CN113987792B - Method for realizing accurate mode source input in FDTD algorithm - Google Patents
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Abstract
The invention belongs to the technical field of electromagnetic field numerical simulation, and particularly relates to a method for realizing accurate mode source input in an electromagnetic field time domain finite difference method. A method for realizing accurate mode source input in FDTD algorithm comprises the following steps: setting medium parameter distribution, selecting a mode source input surface in an FDTD simulation space and carrying out mode solving; performing phase correction on a time domain function of an electric field and a magnetic field of a mode source on an input surface; the phase-corrected electric and magnetic field time domain functions are input into an iterative calculation of FDTD. According to the method for realizing accurate input of the mode source in the FDTD algorithm, when the mode source time domain function is coupled into the calculation space of the FDTD, the mode source time domain function is subjected to phase correction according to the medium coefficient and the grid setting of the calculation space, so that the coupling efficiency of the mode source is improved, and the accuracy of the mode source input is improved.
Description
Technical Field
The invention belongs to the technical field of electromagnetic field numerical simulation, and particularly relates to a method for accurately inputting a mode source in an electromagnetic field time domain finite difference method.
Background
The method for calculating the propagation of electromagnetic waves in a complex model mainly depends on numerical calculation, and in the process of designing and simulating optical chips and microwave devices, the numerical calculation of the electromagnetic waves by adopting a computer can greatly improve the design and the efficiency of related products, and the common methods comprise a finite element method, a spectrum decomposition method and a time-finite difference method.
The finite difference method (FDTD) of the time domain of the electromagnetic field directly solves Maxwell's equations or wave equations in the time domain to obtain the magnetic field intensity of the electric field on the spatial nodes in the time domain, and the response rule of the electromagnetic wave in the specific waveguide structure can be accurately reflected. In FDTD, the spatial arrangement of discrete electric and magnetic field computation nodes is as shown in FIG. 1, with each electric field component surrounded by four magnetic field components orthogonal thereto, as are the magnetic fields; the arrangement mode is very consistent with the formula structure of Faraday 'law and Ampere' law, and is very suitable for calculating a Maxwell FDTD formula.
As the time domain finite difference method adopts the Yee grid structure, whether the Yee grid structure is a sub-grid with non-uniform thickness or a common grid, when the Maxwell equation set differential form is adopted to sequentially and alternately solve the electric field and the magnetic field, the electric field and the magnetic field have differences in time and space, so that when a mode source is input, the electric field and the magnetic field have phase differences, and the phase differences are determined by the differences in time and space. In the FDTD calculation method, if a mode source is added to simulate an incident source, the electric field and the magnetic field are not in phase. As shown in fig. 3, assuming that a TEM wave mode source pulse propagating in the z direction is added to the three-dimensional grid to simulate the mode after light is stably present in the scatterer, the added Ex, ey and Hx, hy are respectively at k=k A Plane sum k=k A -1/2 plane, and k=k A K=k A The 1/2 planes also form the inner and outer boundaries of the total field, respectively. In order to ensure the validity of the total field boundary and that the simulated incident wave or light propagates only in a defined direction, no reflection of the incident wave occurs at the other end of the total field boundary, and therefore, when the incident wave is introduced at the corresponding spatial position and instant, the introduced incident field component must also satisfy the corresponding position and phase at the total field boundary. If the position and phase errors of the simulated incident electromagnetic field and the actual incident electromagnetic field are larger, the boundary of the total field is greatly affected, for example, the incident field on the boundary of the total field is seriously leaked, that is, a part of the incident wave directly propagates out of the total field, when the FDTD calculates the field distribution, the calculation area is generally divided into a total field area and a scattering field area, since the matching layer arranged on the cut-off boundary only can absorb the outward traveling wave, the leakage of the wave source can interfere the field of the calculation space to reduce the accuracy of the simulation degree, so that the incident wave is limited in the total field area, and the division mode of the closed total field space is introduced in fig. 2.
In view of the above, when a certain known incident wave field value is set at the boundary of the total field, the electric field and magnetic field components of the incident wave are superimposed on the electric field and magnetic field components of the boundary of the total field in a portion as time steps progress, and in the electromagnetic scattering problem, the spatial field can be regarded as a superposition of the scattered field and the incident field. One common method is shown in fig. 3, where the total field becomes a half-space field by multiplying the plane of the input light wave source by a mode field corresponding to the waveguide structure, and a closed total field space is not required. However, whether the total field-scattering scheme or the mode source input scheme is used, tangential components of an electric field and a magnetic field of an incident electromagnetic wave are added on a total field-scattering field interface or a mode source input interface, and the directions of the electric field, the magnetic field and the wave vector are orthogonal, so that the direction of the wave vector can be uniquely determined. However, in the FDTD grid, the discrete values of the electric field and the magnetic field are often not coincident, and when the electromagnetic fields of the incident wave are superimposed on the coordinates of a pair of adjacent electric fields and magnetic fields, the phases of the applied electric fields and magnetic fields are required to be identical at the same time at the same point in space, so that the phase difference between the electric fields and the magnetic fields must be considered at different points and at different moments.
Disclosure of Invention
In order to solve the above problems, the present invention proposes a method for implementing a mode source input in an FDTD algorithm through Hilbert transformation, based on the function that Hilbert transformation can convert any one real signal into a corresponding analytic signal, extract phase information from amplitude information, and add the phase information to the mode source, so as to increase coupling efficiency of the mode source in FDTD.
The technical scheme adopted for solving the technical problems is as follows: a method for realizing accurate mode source input in FDTD algorithm comprises the following steps:
setting medium parameter distribution, selecting a mode source input surface in an FDTD simulation space and carrying out mode solving;
performing phase correction on a time domain function of an electric field and a magnetic field on a mode source input surface;
the phase-corrected electric and magnetic field time domain functions are added to the iterative computation of FDTD.
As a preferred mode of the present invention, the time domain functions of the electric field and the magnetic field on the input surface are subjected to phase correction by using Hilbert transformation, which is expressed as:wherein f (t) represents a pre-transformation time domain function; />Representing the Hilbert transformed time domain complex function.
Further, the Hilbert transform adopts a convolution operation mode, and an operation formula is expressed as follows:s= -1 when exp (-j (wt- βr)) is defined as forward propagation in the r direction, and s=1 when propagation in the r negative direction.
Further, the method comprises the steps of correcting the mode source electric field time domain function, correcting the magnetic field time domain function, and solving the problem that the mode source electric field time domain function is not the same as the magnetic field time domain function:
setting simulation space parameters and medium parameter distribution, and solving space distribution function of mode source input surfaceAnd a propagation constant beta, and setting a time domain function f m (t)；
Let the mode source electric field time domain function f E (t)＝f m (t)，Wherein f m (t) represents a normalized amplitude time domain function, < ->Indicating the phase difference of the electric and magnetic fields;
through Hilbert transformation, obtain
Substituting the obtained electric field time domain complex function into a formula:
and (3) obtaining a mode source magnetic field time domain complex function:
taking the real parts of the electric field and magnetic field time domain complex functions to obtain corrected electric field and magnetic field time domain functions:
wherein Deltat is the actual time step, t is the actual time, due to f E Ratio f H Early half a time step, so f E Where t=nΔt; f (f) H In (a)
Further, the magnetic field time domain function is corrected first, then the electric field time domain function is corrected, and the solving method comprises the following steps:
setting simulation space parameters and medium parameter distribution, and solving space distribution function of mode source input surfaceAnd a propagation constant beta, and setting a time domain function f m (t)；
Let the mode source magnetic field time domain function f H (t)＝f m (t)，Wherein f m (t) represents a normalized amplitude time domain function, < ->Representing the phase difference of the magnetic field and the electric field, and obtaining through a mode source propagation constant and simulation space parameters;
through Hilbert transformation, obtain
Substituting the obtained complex function of the magnetic field time domain into a formula:
obtaining a mode source electric field time domain complex function:
taking the real parts of the electric field and magnetic field time domain complex functions to obtain corrected electric field and magnetic field time domain functions:
wherein Deltat is the actual time step, t is the actual time, due to f E Ratio f H First half a time step occurs, at f E Where t=nΔt; f (f) H In (a)
According to the method for realizing accurate input of the mode source in the FDTD algorithm, when the mode source time domain function is coupled into the calculation space of the FDTD, the mode source time domain function is subjected to phase correction according to the medium coefficient and the grid setting of the calculation space, so that the coupling efficiency of the mode source is improved, and the accuracy of the mode source input is improved.
Drawings
FIG. 1 is a schematic diagram of a regular hexahedral Ye lattice;
FIG. 2 is a closed total field space division;
FIG. 3 is a total field space division of half space;
FIG. 4 is a discrete distribution plot of electromagnetic computation points at the boundary of the total field (incident field incident in the z direction);
FIG. 5 is a flow chart of phase correction by Hilbert transform;
FIG. 6 is a flow chart of a simulation of FDTD mode source detection;
FIG. 7 is a dielectric profile of a mode source input face;
FIG. 8 is a view of the position of the detection point;
FIG. 9 is a time domain function or sequence f of E-field and H-field correspondences derived by Hilbert transform E And f H ；
FIG. 10 is an enlarged partial view of the vicinity of the maximum peak in FIG. 9;
FIG. 11 is a graph of normalized intensity after correction of phase using Hilbert transform according to the present invention; the circles in the figure represent the maximum value of the reflected wave;
FIG. 12 is a graph of normalized intensity after correction using direct phase shift;
fig. 13 is a normalized intensity plot of uncorrected phase.
Detailed Description
In order that the invention may be readily understood, a more particular description thereof will be rendered by reference to specific embodiments that are illustrated in the appended drawings. Preferred embodiments of the present invention are shown in the drawings. This invention may, however, be embodied in many different forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete.
The invention provides a method for realizing accurate mode source input in an FDTD algorithm through Hilbert transformation, which is based on the function of converting any one real signal into a corresponding analysis signal by Hilbert transformation, and extracts phase information from amplitude information by Hilbert transformation. The Hilbert transform in the time domain is essentially a convolution operation, and can be usually obtained by converting the Fourier transform into multiplication in the frequency domain and then solving the multiplication by the inverse Fourier transform, or directly solving the multiplication by using the convolution operation. The electric and magnetic fields E and H solved in the mode solution, if their time-space coordinates are the same, when coupled to the FDTD mode input port, the phase difference created by the misalignment of the E and H nodes reduces the coupling efficiency. The present invention proposes to use the Hilbert transform to add phase information to the mode source to increase the coupling efficiency of the mode source in FDTD.
The technical scheme of the invention is further described in detail below with reference to the accompanying drawings.
Before the technical scheme of the invention is introduced, a Hilbert transformation mode adopted in the invention is introduced.
The invention adopts a convolution operation method to realize Hilbert transformation, namely:
can also be expressed as:
the above operation is described asThis operator is directly employed when the Hilbert transform is performed later, s= -1 when exp (-j (wt- βr)) is defined as forward propagation in the r direction, and s=1 when reverse propagation in the r direction.
The method for realizing accurate mode source input in the FDTD algorithm comprises the following specific steps:
firstly setting waveguide structure parameters, generating a cuboid Yee grid based on a rectangular coordinate system, then reading medium parameter distribution at the cross section of the waveguide structure needing to be added with a mode source for solving the mode source, and because the solution of the mode does not contain amplitude information, also needing an amplitude function changing along with time step at the cross section, then superposing the amplitude function on the mode source according to a phase relation corresponding to space and time, and finally carrying out FDTD iterative calculation to solve the field distribution of a calculation space.
By the maxwell rotation equation:
the FDTD differential discrete iteration formula of the total field area and the scattering field area under the rectangular coordinate system can be deduced, the FDTD differential discrete iteration formula is the prior art, the FDTD differential discrete iteration formula is directly listed and used, and before the FDTD formula is listed, the quantity in the formula is explained as follows:
electric field:magnetic field: />
(i, j, k) represents the grid coordinates of the computation nodes in the computation space, dimensionless. (x, y, z) represents the actual physical distance coordinates whose dimensions are length dimensions, e.g., meters, micrometers, etc. Δx, Δy, Δz are the actual lengths of one grid in the x, y, z directions, respectively, under a uniform grid. The relation of i, j, k and x, y, z is that x=iΔx; y=iΔy; z=iΔz. Δt is the actual time step, t is the actual time, and the relationship between Δt and the number of time steps n is: t=nΔt orIn particular according to electric and magnetic fieldsThe iteration sequence and the time step number are determined.
The FDTD iterative equations for the three electric field components also have the same form, and only the iterative equations for the x-direction electric field and the magnetic field are listed here as examples, and the magnetic field is iterated first, see the time step terms on equation (5) and equation (11).
The x-axis direction electric field FDTD iterative formula:
in the formula (5), the amino acid sequence of the compound,
under the ladder grid, the equivalent dielectric constant calculation formula is as follows:
in the above formulas (8) and (9), ε represents the dielectric constant of the medium, σ represents the electrical conductivity, ε eff For effective dielectric constant, σ eff Is effective conductivity.
For simplicity of description, this formula is written later on as:
similarly, the X-axis magnetic field FDTD iterates the formula:
in the formula (11), the amino acid sequence of the compound,
mu in the formulas (14) and (15) represents the permeability coefficient, sigma of the medium m Is magnetic permeability, mu eff For effective permeability coefficient, σ meff Is effective magnetic permeability. For simplicity of description, when the formula is described later, it is written as the first formula of the following formula, and the iterative formula of the electric field is also listed, and the FDTD iterative formula of 6 components of the electromagnetic field is the existing algorithm, so it is simply listed and used as follows.
Considering the plurant stability condition, Δt also needs to satisfy the following condition, namely:
according to the equivalent principle, an equivalent plane electromagnetic current is introduced at the boundary of the total field region and the scattering field region,
in the above formulae (19) and (20),is the normal vector pointing in the wave propagation direction at the interface, < >>Represents the electromagnetic field equivalent to the electromagnetic flow, which is parallel to the total field region-scattering field region interface. In general, three-dimensional case total field-fringe field boundaries, only one total field boundary is considered, z=k c Δz total field boundary as an example; as shown in fig. 4, the mode source is shown at the total field boundary k=k A Plane incident in z direction, at a (i A ,j A ,k A ) Of the four magnetic field components around the point, onlyIn the scattering field region, then, after introducing an incident mode source propagating in the z-direction (incidence from other directions is also possible, only propagation in the z-direction will be discussed here), a tangential FDTD formula at the total field boundary and the outer boundary is derived, and the mode source addition formula is also an existing algorithm formula, which is only listed and used by the present invention:
the above pattern source joins the second term to the right of equation set (21), i.e., the pattern source time-space domain function, which can be separated in time and space, the space-time function of the pattern source is expressed as follows:
the first term to the right of the above-mentioned set of mode source time-space domain function formulas (22) is related only to spatial coordinates, which in the present invention is called mode source spatial domain function, which can be defined by z=k A The Δz plane, i.e., the electrical and magnetic medium coefficients of the total field boundary, are directly derived, as is known, and the solution of the mode source spatial domain function is also prior art. The second term on the right is only time dependent and is referred to herein as the pattern source time domain function. The algorithm of the invention is to carry out phase correction on the mode source time domain function according to the medium coefficient and the grid setting of the calculation space when the mode source time domain function is coupled into the calculation space of the FDTD so as to improve the accuracy of the mode source input, and the mode source time domain complex functions at the boundary of the total field are respectively as follows:
order theThe phase difference is obtained as follows:
since the first two equations of equation set (22) occur first, t E ＝nΔt,The magnetic field incidence plane is behind the electric field r E ＝k A n eff Δr,/>Then equation (25) can be written again as:
r=x or y or z, meaning that the wave described by the formula can propagate in any direction of x, y, z; w is the angular frequency of the light in vacuum, a known quantity; beta is a propagation constant, n eff The equivalent refractive index of the waveguide structure corresponding to the mode can be directly obtained by the electromagnetic parameters of the medium, and the equivalent refractive index is a known quantity; thenOr directly from the parameter settings of the computation space and the propagation constants of the mode sources. />The modulation signal, i.e. the amplitude function mentioned above, is also known as complex function of the modulation signal.
At correction f E (t) and f H At (t), there are two paths: can let f E (t)＝f m (t) and then byAnd f E (t) obtaining f H (t); also let f H (t)＝f m (t) by->And f H (t) obtaining f E (t)。
Detailed description solving for corrected f E (t) and f H The specific steps of (t) can be seen in the flow chart of fig. 5, which is the first path:
let f E (t)＝f m (t), then is equivalent to let n=0, k A =0, thenexp(-j(wt E -βr E ))＝1，The same is true here as in the case of equation (26);
f is then E (t)，Are known, and are obtainable by Hilbert transform: />Further, a complex function of the magnetic field time domain can be derived:
will bePerforming Euler expansion to obtain->And because ofThese two formulae are brought into formula (27. A) to give formula (27. B).
Only the real part of the time domain function needs to be taken, and finally the following can be obtained:
equations (28), (29) are time domain functions of the mode sources of the modified electric and magnetic fields, respectively, because the two time domain functions are interrelated, one determines that the other needs to be calculated and unique, where f E (t) and f H The variable t of (t) being converted to nDeltat orDetermined by equation (22).
Thus, the time-space domain function of the tangential component of the mode source in any direction of x, y and z can be obtained. k (k) A The total field boundary is represented, which boundary plane is graphically represented in fig. 3.
In order to verify the effect of the method provided by the invention, the FDTD simulation algorithm comparison experiment is carried out, the flow is shown in a figure 6, and a step C1 in the figure 6 represents the correction of the phase by the Hilbert transformation mode; step C2 represents not using any phase correction, i.e. let f E (t)＝f H (t)＝f m (t); step C3 represents a direct phase shift correction method, which is to let f E (t)＝f m (t),Or-> f H (t)＝f m (t). In the specific embodiment, these 2 phase correction algorithms are compared with each other, and the phase is not corrected. The correction phase through the Hilbert transformation mode can calculate and store the needed time domain function before the FDTD iterative computation starts, and the data is directly read during iteration, so that the iteration speed is hardly influenced.
Further practice is made on the iterative formula of the total field region and the scattering region in the FDTD described above, as shown in FIG. 7, the background environment of practice is a rectangular waveguide centered at 0.5X0.2 um, the material is Si, the base material is SiO 2 Magnetic permeability coefficient mu, magnetic permeability sigma m And conductivity σ are both set to 0, the dielectric constant taking the dielectric constant of the material (which is the square of the refractive index); the width of the section taken when the fundamental mode is calculated is 3 multiplied by 2um, the rectangular waveguide is positioned at the center of the rectangular waveguide, the requirement on the FDTD calculation area is only required to be larger than a mode source entrance, and the grid parameter of FDTD is deltax=deltay=deltaz=0.02 um; wavelength lambda of light source 0 =1.55 um, modulation function is gaussian pulseSetting the phase difference of the adjacent E node domain H node at the input port of the mode source by the input of the propagation factor mode source of the wave function>As shown in FIG. 8, the mode source input surface is divided into an electric field input surface and a magnetic field input surface, and the distance between the two surfaces is +.>That is, one cell has a dimension of half of the dimension in the propagation direction, and 3 pairs of detection points ((S1, S2), (S3, S4), (S5, S6)) are provided in total at the test, and are located at the front and rear of the electric field input surface, respectively at one wavelength (d) 3 =22Δz), one-half wavelength (d 2 =11Δz) and a quarter wavelength (d 1 =5Δz), where the wavelength is the wavelength in the medium, i.e. +.>n is the refractive index of the medium, 6 monitoring points are all positioned at the right center of the rectangular waveguide on the xy plane, the detection quantity is the electric field intensity Ey in the y direction at the monitoring point, and the light wave propagation direction is along the z axis direction. FIG. 9 shows the E-field and H-field corresponding time domain functions or sequences f obtained by Hilbert transform according to the present invention E And f H Since the phase difference between the two is very small, a partial enlarged view of the vicinity of the maximum peak of fig. 9 is shown in fig. 10. The specific parameters are shown in Table 1, and the non-integer variables in the c program uniformly use the type of float.
Table 1 simulation parameter table
When the time domain signal is a known analytical formula, the time domain function or sequence can be delayed or advanced by using the direct phase shift correction method to obtain the corresponding time domain signal of another field on the time variable t, and the time required for delay or advance is recorded asdtt is negative and phase-retarded, and positive and phase-advanced. When the incident signal is a time domain sequence signal which randomly occurs and cannot be expressed by an analytic expression, a direct phase shift correction method cannot be used, and the signal needs to be stored to realize phase correction through Hilbert transformation provided by the invention.
The data measured at the detection point after the simulation is finished are shown in fig. 11, and the normalized intensity maximum values of the reflected wave at the wavelength of one time, the wavelength of one time and the wavelength of one fourth are 0.00442, 0.00842 and 0.0157 respectively. In the case where no phase correction is used, as shown in fig. 13, the reflected wave normalized intensity maximum values at the one-time wavelength (22Δz), the one-half wavelength (11Δz), and the one-fourth wavelength (5Δz) are 0.0622, 0.0589, and 0.0615, respectively. In contrast, the intensity reflection coefficient at half wavelength was reduced from 5.89% to 0.842%, as shown in table 2. The results obtained by the delay correction are shown in fig. 12, and the corresponding three maximum reflection coefficients are 0.00359, 0.00767 and 0.0132; the intensity reflection coefficient should be 0 in the ideal state, and under the experiment, the direct phase shift correction method of the time domain lead can bring better phase correction than Hilbert transformation.
TABLE 2 intensity reflection coefficient table
/>
Claims (3)
1. A method for implementing accurate mode source input in an FDTD algorithm, comprising:
setting medium parameter distribution, selecting a mode source input surface in an FDTD simulation space and carrying out mode solving;
performing phase correction on a time domain function of an electric field and a magnetic field on a mode source input surface; the time domain function of the electric field and the magnetic field on the input surface is subjected to phase correction by Hilbert transformation, which is expressed as follows:wherein f (t) represents a pre-transformation time domain function; />Representing a Hilbert transformed time domain complex function; the Hilbert transformation adopts a convolution operation mode, and an operation formula is expressed as follows: />S= -1 when exp (-j (wt- βr)) is defined as forward propagation in the r direction, s=1 when it is propagation in the r negative direction;
the phase-corrected electric and magnetic field time domain functions are added to the iterative computation of FDTD.
2. The method for realizing accurate input of a mode source in an FDTD algorithm according to claim 1, wherein the method for solving comprises the steps of:
setting simulation space parameters and medium parameter distribution, and solving space distribution function of mode source input surfaceAnd a propagation constant beta, and setting a time domain function f m (t)；
Let the mode source electric field time domain function f E (t)＝f m (t)，Wherein f m (t) represents a normalized amplitude time domain function, < ->Indicating the phase difference of the electric and magnetic fields;
through Hilbert transformation, obtain
Substituting the obtained electric field time domain complex function into a formula:
and (3) obtaining a mode source magnetic field time domain complex function:
taking the real parts of the electric field and magnetic field time domain complex functions to obtain corrected electric field and magnetic field time domain functions:
wherein Deltat is the actual time step, t is the actual time, due to f E Ratio f H Early half a time step, so f E Where t=nΔt; f (f) H In (a)
3. The method for realizing accurate mode source input in the FDTD algorithm according to claim 1, wherein the method for solving comprises the steps of:
setting simulation space parameters and medium parameter distribution, and solving space distribution function of mode source input surfaceAnd a propagation constant beta, and setting a time domain function f m (t)；
Let the mode source magnetic field time domain function f H (t)＝f m (t)，Wherein f m (t) represents a normalized amplitude time domain function, < ->Representing the phase difference of the magnetic field and the electric field, and obtaining through a mode source propagation constant and simulation space parameters;
through Hilbert transformation, obtain
Substituting the obtained complex function of the magnetic field time domain into a formula:
obtaining a mode source electric field time domain complex function:
taking the real parts of the electric field and magnetic field time domain complex functions to obtain corrected electric field and magnetic field time domain functions:
wherein Deltat is the actual time step, t is the actual time, due to f E Ratio f H First half a time step occurs, at f E Where t=nΔt; f (f) H In (a)
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