A method, device, equipment and medium for calculating light intensity

Abstract

The present invention provides a method, device, equipment, and medium for calculating light intensity. The method utilizes the Abbe imaging principle combined with the refraction and reflection effects of optical thin film stacks to calculate the polarization of light, obtaining a relationship between the electric field intensity of a single point of the light source and the image plane. Based on the relationship, the electric field intensity is integrated within the light source using the polarization of the light source, and the order of multiple integrations is adjusted using the Hopkins imaging formula to obtain a vector expression for the transmission cross coefficient. The vector expression for the transmission cross coefficient is expanded to obtain N scalar transmission cross coefficients. The N scalar transmission cross coefficients are used to simulate and calculate the light intensity of an actual light source, where N is a positive integer. Through derivation based on physical principles, without any additional approximations, while retaining the form of the transmission cross coefficient, the method achieves both accuracy and speed, meeting the computationally intensive requirements for optical proximity effect correction.

Description

Technical Field

[0001] The present invention relates to the field of photolithography analysis, and in particular to a method, device, equipment and medium for calculating light intensity. Background Art

[0002] Polarization is an intrinsic property of light as an electromagnetic wave. In lithography systems, using a properly polarized light source can improve image resolution and is a key enabling technology for advanced manufacturing processes. To simulate the lithography imaging process, computational lithography software must accurately handle the polarization effects of light. Furthermore, due to the complex structures and dense graphics of advanced chip designs, computational lithography software, such as optical proximity correction (OPC), requires extensive calculations of light intensity to simulate the reticle's aerial image. Therefore, ensuring the accuracy and speed of computational lithography software is crucial.

[0003] There are two methods for calculating light intensity in mainstream lithography simulation software: one based on the Abbe imaging principle and the other based on the Hopkins imaging formula. However, the former requires point-by-point calculation of the light source, which becomes extremely computationally intensive when the mask pattern becomes complex, making it difficult to achieve speed. The latter assumes that light from different directions is simply superimposed on the image plane and fails to account for polarization effects. This leads to significant errors in scenarios with large numerical apertures and polarized light sources. This demonstrates that existing calculation methods cannot simultaneously meet application requirements for both speed and accuracy.

[0004] Therefore, a method, device, equipment and medium for calculating light intensity are urgently needed to improve the above problems. Summary of the Invention

[0005] The object of the present invention is to provide a method, device, equipment and medium for calculating light intensity, which can have the advantages of both accuracy and speed and meet the intensive calculation requirements of optical proximity effect correction.

[0006] In a first aspect, the present invention provides a method for calculating light intensity, comprising:

[0007] The polarization of light is calculated using the Abbe imaging principle combined with the refraction and reflection effects of optical thin film stacks, and the relationship between the electric field intensity of a single point of the light source and the image plane is obtained;

[0008] Based on the relationship, the electric field intensity is integrated within the light source range using the polarization of the light source, and the order of multiple integrations is adjusted using the Hopkins imaging formula to obtain a transmission cross coefficient vector expression;

[0009] Expanding the transmission cross coefficient vector expression to obtain N scalar transmission cross coefficients;

[0010] The light intensity of the actual light source is simulated and calculated using the N scalar transmission cross coefficients, where N is a positive integer.

[0011] The method of the present invention has the following beneficial effects: the polarization of light is calculated using the Abbe imaging principle combined with the refraction and reflection effects of optical thin film stacks to obtain a relationship between the electric field intensity of a single point of the light source and the image plane; based on the relationship, the electric field intensity is integrated within the light source range using the polarization of the light source, and the order of multiple integrations is adjusted using the Hopkins imaging formula to obtain a vector expression for the transmission cross coefficient; the vector expression for the transmission cross coefficient is expanded to obtain N scalar transmission cross coefficients; and the N scalar transmission cross coefficients are used to simulate and calculate the light intensity of the actual light source, where N is a positive integer. Through derivation based on physical principles without any additional approximation, while retaining the form of the transmission cross coefficient, the method has the advantages of both accuracy and speed, and can meet the intensive computational requirements of optical proximity effect correction.

[0012] In a second aspect, the present invention provides a device comprising modules / units for executing any one of the possible design methods of the first aspect. These modules / units may be implemented by hardware, or by hardware executing corresponding software implementations.

[0013] In a third aspect, the present invention provides an electronic device comprising a memory and a processor, wherein the memory stores a program that can be run on the processor, and when the program is executed by the processor, the electronic device implements a method for executing any possible design of any of the above aspects.

[0014] In a fourth aspect, the present invention provides a readable storage medium, wherein the readable storage medium stores a program, and when the program is executed, it implements any possible design method of any of the above aspects.

[0015] For the beneficial effects of the second to fourth aspects, please refer to the description of the first aspect. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 A schematic flow chart of a method for calculating light intensity provided by an embodiment of the present invention;

[0017] Figure 2 A schematic structural diagram of a light intensity calculation device provided by an embodiment of the present invention;

[0018] Figure 3 A schematic structural diagram of an electronic device provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0019] In order to make the purpose, technical solutions and advantages of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention. Unless otherwise defined, the technical terms or scientific terms used herein should be the common meanings understood by people with ordinary skills in the field to which the present invention belongs. The words "including" and similar words used in this article mean that the elements or objects appearing before the word cover the elements or objects listed after the word and their equivalents, without excluding other elements or objects.

[0020] In view of the fact that existing calculation methods cannot simultaneously meet the application requirements of speed and accuracy, the calculation method of light intensity provided by the present invention can have the advantages of both accuracy and speed, and meet the intensive calculation requirements of optical proximity effect correction. The technical solutions in the embodiments of the present invention are described below in conjunction with the drawings in the embodiments of the present invention. Among them, in the description of the embodiments of the present invention, the terms used in the following embodiments are only for the purpose of describing specific embodiments, and are not intended to be limiting of the present invention. As used in the specification of the present invention and the appended claims, the singular expressions "a", "said", "above", "the" and "this" are intended to also include expressions such as "one or more", unless there is a clear contrary indication in the context. It should also be understood that in the following embodiments of the present invention, "at least one", "one or more" refer to one or more (including two). The term "and / or" is used to describe the association relationship of associated objects, indicating that three relationships can exist; for example, A and / or B can mean: A exists alone, A and B exist at the same time, and B exists alone, where A and B can be singular or plural. The character " / " generally indicates that the related objects are in an "or" relationship. References to "one embodiment" or "some embodiments" throughout this specification mean that a particular feature, structure, or characteristic described in connection with that embodiment is included in one or more embodiments of the present invention. Thus, the phrases "in one embodiment," "in some embodiments," "in other embodiments," and "in yet other embodiments" appearing in various places throughout this specification do not necessarily refer to the same embodiment, but rather mean "one or more, but not all, embodiments," unless otherwise specifically emphasized. The terms "include," "comprising," "having," and their variations all mean "including but not limited to," unless otherwise specifically emphasized. The term "connected" includes both direct and indirect connections, unless otherwise stated. The terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the quantity of the technical features being referred to. In the embodiments of the present invention, the terms "exemplarily" or "for example" are used to indicate an example, illustration, or description. Any embodiment or design described as "exemplary" or "for example" in the embodiments of the present invention should not be construed as preferred or advantageous over other embodiments or designs. Rather, the use of "exemplarily" or "for example" is intended to present the relevant concepts in a concrete manner.

[0021] like Figure 1 As shown, the present invention provides a method for calculating light intensity, comprising:

[0022] S101, using the Abbe imaging principle combined with the refraction and reflection effects of the optical thin film stack to calculate the polarization of light, and obtain a relationship between the electric field intensity of a single point of the light source and the image plane.

[0023] In some embodiments, the polarization of light is calculated using the Abbe imaging principle in combination with the refraction and reflection effects of the optical thin film stack to obtain a relationship between the electric field intensity of a single point of the light source and the image plane, specifically including: calculating the relationship between the light intensity and the average value of the square of the electric field intensity within one period, and calculating the expression of the light intensity; calculating the spatiotemporal expression of the electric field intensity, and obtaining an expression in complex form; calculating the expression of the light intensity in complex amplitude form; obtaining the light intensity distribution on the image plane in scalar form based on the Abbe imaging formula in scalar form and the complex amplitude distribution; calculating the polarization of light in combination with the refraction and reflection effects of the optical thin film stack to obtain the Abbe imaging formula in vector form; calculating the transmittance and reflectance respectively according to the Fresnel formula to obtain a vector thin film function; obtaining the light intensity distribution on the image plane in vector form based on the Abbe imaging formula in vector form and the vector thin film function.

[0024] S102 , integrating the electric field intensity within the light source range using the polarization of the light source based on the relationship, and adjusting the order of multiple integrations using the Hopkins imaging formula to obtain a transmission cross coefficient vector expression.

[0025] S103 , expanding the transmission cross coefficient vector expression to obtain N scalar transmission cross coefficients.

[0026] S104 , performing simulation calculation on the light intensity of the actual light source using the N scalar transmission cross coefficients, where N is a positive integer.

[0027] In some specific embodiments, the relationship between the light intensity and the average value of the square of the electric field intensity within one period satisfies the following formula:

[0028]

[0029] Where I is the light intensity, T is the time period of electric field vibration, E is the electric field strength, and t is time;

[0030] When the amplitude A is the maximum value of the electric field intensity, the space-time expression of the electric field intensity satisfies the following formula:

[0031]

[0032] Where ω is the angular frequency, k is the spatial angular frequency; z is the spatial coordinate along the direction of light propagation, is the initial phase;

[0033] The expression for calculating light intensity is based on formula (1). The specific process is shown in the following formula:

[0034]

[0035] Rewriting formula (2), the complex expression satisfies the following formula:

[0036]

[0037] In the formula, Re is the real part operator, j is the imaginary unit, e jωt is the time factor. After separating the time factor, we get the complex number This complex number is called the complex amplitude of light, and is also represented by A below;

[0038] In complex amplitude form, the expression of light intensity satisfies the following formula:

[0039]

[0040] When the electric field is decomposed into the x-axis and y-axis directions, the electric field intensity components in the two directions at the same spatial position are:

[0041]

[0042] Where A x , A y They represent the amplitudes of the electric field strength components in the two directions, and δ represents the phase of the y-axis component leading the x-axis component. The total electric field strength is the vectorial superposition of the two components, which can be expressed as:

[0043]

[0044] The magnitude of this vector is the absolute value of the total electric field strength, as shown below:

[0045]

[0046] According to A x , A y Due to the different proportional relationship and the value of δ, polarized light can be divided into three types: linear polarized light, circular polarized light and elliptically polarized light. The relationship between the instantaneous value of the total electric field intensity of the three polarization types and time is different. However, by substituting formula (8) into formula (3), we can get

[0047]

[0048] Based on the above formula, the light intensity is only related to A x , A y It is related to the phase difference δ, that is, the intensity of polarized light is independent of the polarization form; the complex amplitude vector is the vector composed of complex amplitudes in all directions, as shown in the following formula:

[0049]

[0050] Based on the above formula, the light intensity is expressed as the inner product of the complex amplitude vector and its conjugate:

[0051]

[0052] For a beam of coherent light of unit intensity at normal incidence, its complex amplitude distribution on the image plane is as follows:

[0053]

[0054] Where (x, y) is the spatial domain coordinate, A(x, y) is the expression of the complex amplitude in the spatial domain; (μ, v) is the spatial frequency domain coordinate, each spatial frequency domain coordinate corresponds to an angle in the spatial domain, indicating the propagation direction of the diffracted light, P(μ, v) and M(μ, v) are the expressions of the pupil function and the mask in the frequency domain, respectively; is the inverse Fourier transform operator;

[0055] For a beam of coherent light of unit intensity incident at any angle, let the frequency domain coordinate corresponding to its direction be (μ, v), then the complex amplitude distribution of this beam of light on the image plane can be expressed as:

[0056]

[0057] In the formula, (μ1, v1) is the independent variable, and (μ, v) is a constant. To distinguish them, the above formula adds the subscript (μ1, v1)→(x, y) to the inverse Fourier transform operator, and at the same time indicates the coordinate variable names before and after the transformation. After obtaining the complex amplitude distribution, it is substituted into formula (5) to obtain the light intensity distribution on the image plane, and the Abbe imaging formula in scalar form is obtained.

[0058] The Abbe imaging formula in vector form satisfies the following formula:

[0059]

[0060] Where Q is the film function, which is a vector, and P and M are still scalars;

[0061] According to the Fresnel formula, when light is refracted and reflected at the interface, its electric field needs to be decomposed into components perpendicular to and parallel to the incident surface, and the transmittance and reflectance are calculated respectively, which are recorded as TE component and TM component respectively. For a beam of polarized light with unit intensity, the amplitudes of its TE and TM components are recorded as ρ TE and ρ TE , and both satisfy the following formula:

[0062]

[0063] In addition, let the phase difference between the TM direction component and the TE direction component be δ, then the thin film function of the vector is full

[0064] The following formula is enough:

[0065]

[0066] Where, each component can be calculated based on the complex refractive index, image depth and incident angle of each layer of the film; for isotropic films, Q TE and Q TM Each component has a circularly symmetric property; substituting Equation (16) into Equation (14) yields the vector form of the Abbe imaging formula for the components on the x, y, and z axes; finally, after obtaining the distribution of the complex amplitude vector, substituting it into Equation (11) yields the light intensity distribution on the image plane.

[0067] When the light intensities of light from different directions on the image plane are directly superimposed to obtain the total spatial image, the Abbe imaging formula is:

[0068] I(x,y)=∫∫J(μ,v)A(x,y|μ,v)A * (x,y|μ,v)dμdv (Formula 17)

[0069] Where J(μ, v) is the light source function, which represents the light intensity of the light source in the direction of (μ, v); A(x, y|μ, v) means that A is a function of (x, y) and contains the parameter (μ, v), which is the integral variable;

[0070] The Hopkins imaging formula is used to further expand the Abbe imaging formula and adjust the order of integration to obtain:

[0071]

[0072] The second to third rows of the formula are substituted to separate (μ, v) from the mask expression; the integral in the innermost brackets of the fourth row is called the transmission cross coefficient, that is:

[0073] T CC (μ1, v1; μ2, v2)=∫∫J(μ, v)P (μ1+μ, v1+v)P * (μ²+μ, v²+v)dμdv (Formula 19)

[0074] Substituting formula (19) back into formula (18), we can get:

[0075]

[0076] When the transmission cross coefficient is expressed as a separate form of variables (μ1, v1) and (μ2, v2):

[0077] T CC (μ1, v1; μ2, v2) = Φ (μ1, v1) Φ *(μ2,v2) (Formula 21)

[0078] Then formula (20) can be organized as:

[0079]

[0080] Where, is the convolution operator, m(x, y) is the representation of the mask in the spatial domain, that is, the spatial domain function corresponding to M(x, y); the last line of the above formula uses the properties of the inverse Fourier transform to convert the frequency domain multiplication into a convolution operation in the spatial domain.

[0081] The polarization of the light source is calculated using the direction angle α and the phase difference Φ. α and Φ are related to A in formula (6). x , A y The corresponding relationship between and δ is:

[0082]

[0083] When the light source is a polarized light source and the photolithography mode is the polarization mode XY4, the expressions of α and φ are:

[0084]

[0085] In order to calculate the film function Q, it is necessary to convert the x and y components into TE and TM components, and the TE and TM components can be obtained by rotating the x and y components by a certain angle; to represent the angle, the argument function atan2 is first introduced: atan2(v, μ) is the argument value after the rectangular coordinates (μ, v) are converted to polar coordinates; let φ = atan2(v, μ) be the argument of a point (μ, v) in the frequency domain, at which point a plane rectangular coordinate system with TE and TM as the coordinate axes can be established, and the coordinate axes of the coordinate system can be obtained by rotating the x and y coordinate axes counterclockwise by (φ-π / 2) radians, and the complex amplitude vector [A TE , A TM ]By [A x , A y ]Counterclockwise rotation Radians are obtained;

[0086] For an incident light beam of unit intensity, the complex amplitudes of the x and y components are:

[0087]

[0088] Then according to the rotation relationship between x, y and TE, TM, we have

[0089] Expand to get:

[0090]

[0091] ρ in formula (16) TE , ρ TM The expressions of and δ are:

[0092]

[0093] Where Im is the imaginary part operator; to simplify the representation, set ρ TE , ρ TM is a plural number, that is:

[0094]

[0095] At this time ρ TE and ρ TM satisfy:

[0096]

[0097] When the polarization information of the light source is considered, Equation (16) can be rewritten as:

[0098]

[0099] Where Q(μ1, v1|μ, v) means that Q is a function of (μ1, v1) and contains the parameter (μ, v); (μ, v) and (μ1, v1) have different meanings. The former is the frequency domain coordinate corresponding to the angle of the incident light, while the latter is the frequency domain coordinate corresponding to the angle of the diffracted light after the incident light passes through the mask. Since the generalized pupil function is the product of the pupil function and the film function, the generalized pupil function is also a function containing the parameter (μ, v);

[0100] For XY4 polarization mode, ρ TE , ρ TM The expressions of and δ are:

[0101]

[0102] Substitute equation (30) into equation (14) and refer to equation (18) to sort it out, extract the integral corresponding to equation (19), and you can get the transmission cross coefficient vector expression. To save space, the frequency domain coordinate (μ, v) is abbreviated as μ; similarly, (μ+μ1, v+v1) is abbreviated as μ+μ1, and the other symbols are similar. When the XY4 polarization mode is used, ρ TE , ρ TM are all real-valued functions, then:

[0103]

[0104] Where Q TE and Q TM are all vectors and can be further expanded after dot product; according to formula (30), Q TEThere are only two components, x and y, so after expanding the three terms, we get a total of seven scalar expressions:

[0105]

[0106]

[0107]

[0108] Further arranging the expanded x component and y component in formula (35) has the same form, and the x component is:

[0109]

[0110] Substituting equations (33), (34), (35) and (36) back into equation (32), we can obtain seven scalar transmission cross coefficients, which are denoted as T E，x 、T E，y 、T M，x 、T M，y 、T M，z 、T I，x and T I，y , where subscripts E, M, and I correspond to the expanded terms of equations (33), (34), and (35), respectively; T E，y With T E，x Same form, T M，y 、T M，z With T M，x Same form, T I，y With T I，x Same form, T E，x 、T M，x and T I，x The expression is as follows:

[0111]

[0112]

[0113]

[0114] Adding the above 7 scalar transmission cross coefficients, we can get the expression of polarization transmission cross coefficient:

[0115]

[0116] When ρ is a complex-valued function, equation (32) can be rewritten as:

[0117]

[0118] The expanded form of the first two terms after dot product is the same as that of (33) and (34), but the third term cannot separate ρ and Q; in this case, T 1，xThe expression should be written as:

[0119]

[0120] The light intensity of natural light imaging is equal to the average light intensity of all forms of polarized light imaging. When imaging with a unit intensity and a single point natural light source, according to formula (14), we have:

[0121]

[0122] For the entire light source, equation (17) can be rewritten as:

[0123]

[0124] To expand the above formula and extract the transmission cross coefficient, it is necessary to first write the expansion of the amplitude A; according to equations (26), (28) and (30), its expansion is:

[0125]

[0126] Where, ρ TE , ρ TE The expression does not contain the independent variables (μ1, v1), so it can be mentioned outside the integral sign; the last line refers to formula (18) for substitution, and on this basis, substitute formula (45) back into formula (44) and continue to expand to obtain:

[0127]

[0128] The above formula is the addition of three terms. The parameters α and φ can be separated from other variables and parameters. By sorting out the first term and eliminating α and φ, we can get:

[0129]

[0130] Similarly, we sort out the last two terms and eliminate α and φ, and the result is:

[0131]

[0132] I 3(x，y )=0 (Formula 49)

[0133] Thus:

[0134]

[0135] Based on the above formula, when natural light imaging is used, its light intensity is equivalent to the average of the imaging intensity of pure TE polarized light with equal intensity and the imaging intensity of pure TM polarized light with equal intensity. The transmission cross coefficient corresponding to natural light imaging can be decomposed into 5 scalar transmission cross coefficients:

[0136] T CC(μ1;μ2)=T E，x (μ1;μ2)+T E，x (μ1;μ2)+T M，x (μ1;μ2)+T M，x (μ1;μ2)+T M，z (μ1; μ2) (Formula 51)

[0137] T E，x and T M，x The expressions are:

[0138]

[0139]

[0140] T E，y With T E，x Same form, T M，y 、T M，z With T M，x Same format.

[0141] Compared with natural light, unpolarized light has an additional weight coefficient W(α, φ|μ, v). The value of the weight coefficient should be a non-negative real number. To simplify the expression, the weight coefficient is normalized. That is, the weight coefficient W that meets the following conditions is called the normalized weight coefficient:

[0142]

[0143] After adding the normalized weight coefficient, formula (44) is rewritten as:

[0144]

[0145] The transmission cross coefficient corresponding to the non-polarized light imaging of the normalized weight coefficient is calculated, and 7 scalar transmission cross coefficients are obtained, which are denoted as T E，x 、T E，y、 T M，x 、T M，y 、T M，z 、T I，x and T I，y , where T E，x The expression is:

[0146]

[0147] Where W and ρ TE It is a function of (α, φ) and also contains parameter variables (μ, v). After integration, α and φ are eliminated and the integral result is a function of (μ, v).

[0148] Partially polarized light can be decomposed into polarized light and unpolarized light in proportion. The two are imaged separately, and their light intensities are then added proportionally to form the intensity of the partially polarized light image. Let p(μ, v) be the proportional function of the polarized light intensity. Obviously, 0≤p(μ, v)≤1, so the light intensity is:

[0149]

[0150] Where A po is the complex amplitude vector corresponding to the unit intensity polarized light, A un is the complex amplitude vector corresponding to the unit intensity unpolarized light, and the transmission cross coefficients corresponding to the two parts can be written separately, totaling 14 scalar transmission cross coefficients, which are:

[0151]

[0152] Among them, the x component of the transmission cross coefficient of the polarized light part is:

[0153]

[0154]

[0155]

[0156] The term T of the transmission cross coefficient of the unpolarized light part E，x for:

[0157]

[0158] When the unpolarized light in the partially polarized light is natural light, it corresponds to 5 scalar transmission cross coefficients, and the x component is:

[0159]

[0160]

[0161] For ease of understanding, this embodiment further illustrates the specific implementation process of the above method in conjunction with a specific application scenario system, which specifically includes the following steps:

[0162] In computational lithography, studying the polarization effect of light is essentially studying the vector addition of electric field intensities. Therefore, the calculation must consider both the polarization of the light source and optical components such as lenses and thin films that can alter the intensity, propagation direction, and polarization of light. Since the goal of computational lithography is to calculate light intensity, and the Abbe imaging formula gives the electric field intensity, before formally introducing the Abbe imaging formula, we first clarify the relationship between light intensity, electric field intensity, and electric field amplitude (hereinafter referred to as amplitude). Based on this, we analyze imaging from a single point light source using the Abbe imaging formula. This is then extended to a surface light source, and a method for calculating light intensity based on the transmission cross coefficient is presented.

[0163] 1.1 Relationship between light intensity, electric field strength and amplitude

[0164] 1.1.1 Overview

[0165] As light propagates, its electric field intensity E varies with time and space. Light intensity is generally considered a steady-state quantity; that is, for light that is steadily transmitted and unattenuated, its intensity remains constant over time and space. Therefore, this paper defines light intensity I as the average value of the square of the electric field intensity over a period, as shown in the following formula:

[0166]

[0167] Where T is the time period of the electric field oscillation, a constant for light of a given wavelength and propagation medium (hereinafter referred to as light with given propagation conditions), E is the electric field intensity, and t is time. To simplify calculations, we will now define the amplitude for different forms of electric field intensity and provide the relationship between light intensity and amplitude.

[0168] For plane linear polarized light, the electric field direction is constant and does not change with time. When the amplitude A is the maximum value of the electric field intensity, the spatiotemporal expression of the electric field intensity satisfies the following formula:

[0169]

[0170] Where ω is the angular frequency, k is the spatial angular frequency; z is the spatial coordinate along the direction of light propagation, is the initial phase;

[0171] For this type of light, the light intensity expression is calculated according to formula (1). The specific process is shown in the following formula:

[0172]

[0173] The above equation shows that for plane linearly polarized light, the intensity is proportional to the square of the amplitude, with a constant proportionality factor of 1 / 2. In photolithography calculations, relative values ​​of light intensity are generally used; however, the proportionality factor can be ignored and the square of the amplitude can be used to express the intensity.

[0174] Rewriting formula (2), the complex expression satisfies the following formula:

[0175]

[0176] In the formula, Re is the real part operator, j is the imaginary unit, e jωt is the time factor. After separating the time factor, we get the complex number This complex number is called the complex amplitude of light, and is hereinafter also represented by A.

[0177] Later we will study the coherent superposition of light, that is, the superposition of electric field intensities. The time of superposition is of course the same, so the superposition of electric field intensities can be converted into the superposition of complex amplitudes. Among them, for two columns of light with equal wavelengths, collinear propagation directions and electric field directions, the superposition of complex amplitudes is the algebraic addition of complex amplitudes. When light is refracted or reflected, the phase of its electric field intensity will jump. The moments before and after refraction and reflection are considered to be the same moment, so the change in phase can be expressed by multiplying the complex amplitude by a complex number with a modulus of 1. Specifically, advancing the phase by φ is equivalent to multiplying The delayed phase is equivalent to multiplying

[0178] In complex amplitude form, the expression of light intensity satisfies the following formula:

[0179]

[0180] 1.1.3 Amplitude of Arbitrary Polarized Light

[0181] Without loss of generality, assume that light propagates along the positive z-axis (in fact, it is always possible to establish a rectangular coordinate system in space and define the propagation direction of light as the positive z-axis). As a transverse wave, its electric field oscillation direction is perpendicular to the z-axis and therefore parallel to the xOy plane. If amplitude is still defined as the maximum value of the electric field intensity, then for certain polarizations of light, the intensity is not equal to the square of the amplitude. The fundamental reason for this phenomenon is that the direction of the electric field varies with time and is not a simple harmonic oscillation. Therefore, it is necessary to find an alternative definition of amplitude.

[0182] According to the polarization properties of light, its electric field can be decomposed into simple harmonic oscillation components in two mutually perpendicular directions. If the electric field is decomposed into the x-axis and y-axis directions, the electric field intensity components in the two directions at the same spatial position are (here we mainly focus on the phase difference between the x-axis and y-axis components, so the initial phase of the x-axis component is assumed to be 0)

[0183] When the electric field is decomposed into the x-axis and y-axis directions, the electric field intensity components in the two directions at the same spatial position are:

[0184]

[0185] Where A x , A y They represent the amplitudes of the electric field strength components in the two directions, and δ represents the phase of the y-axis component leading the x-axis component. The total electric field strength is the vectorial superposition of the two components, which can be expressed as:

[0186]

[0187] The magnitude of this vector is the absolute value of the total electric field strength, as shown below:

[0188]

[0189] According to A x , A y Due to the different proportional relationship and the value of δ, polarized light can be divided into three types: linear polarized light, circular polarized light and elliptically polarized light. The relationship between the instantaneous value of the total electric field intensity of the three polarization types and time is different. However, by substituting formula (8) into formula (3), we can get

[0190]

[0191] Based on the above formula, the light intensity is only related to A x , A y It is related to the phase difference δ, that is, the intensity of polarized light is independent of the polarization form; the complex amplitude vector is the vector composed of complex amplitudes in all directions, as shown in the following formula:

[0192]

[0193] Based on the above formula, the light intensity is expressed as the inner product of the complex amplitude vector and its conjugate:

[0194]

[0195] 1.2 Abbe imaging formula

[0196] 1.2.1 Scalar form

[0197] The Abbe imaging formula describes the electric field intensity of light irradiated from a single specific direction, passing through the mask and pupil to reach the image plane. For a beam of coherent light of unit intensity at normal incidence (i.e., the direction of light propagation is perpendicular to the mask and image plane), its complex amplitude distribution on the image plane is shown as follows:

[0198]

[0199] Where (x, y) is the spatial domain coordinate, A(x, y) is the expression of the complex amplitude in the spatial domain; (μ, v) is the spatial frequency domain coordinate, each spatial frequency domain coordinate corresponds to an angle in the spatial domain, indicating the propagation direction of the diffracted light, P(μ, v) and M(μ, v) are the expressions of the pupil function and the mask in the frequency domain, respectively; is the inverse Fourier transform operator; all integrals involved are related to the Fourier transform or inverse transform, so the integration region is the entire plane (of course, the actual integration region is subject to the domain of definition of each function). To simplify the notation, the integration interval is omitted below, and only the integral symbol is used to represent the integral over the entire plane.

[0200] It should be pointed out that due to the limited numerical aperture of the pupil, the domain of the pupil function does not cover the entire frequency domain plane. Let the cutoff frequency be fm, then the domain of the pupil function is a circle with the origin as the center and fm as the radius. Therefore, when calculating the inverse Fourier transform in equation (12), it is only necessary to integrate within this range. This results in that even for an ideal pupil (no aberration, that is, P(μ, v) = 1), the electric field intensity distribution is generally not equal to the pattern of the mask in the spatial domain.

[0201] For a beam of coherent light of unit intensity incident at any angle, let the frequency domain coordinate corresponding to its direction be (μ, v), then the complex amplitude distribution of this beam of light on the image plane can be expressed as:

[0202]

[0203] In the formula, (μ1, v1) is the independent variable, and (μ, v) is a constant. To distinguish them, the above formula adds the subscript (μ1, v1)→(x, y) to the inverse Fourier transform operator, and at the same time indicates the coordinate variable names before and after the transformation. After obtaining the complex amplitude distribution, it is substituted into formula (5) to obtain the light intensity distribution on the image plane, and the Abbe imaging formula in scalar form is obtained.

[0204] 1.2.2 Vector form

[0205] Equation (13) only describes the magnitude of the electric field intensity, not its direction. To study polarized light imaging, the refraction and reflection effects of the film must be considered. Refraction and reflection change the propagation direction of light and also the direction of the electric field. Therefore, the scalar Abbe imaging formula needs to be expanded into a vector form to describe the film function. Since both the lens group and the film can change the direction, intensity, and phase of light, this paper refers to their effect functions as the generalized pupil function. The vector Abbe imaging formula satisfies the following formula:

[0206]

[0207] Where Q is the film function, which is a vector, and P and M are still scalars;

[0208] According to the Fresnel formula, when light is refracted and reflected at an interface, its electric field needs to be decomposed into components perpendicular to and parallel to the incident surface, and the transmittance and reflectance are calculated respectively, which are recorded as TE component and TM component (also known as S and P components). For a beam of polarized light with unit intensity, the amplitudes of its TE and TM components are recorded as ρ TE and ρ TE , and both satisfy the following formula:

[0209]

[0210] In addition, let the phase difference between the TM direction component and the TE direction component be δ, then the thin film function of the vector is full

[0211] The following formula is enough:

[0212]

[0213] Where, each component can be calculated based on the complex refractive index, image depth and incident angle of each layer of the film; for isotropic films, Q TE and Q TM Each component has a circularly symmetric property; substituting Equation (16) into Equation (14) yields the vector form of the Abbe imaging formula for the components on the x, y, and z axes; finally, after obtaining the distribution of the complex amplitude vector, substituting it into Equation (11) yields the light intensity distribution on the image plane.

[0214] 1.3 Transmission cross coefficient

[0215] 1.3.1 Scalar Form

[0216] Modern photolithography systems generally use partially coherent light sources, which are characterized by the direction of the light irradiating the mask being within a range, and the light from different directions being incoherent. When the light intensities of light from different directions on the image plane are directly superimposed to obtain the total aerial image, the Abbe imaging formula is:

[0217] I(x,y)=∫∫J(μ,v)A(x,y|μ,v)A * (x,y|μ,v)dμdv (Formula 17)

[0218] Where J(μ, v) is a light source function (the function value is a non-negative real number), which represents the light intensity of the light source in the direction of (μ, v); A(x, y|μ, v) means that A is a function of (x, y) and contains the parameter (μ, v), which is the integral variable;

[0219] The Hopkins imaging formula is used to further expand the Abbe imaging formula and adjust the order of integration to obtain:

[0220]

[0221] The second to third rows of the formula are substituted to separate (μ, v) from the mask expression; the integral in the innermost brackets of the fourth row is called the transmission cross coefficient, that is:

[0222] T CC (μ1, v 1； μ2, v2)=∫∫J(μ, v)P (μ1+μ, v1+v)P * (μ²+μ, v²+v)dμdv (Formula 19)

[0223] Substituting formula (19) back into formula (18), we can get:

[0224]

[0225] The main advantage of the transmission cross coefficient is that it is only related to the imaging conditions (light source and generalized pupil), and is not related to the mask. Lithography simulation usually calculates the aerial images of different patterns for given imaging conditions. In this case, pre-calculating the transmission cross coefficient can greatly improve the overall calculation speed. When the transmission cross coefficient is expressed as a separate form of variables (μ1, v1) and (μ2, v2):

[0226] T CC (μ1, v 1； μ2, v2) = Φ(μ1, v1)Φ * (μ2, v2) (Formula 21)

[0227] Then formula (20) can be organized as:

[0228]

[0229] Where, is the convolution operator, m(x, y) is the representation of the mask in the spatial domain, that is, the spatial domain function corresponding to M(x, y); the last line of the above formula uses the properties of the inverse Fourier transform to convert the frequency domain multiplication into a convolution operation in the spatial domain, thereby further simplifying the calculation process and further improving the calculation speed.

[0230] 1.3.2 Effect of polarization on film function

[0231] Equation (6) gives the expression for decomposing a beam of arbitrarily polarized light into two components in mutually perpendicular directions. For a light source, unless it is normally incident, its propagation direction is not parallel to the z-axis (i.e., perpendicular to the reticle), but it is also not perpendicular to the z-axis. Therefore, the polarization direction of the light emitted by the light source can still be described by the x-axis and y-axis. It should be noted that the x and y components do not indicate that the direction of the electric field is parallel to the x-axis and y-axis, but rather that the projection of the electric field direction on the xOy plane is parallel to the x-axis and y-axis.

[0232] The polarization of the light source is calculated using the direction angle α and the phase difference Φ. α and Φ are related to A in formula (6). x , A y The corresponding relationship between and δ is:

[0233]

[0234] In formula (6), Ax and Ay are both non-negative real numbers, so the range of α is [0, π / 2]. In practical applications, to increase the flexibility of expression, the range of α is usually extended to [0, 2π]. In this case, the absolute value of the tangent of α is equal to Ay / Ax, and φ is equal to δ, or differs from δ by π.

[0235] For the polarized light source actually used, its α and φ are functions in the frequency domain. When the light source is a polarized light source and the lithography mode is the polarization mode XY4, the expressions of its α and φ are:

[0236]

[0237] In order to calculate the film function Q, it is necessary to convert the x and y components into TE and TM components, and the TE and TM components can be obtained by rotating the x and y components by a certain angle; to represent the angle, the argument function atan2 is first introduced: atan2(v, μ) is the argument value after the rectangular coordinates (μ, v) are converted to polar coordinates; let φ = atan2(v, μ) be the argument of a point (μ, v) in the frequency domain, at which point a plane rectangular coordinate system with TE and TM as the coordinate axes can be established, and the coordinate axes of the coordinate system can be obtained by rotating the x and y coordinate axes counterclockwise by (φ-π / 2) radians, and the complex amplitude vector [A TE , A TM ]By [A x , A y ]Counterclockwise rotation Radians are obtained;

[0238] For an incident light beam of unit intensity, the complex amplitudes of the x and y components are:

[0239]

[0240] Then according to the rotation relationship between x, y and TE, TM, we have

[0241]

[0242] Expand to get:

[0243]

[0244] ρ in formula (16) TE , ρ TM The expressions of and δ are:

[0245]

[0246] Where Im is the imaginary part operator; since the above formula contains many piecewise operators (taking absolute value, taking angle), it is more complicated in actual use. To simplify the representation, set ρ TE , ρ TM is a plural number, that is:

[0247]

[0248] At this time ρ TE and ρ TM satisfy:

[0249]

[0250] When the polarization information of the light source is considered, Equation (16) can be rewritten as:

[0251]

[0252] Where Q(μ1, v1|μ, v) means that Q is a function of (μ1, v1) and also contains the parameter (μ, v); (μ, v) and (μ1, v1) have different meanings. The former is the frequency domain coordinate corresponding to the angle of the incident light, while the latter is the frequency domain coordinate corresponding to the angle of the diffracted light after the incident light passes through the mask. The two should not be confused. Since the generalized pupil function is the product of the pupil function and the film function, the generalized pupil function is also a function containing the parameter (μ, v);

[0253] For XY4 polarization mode, ρ TE , ρ TM The expressions of and δ are:

[0254]

[0255] It can be seen from the above formula that in the XY4 polarization mode, the amplitude (real amplitude, non-negative real number) of the TM component at the same angle is always no greater than the TE component.

[0256] For the case of normal incidence. Note that the domain of the atan2 function does not include the point (0, 0), so the TE and TM components cannot be separated for normal incident light. However, since the transmittance and reflectance are equal when the incident angle is 0, there is actually no need to distinguish between TE and TM components. In actual calculations, TE and TM components are usually processed separately and then merged. According to the nature of the integral operation, changing the value of the integrand at a finite number of points does not affect the integral value, so ρ for normal incident light TE , ρ TM It can take any reasonable value, such as ρ TE (μ, v), ρ TMThe limit value of (μ, v) at the point (0, 0) or the limit value under a specific approximation path. Similarly, for the XY4 polarization mode, when |μ| = |v|, α can also take any reasonable value.

[0257] 1.3.3 Vector form

[0258] Similar to the scalar form, substitute Equation (30) into Equation (14), and refer to Equation (18) for arrangement, extract the integral corresponding to Equation (19), and the transmission cross coefficient vector expression can be obtained; to save space, the frequency domain coordinate (μ, v) is abbreviated as μ; similarly, (μ+μ1, v+v1) is abbreviated as μ+μ1, and the other symbols are similar; when the XY4 polarization mode is used, ρ TE , ρ TM are all real-valued functions, then:

[0259]

[0260] Where Q TE and Q TM are all vectors and can be further expanded after dot product; according to formula (30), Q TE There are only two components, x and y, so after expanding the three terms, we get a total of seven scalar expressions:

[0261]

[0262]

[0263]

[0264] Further arranging the expanded x component and y component in formula (35) has the same form, and the x component is:

[0265]

[0266] Substituting equations (33), (34), (35) and (36) back into equation (32), we can obtain seven scalar transmission cross coefficients, which are denoted as T E，x 、T E，y 、T M，x 、T M，y 、T M，z 、T I，x and T I，y , where subscripts E, M, and I correspond to the expanded terms of equations (33), (34), and (35), respectively; T E，y With T E，x Same form, T M，y 、T M，z With T M，x Same form, T I，y With T I，xThe form is the same, and we can infer that T E，x 、T M，x and T I，x The expression is as follows:

[0267]

[0268]

[0269]

[0270] Adding the above 7 scalar transmission cross coefficients, we can get the expression of polarization transmission cross coefficient:

[0271]

[0272] When ρ is a complex-valued function, equation (32) can be rewritten as:

[0273]

[0274] The expanded form of the first two terms after dot product is the same as that of (33) and (34), but the third term cannot separate ρ and Q; in this case, T 1，x The expression should be written as:

[0275]

[0276] Note that the two expanded terms alone do not constitute a transmission cross coefficient due to asymmetry; their sum is symmetric and constitutes an independent transmission cross coefficient. In actual calculations, the two terms are inverse Fourier transformed separately, and the results are summed to obtain the transmission cross coefficient. The subsequent calculation process is the same as for other directly derived transmission cross coefficients.

[0277] 2. Calculation method of light intensity of non-polarized light and partially polarized light

[0278] Unpolarized light can be viewed as an incoherent superposition of polarized light within a certain range. Natural light, in particular, can be viewed as a uniform, incoherent superposition of all polarized light. This section first discusses the vector model for natural light, then the vector model for general unpolarized light, and finally the vector model for partially polarized light.

[0279] 2.1 Natural Light Imaging

[0280] Equation (14) gives the Abbe imaging formula in vector form. According to the content of Section 1.3.2, the incident angle Q of the film function is affected by both the incident light angle (μ, v) and the incident light polarization parameters α and φ. Therefore, this section expresses Q as a function containing parameter variables (μ, v; α, φ). However, to save space, this subsection adopts the same frequency domain coordinate abbreviation method as Section 1.3.3, that is, the film function is expressed as Q(μ1|μ; α, φ). Similarly, other function representation methods containing frequency domain coordinates can be obtained. Since the incoherent superposition of light is equivalent to the superposition of light intensity, the light intensity of natural light imaging is equal to the average of the light intensity of all forms of polarized light imaging. Therefore, when imaging a unit intensity, single-point natural light source, according to Equation (14), we have:

[0281]

[0282] For the entire light source, equation (17) can be rewritten as:

[0283]

[0284] To expand the above formula and extract the transmission cross coefficient, it is necessary to first write the expansion of the amplitude A; according to equations (26), (28) and (30), its expansion is:

[0285]

[0286] Where, ρ TE , ρ TE The expression does not contain the independent variables (μ1, v1), so it can be mentioned outside the integral sign; the last line refers to formula (18) for substitution, and on this basis, substitute formula (45) back into formula (44) and continue to expand to obtain:

[0287]

[0288] The above formula is the addition of three terms. The parameters α and φ can be separated from other variables and parameters. By sorting out the first term and eliminating α and φ, we can get:

[0289]

[0290] Similarly, we sort out the last two terms and eliminate α and φ, and the result is:

[0291]

[0292] I3(x,y)=0(Formula 49)

[0293] Thus:

[0294]

[0295] Based on the above formula, when natural light imaging is used, its light intensity is equivalent to the average of the imaging intensity of pure TE polarized light with equal intensity and the imaging intensity of pure TM polarized light with equal intensity. The transmission cross coefficient corresponding to natural light imaging can be decomposed into 5 scalar transmission cross coefficients:

[0296] T CC (μ1;μ2)=T E，x (μ1;μ2)+T E，x (μ1;μ2)+T M，x (μ1;μ2)+T M，x (μ1;μ2)+T M，z (μ1; μ2) (Formula 51)

[0297] T E，x and T M，x The expressions are:

[0298]

[0299]

[0300] T E，y With T E，x Same form, T M，y 、T M，z With T M，x The form is the same and can be deduced by analogy.

[0301] Appendix: The process of eliminating α and φ in the third term

[0302] (cosαsin atan2(v,μ)-e jφ sinαcos atan2(v,μ))(cosαcos atan2(v,μ)+e -jφ sinαsin atan2(v,μ))

[0303] =cos 2 αsin atan2(v,μ)cosatan2(v,μ)-sin 2 αsinatan2(v,μ)cos atan2(v,μ)+e -jφ sinαcosαsin 2 atan2(v,μ)-e jφ sinαcosαcos 2 atan2(v,μ)

[0304] =(cos 2 α-sin 2 α)sinatan2(v,μ)cosatan2(v,μ)+2cosφsinαcosαsin2 atan2(v,μ)-e jφ sinαcosα(cosαcos atan2(v,μ)+e jφ sinαsin atan2(v,μ))(cosαsin atan2(v,μ)-e -jφ sinαcos atan2(v,μ))

[0305] =cos 2 αsin atan2(v,μ)cos atan2(v,μ)-sin 2 αsin atan2(v,μ)cos atan2(v,μ)+e jφ sinαcosαsin 2 atan2(v,μ)-e -jφ sinαcosαcos 2 atan2(v,μ)

[0306] =(cos 2 α-sin 2 α)sinatan2(v,μ)cosatan2(v,μ)+2cosφsinαcosαsin 2 atan2(v,μ)-e -jφ sinαcosα

[0307] 2.2 Arbitrary non-polarized light imaging

[0308] Compared to natural light, general non-polarized light has an additional weight coefficient W(α, φ|μ, v) in its expression. (In the previous section, natural light defaulted to a weight coefficient W(α, φ|μ, v) = 1, which is a constant function.) The value of the weight coefficient should be a non-negative real number. To simplify the expression, the weight coefficient is normalized. That is, the weight coefficient W that satisfies the following conditions is called the normalized weight coefficient:

[0309]

[0310] After adding the normalized weight coefficient, formula (44) is rewritten as:

[0311]

[0312] Referring to the derivation process in the previous section, the transmission cross coefficient corresponding to non-polarized light imaging with normalized weight coefficients can be written. It can usually be decomposed into 7 scalar transmission cross coefficients, which are denoted as T E，x 、T E，y 、T M，x 、T M，y、 T M，z 、TI，x and T I，y , where T E，x The expression is:

[0313]

[0314] Where W and ρ TE It is a function of (α, φ) and also contains parameter variables (μ, v). After integration, α and φ are eliminated and the integral result is a function of (μ, v).

[0315] 2.3 Partially polarized light imaging

[0316] Partially polarized light can be decomposed into polarized light and unpolarized light in proportion. The two are imaged separately, and their light intensities are then added proportionally to form the intensity of the partially polarized light image. Let p(μ, v) be the proportional function of the polarized light intensity. Obviously, 0≤p(μ, v)≤1, so the light intensity is:

[0317]

[0318] Where A po is the complex amplitude vector corresponding to the unit intensity polarized light, A un is the complex amplitude vector corresponding to the unit intensity unpolarized light, and the transmission cross coefficients corresponding to the two parts can be written separately, totaling 14 scalar transmission cross coefficients, which are:

[0319]

[0320] Among them, the x component of the transmission cross coefficient of the polarized light part is:

[0321]

[0322]

[0323]

[0324] The term T of the transmission cross coefficient of the unpolarized light part E，x for:

[0325]

[0326] When the unpolarized light in the partially polarized light is natural light, it corresponds to 5 scalar transmission cross coefficients, and the x component is:

[0327]

[0328]

[0329] In summary, to address the problem in computational lithography where existing polarized light imaging intensity simulation methods fail to meet both rapid and accurate application requirements, this paper proposes a fast intensity calculation method that accounts for polarization effects. This method, based on the Abbe imaging principle and the partially coherent imaging properties of modern lithography systems, incorporates the refraction and reflection effects of thin films to calculate the decomposition and superposition of the TE and TM polarization components, extracting a vector expression for the transmission cross-coefficient that includes both the light source polarization function and the film function. This vector expression can be broken down into seven scalar expressions: two TE components, three TM components, and two TE / TM cross components. For unpolarized light, this paper treats it as an incoherent superposition of polarized light. By integrating the light source polarization parameter in the imaging formula, a vector expression for the transmission cross-coefficient containing seven scalar expressions is similarly derived. Specifically, for natural light, this paper derives that the TE and TM cross components in the transmission cross-coefficient vector are zero, resulting in only five scalar expressions. For partially polarized light, this paper treats it as a proportional superposition of polarized and unpolarized light, deriving a vector expression for the transmission cross-coefficient containing 14 scalar expressions. In summary, the calculation system proposed in this paper, which takes polarization effects into account, can handle the light intensity obtained from various types of light imaging. At the same time, because the form of the transmission cross coefficient is retained, the imaging system and the mask representation can be calculated separately, which has the advantage of speed.

[0330] like Figure 2 As shown, based on the above-mentioned method for calculating light intensity, the present invention provides a light intensity calculation device, including: a calculation module 201, used to calculate the polarization of light by using the Abbe imaging principle combined with the refraction and reflection effects of the optical thin film stack, and obtain a relationship between the electric field intensity of a single point of the light source on the image plane; an adjustment module 202, used to integrate the electric field intensity within the range of the light source by using the polarization of the light source based on the relationship, and adjust the order of multiple integrations by using the Hopkins imaging formula to obtain a transmission cross coefficient vector expression; an expansion module 203, used to expand the transmission cross coefficient vector expression to obtain N scalar transmission cross coefficients; a simulation module 204, used to use the N scalar transmission cross coefficients to simulate and calculate the light intensity of an actual light source, wherein N is a positive integer.

[0331] It should be understood that all relevant contents of each step involved in the above method embodiment can be referred to the functional description of the corresponding functional module, and will not be repeated here. In addition, the use of suffixes such as "module", "component" or "unit" to represent elements is only for the purpose of facilitating the description of the present invention, and has no specific meaning in itself. Therefore, "module", "component" or "unit" can be used in a mixed manner. The terminal can be implemented in various forms. For example, the terminal described in the present invention may include mobile terminals such as mobile phones, tablet computers, laptop computers, PDAs, portable media players (PMPs), navigation devices, wearable devices, smart bracelets, pedometers, etc., as well as fixed terminals such as digital TVs and desktop computers. The subsequent description will be explained using mobile terminals as an example. It will be understood by those skilled in the art that, in addition to components specifically used for mobile purposes, the construction according to the embodiment of the present invention can also be applied to fixed-type terminals.

[0332] In other embodiments of the present invention, an electronic device 300 is disclosed. Figure 3 As shown, the system may include: one or more processors 301; a memory 302; a display 303; one or more applications (not shown); and one or more computer programs 304. The above components may be connected via one or more communication buses 305. The one or more computer programs 304 are stored in the memory 302 and configured to be executed by the one or more processors 301. The one or more computer programs 304 include instructions, which may be used to execute the following instructions: Figure 1 The various steps in the corresponding embodiments.

[0333] The processor 301 may be a central processing unit (CPU), or other general-purpose processors, digital signal processors (DSP), application-specific integrated circuits (ASIC), field-programmable gate arrays (FPGA), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. The general-purpose processor may be a microprocessor or any conventional processor, etc.

[0334] The memory 302 may be an internal storage unit of the electronic device 300, such as a hard disk or memory of the electronic device 300. The memory 302 may also be an external storage device of the electronic device 300, such as a plug-in hard disk, a Smart Media Card (SMC), a Secure Digital (SD) card, a Flash Card, etc. equipped on the electronic device 300. Furthermore, the memory 302 may include both an internal storage unit of the electronic device 300 and an external storage device. The memory 302 is used to store computer programs and other programs and data required by the electronic device. The memory 302 may also be used to temporarily store data that has been output or is about to be output.

[0335] The computer program 304 may be divided into one or more modules / units. The one or more modules / units may be a series of computer program instruction segments capable of performing specific functions. The instruction segments are used to describe the execution process of the computer program 304 in the electronic device 300 .

[0336] In addition to the above structure, those skilled in the art can understand that Figure 3 This is merely an example of the electronic device 300 and does not constitute a limitation on the electronic device 300. The electronic device 300 may include more or fewer components than shown in the figure, or a combination of certain components, or different components. For example, the electronic device may also include input and output devices, network access devices, buses, etc.

[0337] Those skilled in the art can clearly understand that, for the convenience and brevity of description, only the division of the above-mentioned functional units and modules is used as an example for illustration. In actual applications, the above-mentioned functions can be distributed and completed by different functional units and modules as needed, that is, the internal structure of the device can be divided into different functional units or modules to complete all or part of the functions described above. The functional units and modules in the embodiment can be integrated into one processing unit, or each unit can exist physically alone, or two or more units can be integrated into one unit. The above-mentioned integrated unit can be implemented in the form of hardware or in the form of software functional units. In addition, the specific names of the functional units and modules are only for the convenience of distinguishing each other, and are not used to limit the scope of protection of the present invention. The specific working process of the units and modules in the above-mentioned system can refer to the corresponding process in the aforementioned method embodiment, and will not be repeated here.

[0338] Based on the above embodiments, the present invention further discloses a computer-readable storage medium storing at least one computer program, which implements the light intensity calculation method in the above embodiments when executed by a processor.

[0339] Those skilled in the art will appreciate that all or part of the steps in the method for implementing the above embodiment can be performed by instructing a processor through a program, and the program can be stored in a computer-readable storage medium, which is a non-transitory medium, such as a random access memory, a read-only memory, a flash memory, a hard disk, a solid-state drive, a magnetic tape, a floppy disk, an optical disc, and any combination thereof. The above storage medium can be any available medium that can be accessed by a computer or a data storage device such as a server or a data center that includes one or more available media. The available medium can be a magnetic medium (e.g., a floppy disk, a hard disk, a tape), an optical medium (e.g., a digital video disc (DVD)), or a semiconductor medium (e.g., a solid-state drive (SSD)).

[0340] The descriptions of the processes or structures corresponding to the above figures have different emphases. For parts that are not described in detail in a certain process or structure, please refer to the relevant descriptions of other processes or structures.

[0341] In summary, the light intensity calculation method, device, equipment and medium disclosed in the present invention are derived based on physical principles without any additional approximation, while retaining the form of the transmission cross coefficient. Therefore, it has the advantages of both accuracy and speed, and can meet the intensive calculation requirements of optical proximity effect correction.

[0342] While the embodiments of the present invention have been described in detail above, it will be apparent to those skilled in the art that various modifications and variations of these embodiments are possible. However, it should be understood that such modifications and variations are within the scope and spirit of the present invention as set forth in the claims. Furthermore, the invention described herein is susceptible to other embodiments and may be practiced or implemented in a variety of ways.

Claims

1. A method for calculating light intensity, characterized in that: include: The polarization of light is calculated using the Abbe imaging principle in combination with the refraction and reflection effects of the optical thin film stack to obtain the relationship between the electric field intensity of a single point of the light source and the image plane, including: calculating the relationship between the light intensity and the average value of the square of the electric field intensity within one cycle, and calculating the expression of the light intensity; calculating the spatiotemporal expression of the electric field intensity and obtaining the expression in complex form; calculating the expression of the light intensity in complex amplitude form; based on the scalar form of the Abbe imaging formula and the complex amplitude distribution, the light intensity distribution on the image plane is obtained in scalar form; combining the refraction and reflection effects of the optical thin film stack to calculate the polarization of light, the Abbe imaging formula in vector form is obtained; according to the Fresnel formula, the transmittance and reflectance are calculated respectively to obtain the vector thin film function; based on the vector Abbe imaging formula and the vector thin film function, the light intensity distribution on the image plane is obtained in vector form; Based on the relationship, the electric field intensity is integrated within the light source range using the polarization of the light source, and the order of multiple integrations is adjusted using the Hopkins imaging formula to obtain a transmission cross coefficient vector expression; Expanding the transmission cross coefficient vector expression to obtain N scalar transmission cross coefficients; The light intensity of the actual light source is simulated and calculated using the N scalar transmission cross coefficients, where N is a positive integer.

2. The method according to claim 1, characterized in that The relationship between the light intensity in one period and the average value of the square of the electric field intensity satisfies the following formula: (Formula 1) Where, For light intensity, is the time period of the electric field vibration, is the electric field strength, For time; When the amplitude A is the maximum value of the electric field intensity, the space-time expression of the electric field intensity satisfies the following formula: (Formula 2) Where ω is the angular frequency, k is the spatial angular frequency; z is the spatial coordinate along the direction of light propagation, is the initial phase; The expression for calculating light intensity is based on formula (1). The specific process is shown in the following formula: (Formula 3) Rewriting formula (2), the complex expression satisfies the following formula: (Formula 4) In the formula, Re is the real part operator, j is the imaginary unit, is the time factor. After separating the time factor, we get the complex number , this complex number is called the complex amplitude of light, and is also represented by A below; In complex amplitude form, the expression of light intensity satisfies the following formula: (Formula 5).

3. The method according to claim 2, characterized in that When the electric field is decomposed into the x-axis and y-axis directions, the electric field intensity components in the two directions at the same spatial position are: (Formula 6) Where, , They represent the amplitudes of the electric field strength components in the two directions, and δ represents the phase of the y-axis component leading the x-axis component. The total electric field strength is the vectorial superposition of the two components, which can be expressed as: (Formula 7) The magnitude of this vector is the absolute value of the total electric field strength, as shown below: (Formula 8) according to , Due to the different proportional relationship and the value of δ, polarized light can be divided into three types: linear polarized light, circular polarized light and elliptically polarized light. The relationship between the instantaneous value of the total electric field intensity of the three polarization types and time is different. However, by substituting formula (8) into formula (3), we can get (Formula 9) Based on the above formula, the light intensity is only related to , It is related to the phase difference δ, that is, the intensity of polarized light is independent of the polarization form; the complex amplitude vector is the vector composed of complex amplitudes in all directions, as shown in the following formula: (Formula 10) Based on the above formula, the light intensity is expressed as the inner product of the complex amplitude vector and its conjugate: (Formula 11).

4. The method according to claim 3, characterized in that For a beam of coherent light of unit intensity at normal incidence, its complex amplitude distribution on the image plane is as follows: (Formula 12) Where, is the spatial domain coordinate, is the expression of complex amplitude in the spatial domain; is the spatial frequency domain coordinate, each spatial frequency domain coordinate corresponds to an angle in the spatial domain, indicating the propagation direction of the diffracted light. and Expressions of pupil function and mask in frequency domain respectively; is the inverse Fourier transform operator; For a beam of coherent light of unit intensity incident at any angle, the frequency domain coordinate corresponding to its direction is recorded as , then the complex amplitude distribution of this beam of light on the image plane can be expressed as: (Formula 13) Where ( ) is the independent variable, and is a constant; to distinguish it, the above formula adds a subscript to the inverse Fourier transform operator , and the names of the coordinate variables before and after the transformation are specified. After obtaining the complex amplitude distribution, substituting it into formula (5) can obtain the light intensity distribution on the image plane and obtain the Abbe imaging formula in scalar form.

5. The method according to claim 3, characterized in that The Abbe imaging formula in vector form satisfies the following formula: (Formula 14) Where, is the film function, which is a vector, and P and M are still scalars; According to the Fresnel formula, when light is refracted and reflected at the interface, its electric field needs to be decomposed into components perpendicular to and parallel to the incident surface, and the transmittance and reflectance are calculated respectively, which are recorded as TE component and TM component respectively. For a beam of polarized light with unit intensity, the amplitudes of its TE and TM components are recorded as and , and both satisfy the following formula: (Formula 15) In addition, let the phase difference between the TM direction component and the TE direction component be δ, then the thin film function of the vector satisfies the following formula: (Formula 16) Where, each component can be calculated based on the complex refractive index, image depth and incident angle of each layer of the film; for isotropic films, and Each component has a circularly symmetric property; substituting Equation (16) into Equation (14) yields the vector form of the Abbe imaging formula for the components on the x, y, and z axes; finally, after obtaining the distribution of the complex amplitude vector, substituting it into Equation (11) yields the light intensity distribution on the image plane.

6. The method according to claim 5, characterized in that When the light intensities of light from different directions on the image plane are directly superimposed to obtain the total spatial image, the Abbe imaging formula is: (Formula 17) Where, is the light source function, indicating that the light source is Light intensity in a direction; express yes Function with parameters , the parameter is the integral variable; The Hopkins imaging formula is used to further expand the Abbe imaging formula and adjust the order of integration to obtain: (Formula 18) The second and third rows are exchanged, so that Separated from the mask expression; the integral in the innermost brackets of the fourth row is called the transmission cross coefficient, that is: (Formula 19) Substituting formula (19) into formula (18), we can get: (Formula 20) When the transmission cross coefficient is expressed as a variable (μ and Separation form: (Formula 21) Then formula (20) can be organized as: (Formula 22) Where, is the convolution operator, is the representation of the mask in the spatial domain, that is The corresponding spatial domain function; the last line of the above formula uses the properties of the inverse Fourier transform to convert the frequency domain multiplication into a convolution operation in the spatial domain.

7. The method according to claim 6, characterized in that The polarization of the light source is calculated using the direction angle α and the phase difference Φ. and With formula (6) , The corresponding relationship between and δ is: (Formula 23) When the light source is a polarized light source and the photolithography mode is the polarization mode XY4, the expressions of α and φ are: (Official 24) To calculate the film function Q, it is necessary to convert the x and y components into TE and TM components, which can be obtained by rotating the x and y components by a certain angle. To represent this angle, we first introduce the argument function atan2: is a rectangular coordinate Argument value after conversion to polar coordinates; note φ=atan2 A point in the frequency domain The argument of the angle, at this point, a plane rectangular coordinate system with TE and TM as coordinate axes can be established. The coordinate axes of this coordinate system can be obtained by rotating the x and y coordinate axes counterclockwise by (φ-π / 2) radians. The complex amplitude vector [ ]Depend on Counterclockwise rotation Radians are obtained; For an incident light beam of unit intensity, the complex amplitudes of the x and y components are: (Formula 25) Then according to the rotation relationship between x, y and TE, TM, we have Expand to get: (Formula 26) In formula (16) 、 The expressions of and δ are: (Formula 27) In the formula is the imaginary part operator; to simplify the representation, set 、 is a plural number, that is: (Formula 28) at this time and satisfy: (Formula 29) When the polarization information of the light source is considered, Equation (16) can be rewritten as: (Formula 30) Where, Indicates that Q is Function with parameters ; and The meanings are different. The former is the frequency domain coordinate corresponding to the incident light angle, while the latter is the frequency domain coordinate corresponding to the diffracted light angle after the incident light passes through the mask. Since the generalized pupil function is the product of the pupil function and the film function, the generalized pupil function also contains parameters. function; For XY4 polarization mode, 、 The expressions of and δ are: (Formula 31).

8. The method according to claim 7, characterized in that Substitute equation (30) into equation (14), and refer to equation (18) for arrangement, extract the integral corresponding to equation (19), and you can get the transmission cross coefficient vector expression; to save space, the frequency domain coordinates are Abbreviated as μ; similarly Abbreviated as , the other symbols are the same; when the XY4 polarization mode is used, 、 are all real-valued functions, then: (Formula 32) In the formula and are all vectors and can be further expanded after dot product; according to formula (30), There are only two components, x and y, so after expanding the three terms, we get a total of seven scalar expressions: (Formula 33) (Formula 34) (Formula 35) Further arranging the expanded x component and y component in formula (35) has the same form, and the x component is: (Formula 36) Substituting Equations (33), (34), (35) and (36) back into Equation (32), we can obtain seven scalar transmission cross coefficients, which are denoted as 、 、 、 、 、 and , the subscripts E, M, and I correspond to the expanded terms of Equations (33), (34), and (35), respectively; and Same form, 、 and Same form, and Same form, 、 and The expression is as follows: (Formula 37) (Formula 38) (Formula 39) Adding the above 7 scalar transmission cross coefficients, we can get the expression of polarization transmission cross coefficient: (Formula 40) When ρ is a complex-valued function, equation (32) can be rewritten as: (Formula 41) The expanded form of the first two terms after dot product is the same as that of (33) and (34), but the third term cannot be Separated from Q; at this time The expression should be written as: (Formula 42).

9. The method according to claim 7, characterized in that The light intensity of natural light imaging is equal to the average light intensity of all forms of polarized light imaging. When imaging with a unit intensity and a single point natural light source, according to formula (14), we have: (Formula 43) For the entire light source, Equation (17) can be rewritten as: (Formula 44) To expand the above formula and extract the transmission cross coefficient, it is necessary to first write the expansion of the amplitude A; according to equations (26), (28) and (30), its expansion is: (Formula 45) Where, 、 The expression does not contain any independent variables , so it can be mentioned outside the integral sign; the last line refers to formula (18) for substitution, and on this basis, substitute formula (45) back into formula (44) and continue to expand to obtain: (Formula 46) The above formula is the addition of three terms. 、 The parameter variable can be separated from other variables and parameters, and the first item can be sorted and eliminated. 、 , we can get: (Formula 47) Similarly, sort out and eliminate the last two items 、 , the result is: (Formula 48) (Formula 49) Thus: (Official 50) Based on the above formula, when natural light imaging is used, its light intensity is equivalent to the average of the imaging intensity of pure TE polarized light with equal intensity and the imaging intensity of pure TM polarized light with equal intensity. The transmission cross coefficient corresponding to natural light imaging can be decomposed into 5 scalar transmission cross coefficients: (Formula 51) and The expressions are: (Formula 52) (Formula 53) and Same form, 、 and Same format.

10. The method according to claim 9, characterized in that Compared with natural light, non-polarized light has an additional weight coefficient , the value of the weight coefficient should be a non-negative real number; to simplify the expression, the weight coefficient is normalized, that is, the weight coefficient W that meets the following conditions is called the normalized weight coefficient: (Formula 54) After adding the normalized weight coefficient, formula (44) is rewritten as: (Formula 55) The transmission cross coefficient corresponding to the non-polarized light imaging of the normalized weight coefficient is calculated, and 7 scalar transmission cross coefficients are obtained, which are respectively recorded as 、 、 、 、 、 and ,in, The expression is: (Formula 56) Where W and for Function with parameters , after integration, α and φ are eliminated, and the integral result is function.

11. The method according to claim 6, characterized in that Partially polarized light can be decomposed into polarized light and non-polarized light in proportion. The two are imaged separately, and their light intensities are then added in proportion to form the intensity of the partially polarized light image. The proportional function of the polarized light intensity is recorded as , obviously 0≤ ≤1, then the light intensity is: (Formula 57) Where, is the complex amplitude vector corresponding to the unit intensity polarized light, is the complex amplitude vector corresponding to the unit intensity unpolarized light, and the transmission cross coefficients corresponding to the two parts can be written separately, totaling 14 scalar transmission cross coefficients, which are: (Formula 58) Among them, the x component of the transmission cross coefficient of the polarized light part is: (Formula 59) (Formula 60) (Formula 61) The term of the transmission cross coefficient of the unpolarized light part for: (Formula 62).

12. The method according to claim 11, characterized in that When the unpolarized light in the partially polarized light is natural light, it corresponds to 5 scalar transmission cross coefficients, and the x component is: (Formula 63) (Formula 64).

13. A light intensity calculation device, used in the method according to any one of claims 1 to 12, characterized in that: include: A calculation module is used to calculate the polarization of light using the Abbe imaging principle combined with the refraction and reflection effects of optical thin film stacks to obtain a relationship between the electric field intensity of a single point of the light source and the image plane; An adjustment module is used to integrate the electric field intensity within the light source range using the polarization of the light source based on the relationship, and to adjust the order of multiple integrations using the Hopkins imaging formula to obtain a transmission cross coefficient vector expression; An expansion module, configured to expand the transmission cross coefficient vector expression to obtain N scalar transmission cross coefficients; The simulation module is used to simulate and calculate the light intensity of the actual light source using the N scalar transmission cross coefficients, wherein N is a positive integer.

14. An electronic device, characterized in that: The electronic device comprises a memory and a processor, wherein the memory stores a program that can be run on the processor, and when the program is executed by the processor, the electronic device implements the method according to any one of claims 1 to 12.

15. A readable storage medium having a program stored therein, characterized in that: When the program is executed, the method according to any one of claims 1 to 12 is implemented.

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