are executed
repetitively, the child individuals newly generated from the parent
individuals selected succeed to characters (properties) of their
parent individuals, thereby avoiding wasteful searches. In the
initial stage of the search process, the parent individuals are
apart from each other and the individuals are also scattered in
various spaces, depending upon distances between the parent
individuals. The search is thus done in a wide range. With progress
in searches (with increase in the number of executions of the above
steps), the distances between the parent individuals become
shorter, so that more child individuals will be generated in a
partial space including the optimal solution, depending upon the
distances between the parent individuals.
[0044] The above crossover operator is a genetic operator that
directly handles continuous values, which can be selected, for
example, from UNDX (Ono, I. and Kobayashi, S: A Real-coded Genetic
Algorithm for Function Optimization Using Unimodal Normal
Distribution Crossover, Proceeding of 7th International Conference
on Genetic Algorithms, pp. 246-253 (1997)), BLX-.alpha. (N. J.
Radcliffe: Formal Analysis and Random Respectful Recombination,
Proceeding of the Fourth International Conference on Genetic
Algorithms, pp. 222-229, 1991); NDX (I. Ono, M. Yamamura and S.
Kobayashi: A Genetic Algorithm with Characteristic Preservation for
Function Optimization, Proceedings of IIZUKA '96, pp. 511-514,
1996); or UNDX (Ono, I. and Kobayashi, S: A Real-coded Genetic
Algorithm for Function Optimization Using Unimodal Normal
Distribution Crossover, Proceeding of 7th International Conference
on Genetic Algorithms, pp. 246-253 (1997)).
[0045] Further, the above survival selection step is to select
individuals to be left as individuals in the next generation
population out of the n-th generation population including the
parent individuals and the population of the child individuals
newly generated.
[0046] In the above survival selection step, in order to realize
the multi-objective optimization, individuals satisfying at least
either criterion out of one or two or more evaluation criteria are
selected from the population including the child individuals
generated. This selection of individuals is carried out preferably
so as to select individuals to be left as individuals in the next
generation in order in proportion to fitness values of the
respective individuals from the fittest to a predetermined
evaluation criterion.
[0047] Further, in the designing method of optical system according
to the present invention, the above survival selection step is
preferably a step of generating the next generation population by
replacing individuals not selected in the n-generation population
with the selected individuals.
[0048] This designing method of optical system according to the
present invention can be realized in the form of a program
described in a predetermined language, and this program is carried
out, for example, by a computer having basic components of an
arithmetic circuit with memories (RAM, ROM), input/output sections,
an arithmetic section, a memory section, and a control section.
Particularly, when carried out by the computer, the program for
realizing the designing method of optical system should be
preferably optically or magnetically recorded in a predetermined
recording medium such as CD, MO, FD, a hard disk, a magnetic tape,
or ROM.
[0049] According to the above constitution, the present invention
can automatically generate an optical system as being a globally
optimal solution independent of the initial solution given. Namely,
the invention makes it possible to find an optimal solution or a
near-optimal solution of plural parameters (components of a real
vector) featuring the optimal optical system within a practical
time from the initial solution arbitrarily given.
[0050] In the conventional designing method and modifying method of
optical system, it is also conceivable to employ a method for
obtaining a plurality of local optimal solutions by repeating such
an operation that once a search falls in a local optimal solution
and if little change appears in increase or decrease of the
performance function with repetition of improving procedure, the
local optimal solution of the optical system obtained at this time
is once stored as a solution candidate on the memory and that the
improving procedure is carried out again for a new optical system
to be improved, obtained by arbitrarily changing one or more
parameters out of the plural parameters of the solution, thereby
obtaining another local optimal solution. However, as the optical
system becomes more complex with increase in the number of optical
elements, the number of local optimal solutions becomes so large in
general. It is, therefore, difficult to obtain the globally optimal
solution by this method.
[0051] Since plural optical systems as multi-objective optimal
solutions, which must exist in correspondence to a plurality of
conflicting evaluation criteria, can be selected simultaneously in
the above survival selection step, when a plurality of evaluation
criteria for evaluation of optical system are given, a plurality of
Pareto optimal solutions or a plurality of Pareto near-optimal
solutions can be obtained within a practical time by performing the
multi-objective optimization for simultaneously handling the plural
evaluation criteria.
[0052] Further, according to the present invention, an optimal
solution or a near solution to the optimal solution of the
parameters featuring the optimal optical system can be obtained
within a practical computation time even in a range satisfying
constraints on (plural) specific parameters arbitrarily designated
by the user. For example, even in the case where the constraints
given are that the first surface of the first lens must be convex
and that a refractive index of a glass material of the second lens
must be x (e.g., 1.5266), y (e.g., 1.6010), or z (e.g., 1.7294),
the multi-objective optimization can be done within the range
satisfying such constraints.
BRIEF DESCRIPTION OF THE DRAWINGS
[0053] FIG. 1 is a drawing to show a solution of a movement
sequence (the first example) obtained by the conventional approach
based on the rule of thumb;
[0054] FIG. 2 is a drawing to show a solution of a movement
sequence (the second example) obtained by the conventional approach
based on the rule of thumb;
[0055] FIG. 3 to FIG. 5 are drawings for explaining the
coding/crossover procedures in the conventional designing method of
optical system using the discrete GA;
[0056] FIG. 6 is a drawing to show the overall structure of the
projection-exposure apparatus (stepper) including the alignment
apparatus provided with the arithmetic section for realizing the
determining method of movement sequence according to the present
invention;
[0057] FIG. 7 is a flowchart for explaining the measurement process
of alignment mark position for the EGA arithmetic;
[0058] FIG. 8 is a table to show the performance of the wafer stage
etc.;
[0059] FIG. 9 is a table to show chip areas positions of which are
to be measured, and coordinates of their centers (representing the
positions of the chip areas);
[0060] FIG. 10 is a table to show a movement time management
table;
[0061] FIG. 11 is a drawing to show locations of chip areas and
alignment marks provided on a wafer;
[0062] FIG. 12 is a drawing to show a first arrangement example of
alignment marks in a chip area;
[0063] FIG. 13 is a drawing to show a second arrangement example of
alignment marks in a chip area;
[0064] FIG. 14 is a diagram to show an example of flowchart for
explaining an example of the genetic algorithm applied as a search
technique of movement sequence;
[0065] FIG. 15 is a diagram to show an example of flowchart for
explaining another example of the genetic algorithm applied as a
search technique of movement sequence;
[0066] FIG. 16 is a drawing for explaining a crossover operator in
the genetic algorithm;
[0067] FIG. 17 is a drawing for explaining a mutation operator
(cyclic shift) in the genetic algorithm;
[0068] FIG. 18 is a drawing to show a solution of a movement
sequence obtained by the determining method of movement sequence
according to the present invention;
[0069] FIG. 19 is a drawing to show a solution of a movement
sequence obtained by the determining method of movement sequence
according to the present invention, to which the linear programming
method (NN method) is applied;
[0070] FIG. 20 is a flowchart for explaining the determining method
of movement sequence according to the present invention, to which
the NN method is applied;
[0071] FIG. 21 is a drawing to show a solution of a movement
sequence obtained by the determining method of movement sequence
according to the present invention, to which the LK method
utilizing initial solutions obtained by the NN method is
applied;
[0072] FIG. 22 is a flowchart for explaining the determining method
of movement sequence according to the present invention, to which
the LK method is applied;
[0073] FIG. 23 is a drawing to show the relation between the
computation time of the approach based on the rule of thumb, the NN
method, and the LK method, and the overall movement time of the
movement sequences obtained thereby;
[0074] FIG. 24 is a drawing to show the relation between the
computation time of the GA using randomly generated feasible base
solutions as an initial group and the overall movement time of the
movement sequence obtained;
[0075] FIG. 25 is a drawing to show the relation between the
computation time of the GA using plural solutions obtained by the
NN method + the LK method as an initial group and the overall
movement time of the movement sequence obtained thereby;
[0076] FIG. 26 is a drawing to show a designing apparatus of
optical system according to the present invention and a medium in
which a program for realizing the designing method of optical
system according to the present invention is recorded;
[0077] FIG. 27 is a drawing for explaining the crossover operator
in the designing method of optical system according to the present
invention;
[0078] FIG. 28 is a drawing for explaining a generation alteration
model in the designing method of optical system (the first
embodiment) according to the present invention;
[0079] FIG. 29 is a drawing for explaining an example of the
initial population in the designing method of optical system (the
first embodiment) according to the present invention;
[0080] FIG. 30 is a drawing for explaining another example of the
initial population in the designing method of optical system (the
first embodiment) according to the present invention;
[0081] FIG. 31 is a path diagram of an optical system (a lens
system) obtained in Experiment 1 of the designing method of optical
system (the first embodiment) according to the present
invention;
[0082] FIG. 32 is an aberration diagram of the lens system shown in
FIG. 31;
[0083] FIG. 33 is a drawing to show path diagrams and spot diagrams
of lens systems (focal length 50 mm, brightness F 2.0, field angle
46.degree.) obtained in Experiment 2 of the designing method of
optical system (the first embodiment) according to the present
invention;
[0084] FIG. 34 is a drawing to show path diagrams and spot diagrams
of lens systems (focal length 135 mm, brightness F 2.8, field angle
18.2.degree.) obtained in Experiment 2 of the designing method of
optical system (the first embodiment) according to the present
invention;
[0085] FIG. 35 is a drawing to show path diagrams and spot diagrams
of lens systems (focal length 20 mm, brightness F 5.6, field angle
92.degree.) obtained in Experiment 2 of the designing method of
optical system (the first embodiment) according to the present
invention;
[0086] FIG. 36 is a drawing for explaining a generation alteration
model in the designing method of optical system (the second
embodiment) according to the present invention;
[0087] FIG. 37 is a drawing to show a Pareto solution set obtained
by the designing method of optical system (the second embodiment)
according to the present invention;
[0088] FIG. 38 is a drawing to show a state in which the optimal
solution obtained in Experiment 1 of the designing method of
optical system (the first embodiment) according to the present
invention is plotted on FIG. 37;
[0089] FIG. 39 is a drawing for specifically explaining parameters
in the designing method of optical system (the third embodiment)
according to the present invention;
[0090] FIG. 40 is a drawing to show a converging state of various
beams on the image plane, where the spot diagram is used, as an
example for evaluating the photographic lens of FIG. 39;
[0091] FIG. 41 is a drawing to show an example (a real vector) of
gene representation representing a plurality of continuous
parameters for a lens optical system in the form of continuous
values, where in the case of FIG. 39 the field angle, focal length,
and refractive indices of glass materials for the respective lenses
are preliminarily given and they are constraints;
[0092] FIG. 42 is a flowchart for explaining the designing method
of optical system (the fourth embodiment of the designing method of
optical system according to the present invention);
[0093] FIG. 43 is a drawing to schematically show operations in the
designing method of optical system (the fourth embodiment)
according to the present invention;
[0094] FIG. 44 is a drawing for explaining the crossover operator
where the UNDX is applied;
[0095] FIG. 45 is a flowchart for explaining the operation where
the multi-objective optimization is effected in the designing
method of optical system of the fourth embodiment according to the
present invention;
[0096] FIG. 46 is a drawing to schematically show operations where
the multi-objective optimization is effected in the designing
method of optical system of the fourth embodiment according to the
present invention;
[0097] FIG. 47 is a path diagram of a projection optical system
(No. 1) generated by the designing method of optical system (the
fourth embodiment) according to the present invention;
[0098] FIG. 48 is an aberration diagram of the projection optical
system shown in FIG. 47; and
[0099] FIG. 49 is a path diagram of a projection optical system
(No. 2) generated by the designing method of optical system (the
fourth embodiment) according to the present invention.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0100] The embodiments of the determining method of movement
sequence and alignment apparatus according to the present invention
will be described using FIG. 6 to FIG. 25. The embodiments are
examples to find the optimum movement sequence indicating the order
of measurement of alignment mark positions in the EGA (Enhanced
Global Alignment: aligning by the statistical arithmetic method) in
the step-and-repeat type (one-shot type) projection exposure
apparatus. It should be noted that the present invention can also
be applied to the step-and-scan type exposure apparatus for
synchronously scanning the reticle relative to the wafer.
[0101] FIG. 6 shows the structure of a projection exposure
apparatus (stepper). In this FIG. 6 the reticle R is illuminated
under almost uniform illuminance by exposure light IL emitted from
illumination optical system 1. The reticle R is held on reticle
stage 3 and the reticle stage 3 is supported so as to be movable
and finely rotatable in a two-dimensional plane on base 4. Main
control system 6 for controlling the operation of the whole
apparatus controls the operation of the reticle stage 3 through
driving device 5 provided on the base 4.
[0102] The above main control system 6 has an arithmetic section 61
for carrying out the determining method of movement sequence
according to the present invention and a memory 62 in which a
movement time management table (see FIG. 10) is stored. This main
control system 6 receives instructions from the operator or the
like through input device 63 such as a keyboard or a pointing
device.
[0103] An image of a pattern of the reticle R illuminated by the
exposure light IL is projected through a projection optical system
PL onto each chip area on the wafer W. The wafer W is mounted on
wafer stage 10 through wafer holder 9. The wafer stage 10 is
composed of an XY stage for two-dimensionally aligning the wafer W
on a plane normal to the optic axis of the projection optical
system PL, a Z-stage for aligning the wafer W in a direction
(Z-direction) parallel to the optic axis of the projection optical
system PL, a stage for finely rotating the wafer W, and other
components.
[0104] A moving mirror 11 is fixed on the top surface of the wafer
stage 10 and a laser interferometer 12 is positioned opposite to
the moving mirror 11. Although the moving mirror 11 is illustrated
in a simplified form in FIG. 6, where the orthogonal coordinate
system is defined with the X-axis and Y-axis in the plane normal to
the optic axis of the projection optical system PL, the moving
mirror 11 is composed of a plane mirror having a reflective surface
normal to the X-axis and a plane mirror having a reflective surface
normal to the Y-axis. The laser interferometer 12 is composed of
two X-axis laser interferometers for emitting a laser beam to the
moving mirror 11 along the X-axis and a Y-axis laser interferometer
for emitting a laser beam to the moving mirror 11 along the Y-axis,
and the one X-axis laser interferometer and one Y-axis laser
interferometer measure X-coordinate and Y-coordinate of the wafer
stage 10.
[0105] An angle of rotation of the wafer stage 10 is measured from
a difference between measured values by the two X-axis laser
interferometers. The information about the X-coordinate,
Y-coordinate, and angle of rotation measured by the laser
interferometer 12 is supplied to position measuring circuit 12a and
to the main control system 6. The main control system 6 controls
the alignment operation of the wafer stage 10 through driving
device (linear motor M) 13 while monitoring the coordinate data
supplied. The same interferometer system as on the wafer side is
also provided on the reticle side, though not illustrated in FIG.
6.
[0106] Further, the projection optical system PL of FIG. 6 is
equipped with imaging property controlling device 14. The imaging
property controlling device 14 controls the projection
magnification and distortion of the projection optical system PL,
for example, by adjusting a space between predetermined lenses out
of a group of lenses composing the projection optical system PL or
by adjusting the pressure inside the lens chamber between
predetermined lenses. The main control system 6 also controls the
operation of the imaging property controlling device 14.
[0107] This embodiment uses an off-axis alignment optical system of
an image processing type as a method for obtaining a position of an
alignment mark (coordinates of the center thereof). However, the
position of the alignment mark can also be obtained by receiving
diffracted light by the alignment optical system of the TTL
(through the lens) type.
[0108] The off-axis alignment optical system (device) 15 is placed
beside the projection optical system PL of FIG. 6. This alignment
system 15 forms an image of a cross alignment mark AM on an image
pickup surface of an X-axis image pickup device composed of a
two-dimensional CCD for the X-axis and also forms the image on an
image pickup surface of a Y-axis image pickup device composed of a
two-dimensional CCD for the Y-axis. The image of the alignment mark
AM and an image of an index mark provided on an index screen (not
illustrated) are superimposed on each of the image pickup surfaces
of the image pickup devices. Image signals of the image pickup
devices are supplied to the position measuring circuit 12a, which
calculates amounts of positional deviation in the X-axis direction
and in the Y-axis direction between the image of the alignment mark
AM and the image of the index mark.
[0109] Accordingly, in FIG. 6, the position measuring circuit 12a
obtains coordinates of the alignment mark AM on the stage
coordinate system (X, Y) from the positional relation between the
image of the alignment mark AM on the wafer W and the index mark on
the index screen and the measurement results of the laser
interferometer 12 in that case, and supplies the coordinate data
thus measured to the main control system 6.
[0110] FIG. 7 is a flowchart for explaining the position
measurement of alignment mark for the EGA arithmetic being the
statistical arithmetic method. The movement control of the XY stage
for the position measurement of alignment mark will be described
along this flowchart.
[0111] First, the operator selects a plurality of chip areas for
the EGA measurement from the all chip areas in the wafer W and the
data is supplied to the main control system 6 (step ST101). The
operator is allowed to select the all chip areas.
[0112] Then the operator determines which or how many marks should
be measured out of the plural alignment marks AM mainly provided in
the peripheral region of each chip area, and the information is
supplied to the main control system 6 (step ST103). In this
embodiment there are four cross alignment marks AM (which permit
two-dimensional coordinate measurement) positioned per each chip
area.
[0113] The present invention permits use of an arrangement in which
two alignment marks AM are positioned per each chip area or an
arrangement of a linear alignment mark AM (permitting
one-dimensional coordinate measurement) for each chip area. It is
necessary to obtain coordinates of ten alignment marks AM on a
one-dimensional basis, in order to obtain (1) residual rotation
error .THETA. of the wafer, (2) orthogonality error W of the stage
coordinate system (or shot arrangement), (3) linear expansion or
contraction Rx, Ry of the wafer, (4) offset (translation) OX, OY of
the wafer (the center position), (5) residual rotation error
.theta. of the circuit pattern (chip pattern) on each shot area of
the wafer, (6) orthogonality error w of the coordinate system (chip
pattern) on the wafer, and (7) linear expansion or contraction rx,
ry in two orthogonal directions of the chip pattern (see Japanese
Laid-open Patent Application No. Hei 6-275496).
[0114] Next, the main control system 6 sends a command to a loader
(not illustrated) to make the loader mount (or load) the wafer W to
be exposed on the wafer holder 9 (step ST105). In this case, the XY
stage 10 moves to a predetermined mount position for receiving the
wafer W.
[0115] After the wafer W is mounted on the wafer holder 9 to be
sucked by vacuum, the main control system 6 moves the XY stage 10
so as to locate the measuring point (representing the measuring
area) of the alignment optical system 15 at the start point ST, for
carrying out the EGA (step ST107).
[0116] In the main control system 6, almost at the same time as the
arithmetic section 61 sends the load command of wafer W described
above, the arithmetic section 61 starts finding a sequence of a
short time indicating a measurement order of the chip areas and
alignment marks AM given in steps ST101 and ST103 by a search
technique (a linear programming method, a Lin and Kernighan's
approach, a K-Opt method, or a genetic algorithm) (step ST 109).
The arithmetic executed in this arithmetic section 61 will be
described hereinafter. Designed center coordinates of the
respective chip areas and designed coordinates of the respective
alignment marks AM are preliminarily recorded in the memory 63 in
the main control system 6.
[0117] Next, in step ST111, the main control system 6 moves the XY
stage according to the measurement order of the alignment marks AM
obtained by the genetic algorithm in the arithmetic section 61. A
position of each alignment mark AM is measured by the alignment
optical system, whereby an amount of positional deviation thereof
is measured relative to the index mark. At this time values of the
laser interferometer 12 are also read, whereby coordinates of the
alignment mark AM are obtained on the stage coordinate system (X,
Y) by the position measuring circuit 12a. The coordinate values
measured are supplied to the main control system 6.
[0118] At the time of completion of the measurement of positions of
the all alignment marks as being measurement targets, the main
control system 6 moves the XY stage so as to move the measuring
point of the alignment optical system 15 to the end point EN (step
ST113).
[0119] The arithmetic section of the main control system 6 further
performs the EGA arithmetic, based on the information about the
positions of the plural alignment marks measured, thereby obtaining
the error parameters, expansion or contraction of the wafer,
rotation of each chip area, etc. (step ST115).
[0120] Then the main control system 6 moves the XY stage 10 to
coordinates of each chip area to be exposed, according to the
computation result of EGA or compensates for expansion or
contraction of chip area or the like by the imaging property
controlling device 14, and then performs exposure (step ST117).
[0121] Next described is the optimization of movement sequence for
the position measurement of alignment marks. A path (order) of the
position measurement of alignment marks is obtained in the
arithmetic section 61 in the main control system 6, and
preconditions necessary for the arithmetic will be described
first.
[0122] FIG. 8 to FIG. 13 are drawings to illustrate the
preconditions necessary for obtaining the movement sequence for the
position measurement of alignment marks, in order to compare the
effect of the present invention with the conventional technology.
Of course, this is just an example, and it should be noted that the
conditions including the size of chip area, the number of chips,
the positions of alignment marks, the number of alignment marks,
and so on can be changed freely.
[0123] FIG. 8 shows data of the stepper used in the present
embodiment, including the start point ST and end point EN of the
wafer stage 10.
[0124] Accelerations and maximum speeds of movement of the wafer
stage 10 are as shown in FIG. 8, because the time of movement in
the X-direction is different from that in the Y-direction because
of the difference between the weights of the Y-stage, mechanisms,
and so on.
[0125] In this embodiment, the time necessary for measurement of a
position of one alignment mark is set to 0.5 sec, as shown in FIG.
8.
[0126] FIG. 9 shows coordinates of centers of twenty chips (X, Y in
units of mm) denoted by numerals 12, 13, 14, 15, 21, 22, 25, 26,
31, 36, 41, 46, 51, 52, 55, 56, 62, 63, 64, 65 in the coordinate
system on the wafer, and the twenty chip areas are those measured
in this embodiment.
[0127] FIG. 10 is a drawing to show the movement time management
table preliminarily stored in the memory 62. From this movement
time management table, moving times between alignment marks are
found where four alignment marks are provided for each of the
twenty chip areas being measurement targets. For example, in this
movement time management table, the time indicated by F in the
drawing represents a time necessary for the third alignment mark in
the chip area 12 on the wafer W to move from a position thereof at
the time of completion of the measurement of the position of the
second alignment mark of the chip area 65 on the wafer W to the
measuring point of the alignment optical system 15. This movement
time management table also includes movement prohibition
information for indicating prohibited movement. This movement
prohibition information is indicated by symbol "X" in the
drawing.
[0128] FIG. 11 is a drawing to show an arrangement of the chip
areas on the wafer. Particularly, the chip areas indicated by
hatching are those where the position measurement of the alignment
marks specified in FIG. 9 is carried out. The size of each chip
area is 22 mm (in the X-direction).times.22 mm (in the
Y-direction). Distances between centers of chip areas are 22 mm in
the X-direction and 22 mm in the Y-direction.
[0129] Next, FIG. 12 shows cross alignment marks AM disposed in
each chip area to be a measured object. In this embodiment there
are four alignment marks AM1 to AM4 in each chip area as a measured
object. When the center coordinates A0 of the chip area are defined
by (0, 0), the position measurement of the alignment marks in the
chip area is set to be carried out in the order of AM1 (10, 10),
AM2 (10, -10), AM3 (-10, -10), and AM4 (-10, 10). Of course, this
is just an example, and the alignment marks AM 1' to AM4' can be
positioned as shown in FIG. 13 on the chip area.
[0130] The position measurement of the alignment marks is started
after the wafer stage 10 has been moved so that the optic axis of
the alignment optical system 15 (or the measuring point on the
wafer W by the alignment optical system 15) was aligned with the
start point ST (X=186.5, Y=155.5). The position measurement is
terminated at the time of arrival of the optic axis of the
alignment optical system 15 (the measuring point on the wafer) at
the end point EN (X=215, Y=133).
[0131] Since the measurement time at each alignment mark AM is
constant, the optimization of movement sequence means eliminating
waste movement time from the movement sequence as much as possible.
Let us consider a time from a point of completion of the position
measurement at an arbitrary alignment mark AM (a start point of a
unit movement sequence) to a point of start of the measurement
process at another arbitrary alignment mark AM (an end point of the
unit movement sequence). In the case where there are n alignment
marks AM, the number of cases of arbitrary unit movement sequences
is at most n.times.(n-1). Since the shortest movement times of the
respective unit movement sequences are uniquely determined from the
aforementioned preconditions, these are computed first and the
movement time-management table obtained (see FIG. 10) is stored in
the memory 62. After completion of the operation up to this point,
the optimum movement sequence can be obtained by optimization of
selection and order to determine which unit movement sequences
should be selected out of the unit movement sequences and used in
what order (in the sense of minimizing the final measurement
process sequence of coordinates of alignment marks AM).
Accordingly, to minimize the overall movement time of the movement
sequence means to optimize the movement sequence for the position
measurement of alignment marks AM, which is an object of the
present invention. This method will be described below.
[0132] This embodiment is arranged to automatically obtain the
optimum solution of the order (movement sequence) of position
measurement of alignment marks (a solution of a movement sequence
to minimize the overall movement time), using the genetic algorithm
(GA: Genetic Algorithm), which is a typical example of the
evolutionary computation method. FIG. 14 and FIG. 15 illustrate
examples of the operation of the genetic algorithm.
[0133] The name of the GA was given, because individual operations
thereof could be compared to genes, as will be described below.
[0134] An initial solution in the GA is arbitrary. Accordingly, the
genetic algorithm can be readily combined with another search
method and it is also easy to incorporate only an advantage of
another search method into a genetic operator. Therefore, proper
analysis of problem and programmer's experiences in evolutionary
computation are required to achieve an efficient approach with high
degrees of freedom of design.
[0135] The GA of this embodiment can be classified roughly into GA
as a combinatorial optimization approach placing the main point on
crossover, ES (Evolutional Strategy) as a continuous value
optimization approach placing the main point on mutation, and,
different in classification from these, GP (Genetic Programming)
directed to the source and process procedures of program, but the
essence of the all algorithms is identical.
[0136] In the GA, a group search is carried out with plural (N)
agents called "genes," and the GA thus has such a property that
even if a part of the group falls into a local optimum solution but
if the other gene finds a better solution the search is led to the
better solution. Since it is a multipoint search, it takes some
time, but can efficiently search the optimum solution. A plurality
of pairs (parents) are selected from the gene group consisting of
the N genes and each of the gene pairs P bears children C
resembling their parents. Further, some children C experience
mutation in part of gene. Among these genes, descendents having
genes with higher evaluation are made to survive with higher
probability in the next generation. Since the size of the group is
normally fixed to N throughout generations except for special
cases, descendents having genes with lower evaluation will
gradually become extinct. Repetition of such alteration of
generation will find plural (N) genes with the optimum solution
appearing in the group sooner or later.
[0137] This embodiment adopts the Subtour Exchange Crossover (SXX)
famous as an approach by GA in order to improve the movement
sequence. In this way the optimization problem of movement sequence
for the position measurement of alignment marks in the stepper is
formulated by the GA. Examples of models for the alteration of
generation used in the GA include the MGG (Minimum Generation Gap)
model in which the all genes are paired without any unmarried
person and best two genes out of each family including a pair and
children born are left to the next generation (H. Satoh, M.
Yamamura and S. Kobayashi, Minimum Generation Gap Model for GAs
Considering Both Exploration and Exploitation, Proceedings of
IIZUKA '96, pp. 494-497), and the Elitist model in which good
people out of the all parents and children are left preferentially
(D. E. Goldberg, Genetic Algorithm in Search, Optimization and
Machine Learning, Addison-Wesley Publishing Company Inc., 1989).
The MGG model is used if the true optimum solution is desired to
obtain finally even with some computation time; whereas the Elitist
model is used if a good solution can be found in the early stage of
the computation process and even if a computation time longer than
that in the MGG model is necessary for obtaining the true optimum
solution. Thus, they can be used selectively based on how to make
the trade-off between the computation time and the quality of
solution. In this embodiment, an example using the Elitist model
will be described as an example of such selective use.
[0138] Before describing the GA shown in the flowcharts of FIG. 14
and FIG. 15 in detail, each of operators shown in the algorithms
will be described first.
[0139] In FIG. 16 and FIG. 17 P1, P2, C1, C2, etc. represent
measurement orders of alignment mark positions, imitating the
genes. One unit is defined as a position of one measurement of an
alignment mark. One gene is expressed by a string of n units
corresponding to n alignment marks. Since in this embodiment the
number of chip areas position-measured is 20 and the number of
alignment marks in each chip area is 4, one gene is composed of
20.times.4=80 units. For simplification of description, the
description will be given as to an example in which eight alignment
marks are measured and denoted by A to H. Namely, each operator
will be explained using an example in which one gene is composed of
eight units.
[0140] FIG. 16 is a drawing to illustrate a crossover operator for
subtour exchange crossover, which is one of the genetic operators.
The concept of the crossover operator is to make from two parents
two or more children having characteristics of the both parents.
The subtour exchange crossover operator is applied to a pair of
genes P1 and P2 randomly selected out of plural genes. At this time
the crossover operator searches portions having a common partial
set of alignment marks in the genes P1 and P2. The third unit to
the sixth unit of the gene P1 are a set of alignment marks C, D, E,
and F. Further, the fourth unit to the seventh unit of the gene P2
are also a set of alignment marks C, D, E, and F.
[0141] These common portions (subtours) are exchanged between the
gene P1 and the gene P2. Namely, new genes of C1 and C2 are
generated. In this embodiment genes C3 and C4 are further generated
by inversion of the partial information of the exchanged portions
from the genes C1 and C2. Namely, the two parents have four
children.
[0142] FIG. 17 is a drawing to illustrate a mutation operator,
which is one of the genetic operators. The concept of the mutation
operator is to generate children having characteristics that their
parents do not have. FIG. 17 illustrates the mutation operator,
particularly, for improvement in the movement order of object (the
order of position measurement of alignment marks: cyclic shift).
The mutation operator is applied to a pair of genes P3 and P4
randomly selected out of plural genes.
[0143] This mutation operator first searches portions having a
common partial set of alignment marks in the genes P3 and P4. In
FIG. 17, the second unit to the fifth unit of the gene P3 are a set
of alignment marks B, C, D, and E, while the fourth unit to the
seventh unit of the gene P4 are a set of alignment marks B, C, D,
and E. Thus, the two sets are identical.
[0144] A difference of units between such partial strings appearing
in the genes P3 and P4 is defined as a number of shifts. In FIG. 17
the partial string starts from the second unit in the gene P3 while
the partial string starts from the fourth unit in the gene P4;
therefore, there is the difference of two units and thus the number
of shifts is 2. Then the units appearing in the partial set of
alignment marks are shifted in order and in cycle by the number of
shifts, thus generating new genes C5 and C6. This means that the
gene C5 is obtained in such a way that the partial set of BCDE in
the gene P3 is shifted once to be CDEB and then it is shifted once
more to be DEBC. The gene C6 is also obtained in the same way.
[0145] The contents of each operator were detailed above and now,
returning to the flowcharts of FIG. 14 and FIG. 15, the procedures
of application of each operator will be described.
[0146] First, genes of a gene group consisting of genes numbering N
preliminarily set (for example, N=80) are generated randomly (step
ST12). Each gene in this group is determined as one for measuring
each of the all positions of the alignment marks numbering n (for
example, n=80) once. Each of the genes can be a solution generated
(or updated) by an approach based on the rule of thumb, an approach
based on the linear programming method (for example, the nearest
neighbor method which will be called the NN method and the details
of which will be described hereinafter, or the like), or the Lin
and Kernighan's approach (hereinafter referred to as the LK method
the details of which will be described hereinafter) with an initial
solution being a constraint satisfying solution generated
arbitrarily; or a solution generated (or updated) by the LK method
with an initial solution being a constraint satisfying solution
generated by the approach based on the linear programming method
(for example, the NN method or the like). Since the generation t at
this time is t=1, the gene group of the first generation will be
called G1. The overall movement times of movement sequences are
computed for the position measurement of alignment marks of
individual genes in the gene group G1 of the first generation. For
example, the overall movement times are 64.278 sec for the first
gene, 63.448 sec for the second gene, . . . , 56.163 sec for the
twenty fifth gene, and 53.046 sec for the twenty sixth gene.
[0147] In one alteration of generation, first, the crossover is
carried out in the number of crossovers according to a crossover
rate preliminarily set (Pc=0.4): N.times.Pc times (80.times.0.4=32
times) out of the group of the N (=80) genes. A pair of (two) genes
as objects of one crossover are sampled without replacement (or
without permitting crossover of a same gene) preferentially from
the shortest measurement time out of the gene group. This selection
of the paired genes is carried out the number of crossovers times
by sampling with replacement (permitting one gene to be selected
plural times for different pairs) out of the gene group. One of
specific, applicable methods for preferentially selecting the genes
of short measurement time is the roulette wheel selection, which
will be described in step ST36. Another applicable method is a
method for selecting N.times.Pc.times.2 genes in order from the
shortest overall movement time of movement sequence out of the gene
group and randomly selecting pairs of N.times.Pc genes
therefrom.
[0148] When the crossover operator is applied to each of these gene
pairs (step ST16), new genes numbering N.times.Pc.times.2
(80.times.0.4.times.2=64) are generated. This operator produces a
group of new-born genes A (=N.times.Pc.times.2). (step ST18).
[0149] Next, the "mutation of ordinal shift of position measurement
of alignment marks" is carried out A.times.Pm times
(64.times.0.4=25.6 times: which is raised to an integer, 26)
according to the rate of "mutation of ordinal shift of position
measurement of alignment marks" preliminarily set (Pm=0.4) out of
these new-born genes. Selection of one gene as an object of one
"mutation of ordinal shift of position measurement of alignment
marks" is carried out by random sampling with replacement
(permitting one gene to be selected as a mutation object plural
times) out of the gene group (step ST20: selection 2). When the
"mutation operator of ordinal shift of position measurement of
alignment marks" is applied to the totally twenty six genes
selected (step ST22), new genes numbering A.times.Pm (twenty six
genes) are generated. This operator produces a group of new-born
genes B (=A.times.Pm=N.times.Pc.tim- es.2.times.Pm) (step
ST24).
[0150] By the process heretofore, there are the gene group G1 of
the first generation (N=80), the new-born gene group A (64), and
the new-born gene group B (26). Totally, there are the genes
numbering N.times.{1+2.times.Pc.times.(1+Pm)} (170 genes) (step
ST26). In the case of the GA in the flowchart shown in FIG. 14, the
steps up to this point correspond to multiplication of new-born
genes by the genetic operators. In the case of the GA in the
flowchart shown in FIG. 15, multiplication by the LK operator
detailed hereinafter is introduced immediately after this
point.
[0151] After completion of multiplication of new-born genes in each
generation (i.e., in one loop in the flowchart shown in FIG. 14 or
FIG. 15), in the final step of the loop a survival process is
carried out to select genes to be left as a gene group of the next
generation (i.e., an initial gene group in the next loop) and to
dismiss the other genes (step ST36). Namely, the movement times are
checked for the all movement sequences corresponding to the
individual genes included in the new-born gene group A and the
new-born gene group B generated by the genetic operators (the
new-born gene group C is further added in the case of the GA in the
flowchart shown in FIG. 15) in addition to the initial gene group
Gt. Then the best gene, which is a gene to achieve a movement
sequence having the shortest movement time among them, is
outputted. At this time, if S different genes all are the best,
these S genes are left unconditionally in the next generation.
Further, (N-S) genes are also selected out of the remaining (170-S)
genes by such a selecting method as to select a gene corresponding
to a movement sequence of a shorter movement time more
preferentially. These genes are also left in the next generation
(i.e., in the (t+1)-th generation). Namely, the survival process is
carried out so as to select totally N genes including those of the
movement sequences of the shortest movement time and the other
genes (step ST36). This selection method is not one for selecting N
movement sequences from the shortest movement time, but is one for
selecting epistatic movement sequences with a higher priority but
still leaving a possibility of selecting even a hypostatic movement
sequence (of longer movement time), though low. The reason why the
possibility of selecting a hypostatic movement sequence is left is
that the true optimum solution is not always derived from a group
of epistatic movement sequences, but is also possibly derived from
a group of hypostatic movement sequences. Meanwhile, if a solution
derived from a hypostatic movement sequence is also still
hypostatic in the next generation, it will be naturally dismissed.
Therefore, it will not be a hindrance against finding of the true
optimum solution. The roulette wheel selection can be applied as a
selection method of this type. For example, supposing in the
optimization problem of movement sequence there are three
candidates X, Y, Z as solutions of movement sequences and their
movement times are 10 sec, 20 sec, and 20 sec, respectively, it is
contemplated that the inverse of the time is set as a fitness value
of each solution. At this time, supposing one solution is selected
out of these three solutions by roulette wheel selection,
probabilities of selection of the solutions X, Y, and Z are 0.5,
0.25, and 0.25, respectively. The eighty new genes thus selected
compose a set of genes of the next generation, Gt+1. Then,
returning to step ST14, the same process with the genetic operators
is repeated.
[0152] As apparent from the algorithm to repeat this alteration of
generation, for example, supposing exchange of reticle takes 10 sec
and within that period the computation is completed, for example,
up to the fortieth generation, the shortest movement sequence for
position measurement of alignment marks can be obtained among the
group of genes (solutions) of the fortieth generation. Supposing
exchange of reticle takes 11 sec and within that period the
computation is completed up to the forty first generation, the
shortest movement sequence for position measurement of alignment
marks can be obtained among the group of genes (solutions) of the
forty first generation.
[0153] Further, as shown in FIG. 15, the LK method can be
incorporated as one of the genetic operators into the GA in order
to obtain the optimum solution or the best solution more
efficiently. In this case, the LK method (hereinafter referred to
as an LK operator) is applied to plural genes (for example, to the
all genes) in the gene group including the new-born genes obtained
after the application of the crossover operator and mutation
operator (step ST30). Thus, a new-born gene group C (the number of
new-born genes included in C is N.times.2.times.Pc.times.(1+Pm)- )
is newly produced herein (step ST32). Namely, immediately before
the survival, there exist the genes numbering
2.times.N.times.2.times.Pc.time- s.(1+Pm) (Step ST34). The preset
crossover rate Pc, the preset mutation rate Pm, the GA, etc. are
stored in the memory 62 of the main control system 6 (FIG. 6).
[0154] Since the above embodiment minimizes the overall movement
time of the movement sequence for position measurement of alignment
marks by applying the GA to the projection exposure apparatus, it
makes possible scheduling of the order (path) of position
measurement of each alignment mark.
[0155] The present embodiment also makes possible the optimization
in which the local search by the crossover operator is merged with
the global search by the mutation operator. In addition, since the
best solution of each generation is presented every alteration of
generation by the GA, a near-optimum solution consistent with a
computation time can be outputted even with interruption of
computation within the allowed computation time.
[0156] The crossover operator, SXX, in the present embodiment can
be replaced by another crossover operator used in approaches to
TSP, such as OX: Order Crossover (L. Davis, "Applying Adaptive
Algorithms to Epistatic Domains," Proceedings of International
Joint Conference on Artificial Intelligence (IJCAI), 1985), PMX:
Partially Mapped Crossover (D. E. Goldberg and R. Linge, "Alleles,
Loci, and the Traveling Salesman Problem," Proceedings of
International Conference on Genetic Algorithms (ICGA), 1985), or
EAX: Edge Assembly Crossover (Nagata, Ono, and Kobayashi: "Proposal
and evaluation of edge assembly crossover: New crossover of TSP
considering tradeoff between character heredity and degree of
freedom of crossover," System and Information Joint Symposium '96,
Keisoku Jido Seigyo Gakkai, 1996).
[0157] Next described is the difference between the determining
method of movement sequence according to the present invention and
the conventional-determining method of movement sequence.
Specifically, for clearly expressing the difference between the
solution of movement sequence obtained by the present invention
(the result of execution of the determining method according to the
present invention) and the solution of movement sequence obtained
by the conventional technology (the result of execution of the
conventional determining method), the following description shows
the difference of measurement path in the serial movement sequence,
i.e., the difference in a path of a locus drawn by measuring light
on one wafer (a locus of the point where the measuring light
arrives), and the difference in the overall movement time of
movement sequence necessary for measuring positions of alignment
marks on one wafer.
[0158] For guaranteeing the true optimum solution, all constraint
satisfying solutions (feasible basic solutions) that can be
generated must be checked and it requires enormous computational
complexity, which is not practical. For examination, alterations of
generation, which seemed sufficient to converge, were made in the
GA using the MGG model under such preset conditions that the group
size was somewhat large, the crossover rate was somewhat high, and
the mutation rate was sufficiently high in comparison with the
group size. When the alterations of generation were made with a
sufficient time from an initial group generated at random, the
convergent solution (the true optimum solution) was the movement
sequence as shown in FIG. 18. The route passed the nearest
alignment marks after or before the start point ST and the end
point EN and there existed no seemingly long path. The overall
movement time was 49.052 sec. After that, the convergent solution
this time was handled as a substitute for the "true optimum
solution." In this GA the computation was continued up to the 500th
generation (for about thirty hours) under the setting conditions
that the group size N was N=160, the crossover rate Pc was Pc=0.5,
and the mutation rate Pm was Pm=0.4. A solution regarded as the
convergent solution first appeared near the 300th generation and
approximately twenty hours were consumed up thereto. This GA can
finally reach the optimum solution, with the sufficient time
consumed, but it is not practical to apply the GA as it is. Thus,
the approaches by the GAs shown in FIG. 14 and FIG. 15 above
involve some means to find the optimum solution quickly.
[0159] FIG. 1 and FIG. 2 show solutions of movement sequences for
the position measurement of alignment marks being measurement
targets, which were obtained by the prior-art determining method
described previously. These movement sequences obtained by the
prior-art approach are utilized from an empirical hope that they
might not be the optimum solution, but may be satisfactorily good
solutions. In the following description this approach is thus
called "approach based on the rule of thumb."
[0160] When the approach based on the rule of thumb is applied, the
operator determines the movement sequence that seems empirically
good, according to the arrangement of chip areas to be measured.
For example, a movement sequence obtained by the approach based on
the rule of thumb is as follows. First, when the chip area 64
closest to the start point ST is selected as a first chip area to
be measured, the measurement is carried out clockwise from the
alignment mark AM1 (see FIG. 12) of this chip area via AM2 and AM3
to AM4 in order (of course, the measurement can be carried out
counterclockwise in the order of AM1, AM4, AM3, and AM2). After
completion of the position measurement of the all alignment marks
given to the chip area 64, the measurement is continued to measure
the positions of the all alignment marks of the chip area 63.
Subsequently, the positions of the all alignment marks of the chip
area 62 are measured in the same manner and then the positions of
the all alignment marks in the all chip areas are successively
measured.
[0161] With the approach based on the rule of thumb used herein,
the optimization of movement sequence for the position measurement
of alignment marks was not effected taking account of such a route
that before completion of the position measurement of the all
alignment marks of one chip area, the measuring point visited an
alignment mark of another chip area and thereafter again visited
the remaining alignment marks of the first chip area. Therefore,
the approach based on the rule of thumb takes a short time for
computation, but it fails to effect the optimization taking full
account of the various conditions of the positions of the
respective alignment marks, except for restriction conditions such
as the performance of the stage or the chip areas.
[0162] The solution obtained by the approach based on the rule of
thumb was as shown in FIG. 1 where the measurement was conducted
clockwise for the positions of the alignment marks in each chip
area. The overall movement time of the resultant movement sequence
(from the start point ST to the end point EN) was 53.149 sec. When
the measurement was conducted counterclockwise for the positions of
the alignment marks in each chip area, the overall movement time of
the resultant movement sequence (from the start point ST to the end
point EN) was 52.757 sec. In contrast, the overall movement time of
the resultant movement sequence according to the optimum solution
obtained by the GA of the MGG model described above was 49.052 sec.
It is thus seen that the overall movement time can be shortened by
about 3 sec from that of the movement sequence obtained by the
approach based on the rule of thumb.
[0163] Linear Programming Method: Nearest Neighbor Method (NN
Method)
[0164] In general, even if a solution of local partial movement
sequences is optimum, a movement sequence having such partial
movement sequences will not be always an optimum solution. When
this problem is solved by the linear programming method to obtain a
solution, the resultant solution is not always the (globally)
optimum solution. However, since the determining method of movement
sequence according to the present invention is a method for
generating a good solution, or a locally optimum solution very
quickly, it is used as an effective near-optimum obtaining method
where the point is to obtain a solution in a short time.
[0165] The nearest neighbor method (the NN method), which is one of
most popular search methods, will be described herein as an
embodiment by the linear programming method. The NN method is a
method for repeating such an operation as to arbitrarily select one
starting point and choose a nearest neighbor point thereto as a
next point.
[0166] A point selected once will be excluded in order from a
candidate group of next selection. In the search process of the
best unit movement sequence, when there are plural candidates for
the unit movement sequence selected next (or when there are plural
candidates having an identical and shortest movement time of unit
movement sequence for movement to the start point of the next
position measurement of alignment mark) and if the all cases are
searched, the search could be a search of all solutions in the
worst case. For the purpose of decreasing the computation time by
avoiding it, an idea to randomly select several points out of the
next candidate points at each point (for example, in such a way as
to preliminarily assume that p points are selected in the highest
case and arbitrarily determine how many points should be selected
between 1 and p, at each point) can be introduced.
[0167] In this embodiment the above number p is set to 1 and a
solution is sought by the NN method while a search start position
is selected from the end point EN and the all alignment marks. The
reason is that when the above number p is 1 and if the search start
position is fixed at the start point ST, we can find merely one
solution. In this embodiment, however, the measurement start
position is set to be the start point ST and the measurement end
position to be the end point EN. In order to satisfy this
constraint, the start point ST must follow the end point EN.
Therefore, after the solution is obtained by the NN method, the
units of the gene need to be shifted so as to start from the start
point ST and end at the end point EN.
[0168] For quickly carrying out the algorithm of this type, it is
preferable to make a movement time management table of movement
patterns preliminarily contemplated and to write unit movement
times of the respective patterns in the memory or the like. With a
list of candidates for movement from a certain point to a next
point, it is effective to sort the candidates in the order of unit
movement times from the smallest. This operation makes it possible
to select points not having passed yet one by one from the top of
the candidate list. The same table of movement patterns is also
utilized for increasing the efficiency of calculation of movement
time similarly in the LK method and the evolutionary computation
method.
[0169] By employing such a method that every time a new solution is
generated by the NN method, the best solution at that point is
recorded in the memory 62 of the main control system 6, the best
solution by the NN method at that time can be obtained even with
interruption of computation on the way. For example, the operation
in this embodiment is carried out according to the flowchart shown
in FIG. 20. Eighty one solutions were generated by the NN method
and the computation time for generation of the eighty one solutions
was about 0.03 sec. As a result, the best solution by the NN method
in this embodiment was as shown in FIG. 19, and the overall
movement time of the movement sequence for the position measurement
of the alignment marks (from the start point-ST to the end point
EN) was 49.614 sec.
[0170] Lin and Kernighan's Approach (The LK Method)
[0171] The LK method is famous as a quick near-optimum obtaining
method for symmetric TSP (Traveling Salesman Problem), i.e., as a
technique capable of obtaining a near-optimum solution in a very
short turnaround time of computation and is a technique of
development of the k-OPT method, which is the general name of the
2-OPT method and the 3-OPT method. Details of these techniques will
be described hereinafter. The TSP is a problem: given an arbitrary
city matrix for n (n>1) cities, find a minimum-length tour that
visits each city only once and the all cities. There are symmetric
and asymmetric TSPs. The symmetric TSP is a TSP for an object in
which the overall pathlength is invariant even with inversion of
the tour order from the sequential city visit order given for the
matrix of n cities. Conversely, the asymmetric TSP is a TSP for an
object in which the overall pathlength will vary with inversion of
the tour order from the above city visit order.
[0172] The TSP was originally the problem considered in order to
minimize the length of the tour, for example, assuming that a
matrix of cities are located randomly on a two-dimensional plane or
the like. In such a city matrix space the shortest path between two
arbitrary cities (i.e., an arbitrary unit path) is given uniquely
by connecting the two cities by a straight line; therefore, once
the visit order of the cities is determined, the minimum-length
tour path according to the visit order can be obtained uniquely.
Accordingly, finding a city visit order (city permutation) to
minimize the length of the path to visit the all cities is
equivalent to obtaining a solution to the TSP. Therefore, even if
the objective problem is not directed to minimization of the path
itself (i.e., a distance in a physical space) and if it is
converted to one equivalent to the optimization of permutation, the
approach of TSP can be applied as it is. For example, the technique
of TSP can be applied as it is, to the problem to determine the
movement sequence so as to minimize the overall movement time for
the position measurement of the all alignment marks in the one-shot
exposure type stepper. The movement times of unit movement
sequences in this case are not distances between two arbitrary
alignment marks, but they are the times as the memory contents in
the movement time management table stored in the memory 62 (because
the movement along the X-direction and the movement along the
Y-direction of the wafer stage can be made independently and
simultaneously). Let us consider below application of the "LK
method" which can be regarded as an improvement of the k-OPT
method.
[0173] The NN method is a "generating method" to generate a
solution from nothing, while the k-OPT method and LK method are
so-called "improving methods" for initially giving a certain
initial solution (here, in the case of the "constraint satisfying
problem" to require an output solution to satisfy a specific
constraint, a necessary condition is that the initial solution is a
feasible basic solution) and successively improving the solution.
Particularly, the LK method is a method for repetitively performing
such an operation as to extract a part of a tour sequence of the
initial solution and to invert the partial order, thereby effecting
repetitive improvements even in a solution after improved, as long
as an improvement is possible.
[0174] In one improvement, if the sum of lengths of paths (two cut
paths) at the both ends of the partial sequence extracted is
smaller than that before the inversion, the length of the entire
path will be shortened by the difference between them. Therefore,
this method is efficient, because attention is focused only on the
lengths of the two cut paths and computation is unnecessary for the
total pathlength. This single improvement is called 2-OPT. After
accomplishment of 2-OPT, when 2-OPT is again carried out in a
combination of one end of the partial path extracted with another
end this time, three ends are changed as a whole. This is called
3-OPT. These are generally called k-OPT. In the k-OPT the value of
k is preliminarily given. Therefore, the k-OPT had a problem that
it was not sufficiently adaptable to the case wherein the optimum k
value varied depending upon the initial solution to be improved.
However, the LK method updates k in k-OPT (from 2) one by one as
long as an improvement is made. Thus the LK method overcame the
problem in the k-OPT method.
[0175] If the method is so arranged that every generation of a
solution by the LK method the best solution at that point is stored
in the memory 62 of the main control system 6, the best solution by
the LK method at that time can be obtained even with interruption
of computation on the way. For example, the operation of this
embodiment is carried out according to the flowchart shown in FIG.
22.
[0176] In this embodiment the LK method is applied to near-optimum
solutions obtained by the NN method. The LK method generates one
improved solution per initial solution. Since one improved solution
is generated per initial solution, the number of initial solutions
should be determined preferably as high as possible. This
embodiment employs an approach method for using eighty one
solutions obtained by the NN method as initial solutions and
improving them.
[0177] The computation time was about 0.14 sec for obtaining eighty
one solutions by applying the LK method to the all eighty one
solutions resulting from the forward search of the NN method. Since
the computation time becomes longer with more improved portions by
the LK method, it is not simply proportional to the number of
initial solutions to be improved. The result of this experiment
showed that when the solutions by the NN method were used as
initial solutions of the LK method, the best solution was the
solution of the movement sequence shown in FIG. 21 and the overall
movement time of the movement sequence was 49.117 sec.
[0178] Approach Based on the Evolutionary Computation Method
[0179] Next described is the approach based on the evolutionary
computation method according to the present invention. An initial
group is first generated. Since the shortest movement times of the
above unit movement sequences are already recorded in the
two-dimensional table (see FIG. 10), for example, if a movement
sequence is arbitrarily produced so as to pass each alignment mark
only once without duplication with reference to this table, a gene
indicating an arbitrary feasible basic solution can be generated.
Using plural genes (each of which is a feasible basic solution)
arbitrarily generated as described, as an initial group,
preliminary experiments were repetitively conducted with different
group sizes, crossover rates, and mutation rates. The preliminary
experiments verified that the same solution as the optimum
solution, which was obtained by 20-hour computation by the GA of
the MGG model described above, was generated within about 20
seconds under such setting conditions that the group size N=the
number of alignment marks, the crossover rate Pc=0.4, and the shift
mutation rate Pm=0.4 and that the LK operator was applied to the
all genes.
[0180] Further, experiments were conducted using the solutions
obtained by the above LK method as initial solutions. It is highly
possible that a solution obtained by the NN method + the LK method
(i.e., by the LK method using the solutions obtained by the NN
method, as initial solutions) is a local solution in the
optimization problem of measurement process sequence of coordinates
of the alignment marks (which is a locally optimum solution, but is
not a true optimum solution or a globally optimum solution). This
thus raises a risk of biasing the solution search space in the
global search by the GA (i.e., a risk of search biased to near the
local solution). In order to avoid it, the initial group should
desirably include a solution of good quality representing a
completely different movement sequence, as well as the best
solution by the NN method + the LK method. If the added solution,
though representing the completely different movement sequence, is
not of good quality, it will be dismissed to be extinct soon in the
GA. Therefore, this example employed a method for sampling
solutions without replacement by the number equivalent to the group
size (N) at predetermined probabilities, each at a probability
consistent with an evaluation value of each solution, out of the
solutions obtained by the NN method and the solutions obtained by
applying the LK method to the all solutions obtained by the NN
method, in addition to the best solution obtained by the NN method
+ the LK method (the roulette wheel selection). This group of
solutions was used as an initial gene group in the GA.
[0181] With the above group being used as an initial gene group and
under such setting conditions that the group size N=the number of
alignment marks, the crossover rate Pc=0.4, and the shift mutation
rate Pm=0.4 and that the LK operator was applied to the all genes,
the GA stably generated a solution superior to that by the LK
method within 2 sec and also generated the same solution as the
optimum solution obtained after 20-hour computation by the GA of
the MGG model, within 10 sec.
[0182] FIG. 23 to FIG. 25 are drawings to show relations among the
various approaches described above, computation times necessary for
such approaches, and overall movement times of movement sequences
obtained by the approaches. The ordinate indicates the overall
movement time of movement sequence obtained and the abscissa the
computation time. The units are sec.
[0183] FIG. 23 is a drawing to illustrate the overall movement time
of the movement sequence of the solution obtained by the approach
based on the rule of thumb, the time necessary for obtaining the
best solution by the NN method and the overall movement time of the
movement sequence being the solution that time, and the computation
time necessary for obtaining the best solution by the LK method
using the solution set by the NN method as initial solutions and
the overall movement time of the movement sequence being the
solution that time.
[0184] FIG. 24 is a drawing to show the result achieved when the GA
was applied to an initial group as being a set of solutions
consisting of the feasible basic solutions randomly generated as
described above. Further, FIG. 25 is a drawing to show the result
achieved when the GA was applied to an initial group as being a set
of solutions consisting of the solutions obtained by the NN method
+ the LK method. It is apparent from either figure that the best
solution is updated with progress in alteration of generation. For
comparison's sake, FIG. 24 and FIG. 25 also show the overall
movement time of the movement sequence of the solution obtained by
the approach based on the rule of thumb and the overall movement
time of the movement sequence of the best solution obtained by the
LK method using the solution set by the NN method as an initial
solution group. The best solution obtained by the GA shown in FIG.
24 was obtained in the computation time of 6.18 sec and the overall
movement time of the movement sequence being the best solution
obtained was 49.052 sec. Similarly, the best solution obtained by
the GA shown in FIG. 25 was obtained in the computation time of
5.00 sec and the overall movement time of the movement sequence
obtained was 49.052 sec.
[0185] The above experiments were conducted on a workstation
provided with the CPU of 200 MHz. It should be noted that the above
computation times naturally differ depending upon the performance
of the CPU or the like.
[0186] Next described referring to FIG. 26 to FIG. 49 are
embodiments of the designing method and apparatus of optical system
according to the present invention, and the recording medium in
which the program for realizing the designing method is recorded.
In the drawings the same portions will be denoted by the same
reference symbols and the description thereof will be omitted.
[0187] The designing method of optical system according to the
present invention can also be realized by a program described in
predetermined language. This program is executed, for example, by
an arithmetic circuit having memories (RAM, ROM) or by a computer 1
as shown in FIG. 26. Particularly, the computer 1 shown in FIG. 26
has a main control system 100 composed of control section 101,
arithmetic section 102, and storage section 103 (memory) and also
has peripheral devices, including input section 104, output section
105, and external storage device 106 such as a hard disk. Examples
of the output section 105 include display devices such as CRT 115
or a liquid-crystal display. When the designing method of optical
system according to the present invention is executed by the
computer 1 as shown in FIG. 26, it is preferred to record the
program for realizing the designing method of optical system in the
above external storage device 106 or to optically or magnetically
record the program in a predetermined recording medium such as CD
115, MO, FD, a magnetic tape, or ROM.
[0188] For easier understanding, definitions of terms used in this
specification will be described below.
[0189] Individual: autonomous individual featured by genes;
[0190] Population: set of individuals;
[0191] Population size: number of individuals allowed to exist in a
population;
[0192] Gene: basic component representing genetic information;
[0193] Allele: value that each gene can take;
[0194] Chromosome: genes expressed in the form of a string (a
string of characters), a vector (a string of numerals), or a
mixture thereof;
[0195] Locus: position of a gene on a chromosome;
[0196] Phenotype: external expression of a character revealed by a
chromosome;
[0197] Genotype: gene expression (internal expression) for defining
a character;
[0198] Coding: mapping from phenotype to genotype;
[0199] Decoding: inverse mapping from genotype to phenotype;
[0200] Fitness value: scalar evaluation value which indicates a
degree of fitness of an individual to an environment (in the case
where the object is the optimization problem, a fitness value is a
value of an objective function itself or a value of a penalty
function taking account of constraints);
[0201] Genetic operations: operations of crossover, mutation, and
selection;
[0202] Selection: selection for survival of individuals according
to their fitness values (to make a pair of individuals for mating,
taking account of difference in fitness value);
[0203] Crossover: recombination of genes between individuals (to
make new individuals by mutual recombination of chromosomes of a
pair of individuals between them);
[0204] Mutation: replacement with an allele (to replace a value of
a certain locus with another allele);
[0205] Generation: one cycle of genetic operations;
[0206] Diversity: degree of preserving diversity of gene;
[0207] Parallel algorithms: model in a distributed population;
[0208] Migration: exchange of individuals between plural
populations.
[0209] In the following description we will consider two evaluation
criteria and first check the global search capability of the
designing method where the designing method of optical system
according to the present invention is applied to design of a lens
system of three lenses and a lens system of four lenses while the
two evaluation criteria are converted to a single objective (the
first embodiment). Next, we will handle the two evaluation criteria
as multiple objectives to show excellent multi-objective
optimization capability of the designing method of optical system
according to the present invention as to design of a lens system of
three lenses (the second embodiment).
[0210] In the embodiments of the designing method of optical system
according to the present invention, supposing the user sets the
focal length f, brightness F, and field angle 2w as design
specifications of lens system, let us consider optimization of
radii of curvature ri of respective lens surfaces (boundary
surfaces of optical elements composing the optical system),
thicknesses of lenses, and distances di between the lenses. It is
assumed that the refractive indices of glasses and the number N of
lenses are preliminarily given. A radius of curvature of a lens
surface located closest to the film surface (the image plane) and a
distance from the lens surface to the film surface will be
corrected so as to satisfy the focal length given by tracing of
paraxial rays. Accordingly, design of a lens system of N lenses is
equivalent to optimization of a (4N-2)-dimensional function.
[0211] A lens system designed is evaluated based on three spot
diagrams produced by letting totally eleven rays around the
principal ray pass through the lens system at three angles of
incidence of 0.degree., 0.65w.degree., and w.degree.. The two
evaluation criteria are distortion (D) and resolution (R). The
distortion D is given as a distance between an ideal image position
(f.multidot.tan w) and an image point of the principal ray. The
resolution R is given as a standard deviation to indicate a degree
of dispersion of image points of the other ten rays than the
principal ray from the image point of the principal ray, as shown
in the following Equation 2. 1 D = W E ( 0 , 0.65 W , W ) ( x 0 - x
ideal ) 2 + ( y 0 - y ideal ) 2 R = W E ( 0 , 0.65 W , W ) k = 1 10
[ ( x k - x 0 ) 2 + ( y k - y 0 ) 2 ] / 10
[0212] In the equations, (xideal, yideal) represents the ideal
image position, (x0, y0) the position of the image point of the
principal ray, and (xk, yk) positions of the respective image
points of the other ten rays.
[0213] Next, the genetic algorithm (GA) will be explained.
[0214] The GA is an engineering model imitating the evolutionary
process of organism, and a general algorithm thereof is formed as
follows. Specifically, it is composed of the following
operations.
[0215] (1) Generation of Initial Population
[0216] A plurality of individuals (solution candidates) are
generated and are used as an initial population.
[0217] (2) Selection for Reproduction
[0218] Individuals to be parents (parent individuals) are selected
from the population.
[0219] (3) Generation of Children
[0220] Child individuals (new solution candidates) are generated by
applying the crossover operator and/or the mutation operator to the
parent individuals.
[0221] (4) Selection for Survival
[0222] Individuals to form a next-generation population are
selected from the population of the individuals in the current
generation population and the child individuals generated in above
step (3).
[0223] (5) The above steps (2) to (4) are repeated before a
predetermined end condition is satisfied.
[0224] Design items of the GA are classified roughly into design of
coding/crossover-mutation and design of generation model. The
design of coding/crossover-mutation involves the design of a
representation method for representing individuals and the design
of a generating method for generating new individuals, and is
dependent on a problem region. Since the performance of the GA is
greatly affected by the design of coding/crossover-mutation, it is
a very important design item.
[0225] On the other hand, the design of generation alteration model
involves the design of a selecting method for selecting parents to
generate children and the design of a selecting method for
selecting individuals to be left in the next generation population.
By well managing this design, it becomes possible to search at a
time a set of solutions in plural tradeoff relations in the
multi-objective optimization.
[0226] The design of coding/crossover will be described below.
[0227] In the embodiments of the designing method of optical system
according to the present invention, real vector representation is
employed as coding. An individual is expressed by a real vector
(r1, r2, . . . , rn, d1, d2, . . . , dn) components of which are
radii of curvature and distances between surfaces (parameters
featuring each candidate of optical system to be designed).
[0228] In this embodiment UNDX (Ono, I. and Kobayashi, S: A
Real-coded Genetic Algorithm for Function Optimization Using
Unimodal Normal Distribution Crossover, Proceeding of 7th
International Conference on Genetic Algorithms, pp. 246-253 (1997))
is adopted as a crossover operator. The UNDX generates, from two
parents of Parent 1 and Parent 2 out of selected parents, two
children according to a normal distribution set around them, as
shown in FIG. 27. The standard deviation of the normal distribution
is set so that a component .sigma.1 along the major-axis direction
connecting the both parents is proportional to a distance between
the parents (.sigma.1=.alpha.d1 where d1: the distance between
Parent 1 and Parent 2) and so that a component .sigma.2 along the
other axis is proportional to a distance between the major axis and
Parent 3 (.sigma.2=.beta.d3, where d3: the distance between Parent
3 and the axis connecting Parent 1 with Parent 2). FIG. 27
illustrates an example of two variables.
[0229] This crossover operator is effective to functions wherein
strong dependence exists between variables, and multimodal
functions. This is conceivably because the UNDX undergoes the
global search in the early stage of search where distances are
large between parents and undergoes the local search in the final
stage where the distances are close between parents and it can
accomplish the search little dependent upon the coordinate
system.
[0230] First Embodiment by the Designing Method of Optical
System
[0231] In the designing method of optical system of the first
embodiment according to the present invention, the effectiveness
concerning the global search capability of the GA will be verified
by experiments with scalar evaluation values being set by linear
combination under a given tradeoff ratio of resolution R and
distortion D.
[0232] Used as a generation alteration model of the GA in
Embodiment 1 is the MGG model shown in FIG. 28 (Satoh, H.,
Yamamura, M. and Kobayashi, S.: Minimal Generation Gap Model for
GAs Considering Both Exploration and Exploitation, Proceedings of
IIZUKA '96, pp. 494-497 (1996)). In this generation alteration
model, n crossovers are carried out each with two individuals
randomly selected as parents from a population. Then alteration of
generation is done by returning one elitist individual and one
individual selected by roulette selection out of the parent and
child individuals into the population.
[0233] For verifying the effectiveness of the designing method in
this first embodiment, optimization was done for the lens system of
three lenses and the lens system of four lenses. For the lens
system of four lenses, three experiments were conducted with
different design specifications. In the all experiments, the number
of individuals in the population was 100, the number of crossovers
n was 100, a of UNDX was 0.5, and .beta. thereof was 0.35. The
tradeoff ratio of distortion D and resolution R was 1:1.
[0234] Experiment 1
[0235] The optimization of the lens system of three lenses was
conducted as Experiment 1 of the first embodiment. The design
specifications were as follows; the focal length f=100 mm, the
brightness F=3.0, and the field angle 2w=38.0.degree.. This
experiment was started from the initial population consisting of
random lenses L1 to L3 as shown in FIG. 29 and FIG. 30 and was
ended in the stage after evaluation of four million lenses. In
these FIG. 29 and FIG. 30, S represents a stop and I the image
plane. A lens configuration and spot diagrams of the resultant
optical system are shown in FIG. 31 and an aberration diagram
thereof in FIG. 32. The spot diagrams are indicated in the range of
.+-.0.5 mm.
[0236] Experiment 2
[0237] The optimization of the lens system of four lenses was
conducted as Experiment 2 of the first embodiment. The design
specifications were set in the following three types; the standard
lens specifications were f=50 mm, F=3.0, and 2w=46.degree.; the
telephoto lens specifications were f=135 mm, F=2.8, and
2w=18.2.degree.; the wideangle lens specifications were-f=20 mm,
F=5.6, and 2w=92.degree.. As in the case of the lens system of
three lenses, the search was started from lenses L1 to L4 of random
configuration and the search was ended after one million
evaluations. Lens configurations and spot diagrams of resultant
optical systems are shown in each of FIG. 33 to FIG. 35. FIG. 33
shows optical systems obtained in the case of the standard lens
specifications, FIG. 34 optical systems obtained in the case of the
telephoto lens specifications, and FIG. 35 optical systems obtained
in the case of the wideangle lens specifications.
[0238] In Experiment 1 of the first embodiment, ten trials all
resulted in obtaining the lens system having the lenses, i.e., a
triplet in the configuration as shown in FIG. 31. The triplet
considered to be empirically optimal was obtained by starting the
search from the fairly random lens systems without relying on
knowledge. It can be said from this result that the effectiveness
of the global search of lens system by the GA was verified.
[0239] In Experiment 2 the lens systems were also achieved with
relatively high performance under the respective design
specifications. For either of the telephoto lens and wideangle
lens, almost identical configurations were obtained in each trial
(two patterns each), as shown in FIG. 34 and FIG. 35. For the
standard lens, lenses of various configurations (six patterns) were
obtained as shown in FIG. 33.
[0240] Second Embodiment by the Designing Method of Optical
System
[0241] Next, the multi-objective optimization by the GA will be
described as the second embodiment according to the present
invention.
[0242] In this second embodiment two objectives of the distortion
and resolution are set explicitly and effectiveness will be
verified concerning the multi-objective optimization capability of
the GA. In this second embodiment the generation alteration model
is a model based on the non-Pareto solution selection strategy
(Shigenobu Kobayashi, Koji Yoshida, and Masayuki Yamamura:
Generation of Pareto optimal setting tree set by GA, Jinkochinou
Gakkai Shi, vol. 11, No. 5, pp. 778-785 (1996)) (see FIG. 36).
[0243] A crossover is carried out with n pairs randomly selected
from a population. Reproduced children are merged with the existing
population and individuals not being Pareto solutions are weeded
out, so that a population of the next generation is composed of
only Pareto solutions. A Pareto solution is a solution superior in
at least one evaluation criterion to the all other solutions. This
operation allows the multi-objective optimization to be conducted
explicitly without setting the tradeoff ratio between the
resolution and distortion.
[0244] In this second embodiment the effectiveness of the
multi-objective optimization by the GA will be verified by an
experiment with an object of the lens system of three lenses. An
initial population prepared includes five Pareto solutions obtained
in the search by the single-objective GA, the number of crossovers
n is 40, and the number of evaluations is 1,600,000 (the search is
carried on up to the evaluation number of 1,600,000). The design
specifications of the lens system are the same as in Experiment 1
of the first embodiment described above. FIG. 37 shows plots of the
resultant Pareto solution set on the distortion-resolution plane.
The number of resultant Pareto solutions is 750.
[0245] Also in this second embodiment, if the search is further
carried on from the Pareto solution set shown in FIG. 37, the
Pareto solution curve will be evolved toward the left bottom
corner. However, the rate of evolution is already slow, and it is
thus assumed that the curve is sufficiently close to a true Pareto
optimal solution curve. It also became apparent that the resolution
was improved up to a certain value with no penalty of distortion
but a further improvement in the resolution over that value was
hard even with a penalty of distortion. Configurations of resultant
solutions all are triplets and it is thus considered that the
triplets are optimal in the entire region under the design
specifications.
[0246] FIG. 38 shows a state in which the best solution P (the lens
system shown in FIG. 31) obtained in Experiment 1 of the first
embodiment described above is plotted on the enlarged view of FIG.
37. In the drawing letter S indicates lens systems dominating the
solution found by the single-objective optimization of the
evaluation criteria. As also apparent from this FIG. 38, it is
clearly seen that the second embodiment (multi-objective
optimization) obtains many more excellent solutions than that
obtained by the single-objective GA. This conceivably suggests that
there is a possibility of making the problem harder if the
multi-objective problem is forced to be the single-objective
problem.
[0247] It seems more natural that a plurality of lenses in the
tradeoff relation are first obtained by a search and an appropriate
lens is selected therefrom according to judgment of selection
favored by the designer, rather than the way of converting the
problem to the single-objective problem by linear combination of
plural evaluation criteria and obtaining one lens by a search.
[0248] As described above, the embodiments of the designing method
of optical system according to the present invention clarified the
effectiveness thereof by the above experiments. Particularly, the
multi-objective GA handling the evaluation criteria explicitly
permits us to search lenses with different performances at a
time.
[0249] Since the designing method of optical system according to
the present invention employs the principle of successively
selecting near-optimal solutions and finally outputting the optimal
solution from the two features of the principle of the genetic
algorithm, i.e., (1) or (1a), and (2) below, a near solution or the
optimal solution can be generated efficiently according to the
length of turnaround computation time.
[0250] (1) Simultaneous progress of local search and global search
in a problem space by only the crossover operator
[0251] (1a) Simultaneous progress of local search and global search
in a problem space by the combination of the crossover operator
with the mutation operator
[0252] (2) Alteration of generation for defining a series of
operations with genetic operators as one generation and
repetitively effecting the series of operations on finite genes
including the elitist gene of each generation
[0253] A chromosome in the genetic algorithm is represented by a
string having a length corresponding to the number of parameters (a
variable string consisting of variables of genes), in which a gene
at a position (locus) of occurrence of each parameter corresponds
to each parameter featuring the lens system to be designed.
[0254] The crossover operator acts to make a gene of a child from
genes of a pair of two parents in such a manner that, as to each
parameter, the parameter of the gene of the child takes a value
occurring at a predetermined probability in a partial space defined
by parameters of the two parents.
[0255] When the mutation operator is also used in addition, a gene
of a child individual is generated in such a manner that, as to
each parameter, a parameter value is a value occurring according to
a predetermined continuous occurrence probability distribution of
occurrence probabilities increasing with approaching a parameter
value of one parent arbitrarily selected.
[0256] Since candidates of solutions generated internally can be
always evaluated directly, a solution not satisfying the constraint
arbitrarily given is allowed to become extinct. Thus, the
multi-objective evaluation can be done even in the case of plural
objective functions corresponding to the evaluation criteria.
[0257] Further, during execution of the GA, the alteration of
generation is made so that the elitist solution (or Pareto optimal
solution) out of a plurality of genes (satisfying the arbitrary
constraint) stored is always preserved and selection is made so
that better solutions are more unlikely to become extinct and thus
are left in the next generation, thus enabling multi-point
simultaneous searches with high efficiency.
[0258] Third Embodiment by the Designing Method of Optical
System
[0259] The third embodiment in the designing method of optical
system according to the present invention will be described
referring to FIG. 39 to FIG. 41. This third embodiment is an
example for obtaining the optimal solution of a lens system for
photography, which is a lens system of three lenses the refractive
indices of which are known. Namely, the refractive indices of the
three respective lenses are set by the user to be constraints.
[0260] FIG. 39 is a schematic diagram of the structure of the
photographic lens system. In this figure g designates the image
plane. The photographic lens system of this figure is an example of
the three-lens configuration, in which there are six boundary
surfaces of a to g having their respective curvatures, and six
distances of d1 to d6 between the boundary surfaces (d1 between A
and B, d2 between B and C, d3 between C and D, d4 between D and E,
d5 between E and F, and d6 between F and G).
[0261] In the case of the photographic lens system of FIG. 39 the
refractive indices of spaces between the lens boundary surfaces are
preliminarily given, but if degrees of freedom are also given to
the refractive indices of the spaces between the lens boundary
surfaces the number of parameters will increase by the degrees.
Since the field angle and overall focal length of the lens optical
system are normally constraints given by the designer, once six
(e.g., a-f) are determined out of a-g and five (e.g., d1-d5) out of
d1-d6, the two remaining parameters (e.g., g and d6) are determined
naturally. This third embodiment becomes a simultaneous
optimization problem of ten parameters accordingly.
[0262] The distances between the boundary surfaces themselves are
used for d1 to d6 (i.e., the necessary conditions are d1-d6>0 in
order to avoid physical interference) and an inverse of a radius of
curvature is used for indicating each curvature of a-g. The reason
why the inverse numbers are used is that use of inverse allows us
to handle the change in curvature as continuous change on the
parameter space, thereby being ready for transition of surface
configuration into the both convex and concave spaces with small
change of a flat surface (having the radius of curvature of the
infinity).
[0263] There are a variety of evaluation criteria for lens system
and, corresponding thereto, a variety of performance functions can
be defined. Famous criteria are five evaluation criteria
represented by the Seidel's five aberrations, which can also be
applied to this embodiment. For brevity of description, however,
this embodiment shows an example using two conflicting evaluation
criteria, using the ray tracing by spot diagrams.
[0264] FIG. 40 illustrates a photographic lens system generally
called a triplet type. In the drawing the left side is the object
side while the right side is the image side. When light is assumed
to come from the infinity on the object side and to be represented
by a parallel beam, this beam will be converged at one point on the
image plane g. An image formed on the entire image plane can be
evaluated by consideration on typical beams coming from the left of
the same drawing, including a parallel beam parallel to the optic
axis of the lens system (rays indicated by solid lines in the
drawing), a parallel beam an angle of which to the optic axis is
the maximum field angle in the specifications of the lens system
(rays indicated by dashed lines in the drawing), and a parallel
beam an angle of which to the optic axis is approximately a half of
the maximum field angle in the specifications of the lens system
(rays indicated by chain lines in the drawing).
[0265] For example, let us consider the degree of this convergence
as an evaluation criterion and evaluate the degree of convergence
by dispersion of image-plane arrival points of three rays, the
paraxial ray of the above beams, a ray on the maximum outside
diameter (normally, a diameter corresponding to the diameter of the
entrance pupil) of the above beams, and rays on intermediate
diameters (for example, diameters equal to approximately 70% and
80% of the maximum outside diameter) between the two rays, from the
image-plane arrival position of the paraxial ray. This evaluation
criterion approximately stands for the resolution of the image
formed on the image plane, if the sagittal aberration, coma, and
flare are ignored. When this dispersion is checked for the above
three types of parallel beams, approximate evaluation can be made
as to the resolution of the formed image.
[0266] For example, let us consider another evaluation criterion of
measuring a deviation between a position where the principal ray
actually reaches the image plane and a position where it should
impinge theoretically, for each of the above three types of
parallel beams. An evaluation value is obtained by overall
evaluation of deviations from the theoretical values of the three
principal rays obtained corresponding to the respective principal
rays (for example, by measuring the standard deviation from the
mean value of the deviations) and it approximately corresponds to
the distortion of the formed image.
[0267] When the multi-objective optimization is carried out by
using two different evaluation criteria and minimizing the both two
objective functions corresponding to the two evaluation criteria
(resolution and distortion) as described above, it will result in
obtaining many multi-objective optimal solutions (Pareto optimal
solutions), including solutions in which one of values of the two
objective functions is very small while the other is large, and
solutions in which the both are moderately small.
[0268] In the embodiment of the designing method of optical system
according to the present invention, the genetic operator in the
genetic algorithm being one of the evolutionary computation methods
is the crossover operator for directly operating continuous values,
or the combination of the crossover operator for directly operating
continuous values with the mutation operator for directly handling
continuous values.
[0269] FIG. 41 illustrates a gene representation of ten parameters
of continuous values featuring the lens system in the three-lens
configuration shown in FIG. 39. In each of a-g and d1-d5 in the
same drawing a parameter of the corresponding lens system is stored
in the form of continuous value. Among such genes, n (n>1) genes
satisfying the minimum constraints are reproduced arbitrarily.
[0270] Then the crossover operator for directly handling continuous
values is applied to a pair of two genes selected properly. With
the crossover operator, an occurrence probability distribution of
new-born genes is set on a space of a parameter (a) from
corresponding parameters in two genes (for example, let av1 be a
value of the parameter a of one gene and av2 be a value of the
parameter a of the other gene).
[0271] This occurrence probability distribution normally has a
configuration in which occurrence probabilities become higher with
getting near av1 and av2. Then selected from the partial space
defined by this occurrence probability distribution are two values
of the parameter a (for example, av3 and av4) that occur according
to the occurrence probabilities and that a new-born gene should
have. This operation is effected on the all parameters. (the ten
parameters in the example of FIG. 41).
[0272] This third embodiment can also employ a method for handling
the individual parameters independently of each other or a method
for simultaneously selecting some parameters or the all parameters
in consideration of correlation between parameters as in the NDX.
In this way a pair of two genes are sampled with replacement
Pc.times.n times according to a preset crossover rate Pc from a
population of n genes so as to be modified so that a gene of better
evaluation result is more likely to be selected. The crossover
operator is again applied to the sampled gene population.
[0273] This operation increases the number of genes in the gene
group from n to n.times.(1+2Pc). In the case of a single
performance function (i.e., in the case of a single evaluation
criterion or in the case of a plurality of evaluation criteria
being combined so as to be expressed by one performance function),
a gene of the best evaluation value is always left and sampling
without replacement is carried on so that genes of better
evaluation values are more likely to be selected and left, before
the number of genes in the gene group reaches n. The other
2.times.n.times.Pc genes are eliminated. A population of n new
genes is reproduced in this way. The population of genes reproduced
in this way is called the first generation of genetic algorithm.
Repetition of such generation alteration will find the optimal
solution sooner or later if a sufficient computation time is
permitted. Unless a sufficient computation time is given, a
solution with good quality consistent with the computation time can
be generated efficiently.
[0274] When the multi-objective optimization is carried out with
plural performance functions, only multi-objective optimal genes
are left out of the population of the all (n.times.(1+2Pc)) genes
after generation of new-born genes (or the multi-objective optimal
genes are always left while genes being not multi-objective optimal
and numbering k (k>1) times the number of the multi-objective
optimal genes are also-left), and the other genes are allowed to
become extinct, thus forming a gene population of a new generation.
In this case, however, care is needed, because the number of genes
n in the gene population varies generation from generation.
[0275] As the genetic operator for directly handling continuous
values, the mutation operator for directly handling continuous
values can also be used, as well as the crossover operator for
directly handling continuous values. In this case, genes are
randomly sampled with replacement Pm.times.n.times.(1+2Pc) times
according to the mutation rate Pm preliminarily set, from the
population of (n.times.(1+2Pc)) genes after the application of the
crossover operator. Then new-born genes being mutants are
reproduced so that an arbitrary (one or more) parameter of the
genes is altered arbitrarily (or in a probability distribution
biased to the neighborhood of its original value). Immediately
after the application of the mutation operator, the number of genes
in the gene group is n.times.(1+Pm).times.(1+2Pc). Then genes to be
left in the next generation are selected out of this gene
population in the same manner as above, thus generating the gene
population of the next generation.
[0276] The above embodiments were described using the typical
parameters represented by the radii of curvatures of the boundary
surfaces and the distances between the boundary surfaces, but it is
needless to mention that the optimal solution can be gained with
additional parameters of continuous values related to the lens
designing, such as the refractive indices of the three respective
lenses and the pressure between the lenses.
[0277] As described above, the designing method of optical system
according to the present invention employs the principle of
successively selecting near-optimal solutions and finally
presenting the optimal solution from the two features of the
principle of the genetic algorithm, and, therefore, a near solution
or the optimal solution can be reproduced efficiently according to
the length of turnaround computation time.
[0278] Fourth Embodiment by the Designing Method of Optical
System
[0279] The fourth embodiment in the designing method of optical
system according to the present invention will be described
referring to FIG. 42 and FIG. 43.
[0280] In this fourth embodiment FIG. 39 explained previously will
be used as a figure for defining parameters. FIG. 42 is a drawing
to show the flowchart for explaining procedures of the designing
method of optical system according to the fourth embodiment and
FIG. 43 is a drawing to schematically show the operations in the
fourth embodiment.
[0281] The individual of the optical system shown in FIG. 39 has
the parameters including the curvatures a-f of the lens surfaces
(boundary surfaces) of the respective lenses L1-L3, the
surface-to-surface distances (distances between the boundary
surfaces) AB=d1, BC=d2, CD=d3, DE=d4, EF=d5, FG=d6, the refractive
indices of media between the lens surfaces nab=n1, nbc=n2, ncd=n3,
nde=n4, nef=n5, nfg=n6, and Abbe's numbers .nu.ab=.nu.1,
.nu.bc=.nu.2, .nu.cd=.nu.3, .nu.de=.nu.4, .nu.ef=.nu.5,
.nu.fg=.nu.6.
[0282] A chromosome of this optical system is represented by a
string of genes of
-a-b-c-d-e-f-d1-d2-d3-d4-d5-d6-n1-n2-n3-n4-n5-n6-.nu.1-.nu.2-.nu-
.3-.nu.4-.nu.5-.nu.6-.
[0283] In this example the curvatures a-f of the respective lens
surfaces are continuous values and the surface-to-surface distances
d1-d6 are also continuous values. In the above chromosome of the
fourth embodiment values (alleles) of the genes corresponding to
the curvatures and surface distances are continuous values.
[0284] Next, the operation of the fourth embodiment will be
described referring to FIG. 42 and FIG. 43. Here is described an
example in which, for handling the curvatures a-f of the respective
lens surfaces and the surface-to-surface distances d1-d6 in common
dimensions, each parameter is first normalized in the range of
-2.048 to 2.048. This makes influence on each ray continuously
equivalent. For example, let us consider a lens system
characterized by the following parameters.
1 Curvature Lens distance Refractive index (L1) 0.0100 8 1.5 (L2)
-0.0083 5 1.8 (L3) -0.0077
[0285] When in the above lens system the curvature of the lens
surface (the final surface) located closest to the image plane is
changed (bent) in order to obtain a desired focal length, the above
lens system is represented in the form of a vector of at most four
dimensions or less. When the final surface is not bent, the above
lens system is represented in the form of a vector of at most five
dimensions or less.
[0286] When the above lens system is expressed in the form of a
five-dimensional vector without normalization (note that the
refractive indices are fixed values), the vector is (0.0100,
-0.0083, -0.0077, 8, 5). For normalizing each parameter (each
vector component), the maximum and minimum need to be set as to at
least the curvatures and lens distances (glass, air).
[0287] This setting of maximum and minimum is done by the user
(designer). For example, the curvatures are set in the range of
-0.5 to +0.5, the surface distances with glass in between as a
medium (corresponding to the thicknesses of lenses) are set in the
range of 0.1 to 50, and the surface distances with air in between
as a medium (the aerial distances between the lenses) are set in
the range of 0.1 to 100. In this case, a normalized value of each
parameter is defined as follows.
[0288] (normalized value)=4.096.times.(value of parameter)/(maximum
of parameter-minimum of parameter)
[0289] Accordingly, the normalized vector is given as (0.04096,
-0.0339968, -0.0315392, 0.6567, 0.41042).
[0290] In this fourth embodiment 0 is first set as an initial value
into the parameter t indicating the generation (step ST0).
[0291] Then a plurality of lens data for the components of the
above parameters a-f, d1-d6, n1-n6, and .nu.1-.nu.6 are reproduced
to form an initial population (step ST1). In this step random
numbers are used for generation of plural parameters to form the
initial population. On this occasion each parameter made of a
random number is normalized as described above. If an optical
system characterized by these parameters (a candidate of an optical
system to be designed) satisfies predetermined constraints and if
all rays incident into this optical system can reach the image
plane, the parameters for this optical system are used as an
initial population. The above constraints are preferably, for
example, conditions for eliminating optical systems that cannot
exist physically (for example, optical systems having a negative
surface-to-surface distance) (for efficient search). Such
constraints can be set by the designer himself or herself on the
occasion of setting the maximums and minimums of the curvatures and
the other parameters.
[0292] Since in this fourth embodiment the field angle and overall
focal length of the optical system are the constraints given by the
designer, values satisfying the above constraints, obtained by
tracing of paraxial rays in the optical system, are used for the
curvature f of the lens surface closest to the image plane in the
optical system and the distance d6 from the lens surface closest to
the image plane thereto. In this fourth embodiment the refractive
indices of the respective lenses L1 to L3 are given preliminarily
and the medium between the lenses L1-L3 is the air. Thus, the
chromosome of the optical system in this embodiment is given as
-a-b-c-d-e-d1-d2-d3-d4-d5-.
[0293] In this case, the above initial population includes a
plurality of individuals having different chromosomes.
[0294] Subsequently, 1 is added to the generation parameter t (step
ST2), and thereafter, as shown in FIG. 43, a set of three parents
Pa1, Pa2, Pa3 are selected from the above initial population (step
ST3). If t>2, a set of three parents Pa1, Pa2, Pa3, are not
selected from the initial population, but from a population of the
generation. (t-1) one before the current generation (t).
[0295] In step ST4 new genes are reproduced from the parents Pa1,
Pa2, Pa3 selected in above step 3. Namely, the crossover is
effected. The crossover operator in this fourth embodiment is the
UNDX (Uni-modal Normal Distribution Crossover) out of the genetic
operators for directly handling continuous values. Therefore, in
the following description, the operation in step ST4 with
application of UNDX will be described referring to FIG. 44.
[0296] First, considering a space of n dimensions corresponding to
the number of genes that the chromosomes of individuals have, each
individual is expressed by a vector of the n dimensions. In this
fourth embodiment, since a chromosome is composed of the ten genes
-a-b-c-d-e-d1-d2-d3-d4-d5- - as described above, n=10 and each
individual is expressed by a vector of the ten dimensions (a, b, c,
d, e, d1, d2, d3, d4, d5). In FIG. 44 each individual is
represented as a three-dimensional vector for illustration's
sake.
[0297] Substep ST4-1
[0298] First, as shown by 441 in FIG. 44, substep ST4-1 is carried
out to set points VC1, VC2, VC3 on the space., corresponding to the
parents Pa1, Pa2, Pa3 selected in step ST3. Here, let M be a middle
point between the point VC1 and the point VC2 and H be the length
of a perpendicular from the point VC3 to a line segment
VC1-VC2.
[0299] Substep ST4-2
[0300] Next, as shown by 442 in FIG. 44, substep ST4-2 is carried
out to reproduce normal random deviates with the middle point as an
expectation and .sigma.a as a standard deviation. According to the
normal random deviates, a point P1 is reproduced on the segment
VC1-VC2 (the point P1 is generated on the segment VC1-VC2 in a
probability according to such a normal distribution that the middle
point M is at the vertex).
[0301] Substep ST4-3
[0302] Further, as shown by 443 in FIG. 44, substep ST4-3 is
carried out to generate a probability space Ex of an n-dimensional
normal distribution according to n normal random deviates with the
point P1 as an expectation and .sigma.b as a standard deviation. In
this embodiment the probability space is the one of ten-dimensional
normal distribution.
[0303] Substep ST4-4
[0304] As shown by 444 in FIG. 44, this substep ST4-4 is carried
out to reproduce a point P2 according to the n-dimensional normal
random deviates having the probability space Ex of the
n-dimensional normal distribution reproduced in above substep
ST4-3. Namely, the point P2 is reproduced in the probability
according to the n-dimensional normal distribution.
[0305] Substep ST4-5
[0306] As shown by 445 in FIG. 44, this substep ST4-5 is carried
out to set a hyperplane .pi. perpendicular to the vector VC1-VC2,
including the point P1. The hyperplane .pi. is a plane of the (n-1)
dimensions obtained by subtracting 1 from the number of dimensions
of the space. In this embodiment the hyperplane is of nine
dimensions. A point P3 is defined at a point where a perpendicular
from the point P2 to the hyperplane .pi. falls.
[0307] Substep ST4-6
[0308] As shown by 446 in FIG. 44, this substep ST4-6 is carried
out to generate a point P4 capable of constituting a vector P1-P4
having a component parallel to a vector P1-P3 starting from the
point P1 and having a length according to normal random deviates
with the point P1 as an expectation and .sigma.c as a standard
deviation. Here, .sigma.c is proportional to the n-th power root of
a distance H-VC3 between the point H and the point VC3 in step
ST4-1.
[0309] The n-dimensional coordinates of the point P4 reproduced by
above steps ST4-1 to ST4-6 correspond to the n parameters a, b, c,
d, e, f, d1, d2, d3, d4, d5 of a chromosome of a new-born gene or a
child. In this step ST4 of the fourth embodiment the substeps ST4-1
to ST4-6 described above are repeated m times, whereby m new genes
are reproduced from the three parents Pa1, Pa2, Pa3.
[0310] After completion of above step ST4, evaluation values are
calculated for the set (family set) of the m new-born genes
reproduced in this step ST4 and the two parents Pa1, Pa2. In this
fourth embodiment the evaluation values (fitness values) are
expressed by a single evaluation criterion or by one performance
function .phi. consisting of a set of plural evaluation criteria.
For example, when there are k evaluation criteria abe(1), abe(2), .
. . , abe(k), they can be linearly combined with weights on the
respective evaluation criteria abe(1), abe(2), . . . , abe(k).
Namely, the performance function .phi. is given as
.phi.=W1*abe(1)+W2*abe(2)+ . . . +Wk*abe(k). Then an individual
with the best evaluation value and an individual selected by
roulette selection are selected from the above family set (step
ST5).
[0311] The roulette selection is a technique for selecting an
individual in a predetermined probability in proportion to a
fitness value (evaluation value) of each individual or in
proportion to the ranking of the fitness value. For example, let us
consider an example of four individuals A, B, C, D. In the former
fitness value proportion method, supposing the fitness value of
individual A is 40, that of individual B 60, that of individual C
100, and that of individual D 200, the probability of selection of
individual A is 10%, that of individual B 15%, that of individual C
25%, and that of individual D 50% (A:B:C:D=10:15:25:50). In the
latter fitness value ranking proportion method, the probability of
selection of individual A is 10%, that of individual B 20%, that of
individual C 30%, and that of individual D 40% (A:B:C:D=1:2:3:4).
The fourth embodiment employs the latter fitness value ranking
proportion method (ranking method), but it should be noted that the
invention is not limited to this method.
[0312] Further, in step S6 the individuals except for the two
individuals selected in above step S5 are allowed to become extinct
(selection). Then the selected individuals replace the parents Pa1,
Pa2 in the initial population. This replacement of the parents in
the population by the selected individuals is called the alteration
of generation.
[0313] Subsequently in step ST7, an individual having the best
evaluation value is outputted out of the individuals in the
population after the alteration of generation (presentation of the
best solution) and the operation of step ST2 is repeated.
[0314] Here, above steps ST2 to ST7 are called the first generation
of the genetic algorithm. In the second generation and after, three
parents Pa1, Pa2, Pa3 are selected from the population after the
alteration of generation.
[0315] Repetitive operations of such alteration of generation will
automatically reproduce an optical system as a globally optimal
solution independent of the initial solution given.
[0316] The fourth embodiment described above uses only the
crossover operator, but the mutation operator may also be used in
addition to this crossover operator.
[0317] An example of application of the mutation operator in
addition to the crossover operator will be described. In this
fourth embodiment above steps ST5 and ST6 select genes to be left
in the next generation from the set (family set) of the m new-born
genes and the two parents Pa1, Pa2.
[0318] In the case of the mutation operator being applied, new
chromosomes are sampled at random Pm.times.(m+2) times according to
the preset mutation rate Pm out of the set (family set) of the m
new-born genes and two parents Pa1, Pa2. Then new genes of mutants
are reproduced by changing an arbitrary (one or more) chromosome
out of the genes in the sampled chromosomes on an arbitrary basis
(or according to a probability distribution biased to the
neighborhood of its original value). After that, genes to be left
in the next generation are selected from the set of the m new-born
genes, the new-born genes reproduced as mutants, and the two
parents Pa1, Pa2. Namely, the two parents Pa1, Pa2 in the
population are replaced by an individual of the best evaluation
value and an individual selected by roulette selection from this
set.
[0319] In this fourth embodiment it is also possible to apply only
the mutation operator without application of the crossover
operator. In this case, instead of above steps ST3 to ST6, new
chromosomes are sampled at random by the number of times according
to the mutation rate Pm out of the population consisting of the
chromosomes corresponding to the plural lens data. Then mutants are
generated by changing an arbitrary (one or more) gene of the
sampled chromosomes according to a probability distribution biased
to the vicinity of its original value. Then chromosomes to be left
in the next generation are selected from this set of mutants.
[0320] Next described is an example of the multi-objective
optimization in the fourth embodiment. In steps ST5 and ST6 of the
fourth embodiment, one performance function .phi. is used at the
time of selection of the individual of the best evaluation value
and selection of the individual by roulette selection from the set
(family set) of the m new-born genes and two parents Pa1, Pa2, as
described above. These two individuals selected then replace the
two parents Pa1, Pa2 in the population. This means that even cases
of presence of plural evaluation criteria are handled in the form
of a single performance function by linearly combining these
evaluation criteria.
[0321] In the case of the multi-objective optimization, steps S11
and S15-S17 below are executed in place of above steps ST1 and
ST5-ST7.
[0322] The operation of the multi-objective optimization will be
described using the flowchart shown in FIG. 45 and the schematic
diagram of operations shown in FIG. 46.
[0323] Step ST110 is different from above step ST1 in that
evaluation values of plural performance functions .phi.1-.phi.k are
calculated according to a plurality of (k) evaluation criteria of
plural individuals (lens data) in the initial population.
[0324] Next, step ST150 is a step of calculating each of evaluation
values of plural performance functions .phi.1-.phi.k for each
individual in the set (family set) of the m new-born genes
reproduced in above step ST4 and the two parents Pa1, Pa2. Then
individuals not being Pareto optimal (non-Pareto solutions) are
selected from these evaluation values. Here, a Pareto optimal
solution means, under presence of plural evaluation criteria, a
solution superior in at least one evaluation criterion out of the
plural evaluation criteria to the other solutions.
[0325] Subsequently, step ST160 is carried out to weed out
individuals not-being Pareto optimal. This step eliminates
individuals inferior to the other individuals in the all evaluation
functions among the evaluation values of the plural performance
functions .phi.1-.phi.k. The Pareto optimal individuals left herein
compose a population of the next generation and in the next
generation parents for the crossover operator are selected from
this population.
[0326] After completion of step ST160, step ST170 is carried out to
output the Pareto optimal individuals obtained in above step ST160,
and then the steps are repeated in order from step ST2.
[0327] After that, steps ST2-ST4 and steps S150-S160 are repeated
in the same manner, thereby effecting the multi-objective
optimization of individuals in the population.
[0328] Also in the case of the multi-objective optimization as
described, the mutation parameter described above can also be
applied in addition to the crossover operator. Further, the
mutation parameter can also be applied instead of the crossover
operator.
[0329] The execution of the above steps (steps ST110, ST2-ST4, and
ST150-ST170) permits the multi-objective optimization to be
effected even with presence of plural evaluation criteria for
optical system and without placing the tradeoff ratio (weights) on
these evaluation criteria. This results in always preserving the
elitist individual out of plural chromosomes in the population and
selecting better individuals to be left in the next generation
according to such predetermined probabilities that better
individuals are more unlikely to be weeded out, thus enabling the
multi-point simultaneous search with efficiency.
[0330] In the Pareto optimization described above, the number of
individuals in the population differs depending upon the
generation. The Pareto optimization described above has such an
advantage that the population size does not become too large,
because non-Pareto solutions are weeded out in order.
[0331] The fourth embodiment described above employed the UNDX as a
crossover operator, but, instead thereof, BLX-.alpha. or NDX can
also be applied.
[0332] It can also be contemplated that, for example, some of
plural evaluation criteria are linearly combined to form plural
performance functions and these performance functions are subjected
to the multi-objective optimization.
[0333] Described below are application examples where the designing
method of optical system according to the fourth embodiment is
applied to an optical system consisting of fifteen lenses.
[0334] Application 1
[0335] Application 1 of the fourth embodiment is to design a
projection optical system for transferring a circuit pattern on a
mask onto a wafer at a demagnification ratio.
[0336] In this Application 1, the evaluation criteria of the
optical system are those (1) to (16) listed below and the
performance functions are those obtained by linearly combining some
evaluation criteria.
[0337] (1) Spherical aberration at the maximum numerical
aperture
[0338] (2) Spherical aberration at 75% of the maximum numerical
aperture
[0339] (3) Sagittal image surface at the maximum object height
[0340] (4) Meridional coma of upper rays at the maximum object
height and at 70% of the maximum numerical aperture
[0341] (5) Meridional coma of lower rays at the maximum object
height and at 70% of the maximum numerical aperture
[0342] (6) Meridional coma of upper rays at the maximum object
height and at 50% of the maximum numerical aperture
[0343] (7) Meridional coma of lower rays at the maximum object
height and at 50% of the maximum numerical aperture
[0344] (8) Meridional coma of upper rays at 75% of the maximum
object height and at 70% of the maximum numerical aperture
[0345] (9) Meridional coma of lower rays at 75% of the maximum
object height and at 70% of the maximum numerical aperture
[0346] (10) Meridional coma of upper rays at 75% of the maximum
object height and at 50% of the maximum numerical aperture
[0347] (11) Meridional coma of lower rays at 75% of the maximum
object height and at 50% of the maximum numerical aperture
[0348] (12) Meridional coma of upper rays at 50% of the maximum
object height and at 70% of the maximum numerical aperture
[0349] (13) Meridional coma of lower rays at 50% of the maximum
object height and at 70% of the maximum numerical aperture
[0350] (14) Meridional coma of upper rays at 50% of the maximum
object height and at 50% of the maximum numerical aperture
[0351] (15) Meridional coma of lower rays at 50% of the maximum
object height and at 50% of the maximum numerical aperture
[0352] (16) Distortion at the maximum object height
[0353] In this Application 1 constraints (restraint conditions) of
(17) to (19) listed below are given.
[0354] (17) A distance from the lens surface closest to the reticle
thereto, WD>25
[0355] (18) Magnification of the overall system, .beta.=-3.0 (in
the ray tracing from the wafer side to the reticle side)
[0356] (19) Distances between the lens surfaces, d<25
[0357] In this Application 1, parameters to be altered in designing
of the optical system are the curvatures of the respective lens
surfaces and surface distances. The refractive index n of glass
materials of the lens elements is preliminarily given as
n=1.56384.
[0358] Under the above preconditions, the parameters in the
designing method of optical system according to the fourth
embodiment are set as listed below.
[0359] Size of initial population: 50
[0360] Number of crossovers: 300,000
[0361] Number of children generated by crossover operator: 20
[0362] .sigma.a of UNDX: 0.5.times.VC1VC
[0363] .sigma.b of UNDX: 1
[0364] .sigma.c of UNDX: 0.35.times.(VC1VC2).sup.1/n
[0365] The projection optical system represented by Table 1 below
was generated by above Application 1. In Table 1, .beta. is the
magnification from the reduced side (the wafer side) to the
enlarged side (the reticle side), L the conjugate length
(object-image distance), NA the reduced-side numerical aperture,
and WD the distance from the lens surface closest to the reticle
thereto. In Table 1 below, numerals in the left end column
represent surface numbers from the wafer surface, r radii of
curvature, d surface distances, and n indices of refraction.
2TABLE 1 .beta. = -3.0 L = 380.52758 NA = 0.30 WD = 24.96409 r d n
0 .infin. 12.49751 1.000000 (conjugate plane on wafer side) 1
-120.85562 18.23479 1.563840 2 -70.82037 12.67890 1.000000 3
-92.19251 7.16914 1.563840 4 -59.21449 17.55595 1.000000 5
-2708.99510 18.44782 1.563840 6 -132.78761 6.51286 1.000000 7
.infin. 8.63269 1.000000 (aperture stop) 8 173.72702 17.51225
1.563840 9 -238.22683 12.06856 1.000000 10 -101.43114 17.11040
1.563840 11 -203.60368 15.33085 1.000000 12 166.15887 5.69882
1.563840 13 -238.28227 3.88554 1.000000 14 -81.50201 6.11086
1.563840 15 -78.07944 6.05381 1.000000 16 74.48074 12.50264
1.563840 17 -220.48073 11.91135 1.000000 18 -82.31072 8.08050
1.563840 19 275.42052 19.38991 1.000000 20 -67.68594 9.24998
1.563840 21 -210.48779 7.57457 1.000000 22 -244.62903 15.02358
1.563840 23 -48.44360 9.91096 1.000000 24 69.69555 8.01551 1.563840
25 97.21932 10.30374 1.000000 26 -28.90898 14.32626 1.563840 27
102.56872 7.92284 1.000000 28 43.32668 8.22930 1.563840 29
2828.72964 10.81079 1.000000 30 243.23452 16.81079 1.563840 31
39.78134 24.96409 1.000000 32 .infin. (conjugate plane on reticle
side)
[0366] FIG. 47 is a cross-sectional view of the projection optical
system represented in above Table 1 and FIG. 48 is an aberration
diagram to show aberrations thereof. Here, 481 in FIG. 48 is a
spherical aberration diagram, 482 in FIG. 48 is an astigmatism
diagram, 483 in FIG. 48 shows the meridional coma at the maximum
object height, 484 in FIG. 48 the meridional coma at 75% of the
maximum object height, 485 in FIG. 48 the meridional coma at 50% of
the maximum object height, 486 in FIG. 48 the meridional coma on
the optic axis, 487 in FIG. 48 the sagittal coma at the maximum
object height, 488 in FIG. 48 the sagittal coma at 75% of the
maximum object height, 490 in FIG. 48 the sagittal coma at 50% of
the maximum object height, 491 in FIG. 48 the sagittal coma on the
optic axis, and 492 in FIG. 48 a distortion diagram. In the
astigmatism diagram of 482 in FIG. 48 the dashed line indicates the
meridional image surface and the solid line the sagittal image
surface. In FIG. 48 NA denotes the numerical aperture and Y the
object height.
[0367] FIG. 49 is a cross-sectional view of the projection optical
system as Application 2 where twenty four thousand crossovers were
carried out under conditions that the size of initial population
was 400 and fifty children were generated.
[0368] By the designing method of optical system in each of
Applications 1 and 2 as described above, a good solution
(individual) can be obtained even though the number of lens
elements is large.
[0369] Although the glass materials for the respective lenses were
predetermined in above Application 1 and Application 2, it is
needless to mention that they can be defined as parameters. On this
occasion, since the glass materials often exist discretely, the
genetic parameters used should be those for handling discrete
values.
[0370] As described above, the designing method of optical system
according to the present invention has the effect of generating the
optical system as a globally optimal solution independent of the
initial solution given. Since the present invention utilizes the
genetic parameters for handling the continuous values explicitly,
child individuals of the next generation generated from parent
individuals can succeed to characters (properties) of the parent
individuals and this succession to characters is repeated, thus
achieving the effect of eliminating wasteful searches and thus
obtaining the optimal solution or the near-optimal solution within
a shorter time. Further, the designing method of optical system of
the present invention also has the effect that a plurality of
optical systems can be generated simultaneously as multi-objective
optimal solutions that must exist corresponding to a plurality of
conflicting evaluation criteria.
* * * * *