CN109977933B - Calculate the processing method of aliasing in lithography system model - Google Patents
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Abstract
This application discloses a kind of processing methods of aliasing in calculating lithography system model, this method comprises: according to effective sample area and preset spatial sampling interval, determine the sampled point quantity to mask function discrete sampling, mask function is used to indicate the geometry of mask；According to spatial sampling interval, anti-aliasing filter function is established, anti-aliasing filter function is used for bandwidth of the limit mask function in domain space；According to anti-aliasing filter function and sampled point quantity, convolution algorithm is taken for anti-aliasing filter function and mask function, is obtained to the filtered spatial sampling signal of mask function；Convolution algorithm spatial sampling signal and kernel function, determine light distribution.In this application, by anti-aliasing filter function, the bandwidth that sampling carrys out limit mask function is being filtered to mask function, so that avoiding the occurrence of aliasing leads to light intensity calculated distortion, can be improved the accuracy for calculating lithography model.
Description
Technical Field
The application belongs to the technical field of semiconductor lithography, and particularly relates to a processing method for calculating aliasing phenomenon in a lithography system model.
Background
Smart devices are inseparable from the present life, smart phones, smart homes and intelligent wearing devices are generally related to everyone. The operation of the intelligent device is more the operation of the chip which is silent behind. At present, in the production process of chips, the photolithography technique, which is a key technique in chip production, transfers a pattern designed in advance on a mask to a wafer substrate by using the principle of photochemical reaction, thereby making etching and ion implantation possible. The light irradiates on the mask to be diffracted, and the light with different diffraction orders is converged on the surface of the photoresist, and the process is an optical process; the image on the photoresist excites a chemical reaction, which upon baking results in the photoresist being locally soluble in a developer solution.
The photoetching system model mainly comprises a light source, a pupil lens, a mask and an imaging plane, namely, the light source emits incident light to pass through the pupil lens, and a graph corresponding to the mask is transferred to the imaging plane. Wherein the imaging plane refers to a plane on the wafer substrate. Lithography calculation refers to the technical staff describing the optical process through calculation and mathematical formulas to simulate the whole lithography process in order to achieve the purpose of auxiliary design. Based on the Hopkins diffraction optics theory, the optical part of the computational lithography model is represented as a cross-transfer function. The cross transfer function is described collectively by the corresponding functions of the light source and the pupil lens. And decomposing the cross transfer function by adopting a matrix eigenvalue to obtain a group of kernel functions for describing optical characteristics. Finally, the light intensity distribution on the imaging plane is calculated by convolution of the kernel function and the mask function. The convolution can directly adopt an analytic method, and is the integral operation of a kernel function in an effective area corresponding to the mask pattern. The analytical method can obtain a calculation result with higher precision for light intensity operation at a few positions. If more unknown light intensities need to be calculated, the light intensity calculated by the analytical method cannot meet the requirements of production design. Theoretically, the convolution can be translated to a multiplication operation on frequency, and then the convolution result is obtained using a fast fourier transform.
With the fast fourier method, discrete sampling of the signal in the spatial domain is required. According to nyquist's sampling law, the sampling frequency should be at least 2 times the highest frequency of the analyzed signal, otherwise it may happen that the high frequency signal in the signal is folded to the low frequency band, and there is a phenomenon of spurious frequency components, i.e. frequency aliasing. In the photoetching system, if the frequency aliasing phenomenon occurs, the light intensity calculation is distorted, and the accuracy of calculating the photoetching model is further reduced.
Disclosure of Invention
The application provides a processing method for calculating aliasing phenomena in a photoetching system model, which can be used for solving the problem that the aliasing phenomena in the related technology can cause distortion of light intensity calculation, and further the accuracy of calculating the photoetching model is reduced.
In a first aspect, the present application provides a method for processing aliasing in a computational lithography system model, the method comprising:
determining the number of sampling points for discretely sampling a mask Function according to an effective sampling range and a preset space sampling interval, wherein the effective sampling range is the ninth zero root of a first-order Bessel Function (Bessel Function), and the mask Function is used for representing the geometric shape of a mask;
according to the spatial sampling intervalEstablishing an anti-aliasing filter function, wherein the anti-aliasing filter function is used for limiting the bandwidth of the mask function on a frequency domain space; allowing frequencies below that of the computational lithography model lens design limits/Passes through the lens;
and obtaining a spatial sampling signal after the mask function and the anti-aliasing filter function are processed by adopting convolution operation on the anti-aliasing filter function and the mask function according to the anti-aliasing filter function and the number of the sampling points.
In the application, after acquiring the spatial sampling signal, the terminal performs convolution operation on the spatial sampling signal and the kernel function to determine the light intensity distribution. Kernel functionIn connection with the lenses of the lithography system, are inherently frequency limited with a maximum cut-off frequency of，Where is the numerical aperture size of the pupil lens,is the wavelength of the incident light. The bandwidth of the frequency response for the kernel function is limited, while the mask functionThe frequency bandwidth corresponding to direct discrete sampling is infinite. Aiming at aliasing display brought by discrete sampling in a fast Fourier method, an anti-aliasing filtering function is provided, convolution operation is carried out on the anti-aliasing filtering function and the discrete sampling of a mask function on a spatial domain, a spatial sampling signal obtained after the mask function is processed by the filtering function is obtained, and the spatial sampling signal is limited in frequency and used for determining light intensity distribution.
Optionally, the establishing an anti-aliasing filtering function according to the spatial sampling interval includes:
establishing the anti-aliasing filter function according to the following relation
Wherein,representing the anti-aliasing filter function in a single pass,is at a sampling pointThe coordinates of the two points in a cartesian direction,is the polar coordinates of the sample point(s),is the aperture value of the pupil lens,is the wavelength of the light source and,for the purpose of the first-order bezier function,is a window function of
Is the ninth zero-root of the first order bessel function.
Optionally, the method further comprises:
decomposing the mask function into elementary graphics units, wherein:
establishing the basic graphic unit according to the following relational expression
Wherein,a three-eighths infinite plane representing the basic graphical unit description,for defining said three-eighths infinite plane;
decomposing the mask function according to the following relation
Wherein,is thatThe coefficients corresponding to the represented decomposed graphic elements,is thatThe coefficients corresponding to the represented decomposed graphic elements,is thatCoefficients corresponding to the represented decomposition graph elements;the graphical elements are such that said basic graphical elements are rotated 90 degrees with respect to a cartesian coordinate system,graphic unit and the basic graphic unitSymmetrical about the x-axis.
Optionally, the performing a convolution operation on the anti-aliasing filtering function and the mask function includes:
determining a convolution operation of the anti-aliasing filter function and the mask function according to the following relation
Wherein,is an integral variable;
the method further comprises the following steps:
determining a convolution operation of the anti-aliasing filter function and the primitive graphics unit according to the following relation
Wherein,the mask function is represented as a function of the mask,the basic graphic unit is represented by a graphic unit,andrepresenting the corresponding integration area.
Optionally, the method further comprises:
dividing the integration area into a plurality of integrated sub-areas;
establishing the effective area of the basic graphic unit and the anti-aliasing filter function according to the following relation
Determining the divided sub-regions according to the following relation
Wherein,andis a sampling serial number and has a value range of，，For the number of the sampling points in question,is the sampling interval.
Optionally, the method further comprises:
calculating the integral operation of the sub-region according to the following relational expression
Wherein,，andperforming numerical integration operation;
according to the relational expressionDetermining a convolution result of the antialiasing function and the primitive graphics unit.
Optionally, the method further comprises:
and pre-calculating the convolution result of the anti-aliasing function and the basic graphic unit to generate a pre-calculation data table.
Optionally, the method further comprises:
calculating a convolution result of the anti-aliasing function and the mask function according to the pre-calculation data table, wherein the convolution result of the anti-aliasing function and the mask function is obtained according to the following relational expression
Wherein,by looking up said pre-calculated data tableAnd then the step of determining the number of the first time,androtated 90 degrees with respect to the cartesian coordinate system,andsymmetrical about the x-axis.
According to the scheme provided by the application, the terminal carries out filtering sampling on the mask function to limit the bandwidth of the mask function before calculating the light intensity distribution through the established anti-aliasing filtering function, so that the highest frequency of the mask function is less than one half of the sampling frequency, the light intensity calculation distortion caused by the aliasing effect is avoided, and the accuracy of calculating the photoetching model can be improved.
In addition, the anti-aliasing filtering function provided by the application effectively limits the bandwidth of the mask function in the frequency domain space, and effectively recovers a continuous signal in the space and eliminates the aliasing effect. The anti-aliasing filter function is a low-pass filter, limits high-frequency signals in optical phenomena and reasonably corresponds to the design of the lens. In addition, the anti-aliasing filter is used in the application, which is a two-dimensional integral fast calculation method, and the pre-calculation data table obtained by inquiring pre-calculation is utilized, so that the final signal is fast and accurate to obtain.
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In order to more clearly illustrate the technical solutions in the embodiments of the present application, the drawings needed to be used in the description of the embodiments are briefly introduced below, and it is obvious that the drawings in the following description are only some embodiments of the present application, and it is obvious for those skilled in the art to obtain other drawings based on these drawings without creative efforts.
FIG. 1 is a schematic diagram illustrating the structure of a lithography model according to an exemplary embodiment;
FIG. 2 is a flow diagram illustrating a method of processing to compute aliasing in a model of a lithography system in accordance with an exemplary embodiment;
FIG. 3a is a diagram illustrating a circular function in the frequency domain in accordance with an exemplary embodiment;
FIG. 3b is a schematic diagram illustrating a window function in accordance with an exemplary embodiment;
FIG. 3c is a schematic diagram illustrating an anti-aliasing filtering function according to an exemplary embodiment;
FIG. 4 is a diagram illustrating sample signals of a mask function according to an exemplary embodiment;
FIG. 5 is a schematic diagram illustrating an integration region according to an exemplary embodiment;
FIG. 6 is a diagram illustrating a pre-calculation table according to an exemplary embodiment.
Detailed Description
In order to make the technical solutions in the embodiments of the present application better understood and make the above objects, features and advantages of the embodiments of the present application more comprehensible, the technical solutions in the embodiments of the present application are described in further detail below with reference to the accompanying drawings.
In the method provided by the embodiment of the application, the execution main body of each step may be a terminal. The terminal is used for calculating relevant data of a photoetching system in the photoetching process and photoetching according to the photoetching model. As described in fig. 1. Which shows a schematic structural view of a lithography model. A light source 101, a condenser lens 102, a reticle 103, a projection pupil 104, a projection lens 105, and a wafer 106. Each light emitted from the light source 101 is collimated by the condenser lens 102, and the collimated light is irradiated onto the reticle 103, so that the pattern on the reticle is imaged on the surface of the wafer 106 through the projection pupil 104 and the projection lens 105, and thus the surface of the wafer is also referred to as an imaging plane. For frequency aliasing, in other fields, an anti-aliasing filter is usually designed before the discrete sampled signal to limit the bandwidth of the signal in the important band. When the frequency is above the nyquist frequency, its frequency response is zero. If a fast Fourier method is adopted to calculate the light intensity, an anti-aliasing filter function for production needs to be designed aiming at the aliasing phenomenon so as to improve the precision of the design of the photoetching process.
FIG. 2 is a process diagram illustrating a method of computing aliasing in a model of a lithography system, according to an example embodiment. The method may include several steps as follows.
Step 201, determining the number of sampling points for discretely sampling the mask function according to the effective sampling range and the preset spatial sampling interval.
In the processing process of the light intensity subsystem, the terminal determines the light intensity distribution on the imaging plane and needs to perform discrete sampling on the kernel function and the mask function. The kernel function is a set of functions used to describe optical characteristics in the lithography system, and is obtained by matrix eigenvalue decomposition of the cross-transfer function. The cross-transfer function is described by a source function and a pupil function. Hence, according to the aperture value of the pupil lensAnd the wavelength of the light sourceThe terminal determines the maximum allowable cutoff frequency for the kernel function during sampling as/. The cut-off frequency selected by the terminal during discrete sampling is greater than/. Thus, for the kernel function, the terminal does not need to limit the bandwidth of its samples. However, for the mask function, since the signal in the spatial domain of the mask function is a step function, the bandwidth of the mask function in the frequency domain is infinite, so that the bandwidth of the mask function in the spatial domain of the frequency domain needs to be limited. Therefore, the terminal determines the number of sampling points for discretely sampling the mask function according to the effective sampling range and the preset space sampling interval, and further determines a discrete sampling matrix.
Wherein, the valid sampling range is the ninth zero root of the first-order Bessel Function (Bessel Function). Optionally, the valid sampling range may also be the eleventh or thirteenth zero-root of a first order Bessel Function (Bessel Function). The spatial sampling interval may be preset by the skilled person depending on the interpolation accuracy, e.g. 4nm or 6 nm.
Optionally, the terminal determines the number of sampling points for discrete sampling of the mask function according to the following relation:
wherein,it is shown that the effective sampling range,which represents the interval of the spatial sampling,indicating the number of sample points. In the embodiment of the present application,the effective sampling range represented is the ninth zero-root of the first-order Bessel function, also denoted as. For example,is the wavelength of the light source with the wavelength of 1024nm,is the nano-particles with the particle size of 4nm,then it is 257.
Step 202, establishing an anti-aliasing filtering function according to the space sampling interval.
In order to limit the bandwidth of the mask function in the frequency domain space, the terminal establishes an anti-aliasing filter function. The anti-aliasing filter function is a filter function in the spatial domain, which may also be referred to as an anti-aliasing filter. The terminal establishes the anti-aliasing filtering function according to the following relation:
wherein,representing an anti-aliasing filter function that is,is the cartesian coordinates of the sample point(s),is the polar coordinates of the sample point(s),in the form of a first order bezier function,is a window function of
Is the ninth zero-root of the first order bessel function. Function(s)Fourier transform of (1) as a function of circle, radius of/. Combining the above window functionsActing on first-order Bessel functions to limit the effective range of sampled information。
Exemplarily, as shown in fig. 3a, 3b and 3c, which show a circle function in the frequency domain, a window function in the spatial domain and an anti-aliasing filtering function, respectively. The cylinder 301 and the trapezoid 302 are in the form of a circular function in a three-dimensional coordinate system and a two-dimensional coordinate system, respectively. The curved surface 303 and the curve 304 are in the form of a window function in a three-dimensional coordinate system and a two-dimensional coordinate system, respectively. The curved surface 305 and the curved line 306 are forms of the anti-aliasing filter function in a three-dimensional coordinate system and a two-dimensional coordinate system, respectively.
And 203, acquiring the spatial sampling signals processed by the mask function and the anti-aliasing filter function by adopting convolution operation according to the anti-aliasing filter function and the number of the sampling points.
And in order to avoid frequency aliasing, the terminal directly performs discrete sampling on a spatial domain according to the sampling point, and then processes the sampled signal by using the established anti-aliasing filter function to obtain a spatial sampling signal after the mask function is filtered. The terminal acquires the spatial sampling signal filtered by the mask function, and actually performs convolution operation on the anti-aliasing filtering function and the mask function. The convolution operation is a double integration operation over a complex area. Therefore, the terminal can perform convolution operation by the following formula:
wherein,as a function of the mask, the mask is,the polygonal effective area of the mask plate is the effective sampling range,is an integral variable; .
Alternatively, for the convolution operation of the anti-aliasing filter function and the mask function, the terminal performs the calculation by decomposing the mask function. The mask function decomposition means that the mask function is decomposed into functions corresponding to the respective basic pattern units obtained by dividing the mask pattern. Firstly, the terminal determines the basic graphic unit according to the following relation:
wherein,representing the three-eighths infinite plane of the basic graphical unit description,for defining three eighths of an infinite plane. The terminal divides the mask graph into a group of basic graph units, and the mask function is decomposed according to the following relational expression:
wherein,is thatThe coefficients corresponding to the represented decomposition pattern,is thatThe coefficients corresponding to the represented decomposition pattern,is thatThe coefficients corresponding to the represented decomposition pattern.，Andis 1 or-1.The graphic elements are basic graphic elements rotated 90 degrees with respect to the cartesian coordinate system,the graphic element and the basic graphic element are symmetrical about an x-axis. Finally, the terminal can convert the convolution operation of the anti-aliasing filter function and the mask function, and calculate according to the following relation:
wherein,androtate ninety degrees coincidence clockwise about the cartesian coordinate system;andsymmetric about the x-axis of the cartesian coordinate system. Therefore, the terminal only needs to determineThe value of (A) can be determined according to the symmetry relationshipAndand determining the final value of the convolution operation, i.e. the spatial sampling signal.
Illustratively, as shown in FIG. 4, a graph 401 represents a sample signal without a filtered mask function. Graph 402 represents a spatially sampled signal after being processed by an anti-aliasing filtering function.
Alternatively, for the way that the terminal performs convolution operation by decomposing the mask function, the terminal needs to calculateThe value of (c). In response to this, the present inventors have conducted extensive studies on,can be expressed as the following relation:
wherein,represents the effective area of integration, an. The terminal divides the integration effective area into four integration sub-areas according to the following relational expression:
wherein,andis a sampling serial number and has a value range ofAnd is and。for the number of sample points mentioned above,the above-mentioned sampling interval. Illustratively, as shown in fig. 5, the shaded area 501 represents the above-described integration effective area. The shaded area 502 represents the integrating sub-area. The integrating sub-regionIs the definition area of the basic graphic unit. The shaded area 503 represents the integrating sub-area. Shaded region 504 represents the integrating sub-region. Integration sub-regionAnd an integration sub-regionIs a belt-mounted infinite area. Shaded region 505 represents an integrating sub-region. Integration sub-regionIs a bounded area. And according to the integration results corresponding to the four integration subregions, determining an anti-aliasing filter function and a convolution result of the three-eighths infinite plane. The terminal may calculate the integration result of the sub-region according to the following relation:
wherein,，andby numerical integration. Determining a convolution result of the antialiasing function and the primitive graphics unit according to the following relation:
。
thus, terminal computingOnly the integral result of the integral sub-area needs to be determined, and then the integral result is determinedFinally, determine. The terminal calculates the convolution of the anti-aliasing filter function and the mask function by using the division of the integral region and the division of the mask pattern, and for the integral operation of the divided integral sub-region and the convolution operation of the divided basic pattern, the terminal may calculate the basic data having these invariant properties in advance and generate a pre-calculation data table to store these data. Illustratively, as shown in fig. 6, it shows the data format and partial content of the pre-calculation data table in the case that the effective sampling range is 1024nm, the spatial sampling interval is 4nm, and the number of sampling points is 257. Wherein the first row 601 is a value of X and the first column 602 is a value of Y. When convolution operation is actually adopted for the mask function and the anti-aliasing filtering function, the pre-stored pre-calculation data table is inquired, and the filtered space sampling signal can be rapidly determined. The computing power and processing overhead required by the double integral operation during the convolution operation of the anti-aliasing filtering function and the mask function are reduced, the computing time can be reduced, and the photoetching efficiency is improved.
After acquiring the space sampling signal, the terminal is used for calculating the light intensity distribution so as to determine the light intensity distribution and calculate the photoetching model. For calculating the light intensity distribution, it is actually a convolution operation of the kernel function and the mask function. Thus, the terminal performs a convolution operation on the kernel function and the filtered spatially sampled signal to determine the light intensity distribution.
In the embodiment of the application, the terminal limits the bandwidth of the mask function by filtering and sampling the mask function before calculating the light intensity distribution through the established anti-aliasing filtering function, so that the highest frequency of the mask function is less than one half of the sampling frequency, the light intensity calculation distortion caused by the aliasing effect is avoided, and the accuracy of calculating the photoetching model can be improved.
In addition, the present application provides an anti-aliasing filter function for spatially sampled signals to limit the bandwidth in the frequency domain and to reconstruct the emphasis frequency. The invention discloses a method for quickly calculating, which uses convolution operation as an anti-aliasing filter, directly convolves the convolution operation into double integral operation, and efficiently calculates basic data with invariable property by utilizing the division of integral areas and the convolution operation of basic graphs aiming at the defect that the integral operation wastes most of calculation capacity. And (4) quickly obtaining the filtered space sampling signal by considering a graphic decomposition method corresponding to the mask plate and a table look-up function. The embodiment of the invention well embodies the beneficial effects, and the technical personnel in the field can easily verify the beneficial effects of the invention according to the embodiment.
Further, in the description of the present application, "a plurality" means two or more than two unless otherwise specified. In addition, in order to facilitate clear description of technical solutions of the embodiments of the present application, in the embodiments of the present application, terms such as "first" and "second" are used to distinguish the same items or similar items having substantially the same functions and actions. Those skilled in the art will appreciate that the terms "first," "second," etc. do not denote any order or quantity, nor do the terms "first," "second," etc. denote any order or importance.
The above-described embodiments of the present application do not limit the scope of the present application.
Claims (6)
1. A method of processing aliasing in a computational lithography system model, the method comprising:
determining the number of sampling points for discretely sampling a mask Function according to an effective sampling range and a preset space sampling interval, wherein the effective sampling range is the ninth zero root of a first-order Bessel Function (Bessel Function), and the mask Function is used for representing the geometric shape of a mask;
establishing an anti-aliasing filter function according to the spatial sampling interval, wherein the anti-aliasing filter function is used for limiting the bandwidth of the mask function on a frequency domain space;
according to the anti-aliasing filter function and the number of the sampling points, performing convolution operation on the anti-aliasing filter function and the mask function to obtain a spatial sampling signal after the mask function and the anti-aliasing filter function are processed;
wherein said establishing an anti-aliasing filter function according to said spatial sampling interval comprises:
establishing the anti-aliasing filter function according to the following relation
AAF(x,y)＝AAF(r,θ)＝W(r)×J1(2πrNA/λ)/r
Wherein AAF (x, y) represents the anti-aliasing filter function, (x, y) is the Cartesian coordinates of the sampling point, (r, θ) is the polar coordinates of the sampling point, NA is the aperture value of the pupil lens, λ is the wavelength of the light source, J1(2 π rNA/λ) is the first order Bessel function, and W (r) is a window function
γ1,9Is the ninth zero-root of the first order Bessel function;
wherein the performing a convolution operation on the anti-aliasing filtering function and the mask function comprises:
decomposing the mask function into basic graphic units, wherein:
establishing the basic graphic unit according to the following relational expression
m0(x,y)＝1、x,y∈{y≥0,x+y≥0}
Wherein m is0(x, y) represents a three-eighths infinite plane described by the primitive graphic unit, and x, y belongs to { y is larger than or equal to 0, and x + y is larger than or equal to 0} and is used for defining the three-eighths infinite plane;
decomposing the mask function according to the following relation
m(x,y)＝∑α×m0(x,y)+∑β×m0(y,-x)+∑γ×m0(x,-y)
Wherein α is m0The coefficient corresponding to the decomposed graphic element represented by (x, y) β is m0(y, -x) and gamma is m0Coefficient, m, corresponding to the decomposition graphic unit represented by (x, -y)0(y, -x) the basic graphical unit is rotated 90 degrees, m, with respect to a Cartesian coordinate system0(x, -y) the graphics unit is symmetric about the x-axis with the base graphics unit;
and performing convolution operation on the anti-aliasing filtering function and the mask function according to the decomposition result of the mask function.
2. The method of claim 1, wherein said convolving said anti-aliasing filtering function and said masking function comprises:
determining a convolution operation of the anti-aliasing filter function and the mask function according to the following relation
Wherein, tau1,τ2Is an integral variable;
the method further comprises the following steps:
determining a convolution operation of the anti-aliasing filter function and the primitive graphics unit according to the following relation
Wherein m (x, y) represents said mask function, m0(x, y) represents the basic graphic units, Ω and Ω0Representing the corresponding integration area.
3. The method of claim 2, further comprising:
dividing the integration area into a plurality of integrated sub-areas;
establishing the effective area of the basic graphic unit and the anti-aliasing filter function according to the following relation
Ω0＝{τ1∈[-τ2-x-y,γ1,9],τ2∈[-y,γ1,9]}
Determining the divided sub-regions according to the following relation
Ω′＝{τ1∈[-τ2,γ1,9],τ2∈[0,γ1,9]}
Wherein i, j and l are sampling serial numbers, and the value ranges are allN is the number of sampling points, and delta is the sampling interval.
4. The method of claim 3, further comprising:
calculating the integral operation of the sub-region according to the following relational expression
I0＝∫∫Ω′AAF(τ1,τ2)dτ1dτ2＝0.375
Wherein,andperforming numerical integration operation;
according to the relational expressionDetermining a convolution result of the antialiasing filter function and the primitive graphics unit.
5. The method of claim 4, further comprising:
and pre-calculating the convolution result of the anti-aliasing filter function and the basic graphic unit to generate a pre-calculation data table.
6. The method of claim 5, further comprising:
calculating a convolution result of the anti-aliasing filter function and the mask function according to the pre-calculation data table, wherein the convolution result of the anti-aliasing filter function and the mask function is obtained according to the following relational expression
Wherein,as determined by looking up the pre-calculated data table,androtated 90 degrees about the cartesian coordinate system to coincide,andsymmetrical about the x-axis.
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