Calculation device and calculation method

Abstract

To provide a calculation device and a calculation method capable of reducing a calculation time cost by improving a calculation speed.SOLUTION: A calculation device 100 in one embodiment includes a calculation unit 110 that divides a sample 10 having a numerical solution application part 11, a connection part 12, and an analysis solution application part 13 into a plurality of cells and calculates an electromagnetic field of each cell on the basis of a physical property value applied to each cell. The calculation unit 110 calculates electromagnetic fields of the numerical solution application part 11 and the connection part 12 by numerical calculation using a Maxwell equation, and performs Fourier transformation on the calculated electromagnetic field of the connection part 12, thereby converting the electromagnetic field into a plane wave expressed by a wave number vector. The electromagnetic wave in a wave number space in a case where the converted plane wave is used as an initial value and the plane wave propagates in the analysis solution application part 13 in a propagating direction is calculated by the analysis solution using a Helmholtz equation. By performing the inverse Fourier transform on the calculated electromagnetic wave in the calculated wave number space, the electromagnetic wave in the analysis solution application part 13 is calculated.SELECTED DRAWING: Figure 1

Description

[Technical Field]

[0001] The present disclosure relates to a calculation device and a calculation method. [Background technology]

[0002] Patent Document 1 describes replacing the waveguide analysis method from the Beam Propagation Method (hereinafter referred to as BPM) with the Finite Difference Time Domain (hereinafter referred to as FDTD) method to calculate the optical coupling efficiency between waveguides more accurately. Specifically, Patent Document 1 improves accuracy by replacing the solution method for waveguides with high refractive index differences that cannot be accurately calculated using BPM with the FDTD method. Furthermore, Patent Document 1 numerically solves the Helmholtz equation using the finite element method to determine the eigenmode of the waveguide, and then performs a Fourier transform from the time domain to the frequency domain to calculate the coupling efficiency between waveguides.

[0003] Patent Document 2 describes that when a special material such as metal, which BPM is not good at handling, is detected during a BPM-based calculation process, the FDTD method is applied to the part containing that material to solve the problem, thereby improving accuracy without reducing speed as much as possible.

[0004] Patent Document 3 describes the partial application of the FDTD method for the purpose of accurately calculating wave optics effects such as diffraction, interference, and polarization when an imaging optical system includes an object of a size comparable to the wavelength. [Prior art documents] [Patent documents]

[0005] [Patent Document 1] Japanese Patent Publication No. 2020-134386 [Patent Document 2] Japanese Patent Application Laid-Open No. 2004-239784 [Patent Document 3] Japanese Patent Application Laid-Open No. 2002-196230 Summary of the Invention [Problem to be solved by the invention]

[0006] As shown in Patent Documents 1 to 3, the FDTD method is one of the numerical calculation techniques used in optical simulations. The FDTD method divides the structure to be simulated into small rectangular parallelepiped cells, assigns a refractive index and an electromagnetic field to each cell, and solves discretized Maxwell's equations. While the FDTD method is highly versatile, it has a slow calculation speed. The calculation time of the FDTD method is proportional to the number of rectangular parallelepiped cells. The rectangular parallelepiped cells must be sufficiently smaller than the wavelength of the incident light. The larger the structure to be simulated, the more rectangular parallelepiped cells the FDTD method requires. In recent years, optical simulation targets have become increasingly miniaturized and large-scale. Improving calculation speed and reducing calculation time and costs are major challenges for the FDTD method.

[0007] The present disclosure has been made in consideration of the above problems, and provides a calculation device and a calculation method that can improve calculation speed and reduce calculation time costs. [Means for solving the problem]

[0008] A calculation device according to the present disclosure includes: a calculation unit that divides a sample having a numerical solution application portion, a connection portion, and an analytical solution application portion into a plurality of cells in the propagation direction of incident light, and calculates the electromagnetic field of each cell based on physical property values ​​assigned to each cell, thereby calculating the electromagnetic field of the sample; the connecting portion and the analytical solution application portion are configured such that the incident light propagates uniformly in the propagation direction, The calculation unit calculating the electromagnetic fields of the numerical solution application portion and the connection portion by numerical calculation using Maxwell's equations; converting the calculated electromagnetic field of the connection portion into a plane wave expressed by a wave vector by Fourier transforming the electromagnetic field of the connection portion; The plane wave of the converted connection portion is used as an initial value, and the electromagnetic field in the wave number space of the analytical solution application portion when the plane wave propagates through the analytical solution application portion in the propagation direction is calculated by an analytical solution using the Helmholtz equation; The electromagnetic field in the analytical solution application portion is calculated by performing an inverse Fourier transform on the electromagnetic field in the wave number space of the calculated analytical solution application portion.

[0009] In the calculation device, the calculation unit may calculate only an electric field, only a magnetic field, or both the electric field and the magnetic field, depending on a predetermined application, in the analytical solution application portion.

[0010] In the calculation device, in the analytical solution application portion, the calculation unit may perform a Fourier transform on two of the three components of the electromagnetic field in three directions including the propagation direction and two directions orthogonal to the propagation direction, and calculate the remaining component from a transverse wave condition of the electromagnetic wave.

[0011] In the calculation device, the calculation unit may calculate the electromagnetic field in the analytical solution application portion by limiting the calculation to a first portion including a predetermined plane perpendicular to the propagation direction, or a second portion sandwiched between two predetermined planes perpendicular to the propagation direction.

[0012] In the calculation device, the calculation unit may determine the numerical solution application portion, the connection portion, and the analytical solution application portion by determining whether the complex refractive index of the cell is within a certain range from a direction opposite to the propagation direction.

[0013] In the calculation device, the calculation unit may calculate the electromagnetic field in the analytical solution application portion by expanding the electromagnetic field in the connection portion to the analytical solution application portion using a symmetric boundary condition and an antisymmetric boundary condition.

[0014] In the calculation device, the sample includes a plurality of the numerical solution application portions and a plurality of the analytical solution application portions; The apparatus may further include a storage device that stores data of the electromagnetic field for a plurality of the numerical solution application parts and a plurality of the analytical solution application parts.

[0015] In the calculation device, the analytical solution application portion and the connection portion may be in a range where the complex refractive index of the cell is constant.

[0016] In the computing device, the sample includes a substrate and an element formed on the substrate; the analytical solution application portion includes the substrate; the numerical solution application part includes the element, The connection portion may include a portion of the substrate to which the element is connected.

[0017] In the calculation device, the numerical calculation may be performed by at least one of the FDTD method, the FEM method, the BEM method, the CIP method, the FIT method, and a method of solving Maxwell's equations for a model in which the material is discretized by a grid.

[0018] The calculation method according to the present disclosure is 1. A method for calculating an electromagnetic field of a sample, comprising: Dividing the sample into a plurality of cells in a propagation direction of the incident light, the cells having a numerical solution application portion, a connection portion, and an analytical solution application portion in that order; calculating the electromagnetic field of the sample by calculating the electromagnetic field of each cell based on the physical property values ​​assigned to each divided cell; Equipped with the connecting portion and the analytical solution application portion are configured such that the incident light propagates uniformly in the propagation direction, In the step of calculating the electromagnetic field of the sample, calculating the electromagnetic fields of the numerical solution application portion and the connection portion by numerical calculation using Maxwell's equations; converting the calculated electromagnetic field of the connection portion into a plane wave expressed by a wave vector by Fourier transforming the electromagnetic field of the connection portion; The plane wave of the converted connection portion is used as an initial value, and the electromagnetic field in the wave number space of the analytical solution application portion when the plane wave propagates through the analytical solution application portion in the propagation direction is calculated by an analytical solution using the Helmholtz equation; The electromagnetic field in the analytical solution application portion is calculated by performing an inverse Fourier transform on the electromagnetic field in the wave number space of the calculated analytical solution application portion.

[0019] In the step of calculating the electromagnetic field of the sample, In the analytical solution application part, depending on the given application, either the electric field only, the magnetic field only, or both the electric field and the magnetic field may be calculated.

[0020] In the step of calculating the electromagnetic field of the sample, In the analytical solution application part, two of the three components of the electromagnetic field in three directions including the propagation direction and two directions perpendicular to the propagation direction may be Fourier transformed, and the remaining component may be calculated from the transverse wave condition of the electromagnetic wave.

[0021] In the step of calculating the electromagnetic field of the sample, In the analytical solution application portion, the electromagnetic field may be calculated by limiting it to a first portion including a predetermined plane perpendicular to the propagation direction, or a second portion sandwiched between two predetermined planes perpendicular to the propagation direction.

[0022] In the step of dividing into a plurality of cells, The numerical solution application portion, the connection portion, and the analytical solution application portion may be determined by determining whether the complex refractive index of the cell is within a certain range from the direction opposite to the propagation direction.

[0023] In the step of calculating the electromagnetic field of the sample, The electromagnetic field in the analytical solution application portion may be calculated by expanding the electromagnetic field in the connection portion to the analytical solution application portion using a symmetric boundary condition and an antisymmetric boundary condition.

[0024] In the calculation method, the sample includes a plurality of the numerical solution application portions and a plurality of the analytical solution application portions; The method may further comprise the step of storing the electromagnetic field data of a plurality of the numerical solution application parts and a plurality of the analytical solution application parts.

[0025] In the calculation method, the analytical solution application portion and the connection portion may be in a range where the complex refractive index of the cell is constant.

[0026] In the calculation method, the sample includes a substrate and an element formed on the substrate; the analytical solution application portion includes the substrate; the numerical solution application part includes the element, The connection portion may include a portion of the substrate to which the element is connected.

[0027] In the above calculation method, the numerical calculation may be performed by at least one of the FDTD method, the FEM method, the BEM method, the CIP method, the FIT method, and a method of solving Maxwell's equations for a model in which the material is discretized by a grid. [Effects of the Invention]

[0028] The present disclosure provides a calculation device and a calculation method that can improve calculation speed and reduce calculation time costs. [Brief explanation of the drawings]

[0029] [Figure 1] FIG. 1 is a perspective view illustrating a sample according to the first embodiment. [Figure 2] FIG. 2 is a perspective view illustrating a plurality of cells obtained by dividing the sample according to the first embodiment. [Figure 3] 3 is an enlarged perspective view illustrating a cell according to the first embodiment, and is an enlarged perspective view of a portion III in FIG. 2. FIG. [Figure 4] 1 is a block diagram illustrating a calculation device according to a first embodiment. [Figure 5]FIG. 3 is a flowchart illustrating a method for calculating an electromagnetic field according to the first embodiment. [Figure 6] FIG. 3 is a flowchart illustrating a method for calculating an electromagnetic field according to the first embodiment. [Figure 7] 4 is a flowchart illustrating a calculation method for a numerical solution application part in the method for calculating an electromagnetic field according to the first embodiment. FIG. [Figure 8] 1 is a graph illustrating a method for determining a numerical solution application portion, a connection portion, and an analytical solution application portion of a sample according to embodiment 1, in which the horizontal axis indicates the position of the sample in the Z-axis direction, and the vertical axis indicates the complex refractive index. [Figure 9] 1 is a diagram illustrating an XY plane perpendicular to the propagation direction of incident light in the sample 10 according to the first embodiment. [Figure 10] 1 is a diagram illustrating an electromagnetic field in the first quadrant of an XY plane perpendicular to the propagation direction of incident light in a sample according to embodiment 1, showing an electric field Ex. [Figure 11] 1 is a diagram illustrating an electromagnetic field in the first quadrant of an XY plane perpendicular to the propagation direction of incident light in a sample according to embodiment 1, showing an electric field Ey. [Figure 12] 1 is a diagram illustrating an electromagnetic field in the first quadrant of an XY plane perpendicular to the propagation direction of incident light in a sample according to embodiment 1, showing an electric field Ez. [Figure 13] 1 is a diagram illustrating an electromagnetic field in an XY plane perpendicular to the propagation direction of incident light in a sample according to embodiment 1, showing an electric field Ex. [Figure 14] 1 is a diagram illustrating an electromagnetic field on an XY plane perpendicular to the propagation direction of incident light in a sample according to embodiment 1, showing an electric field Ey. [Figure 15] 1 is a diagram illustrating an electromagnetic field on an XY plane perpendicular to the propagation direction of incident light in a sample according to embodiment 1, showing an electric field Ez. [Figure 16] FIG. 1 is a perspective view illustrating a sample according to the first embodiment. [Figure 17] FIG. 1 is a perspective view illustrating a sample to which the calculation method according to the first embodiment is applied. [Figure 18]FIG. 10 is a perspective view illustrating a sample to which a calculation method according to a comparative example is applied. [Figure 19] FIG. 10 is a diagram comparing the integral value of the electric field calculated by the calculation method according to the first embodiment with the integral value of the electric field calculated by the calculation method according to the comparative example. [Figure 20] FIG. 1 is a perspective view of a plurality of samples arranged side by side, each having a different length in the propagation direction of the analytical solution application portion according to the first embodiment. [Figure 21] FIG. 10 is a diagram comparing the calculation time of the calculation method according to the first embodiment with the calculation time of the calculation method according to the comparative example. [Figure 22] 1 is a graph comparing the calculation time of the calculation method according to the first embodiment with the calculation time of the calculation method according to the comparative example, where the horizontal axis indicates each sample and the vertical axis indicates the calculation time. [Figure 23] FIG. 10 is a perspective view illustrating a sample according to a second embodiment. [Figure 24] FIG. 10 is a flowchart illustrating a method for calculating an electromagnetic field according to the second embodiment. [Figure 25] FIG. 10 is a flowchart illustrating a method for calculating an electromagnetic field according to the second embodiment. [Figure 26] FIG. 10 is a flowchart illustrating a method for calculating an electromagnetic field according to the second embodiment. [Figure 27] FIG. 10 is a flowchart illustrating a method for calculating an electromagnetic field according to the second embodiment. DETAILED DESCRIPTION OF THE INVENTION

[0030] For clarity of explanation, the following description and drawings have been omitted and simplified as appropriate. In addition, the same elements in each drawing are given the same reference numerals, and duplicate explanations are omitted as necessary.

[0031] (Outline of the embodiment) An overview of a calculation device and a calculation method according to an embodiment will be described. The calculation device according to this embodiment is, for example, a device that calculates an electromagnetic field. The calculation device calculates an electromagnetic field using the calculation method according to this embodiment. The calculation method performs calculations by partially applying analytical solutions in a numerical solution such as the FDTD method. First, an overview of this embodiment will be described, divided into <application modes and effects of numerical solution>, <application modes and targets of Fourier transform>, and <solution method and targets of Helmholtz equation>. At that time, differences from Patent Document 1 will also be explained.

[0032] <Applications and Effects of Numerical Solutions> The numerical solution in this embodiment will be described using the FDTD method as an example. 1. The FDTD method performs numerical calculations by dividing a sample that is the target of electromagnetic field analysis simulation into small cells. The amount of calculations in the FDTD method increases in proportion to the number of cells used. Therefore, the analysis method of this embodiment detects parts of the sample where the electromagnetic field can be analytically calculated. Then, by applying analytical solutions to parts where the electromagnetic field can be analytically calculated, the parts to which the FDTD method is applied are limited. In other words, by reducing the number of cells to which the FDTD method is applied, the time required for numerical calculations using the FDTD method is shortened.

[0033] In contrast, in Patent Document 1, when the refractive index difference between the core and cladding of a waveguide is large, the FDTD method is applied instead of BPM to suppress a decrease in the calculation accuracy of BPM. This improves the calculation accuracy. Therefore, the numerical calculation using the FDTD method in Patent Document 1 differs from the numerical solution method of this embodiment in terms of the physical quantities to be targeted, the applications, and the effects obtained thereby.

[0034] <Applications and Objects of the Fourier Transform> The Fourier transform in this embodiment will be described. 1. The results of the numerical calculation using the FDTD method are converted into electromagnetic fields in wave number space by Fourier transform of the real space. 2. Applying the analytical solution to the above transformation result. By converting the calculated analytical solution back into real space using an inverse Fourier transform, it is possible to quickly obtain the same results as when applying numerical calculations using the FDTD method.

[0035] In contrast, in Patent Document 1, the calculation results obtained by the FDTD method are converted into the frequency domain by a Fourier transform with respect to time. The conversion results are then used to calculate the optical coupling efficiency between waveguides. Therefore, the Fourier transform in Patent Document 1 differs from the Fourier transform of this embodiment in terms of the physical quantity, application, and resulting effect. For example, the Fourier transform in Patent Document 1 is with respect to time and frequency. Generally, the Fourier transform of time and frequency during numerical calculations using the FDTD method is performed in a form in which the electromagnetic field multiplied by a phase factor is added at each time step.

[0036] On the other hand, the Fourier transform of this embodiment transforms an electromagnetic field in real space into one in wave number space. Therefore, a Fourier transform is performed only once at a predetermined portion (connection portion) of the sample at the end of the numerical calculation using the FDTD method. As such, the Fourier transform of this embodiment differs from that of Patent Document 1 not only in the physical quantities to be transformed, but also in the configuration itself that realizes the Fourier transform.

[0037] <Method and object of solving the Helmholtz equation> The Helmholtz equation in this embodiment will be described. 1. Apply the Helmholtz equation to the parts that can be calculated analytically to obtain an analytical solution for the electromagnetic field. 2. The Helmholtz equation is for the Fourier transformed electromagnetic field. 3. The obtained solution is subjected to an inverse Fourier transform to obtain the calculation result of the electromagnetic field in real space.

[0038] In contrast, in Patent Document 1, the Helmholtz equation is numerically solved using the finite element method. The Helmholtz equation in Patent Document 1 is for electromagnetic potential. The obtained solution is used as input for numerical calculation in the FDTD method. Patent Document 1 also uses the Helmholtz equation. However, the Helmholtz equation in Patent Document 1 differs from the calculation method of this embodiment in terms of the physical quantities it targets, its applications, and the effects it achieves. In this way, the method of connecting the parts applying the numerical solution using Fourier transform and the parts applying the analytical solution, which is the key point of this embodiment, can be said to be completely different from that of Patent Document 1.

[0039] The calculation method of this embodiment can shorten calculation time by utilizing the characteristics of the sample to be simulated for calculating the electromagnetic field (including optics). An example of a characteristic structure in a sample is a structure in which elements with complex microstructures on the order of the incident wavelength are formed on a thick silicon substrate. It is desirable to solve the electromagnetic field in a complex structure on a silicon substrate using numerical calculations such as the FDTD method. However, a thick silicon substrate is a uniform medium. Therefore, the electromagnetic field in a thick silicon substrate can be calculated using a plane wave expansion and an analytical solution of the Helmholtz equation.

[0040] In this way, this embodiment limits the slow calculation speed of the FDTD method to the complex-shaped portion of the sample, and applies the plane wave expansion and analytical solution of the Helmholtz equation to the uniform medium portion of the sample. This allows for high-speed calculation of the electromagnetic field of the sample. Therefore, the calculation speed can be improved compared to when applying numerical calculations using the FDTD method to the entire sample.

[0041] Specifically, analytical solutions using plane wave expansions and analytical solutions of the Helmholtz equation are significantly faster than numerical calculations using the FDTD method. Therefore, if the number of cells in the entire sample when the FDTD method is applied to the analytical solution portion of the sample is M, and the number of cells when the FDTD method is limited to the numerical solution portion of the complex shape is N, the calculation time using the FDTD method is reduced to N / M.

[0042] Furthermore, if the convergence of the FDTD method applied to a complex-shaped portion is faster than the convergence of the FDTD method applied to a uniform medium portion, the calculation time is reduced to N / M or less. Therefore, it can be said that the analysis method of this embodiment reduces the calculation time to at least N / M, where M and N are the numbers of cells.

[0043] (Embodiment 1) Next, the details of the calculation device and calculation method according to this embodiment will be described. First, the "sample" for which the electromagnetic field is calculated will be described. Then, the "calculation device" for calculating the electromagnetic field of the sample will be described, and then the "calculation method" for calculating the electromagnetic field of the sample will be described.

[0044] <Sample> FIG. 1 is a perspective view illustrating a sample according to the first embodiment. As shown in FIG. 1, the sample 10 of this embodiment has a numerical solution application portion 11, a connection portion 12, and an analytical solution application portion 13. Here, for convenience of explanation of the sample 10, an XYZ Cartesian coordinate system is introduced. The optical axis of the incident light entering the sample 10 is defined as the Z-axis direction, and the plane perpendicular to the incident light is defined as the XY plane. The incident light propagates in the −Z-axis direction. The −Z-axis direction in which the incident light propagates is referred to as the propagation direction. For convenience of explanation, the +Z-axis direction is referred to as the upward direction, and the −Z-axis direction is referred to as the downward direction. Note that the terms upward and downward are used for convenience of explanation and do not indicate the actual orientation of the sample 10.

[0045] The sample 10 has a numerical solution application portion 11, a connection portion 12, and an analytical solution application portion 13, in that order in the propagation direction of the incident light. That is, the numerical solution application portion 11, the connection portion 12, and the analytical solution application portion 13 are lined up in the -Z axis direction. The sample 10 includes, for example, a substrate and an element formed on the substrate. The analytical solution application portion 13 includes the substrate. The numerical solution application portion 11 includes the element. The connection portion 12 includes a portion of the substrate to which the element is connected. In this embodiment, the sample 10 is divided into multiple cells in order to analyze the electromagnetic field of the sample 10.

[0046] FIG. 2 is a perspective view illustrating a plurality of cells obtained by dividing the sample according to the first embodiment. FIG. 3 is an enlarged perspective view of a cell according to the first embodiment, illustrating a portion III in FIG. 2. As shown in FIGS. 2 and 3, the sample 10, which is the structure to be simulated, is expressed as a collection of a plurality of small rectangular parallelepiped cells. Each cell is assigned a physical property value such as a dielectric constant. An electric field E and a magnetic field H are disposed on the sides and faces of each cell. In this embodiment, the electromagnetic field of the sample 10 is calculated by calculating the electric field E and the magnetic field H of each cell based on the physical property values ​​assigned to each cell. The connecting portion 12 and the analytical solution application portion 13 allow incident light to propagate uniformly in the propagation direction. The analytical solution application portion 13 and the connecting portion 12 may have a constant range of complex refractive index.

[0047] <Calculation device> Next, the calculation device will be described. FIG. 4 is a block diagram illustrating a calculation device according to the first embodiment. As shown in FIG. 1, the calculation device 100 includes a calculation unit 110. The calculation unit 110 functions as a calculation means. The calculation unit 110 divides a sample 10, which has a numerical solution application portion 11, a connection portion 12, and an analytical solution application portion 13, into a plurality of cells in the propagation direction of incident light. The calculation unit 110 calculates the electromagnetic field of each cell based on the physical property values ​​assigned to each cell. In this way, the calculation unit 110 calculates the electromagnetic field of the sample 10.

[0048] The calculation unit 110 also calculates the electromagnetic fields of the numerical solution application portion 11 and the connection portion 12 by numerical calculation using Maxwell's equations. The numerical calculation is, for example, calculation by at least one of the following: the FDTD method, the FEM method (Finite Element Method), the BEM method (Boundary Element Method), the CIP (Cubic Interpolation Profile or Constrained Interpolation Profile) method, the FIT (Finite Integral Technique), and a method of solving Maxwell's equations for a model in which materials are discretized using a general grid. The calculation unit 110 converts the calculated electromagnetic field of the connection portion 12 into a plane wave expressed by a wave vector by Fourier transforming it. The calculation unit 110 uses the transformed plane wave of the connection portion 12 as an initial value and calculates, by an analytical solution using the Helmholtz equation, the electromagnetic field in the wave number space of the analytical solution application portion 13 when the plane wave propagates in the propagation direction through the analytical solution application portion 13. The calculation unit 110 calculates the electromagnetic field in the real space of the analytical solution application portion 13 by performing an inverse Fourier transform on the calculated electromagnetic field in the wave number space of the analytical solution application portion 13.

[0049] The computing device 100 may be, for example, an information processing device such as a microcomputer, a personal computer, or a server. The computing device 100 may include a processor PRC, a memory MMR, a storage device STR, and an interface INT. The storage device STR may store programs that describe the processes performed by each component of the computing device 100, such as the computing unit 110. The processor PRC may also load the programs from the storage device STR into the memory MMR and execute the programs. In this way, the processor PRC realizes the functions of each component of the computing device 100. The computing device 100 may realize the computing device 100 shown in FIG. 4 by having the processor PRC execute a program corresponding to the computing unit 110 while referring to the memory MMR and the storage device STR.

[0050] Each component of the computing device 100, such as the computing unit 110, may be realized by dedicated hardware. Furthermore, some or all of the components may be realized by a general-purpose or dedicated circuit, a processor PRC, or a combination of these. These may be configured by a single chip, or by multiple chips connected via a bus. Some or all of the components may be realized by a combination of the above-mentioned circuits and a program. Furthermore, a CPU (Central Processing Unit), a GPU (Graphics Processing Unit), an FPGA (Field-Programmable Gate Array), a quantum processor (quantum computer control chip), or the like may be used as the processor PRC.

[0051] Furthermore, when some or all of the components of the computing device 100 are realized by multiple information processing devices, circuits, etc., the multiple information processing devices, circuits, etc. may be centrally or distributed. For example, the information processing devices, circuits, etc. may be realized in a form in which they are connected to each other via a communication network using a client-server system, a cloud computing system, etc. Furthermore, the functions of the computing device 100 may be provided in a SaaS (Software as a Service) format.

[0052] <Calculation method> Next, a method for calculating the electromagnetic field of the sample 10 performed by the calculation unit 110 will be described. FIGS. 5 and 6 are flow charts illustrating an example of a method for calculating the electromagnetic field according to the first embodiment. As shown in step S10 of FIG. 5, the sample 10, which has a numerical solution application portion 11, a connection portion 12, and an analytical solution application portion 13, is divided into a plurality of cells in the propagation direction of the incident light. Next, as shown in step S20, the electromagnetic field of the sample is calculated by calculating the electromagnetic field of each cell. Specifically, the calculation unit 110 calculates the electromagnetic field of the sample 10 by calculating the electromagnetic field of each cell based on the physical property values ​​assigned to each divided cell.

[0053] When calculating the electromagnetic field of the sample 10 in step S20, first, as shown in step S21 of Fig. 6, the electromagnetic fields of the numerical solution application portion 11 and the connection portion 12 are calculated by numerical calculation using Maxwell's equations. The numerical calculation is, for example, calculation by at least one of the FDTD method, FEM method, BEM method, CIP method, FIT method, and a method of solving Maxwell's equations for a model in which materials are discretized using a general grid. For example, the calculation unit 110 calculates the electromagnetic fields of the numerical solution application portion 11 and the connection portion 12 by numerically solving the electromagnetic fields of each cell in the numerical solution application portion 11 and the connection portion 12 using Maxwell's equations.

[0054] Next, as shown in step S22, the calculated electromagnetic field of the connection part 12 is Fourier transformed to convert it into a plane wave expressed by a wave vector. Specifically, the calculation unit 110 Fourier transforms the electromagnetic field of each cell of the connection part 12 to convert it into a plane wave in wave number space.

[0055] Next, as shown in step S23, the converted plane wave of the connection portion 12 is used as an initial value, and the electromagnetic field in the wave number space of the analytical solution application portion 13 when the plane wave propagates in the propagation direction through the analytical solution application portion 13 is calculated by an analytical solution using the Helmholtz equation. Specifically, the calculation unit 110 calculates the analytical solution in the wave number space using the Helmholtz equation with the converted plane wave as an initial value.

[0056] Next, as shown in step S24, the calculated electromagnetic field in wavenumber space is subjected to an inverse Fourier transform to calculate the electromagnetic field of analytical solution application portion 13. Below, a method for calculating the electromagnetic field of each portion of <numerical solution application portion>, <connection portion>, and <analytical solution application portion> of sample 10 will be described.

[0057] <Numerical solution application part> The numerical solution application portion 11 is a portion including the complex structure of the sample 10. For example, the numerical solution application portion 11 is a portion including an element having a microstructure of about the size of the incident wavelength formed on a substrate. The numerical solution application portion 11 analyzes the electromagnetic field by numerical calculation. The numerical calculation includes, for example, calculation by the FDTD method. Note that the numerical calculation is not limited to calculation by the FDTD method, but may be calculation by any of the FDTD method, FEM method, BEM method, CIP method, FIT method, and a method of solving Maxwell's equations for a model in which the material is discretized using a general grid.

[0058] The analysis method of this embodiment analyzes the electromagnetic field of the numerical solution application portion 11 by numerically solving the electric field and magnetic field of each cell in the numerical solution application portion 11 using Maxwell's equations. That is, the analysis method of this embodiment analyzes the electromagnetic field by numerically solving Maxwell's equations for the numerical solution application portion 11 in the time domain. Specifically, the Maxwell's equations of the following equations (1) and (2) are solved using the FDTD method to obtain a steady-state solution. Here, E represents the electric field strength, H represents the magnetic field strength, D represents the electric flux density, B represents the magnetic flux density, and J represents the current density. Incident light from a light source is given by the current density J.

[0059]

number

number

[0060] For example, when the sample 10 includes a substrate and an element (e.g., a semiconductor device) having a microstructure formed on the substrate and having a size approximately equal to the incident wavelength, the calculation unit 110 selects the portion including the semiconductor device as the numerical solution application portion 11. Then, the calculation unit 110 reads parameters of physical properties required to calculate the electric field E and magnetic field H of each cell in the selected numerical solution application portion 11. Then, the calculation unit 110 continues to update the electromagnetic field by numerical calculation using the FDTD method using Maxwell's equations until the electromagnetic field in the selected numerical solution application portion 11 reaches a steady state.

[0061] 7 is a flowchart illustrating a calculation method of the numerical solution application part 11 in the calculation method of the electromagnetic field according to the first embodiment. As shown in step S30 in FIG. 7, the calculation method of the numerical solution application part 11 in the calculation method of this embodiment includes updating of the electromagnetic field. Specifically, the electromagnetic field is updated over time using Maxwell's equations.

[0062] For example, as shown in step S31, the electric field E is updated. Next, as shown in step S32, a boundary condition for the electric field E is applied. Next, as shown in step S33, the magnetic field H is updated. Next, as shown in step S34, a boundary condition for the magnetic field H is applied. Then, as shown in step S35, convergence is determined. If no convergence has occurred in step S35 (NO), the process returns to step S31, and steps S31 to S35 are repeated. If convergence has occurred in step S35, the calculation results are output. Examples of update equations for the electric field E and the magnetic field H are the following equations (3) and (4).

[0063]

number

number

[0064] When time advances by t, the current t becomes the next step t-Δt. By repeating this process, the electromagnetic field propagates through the sample 10 over time. The above equations (3) and (4) are electromagnetic field update equations for a lossless dielectric medium. For lossy or dispersive media, the equations become more complicated. The above update equations (3) and (4) are applied to all rectangular parallelepiped cells, so the calculation time increases in proportion to the number of cells.

[0065] <Connection part> The connection portion 12 is a portion between the numerical solution application portion 11 and the analytical solution application portion 13. For example, the connection portion 12 includes a portion of the bottom surface of the numerical solution application portion 11 on the analytical solution application portion 13 side. The connection portion 12 and the analytical solution application portion 13 are portions where the incident light propagates uniformly in the propagation direction. Specifically, the complex refractive index of each cell in the connection portion 12 and the analytical solution application portion 13 includes a portion within a certain range. Note that "constant" does not only mean strictly constant, but also constant within a range that includes unavoidable measurement errors. When the sample 10 includes a substrate and an element having a microstructure formed on the substrate and having a size approximately equal to the incident wavelength, the connection portion 12 is a portion including the top surface of the substrate to which the element is connected. The calculation unit 110 calculates the electromagnetic field of the connection portion 12 by numerical calculation.

[0066] The electromagnetic field calculation method of this embodiment analyzes the electromagnetic field of the connection part 12 by numerically solving the electric field E and magnetic field H of each cell in the connection part 12 using Maxwell's equations. That is, the calculation method of this embodiment calculates the electromagnetic field by numerically solving Maxwell's equations for the connection part 12 in the time domain.

[0067] In addition, the calculation method of this embodiment converts the calculated electric field E and magnetic field H of each cell of the connection part 12 into plane waves in wave number space by Fourier transform. Then, an analytical solution in wave number space is calculated using the Helmholtz equation with the converted plane waves as initial values. The calculated analytical solution is used to calculate the electromagnetic field in the analytical solution application part 13.

[0068] For example, the calculation unit 110 may calculate the electromagnetic field at the connection part 12 as follows. That is, the calculation unit 110 performs a Fourier transform on the electromagnetic field at the connection part 12, calculated by numerical calculation using the FDTD method, for example, from the XY plane to the kxky plane, which is wave number space. The calculation unit 110 can apply an analytical solution to the Helmholtz equation by expanding any electromagnetic field distribution at the connection part 12 into plane waves.

[0069] Specifically, the calculation unit 110 performs a Fourier transform on the XY plane of the electromagnetic field distribution of the connection portion 12 calculated by the FDTD method, and expands the analytical solution into applicable plane waves. This is used as the initial value when solving the Helmholtz equation. Next, the calculation unit 110 generates wavenumber space cells included in the analytical solution of the Helmholtz equation. When using a fast Fourier transform as the Fourier transform, the cells on the XY plane must be uniformly distributed, so uniform cells are generated and the electromagnetic field on those cells is prepared by interpolation or the like.

[0070] <Analysis solution application part> The analytical solution application portion 13 is a portion below the connection portion 12. When the sample 10 includes a substrate and an element having a microstructure formed on the substrate and having a size on the order of the wavelength of incident light, the analytical solution application portion 13 includes a portion of the substrate.

[0071] The method for calculating the electromagnetic field of the analytical solution application portion 13 is as follows: the calculation unit 110 converts the electric field E and magnetic field H of each cell of the connection portion 12 into plane waves in wave number space by Fourier transform. Then, the calculation unit 110 calculates the electromagnetic field in wave number space of the analytical solution application portion 13 using the Helmholtz equation with the converted plane waves as initial values. Specifically, the calculation unit 110 calculates an analytical solution of the Helmholtz equation assuming that a plane wave propagates through the analytical solution application portion 13 in the propagation direction. The analytical solution is calculated analytically. The calculated analytical solution includes the electromagnetic field of the analytical solution application portion 13 in wave number space. Then, the calculation unit 110 calculates the electromagnetic field in real space of the analytical solution application portion 13 by performing an inverse Fourier transform on the calculated analytical solution, i.e., the electromagnetic field in wave number space.

[0072] The calculation unit 110 can quickly calculate the electromagnetic field of a substrate such as silicon as an analytical solution of the Helmholtz equation, with the plane wave in wave number space at the connection portion 12 as the initial value. A plane wave propagating through a uniform medium is given as an analytical solution of the Helmholtz equation of the following equation (5):

[0073]

number

[0074] When the electromagnetic field is in a steady state, the solution of the Helmholtz equation is obtained as shown in the following equations (6) and (7).

[0075]

number

number

[0076] Since the optical axis of the incident light is in the Z-axis direction, when the electromagnetic field is Fourier transformed on the XY plane, it becomes a plane wave expressed by the wave vector (kx, ky). Since these are expressed as exp[-i(kxX+kyY+kzZ)], the above equations can be written as the following equations (8) to (11).

[0077]

number

number

number

number

[0078] Using the electromagnetic field at the connection part 12 obtained by the FDTD method or the like as the initial value, equation (11) is solved in the Z-axis direction, and the results are converted back to real space by inverse Fourier transform. This allows the calculation part 110 to quickly obtain the results of the electromagnetic field in the analytical solution application part 13 as the same results as when calculated by the FDTD method.

[0079] Here, the calculation flow is shown for the electric field E, but the same applies to the magnetic field H, and independent calculations are possible for each. Therefore, the calculation unit 110 of this embodiment can calculate only the electric field E, only the magnetic field H, or both the electric field E and the magnetic field H in the analytical solution application part 13, depending on the predetermined application. This makes it possible to reduce the amount of calculation and memory usage in the analytical solution application part 13.

[0080] Furthermore, since the Fourier transformed electromagnetic wave is a plane wave, each component is perpendicular to the propagation direction. In other words, the transverse wave conditions of the following equations (12) and (13) hold true:

[0081]

number

number

[0082] In the analytical solution application section 13, the calculation unit 110 may perform a Fourier transform on two of the three components of the electric field E in three directions including the propagation direction of the incident light and two directions orthogonal to the propagation direction, and calculate the remaining component from the transverse wave condition of the electromagnetic wave. For example, only Ex and Ey may be Fourier transformed and the analytical solution may be applied, and based on the results, Ez in the following equation (14) may be calculated without a Fourier transform.

[0083]

number

[0084] The same is true for the magnetic field H. Therefore, the calculation unit 110 performs a Fourier transform on only two of the three components of the electromagnetic field in the analytical solution application section 13, and can determine the remaining component from the transverse wave condition of the electromagnetic wave. By utilizing this feature, when the structure of the sample 10 is large in the transverse direction, i.e., in the X-axis direction or Y-axis direction, it is possible to reduce the calculation time and memory usage for the Fourier transform.

[0085] Furthermore, in the above equation (11), there is no particular limit to the magnitude of the value of ΔZ. If it is possible to calculate the electromagnetic field of the connection portion 12, which is the connection portion between the numerical solution application portion 11, which includes a complex structure, and the analytical solution application portion 13, which is a uniform medium, it is possible to obtain the electromagnetic field for any XY plane within the analytical solution application portion 13. This allows the calculation unit 110 to analyze the electromagnetic field within the analytical solution application portion 13, limited to a portion including a predetermined plane perpendicular to the propagation direction, or a portion sandwiched between two predetermined planes perpendicular to the propagation direction. For example, when only the electromagnetic field deep within the substrate of the sample 10 is required, calculations for unnecessary portions can be omitted, thereby reducing calculation time and memory usage.

[0086] 8 is a graph illustrating a method for determining the numerical solution application portion 11, the connection portion 12, and the analytical solution application portion 13 of the sample 10 according to embodiment 1, where the horizontal axis represents the position in the Z-axis direction of the sample 10 and the vertical axis represents the complex refractive index. As shown in Fig. 8, the calculation unit 110 may determine the numerical solution application portion 11, the connection portion 12, and the analytical solution application portion 13 by determining whether the complex refractive index of the cell is within a certain range from the direction opposite to the propagation direction.

[0087] The calculation unit 110 pre-sets conditions under which a medium can be considered to be a homogeneous medium. For example, the calculation unit 110 determines that the medium is a homogeneous medium when the complex refractive index is within a certain range. The calculation unit 110 determines the complex refractive index from the direction opposite to the propagation direction of the incident light. When the complex refractive index of the sample 10 changes smoothly within a certain range, the calculation unit 110 determines that portion as the analytical solution application portion 13. When the complex refractive index is outside the certain range, the calculation unit 110 determines that portion as the connection portion 12. The calculation unit 110 determines the portion in the opposite direction from the connection portion 12 as the numerical solution application portion 11. In this way, the calculation unit 110 can automatically determine the numerical solution application portion 11, the connection portion 12, and the analytical solution application portion 13 by scanning and determining from the direction opposite to the propagation direction of the incident light. For example, let i, j, and k be cell numbers in the X-, Y-, and Z-axis directions, respectively, and let the complex refractive index of the material occupied by the i-th, j-th, and k-th cells be given by the following equation (15).

[0088]

number

[0089] If the reflectance of the bottom cell of the structure is smaller than a set threshold, it is considered to be a uniform medium, as defined by the following equations (16) and (17).

number

number

[0090] 9 is a diagram illustrating an XY plane perpendicular to the propagation direction of the incident light in the sample 10 according to the first embodiment. As shown in FIG. 9, if the structure of the sample 10 is symmetrical with respect to the propagation direction of the incident light, and the incident light is also symmetrical with respect to the propagation direction, the electromagnetic field will also be symmetrical. Therefore, by calculating the electromagnetic field in the first quadrant, it is possible to calculate the distribution of the electromagnetic field in the entire sample 10 from the second to fourth quadrants.

[0091] In the numerical solution application portion 11, if the structure of the sample 10 and the intensity of the incident light are both symmetric with respect to the propagation direction, symmetric boundary conditions and antisymmetric boundary conditions may be used. In this embodiment, the calculation unit 110 calculates the electromagnetic field of the connection portion 12 between the numerical solution application portion 11 and the analytical solution application portion 13 using a plane wave expansion of the electromagnetic field obtained by Fourier transform. In this way, to use a plane wave expansion of the electromagnetic field obtained by Fourier transform, the distribution of the electromagnetic field must be periodic. However, the analytical solution application portion 13 is a homogeneous medium. Therefore, if the electromagnetic field in the first quadrant can be calculated, the original periodic electromagnetic field can be reproduced by appropriately inverting and copying the electromagnetic fields in the second to fourth quadrants, which are omitted due to symmetry. In this way, the calculation unit 110 in this embodiment can calculate the analytical solution application portion 13 using symmetric boundary conditions and antisymmetric boundary conditions.

[0092] 10 to 12 are diagrams illustrating the electromagnetic field in the first quadrant of the XY plane orthogonal to the propagation direction of the incident light in sample 10 according to embodiment 1, with Fig. 10 showing the electric field Ex, Fig. 11 showing the electric field Ey, and Fig. 12 showing the electric field Ez. As shown in Figs. 10 to 12, the electromagnetic field obtained in the first quadrant does not have periodicity.

[0093] 13 to 15 are diagrams illustrating the electromagnetic field in the XY plane perpendicular to the propagation direction of the incident light in the sample 10 according to the first embodiment. FIG. 13 shows the electric field Ex, FIG. 14 shows the electric field Ey, and FIG. 15 shows the electric field Ez. As shown in FIGS. 13 to 15, the original electromagnetic field can be reproduced by expanding, inverting, and copying the regions of FIGS. 10 to 12. Specifically, the calculation unit 110 may calculate the electromagnetic field in the analytical solution application portion 13 by expanding the electromagnetic field in the connection portion 12 to the analytical solution application portion 13 using symmetric boundary conditions and antisymmetric boundary conditions. In this example, only the X component of the electric field E of the incident light is nonzero. The X component of the internal electric field E of the sample 10 is symmetric in the X-axis and Y-axis directions. The Y component is antisymmetric in the X-axis and Y-axis directions, and the Z component is antisymmetric and symmetric in the X-axis and Y-axis directions, respectively. The calculation unit 110 performs a Fourier transform on the reproduced electromagnetic field and applies the analytical solution of the Helmholtz equation. The electromagnetic field in the first quadrant obtained in this way is the same as when a calculation method using numerical calculations is applied to the entire area.

[0094] As described above, the calculation method of this embodiment includes a part for calculating the Helmholtz equation for the electromagnetic field in wavenumber space using analytical solutions. Since calculations using analytical solutions are faster than numerical calculations using the FDTD method, it is possible to speed up calculation of the electromagnetic field over the entire sample 10.

[0095] By calculating the analytical solution for the entire substrate and then converting it back to real space using an inverse Fourier transform, the same results as the FDTD method can be obtained. In other words, by performing an inverse Fourier transform on the electromagnetic field in wavenumber space, it is possible to calculate the electromagnetic field in real space on a substrate such as a silicon substrate.

[0096] Next, the effects of this embodiment will be described. The calculation device and calculation method of this embodiment calculates the sample 10 by dividing it into a numerical solution application part 11, a connection part 12, and an analytical solution application part 13. This improves the calculation speed and reduces the calculation time cost.

[0097] Here, we compare the case where the electromagnetic field is calculated by performing numerical calculations on the entire sample 10 as the numerical solution application portion 11 with the case where the electromagnetic field is calculated by dividing the sample 10 into the numerical solution application portion 11, the connection portion 12, and the analytical solution application portion 13, as in this embodiment. Here, the example where the numerical calculation is performed on the entire sample 10 is called a comparative example.

[0098] Fig. 16 is a perspective view illustrating the sample 10 according to embodiment 1. As shown in Fig. 16, the sample 10 has a layer portion LY, post portions P1 to P4, a space portion AR, and a substrate portion SB. The integral values ​​in each region of the electric field obtained by irradiating the sample 10 with incident light from above the sample 10 are compared between this embodiment and a comparative example.

[0099] Fig. 17 is a perspective view illustrating a sample 10 to which the calculation method according to embodiment 1 is applied. Fig. 18 is a perspective view illustrating a sample 10 to which a calculation method according to a comparative example is applied. As shown in Fig. 17, in the calculation method of this embodiment, the layer portion LY, post portions P1 to P4, and space portion AR other than the substrate portion SB are the numerical solution application portion 11, and the substrate portion SB is the analytical solution application portion 13. On the other hand, as shown in Fig. 18, in the calculation method of the comparative example, the layer portion LY, post portions P1 to P4, space portion AR, and substrate portion SB are the numerical solution application portion 11.

[0100] FIG. 19 is a diagram comparing the integral value of the electric field calculated by the calculation method according to the first embodiment with the integral value of the electric field calculated by the calculation method according to the comparative example. As shown in FIG. 19, the error between the values ​​of this embodiment and the values ​​of the comparative example is 1% or less at each portion. Therefore, the calculation method according to this embodiment can suppress a decrease in the calculation accuracy of the electromagnetic field. In general, the FDTD method is a finite difference method, and therefore includes numerical errors when the electromagnetic field propagates. Furthermore, when the absorbing boundary condition is applied, a small amount of error reflection from the edge of the analytical solution application portion 13 is included. On the other hand, when the analytical solution is applied, such errors caused by numerical calculation are not included. Therefore, the calculation method according to this embodiment can improve the calculation accuracy compared to the calculation method according to the comparative example.

[0101] FIG. 20 is a perspective view showing a plurality of samples 10 arranged with different lengths in the propagation direction of the analysis solution application part 13 according to Embodiment 1. FIG. 21 is a diagram comparing the calculation time of the calculation method according to Embodiment 1 and the calculation time of the calculation method according to the comparative example. FIG. 22 is a graph comparing the calculation time of the calculation method according to Embodiment 1 and the calculation time of the calculation method according to the comparative example, where the horizontal axis represents each sample and the vertical axis represents the calculation time.

[0102] As shown in FIG. 20, for the structures C1 to C4 in which the thickness of the substrate portion SB, which is the analysis solution application part 13 of the sample 10, is changed, the calculation time of the calculation method of the present embodiment and the calculation time of the calculation method of the comparative example are compared. As shown in FIGS. 21 and 22, in the calculation method of the comparative example, the calculation time increases in proportion to the thickness of the substrate portion SB. However, in the calculation method of the present embodiment, the calculation speed does not increase regardless of the thickness of the substrate portion SB. Therefore, it can be said that the calculation method of the present embodiment has a greater effect of reducing the calculation time as the analysis solution application part 13 is larger compared to the numerical solution application part 11.

[0103] When the FDTD method is applied to the entire sample 10 as in the calculation method of the comparative example, the time until the electromagnetic field reaches a saturated state, that is, the time required for convergence, is set to T seconds in the numerical solution application part 11 and S seconds in the analysis solution application part 13. When the relationship between these calculation times is T > S, the convergence speed in the numerical solution application parts 11 of the present embodiment and the comparative example does not change. Therefore, simply, the effect of reducing the calculation time corresponding to the reduction in cells can be obtained. However, when T < S, the calculation method of the present embodiment replaces the calculation of the electromagnetic field in the substrate portion with a slow convergence with a faster analysis solution method from the FDTD method. For this reason, the calculation method of the present embodiment not only simply reduces the calculation time corresponding to the reduction in cells, but also speeds up the convergence of the electromagnetic field itself. Therefore, it can be said that the analysis method of the present embodiment has the effect of shortening the calculation time by at least the amount of cell reduction.

[0104] (Embodiment 2) Next, a second embodiment will be described. In this embodiment, there are a plurality of at least one of the numerical solution application portions 11 and the analytical solution application portions 13. FIG. 23 is a perspective view illustrating a sample according to the second embodiment. As shown in FIG. 23, the sample 30 of this embodiment has a configuration in which, for example, two samples 10 and 20 are stacked in the propagation direction of the incident light. Therefore, the sample 30 includes two numerical solution application portions 11 and 21, two connection portions 12 and 22, and two analytical solution application portions 13 and 23.

[0105] In this case, the analytical solution application portion 13, which is sandwiched between the numerical solution application portion 11 and the numerical solution application portion 21, contains both a transmitted wave from the numerical solution application portion 11 and a reflected wave from the numerical solution application portion 21. For this reason, calculation cannot be completed with only one direction. Therefore, the calculation unit 110 calculates multiple reflections. The calculation device 100 also includes a storage device STR that stores electromagnetic fields. The storage device STR stores electromagnetic field data for multiple numerical solution application portions 11 and 21 and multiple analytical solution application portions 13 and 23. With this configuration, the calculation unit 110 can obtain calculation results equivalent to those obtained when the FDTD method is applied to the entire area of ​​the sample 30.

[0106] 24 to 27 are flow charts illustrating an example of a method for calculating an electromagnetic field according to embodiment 2. As shown in step S101 of Fig. 24, predetermined portions of the storage device STR for individually storing the electromagnetic fields calculated by the numerical solution application parts 11 and 21 are initialized to zero. Next, as shown in step S102, predetermined portions of the storage device STR for individually storing the electromagnetic fields calculated by the analytical solution application parts 13 and 23 are initialized to zero.

[0107] 25, the electromagnetic field obtained by solving Maxwell's equations by numerical calculation such as the FDTD method is added to the corresponding storage device STR for the numerical solution application portion 11. Next, as shown in step S112, the electromagnetic field of the connection portion 12 including the bottom surface of the numerical solution application portion 11 is Fourier transformed to obtain a plane wave expressed by the transformed wave vector.

[0108] Next, as shown in step S113, a plane wave propagating in the propagation direction within analytical solution application portion 13 is obtained as an analytical solution of the Helmholtz equation. Next, as shown in step S114, the analytical solution is calculated in the entire analytical solution application portion 13, and the electromagnetic field obtained by returning it to real space by inverse Fourier transform is added to the corresponding storage device STR.

[0109] 26, the electromagnetic field at the bottom surface of analytical solution application portion 13 is used as input, and the electromagnetic field obtained by solving Maxwell's equations by the FDTD method for numerical solution application portion 21 is added to the corresponding storage device STR. Next, as shown in step S122, the electromagnetic field of connection portion 22 including the bottom surface of numerical solution application portion 21 is Fourier transformed to obtain a plane wave expressed by the transformed wave vector.

[0110] Next, as shown in step S123, a plane wave propagating in the propagation direction within analytical solution application portion 23 is obtained as an analytical solution of the Helmholtz equation. Next, as shown in step S124, the analytical solution is calculated in the entire analytical solution application portion 23, and the electromagnetic field obtained by returning it to real space by inverse Fourier transform is added to the corresponding storage device STR.

[0111] 27, the electromagnetic field of the portion including the upper surface of the numerical solution application portion 21 is Fourier transformed to obtain a plane wave represented by the transformed wave vector. Next, as shown in step S132, a plane wave propagating in the +Z-axis direction within the analytical solution application portion 13 is obtained as an analytical solution of the Helmholtz equation. Next, as shown in step S133, the analytical solution is calculated over the entire analytical solution application portion 13, and the electromagnetic field obtained by converting it back to real space by inverse Fourier transform is added to the corresponding storage device STR.

[0112] Next, as shown in step S134, the electromagnetic field on the top surface of the analytical solution application portion 13 is used as input, and the electromagnetic field obtained by solving Maxwell's equations for the numerical solution application portion 11 using the FDTD method is added to the corresponding storage device STR. Next, as shown in step S135, the process returns to step S112. That is, the electromagnetic field on the bottom surface of the numerical solution application portion 11 is Fourier transformed to obtain a plane wave represented by the transformed wave vector. Then, the electromagnetic field of the analytical solution application portion 13 in real space calculated by the inverse Fourier transform is added to the storage device STR.

[0113] By repeating the calculation until the stored electromagnetic field stops changing, or a specified number of times, it is possible to obtain results equivalent to those obtained by applying a numerical solution to the entire sample 30. By continuing to update the electromagnetic field until it saturates, a steady state is obtained. Based on this electromagnetic field, it is possible to calculate the absorbed energy, etc.

[0114] In this way, when the sample 30 has multiple numerical solution application portions 11 and 21, multiple connection portions 12 and 22, and multiple analytical solution application portions 13 and 23, multiple numerical calculations using the numerical solution method are required due to multiple reflections. However, in cases where the analytical solution application portion 13 is sufficiently smaller than the numerical solution application portion 11, the electromagnetic field can be calculated in a shorter time than when the numerical solution method is applied to the entire sample 30. Note that even if the sample 30 has three or more numerical solution application portions and three or more analytical solution application portions, the electromagnetic field can be calculated using the above-mentioned calculation method.

[0115] The present disclosure is not limited to the above-described first and second embodiments and may be modified as appropriate without departing from the spirit of the present disclosure. For example, the configurations of the first and second embodiments may be combined with each other. Furthermore, the following calculation program, which causes a computer to load and execute the above-described calculation method, is also within the scope of the technical concept of the embodiments. The calculation program may be stored on a non-transitory computer-readable medium or a tangible storage medium. By way of example and not limitation, computer-readable mediums or tangible storage media include random-access memory (RAM), read-only memory (ROM), flash memory, solid-state drive (SSD) or other memory technologies, CD-ROM, digital versatile disc (DVD), Blu-ray (registered trademark) disc or other optical disk storage, magnetic cassette, magnetic tape, magnetic disk storage or other magnetic storage devices.

[0116] (Appendix 1) A calculation program for calculating an electromagnetic field of a sample, Dividing the sample into a plurality of cells in the propagation direction of the incident light, the cells having a numerical solution application portion, a connection portion, and an analytical solution application portion in this order; calculating the electromagnetic field of the sample by calculating the electromagnetic field of each cell based on the physical property values ​​assigned to each divided cell; A calculation program that causes a computer to execute the following: the connecting portion and the analytical solution application portion are configured such that the incident light propagates uniformly in the propagation direction, When calculating the electromagnetic field of the sample, calculating the electromagnetic fields of the numerical solution application portion and the connection portion by numerical calculation using Maxwell's equations; converting the calculated electromagnetic field of the connection portion into a plane wave expressed by a wave vector by Fourier transforming the electromagnetic field of the connection portion; The converted plane wave of the connection portion is used as an initial value, and the electromagnetic field in the wave number space of the analytical solution application portion when the plane wave propagates through the analytical solution application portion in the propagation direction is calculated by an analytical solution using the Helmholtz equation; calculating the electromagnetic field of the analytical solution application portion by performing an inverse Fourier transform on the electromagnetic field of the calculated analytical solution application portion in the wave number space; Calculation program. (Appendix 2) When calculating the electromagnetic field of the sample, In the analytical solution application part, either the electric field only, the magnetic field only, or both the electric field and the magnetic field are calculated depending on a predetermined application. Calculation program described in Appendix 1. (Appendix 3) When calculating the electromagnetic field of the sample, In the analytical solution application part, two of the three components of the electromagnetic field in three directions including the propagation direction and two directions orthogonal to the propagation direction are Fourier transformed, and the remaining component is calculated from a transverse wave condition of the electromagnetic wave. Calculation program described in Appendix 1. (Appendix 4) When calculating the electromagnetic field of the sample, In the analytical solution application part, the electromagnetic field is calculated by limiting the calculation to a first part including a predetermined plane perpendicular to the propagation direction or a second part sandwiched between two predetermined planes perpendicular to the propagation direction. Calculation program described in Appendix 1. (Appendix 5) When dividing into the plurality of cells, determining whether the complex refractive index of the cell is within a certain range from a direction opposite to the propagation direction, thereby determining the numerical solution application portion, the connection portion, and the analytical solution application portion; Calculation program described in Appendix 1. (Appendix 6) When calculating the electromagnetic field of the sample, Calculating the electromagnetic field in the analytical solution application portion by expanding the electromagnetic field in the connection portion to the analytical solution application portion using a symmetric boundary condition and an antisymmetric boundary condition. Calculation program described in Appendix 1. (Appendix 7) the sample includes a plurality of the numerical solution application portions and a plurality of the analytical solution application portions; further causing the computer to store the electromagnetic field data of the plurality of numerical solution application portions and the plurality of analytical solution application portions. Calculation program described in Appendix 1. (Appendix 8) the analytical solution application portion and the connection portion have a complex refractive index of the cell in a certain range; Calculation program described in Appendix 1. (Appendix 9) the sample includes a substrate and an element formed on the substrate; the analytical solution application portion includes the substrate; the numerical solution application part includes the element, the connection portion includes a portion of the substrate to which the element is connected. Calculation program described in Appendix 1. (Appendix 10) The numerical calculation is performed by at least one of the FDTD method, the FEM method, the BEM method, the CIP method, the FIT method, and a method of solving Maxwell's equations for a model in which the material is discretized by a grid. Calculation program described in Appendix 1. [Explanation of symbols]

[0117] 10, 20, 30 samples 11, 21 Numerical solution application part 12, 22 Connection part 13, 23 Analytical solution application part 100 Calculation Device 110 Calculation Unit AR spatial part C1, C2, C3, C4 structure INT interface LY layer part MMR Memory PRC Processor P1, P2, P3, P4 post parts SB board part STR storage

Claims

1. a calculation unit that divides a sample having a numerical solution application portion, a connection portion, and an analytical solution application portion into a plurality of cells in the propagation direction of incident light, and calculates the electromagnetic field of each cell based on physical property values ​​assigned to each cell, thereby calculating the electromagnetic field of the sample; the connecting portion and the analytical solution application portion are configured such that the incident light propagates uniformly in the propagation direction, The calculation unit calculating the electromagnetic fields of the numerical solution application portion and the connection portion by numerical calculation using Maxwell's equations; converting the calculated electromagnetic field of the connection portion into a plane wave expressed by a wave vector by Fourier transforming the electromagnetic field of the connection portion; The plane wave of the converted connection portion is used as an initial value, and the electromagnetic field in the wave number space of the analytical solution application portion when the plane wave propagates through the analytical solution application portion in the propagation direction is calculated by an analytical solution using the Helmholtz equation; calculating the electromagnetic field of the analytical solution application portion by performing an inverse Fourier transform on the electromagnetic field of the calculated analytical solution application portion in the wave number space; Calculation device.

2. The calculation unit calculates, in the analytical solution application portion, any of an electric field only, a magnetic field only, or both the electric field and the magnetic field according to a predetermined application. The computing device of claim 1 .

3. In the analytical solution application portion, the calculation unit performs a Fourier transform on two of the three components of the electromagnetic field in three directions including the propagation direction and two directions orthogonal to the propagation direction, and calculates the remaining component from a transverse wave condition of the electromagnetic wave. The computing device of claim 1 .

4. The calculation unit calculates the electromagnetic field in the analytical solution application portion by limiting the electromagnetic field to a first portion including a predetermined plane perpendicular to the propagation direction or a second portion sandwiched between two predetermined planes perpendicular to the propagation direction. The computing device of claim 1 .

5. the calculation unit determines whether the complex refractive index of the cell is within a certain range from a direction opposite to the propagation direction, thereby determining the numerical solution application portion, the connection portion, and the analytical solution application portion. The computing device of claim 1 .

6. the calculation unit calculates the electromagnetic field in the analytical solution application portion by expanding the electromagnetic field in the connection portion to the analytical solution application portion using a symmetric boundary condition and an antisymmetric boundary condition. The computing device of claim 1 .

7. the sample includes a plurality of the numerical solution application portions and a plurality of the analytical solution application portions; Further comprising a storage device that stores data of the electromagnetic field of the plurality of numerical solution application parts and the plurality of analytical solution application parts; The computing device of claim 1 .

8. the analytical solution application portion and the connection portion have a complex refractive index of the cell in a certain range; The computing device of claim 1 .

9. the sample includes a substrate and an element formed on the substrate; the analytical solution application portion includes the substrate; the numerical solution application part includes the element, the connection portion includes a portion of the substrate to which the element is connected. The computing device of claim 1 .

10. The numerical calculation is a calculation by at least one of the FDTD method, the FEM method, the BEM method, the CIP method, the FIT method, and a method of solving Maxwell's equations for a model in which the material is discretized by a grid. The computing device of claim 1 .

11. 1. A method for calculating an electromagnetic field of a sample, comprising: Dividing the sample into a plurality of cells in a propagation direction of the incident light, the cells having a numerical solution application portion, a connection portion, and an analytical solution application portion in that order; calculating the electromagnetic field of the sample by calculating the electromagnetic field of each cell based on the physical property values ​​assigned to each divided cell; Equipped with the connecting portion and the analytical solution application portion are configured such that the incident light propagates uniformly in the propagation direction, In the step of calculating the electromagnetic field of the sample, calculating the electromagnetic fields of the numerical solution application portion and the connection portion by numerical calculation using Maxwell's equations; converting the calculated electromagnetic field of the connection portion into a plane wave expressed by a wave vector by Fourier transforming the electromagnetic field of the connection portion; The plane wave of the converted connection portion is used as an initial value, and the electromagnetic field in the wave number space of the analytical solution application portion when the plane wave propagates through the analytical solution application portion in the propagation direction is calculated by an analytical solution using the Helmholtz equation; calculating the electromagnetic field of the analytical solution application portion by performing an inverse Fourier transform on the electromagnetic field of the calculated analytical solution application portion in the wave number space; Calculation method.

12. In the step of calculating the electromagnetic field of the sample, In the analytical solution application part, either the electric field only, the magnetic field only, or both the electric field and the magnetic field are calculated depending on a predetermined application. The calculation method according to claim 11.

13. In the step of calculating the electromagnetic field of the sample, In the analytical solution application part, two of the three components of the electromagnetic field in three directions including the propagation direction and two directions orthogonal to the propagation direction are Fourier transformed, and the remaining component is calculated from a transverse wave condition of the electromagnetic wave. The calculation method according to claim 11.

14. In the step of calculating the electromagnetic field of the sample, In the analytical solution application portion, the electromagnetic field is calculated by limiting the first portion including a predetermined plane perpendicular to the propagation direction or the second portion sandwiched between two predetermined planes perpendicular to the propagation direction. The calculation method according to claim 11.

15. In the step of dividing into a plurality of cells, determining whether the complex refractive index of the cell is within a certain range from a direction opposite to the propagation direction, thereby determining the numerical solution application portion, the connection portion, and the analytical solution application portion; The calculation method according to claim 11.

16. In the step of calculating the electromagnetic field of the sample, Calculating the electromagnetic field in the analytical solution application portion by expanding the electromagnetic field in the connection portion to the analytical solution application portion using a symmetric boundary condition and an antisymmetric boundary condition. The calculation method according to claim 11.

17. the sample includes a plurality of the numerical solution application portions and a plurality of the analytical solution application portions; further comprising a step of storing the electromagnetic field data of the plurality of numerical solution application portions and the plurality of analytical solution application portions; The calculation method according to claim 11.

18. the analytical solution application portion and the connection portion have a complex refractive index of the cell in a certain range; The calculation method according to claim 11.

19. the sample includes a substrate and an element formed on the substrate; the analytical solution application portion includes the substrate; the numerical solution application part includes the element, the connection portion includes a portion of the substrate to which the element is connected. The calculation method according to claim 11.

20. The numerical calculation is a calculation by at least one of the FDTD method, the FEM method, the BEM method, the CIP method, the FIT method, and a method of solving Maxwell's equations for a model in which the material is discretized by a grid. The calculation method according to claim 11.

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