CN111897188B - Method for detecting polarization aberration of ultra-large numerical aperture strict vector imaging system - Google Patents

Method for detecting polarization aberration of ultra-large numerical aperture strict vector imaging system Download PDF

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CN111897188B
CN111897188B CN202010597775.7A CN202010597775A CN111897188B CN 111897188 B CN111897188 B CN 111897188B CN 202010597775 A CN202010597775 A CN 202010597775A CN 111897188 B CN111897188 B CN 111897188B
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CN111897188A (en
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李艳秋
李恩泽
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Beijing Institute of Technology BIT
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    • G—PHYSICS
    • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
    • G03F—PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR
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Abstract

The invention provides a method for detecting polarization aberration of an ultra-large numerical aperture strict vector imaging system, which comprises the following steps: creating new tables for strict thick mask vector imagingCharacterization theory and model, polarization aberration J of thick mask effectMDesigning a group of test masks with specific period range in a theoretical form of multiplication product with a traditional thin mask vector imaging model, and calculating J of the group of masks by using a thick mask strict electromagnetic field theory under the illumination of a specific rotationally symmetrical (with rotational invariance) light sourceMEstablishing a library, establishing a strict vector measurement model of the polarization aberration of the projection objective according to a new characterization theory of strict thick mask vector imaging, obtaining a nonlinear relation between the expansion coefficient of the polarization aberration J and the strict analysis of the space image frequency spectrum, and establishing a synchronous measurement method by analyzing the property of the thick mask effect to improve the JMUtilization and detection speed of the library; and finally obtaining an overdetermined measurement equation set, and solving the overdetermined measurement equation set reversely by adopting a neural network algorithm to realize the high-precision detection of the polarization aberration of the high-numerical-aperture imaging system.

Description

Method for detecting polarization aberration of ultra-large numerical aperture strict vector imaging system
Technical Field
The invention relates to the technical field of aberration detection of an ultra-large numerical aperture imaging system, in particular to an online detection method for polarization aberration of an ultra-large numerical aperture lithography imaging system.
Background
The present imaging technology of very large Numerical Aperture (NA) is widely applied to microscopes, telescopes and immersion lithography machines for preparing very large scale integrated circuits. With the trend of the preparation process towards nodes of 22nm and below, the NA value of a photoetching machine is mostly 1.35 or above, and resolution enhancement technologies such as a vector imaging model, polarized illumination, light source mask optimization and the like are developed to improve the imaging quality. The degrading effect of polarization aberration (NA) on imaging is not negligible, and these techniques all require the introduction of PA parameters of the imaging system to optimize the results that are practically usable. Therefore, a method for rapidly and accurately detecting the projection objective PA of the imaging system on line needs to be developed, which is a premise for carrying out real-time control or compensation on the projection objective PA and has important significance for improving the imaging quality.
Related patents (chinese patent CN104281011A, chinese patent CN108828901A) all disclose methods of Polarization Aberration (PA) of over-high numerical aperture imaging systems. The method is based on a mask imaging method, establishes a linear or nonlinear relation between a PA expansion coefficient and some measured values of the space image, and obtains the PA of the imaging system by inverse extrapolation. However, the method adopts a thin mask imaging model at present, and ignores the thick mask effect of the mask. However, the current theory finds that the thick mask effect of the test mask has a considerable polarization effect, which causes an extra PA, if not considered, which is doped in the measurement result together with the PA of the imaging system, and reduces the measurement accuracy of the PA of the imaging system.
Disclosure of Invention
The invention provides a method for detecting polarization aberration of an ultra-large numerical aperture strict vector imaging system, which is used for establishing a PA (power amplifier) measuring method under a strict thick mask imaging model, eliminating PA (power amplifier) caused by a doped thick mask effect in a measuring result and measuring all information of PA of the imaging system with high precision.
The method for detecting the polarization aberration of the ultrahigh numerical aperture strict vector imaging system comprises the following steps:
step one, introducing polarization aberration J of thick mask effect into a traditional thin mask vector imaging modelMTo characterize a strictly thick mask vector imaging model;
designing a group of test masks, and calculating the polarization aberration J of the group of masks by using the strict electromagnetic field theory of the thick maskMAnd establishing a library for the database;
step three, establishing an imaging system polarization aberration measurement model corresponding to the strict thick mask vector imaging model according to the strict thick mask vector imaging model in the step one, and obtaining a strict nonlinear analytical relation between the expansion coefficient of the polarization aberration J of the imaging system and the space image frequency spectrum:
F{I(x，y，z))＝∫∫S(fs，gs)P·Sm·PHdfsdgswherein Sm is a group containing JMA polarization aberration sensitivity matrix of information, wherein P is a vector formed by polarization aberration expansion coefficients; i (x, y, z) is an aerial image, (f)s,gs) Coordinates representing the effective light source surface, S (f)s,gs) Is the intensity distribution of the effective light source;
step four, the polarization aberration J of each mask in the library of the step twoMAnd (4) importing the data into the imaging system polarization aberration measurement model in the third step, establishing an overdetermined measurement equation set and solving the overdetermined measurement equation set to obtain a vector P formed by expansion coefficients, and realizing the detection of the polarization aberration.
Further, in the second step, the library establishing method comprises:
selecting n angles theta is 0 DEG and theta1,θ2...θn(ii) a Theta epsilon (0,180 DEG), each of which isRespectively rotating the mask according to the n angles to obtain sub-masks, then placing all the sub-masks on one mask plate, and establishing a test mask matrix; setting the light source to be a light source with rotation invariance, for each sub-mask in a test mask matrix, the polarization aberration J for the sub-mask at a first angleMCarrying out n-angle rotation transformation to correspondingly obtain the polarization aberration J of the sub-mask under each angleM-q(ii) a Thereby completing the library construction; the number of the masks and the number of n are required to satisfy that the number of the established over-determined equation sets is larger than the number of the expansion coefficients.
Preferably, the test mask includes, but is not limited to, a one-dimensional dense line binary mask or a phase shift mask.
Preferably, the light source is annular illumination, conventional coherent illumination, TE polarized illumination or TM polarized illumination.
Preferably, the specific process of the first step is as follows: the strict thick mask vector imaging model is characterized by:
Figure GDA0002678774370000021
wherein (x, y, z), (f, g) and (f)s,gs) Coordinates of an image surface, a pupil surface and an effective light source surface are respectively; kp(f, g) is the modeling of the projection objective part in the optical system, and the concrete form is as follows: k (f, g) × a (f, g) × Mo (f, g) × J (f, g), where a (f, g) is the radiance correction factor and M is the radiance correction factoro(f, g) is the correction of the in-and-out pupil electric vector direction for large NA objectives, and J (f, g) is the objective PA function characterized by Jones pupil, with each element being a 2 x 2 complex matrix expressed as
Figure GDA0002678774370000022
From pesudo-zernike basis functions
Figure GDA0002678774370000031
Can be unfolded into
Figure GDA0002678774370000032
Wherein
Figure GDA0002678774370000033
Is the PA expansion coefficient to be found, and can also be expressed in the form of a fringe ordinal number { ai，bi，a′i，b′i|i∈{1，2，3，…}}；S(fs,gs) Intensity distribution of effective light source, E (f)s,gs) Is the polarization distribution of the effective light source; t (F, g) is the mask spectral far field, the Fourier transform of the mask near field, F { M } which is solved for strict electromagnetic field theorynear(x, y) }, each element being a 2 × 2 complex matrix; and in the thin mask imaging model T (F, g) is the Fourier transform of the mask transmittance function F { T (x, y) }, which is a scalar matrix; so can be selected from F { Mnear(x, y) } separates a scalar matrix equal to F { t (x, y) } and the rest is the thick mask effect:
Figure GDA0002678774370000034
and the second term has the same form as J (f, g), so it is defined as a thick mask effect polarization aberration JMIt can be obtained from the element-to-element ratio of the spectral far-field matrix of the two models
Figure GDA0002678774370000035
Will JMThe strict thick mask imaging model can be represented as a traditional thin mask vector imaging model by adding thick mask effect polarization aberration JMThe theoretical form of (1):
Figure GDA0002678774370000036
the invention has the following beneficial effects:
1) the method establishes a new representation theory of a strict thick mask model, and deduces the relation between the projection objective PA and the space image frequency spectrum strict analysis according to the new representation theory, so that the method has high detection precision.
2) The imaging system PA measuring method established by the invention utilizes the imaging system to directly measure, does not need to add extra optical elements, and can measure the on-line PA information after the projection objective is assembled.
3) The invention establishes a synchronous measurement method and can effectively improve JMThe utilization rate of the library is improved, and the detection speed is increased.
4) The method of the present invention is applicable to any high NA polarization (vector) imaging system including, but not limited to, high resolution microscopes, telescopes, and lithography systems for the fabrication of very large integrated circuits.
Drawings
FIG. 1 is a schematic block diagram of a lithographic projection system useful in the present invention.
FIG. 2 is a PA test mask matrix designed according to the present invention.
Fig. 3 shows design values of the lithographic projection objective PA in this embodiment.
Fig. 4 is a comparison of the measurement error of the PA difference detection method of the thick mask imaging model in the present embodiment and the PA detection method of the conventional thin mask imaging model.
101-light source, 102-test mask, 103-mask stage, 104-high numerical aperture projection objective, 105-spatial image intensity distribution sensor, 106-workpiece stage, 107-data processing system.
Detailed Description
The invention is described in detail below by way of example with reference to the accompanying drawings.
Taking a lithography imaging system as an example, the lithography projection objective PA detection system used in the invention is shown in FIG. 1, that is, the PA detection method provided by the invention only utilizes the imaging system itself, does not need to add extra optical elements, and is a method capable of realizing online detection of PA at low cost.
The method comprises the following specific steps:
step one, establishing a new characterization theory and model for strict thick mask vector imaging, which is characterized by the polarization aberration J of the thick mask effectMThe theoretical form of the product with a conventional thin mask vector imaging model. The vector imaging model of the thick mask can be characterized as:
Figure GDA0002678774370000041
wherein (x, y, z), (f, g) and (f)s,gs) The coordinates of the image plane, the pupil plane and the effective light source plane are respectively. I (x, y, z) is an aerial image, S (f)s,gs) Intensity distribution of effective light source, E (f)s,gs) Is the polarization distribution of the effective light source. K is the modeling of the projection objective part in the photoetching system, and the concrete form is as follows:
K(f,g)＝A(f,g)·Mo(f,g)·J(f,g) (2)
a is the radiance correction factor, M0 is the correction of the in-and-out pupil electric vector direction for large NA objectives, J (f, g) is the objective PA function characterized by jones pupil, each element is a 2 × 2 complex matrix, which can be expressed as:
Figure GDA0002678774370000042
from pesudo-zernik basis functions
Figure GDA0002678774370000043
The expansion is as follows:
Figure GDA0002678774370000051
wherein
Figure GDA0002678774370000052
Is that the PA expansion coefficient to be solved can be expressed in the form of a fringe ordinal numberi，bi，a′i，b′i|i∈{1，2，3，…}}。
Figure GDA0002678774370000053
Is the polarization distribution of the effective light source.
T (F, g) is the mask spectral far field, the Fourier transform of the mask near field, F { M } which is solved for strict electromagnetic field theorynear(x, y) }, each element being a 2 x 2 complex matrix. And in the thin mask imaging model T (F, g) is the Fourier transform of the mask transmittance function F { T (x, y) }, which isA scalar matrix. So can be selected from F { Mnear(x, y) } separates a scalar matrix equal to F { t (x, y) } and the rest is the thick mask effect:
Figure GDA0002678774370000054
the second term has the same form as J (f, g), so it is defined as the thick mask effect polarization aberration JMIt can be obtained from the ratio of the spectral farfields of the two models:
Figure GDA0002678774370000055
will JMSubstituting equation (1), a strict thick mask imaging model can be characterized as a traditional thin mask vector imaging model with thick mask effect polarization aberration J addedMThe theoretical form of (1):
Figure GDA0002678774370000056
as can be seen from the formula (7), the polarization aberration J is removedMBesides the import, other terms of the model are analyzed, and a basis is provided for establishing a strict analysis relation between the PA and the aerial image under a strict thick mask model.
Step two, designing a group of test masks with specific period ranges, including but not limited to binary masks and phase shift masks. Under the illumination of a specific rotation symmetrical (with rotation invariance) light source, J of the set of masks is calculated by using a thick mask strict electromagnetic field theoryMAnd establishing a library for the same. To reduce the complexity of thick mask imaging models, test masks are designed to satisfy a three-beam interference condition, i.e.
Figure GDA0002678774370000061
When the coherence factor sigma of the light source is small, NA is 1.35 deep ultraviolet for 193nmThe period p of the test mask is between about 660nm and 1520nm in a lithography system. The structure of the mark mask designed by the inspection method is shown in 201 of fig. 2. Selecting illumination structures and polarization states with rotational invariance, including but not limited to: conventional coherent illumination, annular illumination, TE/TM polarization, using thick masks with strict electromagnetic field theory to solve for F { M } for each masknear(x, y) } and F { t (x, y) }, and solving J according to the formula (6)MA mixture of J andMand merging the image data with the corresponding imaging conditions to build a library.
And step three, establishing an imaging system polarization aberration measurement model corresponding to the strict thick mask vector imaging model, wherein the imaging system polarization aberration measurement model is characterized in that a strict nonlinear analytical relation between the expansion coefficient of the imaging system polarization aberration J and the spatial image frequency spectrum is established. Taking a binary mask as an example, its diffracted far field under kirchhoff approximation can be written as:
Figure GDA0002678774370000062
where t is the cutoff frequency, when we choose a mask to satisfy equation (8), t is 1, and then equation (9) is substituted into equation (7), resulting in:
Figure GDA0002678774370000063
wherein Is(x, y, z) is the imaging of the mask under each source point. According to the expanded form of PA in equation (4), Is(x, y, z) is derived as follows:
Figure GDA0002678774370000064
wherein
Figure GDA0002678774370000071
jmax is the maximum expansion ordinal number, (12)
C11,C12,C21,C22,C31,C32And C'11,C'12,C'21,C'22,C'31C'32Is a row vector with length jmax, which can be obtained by the joint expression of the formula (4) and the formula (10). Bringing equation (11) back to (10) and taking a fourier transform on both sides of the equal number yields:
Figure GDA0002678774370000072
equation (13) is a strict analytic nonlinear relationship between the PA and the aerial image obtained by a strict thick mask imaging model, and is a quadratic form, Sm is a sensitivity matrix of the quadratic form, and J is included in the sensitivity matrixMAnd (4) information. P ═ aH,bH，a'H,b'H]A, b, a ', b' are vectors formed by the expansion coefficients of PA, and a is ═ a1，a2...ajmax]T，b＝[b1,b2...bjmax]T，a'＝[a'1，a'2...a'jmax]T，b'＝[b'1,b'2...b'jmax]TJmax is a truncation coefficient.
And step four, establishing an overdetermined measurement equation set and solving the overdetermined measurement equation set reversely by adopting a neural network algorithm to realize the high-precision detection of the polarization aberration. Substituting the test mask matrices (M) for different periods into equation (15) can create an overdetermined system of equations:
Figure GDA0002678774370000073
where θ is 0 ° and θ1,θ2...θn；θ∈(0,180°)，p＝p1,p2...pm(ii) a p.epsilon. (660,1520 nm). Equation (16) establishes an overdetermined system of M × N equations, and PA can be solved in reverse as long as M × N is ensured to be greater than the number of PA expansion coefficients (when the maximum expansion number jmax is 37, the expansion coefficients total 148). Because the equation system is quadratic, the traditional solution method based on gradient is easy to fall into a minimum value; so that the present invention isObviously, a neural network model is trained aiming at the problem, and the problem can be solved quickly and reversely with high precision.
Among them, the present invention is to increase JMThe utilization rate of each J is reduced, the library building cost and the measurement cost are reduced, a synchronous measurement method is established, and each J is connected with the corresponding JMThe masks in the library are arranged in different orientations into a plurality of test mask matrices, and imaging information of a plurality of masks in the mask matrices is synchronously obtained through single exposure. For Sm in each imaging relationship, a specific J needs to be introducedMThe library cost is high, so according to the characteristics of the thick mask effect, the shape of an effective light source, the illumination polarization state and the structure of the mask, the specific characteristics are as follows:
step 1, selecting N mask orientations, wherein theta is 0 DEG and theta1,θ2...θn(ii) a θ ∈ (0,180 °), each sample mask in the library is individually mapped into its own test mask matrix (M) in the above orientation, as shown in fig. 2.
Step 2, configuring mask effective light source and illumination polarization state as sample values, and for the first sub-mask of each test mask, connecting JMImporting a measurement model to obtain: f { I (x, y, z) } ═ s (F ═ s-s，gs)·P·Sm·PHdfsdgs
Step 3, because the illumination is not changed in rotation, the rotation of the mask is equivalent to the rotation of the whole imaging system, so that the second placing angle is theta1Of sub-masks
Figure GDA0002678774370000081
Can be obtained from the previous JMAngle of rotation theta1Obtaining, namely:
Figure GDA0002678774370000082
importing a measurement model to obtain:
Figure GDA0002678774370000083
step 4, similarly, for each sub-maskDies, corresponding thereto
Figure GDA0002678774370000084
All can be formed by J for the first timeMObtained by using a JMThe sample and one test mask are imaged, and N measurements can be made:
F{Iθ(x,y,z)}＝∫∫Sθ(fs,gs)·P·Smθ·PHdfsdgs (15)
where θ is 0 ° or θ1,θ2...θn(ii) a Theta epsilon (0,180 DEG). It can be seen that this greatly increases JMThe utilization rate of the method reduces the warehouse building cost and the measurement cost.
As shown in fig. 3, the design value of the polarization aberration at a certain field point of the projection objective used in the simulation was obtained by a ray tracing method for a projection objective designed autonomously in a laboratory. Graphs 301-308 are the real and imaginary parts of Jxx, Jxy, Jyx, Jyy, respectively. And (4) expanding the obtained product according to the formula (3) to obtain the expansion coefficient of the PA.
On the basis of the design value shown in fig. 3, a true value of a projection objective PA in simulation is added as a random error, and the PA difference detection method adopting the thick mask imaging model and the PA detection method adopting the thin mask imaging model provided by the invention are respectively used for detection. The error between the two obtained results and the true value is shown in FIG. 4, where 401 and 404 are ai,bi,a'i,b'i405-i,bi,a'i,b'iThe imaginary part of (c). It can be seen that, the method of the present invention considers the effect of the thick mask effect on the PA, so that the errors of the detection results are all less than 0.003, and the accuracy is improved by 3 times compared with the conventional method (about 0.009).
In summary, the above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (4)

1. The method for detecting the polarization aberration of the ultrahigh numerical aperture strict vector imaging system is characterized by comprising the following steps of:
step one, introducing polarization aberration J of thick mask effect into a traditional thin mask vector imaging modelMTo characterize a strictly thick mask vector imaging model;
Figure FDA0003027420060000011
wherein (x, y, z), (f, g) and (f)s,gs) Coordinates of an image surface, a pupil surface and an effective light source surface are respectively; kp(f, g) is the modeling of the projection objective part in the lithography system, in the following specific form: kp(f,g)＝A(f,g)·Mo(f, g) J (f, g), wherein A (f, g) is a radiance correction factor, Mo(f, g) is the correction of the in-and-out pupil electric vector direction for large NA objectives, and J (f, g) is the objective PA function characterized by Jones pupil, with each element being a 2 x 2 complex matrix expressed as
Figure FDA0003027420060000012
From pesudo-zernike basis functions
Figure FDA0003027420060000013
Can be unfolded into
Figure FDA0003027420060000014
Wherein
Figure FDA0003027420060000015
Is the PA expansion coefficient to be found, expressed in the form of a fringe ordinal number { a }i,bi,a′i,b′i|i∈{1,2,3,；
Figure FDA0003027420060000016
Intensity distribution of effective light source, E (f)s,gs) Is the polarization distribution of the effective light source; t (F, g) is the mask spectral far field, the Fourier transform of the mask near field, F { M } which is solved for strict electromagnetic field theorynear(x,y)-each element is a 2 x 2 complex matrix; and in the thin mask imaging model T (F, g) is the Fourier transform of the mask transmittance function F { T (x, y) }, which is a scalar matrix; from F { Mnear(x, y) } separates a scalar matrix equal to F { t (x, y) } and the rest is the thick mask effect:
Figure FDA0003027420060000017
and the second term has the same form as J (f, g), which is defined as the thick-mask effect polarization aberration JMIt is obtained from the element-to-element ratio of the spectral far-field matrix of the two models
Figure FDA0003027420060000018
Will JMThe strict thick mask imaging model can be represented as a traditional thin mask vector imaging model by adding thick mask effect polarization aberration JMThe theoretical form of (1):
Figure FDA0003027420060000019
Figure FDA0003027420060000021
designing a group of test masks, and calculating the polarization aberration J of the group of masks by using the strict electromagnetic field theory of the thick maskMAnd establishing a library for the database;
step three, establishing an imaging system polarization aberration measurement model corresponding to the strict thick mask vector imaging model according to the strict thick mask vector imaging model in the step one, and obtaining a strict nonlinear analysis relation between the expansion coefficient of the polarization aberration J (f, g) of the imaging system and the space image frequency spectrum:
F{I(x，y，z))＝∫∫S(fs，gs)PSm·PHdfsdg, wherein SmTo comprise JMA polarization aberration sensitivity matrix of information, wherein P is a vector formed by polarization aberration expansion coefficients; i (x, y, z) is an aerial image, (f)s,gs) Indicate validityCoordinates of the light source surface, S (f)s,gs) Is the intensity distribution of the effective light source;
step four, the polarization aberration J of each mask in the library of the step twoMAnd (4) importing the data into the imaging system polarization aberration measurement model in the third step, establishing an overdetermined measurement equation set and solving the overdetermined measurement equation set to obtain a vector P formed by expansion coefficients, and realizing the detection of the polarization aberration.
2. The method for detecting polarization aberration of an extra-large numerical aperture strict vector imaging system according to claim 1, wherein in the second step, the library establishing method comprises the following steps:
selecting n angles theta is 0 DEG and theta1,θ2...θn(ii) a Theta belongs to (0,180 degrees), each mask is respectively rotated according to the angle n to obtain sub-masks, and then all the sub-masks are placed on one mask plate to establish a test mask matrix; setting the light source to be a light source with rotation invariance, for each sub-mask in a test mask matrix, the polarization aberration J for the sub-mask at a first angleMCarrying out n-angle rotation transformation to correspondingly obtain the polarization aberration J of the sub-mask under each angleM-θ(ii) a Thereby completing the library construction; the number of the masks and the number of n are required to satisfy that the number of the established over-determined equation sets is larger than the number of the expansion coefficients.
3. The method of claim 2, wherein the test mask is a one-dimensional dense line binary mask or a phase shift mask.
4. The method of claim 2, wherein the light source is annular illumination, conventional coherent illumination, TE polarized illumination, or TM polarized illumination.
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